multi-reference systems in chemistry; unconventional

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Chemistry Theses and Dissertations Chemistry

Spring 5-19-2018

Multi-Reference Systems in Chemistry; Unconventional Bonding in Multi-Reference Systems in Chemistry; Unconventional Bonding in

Organic Chemistry; Covalent Bonding in Transition Metal Clusters Organic Chemistry; Covalent Bonding in Transition Metal Clusters

Alan Wilfred Humason Southern Methodist University, [email protected]

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MULTI-REFERENCE SYSTEMS IN CHEMISTRY

UNCONVENTIONAL BONDING IN ORGANIC CHEMISTRY

COVALENT BONDING IN TRANSITION METAL CLUSTERS

Approved by:

Dr. Elfriede Kraka

Professor and Chair of Chemistry

Dr. Werner Horsthemke

Professor of Chemistry

Dr. Peng Tao

Assistant Professor of Chemistry

Dr. John Wise

Associate Professor of Biology

MULTI-REFERENCE SYSTEMS IN CHEMISTRY

UNCONVENTIONAL BONDING IN ORGANIC CHEMISTRY

COVALENT BONDING IN TRANSITION METAL CLUSTERS

A Dissertation Presented to the Graduate Faculty of the

Dedman College

Southern Methodist University

in

Partial Fulfillment of the Requirements

for the degree of

Doctor of Philosophy

with a

Major in Chemistry

by

Alan Humason

Bachelor of Science, Chemistry, University of Massachusetts, Amherst

Master of Science, Chemistry, Southern Methodist University, Dallas, TX

May 19, 2018

Copyright (2018)

Alan Humason

All Rights Reserved

ACKNOWLEDGMENTS

It requires four scientists to do computational chemistry; the chemist, the physicist, the

mathematician, and the computer scientist. Having been for many years merely a chemist,

I must thank the many fellow scientists who have put their work aside for mine. I thank

Dr. Thomas Sexton, for his clarity in explaining the physics that I had long forgotten, and

the mathematics that I never knew. I thank Dr. Vytor Pinerio Oliveria, for many fruitful

scholarly discussions, always freely granted without bravado and (I hope) to our mutual

enrichment. I thank Drs. Robert John Brown Kalescky and Marek Freindorf for their

expertise, their instruction, and their efforts with and against the unforgiving computer

clusters.

But, mostly I thank Professor Dr. Dieter Cremer, for being the true embodiment of the

four scientists. He could bring the chemistry, physics, mathematics and computer sciences

together, and shared that knowledge and expertise literally to the end of his days. My

proudest academic accolade came at the completion of the annulene project, when he said,

“I now see that you have the intelligence to make a Ph.D.”

I thank and wish to praise my current research advisor, Dr. Elfi Kraka, who, after the

sudden death of my advisor Dr. Cremer picked up the pieces of my academic career and

carried me to this finish line. The strength, dedication, and love that that required was

beyond anything that I have seen before or will probably ever see again.

I wish to thank Dr. Michael Lattman, for guiding me through the intricacies of the

graduate school process, and Dr. Patty Wisian-Neilson for being his right arm. Sometimes

you just need friends.

iv

Humason, Alan Bachelor of Science, Chemistry, University of Massachusetts, AmherstMaster of Science, Chemistry, Southern Methodist University, Dallas, TX

Multi-Reference Systems in Chemistry

Unconventional Bonding in Organic Chemistry

Covalent Bonding in Transition Metal Clusters

Advisor: Dr. Elfriede Kraka

Doctor of Philosophy degree conferred May 19, 2018

Dissertation completed April 19, 2018

The geometries, chemical properties, and reactivities of molecules are determined by

their electronic structure. The field of ab initio computational chemistry works to calculate

the kinetic and potential energies between the nuclei and electrons of a molecule. These

calculations usually begin with the determination the electronic ground state.

Molecules that cannot be adequately described with a single, ground state configuration

are called multi-reference systems, which require the calculation of a linear combination

of all pertinent electronic configurations, with a corresponding increase in computational

cost. This is not ‘black box’ methodology, because solving these systems requires a good

understanding of the chemistry being described, so that the important configurations among

millions of possibilities can be selected. Their multi-reference character also makes them

some of the most interesting molecules in chemistry.

In this dissertation, we have studied ultra-long CC bonds in simple and unique organic

molecules, biradical pancake bonded species, fluxional bridged annulenes, and covalently

bonded transition metal diatoms.

We find that CC ultra-long bonds and electrostatic pancake bonding interactions can be

described by single-reference methods, but that fluxional bridged annulenes require multi-

reference methods.

Transition metal diatoms can only be described by multi-reference methods. We deter-

mined which methods, basis sets, and active spaces work best in each of the 30 cases.

v

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

LIST OF SYMBOLS AND ACRONYMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

CHAPTER

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Multireference Systems - What, Why and How? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Characterization of Carbon-Carbon Single Bond Strength . . . . . . . . . . . . . . . . . . . . . . 4

2.1. Single-Reference Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2. Application of Vibrational Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. General Characterization of Carbon-Carbon Bond Strength . . . . . . . . . . . . . . . . . . . . 13

3.1. Extension of the Single-Reference Description to Unconventional Systems . . 13

3.2. Refinement of Single-Reference Descriptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3. The Shortest CC Single Bonds in Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4. The Longest CC Single Bonds in Chemistry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4. Characterization of Multi-reference Systems by Single-reference Density Func-

tional Theory - Pancake Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1. Refinement of Single-Reference Calculations - Broken Symmetry . . . . . . . . . . . 23

4.2. Characterization of Pancake Bonding Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 25

5. Bridged Annulenes; The Longest CC Bonds? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.1. The Puzzle of 11,11-Dimethyl-methano[10]annulene . . . . . . . . . . . . . . . . . . . . . . . 36

5.2. Analysis of the Annulene Systems by Multiple Levels of Theory . . . . . . . . . . . 38

5.3. Does 11,11-dimethyl-methano[10]annulene possess the longest homoaro-matic CC bond of neutral hydrocarbons? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

vi

6. Multi-Reference Systems in Inorganic Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.1. Transition Metal Diatoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2. A Survey of All Transition Metal Diatoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.3. Integration of all Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7. Transition Metal Diatoms - Maximum Bond Multiplicity . . . . . . . . . . . . . . . . . . . . . . . 80

8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.1. Single-Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.2. Single-Reference Methods on Multi-Reference Systems . . . . . . . . . . . . . . . . . . . . 83

8.3. Multi-Reference Methods in Organic Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.4. Multi-Reference Methods in Inorganic Chemistry. . . . . . . . . . . . . . . . . . . . . . . . . . 84

8.5. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

9. Calculations and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

9.1. Single-Reference Computational Methods - Organic Chemistry . . . . . . . . . . . . 86

9.2. Single-Reference Computational Methods Beyond Energies . . . . . . . . . . . . . . . . 89

9.3. Multi-Reference Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

APPENDIX

A. Publications, Supporting Information and Manuscripts . . . . . . . . . . . . . . . . . . . . . . . . . 95

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

vii

LIST OF FIGURES

Figure Page

2.1 Local Mode Stretching Force Constants (ka) to Bond Strength Order (ωB97X-D/aug-cc-pVTZ), Single Bonds. This plot serves as a conversion chartbetween the two parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Organic molecules investigated in this work. The single and multiple bondsreported are in red. All molecules have singlet ground states. . . . . . . . . . . . . . . 10

3.1 Organic molecules investigated in this work. The single and multiple bondsreported are in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Local Mode Stretching Force Constants (ka) to Bond Strength Order (ωB97X-D/aug-cc-pVTZ), Single, Multiple and Aromatic Bonds. This plotserves as a conversion chart between the two parameters. . . . . . . . . . . . . . . . . . . 16

3.3 Bond Length (ωB97X-D/aug-cc-pVTZ) to Bond Strength Order for CCsingle bonds. [139,140,239–241] (R2 = 0.9955) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Pancake bonded molecules investigated in this work. 4.1) HCNSSN dimer.4.2) HCNSeSeN dimer. 4.3) HCNTeTeN dimer. 4.4) phenalenyl dimer.4.5) 2,5,8-trimethylphenalenyl dimer. 4.6) 2,5,8-tri-t-butylphenalenyldimer. Pancake Bonding Interactions are displayed in red. . . . . . . . . . . . . . . . . . 24

4.2 C2 Symmetry geometries for the HCNTeTeN dimer. a) Singlet. b) Triplet. . . . . 27

4.3 Triplet state geometries for the phenalenyl dimer. a) Staggered. b) Eclipsed.c) Minimum energy geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4 Dissociation Curves for sytems 1 and 2 (BS-UM06/6-311G(d,p), 3 (BS-UM06/SDD), 4 and 5. (BS-UM05-2X/6-31++G(d,p).) . . . . . . . . . . . . . . . . . . . . 29

4.5 Bond Strength Orders (BSO) and Optimized Bond Lengths (in parentheses,A) for the Phenalenyl, Trimethylphenalenyl and tri-tert-ButylphenalenylRadical Monomers and Dimers (4.4 through 4.6.) The Aromaticity In-dices (AI), Bond Weakening/Strengthening parameters (WS) and BondAlteration parameters (ALT) for the full carbon ring structures (FULL)and the outer ring structure (OUTER) are indicated in boxes. . . . . . . . . . . . . . 33

5.1 Annulene species investigated in this work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

viii

5.2 Representation of the energy as a function of the 1,6-distance obtainedat multiple levels of theory. a)11,11-dimethyl-1,6-methano[10]annuleneb)1,6-methano[10]annulene and c) 1,3,5-cycloheptatriene. . . . . . . . . . . . . . . . . . . 39

5.3 Representation of the energy and enthapy as a function of the 1,6-distance ofa)11,11-dimethyl-1,6-methano[10]annulene b)1,6-methano[10]annulene andc) 1,3,5-cycloheptatriene. a) was obtained by CASPT2(14,14)/6-311G(d,p)//B3LYP/6-311G(d,p) and CCSD(T)/6-311G(d,p)// B3LYP/6/311G(d,p),b) by CASPT2(14,14)/6-311G(d,p)// B3LYP/6/311G(d,p), and c) byCASPT2(10,10)/6-311G(d,p)// B3LYP/6-311G(d,p). In all cases, therelative energy and enthalpy (∆H(298K)) are included. . . . . . . . . . . . . . . . . . . . . 41

5.4 Dimer and tetramer configurations that were calculated for this work. Thegreen structure shows how two unit cells fit together in the crystal struc-ture. The numbers on this structure are taken from Bianchi’s x-raystructure. The blue structure shows the arrangement that was calcu-lated and optimized searching for packing effects. The orange structureshows the geometry and hydrogen-to-ring distance for the two optimizedstructures, ωB97X-D/6-311G(d,p). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.5 a) Dependence of calculated NMR chemical shifts [ppm] as a function of theC1,C6 interatomic distance of 11,11-dimethyl-1,6-methano[10]annuleneas obtained at the B3LYP/GIAO/6-311G(d,p) level of theory. b) NMRab inito analysis of the mean deviation between measured and calculated13C NMR chemical shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.6 Dependence of calculated NMR spin-spin coupling constants [Hz] as a func-tion of the C1,C6 interatomic distance of 11,11-dimethyl-1,6-methano[10]annuleneas obtained at the B3LYP/GIAO/6-311G(d,p) level of theory. . . . . . . . . . . . . . 56

5.7 Bond strength orders for carbon-carbon bonding interactions in 11,11-dimethyl-1,6-methano[10]annulene system as a function of C1,C6 interatomic dis-tance, analyzed by adiabatic mode analysis, B3LYP/6-311G(d,p). . . . . . . . . . . 58

5.8 Dependence of a) electron density distribution and b) energy density distri-bution on the C1,C6 interatomic distance of 11,11-dimethyl-1,6-methano[10]annuleneas obtained from the AIMAll program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.9 B3LYP geometries of 5.1 - 5.6. Bond lengths in A and bond angles in degrees. 63

6.1 Calculated Bond Length by Periodic Table Row, All Diatoms. . . . . . . . . . . . . . . . . . 75

6.2 Calculated Bond Dissociation Energy by Periodic Table Row, All Diatoms. . . . . 76

6.3 Calculated Stretching Force Constant by Periodic Table Row, All Diatoms. . . . . 77

ix

6.4 Calculated Stretching Force Constant versus Bond Strength Order, All Di-atoms. This plot serves as a conversion chart between the two parameters. . 78

8.1 Local Mode Stretching Force Contants (ka) versus Number of Methyl Groupsfor a Series of Organic Compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

x

LIST OF TABLES

Table Page

2.1 Calculated and experimental Carbon-Carbon Bond Lengths R [A], BondDissociation Enthalpies BDH [kcal/mol], local mode stretching con-stants ka [mdyn/A], local mode frequencies ωa [cm−1], Bond StrengthOrdersBSO, and Energy Densities [Hartree/A3] for all molecules, ωB97X-D/aug-cc-pVTZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Calculated and experimental Carbon-Carbon Bond Lengths R [A], BondDissociation Enthalpies BDH [kcal/mol], local mode stretching con-stants ka [mdyn/A], local mode frequencies ωa [cm−1], Bond StrengthOrdersBSO, and Energy Densities [Hartree/A3] for all molecules, ωB97X-D/aug-cc-pVTZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Calculated Carbon-Carbon Bond Lengths (re) for various 1,1,1-Trihaloethanes,ωB97X-D/3-21G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Calculated Carbon-Carbon Bond Lengths (re) for C-C Single Bonds hy-bridized by multiple or aromatic carbons, ωB97X-D/aug-cc-pVTZ . . . . . . . . . 21

4.1 Energy Gap results for all Species. CC Distance [A], Triplet/Singlet Gap[kcal/mol] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Calculated and experimental Carbon-Carbon interatomic distances R [A],Dissociation Energies, Counterpoise Complexation Energies and BondDissociation Energies DEcalc, CECP and BDEexp [kcal/mol], local modestretching constants ka [mdyn/A], local mode frequencies ωa [cm−1],Bond Strength Orders BSO, and Electron and Energy Densities, ρ andHb [electron/A3 and Hartree/A3, respectively] for all molecules. Levelof Theory is UM06/6-311G(d,p) for 4.1 and 4.2, UM06/SDD for 4.3,and UM05-2X/6-31++G(d,p) for 4.4, 4.5 and 4.6. . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Aromaticity Indices for the Phenalenyl, Trimethylphenalenyl and tri-tert-Butylphenalenyl Free Radical Dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1 Relative Energies and Tautomerization Barriers of 11,11-Dimethyl-1,6-methano[10]annulene. 42

5.2 Relative Energies and Tautomerization Barriers of 1,6-Methano[10]annulene. . . . 47

5.3 Relative Energies and Tautomerization Barriers of 1,3,5-Cycloheptadieneand Norcaradiene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

xi

5.4 Bond Strength Orders for all pertinent C,C interactions of 11,11-Dimethyl-1,6-methano[10]annulene at the stationary geometries (ethane = 1, ethene= 2.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.1 AQCC, CASPT2, RASPT2 Calculations by aug-cc-pVQZ basis sets andRASPT2 Calculations by ANO-RCC basis set. Summary of all results. . . . . . 73

6.2 Summary of Bond Strength Orders (BSO) and the methods which producedthe best results for each transition metal diatom. . . . . . . . . . . . . . . . . . . . . . . . . . . 75

xii

LIST OF SYMBOLS AND ACRONYMS

ke - Equilibrium Stretching Force Constant

ka - Local Mode (Adiabatic) Stretching Force Constant

ωa - Local Mode (Adiabatic) Vibrational Frequency

rCC - Carbon-Carbon Interatomic Distance

R - rC1,C6 for Annulene Systems

re - Equilibrium Bond Length

re - Bond Length by electron diffraction

rb - Bond Critical Point

rr - Ring Critical Point

ρr - Electron Density

Hb - Energy Density

me - millielectrons

ε - dielectric constant

µ - dipole moment

∆E - Energy Change

∆H - Enthalpy Change

∆G - Gibbs Free Energy Change

BDE - Bond Dissociation Energy

BDH - Bond Dissociation Enthalpy

BSO - Bond Strength Order

NBO - Natural Bond Order

PEC - Potential Energy Curve

xiii

PHC - Potential Enthalpy Curve

PGC - Potential Gibbs Free Energy Curve

ZPE - Zero Point Energy

TS - Transition State

LCAO - Linear Combination of Atomic Orbitals

HOMO - Highest Occupied Molecular Orbital

LUMO - Lowest Unoccupied Molecular Orbital

HF - Hartree Fock Theory

WFT - Wave Function Theory

FCI - Full Configuration Interaction

CAS - Complete Active Space

RAS - Restricted Active Space

SCF - Self Consistent Field

CSF - Configuration State Function

DFT - Density Functional Theory

AIM - Atoms-in-Molecules (Topological Analysis)

BCP - Bond Critical Point

RCP - Ring Critical Point

CCP - Cage Critical Point

AI - Aromaticity Index

WS - Bong Weakening/Strengthening Parameter

ALT - Bond Strength Alteration Parameter

xiv

This is dedicated to my family, for their support, and to my students, for their patience.

But mostly this is dedicated to my wife, Melissa McNamara Humason, B.M.E., M.S.,

for her unflagging belief that I would succeed.

Chapter 1

Introduction

Multireference Systems in Computational Chemistry.

1.1. Multireference Systems - What, Why and How?

Systems that can be described with good precision by determining the electronic config-

uration in the ground state are called single reference systems. From the ground state the

gap between the highest energy occupied molecular orbital (HOMO) and the lowest energy

unoccupied molecular orbital (LUMO) can be determined, and if that energy is reasonably

large, then a single reference description can be used. Much of computational chemistry is

done by single reference calculations, working with single molecules, in vacuum and at 0K.

The energy differences between a molecule at 0K in vacuum and a system of molecules at

298K at one atmosphere are significant, but when making comparisons between two sys-

tems (different constitutional or rotational isomers, molecules before and after a reaction,

larger versus smaller systems), the relative energy differences are usually comparable, and

this approximation is usually valid.

Using a single reference description is also an approximation, and not all systems can

be adequately described this way. Some systems that cannot be described by their ground

electronic state are:

• Molecular systems with low-lying excited states (a narrow HOMO-LUMO gap.)

• Molecules with significant biradicaloid character.

• Fluxional molecules, that undergo interchange of atoms between symmetry-equivalent

positions.

• Metallic systems, which participate in metallic bonding.

1

• Small metal clusters, which participate in covalent bonding, but with many low-lying

excited states.

• Chemically reacting molecules, in the process of bond breaking and/or bond formation.

In these cases, the ground state configuration of the molecule does not fully or accu-

rately describe the electronic state of the molecule, and the inclusion of additional electronic

configurations, known as configuration state functions (CSF’s) is necessary. These cases are

called multi-reference systems.

The calculation of all likely electronic states for multi-reference systems is expensive,

as each additional CSF increases the cost. The best results are obtained by including all

possible electronic states, with each electron placed into every occupied orbital in turn,

which is known as a ‘full configuration interaction’ (FCI) calculation. Although this method

is proven to be the best possible, the number of CSF’s increases so rapidly that FCI becomes

unfeasible for systems greater than 10 electrons (as in water.) Therefore, to analyze a system

of larger size, additional approximations are required.

The chemistry of a molecule usually depends upon the valence shell electrons. Cost can

be reduced by excluding CSF’s which involve the promotion of lower energy core electrons

into valence and excited state orbitals. Excitations into very high excited states will make

only small contributions, and can be excluded to save cost. These are the first steps in

defining an ‘active space’ to describe a molecule.

When the task is to define a certain chemistry, the calculation can be reduced further by

including only the electrons and basis functions that describe that chemistry. For example,

the Walsh orbital description of cyclopropane puts six bonding electrons into three bond-

ing and three antibonding molecular orbitals. [52] Therefore, the bonding in this molecule

can be described by placing the six electrons into the six Walsh orbitals in every possible

arrangement. In shorthand, this is known as a (6,6) active space.

Selecting an active space requires a very good knowledge of the chemistry being described.

In the case of cyclopropane, the six Walsh orbitals may not be the three highest energy

occupied orbitals and the three lowest energy unoccupied (virtual) orbitals. Orbitals must

2

be selected to have the correct spacial arrangement and symmetry. These methods anr not

‘black box’ applications.

Much research effort had been placed into designing single-reference methods that ap-

proximate the solutions to multi-reference systems, because of the cost and complexity of the

multi-reference calculations. Generally, these efforts require the inclusion of semi-empirical

parameters to obtain reasonable results within a given training set.

This dissertation is about multi-reference systems. This work will examine first organic

systems, and then inorganic, transition metal clusters. In chapters 2 and 3, we will study

CC bonding in simple and unique organic molecules, to see how single-reference systems can

be best described with single-reference methods. In chapter 4, we will study multireference

pancake bonded organic species using single-reference methods. In chapter 5, we will study

fluxional bridged annulene systems, which can only be described with multi-reference meth-

ods. Finally, in chapters 6 and 7, we will look at covalently bonded inorganic transition

metal diatoms, and their multi-reference descriptions.

3

Chapter 2

Characterization of Carbon-Carbon Single Bond Strength

2.1. Single-Reference Descriptions

Breaking a carbon-carbon bond is likely to go by homolytic cleavage, resulting in two

free radical fragments. This bond breaking would be correctly described by multi-reference

calculations. The same bond stretched to its longest possible point might be expected to

have biradical character, any may require multi-reference treatment.

To save the cost of multi-reference calculations, a well defined single-reference method

was tested. For this, we calibrated a double-hybrid density functional theory (DFT) method

with a large basis set, using local mode vibrational analysis.

The foundation of density functional theory is that the energetics, and all other param-

eters for a molecule, can be described by calculating the electron density at every point in

a molecule, rather than solving for each orbital basis function. Empirical adjustments can

then be made for electron correlated motion, exchange repulsion, and long-range dispersion

effects.

The problem with any empirical method is that it is dependent upon a ‘training set’ of

molecules. The parameters are adjusted to obtain results in good agreement with exper-

imental results, or the results of higher level methods (often multi-reference methods) for

the entire training set of molecules. The method can then be expected to return accurate

results for molecules that are chemically similar to the species in the training set. To obtain

results which will be accurate for a wide range of systems, we need a set of parameters which

includes the extreme cases. In this first study, we defined the extreme cases as those that

result in the longest and shortest CC bonds in organic chemistry, and employed vibrational

spectroscopy methods to see if they can be parameterized to give consistent results for this

4

entire range.

There are many factors that can lead to the lengthening of a bond.

(i) Exchange (steric) repulsion between bulky substitutents has long been known to

lengthen interatomic distances. [65,94,128,187,206] The central bond of 2,2,3,3-tetramethylbutane

(2.6, r = 1.582 A by electron diffraction, Figure 2.2) is one of the first molecules thus inves-

tigated. [19]

(ii) Loss of bonding electrons or electron density can lead to electron deficient bonding.

[51, 52, 59, 74, 208, 210] Here, the ethane cation (3.37, re = 1.935 A by calculation, ωB97X-

D/aug-cc-pVTZ, Figure 3.1) is a classic example. [91,180]

(iii) By using bridging atoms which enforce a “cage”-topology, known as clamped bonds.

[138,171] Light, heat and pressure sensitive bi(anthracene-9,10-dimethylene) (3.31 is a much

investigated example. [196]

(vi) Sterically hindered systems where decoupled spin paired electrons lead to open singlet

states, or pancake bonds, [25,108,120,132,157,221] of which the 2,5,8-tri-tert-butylphenalenyl

neutral free radical dimer (4.5) and the dichalchodiazolyl free radical dimers (4.1 and 4.2)

are examples [204] (see Chapter 4.)

(v) Electrostatically hindered ionic free radical species, [35, 82, 115, 164, 165, 222] where

electrostatic repulsion is overcome by bonding orbital overlap. This is a special case of

pancake bonding. The tetracyanoethylene anion dimer is an example. [157]

Forces which affect the length of a bond necessarily affect the strength of the bond.

This leads to an empirical observation that “longer bonds are weaker, and shorter bonds are

stronger.” This simple model has been refuted in the literature. [60] Some past studies of

this relationship have depended on compiling the work of many researchers, using different

measurement techniques, [93, 152, 154, 236] so that it is difficult to place confidence in the

comparability of the data set. Bond lengths, bond dissociation energies and spectroscopic

vibrational analysis are determined by several different methods, at different temperatures

and different pressures. The work is necessarily conducted on a set of similar molecules (i.e.:

CC single bonds), to ensure that there is any comparability at all. The result is that the

5

differences between the measured parameters is relatively small.

To get a consistent set of bond length and bond strength values for a large range of

molecules under identical conditions, ab initio computational chemistry is ideal. By analyzing

a training set of molecules at identical levels of computational theory, under the identical

conditions of 0K and vacuum, the results will be consistent, and even small variations of less

than 0.01 A can be considered as significant.

The factors that shorten a CC bond are relatively straight forward. It is known that

double bonds are shorter than single bonds, and triple bonds are shorter still. It is also

known that the bonds in an aromatic system, with a bond order of 1.5, are intermediate

length between single and double bonds. As our study will show, the variations between

sets of CC double bonds or sets of triple bonds are quite small, compared to the variations

between CC single bonds.

In the intermediate case, for CC single bonds which are flanked by double or triple bonds,

the bond lengths are known to be affected by what is described as increased s-character of

the sp2 or sp-hybridized carbons participating in the internal single bond. Here we seek a

description which will be useful in all of these situations.

2.2. Application of Vibrational Spectroscopy

Accurate determinations of bond length is complicated by the fact that atoms within

molecules are in constant motion, as required by the Heisenberg uncertainty principle. Equi-

librium bond lengths can only be determined computationally. In other cases, such as in

small ring compounds, the bonds as defined by the maximum electron density (MED) path

between atoms are bent. The MED path is longer than the interatomic distance, renewing

the discussion of how the ‘bond length’ should be defined.

Then, there are the limitations of experimental measurement. Methods in use include

electron diffraction, x-ray diffraction, microwave spectroscopy and infrared spectroscopy

methods. For this work, a careful review of the scientific literature was conducted to find all

available experimental data. When multiple bond length values were available, the experi-

6

mental values varied by an average of 0.034 A. Here again, which values are correct?

In this work, to have consistent and comparable parameters, we will use calculated equi-

librium bond lengths, determined at the same level of theory, ωB97X-D/aug-cc-pVTZ. Some

cases where a smaller basis set was necessary are noted in Table 2.1.

Measured bond dissociation enthalpies (BDH’s) or calculated bond dissociation energies

(BDE’s) are generally accepted as methods for assessing bond strength. These methods

determine the difference in energy between an intact molecule and the free radical fragments

resulting from homolytic cleavage of the bond. This means separating the fragments to

well beyond bonding distance, resulting in rehybridization of the atoms at the point of

attachment, changes in electronic state, and relaxing of the molecular geometry, all of which

obscure the intrinsic bond strength.

Using ethane (2.1) as a model, the bonded molecule consists of a σ-bond between two

sp3 carbons, each bonded in turn to three hydrogen atoms. This results in a neutral, singlet

state molecule with D3d symmetry and a CC bond distance of 1.536 A. [211] Upon homolytic

cleavage of this bond, two identical free radical fragments are formed, each in the doublet

state, with sp2 hybridization and D3h symmetry. With loss of the CC bond, the CH bond

energies would change, due to rehybridization and relaxation of angle strain, thereby affecting

the apparent BDH of the target bond. Hence, the BDH or BDE obtained includes the

energies of rehybridization and geometry relaxation, as well as the intrinsic energy of the

bond.

Experimentally, BDH measurements are not attainable for every bond in a molecule.

Values for the BDH of energetically unfavorable bond cleavages are difficult and sometimes

impossible to obtain. For example, cleavage of the CC triple bond in propyne (2.25) while

leaving the CC single bond unperturbed would be difficult. As has been cited in the literature

[60,122,131,144], this gives a limited and skewed, and therefore unreliable indication of the

actual bond strength.

To validate our procedures, we have implemented a set of thirty training molecules (Fig-

ure 2.2.) These molecules were chosen based on the availability of experimental values of

7

bond length and bond dissociation energies. In some cases, experimental parameters are

unavailable, as will be outlined in the text. All molecules are known to have singlet ground

No. Bond ID Rcalc Rexp BDHcalc BDHexp ka ωa BSO ρ Hb

2.1 Me-Me 1.523 1.536 89.01 89.68 4.216 1092.07 1.000 1.659 -1.431

2.2 Me-Et 1.524 1.528 87.43 87.2 4.160 1084.81 0.989 1.671 -1.443

2.3 iPr-Me 1.526 1.535 86.66 88.9 4.092 1075.90 0.976 1.675 -1.442

2.4 tBu-Me 1.530 1.539 87.45 86.0 3.992 1062.70 0.957 1.669 -1.422

2.5 (iPr)2 1.542 1.544 82.79 86.6 3.754 1030.49 0.911 1.641 -1.362

2.6 (tBu)2 1.578 1.582 80.41 76.0 3.195 955.86 0.800 1.544 -1.182

2.7 (PhCH2)2 1.542 1.55 66.88 66.6 3.675 1019.63 0.895 1.601 -1.322

2.8 a (AdMe2C)2 1.629 1.677 53.37 43.7 2.414 826.37 0.639 1.387 -0.925

2.9 (Et2MeC)2 1.597 1.601 73.09 60.2 2.784 903.81 0.716 1.506 -1.118

2.10 (Et3C)2 1.611 1.635 61.21 51.0 2.888 872.66 0.737 1.443 -1.015

2.11 a (PhEt2C)2 1.631 1.635 52.21 44.7 2.285 804.04 0.611 1.277 -0.934

2.12 (Ph3C)2 1.699 N/A N/A 16.6 1.518 677.89 0.439 1.188 -0.663

2.13 b hexakis 1.669 1.67 33.9 [94] N/A 1.919 736.76 0.526 1.275 -0.830

2.14 Et-Et 1.524 1.539 86.04 87.2 4.177 1087.06 0.993 1.683 -1.454

2.15 (iPrMe2C)2 1.599 1.601 70.30 62.2 2.852 898.14 0.730 1.476 -1.070

2.16 (tBuMe2C)2 1.622 1.63 50.00 44.0 2.508 842.25 0.658 1.410 -0.965

2.17 (iBuMe2C)2 1.589 1.606 72.13 57.8 3.050 928.86 0.770 1.506 -1.118

2.18 Me-CH=CH2 1.495 1.501 99.96 100.9 4.575 1137.58 1.068 1.770 -1.637

2.19 H2C=CH2 1.322 1.339 173.88 172.2 9.961 1678.62 2.000 2.449 -3.214

2.20 (CH2=CH)2 1.457 1.467 144.00 116.0 5.119 1203.32 1.169 1.920 -1.934

2.21 Et-CH=CH2 1.497 1.502 98.17 99.6 4.484 1126.24 1.051 1.776 -1.639

2.22 iPr-CH=CH2 1.501 1.5 97.32 99.7 4.382 1113.42 1.032 1.775 -1.628

2.23 tBu-CH=CH2 1.520 1.522 96.05 97.5 4.071 1087.45 0.972 1.749 -1.573

2.24 HC≡CH 1.194 1.208 228.13 229.9 17.777 2242.50 3.190 2.894 -4.700

2.25 Me-C≡CH 1.455 1.45 124.30 123.5 5.254 1219.14 1.194 1.844 -1.895

2.26 Me-C≡N 1.455 1.458 122.88 121.1 5.141 1205.90 1.173 1.827 -1.930

2.27 CH2=CH-C≡CH 1.425 1.431 135.25 133.6 5.778 1278.42 1.289 1.974 -2.151

2.28 CH2=CH-C≡N 1.430 1.429 131.69 132.1 5.586 1257.05 1.255 1.938 -2.141

2.29 HC≡C-C≡CH 1.372 1.383 158.27 155 7.406 1447.44 1.575 2.142 -2.517

2.30 HC≡C-C≡N 1.375 1.379 150.85 152.4 7.348 1441.71 1.565 2.122 -2.491

Table 2.1: Calculated and experimental Carbon-Carbon Bond Lengths R [A], Bond Dissoci-ation Enthalpies BDH [kcal/mol], local mode stretching constants ka [mdyn/A], local modefrequencies ωa [cm−1], Bond Strength Orders BSO, and Energy Densities [Hartree/A3] forall molecules, ωB97X-D/aug-cc-pVTZaωB97X-D/cc-pVTZ. bωB97X-D/6-311++G(d,p). N/A: Not Applicable.

8

1

C C

H

HHH

HH

234

C Me

Me

MeMe

5

6

C C

Me

MeMeMe

MeMe

7

C CH

HHH

8 91011

1213

C C

14

15

16

C C

C

MeMeC

MeMe Me

Me

MeMe

MeMe

17

1820

2122

23

25

C C CH

H

HH

2627

C C

H

HC

CH2

28

29

HC C C CH

30

Single Bonds Single Bonds - sp2, spCH3- Strain

Bond

Stre

ngth

Ord

er

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Local Mode Stretching Force Constants [mDyne/Å]0 1 2 3 4 5 6 7 8

Figure 2.1: Local Mode Stretching Force Constants (ka) to Bond Strength Order (ωB97X-D/aug-cc-pVTZ), Single Bonds. This plot serves as a conversion chart between the twoparameters.

states, meaning that they have no unpaired electrons.

We will use local mode vibrational spectroscopy analysis, which is described in Section

9.2 and elsewhere. [139, 140, 239–241] This method determines the vibrational frequencies

and local stretching force constants (ka) of individual bonds, uncoupled from the bulk of the

molecule.

With the local mode tools, we can investigate any or all bonds in a molecule. With

local modes, we are equipped to discuss the different approaches for lengthening a CC bond,

making comparisons and finding trends. Figure 2.1 gives the relationship between ka and

BSO for C-C single bonds. The calculated results and experimental results obtained from

the scientific literature are shown in Table 2.1.

9

C C

H

HHMe

HH

C C

H

HHH

MeMe

C C

H

MeMeH

MeMe

C CH

HHH

C C

H

HHH

HH

C C

Me

MeMeMe

MeMe

C C

H

HHMe

MeMe

C C

C

CCC

CC

2.10

MeMe

Me

MeMe

Me

C C

C

CMeC

MeC

2.9

Me

Me

Me

Me

2.8

C

C

MeMe

MeMe

C CC

C

CC

2.11

Me

Me

Me

Me

C C

2.12

C C

2.13

2.2 2.3 2.5

2.7

2.1 2.4

2.6

Group I

Group II C C

C

MeMeC

MeMe

2.15H

H

MeMe

MeMe

2.16

C C

C

MeMeC

MeMe Me

Me

MeMe

MeMe

C C

C

MeMeC

MeMe C

C

HH

HH Me

Me

MeMe

H

H

2.17

C C

H

HHC

HH

2.14

Me

H H

Group III

C C

H

H

MeMe

CH2

C C

H

Me

MeMe

CH2

2.22 2.23

C C HH

2.24

C C CH

H

HH

2.25

CN

2.26

C

H

HH

CN C

H

2.28

CH2

HC C C

2.29

CHC C

H

HC

2.27

CH2

HC C CN

2.30

C C

H

H

HH

CH2

C C

H

H

H

H

C

H

CH2

C

H

H2C

2.18 2.19 2.20

C C

H

H

MeH

CH2

2.21

Figure 2.2: Organic molecules investigated in this work. The single and multiple bondsreported are in red. All molecules have singlet ground states.

10

The thirty training molecules investigated in this work are divided into three groups; CC

single bonds, CC single bonds stretched by tert-butyl groups, and CC multiple bonds. We

will examine each group separately, and determine what trends are definable.

Group I: (2.1 - 2.13) CC Single Bonds: The first group of molecules have central CC single

bonds which are subjected to increasing steric repulsion, due to sequential addition of more

and larger hydrocarbon groups (methyl, ethyl, phenyl, and adamantyl.) For all cases in

Group I the CC bonds are formed by sp3-sp3 hybrid orbital overlap.

The simplest case, ethane (2.1), exhibits the shortest bond, at 1.523 A. Bond lengthening

continues in the order 2.1 < 2.2 < 2.3 < 2.4 < 2.5, with the addition of more methyl groups.

Examination of this series gives insight into the steric requirements of the four sub-

stituents. The sequence 2.5 < 2.7 < 2.6 indicates that, while a phenyl group is sterically

larger than a methyl group, we discover that it also creates more steric hinderance than two

methyl groups, but not as much as three.

The sequence 2.9 < 2.10 < 2.11 indicates that a phenyl group is also larger than an ethyl

group. This is counterintuitive, as the phenyl group is flat, or nearly flat, and is expected to

have small steric hindrance.

2.8 has a significantly longer experimental C-C bond length than 2.11 due to the

adamantyl groups being much larger than phenyl groups.

Much of what we report here had been deduced empirically. The agreement between

intuition and calculation verifies the calculations and quantifies the deductions. This analysis

is useful for the development of molecular mechanical modeling methods, which we will not

discuss here.

Although 2.12 has not been isolated, its substituted analog, 2.13, has been synthesized

and characterized. [121] The calculation of an optimized geometry and frequencies for this

212-atom molecule with the ωB97X-D functional and the aug-cc-pVTZ basis set did not

converge after several months. Therefore the calculation was done using the augmented

Pople 6-311++G(d,p) basis set and 1.6 TB of memory dedicated to the calculation, and has

11

the longest CC bond in this sequence. Molecule 2.12 reports a longer CC bond, but this is

unreliable, because 2.12 auto-dissociates into two tri-phenylmethyl free radicals, and cannot

be isolated. Any method that describes this species as a stable molecule is describing an

intermediate species prior to bond fissure. Therefore, this result is excluded from the data

plots.

Group II: (2.14 - 2.17) tert-Butyl groups: While the addition of methyl groups adds steric

strain to a molecule, the addition of tert-butyl groups will add much more. Progressing

through zero, one and two methyl groups (2.1, 2.2, 2.14) results in a bond lengthening of

only 0.001 A, and differences in ka of only 0.056 mdyn/A. Increasing steric strain by starting

with four methyl and sequentially adding two isobutyl groups (2.17), two isopropyl groups

(2.15), and two tert-butyl groups (2.16) increases bond lengths by 0.098 Aand ka’s by 1.669

mdyn/A. The affect is much larger, as predicted.

Group III: (2.18 - 2.30) Multiple Bonds: Two interactions play a role in describing CC

single bonds. The bonds are lengthened by the addition of bulky groups, and are shortened

by changes in hybridization of the carbons of interest, with increased ‘s’ character on the

bonded C’s resulting in shorter bonds. Hence, the shortest C-C bond in this study is the

sp-sp single bond in acetylene (2.24.) There are also several molecules in this series where

-C≡CH bond is replaced with the -C≡N bond of a cyano group. The species with the cyano

triple bonds display slightly longer C-C single bonds, in agreement with experimental results.

The hybridization changes have a greater affect on the final result than the steric hinderance

changes.

With these results, we have a working model of the bond length/local mode stretching

force constant relationship.

12

Chapter 3

General Characterization of Carbon-Carbon Bond Strength

3.1. Extension of the Single-Reference Description to Unconventional Systems

In Chapter 2, we established a single-reference description using a training set of thirty

relatively unperturbed, well-behaved CC single bonds. We are now interested in extending

that work to systems which are not as easily investigated experimentally, and which have

been outliers in other interpretation schemes.

3.2. Refinement of Single-Reference Descriptions

We will examine twenty-three exceptional molecules, to determine the quality of our

methodology. The new species are shown in Figure 3.1 and the associated calculated results

and experimental results from the scientific literature are shown in Table 3.1.

For molecules which have double and/or triple bonds, such as 2.27 and 2.29, the local

mode ka values can be obtained for both the single and multiple bonds. Experimental BDH’s

for the multiple bonds are not available in the scientific literature. For clamped systems, such

as 3.31 and 3.32, ka’s can be determined where BDH determinations may be impossible.

Figure 3.2 give the relationship of ka to BSO, and Figure 3.3 gives the relationship

between the calculated bond length (rCC) and BSO. This shows how this method can be

applied to a wide range of molecules.

Group IV: ( 3.31 - 3.33) Clamped bonds: Three clamped bond cases [77, 138, 171, 196] are

included in this study. Very long bonds are possible for these molecules due to their arrange-

ment of stabilizing, unstressed bonds which hold the molecule into bonding alignment. The

biradicaloid structures which would be formed by homolytic bond cleavage would recombine.

13

C C

Cl

ClClCl

ClCl

3.35

C C

Cl

ClClH

HH

3.34

C C

H

HHH

HH

3.37

C C

H

HHH

H

H

D3d C2h

3.38

C

C

C C

C

C

3.32 3.33

3.36

CNC

C

MeMe

H

H H

3.39 3.40 3.41

3.42 3.43 3.44

Group IV

Group VI

Group VII

Group V

C

H

HH

3.45

C

Me

HH

3.46

C

H

MeMe

3.47

C

Me

MeMe

C

H

3.49

C

CH2

H

3.50

C CH

3.51

CN

3.52 3.53

3.48

3.31

N

FeN

N

N

N

Fe

N

NN

O

O

ClCl

Figure 3.1: Organic molecules investigated in this work. The single and multiple bondsreported are in red. 14

No. Bond ID Rcalc Rexp BDHcalc BDHexp ka ωa BSO ρ Hb

3.31 bianthracene 1.642 1.64 N/A N/A 2.411 825.82 0.637 1.317 -0.857

3.32 tetra-Ph-acenaphthene 1.708 1.754 N/A N/A 1.788 711.09 0.502 1.153 -0.620

3.33 IRON 1.6507 N/A N/R N/A 1.617 676.40 0.462 1.344 -0.866

3.34 H3C-CCl3 1.512 1.555 86.94 88.3 4.154 1084.04 0.988 1.747 -1.615

3.35 (Cl3C)2 1.583 1.564 73.42 70.1 2.944 912.63 0.749 1.574 -1.181

3.36 tBu-C≡N 1.472 1.46 119.2 115.8 4.433 1119.76 1.041 1.787 -1.852

3.37 CH3-CH+3 D3d 1.935 N/A 41.20 N/R 0.894 502.92 0.286 0.523 -0.194

3.38 CH3-CH+3 C2h 1.591 N/A 42.67 N/R 0.971 524.03 0.306 1.232 -0.929

3.39 a diAd-Ad 1.614 1.66 78.63 N/R 2.787 887.94 0.717 1.422 -0.983

3.40 a diAd-diAd 1.648 1.649 71.41 ≈ 71 2.400 842.03 0.636 1.326 -0.843

3.41 a triAd-Ad 1.657 1.661 64.26 N/R 2.245 796.95 0.603 1.305 -0.812

3.42 a triAd-diAd 1.693 1.70 N/R N/R 1.874 728.08 0.521 1.218 -0.694

3.43 a triAd-triAd 1.787 N/A N/R ≈ 36.1 1.142 568.43 0.351 1.014 -0.465

3.44 a tetraAd-diAd 1.695 1.707 N/R ≈ 70.0 1.861 725.59 0.519 1.211 -0.687

3.45 Me-Ph 1.504 1.521 103.36 103.9 4.528 1131.75 1.059 1.745 -1.581

3.46 Et-Ph 1.506 1.524 101.75 102.3 4.446 1121.46 1.044 1.751 -1.585

3.47 iPr-Ph 1.514 1.515 124.95 102.1 4.272 1099.32 1.011 1.732 -1.540

3.48 tBu-Ph 1.530 1.524 99.40 97.4 3.966 1059.21 0.952 1.681 -1.439

3.49 Ph-cycloC3H5 1.486 1.52 138.91 111.9 4.751 1159.30 1.101 1.815 -1.708

3.50 Ph-CH=CH2 1.471 1.475 161.97 116.9 4.918 1179.45 1.132 1.867 -1.819

3.51 Ph-C≡CH 1.430 1.436 174.81 140.7 5.750 1275.41 1.284 1.961 -2.112

3.52 Ph-C≡N 1.433 1.438 166.05 132.7 5.569 1255.09 1.252 1.929 -2.116

3.53 Ph-Ph 1.482 1.48 267.19 118.0 4.850 1171.33 1.120 1.840 -1.752

Table 3.1: Calculated and experimental Carbon-Carbon Bond Lengths R [A], Bond Dissoci-ation Enthalpies BDH [kcal/mol], local mode stretching constants ka [mdyn/A], local modefrequencies ωa [cm−1], Bond Strength Orders BSO, and Energy Densities [Hartree/A3] forall molecules, ωB97X-D/aug-cc-pVTZ.

aωB97X-D/cc-pVTZ N/A: Not Applicable. N/R: Not Reported.

The calculated CC bond length for 3.31 agrees well with experiment, giving confidence

to the result. The ka and BSO values for 3.31 fit neatly with other molecules of this bond

length.

For 3.32, however, the calculated re (1.708 A) is much shorter than the experimental

value of 1.754 A, obtained by low temperature x-ray crystallography. This is evidence of

multi-reference character for this molecule, which is being currently investigated by our

group.

15

All Single BondsDiamondoidsClamped BondsMultiple Bonds Aromatic Bonds e- Deficient Bonds

Aromatic Bonds

Double Bonds

Triple Bonds

Bond

Stre

ngth

Ord

er

0

0.5

1.0

1.5

2.0

2.5

3.0

Local Mode Stretching Force Constants [mDyne/Å]0 2 4 6 8 10 12 14 16 18

Figure 3.2: Local Mode Stretching Force Constants (ka) to Bond Strength Order (ωB97X-D/aug-cc-pVTZ), Single, Multiple and Aromatic Bonds. This plot serves as a conversionchart between the two parameters.

Species 3.33 is modified from a published structure [77] which has eight phenyl groups

attached to the SP2-hybridized carbons of the outer 5-membered rings. Here those phenyl

groups were replaced with methyl groups, to reduce computational cost. The resultant re

(1.651 A) is significantly shorter than the reported length for the octaphenylated compound.

It may be that the steric hindrance and/or electronic contributions of the phenyl rings have

a strong effect on the central CC bond. Our theoretical case fits the curve, showing that

local mode analysis describes these three clamped bond cases well.

Group V: (3.34 - 3.38) Electron Deficient Bonding: The electron density of a bond can

be reduced by 1) attaching electron withdrawing groups or 2) removing electrons to form

a cationic species. In this work (3.34 and 3.35), three and six electronegative Cl atoms

16

1

234 5

67

8

9

1011

12

13

31

32

34

35

3738

14

36

15

16

17

3940

4142

43

4433

Single Bonds Clamped Bondse- DeficientCH3- StrainDiamondoidCH3-CH+

3 C2h

Bond

Stre

ngth

Ord

er

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

C-C Bond Length (rCC) [Å]1.5 1.6 1.7 1.8 1.9

Figure 3.3: Bond Length (ωB97X-D/aug-cc-pVTZ) to Bond Strength Order for CC singlebonds. [139,140,239–241] (R2 = 0.9955)

are added, respectively, to ethane as electron withdrawing groups, to determine if bond

lengthening is observed. However, for 3.34, we find a significantly shortened bond length

of 1.512 A, in excellent agreement with the experimental value, and the empirical rule of

Sugie, et al. [202], who found that addition of Cl atoms will shorten a CC bond. The C-Cl

bonds have partial multiple bond character, resulting in an increase of s-character in the

normally sp3 hybridized C’s, which would be expected to shorten the CC bond. In Table

3.2, CC bond lengths of 1,1,1-trihaloethane species for H, F, Cl, Br and I are calculated at

the ωB97X-D/3-21G level of theory, showing that the CC bond for these species increase

with increasing halogen size. This bond lengthening is likely attributable to steric strain, as

decreased electronegatively should shorten the bond.

17

Species Bond Length [A]

CH3CH3 1.540

CH3CF3 1.496

CH3CCl3 1.506

CH3CBr3 1.517

CH3CI3 1.536

Table 3.2: Calculated Carbon-Carbon Bond Lengths (re) for various 1,1,1-Trihaloethanes,ωB97X-D/3-21G.

In comparison, hexachloroethane (3.35) shows a bond lengthening of 0.06 A, compared

to the parent compound. From what is learned from 3.34, it must be deduced that this

CC bond is being lengthened by the steric bulk of the six Cl atoms, rather than by electron

withdrawing effects.

The cyano group, -C≡N, is known to be an electron withdrawing group. However, 3.36

has a short rCC of 1.472 A. This is attributable to increased s-character of the CC single

bond by the C≡N triple bond. This is in agreement with the bond shortening observed in

2.26, the methyl derivative of 3.36.

The more interesting case is the ethane cation (3.37 and 3.38), [74,91,180] because this

system, being a charged species, differs substantially from the original list. Experimentally,

the ethane cation is found to have two geometries, differing only slightly in energy by 0.3

kcal/mol. [109,116,151] The ethane D3d geometry (3.37) yields an ultra-long bond of 1.935

A. However, the lower energy configuration, symmetry C2h (3.38), has two anti H’s with

C-C-H angles of 83.1◦, indicating H-bridging bonding, and a CC bond length of 1.591 A,

much closer to the value for the neutral compound 2.1.

To determine which of these species is appropriate for inclusion in this study, we applied

the Cremer-Kraka criteria for covalent bonding, as described in section 9.2 and elsewhere [55].

Electron density (ρr) and energy density (Hb) calculations show that 3.37 and 3.38 have

the necessary maximum electron density path and bond critical points, reporting ρ values of

0.523 and 1.232 e/A3 and Hb values of -0.194 and -0.929 Hartree/A3, respectively. That is,

18

the energy densities are negative, and stabilizing. This makes both molecules viable additions

to this study. For 3.37 the ka versus re data point fits neatly onto the curve reported here,

and defines a BSO of 0.286. This extends the curve well beyond Zavitsas’ theoretical CC

bond length limit of 1.75 A [236] to nearly 2.0 A. The H-bridged 3.38 gives a BSO of 0.306,

and the ka versus re data point falls far from the curve defined by the neutral species (see

Figure 3.3.) The much shorter bond is not significantly stronger, indicating that the bond

is shortened by H-bridging, without significantly stabilizing the electron deficient C-C bond.

The molecule in this symmetry was excluded from the ‘electron deficient bond’ Group III,

and the ‘clamped bond’ category of Group II.

Group VI: (3.39 - 3.44) Diamondoid dimers: We established in Chapter 2 that adamantyl

groups significantly lengthen a CC bond. Fokin, Schreiner and other [76] recently synthe-

sized, characterized, and computationally investigated a series of diamondoid homo- and

heterodimeric compounds connected by exceptionally long CC bonds. Along with the ex-

ceptionally long central bonds, the compounds are found to have exceptional stability, being

stable to temperatures greater than 200◦C. They note that free radical fragments resulting

from homolytic cleavage are capable of very little geometry relaxation or rehybridization,

losing what the authors referred to as “radical-stabilizing effects.” [229] This is an excellent

test case for bond-strength determination, as observed by Shreiner, Kaupp, and others. [130]

In the case of 3.44, the bond length of 1.707 A and estimated BDE of 70 kcal/mol is “well

off the line!” of Zavitsas’ empirical relationship. [236] They also propose a compound (3.43)

which has evaded synthesis, but for which they theorize the longest possible CC bond.

One can say that the BDH for these diamondoid dimers approaches the intrinsic bond

dissociation energy for homolytic cleavage, as defined by Cremer and others [60] because of

the lack of geometry relaxation and a lessening of rehybridization and steric strain reduction.

Hence, it is correctly predicted that the near-intrinsic BDE would be much greater than what

is observed with more conventional molecules, which experience steric strain reduction and

rehybridization.

19

We calculated 3.39 through 3.44 by ωB97X-D/cc-pVTZ. This level of theory reproduces

the experimental and computational results well for all six molecules. However, it is discov-

ered for 3.43, that the previously calculated rCC of 1.83 A was reproducible, but is not a

global minimum. A second rotational isomer was discovered, with an rCC-value of 1.787 A ,

which is -17.8 kcal/mol lower in energy.

Once again, the calculated rCC’s and ka’s for 3.39 through 3.44 neatly fit our curve. As

with 2.12, species 3.43 is excluded from the curve, as this species has not been isolated.

Hence, these cases which confound all past relationships work well by local modes analysis.

Group VII: (3.45 - 3.53) Aromatic Bonds: The hybridization of aromatic C-C bonds short-

ens the bond similarly to the trends found with the sp2 hybridized alkene derivatives. The

phenyl group shortening is consistently less than the shortening for a vinyl group. This

could be attributable to aromatic sp2 hybridization being different from isolated sp2 hy-

bridization, or to the vinyl group causing less steric strain than a phenyl group. Bonding

the aromatic carbon with an sp2 or sp hybridized carbon shortens the bond and, once again,

substitution of -C≡CH with -C≡N results in a small lengthening of the bond. It is also

notable that cyclopropylbenzene (3.49) has a significantly shorter aromatic bond (by 0.028

A) than cumene (3.47), but longer than styrene (3.50), showing the partial sp2 character

of the cyclopropyl system. Finally, biphenyl (3.53) exhibits a longer bond than 3.50. This

could be attributable to differences in sp2 hybridization of aromatic versus isolated carbons,

or to increased aromatic character in 3.50, due to the molecule being more nearly planar.

Interference between the vinylic H’s of 3.50 results in a dihedral angle of 11.4◦, versus 41.4◦

for 3.53.

3.3. The Shortest CC Single Bonds in Chemistry

Seeking the longest CC bond in chemistry obviously involves the investigation of all CC

single bonds. CC double and triple bonds will be shorter than CC single bonds, and need

not be considered. Compiling the calculated bond lengths for CC single bonds between C’s

20

hybridized by multiple and/or aromatic bonds (Table 3.3), we find that the shortening effect

on the bonds is strongest for greater s-character, following the order: sp > sp2 > aromatic

> sp3. This quantified result is in agreement with experiment and chemical deduction, and

the shortest CC bonds are those between two sp hybridized carbons, 2.24 and 2.25. This

is a reasonable summarization of the effects which shorten a CC single bond.

Hybridization sp3-sp2 sp3-sp sp2-sp2 sp2-sp sp-sp Ar-sp3 Ar-sp2 Ar-sp Ar-Ar

Bond Length [A] 1.495 1.455 1.457 1.425 1.372 1.504 1.471 1.430 1.482

1.497 1.455 1.430 1.375 1.506 1.433

1.501 1.514

1.520 1.530

Average

Bond Length [A] 1.503 1.455 1.457 1.428 1.374 1.514 1.471 1.432 1.482

Table 3.3: Calculated Carbon-Carbon Bond Lengths (re) for C-C Single Bonds hybridizedby multiple or aromatic carbons, ωB97X-D/aug-cc-pVTZ

3.4. The Longest CC Single Bonds in Chemistry

More interesting are the effects which lengthen a CC bond. Bulky, sterically hindering

groups lengthen bonds. With this study, we discover that steric size runs in the order:

H < methyl < ethyl < phenyl < tert-butyl < adamantyl < di-, tri-, tetra-adamantyl

Adamantyl (diamondoid) substituents create the largest steric lengthening effect, but

a much smaller bond breaking effect, due to the restrictions on the geometry relaxation

and rehybridization of the fragments. This results in long bonds with exceptional stability,

which approach the true intrinsic bond dissociation energy. Bond lengthening by methyl or

tert-butyl groups is consistently smaller in magnitude.

The clamped bond systems investigated report bond lengths comparable to the longest

Diamondoid dimers, but for very different reasons. The restrictions are against bond cleav-

age, rather than against fragment relaxation. Very long bonds are possible here, but care

must be taken not to assign covalent bond character to biradicaloid species. The use of

21

the Cremer-Kraka bond criteria prevents this misrepresentation, where a spin-paired birad-

ical species having a positive Hb, would confirm the presence of an electrostatic interaction,

rather than a covalent bond.

Electron deficiency results in the greatest lengthening of CC bonds, although the intro-

duction of chlorines as electron withdrawing groups does not have this effect. In the case of

the ethane cation, the D3d symmetry gives the longest CC bond observed in this study.

What would create the longest bond possible? Combining the bond lengthening effects

should produce longer CC bonds. An electron deficient diamondoid or clamped bond should

be a longer bond, but which would be the longest? To investigate this phenomenon, cationic

forms of 3.32 and 3.41 were calculated, at the ωB97X-D/cc-pVTZ level of theory.

For the clamped system 3.32, which is highly aromatic and delocalized, the positive

charge from the removal of one electron was shared over the entire molecule, resulting in

an increase in CC bond length from 1.708 A to 1.709 A. A change this small could not

be exploited to achieve significant bond lengthening. Upon removal of an electron, the

diamondoid system (3.41) dissociated into two fragments, rather than lengthening the bond.

Therefore, the effect of electron deficiency is clearly strongest on the smallest molecules,

making the ethane cation the longest bond in chemistry.

22

Chapter 4

Characterization of Multi-reference Systems by Single-reference Density Functional Theory

- Pancake Bonding

4.1. Refinement of Single-Reference Calculations - Broken Symmetry

Having established that DFT local mode analysis works well for the characterization of

all types and lengths of singlet CC bonds, as well as the ethane cation, we then applied this

tool to known organic multi-reference systems.

Open-shell, free radical, pancake bonded dimers are such a system. They have been

extensively studied, due to their applicability to the development of conductive polymers

[25, 108, 120, 133, 157, 221]. Pancake bonding involves the stacking of relatively flat free

radical dimers, which can not react to form a single covalently bonded dimer (σ-dimer), due

to steric hindrance. Pancake bonded systems are often compared to π-stacking interactions,

[10,87,88,96,114,160,198,219–221] but differ in that the interatomic CC distances are shorter

than the sum of the van der Waals radii for carbon (< 3.4 A) and the molecules align in a

specific, minimum energy orientation, where π-stacked species have a low rotational barrier

between the monomers. In both cases, the dissociation enthalpies for the complexes are

estimated to be much less than those of a normal covalent bond.

Mulliken and Muller proposed the existence of these molecules in the 1950’s, [161, 162]

although the term “pancake bonds” did not appear in the literature until 1969. [163] They

first presented the tetracyanoethylene anionic dimer (TCNE2−2 ) as a case of pancake bonding

between carbons. Being ionic, charged counterions are necessary to stabilize this system.

More pertinent to the study of long CC bonds are the neutral dimers. Here, we examine

six species; five known compounds [25,81,87,88,90,96,204], and one theoretical compound.

23

S

N S

NC

H

S

N S

NC

H

Se

N Se

NC

H

Se

N Se

NC

H

Te

N Te

NC

H

Te

N Te

NC

H

4.1 4.2 4.3 (theoretical)

4.4 4.5 4.6

CH3

CH3

H3C

H3C

H3C CH3

Figure 4.1: Pancake bonded molecules investigated in this work. 4.1) HCNSSN dimer.4.2) HCNSeSeN dimer. 4.3) HCNTeTeN dimer. 4.4) phenalenyl dimer. 4.5) 2,5,8-trimethylphenalenyl dimer. 4.6) 2,5,8-tri-t-butylphenalenyl dimer. Pancake Bonding In-teractions are displayed in red.

The known species, for which experimental data is available, are the 1,2,3,5-dithiadiazolyl

(4.1), 1,2,3,5-diselenadiazolyl (4.2), phenalenyl (4.4), [62] 2,5,8-trimethylphenalenyl (4.5),

and 2,5,8-tri-tert-butylphenalenyl (4.6) radical dimers. X-ray crystal structures and spectro-

scopic data are available for these compounds. [25,81,90,204,235] The 1,2,3,5-ditelluradiazolyl

free radical dimer (4.3) is not known, but is a logical extension of the dichalocodiazolyl sys-

tem and is included as a theoretical case. The phenalenyl dimer (4.4) exists as a fluxional

compound between the covalently σ-bonded dimer and the pancake dimer. [160, 197, 219]

This is a problem for experimental parameterization. The addition of alkyl groups to block

σ-dimer formation, as in 4.5 and 4.6, solves this problem.

The known systems were verified experimentally [81, 90, 214] to be diamagnetic, indi-

cating a singlet electronic ground state, with all electrons spin paired. This is positive

indication of the multi-reference character of these systems, making them a useful test case

for single-reference methodology. Others report that the interaction between these molecules

exhibit covalent bonding character, in which the spin-paired singly occupied molecular or-

24

bitals (SOMO) combine to form a filled highest occupied molecular orbital (HOMO) and

a corresponding empty antibonding LUMO. [198, 214, 221] The interactions occur at rigid

rotational geometries, due to SOMO-SOMO overlap, explaining the rotational stability of

the dimers. The importance of dispersion or van der Waals interactions for the overall

stabilization of the dimers have been suggested. [25,157,198]

The three phenalenyl free radical monomers are stabilized by aromaticity, with a to-

tal of 13 delocalized electrons. Thus, the dimers will have a total of 26 electrons, which

suggests a Huckel-allowed [111] (4n + 2) system. We will investigate these characteristics

computationally.

4.2. Characterization of Pancake Bonding Interactions

For the pancake bonded dimer to form, the spin-paired singlet state must be energetically

favored over the triplet state. The total energy of the triplet state can be calculated by two

different methods: 1) The dimer is optimized in the singlet state, and the energy of the

singlet geometry in the triplet state can be calculated, or 2) the dimer can be re-optimized

in the triplet state, to a new stable geometry. This results in two different parameters,

ESOMO and Egap, which are defined by the following equations:

ESOMO = E(singlet) − E(triplet, singlet geometry) (4.1)

Egap = E(singlet) − E(triplet, triplet geometry) (4.2)

Following Mou and Kertesz, [158] ESOMO defines the energy associated with the spin

pairing and bonding overlap of the SOMO’s. The ESOMO values for these molecules show

significant contributions to the stabilization of the pancake bonded systems. Egap quantifies

the difference in energy between the triplet and open shell singlet states for the dimer.

Grafenstein and others [92] have determined that the Egap must be small for broken symmetry

DFT to accurately describe an open shell system. The Egap is small in all cases, as reported

25

in Table 4.1. The energy difference between the electronic states is small in all cases, but for

4.3 in the C2v symmetry common to 4.1 and 4.2, the triplet state was lower energy than

the singlet state, indicating an unstable arrangement. When the singlet state was allowed to

relax from C2v symmetry, a lower energy structure of C2 symmetry was found, with BCP’s

between the N-N, Te-Te and N-Te atoms (Figure 4.3.)

symmetry symmetry singlet triplet OPT

dimer monomer CC Distance CC Distance ESOMO Egap

4.1 HCNSSN C2v C2v 3.036 3.452 -15.61 -2.17

4.2 HCNSeSeN C2v C2v 3.119 3.208 -13.90 -2.09

4.3 HCNTeTeN C2v C2v 3.165 3.362 -13.26 0.96

4.3 HCNTeTeN C2 C2 3.563 3.104 -8.46 -1.35

4.4 Phenalenyl Dimer D3d C3h 3.152 3.622 -12.97 -5.98

4.5 tMP Dimer D3d C3h 3.093 3.744 -19.26 -5.44

4.6 tTBP Dimer S6 C3h 3.281 3.855 -6.11 -3.13

Table 4.1: Energy Gap results for all Species. CC Distance [A], Triplet/Singlet Gap[kcal/mol]

In the triplet state, the optimized geometry of 4.3 was with the monomers rotated

72◦ about the CC interdimer axis, to give a C2 alignment where N-N and N-Te bonding

were prevalent, and the CC interatomic distance decreased from 3.563 to 3.104 A. This

indicates attractive interactions between the monomers which are not related to SOMO-

SOMO overlap. This observation is consistent with the findings of Gleiter and Haberhauer

in their calculations on dithiatriazines. For those six-member ring molecules which can

be reoriented to optimize different interactions, SN and SC chalcogen bonding was found

be stronger than SS bonding. [96] This gives evidence that chalogen bonding, electrostatic

attraction between a chalcogen and a less electronegative atom, plays a significant role in

the stabilization of the dichalcodiazolyl dimers.

26

When 4.5 and 4.6 were in the triplet states, the interatomic distances between the central

carbon atoms once again increased from 3.093 and 3.281 A to 3.744 and 3.855 A, respectively.

There were no changes in rotational alignments between the two species, because they are

hindered by the pendent methyl or tert-butyl groups. However, for molecule 4.4, the triplet

geometry exhibited three minima (Figure 4.3.) Staggered and eclipsed geometries were

observed, with the staggered configuration being -1.7 kcal/mol lower in energy than the

eclipsed conformer. These systems have central CC interatomic distances of 3.62 and 3.89 A,

respectively, which is significantly longer than the sum of the van der Waals radii. However,

the most stable arrangement was found to be an intermediate structure with a rotational

dihedral of 40.5◦. This geometry results in the shortest CC interatomic distance of 3.58 A,

and is -0.4 kcal/mol lower in energy than the staggered rotamer. Thus it is shown that the

triplet phenalenyl dimer is a π-complex, because it has separation greater than the sum of

the van der Waals radii for carbon, and a low rotational barrier.

H

H

H

H 99.2º

88.5º

(a)

(b)Figure 4.2: C2 Symmetry geometries for the HCNTeTeN dimer. a) Singlet. b) Triplet.

27

3.62Å 3.89Å 3.58Å

(a) (b) (c)

40.5°

Figure 4.3: Triplet state geometries for the phenalenyl dimer. a) Staggered. b) Eclipsed. c)Minimum energy geometry.

Dimer Dissociation Energy: The dissociation energies for all species were calculated

by varying the geometry of each pancake bonded dimer from 2.5 A to 8.0 A, applying

counterpoise corrections to the final results, to correct for basis set superposition error, which

may be expected for dimeric systems. [10, 25, 62, 198, 221] For the dichalcodiazoyl systems,

the selected levels of theory yielded smooth Morse potential curves, exhibiting dissociation

energies of 5.8, 4.7 and 6.0 kcal/mol, respectively, in good agreement with the available

experimental bond dissociation enthalpy (BDH) of 5.32 kcal/mol for 4.1 [25] These values

are more similar to the dissociation energy for an electrostatic interaction than a covalent

bond (Figure 4.4). The calculated CC interatomic distances of 3.183, 3.343 and 3.165 A are

again in excellent agreement with experiment (see Table 4.2.)

The dissociation energies of 8.9, 12.3 and 9.8 kcal/mol for 4.4, 4.5 and 4.6, respec-

tively, are in good agreement with the experimentally determined enthalpy change (∆HD)

of 9.5 kcal/mol for 4.6. [198] The difference between the dissociation energies, 4.5 being

3.4 kcal/mol more stable than 4.4, gives a measure of the dispersion stabilization of 4.5

due to the methyl groups. The smaller difference between 4.6 and 4.4 indicates that there

is a trade-off between the dispersion stabilization and the steric repulsion of the tert-butyl

groups.

28

CH3

CH3

H3C

H3C

H3C CH3

HCNSSNHCNTeTeN

HCNSeSeN

Phenalenyl Dimer

tri-tert-Butylphenalenyl Dimer

tri-Methylphenalenyl Dimer

Rel

ativ

e En

ergy

[k

cal/m

ol]

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

Interatomic Distance (C-C) [Å]2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

Figure 4.4: Dissociation Curves for sytems 1 and 2 (BS-UM06/6-311G(d,p), 3 (BS-UM06/SDD), 4 and 5. (BS-UM05-2X/6-31++G(d,p).)

What can be deduced from these calculations is that 1) all dissociation energies are much

lower than the BDE’s for a covalent bond, 2) the higher dissociation energy for 4.5 shows that

the substituted species has a more stable equilibrium geometry, and 3) the highest dissocia-

tion energy for 4.6 shows the effect of reduced steric repulsion. 4) The attractive dispersion

interactions between the tert-butyl groups are greater than the exchange repulsion between

them, and the smaller methyl groups also contribute attractive dispersion interactions.

Cremer-Kraka Bonding Criteria: We applied the Cremer-Kraka covalent bonding criteria

to all species (Section 9.2.) For the dichalcodiazolyl dimers, bond critical points (BCP) are

found between the CC, NN [189, 190] and EE (calcogen-chalcogen) [168] pairs in all three

systems. In all cases, the CC and NN Hb’s are positive. All three systems also display

N-N-chalcogen-chalcogen ring critical points (RCP) and cage critical points (CCP’s) at the

geometric center of the dimeric system. The RCP gives the point of lowest electron density

in the center of a ring of bonded atoms, showing that bonding is continuous around the ring.

29

The CCP gives similar information at the center of a three-dimensional structure. Therefore,

there are interactions between all heavy atoms.

No. Species rcalc rexp DEcalc DHcalc DECP CECP BDEexp ka ωa BSO ρ Hb

1 HCNSSN (ring) 3.071 5.8 3.9 2.3 2.4 5.3 0.657 146.5 0.214 0.0163 0.00464

C-C 3.036 3.18 0.208 242.6 0.083 0.0406 0.00701

N-N 3.034 0.128 176 0.056 0.052 0.00369

S-S 3.125 0.192 142.6 0.078 0.1036 -0.00031

2 HCNSeSeN (ring) 3.21 4.7 1.8 2.9 2.6 N/R 0.302 71.6 0.113 0.0148 0.00372

C-C 3.119 3.31 0.08 150.8 0.038 0.0343 0.00609

N-N 3.152 0.074 133.8 0.036 0.0421 0.00334

Se-Se 3.313 0.151 80 0.064 0.0983 -0.00081

3 HCNTeTeN C2v (ring) 3.514 6.0 3.4 4.3 4.2 N/A 0.049 23.4 0.021 0.013 0.00132

C-C 3.219 N/A 0.029 83.4 0.014 0.0364 0.00555

N-N 3.333 0.032 29 0.016 0.0385 0.00605

Te-Te 3.84 0.045 122.6 0.022 0.0729 0.00151

HCNTeTeN C2 (ring) 3.413 8.4 5.8 6.1 6.2 N/A 0.162 42.8 0.062 0.0176 0.00186

N-N 3.342 0.112 164.8 0.045 0.0461 0.00931

Te-Te 3.82 0.038 65 0.018 0.0864 0.00334

N-Te 3.51 0.045 77.8 0.021 0.0688 0.00684

4 phenalenyl (periphery) 3.11 N/A 11.0 9.8 8.9 9.0 N/A 0.366 122.9 0.136 0.0718 0.00537

center 3.152 N/A 0.293 287.8 0.113 0.0631 0.00607

5 tMP (periphery) 2.997 3.053 14.8 10.8 12.3 12.9 N/R 0.172 63.8 0.074 0.0903 0.00577

center 3.093 3.145 0.167 217.3 0.072 0.0703 0.00657

6 tTBP (periphery) 3.391 3.306 12.4 10.3 9.8 11.5 9.5 0.194 67.6 0.081 0.0467 0.00334

center 3.287 3.201 0.147 204 0.065 0.0501 0.00499

Table 4.2: Calculated and experimental Carbon-Carbon interatomic distances R [A], Dis-sociation Energies, Counterpoise Complexation Energies and Bond Dissociation EnergiesDEcalc, CECP and BDEexp [kcal/mol], local mode stretching constants ka [mdyn/A], localmode frequencies ωa [cm−1], Bond Strength Orders BSO, and Electron and Energy Densi-ties, ρ and Hb [electron/A3 and Hartree/A3, respectively] for all molecules. Level of Theoryis UM06/6-311G(d,p) for 4.1 and 4.2, UM06/SDD for 4.3, and UM05-2X/6-31++G(d,p)for 4.4, 4.5 and 4.6.N/A: Not Applicable. N/R: Not Reported.

In the case of 4.1 and 4.2, the energy density at the BCP of the intermolecular SS

bonds have negative, very small values of -0.0003 and -0.0008 Hartree/A3, respectively. This

indicates the presence of chalcogen-chalcogen bonding between the SS and SeSe species. [168]

As the pancake bonded species are drawn apart, the molecules tip outward, so that the

chalogens are separated at a slower rate than the carbons and nitrogens. This shows that

chalogen bonding plays a stabilizing role in the dimerization of the known species, which is

30

in agreement with Gleiter and Haberhauer. [87, 88, 96] The tellurium system, 4.3, however,

exhibits a positive Hb value at all interatomic distances, so chalogen bonding is not indicated

in the theoretical case. We theorize that the C2v ditellurodiazolyl dimer does not exist,

because of this lack of chalcogen-chalcogen bonding.

BCP analysis for the phenalenyl free radical dimer systems, 4.4 through 4.6, found BCP’s

between all overlapping CC atoms (one at the center, six on the periphery), with positive

Hb’s for all geometries. The BCP’s persist at separations up to 4 A.

The ρ and Hb values obtained for the substituted 4.5 and 4.6 systems, and the unsub-

stituted (4.4) system are nearly identical, indicating that the results for 4.4 are a good

approximation for 4.5, as has been asserted by many researchers. [10,63,198,214]

The phenalenyl dimers exhibit three CCP’s and fifteen RCP’s, located equidistantly

between the monomers. However, none were located near the central inversion point of the

complex, and therefore did not appear to enhance the overall stability of the complex.

Aromaticity Index: To test how the aromaticity of the system is affected by 1) substi-

tution and 2) dimerization, the monomers and dimers of 4.4 through 4.6 were subjected

to aromaticity index (AI) analysis (Section 9.2.) [123, 191] For each structure, the AI was

determined twice; first, for all CC bonds in the dimer and monomer systems, and then for

all CC bonds except those to the central carbon.

The results of this analysis are summarized in Table 4.3 and Figure 4.5. The figure shows

the bond strength order (BSO) for each unique CC bond in the aromatic systems, and reports

the AI of each monomeric and dimeric system. Additionally, the parameters WS, which gives

the bond weakening/strengthening relative to the average bond strength, and ALT, bond

strength alteration parameter, were reported. Both the WS and ALT parameters indicate a

loss of aromaticity, due to increased irregularity of the structures. In other words, the more

regular (rather than symmetrical) the aromatic perimeter, the greater the aromaticity. An

example is benzene, which has six identical sides, is highly regular, and has a high AI.

Looking first at substituent effects, we found that in 4.4 the six outermost CC bonds

are identical, with BSO’s of 1.412, near the BSO value of 1.451 for benzene. (The BSO

31

molecule AI, all carbons AI, outer carbons

Phenalenyl Dimer 0.934 0.938

Phenalenyl Monomer 0.915 0.915

tMP Dimer 0.918 0.914

tMP Monomer 0.906 0.898

tTBP Dimer 0.908 0.894

tTBP Monomer 0.901 0.885

Table 4.3: Aromaticity Indices for the Phenalenyl, Trimethylphenalenyl and tri-tert-Butylphenalenyl Free Radical Dimers

for benzene is based on comparison to the local stretching force constants for ethane and

ethylene, rather than the ‘classical bond order’ of 1.5.) This arrangement is skewed for 4.5,

where the lowest energy rotational isomer for the methyl groups is the orientation where one

H atom is in the plane of the three rings, and the other two extend above and below the ring.

This makes the environment of the outer bonds dissimilar. The bond which is coplanar with

the hydrogen increases in length by only 0.002 A, but exhibits a large reduction in BSO by

0.032. The outer CC bond not attached to the alkyl group has an even lower BSO, being

reduced by 0.065. For 4.6, the same effect is observed, where the bulky tert-butyl groups

cause a lengthening of the outer CC bonds, and reduction of the BSO by 0.037 and 0.061,

respectively, for the side in which the methyl group is coplanar with the aromatic system

and the side which is not.

For the unsubstituted periphery CC bonds, the effect of substitution is much smaller,

with bond lengths varying from 1.412 to 1.415 A, and the BSO’s range from 1.283 to 1.312.

The largest effect on bond weakening for the target bonds is observed in 4.6.

Conversely, the three bonds which radiate from the central C in each case increase in BSO

strength by 0.005 between 4.4 and 4.5, and by 0.039 between 4.5 and 4.6. This indicates

that electron density lost by the deformation of the outer CC bonds redistributes to the

inner bonds of this system. This results in small fluctuations in the aromaticity from 0.915

to 0.918 to 0.901 for 4.4, 4.5, and 4.6, respectively.

32

FULL

AI 0.915WS 0.066ALT 0.018

OUTER

AI 0.915WS 0.062ALT 0.023

FULL

AI 0.901WS 0.089ALT 0.010

OUTER

AI 0.885WS 0.108ALT 0.008

FULL

AI 0.934WS 0.043ALT 0.024

OUTER

AI 0.938WS 0.035ALT 0.027

FULL

AI 0.908WS 0.081ALT 0.011

OUTER

AI 0.894WS 0.096ALT 0.010

1.336(1.401)

1.379(1.418)

1.390(1.393)

1.285(1.418)

1.304(1.412)

1.359(1.401)

1.380(1.418)

1.375(1.393)

1.305(1.418)

1.283(1.415)

1.412(1.390)

1.336(1.425)

1.292(1.415)

1.441(1.390)

1.337(1.425)

1.312(1.415)

MONOMERS DIMERS

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

H

HH

H

H

H

H

H

H

H

H H

FULL

AI 0.918WS 0.072ALT 0.010

OUTER

AI 0.911WS 0.077ALT 0.012

FULL

AI 0.918WS 0.073ALT 0.009

OUTER

AI 0.914WS 0.075ALT 0.011

1.382(1.393)

1.350(1.421)

1.300(1.414)

1.347(1.398)

1.341(1.423)

1.288(1.414)

1.380(1.392)

1.277(1.418)

1.385(1.393)

1.303(1.414)

Figure 4.5: Bond Strength Orders (BSO) and Optimized Bond Lengths (in parentheses, A)for the Phenalenyl, Trimethylphenalenyl and tri-tert-Butylphenalenyl Radical Monomers andDimers (4.4 through 4.6.) The Aromaticity Indices (AI), Bond Weakening/Strengtheningparameters (WS) and Bond Alteration parameters (ALT) for the full carbon ring structures(FULL) and the outer ring structure (OUTER) are indicated in boxes.

The trend is more pronounced for the pancake bonded dimeric systems. However, the

ALT parameter is less for the substituted dimers, 4.5 and 4.6, than for the unsubstituted

species, 4.4. This indicates that bringing the six methyl or tert-butyl groups together,

alternating between the rings, gives an overall more stable and regular arrangement.

The lowest energy rotational isomer of 4.5 has a different orientation of the methyl groups

than the monomer. In the pancake bonded dimer case, six hydrogens are rotated inward,

toward the center of the molecule. In this arrangement, there are three unique bond types,

as in the monomer and dimer of 4.4. This results in a very small ALT parameter of 0.009

33

for the full system, compared to the large WS weakening/strengthening parameter of 0.073,

and an AI much lower than that of the unsubstituted 4.4 dimer. This trend was continued

in dimer 4.6, showing that adding substituents to the molecule, while essential to preventing

σ-dimer formation, reduces the overall regularity and aromaticity of both the monomers and

the dimers.

Most importantly, in all cases, the dimeric systems display a higher AI than the monomeric

systems. This indicates that dimerization enhances the aromaticity of the systems. This ob-

servation is in agreement with the nucleus-independent chemical shift (NICS) NMR analysis

of Suzuki, et al, [204] and indicates that the SOMO-SOMO overlap in the dimerized system

supports and stabilizes the aromaticity of the molecules, overall.

Gleiter and Haberhauer propose that pancake bonded systems are stabilized by the com-

bination of the delocalized electrons in two dimers to create a Huckel-allowed [111] (4n + 2

electron) 3-dimensional aromatic system. In the case of the dichalcodiazoyl systems, each

monomer contributes 7 electrons, for the allowed total of 14 electrons. On the phenalenyl sys-

tems, each monomer contributes 13 electrons, for a total of 26. This conclusion is supported

by the finding that the dimers exhibit higher aromaticity than the monomer.

The structure of 4.6 is affected by the six tert-butyl groups, which lock the dimer in

the staggered configuration. Steric repulsion between these groups also causes concave pyra-

midalization of the central phenalenyl system, [62] such that the central CC interatomic

distance is shorter than that of the outer CC interactions, unlike 4.4 and 4.5. This results

in an increased deviation from the optimum geometry and bond strengths, as reflected by

small increases in the bond alteration (ALT) parameters of 0.001 to 0.002 units.

Ultimately we discover that the substituents which are added to the parent phenalenyl

system to inhibit σ-dimerization reduce the aromaticity of the systems.

We here confirm that these multi-reference systems can be well characterized by bro-

ken symmetry hybrid DFT methods and large basis sets. Analysis of the parameters of

equilibrium geometries, rCC , local stretching force constant, ka, Bond Strength Order and

electron density, ρr and energy density, Hb, are in excellent agreement with experimental

34

observations, and can fully characterize these systems.

In Chapter 5, we study bridged annulene systems, to determine 1) the level of theory

required to reproduce experimental results, and 2) whether bridged annulene systems show

significant multi-reference character.

35

Chapter 5

Bridged Annulenes; The Longest CC Bonds?

5.1. The Puzzle of 11,11-Dimethyl-methano[10]annulene

Some organic systems display biradicaloid character at much closer CC distances than

pancake bonded systems, requiring different computational approaches.

In this work we focused on the interactions between two C atoms, which are separated by

1.8 A and may be covalently bonded: a molecular system that has puzzled chemists for over

50 years. We used the measurable quantities of electron density distribution, vibrational

frequencies, NMR chemical shifts, and indirect spin-spin coupling constants (SSCCs) to

characterize the CC interactions investigated as being a covalent bond or a through-space

interaction.

The target system 5.1 (see Figure 5.1) belongs to the group of bridged [10]annulenes,

the parent member of which, 1,6-methano[10]annulene (5.2) was first synthesized by Roth

and Vogel in 1964. [227] Vogel then reported that there were two possible bond structures

for this molecule which satisfy the valence requirements of all atoms (valence tautomers.)

Annulene 5.2 was experimentally characterized as an aromatic 10π-system fulfilling the

Huckel 4n+2 rule [111], with a slightly distorted ring perimeter and having a C1C6 distance

R of 2.235 A. [27] Therefore, it adopts structure 5.2a rather than that of the bisnorcaradiene

(tricyclo[4,4,1,01,6]undeca-2,4,7,9-tetraene) form 5.2b. This measurement was verified in

dozens of experimental investigations [14,27,28,32,37,43,66,85,119,137,148,226] as well as

computational investigations. [36, 43, 83, 84, 119, 195] Therefore, it was an unexpected result

when in 1973 the 11,11-dimethyl derivative 5.1, also synthesized by the Vogel group, was

found to adopt a tricyclic structure with an R value close to 1.8 A by X-ray diffraction [26]

(1.827 and 1.771 A were observed in the unit cell which contains two slightly different

36

6

1

11

HH

H HHH

CCHH

H HHH

CC

H

H

C

C

C

C

N

N

H

H

H

HH

H

5.1a 5.1b

5.2a 5.2b

5.3a 5.3b

5.4 5.5

5.6

1

6

13

2

Figure 5.1: Annulene species investigated in this work.

geometries [26]) that suggests structure 5.1b rather than 5.1a. The corresponding cyano,

methyl derivative had a similar R value to 5.1 [26] and finally the dicyano derivative with

an R value of 1.558 A [26, 228] clearly confirmed the existence of a bisnorcaradiene form.

Gunther and others carried out nuclear magnetic resonance (NMR) investigations on

valence tautomeric systems such as 5.1 and 5.2 and showed that any shift to the bisnorcara-

diene or annulene form leads to characteristic changes in the 13C chemical shifts. [8,95] These

studies were latter confirmed by Frydman and co-workers [80] and Dorn and co-workers [69]

who also documented a strong temperature dependence of the 13C NMR signals of 5.1. It

was questioned whether bridged annulenes could be considered as fluxional systems which

changed their bonding structure according to the valence tautomeric rearrangement indicated

in Figure 5.1 or whether they possessed long CC bonds.

Since the available experimental evidence for 5.1 favored the bisnocaradiene form, the

molecule was considered as the neutral hydrocarbon with the longest covalent cyclopropane

C(sp2)C(sp2) bond. Thereafter, 5.1 was computationally investigated multiple times to

rationalize the bonding situation, [36, 43, 83, 84, 119, 195] but none gave results which were

37

consistent with the experimental data.

The following questions will be answered in this work: i) Which electronic factors deter-

mine the shape of the potential? For example, are there specific interactions between bridge

and π-perimeter, which may influence structure and dynamics of 5.1? ii) What level of the-

ory is required to guarantee the needed accuracy of the quantum chemical description? iii)

Do temperature, entropy, or environmental effects such as solvation or crystal packing influ-

ence the shape of the potential energy surface? iv) Does 5.1 contain a weak covalent C1C6

bond? Is this the longest uncharged hydrocarbon C(sp2)C(sp2) single bond ever observed?

5.2. Analysis of the Annulene Systems by Multiple Levels of Theory

In Figure 5.2, the calculated potential energy curves (PEC) for a representative number

of different methods applied in this work are shown. These include wave functional theory

(WFT), DFT, and complete active space self consistent field (CASSCF) methods. The final,

most accurate, enthalpy curves are given in Figure 5.3. Relative energies ∆E, ∆H(298),

and ∆G(298) obtained at different levels of theory are listed in Table 5.1. In figures 5.2a

and 5.3a, ∆E = 0 was chosen for R = 2.1 A. In Table 5.1, the reference point of all energy

difference determinations is always the most stable annulene form. When one of the target

geometries was located on a shoulder of the PEC (or the corresponding enthalpy curve), the

R value of the inflection point or the R of the annulene minimum of a closely related method

was taken, as is indicated in Table 5.1.

In the following, we will first analyze the potential energy curves (PEC’s), potential

enthalpy curves (PHC’s), and potential free energy curves (PGC’s) obtained for target system

5.1. They are rather inconclusive since the methods first tried in this work fail to reliably

account for the experimental results. Therefore, we investigated the valence tautomeric

molecules 5.2 and 5.3 to clarify which quantum chemical method can provide a reliable

answer to the common problems presented by these systems. To improve the accuracy of

the PECs, we will also discuss the impact of vibrational, temperature, entropy, solvent and

38

HH

H HHH

CCHH

H HHH

CC

5.1a5.1b

BD(T) CCSD(T)

BD CCSD

ωB97X-D REKS/ωB97X-D

B3LYP

CASSCF(10,10)

CASPT2(10,10)

NEVPT2(10,10)MP2//MP2

MP3

MP4(SDQ)

MP4(SDTQ)B2PLYPD

MP2//B3LYP

Rel

ativ

e En

ergy

ΔE

[k

cal/m

ol]

-10

-5

0

5

10

15

20

25

C1,C6 Interatomic Distance [Å]1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6

(a)

B3LYPBD(T) CCSD(T)

BD CCSD

wB97X-DCASSCF(10,10)CASPT2(10,10)

NEVPT2(10,10)

MP2//MP2

MP4(SDTQ)

5.2a5.2b

B2PLYPD

MP4(SDQ)MP3

MP2//B3LYP

Rel

ativ

e En

ergy

ΔE

[kc

al/m

ol]

0

5

10

15

20

25

30


(b)

B3LYP

ωB97X-DMP2//MP2 MP4(SDTQ)

BD CCSD

BD(T) CCSD(T)

B2PLYPD

CASPT2(8,8)

MP3

MP4(SDQ)

5.3a5.3b

Rel

ativ

e En

ergy

ΔE

[k

cal/m

ol]

-2

0

2

4

6

8

10

12

14

16

18

20


(c)

Figure 5.2: Representation of the energy as a function of the 1,6-distance obtained at multiplelevels of theory. a)11,11-dimethyl-1,6-methano[10]annulene b)1,6-methano[10]annulene andc) 1,3,5-cycloheptatriene.

39

crystal packing effects on the shape of the PECs. We will answer the question whether 5.1

possesses a covalent CC bond with an R value close to 1.8 A.

The potential energy curve of 5.1 as described by WFT and DFT: The PECs calculated

for 5.1 reveal that the following must be considered as important electronic effects: i) π-

electron delocalization in annulene and bisnorcaradiene ii) σ-π interactions in the distorted

ring perimeter iii) homoaromatic through-bond and through-space interactions (as defined

by Cremer and co-workers [52]) iv) the generation of a biracialoid and its stabilization by

electron delocalization, v) strain effects in the bisnorcardiene form, and vi) bridge-ring in-

teractions. The different methods account for these effects in different ways so that the PEC

varies in form. The forms include a single-well with a shoulder at small R values (SW+1S), a

single-well with a shoulder at large R values (SW+2S), a double-well with a local minimum

for small R values (DW-F1M), or a double-well with a local minimum for large R values

(DW-F2M.)

HF and dynamical electron correlation: Hartree Fock (HF) theory makes the approx-

imation of non-interacting electrons, which does not account for their correlated motion.

Therefore, HF will return the lowest energy for the structure which has the greatest number

of CC bonds, 5.1b rather than 5.1a. HF exaggerates bond alternation because the de-

scription of π-delocalization requires a dynamical electron correlation method. Accordingly,

5.1a corresponds to a shoulder of the PEC(HF,SW+2S), which is 9.3 kcal/mol above the

minimum (Table 5.1).

Møller Plesset Perturbation Theory, 2nd, 3rd, and 4th Order: Second order Møller Plesset

Perturbation Theory (MP2) introduces pair electron correlation effects, which is a prereq-

uisite for a description of π-delocalization. [50] The minimum is shifted to form 5.1a (2.154

A), which is stabilized by 6.3 kcal/mol relative to the value at 1.64 A occupied by form

5.1b (asymmetric SW+1S PEC, see Figure 5.2). There is hardly any difference in the shape

of the PEC for MP2, or higher levels of theory, whether geometries are optimized at the

HF, B3LYP, ωB97X-D, or MP2. So the following discussion is based on B3LYP geometries,

because DFT has advantages when calculating vibrational frequencies (see below).

40

HH

H HHH

CCHH

H HHH

CC

5.1a5.1b

ΔE(R) CASPT2(14,14)ΔH(R) CASPT2(14,14)ΔG(R) CASPT2(14,14)

ΔH(R) est

ΔH(R) CCSD(T)ΔG(R) CCSD(T)

Rel

ativ

e En

ergi

es

[kc

al/m

ol]

-2

0

2

4

6

8

10

12

14

16


(a)

2a2b

ΔE = 11.48

ΔH = 10.16

ΔG = 10.30

Rel

ativ

e En

ergi

es

[kc

al/m

ol]

-2

0

2

4

6

8

10

12

14

16

8

20

22

24


5.2a5.2b

(b)

3a3b

ΔEa = 12.20ΔGa = 12.22ΔHa = 11.53

ΔE = 6.05

ΔH = 5.89

ΔG = 6.18

Rel

ativ

e En

ergi

es

[kc

al/m

ol]

-2

0

2

4

6

8

10

12

14

16

18


5.3a5.3b

(c)

Figure 5.3: Representation of the energy and enthapy as a function of the 1,6-distance of a)11,11-dimethyl-1,6-methano[10]annulene b)1,6-methano[10]annulene and c)1,3,5-cycloheptatriene. a) was obtained by CASPT2(14,14)/6-311G(d,p)// B3LYP/6-311G(d,p) and CCSD(T)/6-311G(d,p)// B3LYP/6/311G(d,p), b) by CASPT2(14,14)/6-311G(d,p)// B3LYP/6/311G(d,p), and c) by CASPT2(10,10)/6-311G(d,p)// B3LYP/6-311G(d,p). In all cases, the relative energy and enthalpy (∆H(298K)) are included.

41

Method Curve a R(5.1b)b R(5.1a) ∆E(5.1b)c ∆E(TS)d

HF SW+2S 1.552 (2.120) -9.29 -

SVWN5 SW+1S (1.640) 2.125 3.26 -

BLYP SW+1S (1.640) 2.212 6.99 -

B97 SW+1S (1.685) 2.134 2.84 -

B3LYP SW+1S (1.640) 2.167 3.67 -

ωB97 SW+2S 1.595 (2.040) -4.28 -

ωB97X SW+2S 1.595 (2.040) -7.03 -

ωB97X-D DW-F2M 1.638 2.039 -1.02 0.07 (1.09)

REKS/ωB97X-D DW-F2M 1.637 2.039 -0.98 0.07 (1.05)

B2P-LYP-D DW-F2M 1.605 2.118 -0.08 1.05 (1.13)

MP2 SW+1S (1.640) 2.154 6.31 -

MP3 DW-F2M 1.587 2.065 -3.10 0.04 (3.14)

MP4(SDQ) SW+2S 1.581 (2.120) -5.28 -

MP4(SDTQ) SW+1S (1.640) 2.150 2.92 -

CCSD SW+1S 1.590 (2.120) 4.25 -

BD SW+1S 1.590 (2.120) 4.21 -

CCSD(T) DW-F2M 1.641 2.120 -0.42 0.62 (1.04)

BD(T) DW-F2M 1.641 2.120 -0.42 0.62 (1.04)

CASSCF(10,10) DW-F2M 1.529 2.148 -7.81 0.63 (8.44)

CASPT2(10,10) DW-F2M 1.613 2.159 -0.97 1.07 (2.04)

NEVPT2(10,10) DW-F1M 1.598 2.172 2.49 3.59 (1.10)

CASPT2(14,14) DW-F1M 1.647 2.130 0.91 1.32 (0.41)

Estimate DW-F2M 1.643 2.123 0.26 0.93 (0.67)

∆H(5.1b) ∆H(TS)

B2P-LYP-D DW-F2M 1.694 2.110 0.61 1.14 (0.53)

CCSD(T) DW-F2M 1.680 2.120 -0.68 0.03 (0.71)

BD(T) DW-F2M 1.684 2.120 -0.75 0.02 (0.77)

CASPT2(14,14) DW-F1M 1.640 2.120 0.66 -

Estimate DW-F2M 1.662 2.125 -0.11 0.08 (0.19)

∆G(5.1b) ∆G(TS)

B2P-LYP-D DW-F2M 1.680 2.114 0.16 0.83 (0.67)

CCSD(T) DW-F2M 1.658 2.080 -1.09 0.20 (1.29)

BD(T) DW-F2M 1.658 2.120 -1.12 0.19 (1.31)

CASPT2(14,14) DW-F1M 1.665 2.120 0.22 0.73 (0.51)

Estimate DW-F2M 1.662 2.100 -0.45 0.42 (0.87)

Table 5.1: Relative Energies and Tautomerization Barriers of 11,11-Dimethyl-1,6-methano[10]annulene.

42

aCurve indicates the shape of the potential energy well. (DW-F1M - Double Well with LocalFirst Minimum, DW-F2M - Double Well with Local Second Minimum, SW+1S - Single Wellwith Left Shoulder, SW+2S - Single Well with Right Shoulder.)bR(5.1b) and R(5.1a) indicate the C1C6 distance for each structure in angstroms. (Valuesin parentheses are approximate values taken from shoulders.)c∆E(5.1b) gives the energy difference between tautomers, relative to 5.1b [kcal/mol.]d∆E(TS), ∆H(TS) and ∆G(TS) give the energy barriers for the tautomerization from 5.1ato 5.1b [kcal/mol.] (Values in parentheses are for the reverse reactions.)All calculations were completed using the 6-311G(d,p) basis set. MP3, MP4, CAS andcoupled cluster calculations were computed with B3LYP/6-311G(d,p) optimized geometries.

MP2 is known to overestimate pair correlation and the stabilizing effects of π-delocalization

since MP2 does not include the coupling between different pair-correlation effects. MP3 ac-

counts for some of the pair-pair coupling effects and accordingly the PEC minimum is shifted

back to form 5.1b which becomes 3.1 kcal/mol more stable than form 5.1a with DW-F2M

PEC. MP4(SDQ) leads to a similar result, however with a somewhat larger energy difference

(5.3 kcal/mol, see Table 5.1) between forms 5.1b and 5.1a than calculated with MP3.

The situation changes qualitatively and quantitatively with the inclusion of triple exci-

tation correlation effects at the MP4 level of theory. The energy minimum of the SW+1S

PEC is again shifted to 5.1a, and 5.1b at 1.64 A is now just 2.9 kcal/mol higher in en-

ergy. Three-electron correlation effects are not only important for a balanced description of

π-delocalization, but also for those forms with a long C1C6 bond close to the breaking point

where these effects are uncoupled and might be exaggerated.

Coupled cluster theory provides infinite order correlation effects so that coupling of the

specified 2- or 3-electron correlations is accurately described. The coupled cluster with

with single and double excitations (CCSD) and coupled cluster with Brueckner orbitals and

double excitations (BD) PEC’s are largely identical and appear between PEC(MP4(SDQ))

and PEC(MP3), leading to a SW+1S form. Infinite order correlation effects in the SD-space

provide a correct description of orbital-relaxation and pair correlation effects, which leads

to a 4.2 kcal/mol stabilization of form 5.1b relative to 5.1a. If infinite order 3-electron

correlation effects are added at the CCSD(T) or BD(T) levels of theory, the MP4(SDTQ)

43

PEC is corrected in the way that the destabilization of 5.1b relative to 5.1a is reduced from

2.9 to -0.4 kcal/mol, i.e. the PEC becomes rather flat in the range from 1.6 to 2.2 A with

a slight preference for 5.1b at R = 1.641 A. Almost the same result is obtained by BD(T)

(see Figure 5.1 and Table 5.1.)

DFT descriptions : In DFT, the larger the electron density or both electron density and

electron density gradient are, the larger are the correlation effects. [49, 101] The analysis of

the electron density distribution ρ(r) in 5.1a and 5.1b (see below) reveals that the bond

density is larger in the case of the annulene over the bisnorcaradiene form because 5.1a has 5

CC double bonds rather than just 4 when adopting form 5.1b. Hence, the larger correlation

energy of 5.1a leads to its stabilization relative to 5.1b where the effect for a generalized

gradient approximation (GGA) functional such as BLYP (dependence on density and its

gradient: SW+1S) must be larger (7.0 kcal/mol, Table 5.1) than that of a local-density

approximation (LDA) functional (dependence on just the electron density value) such as

SVWN5 (3.3 kcal/mol, Table 5.1).

The B3LYP potential curve is close to the SVWN5 curve and somewhat above the

MP4(SDTQ) curve (Figure 5.1; Table 5.1: ∆E = 3.7 compared to 2.9 kcal/mol). This

indicates the dominance of the correlation energy of the BLYP functional, although by the

admixture of LDA correlation and exchange as well as HF exchange (leading to some stabi-

lization of 5.1b) the correlation energy is significantly reduced.

Results obtained with an ωB97-type of functional differ significantly from those obtained

with LDA or GGA, or other hybrid functionals as they predict a minimum for 5.1b being 7.0

(ωB97), 4.3 (ωB97X), or 1.0 kcal/mol (ωB97X-D) below the energy of 5.1a, the latter being

located on a shoulder of the PEC close to 2.04 A. B97 performs in a similar way to all other

GGA functionals (PEC: SW+1S, Table 5.1). Hence, the inclusion of long-range exact HF

exchange (used to reduce the self interaction error of B97 [23,57,172,173]) must be considered

as the cause for this preference of 5.1b. These long range HF-exchange effects play a role

for spin-pairing and the formation of the C1C6 bond. For B97, self-exchange is larger than

self-electron repulsion so that in a situation of large exchange interactions the molecule is

44

artificially stabilized. The approximate exchange of B97 also overestimates interelectronic

exchange, which is relevant for a bond-no bond process. For the C1C6 bonding situation

(lowering exchange in the bonding region) the exchange interactions play a smaller role than

in a non-bonding situation. Therefore 5.1a is artificially stabilized relative to 5.1b. This

is parallel to the correlation effects discussed above: The density gradient is larger in the

no bond situation and leads to a larger α- and β-electron separation via correlation, again

stabilizing 5.1a.

By including exact-exchange for the long-range part of ωB97, the stabilizing effect of

the exchange interactions is significantly reduced, thus leading to a relative stabilization of

5.1b. If also the short-range exchange effects are improved, another stabilization of the

bisnorcaradiene form (from -4.3 to -7.0 kcal/mol, Table 5.1) results. ωB97X-D reduces the

energy difference between 5.1a and 5.1b to -1.0 kcal/mol indicating that dispersion forces

may lead to a 5 kcal/mol stabilization of the annulene form 5.1a. This seems to indicate a

large increase of bridge-ring interactions for increasing R. This can however not be justified

in view of the reduction of the exact exchange of this functional from 40% in ωB97X to 20%

in ωB97X-D with regard to the long and the short-range exchange.

A relatively flat C1C6-potential was obtained using the double hybrid functional B2PLYPD.

By mixing in both HF exact exchange, dispersion corrections, and MP2 pair correlation ef-

fects, the annulene form becomes sufficiently stabilized and a DW-F2M results (∆ E = -0.1;

barrier: 1.0 kcal/mol) so that the overall result is somewhat closer to the experimental ob-

servation suggesting a rapid valence-tautomeric rearrangement with an average R-value of

1.86 A.

Non-dynamical electron correlation effects : We also investigated whether a multireference

description with DFT leads to a significant change utilizing the REKS(2,2) method of Filatov.

[49,101] However, no improvements of the PEC were obtained thus indicating that the bond-

no bond process is not an isolated 2-electron problem but involves all 10 π-electrons of the

annulene perimeter.

45

CASSCF includes multireference effects, but is lacking dynamic electron correlation. We

first used a (10,10) active space, to describe the 10-electron aromatic system with 10 electrons

in 5 bonding and 5 antibonding orbitals. CASSCF(10,10) leads to an overestimation of the

stability of 5.1b by 7.8 kcal/mol similar to the HF result (9.3 kcal/mol, Table 5.1). The PEC

takes a DW-2FM form, which is characterized by a flat energy region from 1.9 to 2.2 A, that

is, an extension of the “annulene shoulder” of the HF potential. Obviously, a full CI on a

(10,10) active space including all π-electrons is not sufficient to describe π-delocalization in a

balanced way, especially when σ-π interactions become important. CASSCF(10,10) corrects

the description of the transition state region of the valence tautomeric rearrangement by

adjusting its energy to that of the annulene form. However, it fails to adjust these energies

to those of 5.1b.

CASSCF with electron correlation added by MP2 theory (CASPT2) with the (10,10)

active space leads to significantly improved results, with the difference between 5.1a and

5.1b reduced to 1 kcal/mol, a consequence of the improved description of π-delocalization

of the annulene form. N-Electron Valence State Perturbation Theory (NEVPT2) is similar

to CASPT2, but additionally takes into account the two-electron interactions of the active

space electrons and excludes intruder states. Both effects are relevant for the description

of the annulene form (2.5 kcal/mol more stable than 5.1b) by improving the description of

10π-delocalization along a distorted parameter and by excluding intruder states (more likely

for 5.1a than 5.1b), which changes the PEC form to DW-F1M, indicating that the annulene

form is more stable. This is the opposite of the PEC(CASPT2) result.

Despite the testing of a large number of different WFT and DFT methods (more than

20), none of the results is in line with the experimental X-ray diffraction and NMR findings.

Therefore the question remains whether other effects such as thermochemical corrections,

entropy contributions, crystal packing or solvent contributions may improve the agreement

between theory and experiment.

46


HF DW-F1M 1.562 2.219 0.63 3.27 (2.64)

B3LYP SW+1S (1.610) 2.280 13.60 -

ωB97 DW-F2M 1.560 2.226 -1.12 1.48 (2.60)

ωB97X DW-F1M 1.546 2.226 2.71 3.39 (0.68)

ωB97X-D SW+1S (1.610) 2.240 7.22 -

B2P-LYP-D SW+1S (1.610) 2.250 9.22 -

MP2 SW+1S (1.610) 2.254 13.93 -

MP3 DW-F1M 1.630 2.240 4.20 4.68 (0.48)

MP4(SDQ) DW-F1M 1.610 2.214 1.88 3.00 (1.12)

MP4(SDTQ) SW+1S (1.610) 2.265 10.48 -

CCSD DW-F1M 1.613 2.235 3.21 4.04 (0.83)

BD DW-F1M 1.612 2.234 3.25 4.06 (0.81)

CCSD(T) SW+1S (1.610) 2.254 7.29 -

BD(T) SW+1S (1.610) 2.256 7.32 -

CASSCF(10,10) DW-F1M 1.540 2.268 4.37 7.65 (3.28)

CASPT2(10,10) SW+1S (1.610) 2.254 5.32 -

NEVPT2(10,10) SW+1S (1.610) 2.230 14.86 -

CASPT2(14,14) SW+1S (1.700) 2.256 11.48 -

∆H(5.2b) ∆H(TS)

B2P-LYP-D SW+1S (1.610) 2.267 12.59 -

CCSD(T) SW+1S (1.700) 2.256 5.60 -

BD(T) SW+1S (1.700) 2.258 5.55 -

CASPT2(14,14) SW+1S (1.700) 2.256 10.16 -

∆G(5.2b) ∆G(TS)

B2P-LYP-D SW+1S (1.610) 2.267 12.23 -

CCSD(T) SW+1S (1.700) 2.256 5.77 -

BD(T) SW+1S (1.700) 2.258 5.74 -

CASPT2(14,14) SW+1S (1.700) 2.256 10.30 -

Table 5.2: Relative Energies and Tautomerization Barriers of 1,6-Methano[10]annulene.

a Curve indicates the shape of the potential energy well (DW-F1M - Double Well with LocalFirst Minimum, DW-F2M - Double Well with Local Second Minimum, SW+1S - Single Wellwith Left Shoulder.)bR(5.2b) and R(5.2a) indicate the C1C6 distance for each structure in angstroms. (Valuesin parentheses are approximate values taken from shoulders.)c∆E(5.2b) gives the energy difference between tautomers, relative to 5.2b [kcal/mol.]d∆E(TS), ∆H(TS) and ∆G(TS) give the energy barriers for the tautomerization from 5.2ato 5.2b [kcal/mol.] (Values in parentheses are for the reverse reactions.)All calculations were completed using the 6-311G(d,p) basis set. MP3, MP4, CAS andcoupled cluster calculations were computed with B3LYP/6-311G(d,p) optimized geometries.

47


HF DW-F1M 1.557 2.395 7.03 12.70 (5.67)

B3LYP DW-F1M 1.644 2.351 7.51 7.89 (0.38)

ωB97 DW-F2M 1.576 2.366 -0.80 5.43 (6.23)

ωB97X DW-F1M 1.576 2.350 2.18 6.14 (3.96)

ωB97X-D DW-F1M 1.585 2.355 2.20 6.28 (4.08)

B2P-LYP-D DW-F1M 1.594 2.364 7.13 8.92 (1.79)

MP2 DW-F1M 1.675 2.286 3.02 3.07 (0.05)

MP3 DW-F1M 1.591 2.373 4.38 8.31 (3.93)

MP4(SDQ) DW-F1M 1.582 2.385 4.37 8.89 (4.52)

MP4(SDTQ) DW-F1M 1.623 2.341 4.64 5.85 (1.21)

CCSD DW-F1M 1.583 2.382 4.69 8.76 (4.07)

BD DW-F1M 1.585 2.380 4.62 8.68 (4.06)

CCSD(T) DW-F1M 1.607 2.370 4.99 7.33 (2.34)

BD(T) DW-F1M 1.607 2.366 5.10 7.34 (2.24)

CASPT2(6,6) DW-F1M 1.633 2.310 4.36 4.84 (0.48)

CASPT2(10,10) DW-F1M 1.593 2.334 6.05 12.20 (6.15)

∆H(5.3b) ∆H(TS)

B2P-LYP-D DW-F1M 1.653 2.245 6.33 6.34 (0.01)

CCSD(T) DW-F1M 1.738 2.476 4.95 6.92 (1.97)

BD(T) DW-F1M 1.738 2.476 4.96 6.93 (1.97)

CASPT2(10,10) DW-F1M 1.641 2.334 5.89 11.53 (5.64)

∆G(5.3b) ∆G(TS)

B2P-LYP-D DW-F1M 1.697 2.245 6.47 6.64 (0.17)

CCSD(T) DW-F1M 1.698 2.482 5.35 7.62 (2.27)

BD(T) DW-F1M 1.698 2.482 5.34 7.59 (2.25)

CASPT2(10,10) DW-F1M 1.618 2.334 6.18 12.22 (6.04)

Table 5.3: Relative Energies and Tautomerization Barriers of 1,3,5-Cycloheptadiene andNorcaradiene.

a Curve indicates the shape of the potential energy well (DW-F1M - Double Well with LocalFirst Minimum - Double Well with Local Second Minimum.)bR(5.3b) and R(5.3a) indicate the C1C6 distance for each structure in angstroms. (Valuesin parentheses are approximate values taken from shoulders.)c∆E(5.3b) gives the energy difference between tautomers, relative to 5.3b [kcal/mol.]d∆E(TS), ∆H(TS) and ∆G(TS) give the energy barriers for the tautomerization from 5.3ato 5.3b [kcal/mol.] (Values in parentheses are for the reverse reactions.)All calculations were completed using the 6-311G(d,p) basis set. MP3, MP4, CAS andcoupled cluster calculations were computed with B3LYP/6-311G(d,p) optimized geometries.

48

Valence tautomerism of 1,6-methano[10]annulene and cycloheptatriene: Extensive spec-

troscopic and diffraction measurements were carried out in the case of 5.2. [27,80,85,95] The

difference between 5.2a (being more stable) and 5.2b is estimated to be smaller than 10

kcal/mol, and all measured NMR, infrared, Raman or UV values exclude a second min-

imum for the bisnorcaradiene form 5.2b. Hence, a SW+1S form of the PEC is most

likely. [27, 80,85,95]

In the case of the cycloheptatriene-norcaradiene system, a DW PEC has been verified

by both NMR and kinetic studies. [18, 38, 186] Experiments conducted by Rubin [186] led

to a free activation energy for the transition 5.3b → 5.3a of ∆Ga(298) = 7.2 kcal/mol

and estimated the free energy of 5.3b to be 4 ± 2 kcal/mol less stable than 5.3a with a

zero-entropy change assumed [186]. The estimates were based on NMR results of Gunther

and co-workers, who estimated that 5.3b should have a finite concentration of 0.1% at room

temperature. [89] In this work, we find that concentration of 5.3b to be just 0.003%.

As in the case of 5.2, most of the methods applied failed to provide PECs and relative

energies that are in line with these experimental observations (see Tables 5.2 and 5.3 as

well as Figure 5.3.) The rationalization of these shortcomings in terms of dynamical and

non-dynamical electron correlation effects are similar to those given for 5.1. ωB97X-D,

CCSD(T), BD(T), MP4(SDTQ), and B2PLYPD lead to reasonable PECs for 5.2 suggesting

that the bisnorcaradiene form 5.2b is located on a shoulder of the PEC. However, only

MP4(SDTQ), and B2PLYPD provided a relative energy of 5.1b to be close to 10 kcal/mol,

and unexpectedly the multi-reference CASPT2(6,6), CCSD(T), and BD(T) methods severely

underestimate the destabilization of 5.3b. Particularly, the PEC of CASPT2(6,6) leads to

a rearrangement barrier 5.3a → 5.3b, which is 4.8 kcal/mol, more than 6 kcal/mol below

the experimentally estimated ∆Ga(298) of 11 kcal/mol.

The CASPT2(6,6) active space is too small to provide a reliable description of the valence

tautomeric rearrangement. Previous work by Cremer and co-workers has emphasized the

homoaromatic interactions of the two π-bonds of norcaradiene with the three σ-bonds of

the cyclopropyl group. [57] We found that these are essential for a correct description of the

49

process 5.3a→ 5.3b. Accordingly, we enlarged the (6,6) active space to (10,10) by including

six rather than two Walsh orbitals for the cyclopropyl group. With this active space, the

barrier increases to 12.2 kcal/mol and the relative energy of norcaradiene 5.3b adopts a

value of 6.0 kcal/mol (Table 3.2).

Previous investigations of 5.3 by Jarzeki and co-workers [118] led to much different ∆E

values (CASSCF: 21.6 kcal/mol; multireference perturbation corrections (MROPT2); 8.9

kcal/mol) based on the smaller (6,6) active space. Similarly all single reference calcula-

tions provided poor results. [53, 54] However, the CASPT2(10,10) results, if converted into

∆G(298) with the help of B3LYP-calculated ZPE, entropy and thermochemical corrections,

lead to ∆Ga(298)5.3b→ 5.3a = 6.0 kcal/mol in good agreement with the corresponding ex-

perimental value of 7.2 kcal/mol (6.1 kcal/mol at 100 K.) [186] The potential curve obtained

for ∆H(298) is shown in Figure 5.3 and probably presents the most accurate description of

the energetics of the valence tautomeric system 5.3.

Based on our experience with 5.3 we extended the (10,10) active spaces of 5.1 and 5.2

to (14,14) spaces, which include the C1C11, and C1C6 σ and σ? Walsh orbitals. For 5.2, the

PEC is still obtained in a SW+1S form, where 5.2b is now 11.5 rather than 5.3 kcal/mol

higher in energy than the annulene form at the potential minimum at R = 2.256 A (X-ray:

2.235 A [27]). The corresponding ∆H(298) and ∆G(298) values are 10.2 and 10.3 kcal/mol,

respectively, in line with experimental observations. [8, 14, 27, 28, 32, 37, 43, 66, 85, 95, 119,

137, 148, 226] Clearly, 5.2 is a [10]annulene rather than a bicyclic, homoaromatic π-system

because R = 2.256 A is too large to lead to sizable through-space interactions. [51]

These results show that CCSD(T) and BD(T), although they may account for some

non-dynamical effects beside the dynamical electron correlation effects, fail to describe the

transition state region and 5.3b correctly. This can also be observed for 5.2b where too

much stability is found in the norcaradiene forms. Therefore, we repeated the CASPT2

calculations for 5.1 with the larger (14,14) active space.

Additional CASPT2 results and the consideration of thermochemical corrections: With

the CASPT2(14,14) active space, 5.1a becomes the minimum of a relative flat potential

50

(0.91 kcal/mol below 5.1b) with activation energies of 1.32 and 0.41 kcal/mol for the valence

tautomeric rearrangement (Table 5.1). Small changes in the ZPE and thermal correction

values lead to a slight stabilization of 5.1b and a vanishing of the rearrangement barrier for

the enthalpy curve (see Figure 5.3). For CCSD(T) and BD(T), the conversion into enthalpies

has a similar effect on the potential (Table 5.1). ZPE and thermochemical corrections are

based on harmonic frequencies, which may change differently with R than anharmonically

corrected frequencies. We recalculated all vibrational corrections with scaled frequencies

employing the scaling factors suggested by Scott and Radom [188] for DFT, resulting in no

significant changes to the final result.

As mentioned above, even a (14,14) active space may not be sufficient to describe all non-

dynamical electron correlation effects resulting from the interactions of σ- and π-electrons,

the stabilizing interactions in the intermediate biradicals and the bridge-ring interactions.

Apart from this, the amount of dynamical electron correlation provided by CASPT2 is

much too small and biased to prefer the annulene. Conversely, CCSD(T) or BD(T) are not

suitable of correctly describing non-dynamical correlation. Since a combined method is not

available, we constructed a model PEC by averaging the enthalpy curves for CASPT2(14) and

CCSD(T), which gives a broad single well potential slightly preferring the bisnorcaradiene

form by -0.1 kcal/mol (activation enthalpies: 0.1 and 0.2 kcal/mol.) The ∆G(R) curve

increases the preference of 5.1b and introduces a somewhat stronger asymmetry of the

potential (Figure 5.3). The degree of asymmetry of the potential curve increases with the

admixture of additional non-dynamical electron correlation. Such a broad asymmetric SW

potential explains the observed strong temperature-dependence of the 13C chemical shifts of

5.1 (they indicate a stronger population of the annulene form at higher temperature [69,80])

and the fact that 5.1 adopts two different forms in the unit cell, probably influenced by crystal

packing effects (R = 1.836 and 1.780 A [27]). Therefore, we will investigate environmental

influences on the PECs shown in Figure 5.2 in the following subsection.

Consideration of solvent and crystal packing effects: The dipole moment (µ) of annulene

5.1a is 0.35 Debye at R = 2.039 and 0.11 Debye for R = 1.638 A (ωB97X-D calculations)

51

where the orientation is along the C2 axis (bridge: positive, center of the perimeter: neg-

ative end). There is a charge transfer from the CMe2 bridge to the annulene perimeter,

which changes, contrary to the changes in the dipole moment, from 13 (R = 2.039 A) to 9

millielectrons (R= 1.638 A).

Experimental work with 5.1 was carried out in nonpolar solvents such as cyclohexane,

CS2, CCl4, or in the polar solvent methanol. [69,85,227] Therefore, we calculated the solvent

influence by increasing the dielectric constant ε from 2 to 32.7 [100] using Tomasi’s polarizable

continuum method (PCM.) [223] In all calculations, changes in the relative free energies

∆G(298) were 1 kcal/mol or smaller, always in favor of the annulene form (in line with the

calculated dipole moments), which in the case of the of the estimated potential of Figure 5.3

(see also Table 5.1) would reduce the ∆G(298) difference between 5.1a and 5.1b favoring

the annulene form with increasing temperature as found in the NMR experiments. [69, 80].

Hence, environmental effects cannot be ignored if free energy differences smaller than 0.6

kcal/mol are being considered.

There is also the possibility that the unusual C1C6 distances observed in the crystal

structure analysis [26] are the result of packing effects. Clusters of molecule 5.1 in parallel

arrangements form molecular sheets, [26] where molecules which are on top of each other

in different sheets could widen the C12C11C13 bridge angle via exchange repulsion, causing

stronger bridge-perimeter interactions. It is shown in Figure 5.4 that the downward oriented

methyl hydrogens are just 2.4 A away from the center of the C3C4 and C8C9 bond, respec-

tively (Figure 5.4). Considering that these H atoms carry a small positive charge and that

the sum of the van der Waals distances for H and C is 1.2 + 1.6 = 2.8 A, [100] the downward

oriented methyl H atoms should be attracted by the π-density of the [10]annulene perimeter.

There will be a stabilizing H-π-interaction at a distance of 2.4 A, as is qualitatively confirmed

by the increase in stabilization of the annulene form when comparing ωB97X-D and ωB97X

results.

However, the hypothesis of an increased bridge-perimeter attraction caused by bridge

angle widening could not be confirmed by widening of the CCC-bridge angle, as it did not

52

have a significant impact on the parameter R. Widening the bridge angle C12C11C13 by

10◦ leads to a increase in R of only 0.048 A.

2.383

2.486 2.389

2.420

2.148 1.951

1.836(7) 1.780(7)

1.836(7) 1.780(7) 1.836

1.836

5.1a (2.039 Å) 5.1b (1.636 Å)

Figure 5.4: Dimer and tetramer configurations that were calculated for this work. Thegreen structure shows how two unit cells fit together in the crystal structure. The numberson this structure are taken from Bianchi’s x-ray structure. The blue structure shows thearrangement that was calculated and optimized searching for packing effects. The orangestructure shows the geometry and hydrogen-to-ring distance for the two optimized structures,ωB97X-D/6-311G(d,p).

Next, we calculated the geometry of the dimer and the tetramer shown in Figure 5.4

applying a constrained optimization, in which the distance(s) between the monomers and

the relative orientation to each other were frozen (ωB97X-D calculations). The differences

between the monomer and dimer geometries were small, with R being found to be 1.635 A

for the lower monomer and 1.631 A for the upper monomer, which confirms the direction

of changes observed in the crystal (1.834 and 1.787 A, Figure 5.4). Similar changes were

calculated for the tetramer, which suggests that more realistic models comprising at least

53

sixteen monomers (twelve monomers surrounding a tetramer) at fixed distances and orienta-

tions may come close to what has been experimentally observed. However, this calculation

(432 atoms) would be computationally prohibitive.

In another set of calculations, the dimer shown on the left side of Figure 5.4 (with frozen

distance and relative orientation of the monomers) was optimized for fixed R (1.6 < R

< 2.2 A) values of either the upper or the lower monomer, and optimizing the remaining

geometrical parameters. For all these geometry optimizations, R of the second monomer

changed maximally by 0.005 A relative to the corresponding gas phase value where especially

the positions of the methyl H atoms were sensitive. We conclude that, in view of the broad

asymmetric SW potential calculated, crystal packing effects should have an impact on R and

explain the existence of two molecules of 5.1 with different geometries in the unit cell.

NMR investigation - An independent determination of the equilibrium geometry of 5.1:

In previous work, Cremer has shown how an easily changing geometrical parameter of a

flexible system can be determined in solution with the help of measured and calculated

chemical shifts. [59,91] In Figure 5.5a, calculated 13C and 1H chemical shift values are given

as a function of R and compared with the available 13C chemical shifts indicated as dashed

horizontal lines. There are 5 unique C’s and 5 unique H’s, labeled as C1, C3, C3, C11,

C12, H15, H16, H17, H19 and H20. The figure shows that the shift value of C1 is the most

sensitive.

This shift increases from a typical value for vinyl cyclopropane (42.5 ppm [125]) to that

found for the parent annulene 5.2a (114.6 ppm [125]) and directly reflects the changes in R.

At R = 1.763 A, the calculated C1 value becomes equal to the measured one, suggesting that

this R value is the one 5.1 adopts in solution or, alternatively, corresponds to a time-averaged

value if the valence-tautomeric rearrangement of 5.1 is fast on the NMR time-scale.

A similar observation can be made for the NMR chemical shift of C11, which increases

from a value typical of a cyclopropane (13C shift: -2.8 ppm [125]) to the one measured for

5.2a (34.8 ppm [125]) crossing the observed C11 shift at R = 1.782 A. The chemical shifts

of the methyl carbon nuclei coincide at 1.865 A with the corresponding measured value.

54

C2 exp.C3 exp.

C1 exp.

C12 exp.C11 exp.

C1

C11

C2

C3

H19

H20

H15

H17

C12

H19H

H H20HH

C1

C6

C3C2

C11C12C13

H18 H17

H16

H15

13C

Che

mic

al S

hift

[ppm

]

-20

0

20

40

60

80

100

120

140

1H C

hemical Shift [ppm

]

0

1

2

3

4

5

6

7

8

C1,C6 Interatomic Distance [Å]1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6

(a)

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

Mea

n D

evia

tion

of 1

3 C C

hem

ical

Shi

fts [

ppm

]

1.6 1.65 1.7 1.75 1.8 1.85R(C1,C6) Distance [Å]

HH

H HHH

CC

1b

HH

H HHH

CC

1a

1.775 Å

(b)

Figure 5.5: a) Dependence of calculated NMR chemical shifts [ppm] as a function of theC1,C6 interatomic distance of 11,11-dimethyl-1,6-methano[10]annulene as obtained at theB3LYP/GIAO/6-311G(d,p) level of theory. b) NMR ab inito analysis of the mean deviationbetween measured and calculated 13C NMR chemical shifts.

55

However these 13C shifts are less sensitive, as are those of C2 (coincidence at R = 1.788 A)

and C3 (coincidence at R = 2.168 A, Figure 5.5a.) The mean deviation between measured

and calculated 13C chemical shifts for the C atoms of the perimeter adopts a minimum at

R = 1.775 A as is shown in Figure 5.5a. NMR chemical shift calculations are normally less

accurate for conjugated systems, especially if these are nonplanar (the shifts of C2 and C3

are close to the experimental ones in the whole range 1.7 < R < 2.2 A and a specific R value

is difficult to determine). We note that when C11, C12, C13, C1, and C6 are used for the

comparison with experiment a value of R = 1.775 A results (Figure 5.5b), in good agreement

with the X-ray diffraction values of R, which are close to 1.8 A. [26]

H17H18

H16H17

H15H16

C1C6

C1C11

C11C12

C12C13

C1C2

C2C3

C3C4

H19H

H H20HH

C1

C6

C3C2

C11C12C13

H18 H17

H16

H15

J(13

C13

C) S

pin-

Spin

Cou

plin

g C

onst

ants

[H

z]

-20

-10

0

10

20

30

40

50

60

70

80

J( 1H1H

) Spin-Spin Coupling C

onstants [Hz]

-4

-2

0

2

4

6

8

10


Figure 5.6: Dependence of calculated NMR spin-spin coupling constants [Hz] as a functionof the C1,C6 interatomic distance of 11,11-dimethyl-1,6-methano[10]annulene as obtained atthe B3LYP/GIAO/6-311G(d,p) level of theory.

So both the bisnorcaradiene and the annulene can be excluded as clearly dominating the

valence tautomeric rearrangement of 5.1. The remaining possibility is a rapid rearrangement

in solution via a small barrier of a DW PEC or a broad SW PEC with a large-amplitude

56

vibration in solution similar to the one suggested in Figure 5.3a. Accordingly, 5.1 has to be

considered as a fluxional molecule with rapid changes in bonding.

The calculated SSCCs J(13C13C) and J(1H1H) given as a function of R are shown in

Figure 5.6. When R is close to 1.5 A, the values of 1J(C1C6) and 1J(C1C11) adopt the values

typical of cyclopropane (12.4 Hz [125]). For increasing R the former 1J-value decreases to -

12.4 Hz at 1.8A. This is comparable to the geminal 1J(CC) value in a substituted cyclobutane

(-8 Hz [125]) and then increases to a zero value at large R. This indicates that through-bond

or through-space coupling are small because the CCC-angle dependence of 2J implies for

this situation a zero value, 5J coupling along the perimeter is too weak, and/or the C1C6

through-space overlap is too small. Hence, a temperature dependent measuring of J(C1C6)

should provide an excellent possibility for an experimental determination of R.

1J(C1C11) increases from about 16 Hz at R = 1.60 A to 28 Hz typical of a 1J(CC) value

such as cyclobutane. [125] 1J(C11C12) changes from 45 to 39 Hz for the same R values while

3J(H15H16) increases from 5.0 to 7.2 Hz. Since the changes in the J(CC) values are larger,

they should give the more reliable determination of the R value of 5.1 in solution.

5.3. Does 11,11-dimethyl-methano[10]annulene possess the longest homoaro-

matic CC bond of neutral hydrocarbons?

System 5.1 is unusual because of its broad SW potential, which makes a large amplitude

C1C6 vibration, and a barrierless interconversion of the annulene into the bisnorcaradiene

form possible. We clarified the question of the nature of the C1C6 interactions by two

different model approaches utilizing the topological analysis of Bader [11], Cremer and Kraka

[55, 142] and the local vibrational mode approach of Cremer, Zou, and Konkoli. [11, 139,

241] In Figure 5.7, the bond strength orders (BSO(CC)) based on the calculated local CC

stretching force constants are plotted as a function of R. For each set of local vibrational

modes at a given R-value the adiabatic connection scheme [241] is applied to determine

whether a given local mode is still contained in a set of 3N-L modes directly related to the

3N-L normal vibrational modes. As can be seen from Figure 5.7, close to R = 1.7 A the

57

C1C6 stretching mode drops out of the 3N-L set of vibrational modes, indicating that for

larger R values there is no C1C6 covalent bond.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Bo

nd S

tren

gth

Ord

er n

(CC

)

1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6R(C1,C6) Distance [Å]

C1C6

C2C3

C3C4 C1C2

C1C11

HH

H HHHCC

HH

H HHHCC

Figure 5.7: Bond strength orders for carbon-carbon bonding interactions in 11,11-dimethyl-1,6-methano[10]annulene system as a function of C1,C6 interatomic distance, analyzed byadiabatic mode analysis, B3LYP/6-311G(d,p).

C1C6 [A] BSOC1C6 BSOC1C11 BSOC1C2 BSOC2C3 BSOC3C4

1.636 0.307 0.846 1.059 1.786 1.164

2.04 - 0.991 0.977 1.358 1.143

Table 5.4: Bond Strength Orders for all pertinent C,C interactions of 11,11-Dimethyl-1,6-methano[10]annulene at the stationary geometries (ethane = 1, ethene = 2.)

All other BSO(CC) values change smoothly from the bisnorcaradiene form to the an-

nulene form. In the annulene case the alternation of bonds is similar to that found for

naphthalene [123, 240], bonds C2C3, C4C5, etc. are the strongest, followed by bonds C3C4

and C8C9. The CC bridge bonds of 5.1b are weaker than the normal CC bonds with C1C6

being the weakest. The calculated AI (as defined in section 9.2) of 5.1a is 64% of the value

58

obtained for benzene (100%), which is smaller than the value for 5.2a (73%) and signifi-

cantly smaller than the value for naphthalene (86%). [123] This is because 5.2a and 5.3a

experience a strong perturbation of their 10π-perimeter caused by the 1,6-bridge leading to

torsional angles up to 37◦.

In Figure 5.8a, the changes in the bond critical and ring critical points rb and rr of

the electron density distribution ρ(r) are shown as a function of R. Some, but not all,

of the changes are in agreement with those given by the BSO(CC) based on the local CC

stretching force constants (Figure 5.7.). This is attributable to the fact that the electron

density determined at one specific point in the bond region cannot reflect all the density

changes taking place in the zero-flux surface between two bonded atoms apart from the

influences of bond polarity, charge transfer, and other effects given by the changes in the

virial (atomic) spaces of the molecule during change in R. The local CC stretching force

constants account for these effects and therefore are more reliable as CC bond strength

descriptors, as discovered in Chapter 3.

The Cremer-Kraka bond criteria state that covalent bonding requires the existence of a

bond critical point and zero-flux surface between the atoms in question and that the energy

density at this bond critical point, H(rb) must be negative and stabilizing. [55,56,142] This

criterion is fulfilled for all CC bonds in 5.1, except the C1C6 bond, which converts into a

non-covalent interaction at R = 1.696 A (Figure 5.8.) There the C1C6C11 ring critical point

merges with the C1C6 bond critical point, leading to a singularity in the Hessian of ρ(r) and

the C1C6 maximum electron density path connecting these atoms vanishes. We conclude

that a C1C6 covalent bond with R = 1.80 A does not exist, but that a homoaromatic

interaction in the sense of a through-space overlap of π-orbitals exists leading to a small

electron density increase between atoms C1 and C6.

Impact of the bridge on the 10π-perimeter: The influence of electron-withdrawing and

electron-donating substituents of a cyclopropane ring has been amply described in the liter-

ature (for a review, see citation 57). Two cyano groups at C11 lead to a shortening of the

distal bond, and lengthening of the vicinal bonds (see 5.5 as compared to 5.4 in Figure 5.9.)

59

This was exploited to synthesize 11,11-dicyano-bisnorcaradiene, the dicyano analogue

rb(C2C3)

rb(C3C4)

rb(C1C2)

rb(C1C11)

rb(C1C6)

rr(C1C11C6)

C1

C6 C4

C3C2

C11CH3H3C

1.696 Å

Ener

gy D

ensi

ty [

Har

trees

/Å3 ]

-2.5

-2.0

-1.5

-1.0

-0.5


(a)

rb(C2C3)

rb(C3C4)

rb(C1C2)

rb(C1C11)

rb(C1C6)

rr(C1C11C6)

C1

C6 C4

C3C2

C11CH3H3C

1.696 Å

Elec

tron

Den

sity

[e

/Å3 ]

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4


(b)

Figure 5.8: Dependence of a) electron density distribution and b) energy density distributionon the C1,C6 interatomic distance of 11,11-dimethyl-1,6-methano[10]annulene as obtainedfrom the AIMAll program.

of 5.1b. [226] The effect of two methyl groups has been predicted to lead to a slight CC

lengthening of the distal bond (slight shortening of the vicinal CC bonds) [57] as is confirmed

by the bond lengths given for 5.6 in Figure 5.9. This indicates that dimethyl-substitution at

C11 in 5.1 should favor a larger rather R regardless of any other stabilizing bridge-perimeter

interactions.

60

The isodesmic energies (B3LYP, in kcal/mol) for the the formal reactions:

4 + CH3CH2CH3 → 6 + CH4, ∆E = −3.78

2b + CH3CH2CH3 → 1b + CH4 ∆E = −3.38

2a + CH3CH2CH3 → 1a + CH4 ∆E = 4.92

(∆E values in kcal/mol) suggest a similar stabilization by the two methyl groups for cyclo-

propane and 5.2b, but a 5 kcal/mol destabilization for 5.2a. This is in line with a shortening

of two vicinal and one distal CC bond. Molecule 5.6 has a smaller external CCC angle than

the HCH angle in its parent molecule (Figure 5.9), which is the result of steric repulsion

between methyl groups and the ring. For the same reason, 5.1b has an even smaller ex-

ternal angle (110.2◦, Figure 5.9), where the folding back of the two diene units as shown

in the side view at the bottom of Figure 5.4 leads to some reduction of the steric repulsion

between methyl groups and ring perimeter. In the case of 5.1a, steric attraction between

methyl groups and π-perimeter and steric repulsion (for 5.1a the H-center(C3C4) distance

is decreased from 2.148 to 1.951 A, ωB97X-D, see Figure 5.4) must be balanced, which

leads to a small C12C11C13 angle (105.3◦, Figure 5.4), methyl-methyl repulsion, and an

overall destabilization as reflected by the energy of the isodesmic reaction given above (4.9

kcal/mol).

The destabilizing effect of the CMe2 bridge leads to an increase of the relative energy

of 5.1a, which in view of π-delocalization is more stable for the parent system 5.2. π-

Delocalization and bridge-perimeter destabilization lead to 5.1b and 5.1a having comparable

energies.

61

A basic problem of previous studies was that they were based on quantum chemical meth-

ods of low accuracy. Depending on whether HF or MP2 is used, the preference for different

valence tautomeric forms is found. This also holds for the XC functionals of DFT as we have

demonstrated in this work for the first time. This leads to rather limited insight into the

possible cause of a quantum mechanical (electronic structure) effect. For example, Simonetta

and co-workers [194] have used Hoffmann’s approach [57, 104] to rationalize the stability of

substituted cyclopropanes, and explain substituent effects in bridged [10]annulenes where

his arguments were based on low level calculations. As was pointed out by Cremer and

co-workers, [57] the orbital model used does not even explain all substituent effects for the

simpler cyclopropane calculations. Furthermore, it does not consider the impact of bridge-

perimeter interactions, the stabilization of biradicaloid structures for medium-sized R values

by conjugation, or the dynamic aspects of the valence tautomeric rearrangement.

The topological analysis of ρ(r) by Gatti and co-workers, [83] which was interpreted as

proof for a long covalent C1C6 bond, has to be criticized because they were carried out at

the HF/minimal basis set level without using any quantum mechanical criterion for covalent

bonding. Other investigations were based on the natural bond order (NBO) analysis or

heuristic models utilizing the degree of bond length alternation in the ring perimeter. [36]

Whatever property is used, it must be evaluated with regard to the corresponding value of

a suitable reference system.

An interesting rationalization of the carrying dynamic behavior of bridged [10]annulenes

was proposed by Choi and Kertesz, [43] who model the opening of the cyclopropane ring in

5.1 or 5.2 by its conversion into a trimethylene biradical in its triplet ground state. The

stabilization of the triplet trimethylene by substituents as described by DFT provides a basis

to understand whether the bisnorcaradiene or annulene form is more stabilized. The radical

centers are part of a conjugated system and a more realistic model would be the opening of

3-substituted 1,2-divinylcyclopropanes.

62

CH3H3C CH3H3C

5.1a 5.1b

HH

5.2b

HH

5.2a

110.3

99.6

127.7

126.7

116.4

122.5

1.424

1.3901.408

2.279

1.492

1.089

5.3a 5.3b

108.6

102.8

114.9

2.3501.451

1.503

1.352

1.090 1.097

1.380

1.370 1.446

1.645

1.496

1.084 1.084

1.454

H H H H

123.0125.4

121.3

66.7

113.6123.2

119.5121.5

H

H

5.4

C

CN

N

5.5

C

CH

H

H

H H

H

5.6

1.5121.539

114.2 113.3115.8

1.083 1.5141.442

1.4871.508 1.520

116.6

64.3

121.6

117.7

118.2

122.4

1.453

1.3471.475

1.600

1.503

1.089

110.2

65.2

121.8

115.1

120.6

123.3

1.452

1.3421.437

1.636

1.518

1.516

105.3

89.4

125.8

119.4

120.0

124.6

1.425

1.3711.423

2.120

1.514

1.533

Figure 5.9: B3LYP geometries of 5.1 - 5.6. Bond lengths in A and bond angles in degrees.

63

Dorn and co-workers [69] used 13C CPMAS (cross-polarization magic angle spinning)

spectra to refute a rapid valence-tautomeric process for 5.1. Instead, these authors sug-

gested that an asymmetric PEC and a different population of the vibrational levels at higher

temperature would lead to the observed temperature dependence of the NMR spectra. Alter-

natively, temperature dependent intermolecular interactions could cause the observed tem-

perature dependence. Our high level results are in line with the first explanation where

strong intermolecular interactions cannot be confirmed.

Kaupp and Boy [128] analyzed the measured temperature factors of the crystal data and

concluded that a structural disorder in the solid state leads to the coexistence of bisnorcara-

diene and [10]annulene, which implies that the measured R values are just averages. Our

results are not in contradiction with this hypothesis and our study of packing effects is too

limited to provide any arguments for or against structural disorder.

With this comprehensive description of the previously controversial bridged annulene

system 5.1, we demonstrate the solution of an organic multi-reference system. All manner

of density functional (B3LYP, ωB97X-D, B2P-LYP-D, etc.) analysis, coupled cluster analysis

(CCSD, CCSD(T), BD, BD(T)) and Møller Plesset perturbation (MP2, MP3, MP4) analysis

did not provide satisfactory results. Complete active space/self consistent field calculations

still failed, until a carefully selected active space of adequate size was employed. This is

the cost, both in terms of computational resources and thoughtful examination of the active

space, of multi-reference computational chemistry.

64

Chapter 6

Multi-Reference Systems in Inorganic Chemistry

6.1. Transition Metal Diatoms

In the field of organic chemistry, the examples of multi-reference problems are relatively

rare, and often unique and interesting. In inorganic chemistry, multi-reference systems are

ubiquitous.

Bulk metals participate in metallic bonding, rather than covalent bonding, described as

a matrix of metallic nuclei in a sea of electrons. This is what gives metals their distinctive

properties of lustre, malleability and conductivity. The primary reason for this is the wealth

of s-, p- and d-orbitals in large nuclei, resulting in a ‘band’ of low-lying unoccupied orbitals

and low-lying excited states.

However, when considering small clusters of metal atoms, the electronic structure is that

of covalent bonding, with low-lying excited states still present.

For this work, we investigated the ‘d-block’ transition metals, in groups 3-12 and rows

4-6 of the periodic table. We selected the smallest possible clusters of two atoms (diatoms.)

These systems have been extensively studied experimentally and computationally, but com-

pilations of multi-reference calculations for the entire set of transition metals do not exist.

6.2. A Survey of All Transition Metal Diatoms

The results of all calculations are summarized in Table 6.1. Generally, CASSCF/MR-

AQCC and CASSCF/CASPT2 calculations were run using a (2g,12) complete active space,

where g is the group number, 2g the total valence electrons, and 12 the number of orbitals in

the complete active space. The RASSCF/RASPT2 calculations were generally run using a

(2g,18) restricted active space. The active spaces which gave the best results are summarized

65

in Table 6.2.

For each diatom, the electronic state (term symbol) of the ground states was taken from

the literature. Where more than one ground state was reported, the ground state which

gave calculated results nearest the experimental values was used. The basis sets employed

are discussed in Section 9.3.

Group 12; Zn2, Cd2, Hg2: These diatoms have filled valence shells (24 electrons), and are

known to form van der Waals complexes, [24, 64, 141, 201] and are therefore of minimal

interest to this study. They were run at the CASSCF-CASPT2 level of theory, with the

aug-cc-pVQZ-DK (Zn2) and aug-cc-pVQZ-PP (Cd2 and Hg2) basis sets. The (24,14) active

space (24 electrons in 14 orbitals) was used, rather than the (24,12) active space, due to

the near degeneracy of the valence orbitals. This added the p-πu orbitals to the complete

active space. Based on what was learned from the Mn2 van der Waals complex (See Group

7 below), the higher levels of theory were not used. With all orbitals filled, they exhibit the

1Σ+g ground state.

Group 11; Cu2, Ag2, Au2: Spectroscopically, these species are found to be in the 1Σ+g ground

state. With 22 valence electrons, these diatoms have a formal bond order of 1 in the ground

state with all d-electrons existing as unshared electron pairs. For Cu2 and Ag2, selecting an

active space of 14 orbitals (CASPT2(22,14)), using the (n)d, (n+1 )s and (n+1 )p orbitals

to describe the d-σg, d-σu, d-πu, d-πg, d-δg, d-δu, s-σg, s-σu and p-πu molecular orbitals

gave the best orbital description, giving results in close agreement with experimental values.

However, the RASPT2(22,18) restricted active space results also gave reasonable results.

For Au2, a better description was obtained with 18 active orbitals, (RASPT2(22,18)),

including the 5d, 6s and 6p orbitals. This was attributable to the lowering of the excited state

energies from the relativistic contraction of the (n+1 )s and (n+1 )p orbitals. This brings

these orbitals into alignment, by size and energy, with the (n)d orbitals. This phenomenon

is less pronounced in the 3d, 4s and 4p orbitals, owing to the lack of nodal planes in the 3d

66

orbitals, and most prevalent in the 5d, 6s and 6p orbitals, since relativistic contraction is

strongest for the largest nuclei.

These effects are reflected in the equilibrium bond lengths (Re), bond dissociation en-

ergies (BDE’s) and stretching force constants (ke) for each diatom. For Au2, the greater

contributions by the excited states into the 6s and 6p orbitals resulted in the highest BDE

and ke of the sequence, 56.91 (53.06 [156]) kcal/mol and 2.187 (2.12 [106]) mdyn/A, respec-

tively. (Experimental values are in parentheses.) These are much higher than the values

36.05 (38.74 [6]) kcal/mol and 1.212 (1.18 [106]) mdyn/A for Ag2. The BDE values for Cu2,

43.84 (47.97 [174]) kcal/mol and 1.332 (1.33 [103]) mdyn/A are higher than Ag2, which is

attributable to the shorter bond length, and lower than Au2, which is attributable to the

lack of low-lying excited states.

Comparing our BDE results to previously reported theoretical values, our Ag2 results

are in better agreement with experiment than those reported by Andrea and others (27.67

kcal/mol, MRCI(SD)/(22,12) [6]), and the results for Cu2 and Au2 by Roos and oth-

ers (45.43 kcal/mol, CASPT2(22,2) [174].) Barysz and Pyykko’s results (52.35 kcal/mol,

CASPT2(22,12)/BSSE corrected [20]) are slightly better than the numbers reported here.

In a recent paper, Radenkovic and others [177] calculated these diatoms using “breathing

orbital” valence bond theory, and a (2,2) active space, excluding all d-orbitals. We find that

the resultant bond dissociation energies of 40.7, 28.9, and 47.4 kcal/mol for Cu2, Ag2 and

Au2, do not improve on the results obtained with the larger active spaces. Since most of the

bonding is described by s-orbital overlap, these systems were relatively easy to calculate.

Unlike most other groups, the Re’s, BDE’s and ka’s for all group 11 species gave reason-

able results for nearly any of the active spaces selected.

Group 10; Ni2, Pd2, Pt2: Spectroscopically, Ni2 was found to be in the O+g ground state,

while Pd2 and Pt2 are in the 3Σ−g ground state. Ni2 is best described by an active space of

12 orbitals (20,12), using the 3d and 4s orbitals to describe the d-σg, d-σu, d-πu, d-πg, d-δg,

d-δu, s-σg and s-σu molecular orbitals. The aug-cc-pVQZ-DK basis set was used.

67

Pd2, and Pt2, were best described with 18 restricted active orbitals, (RASPT2(20,18)),

using the 4d, 5s and 5p atomic orbitals, and 5d, 6s and 6p atomic orbitals, respectively, to

generate the d-σg, d-σu, d-πu, d-πg, d-δg, d-δu, s-σg, s-σu, p-πu, p-πg, p-σg and p-σu molecular

orbitals. The aug-cc-pVQZ-PP basis set was used.

Agreement between the calculated and experimental values for Re, BDE and ke are

excellent; Ni2 - 2.180 (2.154 [170]) A, 47.49 (47.46 [42]) kcal/mol, 1.156 (1.16 [150]) mdyn/A;

Pd2 - 2.411 (2.48 [110]) A, 23.24 (24.05 [103]) kcal/mol, 1.489 (1.38 [106]) mdyn/A and Pt2

- 2.345 (2.333 [2]) A, 72.64 (72.72 [103]) kcal/mol, 2.848 (2.66 [106]) mdyn/A. Once again,

experimental results are in parentheses.

In particular, the BDE values calculated in this work are improvements on past calcula-

tions reported in the literature: Ni2 45.083 kcal/mol [42], Pd2 22.6 kcal/mol [61], and Pt2

60.0 kcal/mol [61]. Note that the stretching force constants increase going down the group,

as relativistic contraction of the outlying orbitals increases. Although all methods tried gave

generally useful results, some variability was found in the ke descriptions of Pd2, and Pt2,

where the values varied widely by method. The method which gave the best agreement with

the experimental results for Pd2 and Pt2 is RASPT2(20,18) with the Dunning basis set. The

stretching force constant (ke) is the most reliable of the three parameters that we employ,

because it is a second order property, incorporating a range of geometries. Experimental

values for this parameter are also relatively easy to obtain spectroscopically for diatoms,

because there is no mode coupling, and the Badger rule is fully applicable in its original

format. [12,13,146]

Unlike most other groups, but similar to group 11, the Re’s, BDE’s and ka’s for all group

10 species gave reasonable results for nearly any of the active spaces selected.

Group 9; Co2, Rh2, Ir2: These species are all found to be in the 5∆g ground state. In all

cases, the best descriptions were obtained with (18,18) active spaces, which include all (n)d,

(n+1 )s and (n+1 )p orbitals. The best basis sets for Co2, Rh2, and Ir2 were aug-cc-pVQZ-

DK, aug-cc-pVQZ-PP and ANO-RCC, respectively.

68

Agreement between the calculated and experimental values of Re for Rh2 is good; 2.25 A

versus 2.28 A [15]. Experimental Re values for Co2 and Ir2 are not found in the literature.

The comparison between calculated and experimental values of ke’s are excellent for Co2 and

Ir2; 1.580 mdyn/A (1.53 [68]) and 4.471 mdyn/A (4.44 [150]), respectively. This parameter

is reasonable for Rh2, with values of 1.981 versus 2.44 mdyn/A [150], here using the (18,12)

complete active space.

Once again, the BDE values in this work are significant improvements on what was

previously available in the literature. Multi-reference BDE calculations for Co2 and Ir2 have

not been reported. This work reports the following values: Co2, 32.48 versus 38.51 kcal/mol

[71] (by DFT) (39.43 ± 6.42 kcal/mol [126]), Rh2, 32.65 versus 49.3 kcal/mol [15] (32.68

kcal/mol [230]) and Ir2, 82.76 versus 75.84 kcal/mol [71] (by DFT) (79.99 kcal/mol [155],

empirical estimate.)

Group 8; Fe2, Ru2, Os2: These diatomic molecules were all found in septuplet ground states

of 7Σ+g , 7∆u and 7∆u, respectively. The (16,18) active space worked well for all three diatoms.

Fe2 gave the best results with the ANO-RCC basis set, while Ru2 and Os2 worked best with

the aug-cc-pVQZ-PP basis set.

Re values for Fe2 and Ru2 agreed well with experiment: 2.149 versus 2.02 A [175] and

2.270 versus 2.54 A [45], respectively. An experimental Re value is not available for Os2.

Stretching force constants, ke are in less good agreement with experiment for Fe2 and Ru2,

1.713 mdyn/A (1.48 [106]) and 2.511 mdyn/A (3.59 [150].) However, this parameter is in

excellent agreement with experiment for Os2: 6.104 mdyn/A (6.26 [150].)

All BDE values in this work are improvements on what was previously available in the

literature: Fe2, 26.97 versus 27.76 kcal/mol [105] (26.95 kcal/mol [105]), Ru2, 73.45 versus

58.23 kcal/mol [136], (73.56 kcal/mol [136]), and Os2, 108.42 versus 77.25 kcal/mol [135],

(101.00 kcal/mol [135], empirical estimate.)

69

Group 7; Mn2, Tc2, Re2: Mn2 presents the same computational challenge as Zn2, Cd2 and

Hg2, that the diatomic molecule is found to be a van der Waals complex, resulting in a very

large Re and very small BDE and ke. [34] Tc2 being a synthetic element not found in nature,

experimental parameters are difficult to obtain, or trust. Re2 is important to this study,

because the Re2Cl2−8 anion was the first verified quadruple bond, and is one of the reference

species used for the BSO calibration of this work.

Mn2 and Re2 are found to have the 1Σ+g ground state, while Tc2 has the 3Σ−g ground state.

Mn2 was found to be reasonably described with a (14,12) active space, and the larger (14,18)

active space over-bound the molecule, resulting in a description equivalent to a full covalent

bond, with Re’s in the 2.1 to 2.4 A range, and a ke 20 times greater than the reported

experimental value. Tc2 was found to be best described with a (14,12) active space, and

Re2 required a (14,18) restricted active space. These resulted in Re and BDE values in

good agreement with experiment; Mn2 - 3.220 (3.4 [150]) A, 4.16 (3.22 [217]) kcal/mol; Tc2

- 2.073 A (no experimental value available), 87.64 (80.48 [216]) kcal/mol; and Re2 - 2.139

(2.18 [203]) A, 93.66 (92.24 [150]) kcal/mol.

The BDE values were comparable to previously reported calculated values. Comparing

this work to other for the Mn2 van der Waals complex, the results, 4.162 versus 3.05 [216]

kcal/mol were both within 1 kcal/mol of the experimental value. For Tc2, 87.63 versus

82.2 [216] kcal/mol (empirical estimate) were comparable. For Re2 93.66 versus 84.53 [203]

kcal/mol was a significant improvement, which was to be expected, since Sun’s result is from

a single-reference DFT calculation.

Calculations for ke values have proven difficult for Mn2 and Tc2; 0.0671 (0.094 [106])

mdyn/A, and 2.257 (4.37 [150]) mdyn/A, respectively. For Mn2, where the dimer is a van

der Waals complex, it is to be expected that the stretching force constant would be very

low, and so the low numbers are entirely consistent. For Tc2, there are few measurements in

the literature, because this synthetic and radioactive molecule is expensive and difficult to

obtain, leaving some question about accuracy of the available experimental values. For the

critical dimer, Re2, the calculated vale of 6.377 mdyn/A is in excellent agreement with the

70

experimental value of 6.26 mdyn/A [107].

Group 6; Cr2, Mo2, W2: All of these diatoms exhibit 1Σ+g symmetry. Having 12 valence

electrons, exceptionally short bonds and the largest stretching force constants, they are the

candidates for sextuple bonding.

For Cr2, a (12,20) restricted active space proved to be necessary for best results, improving

on the (12,18) restricted active space calculations with the aug-cc-pVQZ-DK basis set. For

Mo2, the (12,12) complete active space with the ANO-RCC basis set gave good results, where

for W2, the (12,18) restricted active space with the aug-cc-pVQZ-PP basis set was best.

Agreement between the calculated and experimental values for Re, BDE and ke are very

good; Cr2 - 1.775 (1.6788 [29]) A, 33.95 (33.95, 35.97 [217]) kcal/mol, 4.060 (3.54 [217])

mdyn/A, Mo2 - 1.943 (1.929 [73]) A, 114.72 (103.17 [181]) kcal/mol, 6.447 (6.43 [150])

mdyn/A and W2 - 2.110 (2.048 [7]) A, 122.54 (115.48 ± 23.48 [181]) kcal/mol, 5.919(6.14

[106]) mdyn/A. These species are the most analyzed of all of the transition metal diatoms,

with a host of calculated results that are in line with the ones reported here.

Group 5; V2, Nb2, Ta2: All of these diatoms were found in the 3Σ−g state. They were best

described with RASPT2 (10,18) active spaces. It was found that for the ANO-RCC basis

set run on the Molcas software [3], the (10,18) complete active space, with up to 9,000,000

configuration state functions (CSF), can be run in a reasonable timeframe.

For V2, agreement between the calculated and experimental values for Re, BDE and ke are

very good; 1.820 A (1.766 A [149]), 52.96 kcal/mol (58.06 kcal/mol [166]), and 3.990 mdyn/A

(4.33 mdyn/A [30]), respectively. We found no calculated values for these parameters in

the literature. Nb2 resulted agree well with experimental values: Re, 2.101 (2.087 [117])

A, BDE 138.48 (128.67 [16]) kcal/mol, ke 3.374 (4.84 [30]) mdyn/A. Agreement was also

very good for Ta2, between the calculated and experimental values for Re, BDE and ke ;

2.374 A (2.374 A [203]), 128.82 kcal/mol (114.38 - 124.52 kcal/mol [30]), and 5.362 mdyn/A

(4.80 mdyn/A [30]), respectively. The BDE, calculated with a (10,18) active space, was in

71

better agreement with experiment than Borin and Gobbo’s calculation with a (10,12) active

space. [30]

Group 4; Ti2, Zr2, Hf2: Ti2 was found to be in the 3∆g state, and was best described with

an (8,12) active space and the aug-cc-pVQZ-DK basis set, while Zr2 and Hf2 are in the 1Σ+g

state. Zr2 and Hf2 were best described with an (8,18) active space. Here, too, it is found

that the ANO-RCC basis set can be run with an (8,18) complete active space. However, in

this case, only the results for Hf2 were improved over the restricted active space calculation

with the aug-cc-pVQZ-PP basis set.

The agreement between the calculations reported here and experimental values is good.

For Ti2, Zr2 and Hf2, the Re values of 1.909 (1.9422 A [70]), 2.265 (2.24 A [150]) and 2.315

(2.46 A [203]), the agreement between the calculations reported here and experimental values

is good. The values for BDE: 38.58 (≥ 31.109 kcal/mol [207]), 64.89 (70.38 kcal/mol [9])

and 92.31 (78.41 kcal/mol [203], empirical estimate) are in reasonable agreement with the

experimental values, as many of the experimental values report large margins of error. For ke

values, the results for Ti2 and Hf2, 2.628 (2.35 mdyn/A [107]) and 1.505 (1.63 mdyn/A [107])

are good, and the results for Zr2, 3.573 (2.51 mdyn/A [107]) are the only computational

results reported in the literature to date.

There are relatively few calculated results reported for these compounds [17,21,207], but

the results of this work are in better agreement with the experimental values than previously

published work.

Group 3; Sc2, Y2, La2: Each of these diatoms are all found to be in the 5Σ−u state. In

all cases, an active space of (6,18) was required for a good description of the molecule. In

these cases, the ANO-RCC (6,18) complete active space did not improve the results over the

restricted active space results with the aug-cc-pVQZ-DK or aug-cc-pVQZ-PP basis sets.

Agreement with experimental values for Re, BDE and ke were good. Sc2: 2.653 A (2.70

A [224]), 22.61 (25.9 kcal/mol [224]), and 0.853 (0.76 mdyn/A. [106]) Y2: 2.841 (2.80 A [224]),

72

Diatom Sc25Σ−

u Ti23∆g V2

3Σ−g Cr2

1Σ+g Mn2

1Σ+g Fe2

7Σ+g Co2

5∆g Ni2 O+g Cu2

1Σ+g Zn2

1Σ+g

Re [A]

AQCC 3.220 2.020 2.350 2.231 2.230

CASPT2 2.572 1.909 1.868 3.220 2.191 2.332 2.180 2.195 5.340

RASPT2 2.787 1.825 1.775 2.073 2.350 2.195 2.176

RAS-ANO 2.653 1.975 1.820 1.876 2.468 2.149 2.195 2.483 2.224

experiment 2.7 1.9422 1.766 1.6788 3.4 2.02 2.154 2.2193 4.49

BDE [kcal/mol]

AQCC 26.97 34.42 46.28 42.59

CASPT2 24.16 38.58 82.19 4.16 39.44 34.40 47.49 43.84 3.20

RASPT2 84.51 52.97 21.24 32.48 46.82 92.54

RAS-ANO 22.61 74.34 44.83 14.02 26.18 26.52 57.36

experiment 25.9 ± 5 ≥31.109 58.06±3.76 33.95, 35.97 3.22 26.95±2.50 39.43±6.42 47.46±0.42 47.97 0.81

ke[mdyn/A]

AQCC 4.630 2.033 1.053 1.428

CASPT2 1.371 2.628 7.027 0.0671 3.224 1.156 1.522 10.107

RASPT2 0.816 12.033 4.060 0.178 0.975 1.151 1.332

RAS-ANO 0.853 1.827 3.990 2.704 1.713 1.580 0.823 1.147

experiment 0.76 2.35 4.33 3.54 0.094 1.48 1.53 1.16 1.33 0.013

Diatom Y25Σ−

u Zr21Σ+

g Nb23Σ−

g Mo21Σ+

g Tc23Σ−

g Ru27∆u Rh2

5∆g Pd23Σ−

g Ag21Σ+

g Cd21Σ+

g

Re [A]

AQCC 2.909 2.330 2.087 1.960 2.277 2.252 2.424 2.548

CASPT2 2.862 2.281 2.101 2.060 2.209 2.220 2.176 2.313 2.498 4.29

RASPT2 2.856 2.265 2.101 1.940 2.073 2.270 2.525 2.411 2.490

RAS-ANO 2.841 2.194 2.316 1.943 1.925 2.180 2.300 2.480 2.471

experiment 2.80 2.24 2.087 1.929 2.54 2.28 2.48 2.33 4.07

BDE [kcal/mol]

AQCC 23.22 52.63 152.91 204.85 78.37 39.51 20.56 36.05

CASPT2 28.49 89.44 57.68 180.11 87.64 73.46 61.97 31.67 43.63 1.82

RASPT2 82.80 64.89 147.85 158.17 74.37 32.65 23.24 30.05

RAS-ANO 87.20 53.53 138.58 114.72 26.81 54.96

experiment 82.16±9.36 70.38 128.67 103.17±0.23 80.48 73.56 32.68±7.31 24.05±0.46 38.74 0.92

ke[mdyn/A]

AQCC 0.950 7.767 2.623 1.981 1.191 1.123

CASPT2 0.943 3.536 3.374 4.156 2.257 2.175 1.557 0.981 1.400 0.633

RASPT2 1.011 3.573 4.745 2.511 1.135 1.489 1.236

RAS-ANO 0.739 1.459 6.556 6.447 6.540 2.475 4.691 1.366

experiment 0.89 2.51 4.84 6.43 4.37 3.59 2.44 1.38 1.18 0.017

Diatom La25Σ−

u Hf21Σ+

g Ta23Σ−

g W21Σ+

g Re21Σ+

g Os25Πg Ir2

5∆g Pt23Σ−

g Au21Σ+

g Hg21Σ+

g

Re [A]

AQCC 2.009 2.550 2.316 1.986 2.009 2.167 2.359 2.313 2.498

CASPT2 3.314 2.567 2.247 2.094 2.009 2.234 3.923 2.344 2.442 3.44

RASPT2 3.175 2.315 2.110 2.073 2.254 2.327 2.345 2.454

RAS-ANO 2.898 2.603 2.374 2.021 2.139 2.219 2.176 2.298 2.475

experiment 2.8 2.46 2.23 2.048 2.18 2.333 2.4715 3.34

BDE [kcal/mol]

AQCC 21.66 42.12 160.43 84.33 93.66 47.11 49.44 66.75 48.11

CASPT2 10.49 57.68 210.71 215.16 78.07 80.59 58.47 1.91

RASPT2 28.49 92.31 122.54 65.21 54.79 72.64 56.91

RAS-ANO 60.58 118.95 128.82 189.83 112.65 108.42 82.77 51.29 65.82

experiment 57.6±5 78.4±14 114.4-124.5 115.5±23.5 92.24 101 79.99 72.72±0.77 53.08±0.73 1.0±0.1

ke[mdyn/A]

AQCC 0.592 1.772 3.162 12.498 3.202 5.859 2.637 3.315 1.971

CASPT2 0.691 1.925 3.877 7.216 2.091 6.796 2.967 1.421 2.520 0.249

RASPT2 0.802 2.053 5.903 6.104 3.463 2.848 2.187

RAS-ANO 1.268 1.505 5.362 5.441 6.377 4.727 4.471 3.249 2.265

experiment 0.76 1.63 4.80 6.14 6.26 6.26 4.44 2.66 2.12 0.02

Table 6.1: AQCC, CASPT2, RASPT2 Calculations by aug-cc-pVQZ basis sets and RASPT2Calculations by ANO-RCC basis set. Summary of all results.

82.80 (82.18 kcal/mol [75]), and 1.011 (0.89 mdyn/A. [150]) La2: 2.898 (2.80 A [224]), 60.58

(57.6 kcal/mol [224]), and 0.802 (0.76 mdyn/A. [150])

73

Computationally, Sc2 has been investigated many times [33, 127, 217], reporting results

from 39.66 to 54.42 kcal/mol. The results of this work are in better agreement with the

experimental values than those previously published results. Results for Y2 improve the

value reported by Tamukong, et al [216]. No computational results are available for La2, so

this work reports that, now.

Bond Strength Descriptors : Having determined the best values for bond length, bond disso-

ciation energy and stretching force constant for all of the transition metal diatoms, we now

consider which parameter gives the best description of bond strength.

The calculated bond lengths for all diatoms are given in Figure 6.1. Here we find a

reasonable pattern, which is disturbed primarily by the van der Waals complexes; Mn2, Zn2,

Cd2 and Hg2. Without speculating about bond order, it can be stated that the third group

represents 6 electron bonding, the forth group represents 8 electron bonding, etc, and that,

through the sixth group the bonds grow shorter with increase of bonding electrons. For the

later groups, additional electrons are going into antibonding orbitals, and the bonds become

longer. Variability is greatest for the forth row elements, where the energy gaps between the

3d and 4s orbitals are larger than in rows 5 and 6.

The calculated bond dissociation energies are given in Figure 6.2. Here it is shown

once again that the 6 electron bond has a lower BDE than the 8 electron bond, and so

forth, and that the rows beyond row six report lower BDE’s due to additional electrons

filling antibonding orbitals. The exception to this pattern, aside from the van der Waals

complexed diatoms, is that the fifth group, V2, Nb2 and Ta2, have larger BDE’s than the

sixth. Each of the fifth group diatoms are found to be in the 3Σ−g state, indicating an equal

number of occupied orbitals for both groups, albeit there are half-filled orbitals in group 5.

The results for stretching force constants, shown sorted by their groups in the periodic

table in Figure 6.3, invert the bond length results. The shortest bond lengths, in group six,

report the largest ke’s. The ke and BSO values for all diatoms are listed in Table 6.2. The

ke’s for Re2 and Os2 are greater than that for W2. This is very significant, because W2 is

considered a sextuple bonding candidate, and the other diatoms are not.

74

Sc2

Y2

La2

Zr2Ti2

Hf2Ta2

V2

Nb2Mo2

Cr2

W2

Mn2

Tc2Re2

Ru2

Os2

Fe2

Rh2

Ir2

Co2

Ni2

Pd2

Pt2 Cu2

Au2

Ag2

Zn2

Cd2

Hg2

Bond

Len

gth

[Å]

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

Periodic Table Group3 4 5 6 7 8 9 10 11 12

Figure 6.1: Calculated Bond Length by Periodic Table Row, All Diatoms.

Diatom Sc25Σ−

u Ti23∆g V2

3Σ−g Cr2

1Σ+g Mn2

1Σ+g Fe2

7Σ+g Co2

5∆g Ni2 O+g Cu2

1Σ+g Zn2

1Σ+g

Method CASPT2 CASPT2 CASPT2 RASPT2 CASPT2 CASPT2 RASPT2 CASPT2 CASPT2 CASPT2

Active (6,18) (8,12) (10,18) (12,20) (14,12) (16,18) (18,18) (20,12) (22,14) (24,14)

Space complete complete complete restricted complete restricted restricted complete complete complete

Basis Set ANO-RCC aVQZ-DK ANO-RCC aVQZ-DK aVQZ-DK aVQZ-DK aVQZ-DK aVQZ-DK aVQZ-DK aVQZ-DK

BSO 0.688 2.141 3.263 3.320 0.053 1.390 1.281 0.935 1.079 0.085

comments vdW all work well all work well vdW

Diatom Y25Σ−

u Zr21Σ+

g Nb23Σ−

g Mo21Σ+

g Tc23Σ−

g Ru27∆u Rh2

5∆g Pd23Σ−

g Ag21Σ+

g Cd21Σ+

g

Method RASPT2 RASPT2 CASPT2 RASPT2 CASPT2 CASPT2 RASPT2 RASPT2 CASPT2 CASPT2

Active (6,18) (8,18) (10,18) (12,12) (14,12) (16,18) (18,18) (20,18) (22,14) (24,14)

Space restricted restricted complete complete complete restricted restricted restricted complete complete

Basis Set aVQZ-PP aVQZ-PP ANO-RCC ANO-RCC aVQZ-PP aVQZ-PP aVQZ-PP aVQZ-PP aVQZ-PP aVQZ-PP

BSO 0.595 2.919 2.755 5.294 1.836 2.045 1.610 1.207 1.000 0.509

comments also ANO also ANO all work well all work well vdW

Diatom La25Σ−

u Hf21Σ+

g Ta23Σ−

g W21Σ+

g Re21Σ+

g Os25Πg Ir2

5∆g Pt23Σ−

g Au21Σ+

g Hg21Σ+

g

Method CASPT2 CASPT2 CASPT2 RASPT2 RASPT2 RASPT2 RASPT2 RASPT2 RASPT2 RASPT2

Active (6,18) (8,18) (10,18) (12,18) (14,18) (16,18) (18,18) (20,18) (22,18) (24,14)

Space complete restricted complete restricted restricted restricted restricted restricted restricted complete

Basis Set ANO-RCC aVQZ-PP ANO-RCC aVQZ-PP ANO-RCC aVQZ-PP ANO-RCC aVQZ-PP aVQZ-PP aVQZ-PP

BSO 0.646 1.220 4.396 4.844 5.236 5.010 3.660 2.321 1.778 0.199

comments also ANO all work well all work well vdW

Table 6.2: Summary of Bond Strength Orders (BSO) and the methods which produced thebest results for each transition metal diatom.

aVQZ-DK = aug-cc-pVQZ-DK; aVQZ-PP = aug-cc-pVQZ-PP; vdW = van der Waals complex; also ANO = ANO-RCC also worked well

75

Sc2

Y2

La2

Zr2

Ti2

Hf2

Ta2

V2

Nb2

Mo2

Cr2

W2

Mn2

Tc2

Re2

Ru2

Os2

Fe2

Rh2

Ir2

Co2

Ni2

Pd2

Pt2

Cu2

Au2

Ag2

Zn2

Cd2

Hg2

Bond

Dis

soci

atio

n En

ergy

[k

cal/m

ol]

20

40

60

80

100

120

140


Figure 6.2: Calculated Bond Dissociation Energy by Periodic Table Row, All Diatoms.

Hence, it is shown that three metrics, bond length, bond dissociation energy, and stretch-

ing force constant do not give a simple correlation, with BDE showing the greatest variability.

In some sense, this is unexpected, because the homolytic cleavage of a diatom should be ener-

getically predictable, as there are no secondary considerations of geometry relaxation. Thus,

with BDE being variable and bond length being an averaged value, it is found that stretching

force constant, which is a second order parameter and thus the best descriptor of a system

in motion, proves also to be the best descriptor of bond strength.

Finally, a plot of all stretching force constants versus bond strength order (BSO) is

shown in Figure 6.4. This leads to a single curve which describes the BSO, and thus the

bond strength, of all diatoms.

76

Sc2

Y2 La2

Zr2

Ti2

Hf2

Ta2

V2

Nb2

Mo2

Cr2

W2

Mn2

Tc2

Re2

Ru2

Os2

Fe2

Rh2

Ir2

Co2Ni2

Pd2

Pt2

Cu2

Au2

Ag2

Zn2

Cd2

Hg2

Stre

tchi

ng F

orce

Con

stan

t [m

dyn/

Å]

0

1

2

3

4

5

6

7


Figure 6.3: Calculated Stretching Force Constant by Periodic Table Row, All Diatoms.

6.3. Integration of all Findings

Analyzing all of the transition metal diatoms by three methods, two basis sets, and two

or more active spaces yielded a wealth of information. To apply this information gener-

ally, observations and explanations of trends are compiled here (Table 6.2.) We define five

subsections that yield the best results:

1) Groups 3, 4 and 5 : Using the Molcas ANO-RCC basis set, these species can be run

with (6,18), (8,18) and (10,18) complete active spaces, respectively, using the (n)d, (n+1 )s

and (n+1 )p atomic orbitals (n = 3 for row 4, 4 for row 5, and 5 for row 6) to generate the d-

σg, d-σu, d-πu, d-πg, d-δg, d-δu, s-σg, s-σu, p-πu, p-πg, p-σg and p-σu molecular orbitals. This

yields the largest number of successful calculations, as judged by agreement with experiment.

In some cases, the (2g,12) (g = group number) active space gives reasonable results, but not

typically better results.

77

Ag2

Re2Cl2-8

Au2

Sc2Y2

La2

Ir2

Zr2

Pd2

Pt2

Ni2Hf2Cu2

Rh2

Zn2Hg2

Cd2

Ru2

Ta2

Mo2

Os2

Cr2

W2

Ti2

V2

Mn2

Fe2

Co2

Nb2

Tc2

Re2Bo

nd S

treng

th O

rder

0

1

2

3

4

5

6

Stretching Force Constant [mdyn/Å]0 1 2 3 4 5 6 7

Figure 6.4: Calculated Stretching Force Constant versus Bond Strength Order, All Diatoms.This plot serves as a conversion chart between the two parameters.

2) Group 6 : The members of group 6, Cr2, Mo2 and W2, are seen as candidates for

sextuple bonding. Each of these diatoms presented a unique challenge, such that they do

not fit into any particular group, or even with each other. The best method for each diatom

(RASPT2(12,20)/ANO-RCC for Cr2, CASPT2(12,12)/ANO-RCC for Mo2, and CASPT2(12,12)/aug-

cc-pVQZ-PP for W2) was determined by extensive trial and error.

3) Groups 7, 8 and 9 : Using the (2g,18) restricted active space yielded the best results

in most cases for these nine diatoms. However, with the larger number of electrons, (14,

16 or 18), a complete active space was no longer practical for either the ANO-RCC or aug-

cc-pVQZ-DK/PP basis sets. Accordingly, both basis sets were useful. In this work, the

ANO-RCC basis set yielded the best results for Re2, Fe2, Os2, Co2 and Ir2, where the aug-

cc-pVQZ-PP basis set gave the best results for Tc2, Ru2, and Rh2. As noted earlier, the

Mn2 van der Waals complex was best described by CASPT2(14,12)/aug-cc-VQZ-DK.

78

4) Groups 10 and 11 : These six diatoms were well behaved by all methods. The group

11 diatoms were often described as singly bonded species, as was done in this work. It

would seem a logical extrapolation that the group 10 diatoms be described as double bonds,

but none of the metrics (Re, BDE, ke) support this model. As the electron count is large

and the analysis straight forward, a (2g,12) active space, with any of the trial basis sets, is

recommended.

MR-AQCC [212] also worked well for the the diatoms in these groups. The method has

the shortcoming of being limited to complete active spaces of 16 orbitals or less, so the

(2g,12) active space was also used here. Restricted active space methods are not available

for this method.

5) Group 12 : Where all of these species are van der Waal complexes, a (2g,12) active

space is recommended. It may be that multi-reference calculations do not improve the

description of these molecules, over the less computationally intensive DFT methods.

79

Chapter 7

Transition Metal Diatoms - Maximum Bond Multiplicity

Is there Sextuple Bonding? In the previous chapter, we completed a comprehensive

survey of the multi-reference description of covalent bonding in transition metal diatoms.

What remains is a standing controversy concerning the maximum possible bond orders in

chemistry.

Setting aside f-orbital bonding trans-lanthanides, because there is little experimental

confirmation of that form of bonding, the five d-orbitals on each of two atoms should the-

oretically combine to form 5 bonding orbitals and 5 anti-bonding orbitals, or a maximum

possible bond order of 5. However, it has been theorized [181] that relativistic contraction of

5s and 6s orbitals, and corresponding relativistic expansion of 4d and 5d orbitals can cause

these shells to be similar in size and energy, creating the possibility of sextuple bonding.

This phenomenon has not been attributed to the 3d/4s orbital mix in the forth row of the

periodic table, because the lack of spherical nodes in the 3d orbitals limits their potential

for relativistic expansion, as described by Kaupp [129] by the term ‘hybridization defect.’

Additionally, to create a sextuple bond, there is the need that each atom donate six

valence electrons. Fewer electrons would be inadequate to produce six bonds, and additional

electrons would occupy anti-bonding orbitals, lowering the bond order. By this formula, the

elements that could potentially create a neutral sextuple bond are limited to molybdenum

(Mo) and tungsten (W). The speculation on the existence of sextuple bonding is supported

by the fact that Cr2, Mo2 and W2 are found to have the shortest bond lengths, and among

the largest stretching force constants (ke) for all transition metal diatoms.

Maximum Bond Orders : Figure 6.4 indicates that Mo2 and W2 have BSO’s of 5.294 and

4.844, respectively, which would indicate that Mo2 exhibits a sextuple bond. However, these

numbers are based on Ag2 and Re2Cl2−8 having bond orders of 1.0 and 4.0, respectively.

80

Although a bond order of 1 for Cu2, Ag2 and Au2 is generally accepted, their properties of

bond length, BDE and ke are dissimilar both experimentally and computationally. Hence,

whichever of the three is chosen will effect the final result. The true bond order of Re2Cl2−8

has been debated. Roos et al., for example, calculate an ‘effective bond order’ of 3.2 for

this species. [181] Using that value lowers the BSO’s for Mo2 and W2 to 4.05 and 3.76,

respectively. Krapp [147] theorizes that bonding in Re2Cl2−8 can be described entirely by σ-

and π-bonding, and has a bond order which is very near 3, and does not have a stronger bond

than the triple-bonded neutral Re2Cl8. Hence, without proving that the historical Re2Cl2−8

quadruple bond is indeed a quadruple bond, the Mo2 sextuple bond cannot be confirmed.

Further, the BSO of Mo2 is not significantly higher (and W2 is lower) than the Re2 and

Os2 diatoms, which clearly cannot exhibit sextuple bonding, due to having greater than 12

valence electrons. Hence, we do not find evidence of sextuple bonding here.

81

Chapter 8

Conclusions

In this dissertation, we have undertaken the study of multi-reference systems in chemistry.

All systems are multi-reference by their nature, but most do not require the inclusion of

multiple electronic states to be described within the models of modern quantum chemistry.

The computational expense of multi-reference calculations, even with the economy of

using complete active space and restricted active space calculations, which reduce cost, but

increase complexity, has led to much effort to develop methods where multi-reference systems

can be described with some degree of confidence with single-reference methods. This work

focuses on the systems that require descriptions beyond the electronic ground state.

In this dissertation, we have looked at some of the methods, and some of the strategies

for the description of multi-reference systems, hoping to establish guidelines that streamline

the use of multi-reference methods, and possibly facilitate their broader use.

8.1. Single-Reference Systems

We demonstrated that single-reference systems can be well characterized by hybrid DFT

methods and large basis sets. Analysis of the parameters of equilibrium geometries, rCC,

local stretching force constant, ka, Bond Strength Order, electron density, ρr and energy

density, Hb can fully characterize these systems.

We calibrated these tools against a training set of well-behaved molecules, to be equipped

to investigate the chemistry of unique systems: extremely strained molecules, clamped

molecules, diamondoid stabilized molecules and electron deficient molecules. This was, in

effect, a search for systems we could not describe.

82

8.2. Single-Reference Methods on Multi-Reference Systems

Moving to adapt our methods to multi-reference systems, we considered pancake bonding.

These species are found experimentally to be spin-paired open-shell dimers, and therefore

have multi-reference character. Using broken symmetry DFT, we were able to reproduce the

experimental observations on a series of dichalcodiazolyl and phenalenyl pancake-bonded

dimers. With DFT methods, topological analysis and local mode vibrational analysis, we

were also able to demonstrate that, in different cases, chalcogen bonding, aromaticity and

dispersion forces all contribute to the stabilization of these molecules. We were able to

establish that pancake bonding is a non-covalent, electrostatic interaction and propose a new,

ditelluradiazolyl pancake bonded dimer, with a unique symmetry. We also found evidence

of 3-dimensional aromaticity in the phenalenyl systems.

8.3. Multi-Reference Methods in Organic Chemistry

We then attempted to describe the bridged annulene systems whose nature had been

debated since they were first synthesized in 1964, and whose character (single or multi-

reference) is in dispute. An extensive survey of single-reference methods succeeded in re-

producing and extending on the work of past researchers, and being equally unsuccessful in

describing and explaining the chemistry of these molecules.

Many multi-reference methods produced more unsupportable results, and FCI was im-

possible for systems of this size. The breakthrough on this analysis was to redefine the

orbital description of the molecules to include the strained chemistry of the cyclopropyl ring

in the ‘norcaradiene’ valence tautomer of each system.

To verify our findings, topological analysis, local mode analysis, aromaticity indices, and

computational analysis of experimental NMR data and crystal packing effects resulted in a

complete description of the target molecule and the answers to the questions that have stood

for over half a decade.

83

8.4. Multi-Reference Methods in Inorganic Chemistry

Many have investigated the covalent bonding descriptions of small transition metal clus-

ters, where covalent bonding, low-lying excited states and clustered σ-, π- and δ-molecular

orbitals guarantee multi-reference behavior. Where hundreds of publications exist, the ones

that attempt to describe every transition metal diatom have used single-reference methods,

primarily DFT.

In our study, we analyzed every transition metal diatom by four different multi-reference

methods and three different basis sets, to establish patterns and guidelines for larger systems.

We succeeded in reproducing or improving on past work, as well as filling some of the

empty spaces in the literature. We have generated an extensive database of results for these

methods, and developed guidelines for method, basis set, and active space selection for all

species.

8.5. Outlook

Extending on the projects we present here, the following studies present themselves:

Long Carbon-Carbon Bonds: The possibility exists of of creating a tool for predicting

local mode stretching force constants for CC single bonds, based on substitutions. This

could be central to designing molecules with longer CC bonds. The relationship would

logically be quadratic, since kas are based on frequencies, and the harmonic approximation.

The calculations of known systems have given enough information to run a curve on methyl

groups (see Figure 8.1.) Many more simple molecules (1,2,3,4,5,6 phenyl groups, for example)

would need to be analyzed to complete the other curves. However, our preliminary work

shows that we need not depend on experimental observations to verify our results. Then,

the extension of these algorithms to mixed substitutions could be investigated.

Clamped Bonds: The calculated geometries for the clamped molecules of Suzuki’s group

(such as 3.32) are not in close agreement with experimental results. These molecules can

be analyzed by multi-reference methods, to determine if there is biradicaloid or fluxional

character in these long bonds.

84

R2 = 0.998

Loca

l Mod

e St

retc

hing

For

ce C

onta

nt k

a [m

dyn/

Å]

3.13.23.33.43.53.63.73.83.94.04.14.24.3

Methyl Groups0 1 2 3 4 5 6

Figure 8.1: Local Mode Stretching Force Contants (ka) versus Number of Methyl Groupsfor a Series of Organic Compounds.

The iron-bridged, clamped molecule calculated in this work (textbf3.33) was simplified

from the molecule published by Moret’s group, by replacing 8 phenyl groups with methyl

groups. This was done to save computational cost, but resulted in a CC distance much

shorter than the experimental result. An analysis of the full molecule may give the correct

geometry, or reveal multi-reference character.

Bridged Annulenes: The molecule 11-cyano-11-methyl-1,6-methano[10]annulene has been

synthesized, and found to have a C1C6 interatomic distance of 1.8 A, similar to 11,11-

dimethyl-1,6-methano[10]annulene, but in disagreement with the result for 11, 11-dicyano-

1,6-methano[10]annulene. CASPT2(14,14) and CCSD(T) calculations would be useful to

discover the chemistry of this molecule.

Transition Metal Diatoms: The sextuple bond question remains for the Mo2 and W2

diatoms. Better reference molecules to define the BSO description can be sought.

85

Chapter 9

Calculations and Methodology

Multi-reference Systems in Computational Chemistry: In this dissertation, we made

use of a significant fraction of all available quantum chemistry methods, both utilizing wave

function theory and density functional theory. Ultimately, true multi-reference systems can

only be reliably described with multi-reference, wave function theory methods, which are

unfortunately the most computationally expensive, are impracticalfor very large molecules,

and requires substantial knowledge of the chemistry being described. Therefore, we weighed

these many methods for their applicability to specific questions.

9.1. Single-Reference Computational Methods - Organic Chemistry

Long Carbon-Carbon Bonds : Many different Density Functional Theory (DFT) methods

are available, which range widely in parameterization, computational cost, and range of

applicability. In the case of the description of long bonds, we are looking for a functional

which will provide consistent results for a wide range of organic compounds, many quite large.

Since we are less concerned about atoms heavier than carbon, we can use relatively expensive

hybrid functionals, which combines Hartree Fock and other DFT functionals to achieve the

widest applicability. The ωB97X-D functional [23] was chosen because it includes Grimme

D2 dispersion correction [94] and Chai-Head-Gordon long-range corrections. [39, 40] As the

CC bonds get longer, long range augmented basis sets with correction for the correlated

motion of electrons are required to accurately describe the extreme cases. For this purpose,

Dunning’s aug-cc-pVTZ basis set [72] was used whenever possible. For some of the largest

molecules, the augmented basis set would not converge, and the cc-pVTZ basis set was

used. Finally, the largest molecule in the sequence, hexakis-(3,5-di-tert-butylphenyl)ethane,

required Pople’s 6-311++G(d,p) [67, 98] basis set to complete the second derivative matrix

86

and frequency calculations which are needed for the analysis of vibrational frequencies to be

completed in a reasonable time frame (under 1 year of wall time.)

Pancake bonding : To describe the spin-paired open shell singlet state of pancake bonded

systems, we applied broken symmetry calculations. [46, 92] This was accomplished in two

steps. First, a single point calculation of the initial guess equilibrium structure was completed

in the triplet state. This was followed by a full optimization in the singlet state, using the

triplet state orbital description. Frequency calculations were conducted on the final singlet

state geometries.

For the dichalodiazolyl molecules 4.1 through 4.3, the unrestricted BS-UM06 DFT

methodology was used [102, 225], following the method of Kertesz and others. [25, 120] The

M06 functional is well parameterized for the chalogens (sulfur, selenium and tellurium), yield-

ing consistent results which were in good agreement with experiments. The Pople triple-ζ

6-311G(d,p) basis set was used [98] for compounds 4.1 and 4.2, again following Kertesz’ pro-

cedure on the description particularly of the selenium compound. [25] The Stuttgart/Dresden

effective core potential SDD basis set was used for (4.3), as tellurium is parameterized in

few other basis sets.

Having established the equilibrium geometry, the dimers were then separated in 0.1 A

increments, spanning the equilibrium distance approximately 1 A in either direction. For

longer distances, 1.0 A increments were used, for a total range of 2.5 to 8.0 A. These values

were plotted to generate potential energy curves were generated, from which dissociation

energies could be deduced. (See Figure 4.4.)

The phenalenyl systems (4.4) through (4.6) were analyzed using M05-2X [159] and M06-

2X DFT functionals. [225] The 6-31++G(d,p) basis set [98] with polarization and diffuse

orbitals was used, to ensure the coverage of dispersion and other long range effects.

For the purposes of comparability with species 4.1 through 4.3, the M06 functional was

considered desirable. However, calculating the system by both M05-2X and M06-2X, M05-

2X BDH’s for these systems agreed well with experiment, where the M06-2X functional gave

results that were too high by a factor of 2. This is in agreement with the findings of Kertesz,

87

et al. [159] Therefore, the M05-2X functional was used for all calculations.

Counterpoise corrections were carried out on all pancake species, to remove basis set

superposition error, and determine the complexation energy between the radical fragments,

which gives an independent evaluation of the dissociation energy of the dimers in the absence

of geometry relaxation.

Annulenes : The C1C6 potential energy curves (PECs) of molecules 5.1, 5.2, and 5.3

were calculated by stepwise increasing of the distance R = R(C1,C6) from 1.4 to 2.6 A in

increments of 0.1A and then, optimizing all other geometrical parameters of the molecule

in question. These energy values were fitted to a suitable analytic function using standard

fitting techniques. For the analytical PECs thus obtained, the R value(s) of the minimum

or minima were determined and then utilized for accurate reoptimization. This procedure

was carried out at the Hartree-Fock (HF) [188] level, density functional theory (DFT) level

using various LDA, GGA, meta-GGA, hybrid, and double-hybrid functionals (see below) as

well as the second, third and forth order Møller-Plesset Perturbation Theory (MP2, MP3,

MP4) level of theory [47,102].

Using the geometries determined in the first step, single point calculations with a variety

of electron correlation WFT (wave function theory) methods were carried out to obtain more

accurate PECs. It turned out that the use of different geometries based on HF, DFT, or

MP2 calculations did not change the resulting PECs significantly. However, the vibrational

frequencies obtained with DFT methods such as B3LYP [22,200] or ωB97X-D [39,40] agree

better with measured values than HF or MP2 frequencies, indicating that the thermochem-

ical corrections calculated with DFT are preferable. Therefore, for reasons of consistency,

we present only the PECs based on geometries obtained with the two XC (exchange and

correlation corrected) functionals ω-B97X-D and B3LYP in this work.

To study the influence of dynamical electron correlation on the shape of the PEC, we

used nth order MP (MPn) theory and Coupled Cluster theory with HF or alternatively

Brueckner (B) orbitals. [97, 178] Correlation effects were determined by considering all S

(single), D (double), and T (triple) or perturbative T (denoted as (T)), and Q (quadruple)

88

excitations. In all, we used MP2, MP3, MP4(SDQ), MP4(SDTQ), [50, 102] CCSD, [176]

BD, [97] CCSD(T), [179] and BD(T). [178]

9.2. Single-Reference Computational Methods Beyond Energies

Beyond the calculation of total energies for all molecules, the following parameters were

determined using single-reference methodologies.

Cremer-Kraka Bond Criterion: The Cremer-Kraka Bond Criteria gives a quantifiable

definition of covalent bonding, and is used extensively in this work. [55] By this criterion,

three conditions are required to define covalent bond:

1. There must be a maximum electron density path (MED) between the targeted atoms.

2. There must a bond critical point (BCP) of minimum electron density (ρr) located along

the MED.

3. The energy density (Hb) at the BCP must be negative and stabilizing.

The determination of bond critical points (BCP’s), and electron densities (ρr) and en-

ergy densities (Hb) at the pertinent BCP’s, using Baders Atoms-In-Molecules analysis, was

completed using the AIMAll software, version 12.6.03 and 17.1.25 [134]

Local Mode Stretching Force Constants : Local Mode Stretching Force Constants (ka) are

determined using the generalized local vibrational mode analysis as developed by Konkoli

and Cremer [139,140] and its extension by Zou and Cremer. [239–241]

For this work, we employ local mode analysis [140,239–241] to calculate local vibrational

frequencies (ωa) and stretching force constants (ka) for the most pertinent CC single, double

and triple bonds. This local CC stretching force constant ka is used as the measure of bond

strength.

Bond Strength Order : To translate the bond strength, as expressed by ka, into a useful

format, a power relationship for the bond strength order, n(CC), was generated [78,122,124,

143] following the treatment of the generalized Badger rule of Kraka, Larsson and Cremer.

[146] The relationship is defined by the equation:

89

n(CC) = a(ka)b (9.1)

This defines bond strength as a bond strength order parameter (BSO) which is recog-

nizable and useful to all chemists. Parameters a and b are determined in this work to be a

= 0.3135 and b = 0.8062, utilizing ethane, ethene and the origin as suitable references with

n(CC) = 1, 2 and 0, respectively.

Aromaticity Index: Employing local mode analysis, the aromaticity index (AI) for species

4.4 through 4.6, 5.1 and 5.2 were determined by the method of Kalescky, Kraka and Cremer.

[123,191] This method characterizes the degree of aromaticity of a delocalized system, based

on the reference values of 1.0 for benzene, and 0.0 for “Kekule” benzene, as modeled by

trans-1,3-butadiene. The AI is defined as:

AI = 1− γ

NB

∑(nopt − ni)2 (9.2)

where nopt is the optimum bond order for benzene, NB is the number of bonds in the aromatic

system, ni is the bond order of each of the “i” bonds in the aromatic system, and γ is a

parameter to set the AI of trans-1,3-butadiene to 0.0. This equation was recast into the

following form:

AI = 1− γ[(nopt − nav)2 +

1

NB

∑(nav − ni)2

](9.3)

= 1− (WS + ALT) (9.4)

where nav is the average bond order, WS gives the weakening/strengthening parameter of

the bonds, relative to the average bond strength (as defined by the average bond order), and

ALT reflects the degree of bond strength alteration, with the greater the bond weakening,

and bond strength alteration, the weaker the aromaticity of the system, and the smaller the

AI.

90

G4 Bond Dissociation Enthalpies: Bond dissociation enthalpies (BDH’s) were calculated

wherever possible for the covalently bonded CC species using Gaussian G4 methodolgy. [41]

These calculations were not feasible for clamped systems, because independent fragments

cound not be defined, aromatic systems, where the results were found in this work and by

others to be unreliable, [44, 237] the largest systems, due to the tremendous computational

cost of the G4 algorithms, and multiple bonds other than ethylene and acetylene.

NMR analysis : To compare with available experimental NMR data for annulene system

5.1, 13C and 1H magnetic shieldings and chemical shifts (the latter with tetramethylsilanes

as reference) were calculated for all systems using guage-invariant atomic orbital (GIAO)

calculations at the B3LYP level of theory [234], as well as the NMR-ab intio-DFT method of

Cremer and co-workers [59]. This latter method is used to determine the most stable form

of a molecular system in solution, especially in the case where one geometrical parameter

strongly depends on relatively small changes in environmental factors as was the case for

system 5.1. Since the 13C-NMR chemical shifts of 5.1 are known [69], we determined

the deviation between measured and calculated 13C chemical shifts along the PEC and

determined the minimum of the mean (absolute) deviation. In addition, all J(13C13C),

J(1H1H), and J(13C1H) indirect spin-spin coupling constants (SSCCs) of 5.1 were calculated

using the method of Cremer and co-workers. [209] Those SSCC which show the strongest

dependence on the valence-tautomeric rearrangement were analyzed as functions of R.

Solvent effects on the experimental NMR data were tested using the PCM (polarizing

continuum model) of Tomasi and co-workers [223] where the dielectric constant ε was in-

creased from 2 to the value of methanol (ε = 32.7 [100]) which was used as the solvent for

the NMR investigations of 5.1. [69, 80]

The single-reference quantum chemical calculations were performed with the program

packages COLOGNE14 [145], CFOUR, [199] and Gaussian [79].

9.3. Multi-Reference Computational Methods

Annulenes: For systems 5.1 through 5.3, the influence of non-dynamical electron corre-

91

lation effects was studied utilizing CASSCF (complete active space self consistent field), [153]

CASPT2 (CASSCF with second order MP perturbation theory), [4] and NEVPT2 (n-electron

valence perturbation theory of second order).) [99] For these multi-reference calculations, ac-

tive spaces of 5.1a and 5.2a with 10 electrons (all π) and 10 orbital (5 occupied π-orbitals

and the lowest 5 virtual π-MOs were selected. In the case of 5.3a the corresponding (6,6)-

active space was chosen. In the valence-tautomeric rearrangement of the annulene (triene)

form to the bisnorcaradiene (norcaradiene) form π-orbitals are converted into C1C6 σ and σ?

orbitals and other σ-orbitals mix with the π-orbitals, changing their character. Apart from

this, one has to consider that π- and σ-orbitals may be ordered in a different way. Therefore,

it was carefully monitored that when changing R the (10,10) or (6,6) active spaces were

consistently maintained along the path defined by R.

This procedure led to unsolvable problems in the case of system 5.3 where the non

planarity of 5.3 (see Figure 5.1) leads to strong σ − π mixing. We found that a consistent

description can only be obtained by a (10,10) active space for the norcaradiene form, which

includes besides the π-system also the 6 electrons and 6 Walsh orbitals of the cyclopropyl

ring. By converting one of the Walsh orbitals into a π-orbital for increasing R, a consistent

(10,10) description could been found for system 5.3a. The following orbitals comprised the

(10,10) space of 5.3b: 10a′, 14a′, 19a′, 8a′′, 12a′′, 16a′′ (Walsh orbital set), 15a′, 16a′, 10a′′,

11a′′ (π-orbital set).

The multi-reference description of systems 5.1 and 5.2 is a similar way, recreating the

PEC calculations with (14,14) active space including the Walsh orbitals of the cyclopropyl

rings of the bisnorcaradiene forms and the σ (σ?) orbitals of the C1C11 and C6C11 bonds.

For 5.1b the following orbitals established the (14,14) space: 14a1, 16a1, 9b1, 12b1, 17b1,

15b2 (Walsh orbital set), 15a1, 17a1, 10b1, 11b1, 12b2, 13b2, 8a2, 10a2 (π-orbital set). In the

case of 5.2b, the (14,14) space comprised: 10a1, 13a1, 17a1, 8b2, 11b2, 14b2 (Walsh orbital

set), 12a1, 15a1, 9b1, 10b1, 9b2, 10b2, 7a2, 9a2 (π-orbital set).

We also applied the smaller active spaces to multi-reference averaged quadratic coupled

cluster theory (MR-AQCC.) [212] In addition, systems 1a/1b and 2a/2b were also investi-

92

gated utilizing the DIP-EOM-CCSD (equation of motion double ionization potential coupled

cluster with S and D excitations) [192] method.

Transition Metal Diatoms: Three multireference methods were used to generate the

transition metal diatom results reported from this study:

1. Complete active space self-consistent field multireference average quadrature coupled

cluster calculations (CASSCF/MR-AQCC.) [212]

2. Complete active space self-consistent field calculations with non-dynamical correlation

by second order perturbation (CASSCF/CASPT2.) [4]

3. Restricted active space self-consistent field calculations with non-dynamical correlation

by second order perturbation (RASSCF/RASPT2.) [153]

Three relativistic basis sets were employed.

1. The augmented, correlation corrected Dunning quadruple-zeta basis set with Douglas-

Kroll-Hess relativistic correction was used for forth period atoms (aug-cc-pVQZ-DK.)

[233]

2. The augmented, correlation corrected Dunning quadruple-zeta basis sets with effective

core potential relativistic correction was used for fifth and sixth period atoms (aug-cc-

pVQZ-PP.) [233]

3. The Roos atomic natural orbital relativistic correlation consistent (ANO-RCC) basis

set was used on all diatoms. [5, 182–184]

For the systematic design of active space, each diatom was first analyzed at the Hartree-

Fock level of theory to determine the order and occupancy of the molecular orbitals. The

active space for each diatom was chosen to include all electrons in the (n)d and (n+1)s

valence shells are considered. Since this work was restricted to homodiatomic species, the

number of valence electrons is twice the number of the column in the periodic table. Either

the orbitals formed by the overlap of all (n)d and (n+1)s atomic orbitals was used, that is d-

σg/d-σu (2), d-πu/d-πg (4), d-δg/d-δu (4), and s-σg/s-σu (2) orbitals, for a total of 12 orbitals,

93

or else the sum of all of these, plus the orbitals formed by the overlap of the (n+1)p atomic

orbitals (p-σg/p-σu (2) and p-πu/p-πg (4)), for a total of 18 active space orbitals. The 18-

orbital active space usually required the use of a restricted active space, to reduce the number

of configuration state functions (CSF’s) and to reduce computational cost. Generally, the

Molpro program [232] can operate on up to 1,000,000 CSF’s. In a few cases, as noted in

Section 3, other active spaces, such as (22,14) and (12,20) were analyzed, too. This is usually

necessitated by having the 12th or 18th orbital being degenerate.

The electronic state of each diatom was taken from the experimental literature. In cases

where more than one electronic state was reported, we calculated both states, and reported

the electronic configuration which yielded results which agreed with the experimental pa-

rameters. For each species, bond lengths (re) are determined by geometry optimization.

Bond dissociation energies (BDE’s) are determined by subtracting twice the energy of the

free atoms from the total energy of the diatom, using one half as many orbitals and valence

electrons as for the molecular species. Hence, Sc2, analyzed with a (6,18) active space, is

compared to the Sc atom with a (3,9) active space. The multiplicity of the atomic species

are taken as reported in the literature, or on the NIST periodic table. [1]

Stretching force constants (ke’s) are determined numerically by calculating the energy

of the molecule at 0.005 A intervals around the optimized geometry, and calculating the

curvature of the resultant potential curve. To convert this parameter into a bond strength

order, we used Ag2, the octachlorodirhenium dianion (Re2Cl2−8 ) and the origin as references,

assigning bond orders of 1, 4, and 0, respectively. These references define the true Badger-

type [12, 13] power relationship, n(MM) = a(ka)b, with variables of a = 0.8075 and b =

1.009.

The Molpro program package [232] was used for all calculations involving the Dunning

basis sets. The calculations using the atomic natural orbital basis sets (ANO-RCC) [183]

were completed using the Molcas program package. [3] The quantum chemical calculations to

analyze the Hartree-Fock molecular orbitals were performed with the Gaussian 09 program

package. [79].

94

Appendix A

Publications, Supporting Information and Manuscripts

11, 11-Dimethyl-1, 6-methano[10]annulene - An Annulene with an Ultralong CC Bond or a

Fluxional Molecule? (2015)

The Longest Covalent CC Bonds - Characterized with Vibrational Spectroscopy

(Manuscript in process)

Local Mode and Aromaticity Index Analysis of Pancake Bonded Systems

(Manuscript in process)

A Study of Various Multi-Reference Methods for the Characterization of 30 Transition

Metal Diatoms (Manuscript in process)

95

11,11-Dimethyl-1,6-methano[10]annuleneAn Annulene with anUltralong CC Bond or a Fluxional Molecule?Alan Humason, Wenli Zou, and Dieter Cremer*

Computational and Theoretical Chemistry Group (CATCO), Department of Chemistry, Southern Methodist University, 3215 DanielAvenue, Dallas, Texas 75275-0314, United States

*S Supporting Information

ABSTRACT: Extensive quantum chemical calculations involving morethan 20 different methods and including vibrational, temperature, entropic,and environmental corrections suggest that 11,11-dimethyl-1,6-methano[10]annulene (1) is characterized by a broad, asymmetric singlewell potential minimum in which the molecule can carry out a large-amplitude vibration. This result is obtained by using CASPT2(14,14) andCCSD(T) together with a VTZ basis set. The average R(C1C6) distance of1 is close to 1.8 Å, in agreement with X-ray diffraction measurements.Lower level methods fail because a reliable account of the electronicstructure of bridged annulenes requires a balanced description ofnondynamical and dynamical electron correlation effects as well as acorrect assessment of bridge−annulene interactions. An independentdetermination of the distance R using the mean deviation between thecalculated and measured 13C NMR chemical shifts of 1 leads to a value of 1.79 Å. By using electron density, energy density, andthe local C1C6 stretching mode, it is demonstrated that the covalent bond ceases to exist at 1.695 Å and that for larger R valuesthrough-space homoaromatic interactions lead to some stabilization. The peculiar potential of 1 is shown to be a result of theinteraction of the methyl groups with the perimeter CC bonds bisected by the symmetry plane of the molecule. CASPT2(14,14),CASPT2(10,10), CCSD(T), and BD(T) calculations were also used to provide for the first time reliable descriptions of thevalence tautomeric potentials for the parent molecule, 1,6-methano[10]annulene (2), and the system 1,3,5-cycloheptatriene−norcardiene (3). In the latter case, calculations confirm a previous kinetic measurement of the free activation energy but correctNMR-based estimates. The methodology described can be applied to other annulenes and fullerenes.

1. INTRODUCTION

For the purpose of getting a better understanding of the natureof the chemical bond, chemists have always been interested inanswers to questions such as “What is the strongest (weakest)or shortest (longest) bond ever observed or may be in generalpossible?”1−3 Since the chemical bond is a concept rather thanan observable quantity, which can be extensively described bysuitable measurements, answers to these questions may haveonly heuristic value if not handled within a well-defined modelof the chemical bond. The typical superlative questions leadonly to useful answers if they are defined as precisely aspossible, thus clarifying definitions and evaluation methodswithin given models and concepts.In this work, we focus on (through-space or through-bond)

interactions between two C atoms, which are separated by 1.8Å and investigate the question whether these interactions leadto a covalent bond. This implies that we contrast theinteraction in question on the background of other long CCbonds, clarify which bonding models and definitions we willuse, and why the investigation is important in connection with amolecular system that has puzzled chemist for almost 50 years(see below).

Typical electron pair bonds between neighboring C atomscan be lengthened as a consequence of (i) exchange (steric)repulsion between bulky substituents,3−8 (ii) a loss of bondingelectrons or electron density leading to electron deficientbonding,9−13 (iii) the occupation of antibonding orbitals, or(iv) decoupling of spin pairs leading to open singlet states (e.g.,pancake bonds).6,14−16 (v) In turn, interacting CC atoms atlarge distances can be clamped together by electrostaticattraction when highly polar or ionic,17−21 by dispersioninteractions,3 or by connecting bridges enforcing a “cage”-topology (clamped bonds22). Packing effects in a solvent cage orin a crystal could also be possible. Often these effects cannot beclearly separated because they support each other. Manyexamples have been given for such long CC bonds or CCinteractions where one has to ask in each case whether a stablemolecule or a labile intermediate was obtained.

Special Issue: 25th Austin Symposium on Molecular Structure andDynamics

Received: August 16, 2014Revised: October 20, 2014Published: October 21, 2014

Article

pubs.acs.org/JPCA

© 2014 American Chemical Society 1666 dx.doi.org/10.1021/jp5082966 | J. Phys. Chem. A 2015, 119, 1666−168296

The existence of a chemical bond can be verified usingexperimental data. Pauling suggested that a chemical bond isgiven when two atoms or f ragments are kept together by some forcesthat it seems to be justif ied to consider the resulting entity as anindependent molecule.1 Less vague definitions of a chemical bondcan be obtained utilizing energy, geometrical, orbital or densitycharacteristics. There is a myriad of bond definitions withinspecific models. We will avoid an explicit discussion of thesedefinitions and models by exclusively focusing in this work onmeasurable quantities such as electron density distribution,vibrational frequencies, NMR chemical shifts, and indirectspin−spin coupling constants (SSCCs). From these propertieswe will determine quantities that make it possible tocharacterize the CC interactions investigated in this work asbeing covalent bonding or just through-space interactions.The target system 1 investigated in this work (see Figure 1)

belongs to the group of bridged [10]annulenes, the parent

member of which, 1,6-methano[10]annulene (2), was firstsynthesized by Roth and Vogel in 1964.23 Annulene 2 wasexperimentally characterized as an aromatic 10π-system ful-filling the Huckel 4n + 2 rule, however with a distorted ringperimeter and a C1C6 distance R of 2.235 Å;24,25 that is, itadopts structure 2a rather than that of the bisnorcaradiene(tricyclo[4.4.1.01,6]undeca-2,4,7,9-tetraene) form 2b (see Fig-ure 1), as was verified in dozens of experimental inves-tigations23−37 as well as computational investigations.32,38−43

Therefore, it was an unexpected result when in the early1970ies the 11,11-dimethyl derivative 1, also synthesized by theVogel group, was found to adopt according to X-ray diffractionstudies44 a “tricyclic” structure with an R value close to 1.8 Å(1.827 and 1.771 due to 2 molecules in the unit cell44) thatspeaks in favor of structure 1b rather than 1a. With thediscovery that the corresponding 11-cyano-11-methyl derivativehad similar R values45 and the dicyano derivative had even an Rvalue of 1.558 Å,44,46 the existence of a bisnorcaradiene formwas fully confirmed.Gunther and others carried out NMR (nuclear magnetic

resonance) investigations on valence tautomeric systems suchas 1 and 2 and showed that any shift to the bisnorcaradiene orannulene form leads to characteristic changes in the 13Cchemical shifts.47,48 These studies were latter confirmed byFrydman and co-workers49 as well as Dorn and co-workers,50

who all documented a strong temperature dependence of the

13C NMR signals of 1. Hence, the basic question was whetherbridged annulenes could be considered as fluxional systemswhich changed their bonding structure according to the valencetautomeric rearrangement indicated in Figure 1 or whether theypossessed long CC bonds.Since the available experimental evidence for 1 favored the

bisnocaradiene form, the molecule was considered as theneutral hydrocarbon with the longest (covalent) cyclopropaneC(sp2)C(sp2) bond. Therefore, 1 was computationallyinvestigated multiple times to rationalize the bondingsituation.32,39−43 However, this work was hampered by thefact that the size of the molecule and the requirements for thebasis set made state-of-the-art computations that guarantee highaccuracy results extremely difficult. In this work, we will closethis gap in the description of 1 by carrying out coupled clusterand multireference calculations and providing a reliabledescription of the electronic structure. The following questionswill be answered in this work: (i) What is the exact shape of theC1C6 potential of 1 (single or double well)? (ii) Whichelectronic factors determine the shape of the potential? Forexample, are there specific interactions between bridge and π-perimeter, which may influence the structure and dynamics of1? (iii) How does π-delocalization in 1 and 2 compare with thatof benzene? (iv) What level of theory is required to guarantee areliable description of the properties of 1 (or 2)? (v) Dotemperature, entropy, or environmental effects such assolvation or crystal packing influence the shape of the potential?(vi) How does the valence tautomeric behavior of 1 relate tothat of other closely related molecules such as 2 or the systemcycloheptatriene-norcaradiene (3, Figure 1)? (vii) Are thereother observations which may lead to a confirmation of theshape of the potential of 1? (viii) Does 1 contain a longcovalent C1C6 bond, as is generally believed today, and if so, isthis the longest C(sp2)C(sp2) single bond of an unchargedhydrocarbon ever observed?We will present the results obtained in this work in the

following way. In section 2, the computational means used inthis work are described. The results and discussion arepresented in section 3. The chemical relevance of this workis outlined in section 4. Conclusions are drawn in section 5.

2. COMPUTATIONAL METHODSThe C1C6 potential energy curves (PECs) of molecules 1, 2,and 3 were calculated by stepwise increasing of the distance R =R(C1,C6) from 1.4 to 2.6 Å in increments of 0.1 Å, and thenoptimizing all other geometrical parameters of the molecule inquestion. The energy values thus obtained were fitted to asuitable analytic function using standard fitting techniques. Forthe analytical PECs thus obtained, the R value(s) of theminimum or minima were determined and then used for anaccurate reoptimization. This procedure was carried out at theHartree−Fock (HF) level, density functional theory (DFT)level using various LDA, GGA, hybrid, and double-hybridfunctionals (see below) as well as the second order Møller−Plesset Perturbation Theory (MP2) level of theory.51,52

Using the geometries determined in the first step, singlepoint calculations with a variety of electron correlation WFT(wave function theory) methods were carried out to obtainmore accurate PECs. It turned out that the use of differentgeometries based on HF, DFT, or MP2 calculations did notchange the resulting PECs significantly. The vibrationalfrequencies obtained with DFT methods such as B3LYP53,54

or ωB97X-D for 2a agree better with measured values34 than

Figure 1. Molecules investigated in this work.

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp5082966 | J. Phys. Chem. A 2015, 119, 1666−1682166797

HF or MP2 frequencies, thus indicating that thermochemicalcorrections calculated with DFT are preferable. Therefore, wepresent in this work for reasons of consistency only the PECsbased on geometries obtained with the two XC functionalsωB97X-D and B3LYP.For the purpose of studying the influence of dynamical

electron correlation on the shape of the PEC, we used nthorder MP (MPn) theory and Coupled Cluster theory with HFor alternatively Brueckner (B) orbitals. Correlation effects weredetermined by considering all S (single), D (double), and T(triple) or perturbative T (denoted as (T)) and Q (quadruple)excitations. In all, we used MP2, MP3, MP4(SDQ), MP4-(SDTQ),51,52 CCSD,55 BD,56 CCSD(T),57 and BD(T).58

The influence of nondynamical electron correlation effectswas studied utilizing CASSCF (complete active space self-consistent field),59 CASPT2 (CASSCF with second order MPperturbation theory),60 and NEVPT2 (n-electron valenceperturbation theory of second order).61 For these multi-reference calculations, active spaces of 1a and 2a with 10electrons (all π) and 10 orbitals (5 occupied π-orbitals and thelowest 5 virtual π-MOs) were selected. In the case of 3a thecorresponding (6,6)-active space was chosen. In the valence-tautomeric rearrangement of the annulene (triene) form to thebisnorcaradiene (norcaradiene) form, π-orbitals are convertedinto C1C6 σ and σ* orbitals and other σ-orbitals mix with theπ-orbitals, changing their character. Apart from this, one has toconsider that π- and σ-orbitals can change their energies andorder. Therefore, it was carefully monitored that whenchanging R the (10,10) or (6,6) active spaces were consistentlymaintained along the path defined by R.This procedure led to unsolvable problems in the case of

system 3 where the nonplanarity of 3 (see Figure 1) leads tostrong σ−π mixing. We found that a consistent description canonly be obtained by a (10,10) active space for the norcaradieneform, which includes besides the π-system also the 6 electronsand 6 Walsh orbitals (for pictorial representations of theseorbitals, see ref 62.) of the cyclopropyl ring. By converting oneof the Walsh orbitals into a π-orbital for increasing R, aconsistent (10,10) description could be found for system 3a.The following orbitals comprised the (10,10) space of 3b: 10a′,14a′, 19a′, 8a″, 12a″, 16a″ (Walsh orbital set), 15a′, 16a′, 10a″,11a″ (π-orbital set).This observation encouraged us to improve the multi-

reference description of systems 1 and 2 in a similar way,repeating the PEC calculations with a (14,14) active spaceincluding the Walsh orbitals of the cyclopropyl rings of thebisnorcaradiene forms and the σ (σ*) orbitals of the C1C11and C6C11 bonds of the annulene. For 1b, the followingorbitals established the (14,14) space: 14a1, 16a1, 9b1, 12b1,17b1, 15b2 (Walsh orbital set); 15a1, 17a1, 10b1, 11b1, 12b2,13b2, 8a2, 10a2 (π-orbital set). In the case of 2b, the (14,14)space comprised the following: 10a1, 13a1, 17a1, 8b2, 11b2, 14b2(Walsh orbital set); 12a1, 15a1, 9b1, 10b1, 9b2, 10b2, 7a2, 9a2 (π-orbital set).Clearly, a (34,34) active space would have been ideal for the

inclusion of all σ−π interactions. However, this approach wasoutside the computational possibilities. We also used thesmaller active space for multireference averaged quadraticcoupled cluster theory (MR-AQCC).63 In addition, systems1a/1b and 2a/2b were also investigated utilizing the DIP-EOM-CCSD (equation of motion double ionization potentialcoupled cluster with S and D excitations) method.64

For the HF, DFT, and MP2 PECs, the correspondingenthalpy and free energy curves were determined by calculatingvibrational, thermal, and entropy corrections. In this way, theenthalpy and free energy curves associated with the PEC couldbe analyzed. At the DFT level, solvation effects were tested withthe PCM (polarizable continuum model) of Tomasi and co-workers65 where the dielectric constant ϵ was increased from 2to the value of methanol (ϵ = 32.766), which was used as asolvent for some of the NMR investigations of 1.49,50

The influence of the methyl groups on the stability of theannulene was determined by utilizing isodesmic reactionenergies. For reasons of comparison, cyclopropane (4), 1,1-dicyanocyclopropane (5), and 1,1-dimethylcyclopropane (6)were investigated. All these calculations were carried out at theB3LYP and ωB97X-D levels of theory. Throughout this workPople’s augmented VTZ basis 6-311G(d,p)67 was employed atall levels of theory. We also tested the necessity of using diffusefunctions by applying Dunning’s aug-cc-pVTZ basis set. In thecase of B2PLYP-D, the inclusion of diffuse functions only led tosmall changes in the relative energies of 1. Because ofcomputational limitations in the case of the WFT methods,we employed the smaller VTZ basis throughout this work.Several molecular properties and their changes along the

PEC were also calculated and analyzed. These included theNBO (natural bond order) charges,68 the charge transferbetween bridge and π-perimeter in the cases of 1 and 2,electron densities, energy densities, NMR (nuclear magneticresonance) properties, and local vibrational mode properties.The last three properties will be shortly discussed in thefollowing.

Electron density analysis. The electron density distribu-tion ρ(r) was analyzed with the help of topological analysis69 todetermine all CC and CH bond critical points rb(CC) andrb(CH) as well as the ring critical points rr of ρ(r). In thisconnection, the Cremer−Kraka definition of covalent bondingwas used: (i) A zero-flux surface and bond critical point rb hasto exist between the atoms in question (necessary condition).(ii) The local energy density H(rb) must be negative andthereby stabilizing (sufficient condition for covalent bonding).A positive H(rb) indicates a dominance of electrostaticinteractions.70,71 The Cremer−Kraka criterion reveals atwhich R value the C1C6 bond ceases to exist (or is formed).

NMR analysis. 13C and 1H magnetic shieldings andchemical shifts (using tetramethylsilane as reference) werecalculated for all systems using the gauge-invariant atomicorbital (GIAO) method72 in connection with B3LYP, whichleads to useful 13C values.73 For the determination of R(1) insolution, the NMR-ab intio method of Cremer and co-workers74,75 was used. Since the 13C-NMR chemical shifts of1 are known,50 the deviation between measured and calculated13C chemical shifts along the PEC was calculated and theminimum of their mean (absolute) deviation was determined,because the latter is a reliable indicator of the R value of 1 insolution. In addition, all J(13C13C), J(1H1H), and J(13C1H)indirect spin−spin coupling constants (SSCCs) of 1 werecalculated using the method of Cremer and co-workers.76

Those SSCCs which show the strongest dependence on thevalence-tautomeric rearrangement were analyzed as functionsof R.

Local vibrational mode analysis. Generalized localvibrational modes are the unique equivalents of the 3N − L(N: number of atoms; L: number of translations and rotations)generalized normal vibrational modes and their properties



(local mode frequency ωa, mass ma, force constant ka, intensityIa) were calculated according to the theory of Konkoli andCremer77−79 and its extension by Zou and Cremer.80−82 Thelocal CC stretching force constant ka is a reliable measure of thestrength of the corresponding bond and can be used tocalculate with the help of a power relationship the bondstrength order (BSO) n:2,83−85 n(CC) = a(ka)

b. Parameters aand b were determined in this work to be a = 0.3116 and b =0.8066, utilizing ethane and ethene as suitable references withn(CC) = 1 and 2, respectively. Additionally, n(CC) = 0 wasimposed for ka = 0. The calculated BSOs provide a measure forthe degree of π-delocalization in aromatic molecules, as wasrecently shown by Kalescky and co-workers.86 These authorsderived an AI (aromaticity index) based on benzene (AI = 1.0)and “Kekule benzene” (1,3,5-cyclohexatriene: AI = 0) where inthe latter case 1,3-butadiene was used to define a suitablereference geometry. A fully aromatic system is predicted tohave all conjugated CC bonds equal to the benzene CC bondlength, whereas deviations from this value indicate the degree ofbond alternation and bond strengthening (weakening).86 AIvalues were calculated for the annulene forms. The AI modelbased on benzene and Kekule benzene cannot be applied tohomoaromatic systems, i.e. systems for which the π-delocalization is interrupted by one or more C atoms with 4formal single bonds, as is the case of 1b and 2b.Analysis of crystal packing effects. To test the

possibility of the unusual R value of 1 being the result ofcrystal packing effects, two different situations were considered.(i) In the unit cell of 1, one molecule sits beside the other.44

We considered the interactions between molecules in differentunit cells, especially that situation where one molecule sits ontop of the other, slightly shifted. Steric repulsion caused bypacking effects could force the widening of the external C12−C11−C13 bridge angle β, and with this widening the R valuecould be forced to decrease. Therefore, the changes in R werecalculated for β values from 102 to 120°; that is, geometryoptimizations were carried out for a series of fixed β values. (ii)Utilizing the crystal data, the dimer geometry was optimizedunder the constraint that the distance between the monomersand the relative orientation to each other does not change.Under the same conditions, R was fixed for one monomer andthe geometry of the dimer reoptimized. (iii) A tetramer definedby the crystal structure was analyzed under the conditionsdescribed in (ii).The quantum chemical calculations were performed with the

program packages MOLPRO,87 COLOGNE14,88 CFOUR,89

and Gaussian.90

3. RESULTS AND DISCUSSIONIn Figure 2, the calculated PECs for a representative number ofthe different methods applied in this work are shown. The mostaccurate enthalpy and free energy curves, ΔH(R) = PHC andΔG(R) = PGC, respectively, are given in Figure 3. Relativeenergies ΔE, ΔH(298), and ΔG(298) obtained at differentlevels of theory are listed in Table 1. In Figures 2 and 3, ΔE = 0or ΔH(298) = 0 was arbitrarily chosen for R = 2.1 Å. In Table1, the reference point of all energy difference determinations isalways the most stable annulene form. However, when one ofthe target geometries was located on a shoulder of the PEC (orthe corresponding PHC and PGC), the R value of the inflectionpoint or the R of the annulene (bisnorcaradiene) minimum of aclosely related method was taken, as is indicated in Table 1. In

Figure 4, some representative geometries of 1 and the referencemolecules investigated in this work are shown.

Figure 2. Representation of the energy changes as a function of the1,6-distance R of (a) 11,11-dimethyl-1,6-methano[10]annulene (1a),(b) 1,6-methano[10]annulene (2a), and (c) 1,3,5-cycloheptatriene(3a) obtained at multiple levels of theory. For 1 and 2, R values of 2.1and 2.25 Å have been used to determine the energy zero level, thusfacilitating the comparison.



In the following, we will first analyze the PECs, PHCs, andPGCs obtained for target system 1. The PECs calculated for 1

reveal that (i) π-electron delocalization in annulene andbisnorcaradiene, (ii) σ−π interactions in the distorted ringperimeter, (iii) homoaromatic through-bond and through-spaceinteractions (as defined by Cremer and co-workers10), (iv) thegeneration of a biradicaloid and its stabilization by electrondelocalization, (v) strain effects in the bisnorcardiene form, and(vi) bridge−ring interactions have to be considered asimportant electronic effects. Different methods account forthese effects in different ways so that the PEC adopts either asingle well (SW) form, a SW with a shoulder at small or large Rvalues (SW + 1S or SW + 2S), as well as a double-well (DW)with a flat minimum for small R (DW-F1M) or large R (DW-F2M).Details of the analysis of the PECs for the different WFT and

DFT methods, especially their account of dynamical andnondynamical electron correlation effects, are given in theSupporting Information. Despite the testing of a large numberof different WFT and DFT methods (more than 20 as shown inTable 1 and Figure 2a), none of them provides a balancedaccount of effects (i) to (vi), and none of the results is in linewith experimental observations made for 1. Therefore, thequestion remains whether other effects such as thermochemicalcorrections, entropy contributions, crystal packing, or solventeffects may improve the agreement between theory andexperiment. Rather than investigating these possibilities first,we pursue an alternative way, which leads to the description ofthe smaller, closely related valence tautomeric systems 2 and 3,for which all experimental information available implies theexistence of a SW potential (2)25,34,47,49 or a DW potentialfavoring the no-bond form 3a (3),91−94 which should be mucheasier to describe.

Valence tautomerism of 1,6-methano[10]annuleneand cycloheptatriene. Extensive spectroscopic and diffrac-tion measurements were carried out in the case of 2.25,34,47,49

The difference between 2a (being more stable) and 2b wasestimated to be ≤10 kcal/mol. All measured NMR, infrared,Raman, or UV values exclude a second minimum for thebisnorcaradiene form 2b. Hence, a SW+1S form of the PEC ismost likely.25,34,47,49

In the case of the cycloheptatriene−norcaradiene system 3, aDW PEC has been verified by both NMR and kineticstudies.91,93 Experiments conducted by Rubin93 led to a freeactivation energy for the transition 3b → 3a of ΔGa(298) = 7.2kcal/mol. Rubin estimated ΔG(298, 3b) to be 4 ± 2 kcal/mol,which implies ΔGa(298, 3a → 3b) = 11 ± 2 kcal/mol where azero-entropy change was assumed.93 The estimates were basedon the NMR results of Gorlitz and Gunther, who estimatedthat 3b should have a finite concentration of 0.1% at roomtemperature.91 In this work, we find the concentration of 3b tobe just 0.003%.As in the case of 1, most of the methods applied failed to

provide PECs and relative energies that are in line with theseexperimental observations (see Tables 2 and 3 as well as Figure2b and 2c), where the rationalization of these shortcomings interms of dynamical and nondynamical electron correlationeffects are similar to those given for 1. ωB97X-D, CCSD(T),BD(T), MP4(SDTQ), and B2PLYPD lead to reasonable PECsfor 2, suggesting that the bisnorcaradiene form 2b is located ona shoulder of the PEC. However, only MP4(SDTQ) andB2PLYPD provide a relative energy of 1b close to 10 kcal/mol,whereas especially CASPT2(10,10), CCSD(T), and BD(T)severely underestimate the destabilization of 2b. Especially, thePEC of CASPT2(6,6) leads to a rearrangement barrier 3a →

Figure 3. Representation of energy, enthapy, and free energy changesas a function of the 1,6-distance R of (a) 11,11-dimethyl-1,6-methano[10]annulene (1a), (b) 1,6-methano[10]annulene (2a), and(c) 1,3,5-cycloheptatriene (3a). Potential energy curves in (a) areobtained at the CASPT2(14,14) and CCSD(T) levels of theory. Forestimated (est) curves, see text. Potential energy curves in (b) and (c)are determined at CASPT2(14,14) and CASPT2(10,10) levels oftheory. See text.



3b, which is 4.8 kcal/mol, more than 6 kcal/mol below theestimated ΔGa(298) of 11 kcal/mol.93

The CASPT2(6,6) results reveal that the active space chosenis too small to provide a reliable description of the valencetautomeric rearrangement. Previous work by Cremer and co-workers has emphasized the homoaromatic interactions of thetwo π-bonds of norcaradiene with the 3 σ bonds of thecyclopropyl group.62 Obviously, these are essential for a correctdescription of the process 3a → 3b. Accordingly, we enlargedthe (6,6) to a (10,10) active space by including all 6 rather thanjust 2 Walsh orbitals of the cyclopropyl group. The result of thisextension of the active space is stunning: The barrier increasesto 12.2 kcal/mol, and the relative energy of norcaradiene 3badopts a value of 6.0 kcal/mol (Table 3).Previous multireference investigations of 3 by Jarzeki and co-

workers95 led to different ΔE values (CASSCF: 21.6 kcal/mol;MROPT2 (multireference with perturbation corrections): 8.9

kcal/mol) because these authors used a (6,6) active space.Similarly, all single reference calculations provided poorresults.38,96 The CASPT2(10,10) results of this work, ifconverted into ΔG(298) with the help of B3LYP-calculatedZPE, entropy, and thermochemical corrections, lead toΔGa(298,3b → 3a) = 6.0 kcal/mol, in good agreement withthe corresponding experimental value of 7.2 kcal/mol (6.1kcal/mol at 110 K).93 The deviation results from anexaggeration of ΔSa by the experiment: −4.5 e.u. (derivedfrom the Arrhenius A factor) vs −1.3 e.u. calculated in thiswork. Rubin’s other estimates are corrected by our calculatedΔGa(298, 3a → 3b) of 12.2 kcal/mol and the relative value of3b: ΔGa(298, 3b) = 6.2 kcal/mol; that is, they show that just0.003% of 3b are in equilibrium with 3a at room temperature,where the previous 0.1% estimate was based on NMRexperiments.91 The potential curve obtained for ΔH(298) isshown in Figure 3c and probably presents the most accurate

Table 1. Relative Energies and Rearrangement Barriers of 11,11-Dimethyl-1,6-methano[10]annulene

Method Curvea R(1b)b R(1a) ΔE(1b)c ΔE(TS)d

HF SW+2S 1.552 (2.120) −9.29SVWN5 SW+1S (1.640) 2.125 3.26BLYP SW+1S (1.640) 2.212 6.99B97 SW+1S (1.685) 2.134 2.84B3LYP SW+1S (1.640) 2.167 3.67ωB97 SW+2S 1.595 (2.040) −7.03ωB97X SW+2S 1.595 (2.040) −4.28ωB97X-D DW-F2M 1.638 2.039 −1.02 0.07 (1.09)REKS/ωB97X-D DW-F2M 1.637 2.039 −0.98 0.07 (1.05)B2P-LYP-D DW-F2M 1.605 2.118 −0.08 1.05 (1.13)MP2 SW+1S (1.640) 2.154 6.31MP3 DW-F2M 1.587 2.065 −3.10 0.04 (3.14)MP4(SDQ) SW+2S 1.581 (2.120) −5.28MP4(SDTQ) SW+1S (1.640) 2.150 2.92CCSD SW+1S 1.590 (2.120) 4.25BD SW+1S 1.590 (2.120) 4.21CCSD(T) DW-F2M 1.641 2.120 −0.42 0.62 (1.04)BD(T) DW-F2M 1.641 2.120 −0.42 0.62 (1.04)CASSCF(10,10) DW-F2M 1.529 2.148 −7.81 0.63 (8.44)CASPT2(10,10) DW-F2M 1.613 2.159 −0.97 1.07 (2.04)NEVPT2(10,10) DW-F1M 1.598 2.172 2.49 3.59 (1.10)CASPT2(14,14) DW-F1M 1.647 2.130 0.91 1.32 (0.41)Estimate DW-F2M 1.643 2.123 0.26 0.93 (0.67)

ΔH(1b) ΔH(TS)B2P-LYP-D DW-F2M 1.694 2.110 0.61 1.14 (0.53)CCSD(T) DW-F2M 1.680 2.120 −0.68 0.03 (0.71)BD(T) DW-F2M 1.684 2.120 −0.75 0.02 (0.77)CASPT2(14,14) DW-F1M 1.640 2.120 0.66Estimate DW-F2M 1.662 2.125 −0.11 0.08 (0.19)

ΔG(1b) ΔG(TS)B2P-LYP-D DW-F2M 1.680 2.114 0.16 0.83 (0.67)CCSD(T) DW-F2M 1.658 2.080 −1.09 0.20 (1.29)BD(T) DW-F2M 1.658 2.120 −1.12 0.19 (1.31)CASPT2(14,14) DW-F1M 1.665 2.120 0.22 0.73 (0.51)Estimate DW-F2M 1.662 2.100 −0.45 0.42 (0.87)

aCurve indicates the shape of the rearrangement potential. DW-F1M, double well with flat first minimum; DW-F2M, double well with flat secondminimum; SW+1S, single well with shoulder at small R; SW+2S, single well with shoulder at large R. bR(1b) and R(1a) indicate the C1C6 distancefor each structure in Å. Values in parentheses are approximate values to determine the position of the shoulder. cΔE(1b) gives the energy differencerelative to 1a in kcal/mol. For the explanation of the estimated values, see text. dΔE(TS), ΔH(TS), and ΔG(TS) give the energy barriers for valencetautomerization from 1a to 1b in kcal/mol. Values in parentheses are for the reverse reactions.



description of the energetics of the valence tautomeric system 3obtained so far.Based on the experience earned for 3, we extended the

(10,10) active spaces of 1 and 2 to (14,14) spaces, whichinclude the C1C11 and C1C6 σ and σ*-orbitals (Walshorbitals). Here we discuss first the results for 2. The PEC is stillobtained in a SW+1S form where 2b is now 11.5 rather than5.3 kcal/mol higher in energy than the annulene form at thepotential minimum at R = 2.256 Å (X-ray: 2.235 Å25). Thecorresponding ΔH(298) and ΔG(298) values are 10.2 and 10.3kcal/mol, respectively, in line with experimental observa-tions.24−27,29−31,33−35,37,47,48 Clearly, 2 has to be consideredas [10]annulene rather than a bicyclic, homoaromatic π-systembecause R = 2.256 Å is too large to lead to sizable through-space interactions.10

These results show clearly that CCSD(T) and BD(T),although they may account for some nondynamical effectsbesides the dynamical electron correlation effects, fail todescribe the TS region and 3b correctly, which can also beobserved for 2b, as too much stability is assigned to thenorcaradiene forms. Therefore, we repeated the CASPT2calculations for 1 with the larger (14,14) active space.

Revised CASPT2 results and the consideration ofthermochemical corrections. CASPT2(14,14) leads to aPEC where the relative energies of 1a and 1b are interchangedcompared to the CASPT2(10,10) PEC; that is, 1a becomes theminimum of a relatively flat potential (0.91 kcal/mol below 1b)with barriers of 1.32 and 0.41 kcal/mol for the valencetautomeric rearrangement (Table 1). Small changes in the ZPEand thermal correction values lead to a slight stabilization of 1band a vanishing of the rearrangement barrier for the PHC (see

Figure 4. B3LYP geometries of 1−6. Bond lengths in Å and bond angles in degrees.



Figure 3a). For CCSD(T) and BD(T), the conversion intoenthalpies has a similar effect on the potential (Table 1).Considering that ZPE and thermochemical corrections arebased in this work on harmonic frequencies, which may changedifferently with R than anharmonically corrected frequencies,we recalculated all vibrational corrections with scaledfrequencies employing the scaling factors suggested by Scottand Radom97 for DFT. However, no significant changes wereobtained in this way.As mentioned above, even a (14,14) active space may not be

sufficient to describe all nondynamical electron correlationeffects resulting from the interactions of σ- and π-electrons, thestabilizing interactions in the intermediate biradicals, and thebridge−ring interactions. Apart from this, the amount ofdynamical electron correlation provided by CASPT2 is muchtoo small and biased to prefer the annulene. Conversely,CCSD(T) or BD(T) is not capable of correctly describing thenondynamical correlation. Since our computational possibilitiesdo not provide a combination of these methods, weconstructed a model PEC, PHC, and PGC by simply averagingthe corresponding curves for CASPT2(14,14) and CCSD(T).Although this model PHC (given in Figure 3a) may be

considered to provide only a semiquantitative insight into theproblem, it suggests a broad single well potential slightly

preferring the bisnorcaradiene form by −0.1 kcal/mol(activation enthalpies: 0.1 and 0.2 kcal/mol). The ΔG(R)curve increases the preference of 1b and introduces a somewhatstronger asymmetry of the potential (Figure 3a). Obviously, thedegree of asymmetry of PHC and PGC increases with theadmixture of an additional nondynamical electron correlation.Such a broad asymmetric SW potential would explain theobserved strong T-dependence of the 13C chemical shifts of 1(they indicate a stronger population of the annulene form athigher T49,50) and the fact that 1 adopts in the unit cell twodifferent forms probably influenced by crystal packing effects (R= 1.836 and 1.780 Å44). Therefore, we will investigateenvironmental influences on the PECs, PHCs, and PGCsshown in Figures 2a and 3a in the following subsection.

Consideration of solvent and crystal packing effects.The dipole moment of annulene 1a is 0.35 D at R = 2.03 and0.11 D for R = 1.638 Å (ωB97X-D calculations), where theorientation is along the C2 axis (bridge, positively charged;center of the perimeter, negatively charged). There is a chargetransfer from the CMe2 bridge to the annulene perimeter,which changes, in agreement with the changes in the dipolemoment, from 13 (R = 2.039 Å) to 9 me (millielectrons; R =1.638 Å).

Table 2. Relative Energies and Rearrangement Barriers of 1,6-Methano[10]annulene


HF DW-F1M 1.562 2.219 0.63 3.27 (2.64)B3LYP SW+1S (1.610) 2.280 13.60ωB97 DW-F2M 1.560 2.226 −1.12 1.48 (2.60)ωB97X DW-F1M 1.546 2.226 2.71 3.39 (0.68)ωB97X-D SW+1S (1.610) 2.240 7.22B2P-LYP-D SW+1S (1.610) 2.250 9.22MP2 SW+1S (1.610) 2.254 13.93MP3 DW-F1M 1.630 2.240 4.20 4.68 (0.48)MP4(SDQ) DW-F1M 1.610 2.214 1.88 3.00 (1.12)MP4(SDTQ) SW+1S (1.610) 2.265 10.48CCSD DW-F1M 1.613 2.235 3.21 4.04 (0.83)BD DW-F1M 1.612 2.234 3.25 4.06 (0.81)CCSD(T) SW+1S (1.610) 2.254 7.29BD(T) SW+1S (1.610) 2.256 7.32CASSCF(10,10) DW-F1M 1.540 2.268 4.37 7.65 (3.28)CASPT2(10,10) SW+1S (1.610) 2.254 5.32NEVPT2(10,10) SW+1S (1.610) 2.230 14.86CASPT2(14,14) SW+1S (1.700) 2.256 11.48

ΔH(1b) ΔH(TS)B2P-LYP-D SW+1S (1.610) 2.267 12.59CCSD(T) SW+1S (1.700) 2.256 5.60BD(T) SW+1S (1.700) 2.258 5.55CASPT2(14,14) SW+1S (1.700) 2.256 10.16

ΔG(1b) ΔG(TS)B2P-LYP-D SW+1S (1.610) 2.267 12.23CCSD(T) SW+1S (1.700) 2.256 5.77BD(T) SW+1S (1.700) 2.258 5.74CASPT2(14,14) SW+1S (1.700) 2.256 10.30

aCurve indicates the shape of the rearrangement potential. DW-F1M, double well with flat first minimum; DW-F2M, double well with flat secondminimum; SW+1S, single well with shoulder at small R; SW+2S, single well with shoulder at large R. bR(2b) and R(2a) indicate the C1C6 distancefor each structure in Å. Values in parentheses are approximate values to determine the position of the shoulder. cΔE(2b) gives the energy differencerelative to 2a in kcal/mol. For the explanation of the estimated values, see text. dΔE(TS), ΔH(TS), and ΔG(TS) give the energy barriers for valencetautomerization from 2a to 2b in kcal/mol. Values in parentheses are for the reverse reactions.



Experimental work with 1 was carried out in nonpolarsolvents such as cyclohexane, CS2, CCl4, or methanol.23,34,50

Therefore, we calculated the solvent influence by increasing thedielectric constant ϵ from 2 to 32.766 and using Tomasi’s PCMmethod.65 In all calculations, changes in the relative freeenergies ΔG(298) were 0.1 kcal/mol or smaller, always in favorof the annulene form (in line with the calculated dipolemoments), which in the case of the estimated potential ofFigure 3a (see also Table 1) would decrease the ΔG(298)difference between 1a and 1b and lead to a larger population ofthe annulene form with increasing T, as found in the NMRexperiments.49,50 Hence, environmental effects cannot beignored if free energy differences smaller than RT = 0.6 kcal/mol have to be discussed.There is also the possibility that the unusual C1C6 distances

observed in the crystal structure analysis44 are the result ofpacking effects. In this connection, it has to be pointed out thatstrings of molecules 1 arranged in parallel form molecularsheets.44 Molecules which are on top of each other in differentsheets could, via exchange repulsion, widen the externalC12C11C13 bridge angle, thus causing stronger bridge−perimeter interactions. As is shown in Figure 5, the downwardoriented methyl hydrogens are just 1.96−2.15 Å away from thecenter of the C3C4 and C8C9 bond, respectively. Consideringthat these H atoms carry a small positive charge and that thesum of the van der Waals distances for H and C is 1.2 + 1.6 =

2.8 Å,66 the downward oriented methyl H atoms should beattracted by the π-density of the [10]annulene perimeter. Thereis a stabilizing H−π interaction at a distance of 2.4 Å, which isqualitatively confirmed by the increase in the stabilization of theannulene form when comparing ωB97X-D and ωB97X results(see Supporting Information).The hypothesis of an increased bridge−perimeter attraction

caused by bridge angle widening in the course of crystal statepacking effects was confirmed as a widening of the CCC-bridgeangle had a significant impact on the parameter R: Wideningthe bridge angle C12C11C13 by 10° leads to an increase in Rby 0.048 Å.Next, we calculated the geometry of the dimer and the

tetramer shown in Figure 5 by applying a constrainedoptimization, in which the distance(s) between the monomersand the relative orientation to each other were frozen (ωB97X-D calculations). The differences in the monomer geometriesare small for the dimer (R: 1.635 for the lower monomer; 1.631Å for the upper monomer). However, for the tetramer, R valuesof 1.636 (top, left), 1.642 (top, right), 1.637 (bottom, left), and1.619 Å (bottom, right) are obtained, leading to a totalvariation of 0.023 Å. These changes are in line with the X-raydiffraction result (ΔR in the unit cell: 0.056 AA44) where onehas to consider that only the lower right monomer (the onewith the short R) has an environment close to that which itwould have in the solid state. For reproducing the experimental

Table 3. Relative Energies and Rearrangement Barriers of 1,3,5-Cycloheptadiene and Norcaradiene


HF DW-F1M 1.557 2.395 7.03 12.70 (5.67)B3LYP DW-F1M 1.644 2.351 7.51 7.89 (0.38)ωB97 DW-F2M 1.576 2.366 −0.80 5.43 (6.23)ωB97X DW-F1M 1.576 2.350 2.18 6.14 (3.96)ωB97X-D DW-F1M 1.585 2.355 2.20 6.28 (4.08)B2P-LYP-D DW-F1M 1.594 2.364 7.13 8.92 (1.79)MP2 DW-F1M 1.675 2.286 3.02 3.07 (0.05)MP3 DW-F1M 1.591 2.373 4.38 8.31 (3.93)MP4(SDQ) DW-F1M 1.582 2.385 4.37 8.89 (4.52)MP4(SDTQ) DW-F1M 1.623 2.341 4.64 5.85 (1.21)CCSD DW-F1M 1.583 2.382 4.69 8.76 (4.07)BD DW-F1M 1.585 2.380 4.62 8.68 (4.06)CCSD(T) DW-F1M 1.607 2.370 4.99 7.33 (2.34)BD(T) DW-F1M 1.607 2.366 5.10 7.34 (2.24)CASPT2(6,6) DW-F1M 1.633 2.310 4.36 4.84 (0.48)CASPT2(10,10) DW-F1M 1.593 2.334 6.05 12.20 (6.15)

ΔH(1b) ΔH(TS)B2P-LYP-D DW-F1M 1.653 2.245 6.33 6.34 (0.01)CCSD(T) DW-F1M 1.738 2.476 4.95 6.92 (1.97)BD(T) DW-F1M 1.738 2.476 4.96 6.93 (1.97)CASPT2(10,10) DW-F1M 1.641 2.334 5.89 11.53 (5.64)

ΔG(1b) ΔG(TS)B2P-LYP-D DW-F1M 1.697 2.245 6.47 6.64 (0.17)CCSD(T) DW-F1M 1.698 2.482 5.35 7.62 (2.27)BD(T) DW-F1M 1.698 2.482 5.34 7.59 (2.25)CASPT2(10,10) DW-F1M 1.618 2.334 6.18 12.22 (6.04)

aCurve indicates the shape of the rearrangement potential. DW-F1M, double well with flat first minimum; DW-F2M, double well with flat secondminimum; SW+1S, single well with shoulder at small R; SW+2S, single well with shoulder at large R. bR(3b) and R(3a) indicate the C1C6 distancefor each structure in Å. cΔE(3b) gives the energy difference relative to 3a in kcal/mol. For the explanation of the estimated values, see text.dΔE(TS), ΔH(TS), and ΔG(TS) give the energy barriers for valence tautomerization from 3a to 3b in kcal/mol. Values in parentheses are for thereverse reactions.



situation, a more realistic model comprising at least 16monomers (12 monomers surrounding a tetramer) at fixeddistances and orientations would be needed. Apart from this,one has to consider that effects would become larger if ωB97X-D could correctly describe the asymmetric SW potentialobtained at higher levels of WFT.In another set of calculations, the dimer shown on the left

side of Figure 5 (with frozen distance and relative orientation ofthe monomers) was optimized for fixed R (1.6 < R < 2.2 Å)values of either the upper or the lower monomer, andoptimizing the remaining geometrical parameters. For allthese geometry optimizations, R of the second monomerchanged maximally by 0.005 Å relative to the correspondingmonomer value where especially the positions of the methyl Hatoms were sensitive. We conclude that, in view of the broadasymmetric SW potential calculated and the results obtained forthe tetramer, crystal packing effects have an impact on R andexplain the existence of two molecules of 1 with differentgeometries in the unit cell.NMR investigation and an independent determina-

tion of the equilibrium geometry of 1. In previous work,one of us has shown how an easily changing geometricalparameter of a flexible system can be determined in solutionwith the help of measured and calculated chemical shifts.74,75 InFigure 6a, calculated 13C and 1H chemical shift values are givenas a function of R and compared with the available 13C chemicalshifts indicated as dashed horizontal lines. Examination of thisfigure reveals that the shift value of C1 is the most sensitive.This shift increases from a typical value for vinyl cyclopropane(42.5 ppm98) to that found for 2a (114.6 ppm98) and directlyreflects the changes in R. At R = 1.763 Å, the calculated C1

value becomes equal to the measured one, thus suggesting thatthis R value is the one 1 adopts in solution or, alternatively,corresponds to a time-averaged value if the valence-tautomericrearrangement of 1 is fast on the NMR time-scale.A similar observation can be made for the NMR chemical

shift of C11, which increases from a value typical of acyclopropane (13C shift: −2.8 ppm98) to the one measured for2a (34.8 ppm98) crossing the observed C11 shift at R = 1.782Å. The chemical shifts of the methyl carbon nuclei coincide at1.865 Å with the corresponding measured value. However,these 13C shifts are less sensitive, as are those of C2(coincidence at R = 1.788 Å) and C3 (coincidence at R =2.168 Å, Figure 6a). The mean deviation Δ between measuredand calculated 13C chemical shifts for the C atoms of theperimeter adopts a minimum at R = 1.775 Å, as is shown inFigure 6. NMR chemical shift calculations are normally lessaccurate for conjugated systems, especially if these arenonplanar (the shifts of C2 and C3 are close to theexperimental ones in the whole range 1.7 < R < 2.2 Å, and aspecific R value is difficult to determine). We note that whenC11, C12, C13, C1, and C6 are used for the comparison withexperiment, a value of R = 1.79 Å results, in good agreementwith the X-ray diffraction values of R, which are close to 1.8Å.44

The bisnorcaradiene and the annulene forms can both beexcluded as clearly dominating the valence tautomericrearrangement of 1 in the sense of a DW potential. Theremaining possibility is a rapid rearrangement in solution via asmall barrier of a DW potential or, more likely in view of ourcalculations, a broad asymmetric SW potential. This speaks for

Figure 5. Dimer and tetramer configurations that were calculated to investigate crystal packing effects. The green structures give a tetramer of 1 andshow how two unit cells are arranged in the crystal.44 The blue structures on the left show the arrangement for the dimer of 1 that was calculated inthe search for packing effects. The red structures give for the ωB97X-D optimized geometry of 1a and 1b the shortest hydrogen-to-ring distances.See text.



a large-amplitude vibration in solution similar to the onesuggested in Figure 3a.Also informative in this connection are the calculated SSCCs

J(13C13C) and J(1H1H) given as a function of R (see Figure 7).When R is close to 1.5 Å, the values of 1J(C1C6) and1J(C1C11) adopt the values typical of cyclopropane (12.4Hz98). For increasing R the former 1J value decreases to −12.4Hz at 1.8 Å. This is comparable to the geminal 1J(CC) value ina substituted cyclobutane (−8 Hz98) and then increases to azero value at large R, indicating that through-bond or through-space coupling are small because the CCC-angle dependence of2J implies for this situation a zero value, 5J coupling along the

perimeter is too weak, and/or the C1C6 through-space overlapis too small. Hence, a measuring of J(C1C6) (best independence of T) should provide an excellent possibility foran experimental determination of R.Also informative should be the measurement of 1J(C1C11),

1J(C11C12), and 3J(H15H16). The former J value increasesfrom about 16 Hz at R = 1.60 Å to 28 Hz, typical of a 1J(CC)value such as that of cyclobutane.98 1J(C11C12) changes from45 to 39 Hz for the same R values while 3J(H15H16) increasesfrom 5.0 to 7.2 Hz. Since the changes in the J(CC) values arelarger, they should be preferably used for an independentdetermination of the R value of 1 in solution.

Figure 6. (a) Dependence of calculated NMR chemical shifts [ppm] of 1 on the distance R(C1C6) as calculated at the B3LYP/GIAO level oftheory. (b) Mean absolute deviation [ppm] between measured and calculated 13C NMR chemical shifts of the ring carbon atoms given as a functionof R.



4. DOES 11,11-DIMETHYL-METHANO[10]ANNULENEPOSSESS THE LONGEST CC BOND OF NEUTRALCYCLOPROPYL DERIVATIVES?

System 1 is unusual because of its broad SW potential, whichmakes a large-amplitude C1C6 vibration possible, making abarrierless interconversion of the annulene into the bisnorcar-adiene form possible. In this work, we clarified the question ofthe nature of the C1C6 interaction by two different modelapproaches utilizing the topological analysis of ρ(r)69 in theway given by Cremer and Kraka70,71,99 and the local vibrationalmode approach of Cremer, Zou, and Konkoli.77,80,100 In Figure8, the BSO values n(CC) based on the calculated local CCstretching force constants are plotted as a function of R. Foreach set of local vibrational modes at a given R value, theadiabatic connection scheme80 is applied to determine whethera given local mode is still contained in a set of 3N-L modesdirectly related to the 3N-L normal vibrational modes. As canbe seen from Figure 8, close to R = 1.7 Å the C1C6 stretching

mode drops out of the 3N-L set of vibrational modes,indicating that for larger R values there is no C1C6 covalentbond.All other BSO(CC) values change smoothly from the

bisnorcaradiene form to the annulene form where in the lattercase the alternation of bonds is similar to that found fornaphthalene;81,86 that is, bonds C2C3, C4C5, etc. are thestrongest, followed by bonds C3C4 and C8C9, whereas theweakest conjugated CC bonds sit at the bridge. The CC bridgebonds of 1b are weaker than the normal CC bonds, with C1C6being the weakest. The calculated AI of 1a is 64% of the valueobtained for benzene (100%), which is smaller than the valuefor 2a (73%) and significantly smaller than the value fornaphthalene (86%).86 This is in line with the fact that 2a and 3aexperience a strong perturbation of their 10π-perimeter causedby the 1,6-bridge leading to torsional angles up to 37°.In Figure 9a, the changes in the bond critical and ring critical

points rb and rr of the electron density distribution ρ(r) areshown as a function of R. Some of the changes are similar andsome are different from those given by the n(CC) based on thelocal CC stretching force constants (Figure 8). This isattributable to the fact that the electron density determinedat one specific point in the bond region cannot reflect all thedensity changes taking place in the zero-flux surface betweentwo bonded atoms apart from the influences of bond polarity,charge transfer, and other effects given by the changes in thevirial (atomic) spaces of the molecule during a change in R.The local CC stretching force constants account for theseeffects and therefore are more reliable CC bond strengthdescriptors.However, it is an accepted fact that covalent bonding

requires the existence of a bond critical-point and zero-fluxsurface between the atoms in question and that the energydensity at this bond critical point, H(rb), must be stabilizing, i.e.smaller than zero (Cremer−Kraka criteria of covalentbonding).70,71,99 This criterion is clearly fulfilled for all CCbonds and R values of 1 investigated, except the C1C6interaction, which converts into a noncovalent interaction at R

Figure 7. Dependence of calculated NMR spin−spin coupling constants J [Hz] on R(C1,C6) of system 1.

Figure 8. Bond strength orders n(CC) derived from local CCstretching force constants given as a function of R(C1,C6) of system 1.



= 1.696 Å. There the C1C6C11 ring critical point merges withthe C1C6 bond critical point, thus leading to a singularity in theHessian of ρ(r) and the C1C6 maximum electron density pathconnecting these atoms vanishes. Applying the Cremer−Krakacriterion, there is no longer a covalent C1C6 bond for R ≥ 1.70.This is in line with the independent observation made inconnection with the local vibrational modes. We draw theconclusion that a C1C6 covalent bond with R = 1.80 Å doesnot exist but that a homoaromatic interaction in the sense of athrough-space overlap of π-orbitals exists leading to a smallelectron density increase between atoms C1 and C6.Impact of the bridge on the 10π-perimeter. The

electronic influence of electron-withdrawing and electron-donating substituents of a cyclopropane ring are well-knownand has been amply described in the literature (for a review, seeref 62). Two cyano groups at C11 lead to a shortening of thedistal bond (lengthening of the vicinal bonds; see 5 ascompared to 4 in Figure 4), which has been exploited tosynthesize 11,11-dicyano-bisnorcaradiene, i.e. the dicyanoanalogue of 1b.35 The effect of two methyl groups has beenpredicted to lead to a slight CC lengthening of the distal bond(slight shortening of the vicinal CC bonds)62 as is confirmed bythe bond lengths given for 6 in Figure 4. This indicates thatdimethyl-substitution at C11 as in 1 should favor a larger rather

than a smaller R, irrespective of any other stabilizing bridge−perimeter interactions.The isodesmic energies for the formal reactions

+ → + Δ = −E4 6CH CH CH CH , 3.783 2 3 4

+ → + Δ = −E2b 1bCH CH CH CH , 3.383 2 3 4

+ → + Δ =E2a 1aCH CH CH CH , 4.923 2 3 4

(B3LYP values in kcal/mol) suggest a similar stabilization bythe two methyl groups for cyclopropane and 2b but a 5 kcal/mol destabilization for 2a. The first two values are in line with ashortening of two vicinal and a lengthening of just one distalCC bond. Molecule 6 has a smaller external CCC angle thanthe HCH angle in its parent molecule (Figure 4), which is theresult of steric repulsion between methyl groups and the ring.For the same reason, 1b has an even smaller external angle(110.2°, Figure 4), where the folding back of the two dieneunits as shown in the side view at the bottom of Figure 5 leadsto some reduction of the steric repulsion between methylgroups and the 6-membered rings. In the case of 1a, stericattraction between methyl groups and the π-perimeter andsteric repulsion (for 1a the H-center(C3C4) distance isdecreased from 2.148 to 1.951 Å, ωB97X-D, see Figure 5)must be balanced, which leads to a small C12C11C13 angle(105.3°, Figure 4), methyl−methyl repulsion, and an overalldestabilization, as reflected by the energy of the isodesmicreaction given above (4.9 kcal/mol).The destabilizing effect of the CMe2 bridge leads to a raising

of the relative energy of 1a. π-Delocalization and bridge−perimeter destabilization lead to the fact that 1b and 1a havecomparable energies.A basic problem of previous studies was that they were based

on quantum chemical methods of low accuracy. Depending onwhether HF or MP2 is used, the preference for differentvalence tautomeric forms is found, which holds also for the XCfunctionals of DFT, as we have demonstrated in this work forthe first time. This leads to rather limited insight into thepossible cause of a quantum mechanical (electronic structure)effect. For example, Simonetta and co-workers101 have usedHoffmann’s approach62,102 to rationalize the stability ofsubstituted cyclopropanes to explain substituent effects inbridged [10]annulenes where their arguments were based onlow level calculations. As was pointed out by Cremer and co-workers,62 the orbital model used does not even explain allsubstituent effects for cyclopropanes. Furthermore, it does notconsider the impact of bridge−perimeter interactions, thestabilization of biradicaloid structures for medium-sized Rvalues by conjugation, or the dynamic aspects of the valencetautomeric rearrangement.Similarly, use of the topological analysis of ρ(r) by Gatti and

co-workers,40 results of which were interpreted as proof for along covalent C1C6 bond, have to be criticized because theywere carried out at the HF/minimal basis set level withoutusing any quantum mechanical criterion for covalent bonding.Other investigations were based on the NBO analysis orheuristic models utilizing the degree of bond length alternationin the ring perimeter.42 These studies were insofar questionableas the properties analyzed were not referenced with regard to asuitable reference system.An interesting rationalization of the carrying dynamic

behavior of bridged [10]annulenes was proposed by Choiand Kertesz,32 who model the opening of the cyclopropane ring

Figure 9. (a) Electron density ρ(r) and (b) energy density H(r) atbond critical and ring critical points given as a function of R(C1,C6) ofsystem 1.



in 1 or 2 by its conversion into a trimethylene biracial in itstriplet ground state. The stabilization of the triplet trimethyleneby substituents as described by DFT provides then a basis tounderstand whether the bisnorcaradiene or annulene form ismore stabilized. We note in this connection that the radicalcenters are part of a conjugated system, and a more realisticmodel would be the opening of 3-substituted 1,2-divinylcyclo-propanes.Dorn and co-workers50 excluded, on the basis of 13C CPMAS

(cross-polarization magic angle spinning) spectra, a rapidvalence−tautomeric process for 1. Instead, these authorssuggested that an asymmetric PEC and a different populationof the vibrational levels at higher temperature would lead to theobserved temperature dependence of the NMR spectra.Alternatively, temperature dependent intermolecular interac-tions could cause the observed temperature dependence.50 Ourhigh level quantum chemical results are in line with the firstexplanation whereas strong intermolecular interactions cannotbe confirmed.Kaupp and Boy103 analyzed the measured temperature

factors of the crystal data and concluded that a structuraldisorder in the solid state leads to the coexistence ofbisnorcaradiene and [10]annulene, which would imply thatthe measured R values are just averages. Our results do notagree with this hypothesis, as the modeling of packing effectsusing a tetramer of 1 leads to changes in R that are in the rangeof R differences measured by X-ray diffraction.44

5. CONCLUSIONSThis investigation provides a convincing explanation for thepuzzling observations made in connection with molecule 1.This explanation is based on extensive calculations with morethan 20 different quantum chemical methods and the analysisof energy, geometry, dipole moment, charge transfer, NMRchemical shifts, indirect spin−spin coupling constants, localvibrational modes, and electron and energy density changesgiven as a function of the critical distance R(C1C6). Our workhas led to a number of methodological insights andexperimentally relevant conclusions, which are of generalrelevance for future experimental or theoretical work on relatedmolecules, especially in connection with materials science.(1) The investigation presented here reveals that system 1

possesses a broad, slightly asymmetric SW potential (or a veryflat DW minimum) in which it carries out a large C1C6amplitude vibration. Forms 1b and 1a have almost identicalrelative energies and enthalpies. A methodologically independ-ent determination of the critical distance R based on acomparison of measured and calculated 13C NMR chemicalshifts leads to a value of 1.775 Å. By using only those shiftvalues strongly dependent on R, the NMR-based determinationof R can be improved to 1.79 Å in line with the X-ray diffractionvalues of 1.780(7) and 1.836(7) Å.44

(2) This work suggests that the R value in solutioncorresponds to a time-averaged value whereas the R values ofthe unit cell are adopted by molecules of 1 being exposed todifferent packing effects. Our investigation could confirm theexistence of packing effects using small models (dimer,tetramer.) For a tetramer model, a ΔR value of 0.023 Å wascalculated suggesting a structure distortion as it was found inthe solid state: ΔR = 0.056 Å determined by the X-raydiffraction analysis.44

(3) The peculiar valence tautomeric potential of 1 is theresult of the destabilization of the annulene form by

introducing steric strain via the geminal dimethyl group. Thisis a direct consequence of balancing methyl-perimeter exchangerepulsion against (electrostatic or dispersion driven) attraction.π-delocalization stabilizes the biradicaloid structure generatedfor medium-sized R values, which otherwise would lead to arelatively high barrier of valence tautomerization. Without themethyl-annulene interactions, π-delocalization and the resultingaromatic 10π-stabilization would shift the global minimumalways to 1a.(4) The analysis of the vibrational modes of 1 as well as its

electron and energy density distribution reveals that covalentC1C6 bonding ceases to exist beyond R = 1.7 Å on to largervalues of R. Hence, at 1.8 Å one can only speak ofhomoaromatic through-space interactions, which will play littlerole for distances larger than 2 Å, i.e. for the annulenic form.(5) In the course of this work, it became necessary to

accurately determine the PEC of the parent system 2.CASPT2(14,14) calculations predict a SW potential with ashoulder between 1.6 and 1.9 Å. For an assumed R value of 1.70Å, the bis norcaradiene form has a relative enthalpy ΔH(298),which is 10.2 kcal/mol higher than that of the annulene form,i.e. at room temperature the percentage of bisnorcaradine isfinite but close to zero.(6) Because of its close relationship to systems 1 and 2, we

investigated also the valence-tautomeric system 3. TheCASPT2(10,10)-based free activation energy ΔGa(298) ofcycloheptatriene rearranging to norcaradiene is 12.2 kcal/molin forward and 6.0 kcal/mol in the reverse reaction where thelatter value is identical to a kinetic value obtained at 110 K, but1.2 kcal/mol smaller than a derived value at 298 K obtained inthe same study.93 We show that the latter deviation is caused byan overly large entropy change ΔSa used in the experimentalstudy. The ΔG(298) value of 3b relative to 3a is 6.2 kcal/mol.The latter value replaces a previous NMR-based estimate of 4 ±2 kcal/mol.93,104 The concentration of 3b at room temperatureis just 0.003% rather than the previously estimated value of0.1%.(7) Although 1, 2, and 3 are closed shell molecules, their

description requires the inclusion of both dynamical andnondynamical electron correlation to a high degree, i.e. for theformer T excitations are absolutely necessary as is a (14,14)active space (because of σ − π mixing and conjugativestabilization of a transient biradicaloid) provided at least secondorder perturbation theory is used. Ideally, these systems wouldbe investigated with MR-AQCCSD based on a large activespace. A basis set of VTZ quality is absolutely needed, becausethe calculations of this work reveal that an augmentation withdiffuse functions is also required together with the multi-reference description and the T dynamical correlation effects.This is currently beyond computational possibilities but will bea target for future studies.(8) This work has also shown the usefulness of long-range

exact exchange and the promising performance of double-hybrid XC functionals. We suggest that the energetics for 1, 2,and 3 obtained in this work will be included in the standardreaction sets for the testing of new XC functionals. We see thepossibility of combining different flavors of MP2 spin scalingand dispersion corrections in a double-hybrid functional to getmore accurate results, as is obtained with the B2PLYPDfunctional in this work.(9) The bridged annulenes synthesized in the Vogel group

have so far evaded thorough quantum chemical investigations,as they can only be accurately described by combining



dynamical and nondynamical electron correlation in a post-HFmethod. This work shows that the active space has to includealso the σ orbitals (or Walsh-orbitals) of the annulene bridge todescribe π-delocalization reliably. A quantum chemicalinvestigation of the bridged annulenes can predict thoseproperties that may be interesting in connection with polymerchemistry, materials chemistry, or nanochemistry. We note inthis connection that the synthesis of bridged annulenes involvescarbene addition, which is a technique more and more used infullerene chemistry where systems with perturbed π-delocaliza-tion and specific material properties are generated in this way.The procedures used in this work would also be useful for thequantum chemical investigation of these systems employingfine-tuned double hybrid functionals.(10) Finally, it is noteworthy that a system such as 1 is the

basis for a molecular switch, which by ring substitution orsuitable interactions with the environment can be pushed intoan on- (e.g., 1b) or off-position (1a).

■ ASSOCIATED CONTENT*S Supporting InformationDiscussion of the calculated PECs as they reveal advantages anddisadvantages of a given method, and calculated geometries,energies, and other properties for all molecules summarized in24 tables. This material is available free of charge via theInternet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected] authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was financially supported by the National ScienceFoundation, Grant CHE 1152357. We thank SMU forproviding computational resources. An early investigation of1 and 2 by Elfi Kraka is acknowledged. Thanks also to MarekFreindorf, Dani Setiawan, and Rob Kalescky for help with someof the calculations.

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11,11-Dimethyl-1,6-methano[10]annulene - An

annulene with a long CC bond or a fluxional

molecule?

Suporting Information

Alan Humason, Wenli Zou, and Dieter Cremer⇤

Computational and Theoretical Chemistry Group (CATCO),

Department of Chemistry, Southern Methodist University

3215 Daniel Ave, Dallas, Texas 75275-0314, USA

E-mail: [email protected]

⇤To whom correspondence should be addressed

113

Tabl

e1:

Car

tesi

anC

oord

inat

es[Å

]for

11,1

1-D

imet

hyl-1

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etha

no[1

0]an

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ne,w

B97

X-D

/6-3

11G

(d,p

)

1.4

Å1.

5Å

1.6

Å1.

636

ÅC

0.00

0000

0.00

0000

1.24

5962

0C

0.00

0000

0.00

0000

1.22

2421

C0.

0000

000.

0000

001.

1955

11C

0.00

0000

0.00

0000

1.18

5334

C-0

.700

000

0.00

0000

-0.1

1456

30C

-0.7

5000

00.

0000

00-0

.106

704

C-0

.800

000

0.00

0000

-0.0

9741

6C

-0.8

1779

70.

0000

00-0

.093

276

C-1

.401

222

1.23

6908

-0.5

7682

90C

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cara

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ntin

ued)

125

Tabl

e7:

Rel

ativ

een

ergi

es[k

cal/m

ol]f

or11

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Dim

ethy

l-1,6

-met

hano

[10]

annu

lene

atal

lGeo

met

ries

and

allL

evel

sofT

heor

y

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C6

HF

SVW

N5

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PB

97B

3LY

Pw

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wB

97X

wB

97X

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S/B

2P-L

YP-

DM

P2M

P3M

P4(S

DQ

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P4(S

DT

Q)

CC

SDB

DC

CSD

(T)

BD

(T)

CA

SSC

FC

ASP

T2

NE

VPT

2C

ASP

T2

Åw

B97

X-D

(10,

10)

(10,

10)

(10,

10)

(14,

14)

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127

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17.6

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415

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Tabl

e8:

The

rmod

ynam

icPa

ram

eter

s[k

cal/m

ol]

for

11,1

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hyl-1

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no[1

0]an

nule

neat

all

Geo

met

ries

and

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her

Lev

elso

fThe

ory

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CC

SD(T

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ASP

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14,1

4)es

timat

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YP-

DC

CSD

(T)

BD

(T)

CA

SPT

2(14

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estim

ate

B2P

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P-D

CC

SD(T

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D(T

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ASP

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14,1

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timat

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DE

DE

DH

DH

DH

DH

DH

DG

DG

DG

DG

DG

1.4

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0.20

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0.33

0.36

2.3

1.56

1.56

1.56

1.04

1.50

1.50

1.49

1.49

1.24

1.66

1.66

1.68

1.68

2.4

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4.37

3.70

4.13

4.13

4.22

4.17

3.93

4.31

4.30

4.43

4.38

2.5

8.73

8.97

8.85

8.06

8.35

8.35

8.58

8.46

8.27

8.52

8.52

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2.6

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15.0

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14.2

314

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14.6

214

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14.3

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14.3

614

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14.5

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126

Tabl

e9:

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ativ

een

ergi

es[k

cal/m

ol]f

or1,

6-M

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no[1

0]an

nule

neat

allG

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etri

esan

dal

lLev

elso

fThe

ory

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C6

HF

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97w

B97

Xw

B97

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P-D

MP2

MP3

MP4

(SD

Q)

MP4

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TQ

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CSD

BD

CC

SD(T

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ASS

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2N

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10)

(10,

10)

(10,

10)

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14)

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24.9

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8.49

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27.1

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0.26

0.04

--

0.36

0.20

0.13

0.17

0.21

0.08

0.18

0.18

0.10

0.10

0.10

0.17

0.22

0.17

2.4

2.61

1.29

2.73

2.92

2.30

2.05

1.76

1.98

2.18

1.43

2.01

2.03

1.59

1.59

1.68

1.95

2.31

1.80

2.5

7.04

4.45

--

6.09

5.92

5.17

5.63

5.92

4.51

5.64

5.68

4.83

4.83

5.32

5.51

6.24

5.21

2.6

13.6

19.

5612

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12.9

011

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11.8

610

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11.2

211

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9.41

11.1

611

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9.92

9.93

11.1

110

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12.0

310

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Tabl

e10

:The

rmod

ynam

icPa

ram

eter

s[kc

al/m

ol]f

or1,

6-M

etha

no[1

0]an

nule

neat

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eom

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esan

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rL

evel

sofT

heor

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C6

B2P

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P-D

CC

SD(T

)B

D(T

)C

ASP

T2(

14,1

4)B

2P-L

YP-

DC

CSD

(T)

BD

(T)

CA

SPT

2(14

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ÅD

HD

HD

HD

HD

GD

GD

GD

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424

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17.8

717

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22.5

124

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17.5

217

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22.1

61.

516

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9.56

9.60

14.3

416

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9.30

9.33

14.0

81.

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6.73

6.76

11.3

312

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6.40

6.42

10.9

91.

710

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5.53

5.55

10.1

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5.67

5.69

10.3

31.

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755.

255.

269.

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831.

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254.

278.

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528.

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281.

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340.

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190.

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160.

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0.00

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0.09

0.09

0.09

0.16

0.09

0.10

0.10

0.16

2.4

1.49

1.51

1.51

1.72

1.51

1.54

1.54

1.75

2.5

4.72

4.62

4.62

5.01

4.72

4.64

4.64

5.03

2.6

9.80

9.50

9.51

10.0

99.

789.

929.

5010

.09

127

Tabl

e11

:Rel

ativ

een

ergi

es[k

cal/m

ol]f

orC

yclo

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atri

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cara

dien

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1.4

14.7

817

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8.15

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113

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11.7

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5.03

5.87

5.73

6.68

6.05

5.98

6.90

6.90

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-7.

721.

585

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.25

2.03

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--

2.66

4.33

4.35

4.60

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6.05

1.6

7.36

7.49

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52.

06-

0.00

6.81

2.49

4.33

4.38

4.50

4.68

4.61

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965.

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367.

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92.

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5.57

5.74

5.45

5.45

0.35

6.77

0.96

7.32

7.87

4.54

7.66

7.59

6.28

6.26

1.98

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11.3

12.

17.

303.

35-

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4.21

0.24

4.88

5.27

2.89

5.15

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982.

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2.4

0.00

0.00

0.00

0.00

0.00

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0.00

0.00

0.00

0.00

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0.00

0.00

0.00

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262.

51.

201.

58-

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61-

1.35

2.24

1.34

1.25

1.57

1.24

1.25

1.33

1.34

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022.

64.

594.

874.

855.

085.

08-

4.82

6.07

4.53

4.34

4.80

4.32

4.35

4.37

4.39

5.74

-5.

45

Tabl

e12

:T

herm

odyn

amic

Para

met

ers

[kca

l/mol

]fo

rC

yclo

hept

atri

ene/

Nor

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ory

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C6

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D(T

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2P-L

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HD

HD

HD

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GD

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417

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14.8

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15.7

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15.0

315

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15.9

21.

59.

136.

866.

867.

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9.43

7.21

7.21

8.16

1.58

5-

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6.04

--

-6.

341.

66.

524.

784.

795.

896.

805.

105.

116.

181.

61-

--

--

--

-1.

645

6.24

--

-6.

41-

--

1.7

6.24

4.52

4.52

6.78

5.99

4.25

4.25

6.47

1.8

5.93

4.70

4.70

7.30

6.48

5.28

5.28

7.84

1.9

5.54

6.00

5.99

11.3

56.

096.

586.

5711

.89

2.0

4.19

6.42

6.42

10.5

64.

726.

986.

9711

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2.1

2.86

6.03

6.02

5.91

2.25

5.46

5.45

5.30

2.2

1.09

4.00

3.99

1.56

1.05

4.00

3.99

1.53

2.3

0.00

1.69

1.68

0.00

0.00

1.73

1.72

0.00

2.4

0.09

0.00

0.00

0.15

0.06

0.00

0.00

0.11

2.5

1.66

-0.4

7-0

.46

1.72

1.57

-0.5

3-0

.52

1.62

2.6

4.89

0.62

0.64

4.92

4.72

0.48

0.51

4.75

128

Table 13: Electron Density, r(r) for 11,11-Dimethyl-1,6-methano[10]annulene [e/Å3]

C1,C6 [Å] C2C7 C1C2 C2C3 C3C4 C4C5 C1C2C71.4 2.005 1.511 1.739 2.304 1.835 1.3481.5 1.628 1.531 1.769 2.291 1.848 1.2881.6 1.325 1.560 1.800 2.275 1.863 1.214

1.636 1.234 1.573 1.812 2.268 1.869 1.1821.67 1.159 1.585 1.824 2.260 1.875 1.145

1.675 1.145 1.587 1.826 2.259 1.876 1.1391.68 1.139 1.589 1.828 2.258 1.877 1.1331.69 1.120 1.593 1.831 2.256 1.878 1.1201.7 - 1.579 1.824 2.237 1.883 -1.8 - 1.636 1.872 2.228 1.900 -2.0 - 1.681 1.947 2.171 1.937 -

2.030 - 1.683 1.958 2.161 1.942 -2.2 - 1.668 1.999 2.121 1.957 -2.4 - 1.589 2.028 2.084 1.960 -2.6 - 1.460 2.042 2.057 1.950 -

Table 14: Energy Density for 11,11-Dimethyl-1,6-methano[10]annulene [Hartree/Å3]

C1,C6 [Å] C2C7 C1C2 C2C3 C3C4 C4C5 C1C2C71.4 -2.062 -1.179 -1.473 -2.630 -1.655 -0.8181.5 -1.364 -1.208 -1.523 -2.601 -1.678 -0.7271.6 -0.900 -1.250 -1.579 -2.564 -1.706 -0.644

1.636 -0.776 -1.270 -1.600 -2.549 -1.717 -0.6171.67 -0.672 -1.288 -1.622 -2.533 -1.729 -0.594

1.675 -0.657 -1.291 -1.625 -2.530 -1.730 -0.5911.68 -0.642 -1.294 -1.628 -2.528 -1.732 -0.5881.69 -0.611 -1.300 -1.635 -2.523 -1.736 -0.5871.7 - -1.270 -1.613 -2.467 -1.734 -1.8 - -1.364 -1.712 -2.463 -1.777 -2.0 - -1.424 -1.857 -2.340 -1.848 -

2.030 - -1.423 -1.880 -2.319 -1.859 -2.2 - -1.383 -1.963 -2.233 -1.889 -2.4 - -1.242 -2.024 -2.155 -1.894 -2.6 - -1.042 -2.053 -2.099 -1.875 -

Table 15: Electron Density, r(r) for 1,6-Methano[10]annulene [e/Å3]

C1,C6 [Å] C2C7 C1C2 C2C3 C3C4 C4C5 C1C2C71.2 3.027 1.513 1.691 2.326 1.816 1.4761.4 1.994 1.552 1.754 2.303 1.836 1.3991.5 1.616 1.582 1.785 2.289 1.848 1.3391.6 1.318 1.621 1.819 2.271 1.863 1.2561.8 - 1.713 1.902 2.220 1.899 -1.9 - 1.752 1.945 2.190 1.918 -2.0 - 1.771 1.982 2.160 1.934 -2.2 - 1.766 2.035 2.109 1.953 -

2.230 - 1.760 2.040 2.103 1.954 -2.4 - 1.693 2.061 2.072 1.953 -2.6 - 1.565 2.072 2.044 1.942 -2.8 - 1.404 2.071 2.021 1.924 -

129

Table 16: Energy Density for 1,6-Methano[10]annulene [Hartree/Å3]

C1,C6 [Å] C2C7 C1C2 C2C3 C3C4 C4C5 C1C2C71.2 -4.584 -1.215 -1.392 -2.679 -1.622 -1.1151.4 -2.047 -1.254 -1.495 -2.629 -1.657 -0.8831.5 -1.351 -1.299 -1.549 -2.597 -1.680 -0.7861.6 -0.888 -1.358 -1.612 -2.557 -1.707 -0.6971.8 - -1.501 -1.767 -2.444 -1.778 -1.9 - -1.559 -1.851 -2.377 -1.815 -2.0 - -1.583 -1.925 -2.312 -1.847 -2.2 - -1.555 -2.033 -2.202 -1.882 -

2.230 - -1.542 -2.044 -2.188 -1.884 -2.4 - -1.415 -2.089 -2.124 -1.884 -2.6 - -1.203 -2.113 -2.065 -1.862 -2.8 - -0.968 -2.117 -2.017 -1.828 -

Table 17: Electron Density, r(r) for Cycloheptatriene/Norcaradiene [e/Å3]

C1,C6 [Å] C2C7 C1C2 C2C3 C3C4 C4C5 C1C2C71.2 2.996 1.557 1.727 2.258 1.767 1.5221.4 1.969 1.586 1.796 2.236 1.786 1.4281.6 1.303 1.631 1.876 2.198 1.827 1.2481.8 - 1.697 1.991 2.126 1.905 -1.9 - 1.728 2.063 2.070 1.962 -2.0 - 1.750 2.134 2.008 2.024 -2.2 - 1.752 2.230 1.903 2.117 -2.4 - 1.683 2.274 1.841 2.161 -

2.435 - 1.702 2.277 1.884 2.216 -2.6 - 1.555 2.292 1.800 2.173 -2.8 - 1.388 2.301 1.769 2.168 -

Table 18: Energy Density for Cycloheptatriene/Norcaradiene [Hartree/Å3]

C1,C6 [Å] C2C7 C1C2 C2C3 C3C4 C4C5 C1C2C71.2 -4.530 -1.280 -1.458 -2.523 -1.546 -1.1691.4 -2.013 -1.302 -1.574 -2.475 -1.581 -0.9091.6 -0.871 -1.368 -1.721 -2.395 -1.655 -0.6931.8 - -1.468 -1.948 -2.239 -1.803 -1.9 - -1.514 -2.097 -2.123 -1.913 -2.0 - -1.544 -2.247 -1.994 -2.034 -2.2 - -1.531 -2.457 -1.787 -2.223 -2.4 - -1.400 -2.554 -1.669 -2.313 -

2.435 - -1.430 -2.565 -1.744 -2.433 -2.6 - -1.188 -2.595 -1.593 -2.338 -2.8 - -0.947 -2.613 -1.539 -2.329

130

Table 19: Dependence of NMR Chemical Shifts [ppm] on C1,C6 Distance in 11,11-Dimethyl-1,6-methano[10]annulene

C1,C6 [Å] C11 C1 C2 C3 C12 H14 H18 H22 H231.4 -14.98 47.12 138.70 127.50 17.11 5.63 5.83 1.26 1.031.5 -3.64 51.18 137.69 127.06 17.82 5.84 5.92 1.11 0.941.6 5.55 59.85 136.34 126.12 17.88 6.03 5.99 0.86 0.78

1.636 8.64 63.67 135.75 125.82 18.00 6.10 6.04 0.77 0.721.7 13.74 70.99 134.65 125.46 18.46 6.26 6.14 0.63 0.631.8 20.13 85.61 133.14 124.96 18.68 6.48 6.28 0.33 0.411.9 25.30 99.45 132.29 125.36 19.32 6.74 6.47 0.14 0.242.0 29.33 112.56 132.28 126.12 19.83 6.97 6.63 -0.06 0.02

2.030 30.68 116.40 132.46 126.47 20.09 7.06 6.68 -0.10 -0.032.1 32.92 122.13 133.07 127.24 20.79 7.22 6.77 -0.14 -0.092.2 36.29 129.34 134.11 128.05 21.53 7.42 6.86 -0.22 -0.242.3 39.70 134.36 135.52 128.98 22.60 7.62 6.94 -0.24 -0.302.4 43.49 138.02 136.65 129.37 23.30 7.77 6.96 -0.30 -0.392.5 47.53 140.95 138.09 130.03 24.39 7.92 7.00 -0.33 -0.412.6 52.58 143.59 139.00 130.22 24.93 8.02 6.99 -0.38 -0.46

Table 20: Dependence of NMR Spin-Spin Coupling Constants [Hz] on C1,C6 Distance in11,11-Dimethyl-1,6-methano[10]annulene

C1,C6 [Å] H14H18 H18H19 H14H17 C1C6 C1C11 C11C12 C11C1C2 C12C11C131.4 8.09 4.67 0.22 27.01 8.96 47.53 -1.68 3.441.5 8.32 4.71 0.14 10.93 12.80 46.10 -1.27 3.171.6 8.43 5.04 0.01 -1.32 16.33 45.24 -0.68 2.99

1.636 8.47 5.14 -0.05 -4.76 17.70 44.86 -0.44 2.931.7 8.58 5.22 -0.17 -9.23 20.10 44.20 -0.35 2.801.8 8.60 5.69 -0.42 -12.41 23.26 43.21 0.69 2.661.9 8.71 6.02 -0.69 -12.12 25.64 42.31 1.22 2.502.0 8.71 6.57 -1.00 -10.02 27.38 40.92 1.60 2.33

2.030 8.73 6.74 -1.11 -9.06 27.81 40.50 1.70 2.272.1 8.83 6.98 -1.29 -7.46 28.41 39.89 1.84 2.152.2 8.88 7.47 -1.56 -5.32 28.64 38.62 1.97 1.932.3 9.03 7.87 -1.78 -3.91 28.15 37.81 2.04 1.722.4 9.14 8.27 -1.97 -2.80 26.80 36.83 2.07 1.472.5 9.28 8.61 -2.11 -1.79 24.86 36.23 2.07 1.252.6 9.40 8.94 -2.28 -0.35 22.25 35.69 1.93 1.05

Table 21: Dependence of NMR Chemical Shifts [ppm] on C1,C6 Distance in 1,6-Methano[10]annulene

C1,C6 [Å] C1 C2 C3 C4 H14 H18 H121.2 -27.64 39.72 141.84 125.16 5.91 5.81 -0.991.4 -3.82 37.33 140.77 125.87 6.21 5.90 -0.211.5 7.65 40.55 139.88 125.63 6.39 5.98 0.121.6 17.87 47.64 138.55 125.29 6.59 6.11 0.341.8 31.37 72.75 135.29 125.81 7.01 6.55 0.261.9 34.03 87.24 134.30 127.06 7.22 6.82 0.012.0 35.24 99.84 134.12 128.65 7.42 7.06 -0.272.2 36.80 116.55 135.28 131.64 7.79 7.43 -0.81

2.23 37.05 118.13 135.56 132.02 7.84 7.47 -0.882.4 39.26 124.07 137.03 133.72 8.09 7.64 -1.272.6 44.01 127.88 138.51 135.34 8.32 7.77 -1.702.8 51.60 131.18 139.73 136.83 8.49 7.84 -2.02

131

Table 22: Dependence of NMR Spin-Spin Coupling Constants [Hz] on C1,C6 Distance in1,6-Methano[10]annulene

C1,C6 [Å] H14H18 H18H19 H14H17 C1C61.2 7.53 4.53 0.26 70.001.4 8.02 4.68 0.18 27.501.5 8.19 4.88 0.09 11.651.6 8.29 5.15 -0.05 -0.961.8 8.28 5.96 -0.51 -12.611.9 8.21 6.46 -0.81 -12.672.0 8.16 6.96 -1.11 -10.802.2 8.14 7.93 -1.59 -6.16

2.23 8.16 8.06 -1.64 -5.692.4 8.24 8.72 -1.83 -3.752.6 8.34 9.39 -1.91 -2.012.8 8.42 9.85 -1.97 0.94

Table 23: Dependence of NMR Chemical Shifts [ppm] on C2,C7 Distance in Cyclohepta-triene/Norcaradiene

C2,C7 [Å] C1 C3 C6 C7 H8 H10 H13 H14 H151.2 57.33 36.93 160.37 205.38 26.03 25.26 29.45 33.42 32.141.4 57.02 69.47 162.07 184.50 25.79 25.15 29.98 33.04 30.951.6 56.19 67.01 154.30 168.62 25.40 25.04 29.67 32.93 29.851.8 53.36 47.67 135.03 160.19 24.83 24.95 28.69 33.08 29.031.9 51.10 49.27 121.45 158.50 24.57 24.93 28.09 33.07 28.782.0 48.86 50.33 107.12 157.82 24.41 24.95 27.56 32.83 28.672.2 46.35 50.79 83.97 157.34 24.48 25.09 26.93 31.79 28.722.4 45.61 49.83 70.12 156.36 24.72 25.17 26.71 30.75 28.89

2.435 51.78 56.50 63.88 158.44 25.06 25.49 26.51 30.38 29.142.6 45.34 48.49 61.04 153.84 24.96 25.16 26.56 30.02 29.072.8 45.18 47.11 53.74 149.07 25.14 25.08 26.40 29.54 29.18

Table 24: Dependence of NMR Spin-Spin Coupling Constants [Hz] on C2,C7 Distance inCycloheptatriene/Norcaradiene

C2,C7 [Å] H8H9 H8H10 H10H13 H13H14 H13H15 C2C71.2 3.84 8.87 5.84 2.43 7.18 65.471.4 4.08 8.83 5.33 3.29 7.41 24.501.6 4.60 8.45 4.98 3.96 7.32 -1.361.8 5.49 7.62 5.24 4.65 7.22 -12.461.9 6.07 7.02 5.70 4.99 7.19 -13.472.0 6.67 6.39 6.30 5.24 7.18 -12.632.2 7.61 5.54 7.23 5.25 7.03 -8.262.4 8.33 5.20 7.61 4.58 6.48 -5.28

2.435 9.80 5.20 8.29 4.14 6.96 -5.072.6 9.05 5.05 7.44 3.59 5.47 -4.162.8 9.78 4.99 7.05 2.60 4.17 -3.69

132

Supporting Information

The Longest Covalent CC Bonds - Characterized

with Vibrational Spectroscopy

Alan Humason, Dieter Cremer, and Elfi Kraka∗





133

Supporting Information

The compilation of experimental bond lengths summarized in Table 4, is from an extensive

literature search of the target compounds.1–42 This data includes electron diffraction, x-ray

diffraction, microwave spectroscopy and infrared spectroscopy methods. The parameters

reported in the literature are obtained by different techniques, at different temperatures and

in different matrices, all of which effects the actual values. Note that when multiple values

are available, the experimental values vary by an average of 0.034 A.

To simplify the on-going analysis, the most accurate bond lengths are selected from

among the literature values by the following criteria:

• Gas phase electron diffraction experiments, yielding ra values, are considered most ac-

curate, because they are taken in the gas phase, making them more comparable to the

theoretical results. These results are also more accurate than x-ray diffraction exper-

iments, because the wavelengths of accelerated electrons are shorter than the wave-

lengths of x-rays, further increasing the accuracy.43 In some experiments, the electron

diffraction results are combined with microwave spectroscopy rotational constants, to

obtain rg values of improved accuracy.

• X-ray diffraction experiments generally have the limitations that 1) structures are de-

termined on solid crystals, and therefore are subject to packing effects, and 2) hydrogen

atoms, which have no core electrons, cannot be accurately located. For this study of

C-C bond lengths, the approximate hydrogen locations are not a concern, but the other

considerations make these values less reliable than electron diffraction.

• Microwave and infrared spectroscopy experiments, yielding rm, ro and rs data, depend

upon comparison of rotational constants for multiple isotopomers. These methods have

the limitations that 1) they are dependent upon the availability of the isotopomers,

and 2) the calculations are based on initial assumed geometries. These results are

considered least accurate, and are used when results from the other techniques are

134

unavailable.

Although bond dissociation energies (BDE’s) are often sited as bond strength descrip-

tors, molecular energies at 0K cannot be measured. Experimentally, only bond dissociation

enthalpies (BDH’s) are accessible. Therefore, calculated (G4) values of BDH at 298K are

used for comparison with reported experimental values.

Several compilations of bond dissociation enthalpies are available,44–49 and are compiled

in Table 5. Here again, the available data are from various methods, and are not comparable.

Where multiple values are available, the BDH’s vary by an average of 4 kcal/mol. We make

no differentiation of data quality here.

IUPAC names for all of the target compounds are included in Table 3, to remove any

ambiguity concerning structure.

Acknowledgments

This work was financially supported by the National Science Foundation, Grant CHE 1152357.

We thank SMU for providing computational resources.

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(2) Hilderbrandt, R.; Wieser, J. The Zero Point Average Structure of Isobutane as Deter-

mined by Electron Diffraction and Microwave Spectroscopy. J. Mol. Struct. 1973, 15,

27–36.

(3) Lide, D. Structure of the Isobutane Molecule; Change of Dipole Moment on Isotopic

Substitution. J. Chem. Phys. 1960, 33, 1519–1522.

135

(4) VanHemelrijk, D.; Van den Enden, L.; Geise, H.; Sellers, H.; Schafer, Structure Deter-

mination of 1-Butene by Gas Electron Diffraction, Microwave Spectroscopy, Molecular

Mechanics, and Molecular Orbital Constrained Electron Diffraction. J. Amer. Chem.

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(5) Bock, C.; Panchenko, Y. An Ab Initio Structural Investigation of 1,3-Butadiene, Iso-

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141

Electron X-Ray Microwave InfraredNo. Bond ID Diffraction Diffraction Spectroscopy Spectroscopy ∆Rd

exp

1 Me-Me 1.53643 1.53632 1.53443 0.0022 Me-Et 1.52838

3 iPr-Me 1.535 (1)2 1.525 (1)3 0.014 tBu-Me 1.53943 1.546(2)11 0.0075 (iPr)2 1.54439

6 (tBu)2 1.58239

7 (PhCH2)2 1.55 (26)25 1.529 (3)25

1.5848 0.0518 (AdMe2C)2 1.677 (30)40 1.63926 0.0389 (Et2MeC)2 1.60148a

10 (Et3C)2 1.63548a

11 (PhEt2C)2 1.63527

12 (Ph3C)2 –c

13 hexakis 1.67 (3)42

14 bianthracene 1.64 (1)28 1.65329

1.648 (3)29

1.7730 0.1315 4-Ph-acenaphthene 1.754 (2)31

16 H3C-CCl3 1.516 (4)12

17 (Cl3C)2 1.564 (14)13

18 CH3-CH+3 D3d –c

19 CH3-CH+3 C2h –c

20 Et-Et 1.539 (3)43

21 tBu-C≡N 1.46 (2)14

22 (iPrMe2C)2 1.60148a

23 (tBuMe2C)2 1.6340

24 (iBuMe2C)2 1.60648a

25 diAd-Ad 1.6641

26 diAd-diAd 1.64740

27 triAd-Ad 1.65940

28 triAd-diAd 1.70440

29 triAd-triAd –c

30 tetraAd-diAd 1.70740

31 Me-CH=CH2 1.50138

32 H2C=CH2 1.33932

33 (CH2=CH)2 1.4675

1.48643 1.48543 0.01834 Et-CH=CH2 1.5024

35 iPr-CH=CH2 1.50010

36 tBu-CH=CH2 1.522b 48

37 HC≡CH 1.20832

38 Me-C≡CH 1.4543 1.45838 1.45943 0.0091.45891

1.45771

39 Me-C≡N 1.45838

1.458361

1.45821 0.00440 CH2=CH-C≡CH 1.431443 1.42643 0.00541 CH2=CH-C≡N 1.429 (5)6

1.4430 (2)6 1.4307

1.45838 1.4297 0.02942 HC≡C-C≡CH 1.38343 1.37419 0.01143 HC≡C-C≡N 1.37968

1.37939

1.20638

1.37751

142

1.3821 0.17644 Me-Ph 1.5214 (65)15 1.512 (1)33

1.52315

1.51 (2)16 0.01345 Et-Ph 1.524 (9)17

46 iPr-Ph 1.50 (5)21

47 tBu-Ph 1.523723

48 Ph-cycloC3H5 1.520 (25)22

49 Ph-CH=CH2 1.475 (23)18

50 Ph-C≡CH 1.436 (4)19 1.4447 (7)19 0.00951 Ph-C≡N 1.438 (5)19 1.44419 0.006

1.434 (3)19 1.4509 (6)19

1.38838

1.4509 (6)20 0.06352 Ph-Ph 1.4824

Table 1: Experimental Carbon-Carbon Bond Lengths R [A]. a) MM2 force fieldcalculation. No experimental value available. b) Average crystallographic valuefor the specific type of bond. c) No experimental result for this compound. d)∆Rexp: The range of experimental R values. The most accurate values are inbold type.

143

Table 2: Experimental Bond Dissociation Enthalpies [kcal/mol].

No. Bond ID NIST46 Luo44 Griller49 Other Refs ∆BDHaexp

1 Me-Me 89.68 90.2 ± 0.2 88.8 88±243 2.22 Me-Et 87.2 88.5 ± 0.5 87.4 1.33 iPr-Me 88.9 88.2 ± 0.9 85.7 87±243 3.24 tBu-Me 86 86.9 ± 0.7 84 8243 4.95 (iPr)2 86.6 84.5 ± 1.1 81 5.66 (Me3C)2 (tBu)2 76 68.3 ± 1.5 72.2 4.97 (PhCH2)2 66.6 62.6 ± 2.2 48 (AdMe2C)2 43.726

9 (Et2MeC)2 60.237

10 (Et3C)2 51.037

11 (PhEt2C)2 44.727

16 H3C-CCl3 88.3 87.6 ± 2.0 87.6 ± 2.036 0.717 (Cl3C)2 70.1 68.3 ± 1.5 70.1 ± 3.536 1.820 Et-Et 87.2 86.8 ± 0.6 86 1.221 tBu-C≡N 115.8 117.7 ± 1.3 109.2 8.522 (iPrMe2C)2 62.237

23 (tBuMe2C)2 44.037

24 (iBuMe2C)2 57.837

31 Me-CH=CH2 100.9 93 7.932 H2C=CH2 172.2 174.1 ± 1.5 2.133 (CH2=CH)2 116 116.9 ± 1.5 0.934 Et-CH=CH2 99.6 100.0 ± 1.0 91.4 8.635 iPr-CH=CH2 99.7 99.2 ± 1.5 89.2 10.536 tBu-CH=CH2 97.5 97.7 ± 1.3 87.3 10.437 HC≡CH 229.9 229.5 ± 1.0 0.438 Me-C≡CH 123.5 126.0 ± 1.0 122.2 3.839 Me-C≡N 121.1 124.7 118.2 122.834 6.440 CH2=CH-C≡CH 133.641 CH2=CH-C≡N 132.1 133.9 ± 1.8 1.842 HC≡C-C≡CH 15543 HC≡C-C≡N 152.4 143.9 8.544 Me-Ph 103.9 102.0 ± 1.0 101 2.945 Et-Ph 102.3 100.2 ± 1.0 99.5 2.846 iPr-Ph 102.1 98.9 ± 1.2 96.8 5.347 tBu-Ph 97.4 93.4 448 Ph-cycloC3H5 109.8 ± 1.2 111.947 2.149 Ph-CH=CH2 116.9 115.2 ± 1.3 1.750 Ph-C≡CH 140.7 141.2 ± 1.435 0.551 Ph-C≡N 132.7 132.8 ± 2.0 0.152 Ph-Ph 118 114.4 ± 1.5 3.6

a) ∆BDHexp: The range of experimental BDH values.

144

Table 3: IUPAC nomenclature for all target analytes.

No. Bond ID IUPAC Name1 Me-Me ethane2 Me-Et propane3 iPr-Me 2-methylpropane4 tBu-Me 2,2-dimethylpropane5 (iPr)2 2,3-dimethylbutane6 (tBu)2 2,2,3,3-tetramethylbutane7 (PhCH2)2 1,2-diphenylethane8 (AdMe2C)2 2,3-diadamantyl-2,3-dimethylbutane9 (Et2MeC)2 3,4-ethyl-3,4-methylhexane

10 (Et3C)2 3,3,4,4-tetraethylhexane11 (PhEt2C)2 3,4-ethyl-3,4-phenylhexane12 (Ph3C)2 hexaphenylethane13 hexakis hexakis-(3,5-di-tert-butylphenyl)ethane14 bianthracene bi(anthracene-9,10-dimethylene)15 4-Ph-acenaphthene acenaphthene-5,6-diyl bis(diphenylmethylium)16 H3C-CCl3 1,1,1-trichloroethane17 (Cl3C)2 hexachloroethane18 CH3CH+

3 D3d ethane cation, D3d

19 CH3CH+3 C2h ethane cation, C2h

20 Et-Et butane21 tBu-C≡N 2-cyano-2-methylpropane22 (iPrMe2C)2 2,3,3,4,4,5-hexamethylhexane23 (tBuMe2C)2 2,2,3,3,4,4,5,5-octamethylhexane24 (iBuMe2C)2 2,4,4,5,5,7-hexamethyloctane25 diAd-Ad 1-(1-adamantyl)diamantane26 diAd-diAd 1-(1-diamantyl)diamantane27 triAd-Ad 2-(1-adamantyl)triamantane28 triAd-diAd 2-(1-diamantyl)triamantane29 triAd-triAd 2-(2-triamantyl)triamantane30 tetraAd-diAd 2-(1-diamantyl)[121]tetramantane31 Me-CH=CH2 propene32 H2C=CH2 ethene33 (CH2=CH)2 1,3-butadiene34 Et-CH=CH2 1-butene35 iPr-CH=CH2 3-methyl-1-butene36 tBu-CH=CH2 3,3-dimethyl-1-butene37 HC≡CH ethyne38 Me-C≡CH propyne39 Me-C≡N acetonitrile40 CH2=CH-C≡CH buta-1-en-3-yne41 CH2=CH-C≡N cyanoethene42 HC≡C-C≡CH 1,3-butadiyne43 HC≡C-C≡N cyanoethyne44 Me-Ph toluene45 Et-Ph ethylbenzene46 iPr-Ph cumene47 tBu-Ph (1,1-dimethylethyl)benzene48 Ph-cycloC3H5 cyclopropylbenzene49 Ph-CH=CH2 styrene50 Ph-C≡CH phenylacetylene51 Ph-C≡N cyanobenzene52 Ph-Ph biphenyl53 iron clamp bis(1,4,5-trimethylimidazolyl)ketone iron complex

145

1 2C 0.000000 0.000000 0.761558 C -1.267554 -0.259242 0.000000C 0.000000 0.000000 -0.761558 C 0.000000 0.586273 0.000000H 0.000000 1.016010 1.157084 C 1.267554 -0.259242 0.000000H 0.000000 -1.016010 -1.157084 H -2.163951 0.361268 0.000000H 0.879891 -0.508005 1.157084 H -1.306571 -0.903329 0.880539H -0.879891 -0.508005 1.157084 H -1.306571 -0.903329 -0.880539H 0.879891 0.508005 -1.157084 H 0.000000 1.242024 0.873618H -0.879891 0.508005 -1.157084 H 0.000000 1.242024 -0.873618

H 2.163951 0.361268 0.000000H 1.306571 -0.903329 -0.880539H 1.306571 -0.903329 0.880539

3 4H 0.000000 0.000000 1.468784 C 0.000000 0.000000 1.530136C 0.000000 0.000000 0.374644 C 0.000000 0.000000 -0.000040C 0.000000 1.451506 -0.095632 C 0.000000 1.442664 -0.510044C 1.257041 -0.725753 -0.095632 C 1.249384 -0.721332 -0.510044C -1.257041 -0.725753 -0.095632 C -1.249384 -0.721332 -0.510044H 0.000000 1.500064 -1.187506 H 0.000000 1.471374 -1.601414H -0.882755 1.984065 0.261208 H -0.882820 1.981006 -0.159722H 0.882755 1.984065 0.261208 H 0.882820 1.981006 -0.159722H 1.299093 -0.750032 -1.187506 H 1.274247 -0.735687 -1.601414H 2.159628 -0.227544 0.261208 H 2.157011 -0.225958 -0.159722H 1.276873 -1.756521 0.261208 H 1.274191 -1.755048 -0.159722H -1.299093 -0.750032 -1.187506 H -1.274247 -0.735687 -1.601414H -1.276873 -1.756521 0.261208 H -1.274191 -1.755048 -0.159722H -2.159628 -0.227544 0.261208 H -2.157011 -0.225958 -0.159722

H -0.882831 0.509703 1.920930H 0.882831 0.509703 1.920930H 0.000000 -1.019405 1.920930

5 6C -0.228808 0.736301 0.000000 C 0.000000 0.000000 0.788266C 0.228808 -0.736301 0.000000 C 0.000000 0.000000 -0.788266H 1.325676 -0.729525 0.000000 C -0.566769 -1.313869 -1.344712C -0.228808 -1.492129 1.247046 C -0.854459 1.147771 -1.344712C -0.228808 -1.492129 -1.247046 C 1.421228 0.166098 -1.344712H 0.112686 -2.527357 1.211402 H -0.680579 -1.235767 -2.427192H 0.172816 -1.045270 2.157391 H 0.094524 -2.156817 -1.145816H -1.316233 -1.507996 1.330515 H -1.547730 -1.547515 -0.929630H 0.112686 -2.527357 -1.211402 H -0.729916 1.207282 -2.427192H -1.316233 -1.507996 -1.330515 H -1.915120 0.996548 -1.145816H 0.172816 -1.045270 -2.157391 H -0.566323 2.114131 -0.929630H -1.325676 0.729525 0.000000 H 1.410495 0.028485 -2.427192C 0.228808 1.492129 1.247046 H 1.820596 1.160269 -1.145816C 0.228808 1.492129 -1.247046 H 2.114052 -0.566615 -0.929630H -0.112686 2.527357 1.211402 C 0.854459 1.147771 1.344712H -0.172816 1.045270 2.157391 C -1.421228 0.166098 1.344712H 1.316233 1.507996 1.330515 C 0.566769 -1.313869 1.344712H -0.112686 2.527357 -1.211402 H 0.729916 1.207282 2.427192H 1.316233 1.507996 -1.330515 H 1.915120 0.996548 1.145816H -0.172816 1.045270 -2.157391 H 0.566323 2.114131 0.929630

H -1.410495 0.028485 2.427192H -1.820596 1.160269 1.145816H -2.114052 -0.566615 0.929630H 0.680579 -1.235767 2.427192

146

H -0.094524 -2.156817 1.145816H 1.547730 -1.547515 0.929630

7 8C 0.525788 0.563983 1.95016 C -0.614146 -0.535114 -0.310626C -0.525788 -0.563983 1.95016 C 0.614146 0.535114 -0.310626H 0.421567 1.135774 2.874033 C 1.635219 0.187727 0.784468H 1.521707 0.117476 1.9595 C 1.346970 0.441579 -1.667207H -0.421567 -1.135774 2.874033 H 2.402287 0.955976 0.857902H -1.521707 -0.117476 1.9595 H 1.176556 0.094781 1.765367C 0.389864 1.483012 0.766559 H 2.153997 -0.740595 0.569211C -0.525788 2.530345 0.788384 H 2.355085 0.844227 -1.590531C 1.126566 1.271939 -0.393649 H 1.444309 -0.579703 -2.008097C -0.702707 3.345712 -0.317998 H 0.831311 0.989572 -2.454605C 0.953799 2.084272 -1.503305 C -1.635219 -0.187727 0.784468C 0.037637 3.123813 -1.469472 C -1.346970 -0.441579 -1.667207H -1.10666 2.709519 1.685557 H -2.402287 -0.955976 0.857902H 1.832762 0.452047 -0.431184 H -1.176556 -0.094781 1.765367H -1.416834 4.157523 -0.280317 H -2.153997 0.740595 0.569211H 1.534604 1.902203 -2.397531 H -2.355085 -0.844227 -1.590531H -0.097444 3.759302 -2.334137 H -1.444309 0.579703 -2.008097C -0.389864 -1.483012 0.766559 H -0.831311 -0.989572 -2.454605C 0.525788 -2.530345 0.788384 C 0.238216 2.110027 -0.086490C -1.126566 -1.271939 -0.393649 C -0.317673 3.959867 1.601523C 0.702707 -3.345712 -0.317998 C -0.043762 2.465645 1.397315C -0.953799 -2.084272 -1.503305 C -0.944305 2.633464 -0.936486C -0.037637 -3.123813 -1.469472 C 1.451351 2.995275 -0.506864H 1.10666 -2.709519 1.685557 C -1.206959 4.129650 -0.717733H -1.832762 -0.452047 -0.431184 C 1.206959 4.491860 -0.280026H 1.416834 -4.157523 -0.280317 C -1.519505 4.385096 0.757273H -1.534604 -1.902203 -2.397531 C 0.913749 4.765738 1.193465H 0.097444 -3.759302 -2.334137 C 0.019025 4.942171 -1.129220

H -0.534254 4.128443 2.658479H -2.063417 4.419659 -1.329938H 2.104041 5.034865 -0.585054H -0.894269 1.911415 1.779422H 0.813319 2.193069 2.012874H -0.736922 2.472831 -1.996941H -1.863846 2.101037 -0.707722H 2.349384 2.704364 0.039272H 1.666180 2.858686 -1.565136H -1.741668 5.443186 0.918448H -2.406136 3.819447 1.056598H 1.770028 4.477065 1.808766H 0.744279 5.833708 1.354133H -0.167308 6.010152 -0.989108H 0.234301 4.786819 -2.189683C -0.238216 -2.110027 -0.086490C 0.317673 -3.959867 1.601523C 0.043762 -2.465645 1.397315C 0.944305 -2.633464 -0.936486C -1.451351 -2.995275 -0.506864C 1.206959 -4.129650 -0.717733C -1.206959 -4.491860 -0.280026C 1.519505 -4.385096 0.757273C -0.913749 -4.765738 1.193465C -0.019025 -4.942171 -1.129220H 0.534254 -4.128443 2.658479H 2.063417 -4.419659 -1.329938

147

H -2.104041 -5.034865 -0.585054H 0.894269 -1.911415 1.779422H -0.813319 -2.193069 2.012874H 0.736922 -2.472831 -1.996941H 1.863846 -2.101037 -0.707722H -2.349384 -2.704364 0.039272H -1.666180 -2.858686 -1.565136H 1.741668 -5.443186 0.918448H 2.406136 -3.819447 1.056598H -1.770028 -4.477065 1.808766H -0.744279 -5.833708 1.354133H 0.167308 -6.010152 -0.989108H -0.234301 -4.786819 -2.189683

9 10C -0.800077 0.021935 -0.172534 C -0.045843 0.804021 0.047056C 0.701910 -0.028837 0.367846 C 0.045843 -0.804021 0.047056C 0.770846 0.631810 1.756303 C -1.319198 -1.501768 0.310406C 1.239713 -1.471235 0.588014 C 0.587969 -1.305351 -1.317318C 1.655807 0.711013 -0.602231 C 1.007466 -1.222293 1.201436H 1.741163 0.454111 2.218495 C -2.500075 -1.291261 -0.635311H 0.624627 1.709705 1.712540 H -1.117818 -2.572555 0.319198H 0.019851 0.219858 2.430814 H -1.653062 -1.276024 1.323265C 1.410902 -2.403659 -0.607464 C 1.007466 -2.769258 -1.422729H 2.218838 -1.363024 1.057562 H -0.165844 -1.123289 -2.082319H 0.628691 -1.968955 1.341436 H 1.443977 -0.702916 -1.608037C 3.103918 0.869426 -0.145759 C 0.864966 -2.606202 1.839207H 1.661281 0.192599 -1.560616 H 2.034474 -1.110952 0.848948H 1.258506 1.701858 -0.809647 H 0.900991 -0.507989 2.017828C -0.869783 -0.169359 -1.696176 C 1.319198 1.501768 0.310406C -1.615099 -1.116702 0.500367 C -0.587969 1.305351 -1.317318C -1.530851 1.349833 0.166361 C -1.007466 1.222293 1.201436H -1.909305 -0.213159 -2.021727 C 2.500075 1.291261 -0.635311H -0.405342 0.653012 -2.236252 H 1.117818 2.572555 0.319198H -0.384842 -1.087186 -2.020117 H 1.653062 1.276024 1.323265C -3.129486 -1.123349 0.298444 C -1.007466 2.769258 -1.422729H -1.232795 -2.072117 0.141740 H 0.165844 1.123289 -2.082319H -1.427109 -1.099497 1.576123 H -1.443977 0.702916 -1.608037C -0.985053 2.682976 -0.340719 C -0.864966 2.606202 1.839207H -2.535732 1.254239 -0.246920 H -2.034474 1.110952 0.848948H -1.667584 1.415824 1.248021 H -0.900991 0.507989 2.017828H 1.946298 -3.301425 -0.296428 H -3.291247 -1.996894 -0.378444H 0.460973 -2.727407 -1.029711 H -2.92806 -0.292946 -0.564871H 1.990054 -1.942641 -1.407948 H -2.237108 -1.471283 -1.67791H 3.671853 1.407416 -0.905123 H 1.286197 -2.992566 -2.453042H 3.183399 1.436930 0.781223 H 1.872907 -2.992299 -0.79955H 3.597779 -0.090471 0.005698 H 0.207413 -3.457291 -1.148816H -3.545150 -2.046884 0.702319 H 1.670509 -2.748465 2.56082H -3.620432 -0.297646 0.811405 H -0.073651 -2.706567 2.382832H -3.409191 -1.077693 -0.754448 H 0.920773 -3.423002 1.123617H -1.721021 3.465997 -0.153978 H 3.291247 1.996894 -0.378444H -0.064507 2.984942 0.155222 H 2.92806 0.292946 -0.564871H -0.796783 2.668508 -1.414400 H 2.237108 1.471283 -1.67791

H -1.286197 2.992566 -2.453042H -1.872907 2.992299 -0.79955H -0.207413 3.457291 -1.148816H -1.670509 2.748465 2.56082H 0.073651 2.706567 2.382832H -0.920773 3.423002 1.123617

148

11 12C 0.680603 -0.449152 -0.765518 C 0.000000 0.000000 0.740768C -0.680603 0.449152 -0.765518 C -0.423303 1.119580 1.454445C -0.856277 1.184681 -2.118108 C 0.423303 -1.119580 1.454445C -1.973576 -0.391973 -0.526237 C -0.423517 1.120306 2.839632C -2.076215 2.098682 -2.223274 C 0.423517 -1.120306 2.839632H 0.035000 1.758031 -2.357808 C 0.000000 0.000000 3.537952H -0.931721 0.441254 -2.908260 H -0.775832 1.990724 0.918093C -2.537309 -1.261058 -1.648632 H 0.775832 -1.990724 0.918093H -1.842697 -1.029102 0.343931 H -0.763061 1.996515 3.375233H -2.751252 0.321129 -0.249069 H 0.763061 -1.996515 3.375233C 0.856277 -1.184681 -2.118108 H 0.000000 0.000000 4.619287C 1.973576 0.391973 -0.526237 C 0.000000 0.000000 -0.740768C 2.076215 -2.098682 -2.223274 C -0.423303 -1.119580 -1.454445H -0.035000 -1.758031 -2.357808 C 0.423303 1.119580 -1.454445H 0.931721 -0.441254 -2.908260 C -0.423517 -1.120306 -2.839632C 2.537309 1.261058 -1.648632 C 0.423517 1.120306 -2.839632H 1.842697 1.029102 0.343931 C 0.000000 0.000000 -3.537952H 2.751252 -0.321129 -0.249069 H -0.775832 -1.990724 -0.918093H -2.066809 2.617365 -3.182307 H 0.775832 1.990724 -0.918093H -3.009800 1.539976 -2.166137 H -0.763061 -1.996515 -3.375233H -2.096350 2.851819 -1.437181 H 0.763061 1.996515 -3.375233H -3.504396 -1.654299 -1.333236 H 0.000000 0.000000 -4.619287H -2.701011 -0.709672 -2.573531H -1.911120 -2.118480 -1.881967H 2.066809 -2.617365 -3.182307H 3.009800 -1.539976 -2.166137H 2.096350 -2.851819 -1.437181H 3.504396 1.654299 -1.333236H 2.701011 0.709672 -2.573531H 1.911120 2.118480 -1.881967C -0.594154 1.471917 0.384863C -1.031788 1.166013 1.673039C -0.055990 2.745721 0.196599C -0.921466 2.071281 2.717835C 0.055990 3.657270 1.232714C -0.374263 3.324125 2.506546H -1.458579 0.199035 1.886925H 0.287502 3.048406 -0.780810H -1.268873 1.788127 3.702554H 0.480049 4.633532 1.038965H -0.289760 4.032896 3.318983C 0.594154 -1.471917 0.384863C 1.031788 -1.166013 1.673039C 0.055990 -2.745721 0.196599C 0.921466 -2.071281 2.717835C -0.055990 -3.657270 1.232714C 0.374263 -3.324125 2.506546H 1.458579 -0.199035 1.886925H -0.287502 -3.048406 -0.780810H 1.268873 -1.788127 3.702554H -0.480049 -4.633532 1.038965H 0.289760 -4.032896 3.318983

13C 0.000000 0.000000 0.835723C 0.000000 0.000000 -0.835723

149

C 0.918211 1.123704 1.418576C 0.718917 2.479589 1.096437C 1.884594 0.844352 2.375648C 1.464225 3.499183 1.674378C 2.675164 1.837875 2.971539C 2.455349 3.157192 2.607580H -0.066263 2.738904 0.406669H 2.044433 -0.176403 2.687360C 1.223514 4.978807 1.342982C 3.748255 1.407088 3.979108H 3.053162 3.944973 3.051791C -0.918211 -1.123704 -1.418576C -1.884594 -0.844352 -2.375648C -0.718917 -2.479589 -1.096437C -2.675164 -1.837875 -2.971539C -1.464225 -3.499183 -1.674378C -2.455349 -3.157192 -2.607580H -2.044433 0.176403 -2.687360H 0.066263 -2.738904 -0.406669C -3.748255 -1.407088 -3.979108C -1.223514 -4.978807 -1.342982H -3.053162 -3.944973 -3.051791C -0.514050 1.357046 -1.418576C 0.211067 2.054282 -2.375648C -1.787929 1.862395 -1.096437C -0.254065 3.235698 -2.971539C -2.298269 3.017648 -1.674378C -1.506534 3.704990 -2.607580H 1.174986 1.682330 -2.687360H -2.405092 1.312067 -0.406669C 0.655553 3.949628 -3.979108C -3.700016 3.548998 -1.342982H -1.889866 4.616602 -3.051791C 1.432261 -0.233342 -1.418576C 1.673527 -1.209930 -2.375648C 2.506846 0.617194 -1.096437C 2.929229 -1.397822 -2.971539C 3.762494 0.481536 -1.674378C 3.961883 -0.547798 -2.607580H 0.869447 -1.858733 -2.687360H 2.338829 1.426837 -0.406669C 3.092702 -2.542540 -3.979108C 4.923530 1.429809 -1.342982H 4.943028 -0.671630 -3.051791C -1.432261 0.233342 1.418576C -2.506846 -0.617194 1.096437C -1.673527 1.209930 2.375648C -3.762494 -0.481536 1.674378C -2.929229 1.397822 2.971539C -3.961883 0.547798 2.607580H -2.338829 -1.426837 0.406669H -0.869447 1.858733 2.687360C -4.923530 -1.429809 1.342982C -3.092702 2.542540 3.979108H -4.943028 0.671630 3.051791C 0.514050 -1.357046 1.418576C 1.787929 -1.862395 1.096437C -0.211067 -2.054282 2.375648C 2.298269 -3.017648 1.674378

150

C 0.254065 -3.235698 2.971539C 1.506534 -3.704990 2.607580H 2.405092 -1.312067 0.406669H -1.174986 -1.682330 2.687360C 3.700016 -3.548998 1.342982C -0.655553 -3.949628 3.979108H 1.889866 -4.616602 3.051791C 0.052369 5.168710 0.372367C 0.904022 5.748244 2.639945C 2.488470 5.574577 0.697117H -0.089696 6.235127 0.166481H 0.222896 4.663322 -0.582368H -0.883221 4.784366 0.789704H 0.700746 6.800992 2.413741H 0.021703 5.329402 3.134765H 1.734888 5.715048 3.351073H 2.344373 6.642215 0.495417H 3.365152 5.468700 1.343790H 2.709918 5.076268 -0.251275C 4.507348 2.602318 4.569864C 3.087799 0.631043 5.135570C 4.763270 0.493266 3.264588H 5.260065 2.244933 5.280126H 5.029659 3.173746 3.795035H 3.837633 3.281041 5.108822H 3.843631 0.332606 5.870807H 2.341821 1.252117 5.642448H 2.587352 -0.276589 4.785054H 5.520566 0.134044 3.970824H 4.272962 -0.379195 2.821745H 5.274597 1.034794 2.461618C -4.507348 -2.602318 -4.569864C -4.763270 -0.493266 -3.264588C -3.087799 -0.631043 -5.135570H -5.260065 -2.244933 -5.280126H -3.837633 -3.281041 -5.108822H -5.029659 -3.173746 -3.795035H -5.520566 -0.134044 -3.970824H -5.274597 -1.034794 -2.461618H -4.272962 0.379195 -2.821745H -3.843631 -0.332606 -5.870807H -2.587352 0.276589 -4.785054H -2.341821 -1.252117 -5.642448C -0.052369 -5.168710 -0.372367C -2.488470 -5.574577 -0.697117C -0.904022 -5.748244 -2.639945H 0.089696 -6.235127 -0.166481H 0.883221 -4.784366 -0.789704H -0.222896 -4.663322 0.582368H -2.344373 -6.642215 -0.495417H -2.709918 -5.076268 0.251275H -3.365152 -5.468700 -1.343790H -0.700746 -6.800992 -2.413741H -1.734888 -5.715048 -3.351073H -0.021703 -5.329402 -3.134765C 0.000000 5.204637 -4.569864C 1.954454 4.371746 -3.264588C 0.997400 2.989634 -5.135570H 0.685864 5.677816 -5.280126

151

H -0.922648 4.964008 -5.108822H -0.233715 5.942686 -3.795035H 2.644197 4.847972 -3.970824H 1.741141 5.085332 -2.461618H 2.464874 3.510896 -2.821745H 1.633770 3.494985 -5.870807H 1.533209 2.102418 -4.785054H 0.086546 2.654135 -5.642448C -4.450050 2.629708 -0.372367C -3.583491 4.942367 -0.697117C -4.526114 3.657028 -2.639945H -5.444626 3.039885 -0.166481H -4.584993 1.627291 -0.789704H -3.927108 2.524695 0.582368H -4.580141 5.351395 -0.495417H -3.041218 4.884992 0.251275H -3.053457 5.648657 -1.343790H -5.539459 4.007360 -2.413741H -4.081933 4.359982 -3.351073H -4.604546 2.683496 -3.134765C 4.507348 -2.602318 -4.569864C 2.808816 -3.878480 -3.264588C 2.090399 -2.358591 -5.135570H 4.574201 -3.432883 -5.280126H 4.760281 -1.682968 -5.108822H 5.263374 -2.768940 -3.795035H 2.876369 -4.713928 -3.970824H 3.533456 -4.050538 -2.461618H 1.808088 -3.890091 -2.821745H 2.209861 -3.162379 -5.870807H 1.054143 -2.379007 -4.785054H 2.255276 -1.402018 -5.642448C 4.502419 2.539002 -0.372367C 6.071960 0.632211 -0.697117C 5.430136 2.091216 -2.639945H 5.354931 3.195243 -0.166481H 3.701772 3.157075 -0.789704H 4.150004 2.138628 0.582368H 6.924514 1.290821 -0.495417H 5.751136 0.191276 0.251275H 6.418609 -0.179957 -1.343790H 6.240205 2.793632 -2.413741H 5.816821 1.355067 -3.351073H 4.626249 2.645905 -3.134765C -4.502419 -2.539002 0.372367C -5.430136 -2.091216 2.639945C -6.071960 -0.632211 0.697117H -5.354931 -3.195243 0.166481H -4.150004 -2.138628 -0.582368H -3.701772 -3.157075 0.789704H -6.240205 -2.793632 2.413741H -4.626249 -2.645905 3.134765H -5.816821 -1.355067 3.351073H -6.924514 -1.290821 0.495417H -6.418609 0.179957 1.343790H -5.751136 -0.191276 -0.251275C -4.507348 2.602318 4.569864C -2.090399 2.358591 5.135570C -2.808816 3.878480 3.264588

152

H -4.574201 3.432883 5.280126H -5.263374 2.768940 3.795035H -4.760281 1.682968 5.108822H -2.209861 3.162379 5.870807H -2.255276 1.402018 5.642448H -1.054143 2.379007 4.785054H -2.876369 4.713928 3.970824H -1.808088 3.890091 2.821745H -3.533456 4.050538 2.461618C 4.450050 -2.629708 0.372367C 4.526114 -3.657028 2.639945C 3.583491 -4.942367 0.697117H 5.444626 -3.039885 0.166481H 3.927108 -2.524695 -0.582368H 4.584993 -1.627291 0.789704H 5.539459 -4.007360 2.413741H 4.604546 -2.683496 3.134765H 4.081933 -4.359982 3.351073H 4.580141 -5.351395 0.495417H 3.053457 -5.648657 1.343790H 3.041218 -4.884992 -0.251275C 0.000000 -5.204637 4.569864C -0.997400 -2.989634 5.135570C -1.954454 -4.371746 3.264588H -0.685864 -5.677816 5.280126H 0.233715 -5.942686 3.795035H 0.922648 -4.964008 5.108822H -1.633770 -3.494985 5.870807H -0.086546 -2.654135 5.642448H -1.533209 -2.102418 4.785054H -2.644197 -4.847972 3.970824H -2.464874 -3.510896 2.821745H -1.741141 -5.085332 2.461618

Table 4: Cartesian Coordinates for Group I - Single Bonds, in angstroms. [A]

153

14 15C 0.000000 0.821141 1.377215 C 3.295040 -0.000264 0.000168C 0.000000 0.821141 -1.377215 C 1.921454 -0.000309 0.000084C 0.000000 -0.821141 1.377215 C 4.033429 -0.151976 -1.181278C 0.000000 -0.821141 -1.377215 C 1.190137 -0.120985 -1.182059C 0.000000 0.766573 2.918096 C 3.330471 -0.310246 -2.356994C 0.000000 0.766573 -2.918096 C 1.892471 -0.297963 -2.354967C 0.000000 -0.766573 2.918096 C 1.189989 0.120632 1.182114C 0.000000 -0.766573 -2.918096 C 4.033274 0.151833 1.181670C 1.220925 1.408207 -0.700695 C 1.892164 0.297829 2.355078C 1.220925 1.408207 0.700695 C 3.330174 0.310287 2.357267C 2.317704 1.912519 -1.377477 C 5.514850 -0.103734 -0.790264C 2.317704 1.912519 1.377477 C 5.514729 0.104053 0.790802C 3.411838 2.419089 -0.689724 C -0.307263 -0.034958 -0.853298C 3.411838 2.419089 0.689724 C -0.307346 0.034783 0.853169H 2.342165 1.919101 -2.456601 H 3.842141 -0.446083 -3.302592H 2.342165 1.919101 2.456601 H 1.366609 -0.430781 -3.293128H 4.258273 2.809072 -1.238362 H 1.366170 0.430878 3.293133H 4.258273 2.809072 1.238362 H 3.841718 0.446462 3.302884C -1.220925 1.408207 0.700695 H 6.027013 0.725568 -1.286599C -1.220925 1.408207 -0.700695 H 6.022580 -1.035647 -1.055022C -2.317704 1.912519 1.377477 H 6.027120 -0.725085 1.287188C -2.317704 1.912519 -1.377477 H 6.022146 1.036118 1.055633C -3.411838 2.419089 0.689724 C -0.955238 -1.314626 -1.406483C -3.411838 2.419089 -0.689724 C -0.375491 -2.547643 -1.079270H -2.342165 1.919101 2.456601 C -2.076735 -1.309259 -2.233235H -2.342165 1.919101 -2.456601 C -0.939055 -3.738425 -1.509194H -4.258273 2.809072 1.238362 C -2.640625 -2.506692 -2.675925H -4.258273 2.809072 -1.238362 C -2.085262 -3.723913 -2.304403C 1.220925 -1.408207 0.700695 H 0.514596 -2.568268 -0.464765C 1.220925 -1.408207 -0.700695 H -2.516250 -0.369964 -2.538507C 2.317704 -1.912519 1.377477 H -0.486806 -4.679712 -1.221925C 2.317704 -1.912519 -1.377477 H -3.515006 -2.479167 -3.315487C 3.411838 -2.419089 0.689724 H -2.529314 -4.653659 -2.639300C 3.411838 -2.419089 -0.689724 C -0.940638 1.234465 -1.427309H 2.342165 -1.919101 2.456601 C -2.272192 1.548694 -1.125642H 2.342165 -1.919101 -2.456601 C -0.214590 2.137358 -2.202235H 4.258273 -2.809072 1.238362 C -2.854968 2.723434 -1.577155H 4.258273 -2.809072 -1.238362 C -0.795785 3.319915 -2.659026C -1.220925 -1.408207 -0.700695 C -2.115505 3.619710 -2.347305C -1.220925 -1.408207 0.700695 H -2.850634 0.887393 -0.495862C -2.317704 -1.912519 -1.377477 H 0.821056 1.929355 -2.432778C -2.317704 -1.912519 1.377477 H -3.880275 2.949510 -1.311213C -3.411838 -2.419089 -0.689724 H -0.206703 4.008229 -3.253484C -3.411838 -2.419089 0.689724 H -2.564771 4.542789 -2.693065H -2.342165 -1.919101 -2.456601 C -0.955404 1.314465 1.406231H -2.342165 -1.919101 2.456601 C -0.375342 2.547447 1.079417H -4.258273 -2.809072 -1.238362 C -2.077481 1.309170 2.232176H -4.258273 -2.809072 1.238362 C -0.939207 3.738262 1.508830H 0.871297 1.208983 3.388218 C -2.641642 2.506645 2.674421H -0.871297 1.208983 3.388218 C -2.086005 3.723826 2.303202H -0.871297 1.208983 -3.388218 H 0.515209 2.568001 0.465586H 0.871297 1.208983 -3.388218 H -2.517293 0.369914 2.537141H -0.871297 -1.208983 3.388218 H -0.486753 4.679523 1.221801H 0.871297 -1.208983 3.388218 H -3.516474 2.479167 3.313368H 0.871297 -1.208983 -3.388218 H -2.530307 4.653607 2.637668H -0.871297 -1.208983 -3.388218 C -0.940880 -1.234503 1.427296

154

C -2.272315 -1.548906 1.125302C -0.215093 -2.136977 2.202962C -2.855266 -2.723372 1.577319C -0.796457 -3.319264 2.660230C -2.116092 -3.619197 2.348268H -2.850510 -0.888007 0.494876H 0.820469 -1.928830 2.433765H -3.880491 -2.949583 1.311170H -0.207582 -4.007251 3.255269H -2.565511 -4.542040 2.694459

53C 0.818421 -0.106748 0.000000C -0.818421 0.106748 0.000000O 1.510057 1.053663 0.000000O -1.510057 -1.053663 0.000000N 0.533094 -2.154488 1.376183N 1.696430 -0.602872 2.374717C 1.067649 -0.956140 1.235676C 0.821575 -2.606466 2.640719C 1.542504 -1.641053 3.276807C 0.367982 -3.938630 3.121420C 2.121359 -1.593078 4.644869C 2.380306 0.638483 2.698024N 1.696430 -0.602872 -2.374717N 0.533094 -2.154488 -1.376183C 1.067649 -0.956140 -1.235676C 1.542504 -1.641053 -3.276807C 0.821575 -2.606466 -2.640719C 2.121359 -1.593078 -4.644869C 0.367982 -3.938630 -3.121420C 2.380306 0.638483 -2.698024N -0.533094 2.154488 1.376183N -1.696430 0.602872 2.374717C -1.067649 0.956140 1.235676C -0.821575 2.606466 2.640719C -1.542504 1.641053 3.276807C -0.367982 3.938630 3.121420C -2.121359 1.593078 4.644869C -2.380306 -0.638483 2.698024N -1.696430 0.602872 -2.374717N -0.533094 2.154488 -1.376183C -1.067649 0.956140 -1.235676C -1.542504 1.641053 -3.276807C -0.821575 2.606466 -2.640719C -2.121359 1.593078 -4.644869C -0.367982 3.938630 -3.121420C -2.380306 -0.638483 -2.698024Fe 0.663452 2.792005 0.000000Cl 1.110878 4.959611 0.000000Fe -0.663452 -2.792005 0.000000Cl -1.110878 -4.959611 0.000000H 0.726292 -4.131955 4.130658H -0.720665 -4.007826 3.121895H 0.726572 -4.728684 2.462794H 1.779903 -0.716346 5.198043H 1.821756 -2.476850 5.202750H 3.212759 -1.567804 4.626089H 2.634834 1.144473 1.776971

155

H 1.727734 1.278379 3.293051H 3.278345 0.411531 3.268018H 1.779903 -0.716346 -5.198043H 3.212759 -1.567804 -4.626089H 1.821756 -2.476850 -5.202750H 0.726292 -4.131955 -4.130658H 0.726572 -4.728684 -2.462794H -0.720665 -4.007826 -3.121895H 2.634834 1.144473 -1.776971H 3.278345 0.411531 -3.268018H 1.727734 1.278379 -3.293051H -0.726292 4.131955 4.130658H 0.720665 4.007826 3.121895H -0.726572 4.728684 2.462794H -1.779903 0.716346 5.198043H -1.821756 2.476850 5.202750H -3.212759 1.567804 4.626089H -2.634834 -1.144473 1.776971H -1.727734 -1.278379 3.293051H -3.278345 -0.411531 3.268018H -1.779903 0.716346 -5.198043H -3.212759 1.567804 -4.626089H -1.821756 2.476850 -5.202750H -0.726292 4.131955 -4.130658H -0.726572 4.728684 -2.462794H 0.720665 4.007826 -3.121895H -2.634834 -1.144473 -1.776971H -3.278345 -0.411531 -3.268018H -1.727734 -1.278379 -3.293051

Table 5: Cartesian Coordinates for Group II - Clamped Bonds, in angstroms.[A]

156

16 17C 0.000000 0.000000 1.766085 C 0.000000 0.000000 0.791295C 0.000000 0.000000 0.254521 C 0.000000 0.000000 -0.791295H 0.000000 -1.026117 2.127011 Cl 0.000000 1.667869 1.395438Cl 0.000000 1.676473 -0.362837 Cl 0.000000 -1.667869 -1.395438H -0.888643 0.513058 2.127011 Cl 1.444417 -0.833935 1.395438H 0.888643 0.513058 2.127011 Cl -1.444417 -0.833935 1.395438Cl -1.451868 -0.838237 -0.362837 Cl 1.444417 0.833935 -1.395438Cl 1.451868 -0.838237 -0.362837 Cl -1.444417 0.833935 -1.395438

18 19C 0.000000 0.000000 0.967574 C 0.00000 0.79554 0.00000C 0.000000 0.000000 -0.967574 C 0.00000 -0.79554 0.00000H 0.000000 1.074029 1.119202 H 1.12789 0.65953 0.00000H 0.000000 -1.074029 -1.119202 H -1.12789 -0.65953 0.00000H 0.930136 -0.537014 1.119202 H -0.34461 1.256125 0.918191H -0.930136 -0.537014 1.119202 H -0.34461 1.256125 -0.918191H 0.930136 0.537014 -1.119202 H 0.34461 -1.256125 0.918191H -0.930136 0.537014 -1.119202 H 0.34461 -1.256125 -0.918191

Table 6: Cartesian Coordinates for Group III - Electron Deficient Bonds, inangstroms. [A]

20 21C 0.701445 1.820985 0.000000 C 0.000000 0.000000 1.193767C 0.701445 0.298133 0.000000 C 0.000000 0.000000 -0.278206C -0.701445 -0.298133 0.000000 C 0.000000 1.454791 -0.767329H 1.715277 2.221508 0.000000 C 1.259886 -0.727395 -0.767329H 0.188195 2.211540 0.880625 C -1.259886 -0.727395 -0.767329H 0.188195 2.211540 -0.880625 H 0.000000 1.466003 -1.857705H 1.246330 -0.069155 0.874251 H -0.883621 1.985943 -0.416077H 1.246330 -0.069155 -0.874251 H 0.883621 1.985943 -0.416077C -0.701445 -1.820985 0.000000 H 1.269596 -0.733002 -1.857705H -1.715277 -2.221508 0.000000 H 2.161688 -0.227734 -0.416077H -1.246330 0.069155 0.874251 H 1.278067 -1.758210 -0.416077H -1.246330 0.069155 -0.874251 H -1.269596 -0.733002 -1.857705H -0.188195 -2.211540 0.880625 H -1.278067 -1.758210 -0.416077H -0.188195 -2.211540 -0.880625 H -2.161688 -0.227734 -0.416077

N 0.000000 0.000000 2.341161

22 23C -0.605097 0.522636 -0.169778 C 0.394622 0.725817 0.000000C 0.605097 -0.522636 -0.169778 C -0.394622 -0.725817 0.000000C 0.091157 -2.006547 -0.023895 C 0.426289 -2.151307 0.000000C 1.374714 -0.375183 -1.496892 C -1.321125 -0.812119 1.227795C 1.629209 -0.254159 0.941239 C -1.321125 -0.812119 -1.227795H 2.315134 -0.922716 -1.456602 H -1.849374 -1.760456 1.245445H 0.813973 -0.748062 -2.351688 H -0.775358 -0.746711 2.162126H 1.622059 0.666882 -1.695586 H -2.079581 -0.037613 1.227793H 2.387723 -1.039102 0.936789 H -1.849374 -1.760456 -1.245445H 2.149045 0.689992 0.784981 H -2.079581 -0.037613 -1.227793H 1.184410 -0.239173 1.934262 H -0.775358 -0.746711 -2.162126C -0.091157 2.006547 -0.023895 C -0.426289 2.151307 0.000000C -1.374714 0.375183 -1.496892 C 1.321125 0.812119 1.227795

157

C -1.629209 0.254159 0.941239 C 1.321125 0.812119 -1.227795H -2.315134 0.922716 -1.456602 H 1.849374 1.760456 1.245445H -0.813973 0.748062 -2.351688 H 0.775358 0.746711 2.162126H -1.622059 -0.666882 -1.695586 H 2.079581 0.037613 1.227793H -2.387723 1.039102 0.936789 H 1.849374 1.760456 -1.245445H -2.149045 -0.689992 0.784981 H 2.079581 0.037613 -1.227793H -1.184410 0.239173 1.934262 H 0.775358 0.746711 -2.162126C 0.976419 -3.042868 -0.725947 C -0.484721 -3.397597 0.000000C -0.091157 -2.492839 1.420876 C 1.321125 -2.357646 -1.240916H -0.885301 -2.062998 -0.507479 C 1.321125 -2.357646 1.240916H 0.587979 -4.043400 -0.531894 H 0.149364 -4.283638 0.000000H 1.014308 -2.910437 -1.804829 H -1.117290 -3.467843 0.882099H 2.000135 -3.015708 -0.345691 H -1.117290 -3.467843 -0.882099H -0.628987 -3.441932 1.417803 H 1.818887 -3.324649 -1.146384H 0.874514 -2.672371 1.895896 H 0.764117 -2.385123 -2.171600H -0.646059 -1.804849 2.050416 H 2.104843 -1.613026 -1.333176C -0.976419 3.042868 -0.725947 H 1.818887 -3.324649 1.146384C 0.091157 2.492839 1.420876 H 2.104843 -1.613026 1.333176H 0.885301 2.062998 -0.507479 H 0.764117 -2.385123 2.171600H -0.587979 4.043400 -0.531894 C 0.484721 3.397597 0.000000H -1.014308 2.910437 -1.804829 C -1.321125 2.357646 -1.240916H -2.000135 3.015708 -0.345691 C -1.321125 2.357646 1.240916H 0.628987 3.441932 1.417803 H -0.149364 4.283638 0.000000H -0.874514 2.672371 1.895896 H 1.117290 3.467843 0.882099H 0.646059 1.804849 2.050416 H 1.117290 3.467843 -0.882099

H -1.818887 3.324649 -1.146384H -0.764117 2.385123 -2.171600H -2.104843 1.613026 -1.333176H -1.818887 3.324649 1.146384H -2.104843 1.613026 1.333176H -0.764117 2.385123 2.171600

24C 0.447290 -0.656641 0.057537C -0.447290 0.656641 0.057537C 0.447290 1.905259 0.263056C -1.230424 0.768433 -1.257805C -1.469584 0.626782 1.204974H -1.945090 1.590027 -1.209056H -0.578105 0.948941 -2.110314H -1.800564 -0.138125 -1.460997H -1.968955 1.592102 1.290604H -2.244993 -0.119845 1.039854H -0.998737 0.419470 2.166497C -0.447290 -1.905259 0.263056C 1.230424 -0.768433 -1.257805C 1.469584 -0.626782 1.204974H 1.945090 -1.590027 -1.209056H 0.578105 -0.948941 -2.110314H 1.800564 0.138125 -1.460997H 1.968955 -1.592102 1.290604H 2.244993 0.119845 1.039854H 0.998737 -0.419470 2.166497C -0.191847 3.291594 0.041745H 1.330194 1.841168 -0.377633H 0.820251 1.867967 1.287523C 0.310482 4.279553 1.092920C 0.080910 3.841349 -1.357654H -1.274857 3.213648 0.163883

158

H 0.049173 3.950013 2.099558H 1.398401 4.371851 1.044257H -0.114738 5.272711 0.939717H -0.402899 4.809278 -1.498196H 1.154436 3.982312 -1.505765H -0.277131 3.172808 -2.138570C 0.191847 -3.291594 0.041745H -1.330194 -1.841168 -0.377633H -0.820251 -1.867967 1.287523C -0.310482 -4.279553 1.092920C -0.080910 -3.841349 -1.357654H 1.274857 -3.213648 0.163883H -0.049173 -3.950013 2.099558H -1.398401 -4.371851 1.044257H 0.114738 -5.272711 0.939717H 0.402899 -4.809278 -1.498196H -1.154436 -3.982312 -1.505765H 0.277131 -3.172808 -2.138570

Table 7: Cartesian Coordinates for Group IV - Methyl-Strained Bonds, inangstroms. [A]

159

25 26H -2.238592 -3.382904 0.005979 C -1.477783 0.633679 1.006774H 3.547617 -0.308562 2.476974 C -0.809000 0.155777 -0.328060C -2.172254 -2.292861 0.006700 C 0.808991 -0.156339 -0.327734C 3.219315 -0.187211 1.442663 C 2.490209 -1.630890 -1.622766C -1.167703 1.766391 1.237361 C 1.033342 -1.251084 -1.401976C -1.281547 1.794631 -1.279134 C 1.477803 -0.631713 1.007970C -3.361069 1.874293 0.056738 C 1.686647 1.080018 -0.742711H -0.709645 -0.150753 2.115726 C 2.945454 -1.068344 0.749632H 1.614940 -2.223187 -0.556042 C 3.154997 0.664521 -0.999496H -0.767409 -0.102182 -2.188884 C 3.047783 -2.162493 -0.306095H 1.202342 2.039191 -0.056989 C 3.261659 -0.400529 -2.081362H -4.433767 0.007722 0.041957 C 3.774790 0.145769 0.301901H 4.907142 0.357149 -1.361382 H 2.537766 -2.408992 -2.387555C -1.183103 0.233000 1.211630 H 3.343469 -1.442047 1.697612C 1.850269 -1.222449 -0.908853 H 3.694867 1.561628 -1.317963C -1.227218 0.255202 -1.261982 H 0.501019 -2.157902 -1.123433C 1.644473 1.203915 -0.593587 H 0.611076 -0.912641 -2.351152C -3.394997 0.350943 0.026727 C 1.543318 0.453108 2.099777C 3.817797 0.270523 -1.390827 H 0.950828 -1.505577 1.391444C -2.797512 -1.731802 1.276544 H 1.324507 1.489039 -1.682702C 3.851118 -1.274090 0.571547 C 1.694949 2.182919 0.323478C -2.856862 -1.729673 -1.234987 H 4.090361 -2.470200 -0.422258C 3.641448 1.189633 0.926804 H 2.483098 -3.045387 0.006168C -0.707598 -1.888972 -0.052745 H 2.846661 -0.030352 -3.022888C 1.690968 -0.291714 1.397372 H 4.311670 -0.644519 -2.263451C -2.709453 -0.212859 -1.232759 H 4.801521 -0.178663 0.107891C 3.169959 1.343186 -0.519395 C 3.774703 1.245999 1.354947C -2.662989 -0.216116 1.251020 C -2.945423 1.069774 0.747653C 3.377550 -1.095532 -0.872984 C -1.686626 -1.081419 -0.740653C -0.462392 -0.357232 -0.042802 C -1.033444 1.248405 -1.404461C 1.138346 -0.149818 -0.039154 C -3.155002 -0.666454 -0.998164H -2.291898 -2.143586 2.154381 C -2.490313 1.627767 -1.625873H 3.565578 -2.263107 0.939674 C -3.774786 -0.145193 0.302245H -2.402188 -2.155140 -2.133644 C -3.047769 2.161894 -0.310167H 3.203041 1.973254 1.550670 C -3.261748 0.396499 -2.082077H -3.851656 -2.012678 1.347937 H -3.343429 1.445299 1.694918H 4.941241 -1.212416 0.623205 H -3.694860 -1.564186 -1.314887H -3.916626 -1.997905 -1.250867 H -2.537894 2.404415 -2.392141H 4.727356 1.300854 0.983711 C -1.543223 -0.449082 2.100621H -0.298480 -2.313167 -0.967445 H -0.950770 1.508242 1.388611H 1.279316 0.491734 2.035329 H -1.324496 -1.492247 -1.679861H -0.171688 -2.348077 0.781746 C -1.694905 -2.182311 0.327553H 1.380034 -1.247051 1.827166 H -0.501168 2.155822 -1.127807H -3.201608 0.198435 -2.118652 H -0.611277 0.908108 -2.353022H 3.441257 2.333814 -0.890325 H -4.801524 0.178856 0.107594H -3.110621 0.203653 2.156931 C -3.774696 -1.243388 1.357418H 3.803633 -1.883407 -1.497655 H -2.483033 3.045353 0.000409H -1.669643 2.108662 2.146888 H -4.090334 2.469423 -0.426872H -0.151197 2.151707 1.288653 H -4.311777 0.640101 -2.264542H -0.313374 2.246295 -1.445282 H -2.846815 0.024514 -3.022924H -1.905204 2.100766 -2.124133 C 2.332342 1.666679 1.615145H -3.906690 2.283674 -0.797432 H 2.030669 0.024300 2.980369H -3.853288 2.239437 0.962268 H 0.562976 0.778424 2.422684C -1.903878 2.327115 0.014045 H 0.686063 2.541124 0.529109H -1.851890 3.417478 0.023260 H 2.260519 3.039701 -0.053453H 1.494370 -1.152574 -1.941248 H 4.368017 2.095811 1.006538

160

H 1.351619 1.295214 -1.639987 H 4.232951 0.887214 2.280414H 3.509978 0.396655 -2.432129 H 2.303096 2.448069 2.376945

C -2.332259 -1.663587 1.618350H -2.030487 -0.018640 2.980452H -0.562877 -0.773866 2.424036H -0.686063 -2.540275 0.533840H -2.260603 -3.039688 -0.047799H -4.368032 -2.093844 1.010626H -4.232936 -0.882790 2.282176H -2.302804 -2.443495 2.381644

27 28C -1.903276 0.685390 -1.146067 C -1.892961 -1.025299 0.724949C -1.345416 -0.035100 0.109477 C -1.031742 0.187095 0.177780C 0.303743 -0.030214 0.266012 C 0.619810 -0.029300 -0.122157C 2.206295 1.274902 1.443089 C 2.437756 0.579029 -1.861177C 0.721821 1.263210 1.051916 C 1.016589 0.868556 -1.347039C 1.207078 -0.003756 -1.035806 C 0.962321 -1.533591 -0.412536C 0.780709 -1.309584 1.051313 C 1.693257 0.380422 0.999037C 2.721037 0.047656 -0.615601 C 2.385594 -1.728629 -0.965535C 2.266113 -1.198304 1.460618 C 3.147448 0.123576 0.441616C 3.052852 1.313415 0.170183 C 2.597790 -0.890612 -2.216465C 2.537745 0.051775 2.281269 C 3.448762 0.980386 -0.786131C 3.130215 -1.179634 0.188710 C 3.396648 -1.341224 0.116135H 2.388554 2.181104 2.028619 H 2.595989 1.190376 -2.754694H 3.291974 0.054242 -1.551088 H 2.500550 -2.788226 -1.212082H 2.513775 -2.082337 2.054511 H 3.827051 0.415577 1.249796C 0.456002 2.538314 0.238620 H 0.353687 0.660061 -2.182193H 0.172140 1.307691 1.988639 C 0.953315 2.370834 -1.021822C 1.026338 -1.291883 -1.848849 C 0.864690 -2.440245 0.828735C 1.019410 1.264369 -1.884894 H 0.292707 -1.898523 -1.184915H 0.185155 -1.397982 1.965141 C 1.740047 1.881351 1.347945C 0.702347 -2.634328 0.259444 C 1.556158 -0.505794 2.244289H 4.113954 1.292542 0.436441 H 3.592315 -1.073023 -2.632481C 2.756854 2.556827 -0.658738 H 1.870489 -1.172935 -2.983256H 1.927654 0.044911 3.188938 C 3.351580 2.462382 -0.451211H 3.584715 0.076310 2.595674 H 4.459838 0.748184 -1.134160H 4.181077 -1.079741 0.476833 H 4.411235 -1.450003 -0.279094C 2.954220 -2.416212 -0.681942 C 3.248289 -2.166962 1.386093C -3.437233 0.730633 -1.178930 C -3.344274 -0.570715 1.064551C -1.989477 0.629808 1.358413 C -1.821488 0.588978 -1.121032C -1.986293 -1.448236 0.048940 C -1.202960 1.317111 1.220651C -3.520744 0.652843 1.308617 C -3.260389 1.054426 -0.787238C -3.519503 -1.438898 -0.004459 C -2.636623 1.739328 1.508281C -3.967500 1.433369 0.071206 C -4.061596 -0.118970 -0.215719C -3.993788 -0.688083 -1.244600 C -3.386773 0.552021 2.087752C -4.071971 -0.767986 1.249228 C -3.272369 2.209788 0.205059H -3.739187 1.287946 -2.068206 H -3.857866 -1.446624 1.470864H -3.883855 1.153842 2.208484 H -3.720254 1.377411 -1.726275H -3.857904 -2.476429 -0.047972 H -2.614866 2.559792 2.228973H -1.546245 1.706434 -1.216741 C -2.084903 -2.205105 -0.248073H -1.562106 0.165461 -2.042060 H -1.463717 -1.371875 1.664366H -1.665105 0.099397 2.258272 H -1.344768 1.434999 -1.606504H -1.666571 1.661770 1.464055 C -1.928346 -0.564124 -2.130426H -1.631427 -1.979127 -0.832519 H -0.746480 1.005867 2.158746H -1.702065 -2.022263 0.928400 H -0.681355 2.203837 0.892357H -5.058123 1.496339 0.035061 H -5.067845 0.223951 0.043108H -3.586402 2.457275 0.117135 C -4.150986 -1.248045 -1.233441H -3.638375 -1.189437 -2.148745 H -2.916635 0.235139 3.022707

161

H -5.085883 -0.672785 -1.289466 H -4.423921 0.811191 2.316970H -5.164826 -0.756067 1.228229 H -4.299013 2.546104 0.372206H -3.770148 -1.325734 2.139678 H -2.714511 3.061465 -0.194552C 1.478899 -2.525996 -1.051245 C 1.830750 -1.978456 1.913570H 1.618458 -1.218347 -2.766676 H 1.119041 -3.460039 0.524532H -0.013501 -1.405349 -2.160275 H -0.131808 -2.490549 1.239067H -0.301693 -2.973945 0.069677 H 0.560348 -0.411980 2.680404H 1.162156 -3.411533 0.876758 H 2.264050 -0.161087 3.004793H 3.270917 -3.317119 -0.150518 H 3.987476 -1.842420 2.123567H 3.580668 -2.326780 -1.573748 H 3.431970 -3.225609 1.185007H 1.310129 -3.421992 -1.651812 H 1.684976 -2.580830 2.812471C 1.281916 2.529321 -1.053183 C -2.736625 -1.725100 -1.545786H 0.753940 3.401562 0.840793 H -2.740447 -2.935835 0.234235H -0.599684 2.675024 0.027777 H -1.169213 -2.734312 -0.469869H 0.048281 1.303798 -2.360569 H -0.943841 -0.910855 -2.441001H 1.747280 1.228308 -2.701967 H -2.423231 -0.195078 -3.033346H 3.390411 2.570617 -1.549408 H -4.648774 -0.895679 -2.141026H 2.982258 3.459469 -0.084543 H -4.747272 -2.072571 -0.834099H 1.039240 3.411209 -1.649253 H -2.769675 -2.546917 -2.263537

C 1.949084 2.733629 0.086042H -0.050987 2.692027 -0.765406H 1.219227 2.926872 -1.925651H 2.599368 2.032398 2.008903H 0.883719 2.214254 1.917579H 3.545823 3.067354 -1.340924H 4.105443 2.727410 0.294526H 1.844040 3.790276 0.339702

29 30C -1.902500 0.282325 -1.028258 C -1.884639 -1.367359 0.239972C -0.900219 -0.091707 0.160172 C -1.394572 0.135032 0.349462C 0.900246 -0.091712 -0.160396 C 0.300359 0.474734 0.226220C 2.777589 0.749042 -1.765065 C 2.609043 -0.371327 1.087441C 1.307122 0.914231 -1.304854 C 1.114662 -0.727515 0.825310C 1.402404 -1.543362 -0.559881 C 0.595286 1.775889 1.074015C 1.902346 0.282424 1.028170 C 1.012188 0.788810 -1.159266C 2.868370 -1.595385 -1.044084 C 2.088770 2.050676 1.286988C 3.404133 0.173467 0.534606 C 2.533369 1.107572 -0.910184C 3.080997 -0.663291 -2.218872 C 2.733107 0.860318 1.969435C 3.711327 1.133845 -0.608106 C 3.262729 -0.065276 -0.274131C 3.794626 -1.231440 0.118811 C 2.739870 2.351135 -0.057810H 2.928779 1.438204 -2.599998 C 3.335308 -1.553589 1.735267H 3.059547 -2.624153 -1.360938 H 2.166671 2.928606 1.935892H 4.009120 0.456631 1.403029 H 2.961977 1.286888 -1.902715H 0.681118 0.731181 -2.177781 H 0.673208 -1.005714 1.788615C 1.234011 2.401355 -0.925045 C 1.124538 -1.973524 -0.086701C 1.381039 -2.599824 0.571616 C -0.011354 3.034558 0.443883H 0.822802 -1.872226 -1.415121 H 0.217151 1.651666 2.079353C 1.818675 1.743342 1.486023 C 1.054190 -0.425307 -2.093840C 1.773199 -0.714952 2.187680 C 0.451644 2.057501 -1.825170H 4.107634 -0.734236 -2.588467 H 3.790213 1.058946 2.171560H 2.419470 -0.945388 -3.043019 H 2.251307 0.675174 2.935088C 3.533088 2.567275 -0.132769 C 3.237219 -1.291990 -1.180184H 4.747688 0.980085 -0.923176 H 4.305363 0.224440 -0.096714H 4.831710 -1.225024 -0.229651 H 3.813229 2.516115 0.075887C 3.663570 -2.150640 1.323374 C 2.108669 3.568137 -0.720623C -3.404152 0.173049 -0.534484 C -3.348863 -1.527659 0.748522C -1.307127 0.914228 1.304749 C -2.396624 0.866690 -0.619846C -1.402114 -1.543315 0.559791 C -1.732897 0.579671 1.803756

162

C -2.777357 0.748857 1.765146 C -3.859362 0.736589 -0.097149C -2.868058 -1.595640 1.044147 C -3.171237 0.394129 2.254204C -3.711463 1.133363 0.608363 C -4.303868 -0.731998 -0.152813C -3.794375 -1.231941 -0.118717 C -3.530015 -1.078468 2.186576C -3.080594 -0.663515 2.218970 C -4.042387 1.237363 1.329446H -4.009359 0.456144 -1.402778 H -3.589985 -2.591625 0.669807H -2.928617 1.438054 2.600057 H -4.489245 1.325286 -0.770565H -3.058964 -2.624417 1.361114 H -3.257471 0.758457 3.280273C -1.819488 1.743339 -1.486084 C -1.917334 -1.952984 -1.185343C -1.773233 -0.714973 -2.187790 H -1.274286 -1.992100 0.891789H -0.680856 0.731269 2.177476 H -2.173864 1.928761 -0.665627C -1.234310 2.401368 0.924734 C -2.415321 0.314728 -2.055374C -1.380476 -2.599803 -0.571667 H -1.071658 0.056296 2.498316H -0.822439 -1.871921 1.415056 H -1.545162 1.640038 1.914795H -4.747669 0.979177 0.923673 H -5.317989 -0.809490 0.250202C -3.533706 2.566820 0.132936 C -4.278326 -1.262875 -1.576780C -3.663156 -2.151097 -1.323312 H -2.878674 -1.650198 2.853368H -4.831475 -1.225744 0.229708 H -4.560091 -1.254053 2.507781H -4.107168 -0.734519 2.588726 H -5.096536 1.169450 1.610496H -2.418928 -0.945644 3.042988 H -3.753689 2.288976 1.409892C 2.205991 -2.128214 1.764906 C 0.622958 3.287563 -0.925570H 1.831406 -3.512417 0.171010 H 0.193702 3.886442 1.098763H 0.403144 -2.893852 0.901949 H -1.092238 2.971007 0.363218H 0.749133 -0.737299 2.563687 H -0.592848 1.953049 -2.088783H 2.401960 -0.378150 3.018024 H 0.990091 2.220235 -2.764491H 4.328197 -1.803803 2.119485 H 2.595168 3.764206 -1.679839H 3.958518 -3.172942 1.074077 H 2.246443 4.456260 -0.098027H 2.058573 -2.811895 2.603249 H 0.144564 4.146557 -1.400231C -2.084518 2.711533 -0.313770 C -2.850283 -1.147999 -2.087504H -2.599402 1.900901 -2.237559 H -2.266883 -2.987458 -1.120705H -0.889179 1.956659 -1.994562 H -0.940587 -1.997705 -1.645847H -0.231769 2.758674 0.815211 H -1.450142 0.409625 -2.535905H -1.641352 2.971447 1.764715 H -3.111214 0.918566 -2.645215H -3.757418 3.276255 0.933408 H -4.960057 -0.687733 -2.209081H -4.225813 2.770354 -0.688411 H -4.611427 -2.303929 -1.598807H -1.890137 3.735184 -0.642045 H -2.782895 -1.526367 -3.109496C 2.083681 2.711683 0.313925 C 1.763929 -1.618396 -1.436366H 0.231398 2.758483 -0.816208 H 0.122071 -2.309359 -0.263416H 1.641416 2.971601 -1.764700 C 1.852296 -3.144074 0.575233H 2.598323 1.901295 2.237706 H 1.598762 -0.143191 -3.000788H 0.888105 1.956312 1.994180 H 0.062890 -0.711622 -2.422626H 3.756566 3.276649 -0.933319 C 3.969713 -2.455222 -0.512900H 4.225007 2.770997 0.688690 H 3.726429 -1.044438 -2.126984H 1.889262 3.735311 0.642125 H 1.691828 -2.485721 -2.100183C -2.205555 -2.128365 -1.764931 C 3.308278 -2.780888 0.828591H -0.749212 -0.737026 -2.563935 H 2.871287 -1.786784 2.698814H -2.402257 -0.378497 -3.018067 H 4.368511 -1.257988 1.942538H -1.830619 -3.512532 -0.171125 H 1.784976 -4.023225 -0.071747H -0.402468 -2.893630 -0.901936 H 1.354092 -3.399284 1.515088H -4.327892 -1.804337 -2.119366 H 5.019510 -2.189261 -0.361740H -3.957908 -3.173462 -1.074054 H 3.950438 -3.335287 -1.161427H -2.058040 -2.811960 -2.603324 H 3.830809 -3.610405 1.308826

Table 8: Cartesian Coordinates for Group V - Diamondoids, in angstroms. [A]

163

31 32C 1.284057 0.150620 0.000000 C 0.000000 0.000000 0.660897C 0.000000 0.472385 0.000000 C 0.000000 0.000000 -0.660897H 1.607135 -0.883964 0.000000 H 0.000000 0.921828 1.228238H -0.275567 1.522643 0.000000 H 0.000000 -0.921828 -1.228238H 2.056628 0.907581 0.000000 H 0.000000 -0.921828 1.228238C -1.131482 -0.504301 0.000000 H 0.000000 0.921828 -1.228238H -0.770111 -1.532283 0.000000H -1.766767 -0.363100 0.876728H -1.766767 -0.363100 -0.876728

33 34C 0.603789 1.736136 0.000000 C -1.715666 -0.248327 -0.291114C 0.603789 0.407323 0.000000 C -0.537413 0.523081 0.303126H -0.322167 2.298560 0.000000 C 0.717240 -0.292743 0.338173H 1.547774 -0.128675 0.000000 H -2.624610 0.352907 -0.276372H 1.524635 2.302211 0.000000 H -1.909234 -1.161876 0.273271C -0.603789 -0.407323 0.000000 H -1.510833 -0.532277 -1.323572C -0.603789 -1.736136 0.000000 H -0.791740 0.829373 1.321809H 0.322167 -2.298560 0.000000 H -0.362887 1.437512 -0.267000H -1.547774 0.128675 0.000000 C 1.848116 0.016165 -0.276766H -1.524635 -2.302211 0.000000 H 2.718495 -0.623351 -0.217836

H 0.664684 -1.215796 0.909921H 1.942466 0.924449 -0.860737

35 36C 1.168988 1.259050 0.029288 C -0.351356 0.001495 0.000000C 0.369509 0.000011 -0.312058 C 1.003747 -0.662920 0.000000C -0.945791 -0.000016 0.410022 C 2.191389 -0.078180 0.000000C -2.138305 -0.000026 -0.163958 C -1.112004 -0.463213 -1.250470C 1.169023 -1.259015 0.029257 C -1.112003 -0.463213 1.250470H 0.620002 2.160119 -0.243315 C -0.256031 1.525285 0.000000H 1.378085 1.303045 1.100622 H -2.121147 -0.047091 -1.259580H 2.124373 1.265301 -0.497268 H -0.600148 -0.142017 -2.158354H 1.378121 -1.303031 1.10059 H -1.196274 -1.551079 -1.276381H 0.620061 -2.160093 -0.243368 H -2.121146 -0.047091 1.259580H 2.124408 -1.265227 -0.497299 H -1.196274 -1.551078 1.276381H -3.048925 -0.000045 0.419806 H -0.600147 -0.142016 2.158354H -0.880528 -0.000029 1.496262 H -1.255834 1.961773 0.000000H -2.244601 -0.000014 -1.242685 H 0.268602 1.890565 0.883985H 0.166257 0.000021 -1.386288 H 0.268602 1.890564 -0.883986

H 3.099499 -0.665766 0.000000H 0.963919 -1.750045 0.000000H 2.307895 0.997757 0.000000

37 38C 0.000000 0.000000 0.596938 C 0.000000 0.000000 1.415247C 0.000000 0.000000 -0.596938 C 0.000000 0.000000 0.218935H 0.000000 0.000000 1.659443 H 0.000000 0.000000 2.477031H 0.000000 0.000000 -1.659443 C 0.000000 0.000000 -1.236320H -0.112686 H 0.000000 1.019386 -1.621399H 1.316233 H 0.882814 -0.509693 -1.621399H -0.172816 H -0.882814 -0.509693 -1.621399

39 40C 0.000000 0.000000 -1.173554 C 0.790342 -1.487847 0.000000C 0.000000 0.000000 0.280952 C -0.313675 -0.748588 0.000000

164

H 0.000000 1.022902 -1.544894 H 1.772628 -1.035934 0.000000H 0.885859 -0.511451 -1.544894 H -1.286377 -1.226417 0.000000H -0.885859 -0.511451 -1.544894 H 0.731517 -2.566972 0.000000N 0.000000 0.000000 1.427185 C -0.313675 0.676608 0.000000

C -0.313675 1.875179 0.000000H -0.313675 2.937215 0.000000

41 42C 0.801389 -1.420218 0.000000 C 0.000000 0.000000 1.885014C -0.318682 -0.709166 0.000000 C 0.000000 0.000000 0.686011H 1.773288 -0.946925 0.000000 H 0.000000 0.000000 2.947280H -1.290383 -1.184525 0.000000 C 0.000000 0.000000 -0.686011H 0.763716 -2.500087 0.000000 C 0.000000 0.000000 -1.885014C -0.318682 0.720494 0.000000 H 0.000000 0.000000 -2.947280N -0.318682 1.869269 0.000000

43C 0.00000 0.00000 -1.830293C 0.00000 0.00000 -0.634306H 0.00000 0.00000 -2.893626C 0.00000 0.00000 0.740968N 0.00000 0.00000 1.890774

Table 9: Cartesian Coordinates for Group IV - Multiple Bonds, in angstroms.[A]

165

44 45C 0.005895 0.907219 0.000000 C -0.433050 -0.000051 -0.333270C 0.005895 0.193312 1.194198 C 0.265053 -1.194491 -0.187670C 0.005895 0.193312 -1.194198 C 0.264977 1.194444 -0.187736C 0.005895 -1.192938 1.197170 C 1.622092 -1.197680 0.095279C 0.005895 -1.192938 -1.197170 C 1.622014 1.197736 0.095213C 0.005895 -1.892108 0.000000 C 2.305724 0.000054 0.238412C -0.024215 2.410576 0.000000 C -1.916552 -0.000099 -0.591332H 0.010429 0.730728 2.134555 H -0.261692 -2.134406 -0.301145H 0.010429 0.730728 -2.134555 H -0.261829 2.134319 -0.301263H 0.009263 -1.728247 2.137229 H 2.147601 -2.137332 0.200965H 0.009263 -1.728247 -2.137229 H 2.147463 2.137428 0.200847H 0.008999 -2.973461 0.000000 H 3.364918 0.000094 0.456405H -1.052882 2.776593 0.000000 H -2.182549 0.875704 -1.185877H 0.468779 2.816647 0.882559 H -2.182533 -0.876070 -1.185637H 0.468779 2.816647 -0.882559 C -2.727405 0.000073 0.705411

H -2.495349 0.880566 1.305455H -3.797813 0.000036 0.498713H -2.495334 -0.880250 1.305698

46 47C -0.294704 -1.663183 0.000000 C -0.516640 1.358042 0.000000H -1.359414 -1.908943 0.000000 C -0.002336 2.081754 1.253974C 0.321996 -2.271653 -1.260938 C -0.002336 2.081754 -1.253974C 0.321996 -2.271653 1.260938 C -2.044967 1.431245 0.000000H 0.181978 -3.353450 -1.272806 H 1.086618 2.108823 1.286170H -0.132166 -1.856193 -2.160363 H -0.362975 3.111916 1.269064H 1.394511 -2.074293 -1.305759 H -0.352505 1.583321 2.158712H 0.181978 -3.353450 1.272806 H -0.352505 1.583321 -2.158712H 1.394511 -2.074293 1.305759 H -0.362975 3.111916 -1.269064H -0.132166 -1.856193 2.160363 H 1.086618 2.108823 -1.286170C -0.187633 -0.153035 0.000000 H -2.472492 0.958016 0.884913C -1.328729 0.640301 0.000000 H -2.358560 2.475823 0.000000C 1.052790 0.480920 0.000000 H -2.472492 0.958016 -0.884913C -1.240078 2.025137 0.000000 C -0.002336 -0.082962 0.000000C 1.148103 1.862400 0.000000 C -0.847634 -1.187374 0.000000C 0.000000 2.641645 0.000000 C 1.372296 -0.326587 0.000000H -2.303243 0.167337 0.000000 C -0.344057 -2.482044 0.000000H 1.959245 -0.111534 0.000000 C 1.880065 -1.613317 0.000000H -2.142650 2.621492 0.000000 C 1.021081 -2.702842 0.000000H 2.121894 2.333701 0.000000 H -1.919326 -1.053871 0.000000H 0.073532 3.720547 0.000000 H 2.065731 0.504170 0.000000

H -1.029055 -3.319459 0.000000H 2.951006 -1.766578 0.000000H 1.414105 -3.710257 0.000000

48 49C 1.576778 0.622655 -0.000014 C 0.509246 -0.219839 -0.042511C 2.583326 -0.213419 -0.747042 C -0.401788 -1.273536 0.00193C 2.583314 -0.213364 0.747092 C 0.009189 1.083007 -0.059739H 1.767918 1.687407 -0.000052 C -1.767094 -1.039557 0.043295H 3.381415 0.301746 -1.261449 C -1.352708 1.319416 -0.019247H 2.212072 -1.091532 -1.256739 C -2.247887 0.259324 0.034876H 3.381395 0.301839 1.261473 C 1.948577 -0.522904 -0.070612H 2.212051 -1.091438 1.256848 H -0.032112 -2.291368 0.00859C 0.132306 0.2759 -0.000011 H 0.689469 1.921834 -0.117028C -0.318536 -1.043056 0.000027 H -2.454913 -1.873343 0.08063

166

C -0.818043 1.292696 -0.000045 H -1.720435 2.336505 -0.035769C -1.672514 -1.332737 0.00003 H -3.31237 0.447212 0.064385C -2.174495 1.006156 -0.000042 C 2.94854 0.334154 0.094592C -2.608825 -0.309279 -0.000004 H 2.19025 -1.568535 -0.232406H 0.392327 -1.860054 0.000054 H 2.788323 1.388191 0.280757H -0.489528 2.324557 -0.000075 H 3.975333 -0.000887 0.055333H -1.998542 -2.364263 0.00006H -2.893018 1.814838 -0.000069H -3.665954 -0.536406 -0.000001

50 51C 0.00000 0.00000 0.585379 C 0.602454 0.000002 0.000000C 0.00000 1.20427 -0.119284 C -0.090406 -1.208569 0.000000C 0.00000 -1.20427 -0.119284 C -0.090406 1.208573 0.000000C 0.00000 1.20047 -1.503159 C -1.473873 -1.202675 0.000000C 0.00000 -1.20047 -1.503159 C -1.473873 1.202678 0.000000C 0.00000 0.00000 -2.198026 C -2.165126 0.000002 0.000000C 0.00000 0.00000 2.015714 C 2.035894 -0.000001 0.000000H 0.00000 2.13685 0.427055 H 0.457971 -2.139570 0.000000H 0.00000 -2.13685 0.427055 H 0.457971 2.139574 0.000000H 0.00000 2.13829 -2.041352 H -2.013710 -2.138998 0.000000H 0.00000 -2.13829 -2.041352 H -2.013710 2.139002 0.000000H 0.00000 0.00000 -3.279335 H -3.246338 0.000002 0.000000C 0.00000 0.00000 3.213828 N 3.184262 -0.000010 0.000000H 0.00000 0.00000 4.275863

52C 0.000000 0.000000 0.740768C -0.423302 1.119580 1.454445C 0.423302 -1.119580 1.454445C -0.423517 1.120306 2.839632C 0.423517 -1.120306 2.839632C 0.000000 0.000000 3.537953H -0.775830 1.990725 0.918094H 0.775830 -1.990725 0.918094H -0.763059 1.996515 3.375233H 0.763059 -1.996515 3.375233H 0.000000 0.000000 4.619288C 0.000000 0.000000 -0.740768C -0.423302 -1.119580 -1.454445C 0.423302 1.119580 -1.454445C -0.423517 -1.120306 -2.839632C 0.423517 1.120306 -2.839632C 0.000000 0.000000 -3.537953H -0.775830 -1.990725 -0.918094H 0.775830 1.990725 -0.918094H -0.763059 -1.996515 -3.375233H 0.763059 1.996515 -3.375233H 0.000000 0.000000 -4.619288

Table 10: Cartesian Coordinates for Group VII - Aromatic Bonds, in angstroms.[A]

167

12

3

4

5

67

89

1011

16

17

20

2223

24

26

Single Bonds e- DeficientCH3- StrainDiamondoid

Bond

Dis

soci

atio

n En

thal

py (e

xper

imen

tal)

[kc

al/m

ol]

40

50

60

70

80

90

Bond Dissociation Enthalpy (G4) [kcal/mol]50 55 60 65 70 75 80 85 90

Figure 1: Correlation of Experimental Bond Dissociation Enthalpy to Bond Dissociationscalculated by Method G4, kcal/mol.

168

6

8

9

10

11

13

14

15

17 2223

24

25 26

27

2830

3242

43

Single BondsClamped Bondse- DeficientCH3- StrainDiamondoidMultiple BondsAromatic Bonds

Expe

rimen

tal B

ond

Leng

th

r CC

,exp

[Å]

1.3

1.4

1.5

1.6

1.7

1.8

Calculated Bond Length rCC,calc [Å]1.3 1.4 1.5 1.6 1.7

Figure 2: Correlation of Experimental Bond Length to Bond Length calculated by MethodωB97X-D, A.

169

Pancake Bonds - Supporting Information

Alan Humason, Dieter Cremer, and Elfi Kraka⇤





170

Key words: long CC bonds, pancake bonds, Cremer-Kraka bond criteria, atoms-in-molecules,

bond critical point analysis, local mode analysis, bond strength order, aromaticity index,

1. Introduction

This work continues our quest for the longest covalent CC bonds.1

This document includes graphic representations of the bond critical point (BCP), electron

density (⇢r), and energy density (Hb) calculations on all systems.2

Also included are cartesian coordinates (A) of the optimized geometries for all species.

References

(1) Humason, A.; Zou, W.; Cremer, D. 11,11-Dimethyl-1,6-methano[10]annulene - An An-

nulene with an Ultralong CC Bond or a Fluxional Molecule? J. Phys. Chem A 2015,

119, 1666–1682.

(2) Keith, T. A. AIMAll (Version 17.01.25); TK Gristmill Software, Overland Park, KS,

USA, 2017.

171

Table 1: Cartesian Coordinates for the 1,2,3,5-Dichalcodiazolyl Radical Dimers1 through 3, in angstroms. [A]

HCNSSN HCNSeSeNC 1.517826 -0.000008 1.495361 C -1.559826 0.000003 1.866395N 1.516829 -1.173422 0.884924 N -1.575798 1.196906 1.306577N 1.516766 1.173421 0.884951 N -1.576149 -1.196863 1.306507S 1.562648 -1.078491 -0.748250 Se -1.656089 1.180793 -0.477202S 1.562590 1.078532 -0.748225 Se -1.656435 -1.180622 -0.477271H 1.495604 -0.000021 2.582706 H -1.514877 -0.000036 2.958512C -1.517833 0.000002 1.495339 C 1.559602 -0.000026 1.866440N -1.516850 1.173433 0.884885 N 1.575760 -1.196940 1.306648N -1.516839 -1.173439 0.884905 N 1.575828 1.196863 1.306596S -1.562601 1.078597 -0.748248 Se 1.656303 -1.180591 -0.477125S -1.562591 -1.078631 -0.748230 Se 1.656371 1.180433 -0.477176H -1.495622 0.000011 2.582680 H 1.513631 -0.000001 2.958502

HCNTeTeNC -1.609662 0.000029 2.150596N -1.666551 1.228122 1.597394N -1.666754 -1.228053 1.597392Te -1.919837 1.428563 -0.370206Te -1.920074 -1.428448 -0.370208H -1.492464 0.000019 3.233692C 1.609594 -0.000055 2.150529N 1.666567 -1.228176 1.597402N 1.666626 1.228068 1.597411Te 1.919933 -1.428507 -0.370195Te 1.920002 1.428402 -0.370185H 1.492357 -0.000056 3.233629

172

Phenalenyl (4) tTMP (5)C -0.00293 -0.001428 1.54638 C 0 0 3.999945C -1.382192 -0.365856 1.543435 C 1.232695 0.711697 4.00309C 0.371079 1.375269 1.546578 C -1.232695 0.711697 4.00309C 1.002304 -1.013694 1.545574 C 0.000000 -1.423394 4.00309C -2.359919 0.658317 1.543119 C -1.202815 2.12692 4.007075C -1.988548 1.997991 1.565692 C 0.000000 2.833064 4.004495C -0.646245 2.360107 1.546197 C 1.202815 2.12692 4.007075C 1.74691 1.709918 1.548338 C -1.240559 -2.105129 4.007075C 2.721402 0.718428 1.569917 C -2.453505 -1.416532 4.004495C 2.363869 -0.625078 1.547363 C -2.443375 -0.021791 4.007075C 0.604211 -2.372516 1.544202 C 2.443375 -0.021791 4.007075C -0.741715 -2.720735 1.563709 C 2.453505 -1.416532 4.004495C -1.726422 -1.739313 1.542117 C 1.240559 -2.105129 4.007075H -3.410147 0.379243 1.540869 H -2.145474 2.669815 4.012381H -0.359855 3.408366 1.546354 H 2.145474 2.669815 4.012381H 2.030341 2.758981 1.548482 H -1.239391 -3.192942 4.012381H 3.128495 -1.397225 1.546751 H -3.384864 0.523127 4.012381H 1.371013 -3.142503 1.543584 H 3.384864 0.523127 4.012381H -2.777432 -2.015422 1.539879 H 1.239391 -3.192942 4.012381H -2.753712 2.768629 1.572239 C 0.000000 4.341439 3.970459H 3.771366 0.995753 1.578092 C -3.759797 -2.17072 3.970459H -1.026542 -3.768705 1.569491 C 3.759797 -2.17072 3.970459C 0.002924 0.001418 -1.547408 H 0.000000 4.704936 2.937377C -1.002325 1.013699 -1.548662 H -0.885696 4.747523 4.465769C 1.382206 0.36585 -1.544747 H 0.885696 4.747523 4.465769C -0.371109 -1.375293 -1.549667 H -4.074594 -2.352468 2.937377C -0.604157 2.372563 -1.546652 H -3.668627 -3.140797 4.465769C 0.741763 2.720794 -1.557505 H -4.554324 -1.606726 4.465769C 1.726392 1.73938 -1.542825 H 4.074594 -2.352468 2.937377C 2.359932 -0.658409 -1.543834 H 4.554324 -1.606726 4.465769C 1.988579 -1.998085 -1.559489 H 3.668627 -3.140797 4.465769C 0.646309 -2.360134 -1.548641 C 0.000000 0.000000 -3.999945C -1.747003 -1.709899 -1.552572 C -1.232695 -0.711697 -4.00309C -2.721491 -0.718414 -1.567222 C 0.000000 1.423394 -4.00309C -2.363929 0.625009 -1.55159 C 1.232695 -0.711697 -4.00309H -1.370907 3.142592 -1.547595 C -1.240559 2.105129 -4.007075H 2.77738 2.015551 -1.540814 C -2.453505 1.416532 -4.004495H 3.41017 -0.379398 -1.541791 C -2.443375 0.021791 -4.007075H 0.359988 -3.408402 -1.55038 C 2.443375 0.021791 -4.007075H -2.030489 -2.758937 -1.554275 C 2.453505 1.416532 -4.004495H -3.128594 1.397109 -1.552557 C 1.240559 2.105129 -4.007075H 1.026997 3.768618 -1.547995 C -1.202815 -2.12692 -4.007075H 2.754143 -2.768285 -1.539714 C 0.000000 -2.383064 -4.004495H -3.771578 -0.995302 -1.561458 C 1.202815 -2.12692 -4.007075

H -1.239391 3.192942 -4.012381H -3.384864 -0.523127 -4.012381H 3.384864 -0.523127 -4.012381H 1.239391 3.192942 -4.012381H -2.145474 -2.669815 -4.012381H 2.145474 -2.669815 -4.012381C -3.759797 2.17072 -3.970459C 3.759797 2.17072 -3.970459C 0.000000 -4.431439 -3.97459H -4.074594 2.352468 -2.937377H -3.668627 3.140797 -4.465769H -4.554324 1.606726 -4.465769

173

H 4.074594 2.352468 -2.937377H 4.554324 1.606726 -4.465769H 3.668627 3.140797 -4.465769H 0.000000 -4.74936 -2.937377H -0.885696 -4.747523 -4.465769H 0.885696 -4.747523 -4.465769

tTBP (6)C 0.000000 0.000000 1.557471C -1.229488 -0.709845 1.561773C 1.229488 -0.709845 1.561773C 0.000000 1.419691 1.561773C 1.199055 -2.125381 1.588032C 0.000000 -2.840430 1.649575C -1.199055 -2.125381 1.588032C 1.241106 2.101102 1.588032C 2.459885 1.420215 1.649575C 2.440161 0.024278 1.588032C -2.440161 0.024279 1.588032C -2.459884 1.420215 1.649575C -1.241106 2.101103 1.588032H 2.150814 -2.648135 1.619508H -2.150815 -2.648135 1.619508H 1.217945 3.186727 1.619508H 3.368759 -0.538593 1.619508H -3.368759 -0.538592 1.619508H -1.217944 3.186728 1.619508C 0.000000 -4.364675 1.829448C 3.779919 2.182337 1.829448C -3.779919 2.182337 1.829448C 1.410201 -4.954305 1.927031C -0.748990 -4.707841 3.130854C -0.720372 -5.032970 0.648146C 3.585453 3.698423 1.927031C 4.451605 1.705276 3.130854C 4.718866 1.892624 0.648146C -4.995655 1.255882 1.927031C -3.702615 3.002565 3.130854C -3.998494 3.140346 0.648146H 1.338424 -6.037889 2.064873H 1.990422 -4.771944 1.017465H 1.963280 -4.545068 2.778621H -0.751031 -5.791753 3.291976H -0.266413 -4.230989 3.989997H -1.788693 -4.369343 3.097789H -0.757614 -6.119013 0.788933H -1.747398 -4.668420 0.554151H -0.198048 -4.827160 -0.292658H 4.559753 4.178053 2.064873H 3.137414 4.109728 1.017465H 2.954504 3.972785 2.778621H 5.391321 2.245465 3.291976H 3.797351 1.884774 3.989997H 4.678308 0.635618 3.097789H 5.678028 2.403394 0.788933H 4.916669 0.820919 0.554151H 4.279467 2.242065 -0.292658H -5.898177 1.859835 2.064873H -5.127836 0.662216 1.017465

174

H -4.917785 0.572283 2.778621H -4.640290 3.546289 3.291976H -3.530938 2.346215 3.989997H -2.889615 3.733725 3.097789H -4.920414 3.715619 0.788933H -3.169271 3.847501 0.554151H -4.081419 2.585095 -0.292658C 0.000000 0.000000 -1.557471C 1.229488 0.709845 -1.561773C 0.000000 -1.419691 -1.561773C -1.229488 0.709845 -1.561773C 2.440161 -0.024278 -1.588032C 2.459885 -1.420215 -1.649575C 1.241106 -2.101102 -1.588032C -1.241106 -2.101103 -1.588032C -2.459884 -1.420215 -1.649575C -2.440161 -0.024279 -1.588032C -1.199055 2.125381 -1.588032C 0.000000 2.840430 -1.649575C 1.199055 2.125381 -1.588032H 3.368759 0.538593 -1.619508H 1.217945 -3.186727 -1.619508H -1.217944 -3.186728 -1.619508H -3.368759 0.538592 -1.619508H -2.150815 2.648135 -1.619508H 2.150814 2.648135 -1.619508C 3.779919 -2.182337 -1.829448C -3.779919 -2.182337 -1.829448C 0.000000 4.364675 -1.829448C 4.995655 -1.255883 -1.927031C 3.702615 -3.002565 -3.130854C 3.998493 -3.140346 -0.648146C -3.585454 -3.698423 -1.927031C -4.451605 -1.705276 -3.130854C -4.718866 -1.892624 -0.648146C -1.410201 4.954306 -1.927031C 0.748990 4.707841 -3.130854C 0.720373 5.032969 -0.648146H 5.898177 -1.859837 -2.064873H 4.917786 -0.572284 -2.778621H 5.127837 -0.662217 -1.017465H 4.640289 -3.546289 -3.291976H 2.889614 -3.733724 -3.097789H 3.530938 -2.346215 -3.989997H 4.920413 -3.715620 -0.788933H 4.081419 -2.585095 0.292658H 3.169270 -3.847501 -0.554151H -4.559754 -4.178053 -2.064873H -2.954505 -3.972785 -2.778621H -3.137415 -4.109728 -1.017465H -5.391321 -2.245463 -3.291976H -4.678307 -0.635617 -3.097789H -3.797350 -1.884774 -3.989997H -5.678028 -2.403393 -0.788933H -4.279467 -2.242065 0.292658H -4.916668 -0.820918 -0.554151H -1.338423 6.037890 -2.064873H -1.963280 4.545069 -2.778621H -1.990422 4.771946 -1.017465

175

H 0.751032 5.791753 -3.291976H 1.788693 4.369342 -3.097789H 0.266413 4.230989 -3.989997H 0.757615 6.119013 -0.788933H 0.198048 4.827160 0.292658H 1.747398 4.668419 -0.554151

Table 2: Cartesian Coordinates for the Phenalenyl, tri-Methylphenalenyl, and,tri-tert-Butylphenalenyl Radical Dimers (4, 5 and 6), in angstroms. [A]

176

NSSN RCP

N-N RCP

10-atom CCP

C-C BCP

NCNNCN RCP

S-S BCP

S-S BCP

Elec

tron

Den

sity,

Inte

r-Rin

g

[e/Å

3 ]

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0.100

0.110

C,C Interatomic Distance [Å]2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

(a)

C-C BCP

N-N BCPS-S BCP

S-S BCP

10-atom CCP

NSSN RCPNCNNCN RCP

Ener

gy D

ensi

ty, In

ter-R

ing

[Har

trees

/Å3 ]

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011


(b)

Figure 1: Critical Point Analysis for the 1,2,3,5-Dithiadiazolyl Radical Dimer (1) (BS-UM06/6-311G(d,p).) a) Electron Density. b) Energy Density. (BCP = bond critical point,RCP = ring critcal point, CCP = cage critical point.)

177

NSeSeN RCP

N-N RCP

10-atom CCP

C-C BCP

NCNNCN RCP

Se-Se BCPEl

ectro

n D

ensi

ty, In

ter-R

ing

[e

/Å3 ]

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0.100

0.110


(a)

10-atom CCP

N-N BCP

NSeSeN RCP

C-C BCP

Se-Se BCP

NCNNCN RCP

Ener

gy D

ensi

ty, In

ter-R

ing

[Har

trees

/Å3 ]

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

0.011


(b)

Figure 2: Critical Point Analysis for the 1,2,3,5-Diselenadiazolyl Radical Dimer (2) (BS-UM06/6-311G(d,p).) a) Electron Density. b) Energy Density. (BCP = bond critical point,RCP = ring critcal point, CCP = cage critical point.)

178

NTeTeN RCP

N-N BCP

10-atom CCP

C-C BCP

Te-Te BCP

NCNNCN RCP

Elec

tron

Den

sity,

Inte

r-Rin

g

[e/Å3 ]

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0.100

C,C Interatomic Distance2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0

(a)

NTeTeN RCP

10-atom CCP

N-N BCP

C-C BCP

Te-Te BCP

NCNNCN RCP

Ener

gy D

ensi

ty, In

ter-R

ing

[Har

trees

/Å3 ]

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

0.011

0.012

0.013


(b)

Figure 3: Critical Point Analysis for the 1,2,3,5-Ditelluradiazolyl Radical Dimer (3) (BS-UM06/SDD.) a) Electron Density. b) Energy Density. (BCP = bond critical point, RCP =ring critcal point, CCP = cage critical point.)

179

CH3

H3CH3C

H3CCH3

CH3

Elec

tron

Den

sity,

Inte

r-Rin

g

[e/Å

3 ]

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22


(a)

CH3

H3CH3C

H3CCH3

CH3

Ener

gy D

ensi

ty, In

ter-R

ing

[Har

trees

/Å3 ]

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

0.011

0.012


(b)

Figure 4: Critical Point Analysis for the Phenalenyl, tri-Methylphenalenyl and tri-tert-Butylphenalenyl Radical Dimers (4 and 5.) Values at the bond critical point (BCP) betweenthe central carbon atoms (BS-UM06-2X/6-31++G(d,p).). a) Electron Density. b) EnergyDensity.

180

Supplemental Information - A Study of Various

Multi-Reference Methods for the

Characterization of 30 Transistion Metal Diatoms

Alan Humason, Dieter Cremer, and Elfi Kraka⇤





181

Literature Search

Tables 1 through 10 list the reported experimental results and multireference calculations

published for the 30 species in the transition metal portion of the periodic table. The results

from this work are also reported.

Computational Methods

The 18 valence orbitals for the 30 species in the transition metal diatom portion of the

periodic table were calculated at the HF/aug-cc-pVQZ level of theory. All are depicted in

Figures 1 through 10.

The multireference method calculations reported here are complete active space self-

consistent field calculations with non-dynamical correlation by second order perturbation

(CASPT2),1–8 restricted active space PT2 (RASPT2),9 N-Electron Valence PT2 (NEVPT2),10,11

multireference Equation-of-Motion Coupled Cluster (MR-EOM-CC),12 Generalized Van Vleck

PT2 (GVVPT2)13,14 and multireference configuration interaction (MRCI)8,12,15–19 Bench-

mark Full Configuration Interaction (FCI) calculations were conducted for C2 and N2.20–22

Results reported from this study are complete active space self-consistent field multirefer-

ence methods. These include average quadrature coupled cluster calculations (CASSCF/MR-

AQCC),23 complete active space self-consistent field calculations with non-dynamical corre-

lation by second order perturbation (CASSCF/CASPT2),1–8,24–26 and restricted active space

self-consistent field calculations with non-dynamical correlation by second order perturbation

(CASSCF/RASPT2).9 The basis sets used are augmented, correlation corrected Dunning

quadruple-zeta basis sets. Douglas-Kroll-Hess relativistic correction (aug-cc-pVQZ-DK) is

used for the lighter forth period atoms, and e↵ective core potential relativistic correction

(aug-cc-pVQZ-PP) for the heavier atoms.19 Additionally, the restricted active space self-

consistent field calculations with non-dynamical correlation by second order perturbation

method was employed using the atom natural orbital - relativistic correlation correction

182

basis set (CASSCF/RAS-ANO) of Roos et at.24–31 was employed.

Acknowledgments

This work was financially supported by the National Science Foundation, Grant CHE 1152357.

We thank SMU for providing computational resources.

183

Table

1:

Experi

menta

land

Com

puta

tionalB

ond

Dis

soci

ation

Energ

ies

[kca

l/m

ol]

and

Str

etc

hin

gForc

eC

on-

stants

[mdyn/A

],G

roup

3.

Gro

up

3D

imer

Scandiu

m

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on

Year

201412

201011

201011

201011

201213

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This

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This

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Sta

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� u5⌃

� u5⌃

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MR

CIS

D(+

Q)

NEV

PT

2N

EV

PT

2N

EV

PT

2G

VV

PT

2C

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

CB

S21s1

5p10d6f4

g2h/

21s1

5p10d6f4

g2h/

21s1

5p10d6f4

g2h/

aug-c

c-p

VT

Zaug-c

c-p

VQ

Z-D

Kaug-c

c-p

VQ

Z-D

KA

NO

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C-

10s9

p8d5f4

g2h

10s9

p8d5f4

g2h

10s9

p8d5f4

g2h

--

--

Acti

ve

Space

(6,1

8)

(6,1

8)

(6,1

4)

(6,1

2)

(6,1

0)

(6,1

2)

(6,1

8)

(6,1

8)

Term

Sym

bol

2D

+2D

2D

g+

4F

g2D

g+

4F

g2D

g+

4F

g-

2D

g+

4F

g2D

g+

4F

g2D

g+

4F

gFro

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none

none

none

none

none

none

none

none

--

--

--

--

Acti

ve

Orb

itals

3d�

g/

u,3d⇡

u/

g3d�

g/

u,3d⇡

u/

g3d�

g/

u,3d⇡

u/

g3d�

g/

u,3d⇡

u/

g3d�

g/

u,3d⇡

u/

g3d�

g/

u,3d⇡

u/

g3d�

g/

u,3d⇡

u/

g3d�

g/

u,3d⇡

u/

g3d�

g/

u,4s�

g/

u3d�

g/

u,4s�

g/

u3d�

g/

u,4s�

g/

u3d�

g/

u,4s�

g/

u3d�

g/

u3d�

g/

u,4s�

g/

u3d�

g/

u,4s�

g/

u3d�

g/

u,4s�

g/

u4p�

g/

u,4p⇡

u/

g4p�

g/

u,4p⇡

u/

g4p�

g/

u-

-4p�

g/

u4p�

g/

u,4p⇡

u/

g4p�

g/

u,4p⇡

u/

gD

e(c

alc

)50.3

741.0

539.6

640.1

254.4

224.1

684.5

122.6

1

De

(exp)

--

-38.0

±2.3

25.9

±532

25.9

±532

25.9

±532

ke

(mdyn/A

)-

--

-0.7

633

1.3

71

0.8

16

0.8

53

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Ytt

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Year

201414

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Sta

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CA

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F/G

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ASSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

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aug-c

c-p

VT

Z-D

Kaug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

PA

NO

-RC

CA

cti

ve

Space

(6,1

2)

(6,1

2)

(6,1

2)

(6,1

8)

(6,1

8)

Term

Sym

bol

-2D

3/2

+4F3

/2

2D

3/2

+4F3

/2

2D

3/2

+4F3

/2

2D

3/2

+4F3

/2

Fro

zen

Orb

itals

-1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

-3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

--

--

-A

cti

ve

Orb

itals

4d�

g/

u,4d⇡

u/

g,

4d�

g/

u,4d⇡

u/

g,

4d�

g/

u,4d⇡

u/

g,

4d�

g/

u,4d⇡

u/

g,

4d�

g/

u,4d⇡

u/

g,

4d�

g/

u,5s�

g/

u4d�

g/

u,5s�

g/

u4d�

g/

u,5s�

g/

u4d�

g/

u,5s�

g/

u4d�

g/

u,5s�

g/

u-

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5p�

g/

u,5p⇡

u/

g5p�

g/

u,5p⇡

u/

gD

e(c

alc

)71.9

523.2

228.4

982.8

087.2

0

De

(exp)

80.7

±9.2

82.1

6±

9.3

634

82.1

6±

9.3

634

82.1

6±

9.3

634

82.1

6±

9.3

634

ke

(mdyn/A

)0.8

933

0.9

50

0.9

43

1.0

11

0.7

39

Dim

er

Lanth

aniu

m

Publicati

on

Year

This

Work

This

Work

This

Work

This

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200233

Sta

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� u5⌃

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CA

SSC

F/M

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F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

experi

ment

Basi

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aug-c

c-p

VQ

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Paug-c

c-p

VQ

Z-P

Paug-c

c-p

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Z-P

PA

NO

-RC

CA

cti

ve

Space

(6,1

2)

(6,1

2)

(6,1

8)

(6,1

8)

Term

Sym

bol

2D

3/2

+2D

3/2

2D

3/2

+4F3

/2

2D

3/2

+2D

3/2

2D

3/2

+2D

3/2

Fro

zen

Orb

itals

-1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

-3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

-4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

--

--

Acti

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itals

5d�

g/

u,5d⇡

u/

g,

5d�

g/

u,5d⇡

u/

g,

5d�

g/

u,5d⇡

u/

g,

5d�

g/

u,5d⇡

u/

g,

5d�

g/

u,6s�

g/

u5d�

g/

u,6s�

g/

u5d�

g/

u,6s�

g/

u,

5d�

g/

u,6s�

g/

u,

--

6p�

g/

u,6p⇡

u/

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g/

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u/

gD

e(c

alc

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610.4

928.4

960.5

8

De

(exp)

57.6

±532

57.6

±532

57.6

±532

57.6

±532

ke

(mdyn/A

)0.5

92

0.6

91

0.8

02

1.2

68

0.7

6

184

Table

2:

Experi

menta

land

Com

puta

tionalB

ond

Dis

soci

ation

Energ

ies

[kca

l/m

ol]

and

Str

etc

hin

gForc

eC

on-

stants

[mdyn/A

],G

roup

4.

Gro

up

4D

imer

Tit

aniu

m

Publicati

on

Year

199935

199136

This

Work

Sta

te3�

g3�

g3�

gM

eth

od

CA

SSC

F/M

RSD

CI+

QC

ASSC

F/M

RC

I+Q

/A

CPF

CA

SSC

F/C

ASPT

2B

asi

sSet

8s5

p3d2f

14s1

1p6d3f/

8s7

p4d1f

aug-c

c-p

VQ

Z-D

KA

cti

ve

Space

(8,1

2)

(8,1

5)

(8,1

2)

Term

Sym

bol

3F

+3F

3F

+3F

3F

+3F

Fro

zen

Orb

itals

none

none

none

--

-A

cti

ve

Orb

itals

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

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g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

4s�

g/

u4s�

g/

u,4p�

g,4p⇡

u4s�

g/

uD

e(c

alc

)22.1

421.6

838.5

8D

e(e

xp)

-28.9

3,33.3

2�

31.1

09

ke

(mdyn/A

)2.3

537

-2.6

28

Dim

er

Zir

coniu

m

Publicati

on

Year

199136

198938

This

Work

This

Work

This

Work

This

Work

Sta

te1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ gM

eth

od

CA

SSC

F/M

RC

I/A

CPF

CA

SSC

F/M

RSD

CI/

RC

IC

ASSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

6s5

p5d3f/

5s4

p4d1f

5s5

p4d1f

aug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

PA

NO

-RC

CA

cti

ve

Space

(8,1

5)

(8,1

2)

(8,1

2)

(8,1

2)

(8,1

8)

(8,1

8)

Term

Sym

bol

3F

+3F

3F

+3F

3F

+3F

3F

+3F

3F

+3F

3F

+3F

Fro

zen

Orb

itals

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

--

--

--

Acti

ve

Orb

itals

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

5s�

g/

u,5p�

g,5p⇡

u5s�

g/

u5s�

g/

u5s�

g/

u5s�

g/

u,5p�

g/

u,5p⇡

u/

g5s�

g/

u,5p�

g/

u,5p⇡

u/

gD

e(c

alc

)50.9

656.5

52.6

389.4

464.8

953.5

3

De

(exp)

-62.2

6-

69.1

870.3

839

70.3

839

70.3

839

70.3

839

ke

(mdyn/A

)2.5

137

--

3.5

36

3.5

73

1.4

59

Dim

er

Hafn

ium

Publicati

on

Year

This

Work

This

Work

This

Work

This

Work

199337

Sta

te1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ gM

eth

od

CA

SSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

experi

ment

Basi

sSet

aug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

PA

NO

-RC

CA

cti

ve

Space

(8,1

2)

(8,1

2)

(8,1

2)

(8,1

2)

Term

Sym

bol

3F2

+3F2

3F2

+3F2

3F2

+3F2

3F2

+3F2

Fro

zen

Orb

itals

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

--

--

Acti

ve

Orb

itals

5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u6s�

g/

u6s�

g/

u6s�

g/

u,6p�

g/

u,6p⇡

u/

g6s�

g/

u,6p�

g/

u,6p⇡

u/

gD

e(c

alc

)42.1

257.6

892.3

1118.9

5D

e(e

xp)

78.4

1pm

14

78.4

1pm

14

78.4

1pm

14

78.4

1pm

14

ke

(mdyn/A

)1.7

72

1.9

25

2.0

53

1.5

05

1.6

3

185

Table

3:

Experi

menta

land

Com

puta

tionalB

ond

Dis

soci

ation

Energ

ies

[kca

l/m

ol]

and

Str

etc

hin

gForc

eC

on-

stants

[mdyn/A

],G

roup

5.

Gro

up

5D

imer

Vanadiu

m

Publicati

on

Year

200040

This

Work

This

Work

Sta

te3⌃

� g3⌃

� g3⌃

� gM

eth

od

CA

SSC

F/M

RC

IC

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

IND

O/S

aug-c

c-p

VQ

Z-D

KA

NO

-RC

CA

cti

ve

Space

(10,1

6)

(10,1

8)

(10,1

8)

Term

Sym

bol

-4F3

/2

+4F3

/2

4F3

/2

+4F3

/2

Fro

zen

Orb

itals

-none

none

--

-A

cti

ve

Orb

itals

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

4s�

g/

u,4p�

g,4p⇡

u4s�

g/

u,4p�

g/

u,4p⇡

u/

g4s�

g/

u,4p�

g/

u,4p⇡

u/

gD

e(c

alc

)-

52.9

774.3

4

De

(exp)

58.0

6±

3.7

658.0

6±

3.7

640

58.0

6±

3.7

640

ke

(mdyn/A

)4.3

341

12.0

33

3.9

90

Dim

er

Nio

biu

m

Publicati

on

Year

200142

200040

This

Work

This

Work

This

Work

This

Work

Sta

te3⌃

� g3⌃

� g3⌃

� g3⌃

� g3⌃

� g3⌃

� gM

eth

od

CA

SSC

F/M

RSD

CI+

QC

ASSC

F/M

RC

IC

ASSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

5s5

p4d4f

IND

O/S

aug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

PA

NO

-RC

C(1

0,1

2)

(10,1

8)

(10,1

2)

(10,1

2)

(10,1

8)

(10,1

8)

Acti

ve

Space

--

6D

1/2

+6D

1/2

6D

1/2

+6D

1/2

6D

1/2

+6D

1/2

6D

1/2

+6D

1/2

Term

Sym

bol

1s�

g/

u,2s�

g/

u,2p�

g/

u,

1s�

g/

u,2s�

g/

u,2p�

g/

u,

1s�

g/

u,2s�

g/

u,2p�

g/

u,

1s�

g/

u,2s�

g/

u,2p�

g/

u,

1s�

g/

u,2s�

g/

u,2p�

g/

u,

1s�

g/

u,2s�

g/

u,2p�

g/

u,

Fro

zen

Orb

itals

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

--

--

--

Acti

ve

Orb

itals

5s�

g/

u5s�

g/

u,5p�

g,5p⇡

u5s�

g/

u5s�

g/

u5s�

g/

u,5p�

g/

u,5p⇡

u/

g5s�

g/

u,5p�

g/

u,5p⇡

u/

gD

e(c

alc

)98.7

-152.9

157.6

8147.8

5138.5

8

De

(exp)

93.4

0-

98.7

0-

128.6

742

128.6

742

128.6

742

128.6

742

ke

(mdyn/A

)4.8

441

--

3.3

74

-6.5

56

Dim

er

Tanta

lum

Publicati

on

Year

200941

This

Work

This

Work

This

Work

Sta

te3⌃

� g3⌃

� g3⌃

� g3⌃

� gM

eth

od

CA

SSC

F/C

ASPT

2C

ASSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

24s2

1p15d11f4

g2h/

aug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

PA

NO

-RC

C9s8

p6d4f3

g2h

(VQ

Z)

--

-A

cti

ve

Space

(10,1

2)

(10,1

2)

(10,1

2)

(10,1

8)

Term

Sym

bol

4F

+4F

4F

+4F

4F

+4F

4F

+4F

Fro

zen

Orb

itals

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

--

--

Acti

ve

Orb

itals

5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u6s�

g/

u6s�

g/

u6s�

g/

u6s�

g/

u,6p�

g/

u,6p⇡

u/

gD

e(c

alc

)107.2

3160.4

3210.7

1128.8

2

De

(exp)

49.1

2114.3

8-

124.5

241

114.3

8-

124.5

241

114.3

8-

124.5

241

ke

(mdyn/A

)4.8

041

3.1

62

3.8

77

5.3

62

186

Table

4:

Experi

menta

land

Com

puta

tionalB

ond

Dis

soci

ation

Energ

ies

[kca

l/m

ol]

and

Str

etc

hin

gForc

eC

on-

stants

[mdyn/A

],G

roup

6.

Gro

up

6D

imer

Chro

miu

m

Publicati

on

Year

201213

200327

199525

This

Work

This

Work

This

Work

Sta

te1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ gM

eth

od

CA

SSC

F/G

VV

PT

2C

ASSC

F/C

ASPT

2C

ASSC

F/LS-C

ASPT

2C

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

cc-p

VT

Z,cc-V

QZ

21s1

5p10d6f4

g8s7

p6d4f

aug-c

c-p

VQ

Z-D

Kaug-c

c-p

VQ

Z-D

KA

NO

-RC

CA

cti

ve

Space

(12,1

2)

(12,1

6)

(12,1

2)

(12,1

2)

(12,1

8)

(12,1

8)

Term

Sym

bol

-7S

+7S

7S

+7S

7S

+7S

7S

+7S

7S

+7S

Fro

zen

Orb

itals

none

none

none

none

none

none

--

--

--

Acti

ve

Orb

itals

3d�

g/

u,3d⇡

u/

g,3d�

g/

u3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u3d�

g/

u,3d⇡

u/

g,3d�

g/

u3d�

g/

u,3d⇡

u/

g,3d�

g/

u3d�

g/

u,3d⇡

u/

g,3d�

g/

u4s�

g/

u4s�

g/

u,4p⇡

u,4p�

g,5s�

g4s�

g/

u4s�

g/

u4s�

g/

u,4p�

g/

u,4p⇡

u/

g4s�

g/

u,4p�

g/

u,4p⇡

u/

gD

e(c

alc

)36.9

-35.5

182.1

921.2

444.3

7

De

(exp)

33.9

5,35.9

7,33.4

4-

33.9

0±

2.0

733.9

5,35.9

713

33.9

5,35.9

713

33.9

5,35.9

713

ke

(mdyn/A

)3.5

4-

-7.0

27

4.0

60

2.7

04

Dim

er

Moly

bdenum

Publicati

on

Year

200726

This

Work

This

Work

This

Work

This

Work

Sta

te1⌃

+ g(a

nd

oth

ers

)1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ gM

eth

od

CA

SSC

F/C

ASPT

2C

ASSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

-aug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

PA

NO

-RC

CA

cti

ve

Space

(12,1

2)

(12,1

2)

(12,1

2)

(12,1

8)

(12,1

8)

Term

Sym

bol

-7S

+7S

7S

+7S

7S

+7S

7S

+7S

Fro

zen

Orb

itals

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

--

--

-A

cti

ve

Orb

itals

4d�

g/

u,4d⇡

u/

g,4d�

g/

u4d�

g/

u,4d⇡

u/

g,4d�

g/

u4d�

g/

u,4d⇡

u/

g,4d�

g/

u4d�

g/

u,4d⇡

u/

g,4d�

g/

u4d�

g/

u,4d⇡

u/

g,4d�

g/

u5s�

g/

u5s�

g/

u5s�

g/

u5s�

g/

u,5p�

g/

u,5p⇡

u/

g5s�

g/

u,5p�

g/

u,5p⇡

u/

gD

e(c

alc

)101.7

204.8

5180.1

1158.1

7114.7

2

De

(exp)

103.1

7±

0.2

343

103.1

7±

0.2

326

103.1

7±

0.2

326

103.1

7±

0.2

326

103.1

7±

0.2

326

ke

(mdyn/A

)6.4

344

7.7

67

4.1

56

4.7

45

6.4

47

Dim

er

Tungst

en

Publicati

on

Year

200726

This

Work

This

Work

This

Work

This

Work

Sta

te1⌃

+ g(a

nd

oth

ers

)1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ gM

eth

od

CA

SSC

F/C

ASPT

2C

ASSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

-aug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

PA

NO

-RC

CA

cti

ve

Space

(12,1

2)

(12,1

2)

(12,1

2)

(12,1

2)

(12,1

2)

Term

Sym

bol

-7S

+7S

7S

+7S

7S

+7S

7S

+7S

Fro

zen

Orb

itals

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

--

--

-A

cti

ve

Orb

itals

5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u6s�

g/

u6s�

g/

u6s�

g/

u6s�

g/

u,6p�

g/

u,6p⇡

u/

g6s�

g/

u,6p�

g/

u,6p⇡

u/

gD

e(c

alc

)123.8

484.3

3-

122.5

4189.8

3

De

(exp)

115.4

8±

23.4

8115.4

8±

23.4

826

115.4

8±

23.4

826

115.4

8±

23.4

826

115.4

8±

23.4

826

ke

(mdyn/A

)6.1

437

12.4

98

7.2

16

5.9

03

5.4

41

187

Table

5:

Experi

menta

land

Com

puta

tionalB

ond

Dis

soci

ation

Energ

ies

[kca

l/m

ol]

and

Str

etc

hin

gForc

eC

on-

stants

[mdyn/A

],G

roup

7.

Gro

up

7D

imer

Manganese

Publicati

on

Year

201213

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199337

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Space

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(14,1

2)

(14,1

8)

(14,1

2)

(14,1

2)

Term

Sym

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--

--

6S5

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none

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none

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e(c

alc

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53.6

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37

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3.2

2-

-3.1

0±

3.1

03.2

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ke

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0.0

671

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Year

201414

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200233

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sSet

aug-c

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Z-P

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NO

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Space

(14,1

2)

(14,1

2)

(14,1

2)

Term

Sym

bol

-6S5

/2

+6S5

/2

6S5

/2

+6S5

/2

Fro

zen

Orb

itals

-1s,2s,2p,

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itals

4d�

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gD

e(c

alc

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187.6

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80.4

880.4

814

80.4

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2.2

57

6.5

40

4.3

7

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er

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Year

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199444

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te1⌃

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ment

Basi

sSet

aug-c

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Z-P

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Z-P

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NO

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cti

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Space

(14,1

2)

(14,1

2)

(14,1

8)

Term

Sym

bol

6S5

/2

+6S5

/2

6S5

/2

+6S5

/2

6S5

/2

+6S5

/2

Fro

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1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

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4s,4p,4d

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5d�

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u,5d⇡

u/

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u6s�

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u,6p�

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alc

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6215.1

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92.2

433

92.2

433

92.2

433

ke

(mdyn/A

)3.2

02

2.0

91

6.3

77

6.2

6

188

Table

6:

Experi

menta

land

Com

puta

tionalB

ond

Dis

soci

ation

Energ

ies

[kca

l/m

ol]

and

Str

etc

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gForc

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stants

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roup

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Year

20149

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Z-D

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NO

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Space

(16,1

6)

(16,1

2)

(16,1

2)

(16,1

8)

Term

Sym

bol

5D

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5D

4+

5F5

5D

4+

5F5

5D

4+

5F5

Fro

zen

Orb

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026.9

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09

26.9

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09

26.9

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09

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837

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30

3.2

24

1.7

13

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Year

201445

199116

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Space

(16,1

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(16,1

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(16,1

2)

(16,1

2)

(16,1

8)

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8)

Term

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5F

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5F

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5F

+5F

5F

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Fro

zen

Orb

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1s,2s,2p,

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1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

3s,3p,3d

3s,3p,3d

3s,3p,3d

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3s,3p,3d

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78.1

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73.5

646

73.5

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73.5

646

73.5

646

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2.6

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75

2.5

11

2.4

75

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Year

20148

20148

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Space

(16,1

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2)

(16,1

2)

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8)

Term

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5D

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1s,2s,2p,

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1s,2s,2p,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

4s,4p,4d

4s,4p,4d

4s,4p,4d

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575.8

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101

101

101.0

08

101.0

08

101.0

08

ke

(mdyn/A

)6.2

633

-5.8

59

6.7

96

6.1

04

189

Table

7:

Experi

menta

land

Com

puta

tionalB

ond

Dis

soci

ation

Energ

ies

[kca

l/m

ol]

and

Str

etc

hin

gForc

eC

on-

stants

[mdyn/A

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roup

9.

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up

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Cobalt

Publicati

on

Year

20096

This

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aug-c

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CA

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ve

Space

(18,1

2)

(18,1

2)

(18,1

2)

(18,1

6)

(18,1

6)

Term

Sym

bol

-4F9

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+4F9

/2

4F9

/2

+4F9

/2

4F9

/2

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39.4

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248

39.4

3±

6.4

248

39.4

3±

6.4

248

39.4

3±

6.4

248

39.4

3±

6.4

248

ke

(mdyn/A

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137

2.0

33

-0.9

75

1.5

80

1.5

3

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Year

201549

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Space

(18,1

2)

(18,1

2)

(18,1

2)

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8)

Term

Sym

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4F9

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/2

4F9

/2

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/2

4F9

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32.6

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150

32.6

8±

7.3

150

32.6

8±

7.3

150

32.6

8±

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433

1.9

81

1.5

57

1.1

35

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200233

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ment

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aug-c

c-p

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PA

NO

-RC

CA

cti

ve

Space

(18,1

2)

(18,1

2)

(18,1

8)

(18,1

8)

Term

Sym

bol

4F9

/2

+4F9

/2

4F9

/2

+4F9

/2

4F9

/2

+4F9

/2

4F9

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1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

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5d�

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u,

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u6s�

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e(c

alc

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578.0

754.7

982.7

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De

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79.9

951

79.9

951

79.9

951

79.9

951

ke

(mdyn/A

)2.6

37

2.9

67

3.4

68

4.4

71

4.4

4

190

Table

8:

Experi

menta

land

Com

puta

tionalB

ond

Dis

soci

ation

Energ

ies

[kca

l/m

ol]

and

Str

etc

hin

gForc

eC

on-

stants

[mdyn/A

],G

roup

10.

Gro

up

10

Dim

er

Nic

kel

Publicati

on

Year

201210

199424

This

Work

This

Work

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This

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R-A

QC

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ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

cc-p

VQ

Z(D

K)

21s1

5p10d6f4

g/

aug-c

c-p

VQ

Z-D

Kaug-c

c-p

VQ

Z-D

Kaug-c

c-p

VQ

Z-D

KA

NO

-RC

C-

6s5

p4d3f2

g-

--

-A

cti

ve

Space

(20,1

2)

(20,1

2)

(20,1

2)

(20,1

2)

(20,1

8)

(20,1

8)

Term

Sym

bol

3D

+3F

3D

+3D

3F4

+3F4

3F4

+3F4

3F4

+3F4

3F4

+3F4

Fro

zen

Orb

itals

none

none

none

none

none

none

--

--

--

Acti

ve

Orb

itals

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

u/

g,3d⇡

u/

g,3d�

g/

u,

3d�

u/

g,3d⇡

u/

g,3d�

g/

u,

4s�

g/

u4s�

g/

u4s�

g/

u4s�

g/

u4s�

g/

u,4p�

g,4p⇡

u4s�

g/

u,4p�

g,4p⇡

u

De

(calc

)45.0

83

48.4

346.2

847.4

946.8

226.5

2

De

(exp)

47.4

6±

0.4

2-

47.4

6±

0.4

210

47.4

6±

0.4

210

47.4

6±

0.4

210

47.4

6±

0.4

210

ke

(mdyn/A

)1.3

652

1.1

633

1.0

53

1.1

56

1.1

51

0.8

23

Dim

er

Pala

diu

m

Publicati

on

Year

19982

This

Work

This

Work

This

Work

Sta

te3⌃

� g3⌃

� g3⌃

� g3⌃

� gM

eth

od

CA

SSC

F/C

ASPT

2C

ASSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2B

asi

sSet

EC

P-T

Z-P

Paug-c

c-p

VQ

Z-D

Kaug-c

c-p

VQ

Z-D

Kaug-c

c-p

VQ

Z-D

KA

cti

ve

Space

(20,1

2)

(20,1

2)

(20,1

2)

(20,1

8)

Term

Sym

bol

1S0

+3D

31S0

+3D

31S0

+3D

3Fro

zen

Orb

itals

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

--

--

Acti

ve

Orb

itals

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u5s�

g/

u5s�

g/

u5s�

g/

u5s�

g/

u,5p�

g/

u,5p⇡

u/

gD

e(c

alc

)22.6

20.5

631.6

723.2

4

De

(exp)

24.0

5±

0.4

652

24.0

5±

0.4

652

24.0

5±

0.4

652

24.0

5±

0.4

652

ke

(mdyn/A

)1.3

837

1.1

91

0.9

81

1.4

89

Dim

er

Pla

tinum

Publicati

on

Year

19982

This

Work

This

Work

This

Work

This

Work

Sta

te3⌃

� g3⌃

� g3⌃

� g3⌃

� g3⌃

� gM

eth

od

CA

SSC

F/C

ASPT

2C

ASSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

pV

TZ-P

Paug-c

c-p

VQ

Z-D

Kaug-c

c-p

VQ

Z-D

Kaug-c

c-p

VQ

Z-D

KA

NO

-RC

CA

cti

ve

Space

(20,1

2)

(20,1

2)

(20,1

2)

(20,1

2)

(20,1

2)

Term

Sym

bol

3D

+3D

3D

+3D

3D

+3D

3D

+3D

3D

+3D

Fro

zen

Orb

itals

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

--

--

-A

cti

ve

Orb

itals

5d�

g/

u,5d⇡

u/

g,5d�

g/

u,

5d�

g/

u,5d⇡

u/

g,5d�

g/

u,

5d�

g/

u,5d⇡

u/

g,5d�

g/

u,

5d�

g/

u,5d⇡

u/

g,5d�

g/

u,

5d�

g/

u,5d⇡

u/

g,5d�

g/

u,

6s�

g/

u6s�

g/

u6s�

g/

u6s�

g/

u,6p�

g/

u,6p⇡

u/

g6s�

g/

u,6p�

g/

u,6p⇡

u/

gD

e(c

alc

)60

66.7

580.5

972.6

451.2

9

De

(exp)

72.7

2±

0.7

752

72.7

2±

0.7

752

72.7

2±

0.7

752

72.7

2±

0.7

752

72.7

2±

0.7

752

ke

(mdyn/A

)2.6

637

3.2

15

1.4

21

2.8

48

3.2

49

191

Table

9:

Experi

menta

land

Com

puta

tionalB

ond

Dis

soci

ation

Energ

ies

[kca

l/m

ol]

and

Str

etc

hin

gForc

eC

on-

stants

[mdyn/A

],G

roup

11.

Gro

up

11

Dim

er

Copper

Publicati

on

Year

201753

199424

This

Work

This

Work

This

Work

This

Work

Sta

te1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ gM

eth

od

SD

-BO

VB

CA

SSC

F/C

ASPT

2C

ASSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

LanL2T

Z(f

)21s1

5p10d6f4

g/

aug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

PA

NO

-RC

C-

6s5

p4d3f2

g-

--

-A

cti

ve

Space

(2,2

)(2

2,1

2)

(22,1

2)

(22,1

2)

(22,1

8)

(22,1

8)

Term

Sym

bol

2S

+2S

2S

+2S

2S1

/2

+2S1

/2

2S1

/2

+2S1

/2

2S1

/2

+2S1

/2

2S1

/2

+2S1

/2

Fro

zen

Orb

itals

--

none

none

none

none

--

--

--

Acti

ve

Orb

itals

4s�

g/

u3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

3d�

g/

u,3d⇡

u/

g,3d�

g/

u,

-4s�

g/

u,4s�

g/

u4s�

g/

u,4s�

g/

u4s�

g/

u,4s�

g/

u4s�

g/

u,4s�

g/

u,

4s�

g/

u,4s�

g/

u,

--

--

4p�

g,4p⇡

u4p�

g,4p⇡

uD

e(c

alc

)40.7

45.4

342.5

943.8

492.5

457.3

6

De

(exp)

47.7

47.9

747.9

724

47.9

724

47.9

724

47.9

724

ke

(mdyn/A

)1.3

352

1.4

28

1.5

22

1.3

32

1.1

47

Dim

er

Silver

Publicati

on

Year

201753

199154

This

Work

This

Work

This

Work

This

Work

Sta

te1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ gM

eth

od

SD

-BO

VB

CA

SSC

F/M

RC

I(SD

)C

ASSC

F/M

R-A

QC

CC

ASSC

F/C

ASPT

2C

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

LanL2T

Z(f

)8s7

p6d2f/

aug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

PA

NO

-RC

C-

6s5

p4d2f

--

--

Acti

ve

Space

(2,2

)(2

2,1

2)

(22,1

2)

(22,1

2)

(22,1

8)

(22,1

8)

Term

Sym

bol

--

2S1

/2

+2S1

/2

2S1

/2

+2S1

/2

2S1

/2

+2S1

/2

2S1

/2

+2S1

/2

Fro

zen

Orb

itals

-1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

-3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

3s,3p,3d

--

--

--

Acti

ve

Orb

itals

5s�

g/

u4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,

4d�

g/

u,4d⇡

u/

g,4d�

g/

u4d�

g/

u,4d⇡

u/

g,4d�

g/

u-

5s�

g/

u5s�

g/

u5s�

g/

u5s�

g/

u,5p�

g/

u,5p⇡

u/

g5s�

g/

u,5p�

g/

u,5p⇡

u/

gD

e(c

alc

)28.9

27.6

736.0

543.6

330.0

554.9

6

De

(exp)

38.3

38.7

438.7

454

38.7

454

38.7

454

38.7

454

ke

(mdyn/A

)-

1.1

837

1.1

23

1.4

00

1.2

36

1.3

66

Dim

er

Gold

Publicati

on

Year

201753

20004

199154

This

Work

This

Work

This

Work

This

Work

Sta

te1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ g1⌃

+ gM

eth

od

SD

-BO

VB

CA

SSC

F/C

ASPT

2C

ASSC

F/M

RC

I(SD

)C

ASSC

F/M

R-A

QC

CC

ASSC

F/M

R-A

QC

CC

ASSC

F/R

ASPT

2C

ASSC

F/R

AS-A

NO

Basi

sSet

LanL2T

Z(f

)21s1

7p11d9f/

8s7

p6d2f/

aug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

Paug-c

c-p

VQ

Z-P

PA

NO

-RC

C-

13s1

1p7d4f

6s5

p3d2f

--

--

Acti

ve

Space

(2,2

)(2

2,1

2)

(22,1

2)

(22,1

2)

(22,1

2)

(22,1

8)

(22,1

8)

Term

Sym

bol

--

-2S1

/2

+2S1

/2

2S1

/2

+2S1

/2

2S1

/2

+2S1

/2

2S1

/2

+2S1

/2

Fro

zen

Orb

itals

-1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

1s,2s,2p,

-3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

3s,3p,3d,

-4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

4s,4p,4d

--

--

--

-A

cti

ve

Orb

itals

6s�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u5d�

g/

u,5d⇡

u/

g,5d�

g/

u-

6s�

g/

u6s�

g/

u6s�

g/

u6s�

g/

u6s�

g/

u,6p�

g/

u,6p⇡

u/

g6s�

g/

u,6p�

g/

u,6p⇡

u/

gD

e(c

alc

)47.4

52.3

5D

e=

38.5

148.1

58.4

656.9

165.8

2

De

(exp)

52.8

53.0

8±

0.7

355

-53.0

8±

0.7

355

53.0

8±

0.7

355

53.0

8±

0.7

355

53.0

8±

0.7

355

ke

(mdyn/A

)-

2.1

237

-1.9

71

2.5

20

2.1

87

2.2

65

192

Table

10:

Experi

menta

land

Com

puta

tionalB

ond

Dis

soci

ation

Energ

ies

[kca

l/m

ol]

and

Str

etc

hin

gForc

eC

on-

stants

[mdyn/A

],G

roup

12.

Gro

up

12

Dim

er

Zin

c

Publicati

on

Year

20055

This

Work

Sta

te1⌃

+ g1⌃

+ gM

eth

od

CA

SSC

F/C

ASPT

2C

ASSC

F/C

ASPT

2B

asi

sSet

17s1

2p9d4f/

aug-c

c-p

VQ

Z-P

P8s8

p8d4f

-A

cti

ve

Space

(4,1

0)

(24,1

2)

Term

Sym

bol

1S

+1S

1S

+1S

Fro

zen

Orb

itals

-none

--

Acti

ve

Orb

itals

3d�

g/

u,3d⇡

u/

g,

3d�

g/

u,3d⇡

u/

g,

3d�

g/

u,4s�

g/

u3d�

g/

u,4s�

g/

uD

e(c

alc

)0.7

83.2

De

(exp)

0.8

10.7

98

ke

(mdyn/A

)0.0

1337

0.1

07

Dim

er

Cadm

ium

Publicati

on

Year

200017

This

Work

Sta

te1⌃

+ g1⌃

+ gM

eth

od

CA

SSC

F/M

RD

CI

CA

SSC

F/C

ASPT

2B

asi

sSet

DZ/3P

aug-c

c-p

VQ

Z-P

PA

cti

ve

Space

-(2

4,1

4)

Term

Sym

bol

1S

+1S

1S

+1S

Fro

zen

Orb

itals

1s,2s,2p,

1s,2s,2p,

3s,3p,3d

3s,3p,3d

--

Acti

ve

Orb

itals

4d�

g/

u,4d⇡

u/

g,4d�

g/

u,5s�

g/

u4d�

g/

u,4d⇡

u/

g,4d�

g/

u,5s�

g/

uD

e(c

alc

)1.3

81.8

2D

e(e

xp)

0.9

20.9

45

ke

(mdyn/A

)0.0

1737

0.6

33

Dim

er

Merc

ury

Publicati

on

Year

199715

This

Work

Sta

teO

g+

1⌃

+ gM

eth

od

CA

SSC

F/M

RC

IC

ASSC

F/C

ASPT

2B

asi

sSet

PP

9s8

p6d/

aug-c

c-p

VQ

Z-P

P8s6

p3d

-A

cti

ve

Space

(20,1

2)

(24,1

2)

Term

Sym

bol

1S

+1S

1S

+1S

Fro

zen

Orb

itals

1s,2s,2p,

1s,2s,2p,

3s,3p,3d,

3s,3p,3d,

4s,4p,4d

4s,4p,4d

4p⇡

u/

g,4d�

g/

u,4d⇡

u/

g,4d�

g/

u4p⇡

u/

g,4d�

g/

u,4d⇡

u/

g,4d�

g/

u-

-A

cti

ve

Orb

itals

5d�

g/

u,5d⇡

u/

g,

5d�

g/

u,5d⇡

u/

g,

5d�

g/

u,6s�

g/

u5d�

g/

u,6s�

g/

uD

e(c

alc

)0.8

51.9

1D

e(e

xp)

1.0

0±

0.0

61.0

86

ke

(mdyn/A

)0.0

2037

0.2

49

193

References

(1) Andersson, K.; Roos, B.; Malmqvist, P. A.; Widmark, P.-O. The Cr2 Potential Energy

Curve Studied with Multiconfigurational Second-order Perturbation Theory. Chem.

Phys. Lett. 1994, 230, 391–397.

(2) Cui, Q.; Musaev, D.; Morokuma, K. Electronic Structure of Asymmetric Metal-Metal

Multiple Bonds: The d2-d6 Complexes X4Mo-Mo(PH3)4 (X = OH, Cl). Inorg. Chem.

1998, 28, 3292–3296.

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199

Sc2 X5Σ-u

πu

δg

δu

πg

σ+g

σ+u

σ+g

πu

πg

σ+u

σ+gσ+

u

Y2 X5Σ-u

πu

δg

δu

πg

σ+g

σ+u

σ+g πu

πg

σ+u

σ-g

σ-u

La2 X5Σ-u

πu

δg

δu

πg

σ+g

σ+u

σ+g

πu

πg

σ+u

σ-g

σ-u

Figure 1: Orbital Diagrams: HF/cc-pVQZ-DK and HF/cc-pVQZ-PP. Group 3.

Ti2 X3Δg

πu

δg

δu

πg

σ+g

σ+u

σ+g

πu

πg

σ+u

σ+g

σ+u

Zr2 X1Σ+g

πuδg

δu

πg

σ+g

σ+u

σ+g

πu

πgσ+

u

σ-g

σ-u

Hf2 X1Σ+g

πu

δg, δu

πg

σ+gσ+

u

σ+g

πu

πgσ+uσ-

g

σ+u


200

V2 X3Σ-g

πu

δg

δu

πg

σ+g

σ+u

σ+g

πu

πg

σ+u

σ+g

σ+u

Nb2 X3Σ-g

πuδg

δu

πg

σ+g

σ+u

σ+g

πu

πgσ+

u

σ-g

σ-u

Ta2 X3Σ-g

πu

δg, δu

πg

σ+g

σ+u

σ+g

πu

πgσ+uσ-

g

σ+u


Cr2 X1Σ+g

πu

δg

σ+g

σ+g

πuπgδuπg

σ+u

σ+g

σ+u

σ-u

Mo2 X1Σ+g

πu

δg

σ+g

σ+g

πu

πgδuπg

σ+u

σ+g, σ+

uσ+

u

W2 X1Σ+g

πu

δg

σ+g

σ+g

πuπgδuπg

σ+u

σ+g, σ+

uσ+

u


201

Mn2 X1Σ+g

δg

δu

πu

σ+g

σ+g

πuπg

πg

σ+u

σ+g

σ+u

σ+u

Tc2 X3Σ-g

πuδg

δu

σ+g

σ+g

πu

πg

πg

σ+u

σ+g, σ+

u

σ+u

Re2 X1Σ+g

πuδg

δu

σ+g

σ+g

πu

πgπgσ+

uσ+

g, σ+u

σ+u


Fe2 X7Σ+g

δg

δu

πu

σ+g

σ+g

πuπg

πg

σ+u

σ+g

σ+u

σ-u

Ru2 X7Δu

πuδg

δu

σ+g

σ+g

πu

πgπg

σ+u

σ+g, σ+

uσ+

u

Os2 X5Πg

δgπu

δu

σ+g

σ+g

πuπgπg

σ+u

σ+g, σ+

u

σ+u

Figure 6: Orbital Diagrams: HF/cc-pVQZ-DK and HF/cc-pVQZ-PP. Group 8 .

202

Ir2 X5Δg

σ+g

δg

δu, πu

σ+g

σ+u

σ+g

σ+u, σ+

uπu, πgπg

Co2 X5Δg

πu

πg

δgδu

σ+g

σ+uσ+

gσ+

uσ+

gσ+

u

πgπu

Rh2 X5Δg

πg

δg

δu

πu

σ+uσ+

uσ+

gσ+

gσ+

u

σ+g

πgπu


Ni2 X1O+g

πuδgδuπg

σ+g

σ+u

πuπg

σ+g

σ+u

σ+g

σ-u

Pd2 X3Σ-g

πu

δgδu

πg

σ+g

σ+u

πuπg

σ+g

σ+u

σ+g

σ+u

Pt2 X3Σ-g

πu

δgδuπg

σ+g

σ+g

πu

πg

σ+u

σ+u

σ+g

σ+u


203

Cu2 X1Σ+g

πuδgδuπg

σ+g

σ+u

σ+g

πu

πgσ+u

σ-g, σ-

u

Ag2 X1Σ+g

πuδgδuπg

σ+g

σ+u

σ+g

πuπg

σ+u

σ+g

σ+u

Au2 X1Σ+g

πu

δgδuπg

σ+g

σ+u

σ+g

πu

πg

σ+u

σ+g, σ+

u


Zn2 X1Σ+g

πu

δg

δu

πg

σ+g

σ+u

σ+g

πuπg

σ+u

σ+g

σ+u

Cd2 X1Σ+g

πu

δg

δu

πg

σ+g

σ+u

σ+g πu

πgσ+

u

σ+g

σ+u

Hg2 X1Σ+g

πu

δg

δu

πg

σ+g

σ+u

σ+g

πuπg

σ+u

σ+g

σ+u


204

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