ab initio studies of a pentacyclo- undecane cage lactam · ab initio studies of a...
TRANSCRIPT
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AB INITIO STUDIES OF A PENTACYCLO-
UNDECANE CAGE LACTAM
by
Thishana Singh
Submitted in partial fulfillment of the requirements for the degree of
Master of Technology
in the Department of Chemistry,
Faculty of Engineering, Science and the Built Environment,
Durban Institute of Technology, Durban, May 2003.
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DECLARATION
I, Thishana Singh, declare that unless indicated, this dissertation is my own work and it
has not been submitted for a degree at another Technikon or University.
________________________
T. Singh
_____day of______________2003
APPROVED FOR FINAL SUBMISSION
________________________ ________________________
Supervisor: Dr. K. Bisetty (Ph.D.) Co-supervisor: Dr. H.G. Kruger (Ph.D.)
________________________
Date
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ACKNOWLEDGEMENTS
I would like to thank my supervisors Dr. K. Bisetty and Dr. H.G. Kruger for their
assistance, guidance, patience and much appreciated words of encouragement
throughout the duration of this research project.
My sincere gratitude also goes to:
(i) My friends and colleagues at the Durban Institute of Technology,
Department of Chemistry, for their support, encouragement and
assistance.
(ii) The National Research Foundation (NRF), M L Sultan Technikon
Research Center for financial assistance.
(iii) De Beers Educational Trust Fund for the purchase of a desk top
computer.
(iv) The University of Natal, Durban for the generous use of computer time
on the DEC Alpha computer workstation.
A special thanks to my parents for their continuous support and encouragement
throughout the duration of my studies.
"Jai Shree Krishna"
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ABSTRACT
The purpose of this study is to utilize computational techniques in the determination of
the mechanistic pathways for the one-pot conversion of a pentacyclo-undecane (PCU)
dione 1.1 to a pentacyclo-undecane cage lactam 1.2.
OO
NaCN
H2O
OH
C O
NHHO
1.1 1.2
In pursuance of this objective, the ab initio quantum mechanical level of theory was
employed. The primary goal of this study was to compute the relative difference in
energies for the reactants, products and transition-states of the proposed mechanistic
pathways. The energy values obtained were used to predict the thermodynamic and
kinetic pathways of the mechanism. All calculations were performed using the
GAUSSIAN 98 series of programs, and GAUSSVIEW was used to visualize the
transition-state structures.
Full geometry optimizations were performed at the Hartree-Fock (HF) level of theory
using the 3-21+G* basis set. In addition, the transition-states were established using a
SCAN technique to obtain a starting structure. Transitions states were verified by using
second-derivative analytical vibrational frequency calculations and the visual inspection
of the movement of atoms associated with the transition.
Hess's Law was applied to compute the heats of formation. It was found that two
transition structures in the gas phase had abnormally high energies. However, these
energies were found to be considerably lower in the presence of a solvent molecule.
Furthermore, it was observed that the one-step conversion of the dione 1.1 to the lactam
1.2 proceeded via a single transition-state.
Previous experimental work found that the reaction proceeds through a cyanohydrin
intermediate which in all likelihood represents the rate determining step. Sound
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arguments exist to demonstrate that the computationally determined rate-determining
step agrees with the experimentally observed rate-determining step.
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TABLE OF CONTENTS
Declaration ...................................................................................................................... ii
Acknowledgements ........................................................................................................ iii
Abstract .......................................................................................................................... iv
Table of contents ............................................................................................................ vi
List of figures ............................................................................................................... viii
List of tables ................................................................................................................. viii
List of Abbreviations ..................................................................................................... ix
Chapter 1 ......................................................................................................................... 1
1. INTRODUCTION ................................................................................................ 1
1.1 Computational Chemistry and Molecular Modeling ....................................... 1
1.2 Pentacyclo-undecane (PCU) Cage Compounds............................................... 2
1.3 Lactam formation ............................................................................................. 4
Chapter 2 ......................................................................................................................... 9
2. THEORETICAL TOOLS FOR MOLECULAR ORBITAL
CALCULATIONS ............................................................................................... 9
2.1 Molecular Orbital Theory ................................................................................ 9
2.2 Molecular Mechanics ....................................................................................... 9
2.3 Electronic Structure Methods ........................................................................ 10
2.3.1 The Ab Initio Method .............................................................................. 11
2.3.2 Semi-Empirical ........................................................................................ 12
2.3.3 Density Functional Theory (DFT) Methods ............................................ 12
2.3.4 Basis Sets ................................................................................................. 13
2.3.4.1 Minimal Basis Sets ............................................................................... 14
2.3.4.2 Split Valence Basis Sets ....................................................................... 14
2.3.4.3 Polarised Basis Sets .............................................................................. 15
2.3.4.4 Basis Sets Incorporating Diffuse Functions ......................................... 16
2.3.5 Hartree-Fock Theory ............................................................................... 16
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Chapter 3 ....................................................................................................................... 19
3. TRANSITION STRUCTURE MODELING .................................................. 19
3.1 Transition-state modeling with empirical force-fields ................................... 20
3.2 Locating minima on the seams of intersecting semi-empirical PES .............. 22
3.3 Transition-structure modeling of a PCU Cage Lactam using ab initio methods
........................................................................................................................ 23
Chapter 4 ....................................................................................................................... 26
4. COMPUTATIONAL DETAILS EMPLOYED ............................................... 26
4.1 The GAUSSIAN 98 Program ........................................................................ 26
4.2 The GaussView Program ............................................................................... 27
4.3 The SCAN Calculation .................................................................................. 27
4.3.1 Commands used during a SCAN or a TS Search .................................... 28
4.4 Calculation Details ......................................................................................... 29
5. RESULTS AND DISCUSSION ......................................................................... 30
5.1 Introduction .................................................................................................... 30
5.2 Local minima on the energy profile ............................................................... 33
5.3 The Transition Structures (TS) ...................................................................... 34
5.3.1 Transition Structure 5.3.1 ........................................................................ 36
5.3.2 Transition Structure 5.5 ........................................................................... 39
5.3.3 Transition Structure 5.7.1 ........................................................................ 41
5.3.4 Transition Structure 5.10 ......................................................................... 47
5.3.5 Transition Structure 5.12 and 5.13 .......................................................... 49
5.4 Transition structure modeling with solvent molecules .................................. 56
5.5 Calculation of Heats of Formation ................................................................. 61
Chapter 6 ....................................................................................................................... 62
6. CONCLUSION ................................................................................................... 62
APPENDIX 1................................................................................................................. 63
Scheme of reaction ...................................................................................................... 64
References ...................................................................................................................... 67
SUPPLEMENTARY MATERIAL: A CD accompanying this thesis includes the following:
Text containing Chapters 1-6 (including References).
Cartesian coordinates of all the 3-D structures from Chapter 5.
Frequency calculations of the TS's.
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LIST OF FIGURES
Figure 3.1 Points on a simple reaction coordinate ....................................................... 19
Figure 5. 1 Modified mechanism for the conversion of the dione 5.1 to the lactam 5.2
.................................................................................................................... 31
Figure 5. 2 Calculated reaction profile for the proposed mechanism. .......................... 32
Figure 5.3 Graphical representation of Energy vs Reaction Coordinate for structure
5.3.1 ............................................................................................................ 37
Figure 5.4 Graphical representation of Energy vs Reaction Coordinate for structure 5.5
.................................................................................................................... 40
Figure 5.5 Graphical representation of Energy vs Reaction Coordinate for structure
5.7.1 ............................................................................................................ 42
Figure 5.6 Graphical representation of Energy vs Reaction Coordinate of the zoomed
in (2.03 Å to 1.78 Å) relaxed scan for structure 5.7.1 ................................ 44
Figure 5.7 Graphical representation of Energy vs Reaction Coordinate of the zoomed
in (1.83 Å to 1.35 Å) relaxed scan for structure 5.7.1 ................................ 45
Figure 5.8 Graphical representation of Energy vs Reaction Coordinate for structure
5.10 ............................................................................................................. 47
Figure 5.9 Graphical representation of Energy vs Reaction Coordinate for structure
5.12 ............................................................................................................. 50
Figure 5.10 Graphical representation of Energy vs Reaction Coordinate for structures
5.12 and 5.13. ............................................................................................. 52
Figure 5.11 Graphical representation of Energy vs Reaction Coordinate for structure
5.13 ............................................................................................................. 53
Figure 5.12 Graphical representation of Energy vs Reaction Coordinate for structure
5.12 ............................................................................................................. 55
Figure 5.13 Calculated reaction profile for the proposed mechanism. ........................... 60
LIST OF TABLES
Table 5.1 Calculated energies of the local minima. ....................................................... 33
Table 5.2 Calculated energies of the transition structures. ............................................ 35
Table 1.1 Summary of Energies ..................................................................................... 63
Table 2.1 Heats of Formation ........................................................................................ 66
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LIST OF ABBREVIATIONS
CPU Central Processing Unit
DEC Digital Equipment Corporation
DFT Density Functional Theory
G98W Gaussian 98 Windows
GTOs Gaussian-Type Orbitals
GUI Gaussian User Interface
HF Hartree Fock
IR Infra Red
LCAO Linear Combination Atomic Orbitals
MM Molecular Mechanics
NMR Nuclear Magnetic Resonance
PCU Pentacyclo-Undecane
PES Potential Energy Surface
RFO Rational Functional Optimization
RHF Restricted Hartree Fock
SCF Self Consistent Field
SE Semi-Empirical
SP Single Point
STOs Slater-Type Orbitals
STQN Synchronous Transit-Guided Quasi Newton
TS Transition-State/Structure
UV Ultra Violet
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CHAPTER 1
1. INTRODUCTION
1.1 Computational Chemistry and Molecular Modeling
The terms “theoretical chemistry”, “computational chemistry” and “molecular
modeling” are used interchangeably and indeed most molecular modelers use all three
concepts to describe various aspects of their research.
“Theoretical chemistry” is often considered synonymous with quantum mechanics,
whereas computational chemistry encompasses not only quantum mechanics but also
molecular mechanics, minimizations, simulations, conformational analysis and other
computer-based methods for the understanding and the prediction of the behaviour of
molecular systems. However theoretical chemistry is a subfield of chemistry where
mathematical methods are used in combination with the fundamental laws of physics to
study chemical processes.1 In particular, this involves the breaking of new ground that
ultimately leads to the writing of new mathematical codes or software that can model
certain aspects of a chemical structure.
Computational chemistry has therefore become one of the mainstays of modern
industrial and academic chemistry that makes use of the established codes or software to
study chemical systems. The ever-increasing power of modern computers coupled with
the development of new theoretical approaches can be used for accurate and precise
prediction of molecular properties.2 It is interesting to note that computational
chemistry accounts for roughly a third of the super computer usage worldwide.
Computer methods are used extensively to solve chemical problems that would be
intractable or even impossible experimentally.3
Computer-aided molecular design became a subject worthy of discussion in the media
around 1981, with the advancement of sophisticated computer graphics hardware.3 The
general aims of computational chemistry is the characterization and the prediction of
chemical structures and their stability; the prediction of NMR, IR and UV spectra, the
prediction of thermodynamic data, and the simulation and modulation of reaction
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courses.4 To achieve this computational chemists are using advanced computer
software that enables them to gain insight into chemical processes and to avoid time-
consuming and expensive experiments.3 This approach does not replace the traditional
wet chemistry experiments but it is a powerful aid in the understanding of experimental
observations and the prediction of new reaction pathways. Some methods can be used
to model not only stable molecules, but also short-lived, unstable intermediates and
even transition-states which are required for kinetic information. In this way, they can
provide information about molecules and reactions that may be impossible to obtain
experimentally. Computational chemistry is therefore both an independent research
area and a vital adjunct to experimental studies.
1.2 Pentacyclo-undecane (PCU) Cage Compounds
During the past half-century,5,6,7,8
many research groups focused on the synthesis and
chemistry of novel polycyclic cage molecules. Davis and co-workers9 were responsible
for discovering that 1-amino-adamantane 1.3, commonly known as amantadine, exhibits
antiviral activity thus realizing that polycyclic cage molecules also have the potential as
biologically active agents.
An unexpected observation10
of the biological activity profile of amantadine revealed
that it could be beneficial to patients with Parkinson‟s disease. The hydrocarbon cage
of amantadine has the ability to cross the blood-brain barrier and to enter the central
nervous system10
due to the hydrophobicity of the "cage" despite the fact that the amino
group is protonated at physiological pH. Drugs containing hydrocarbon cage moieties
promote their transport across cell membranes and increase their affinity for lipophilic
regions in receptor molecules.11
While the hydrophobicity of the cage facilitates transport of the drug across membranes,
the size and stability of the cage inculcate the drug with a structural property which
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results in controlled release of the active ingredients of the drug, namely, stability
towards degradation. In practice this translates into slow metabolism of the drug. The
important implications of this is that the intervals between drug administration is
increased.12
Another factor that was found to influence the rate of release of the active
ingredient and the potency of the drug was the presence of the amantadine substituent in
the drug. It has been shown that the inclusion of amantadine has given rise to longer
time over which the drug is effective, greater potency of the drug and faster drug action.
Furthermore the nature of the substituent influences the specificity of the drug to
antibacterial13
, anabolic14
and analgesic action.15
A number of cases16,17,18
have recently demonstrated the potential therapeutic value of
novel pentacyclic cage compounds. These compounds have promising potential as an
important new class of medicinal and pharmaceutical agents and might extend the
existing range19,20,21
of bio-active pentacyclo-undecane compounds. Further
investigation is required into the influence of the unique steric distribution of important
functional groups around a rigid cage structure on the pharmacological activity.
The Diels-Alder 1.6 adduct of cyclopentadiene 1.4 and p-benzoquinone 1.5 produces
the PCU dione 1.1. The reaction in which the dione 1.1 is synthesized is carried out by
intramolecular photocyclisation.22
h
OO
OO
+
O
O
1.4 1.5 1.6 1.1
Treatment of the dione 1.1 with Strecker reagents (HCN, NH4OH) unexpectedly
produced23
the -lactam compounds 1.2a-1.2c. Strecker reactions normally produce
cyanohydrins or amino nitriles.24
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O
O
C
NH
OH
O
R
C
N
OH
OH
R
Lactam Lactim1.1 1.2 a, R=OH1.2 b, R=CN
1.2 c, R=NH2
811811
The mechanism of this unique one pot conversion23,25,26,27
is not well understood. The
mechanism proposed by the authors23,25,26
was based on basic chemical principles and
is discussed below.
1.3 Lactam formation
In a previous study28
it was shown that the dione 1.1 is easily hydrated to form the
hydrates 1.7 and 1.8 in a 4:1 ratio. Thus it can be expected that this phenomenon should
also play a significant role in the nucleophilic addition reactions on the carbonyl groups
of the dione 1.1 in aqueous media.
It can subsequently be assumed that both 1.1 and 1.9 (in Scheme I shown below) can
participate in the formation of the dihydroxylactam 1.2. Nucleophilic attack is expected
to take place on the exo face of the carbonyl groups27
in the dione 1.1 or the hydrate 1.9
as a result of the proximity of the groups and could lead to the formation of 1.11.
Transannular cyclisation of 1.11 is expected to form the cyclic ether 1.12. Scheme I
shown below was one of the two proposed mechanistic pathways, based on basic
chemical principles and intuition.23,25,26,27
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1.121.11
1.10
1.9
H2O
H2ONaCN
H2O
H2ONaCN
1.1
OHOH
O
CN
OH
O
CN
OH
OHOH
CN
OH
O
H2O
Scheme I: Conversion of the dione to the cyclic ether
It was assumed that the cyclic ether 1.12 plays an intermediate role in the conversion of
the dione 1.1 to the dihydroxylactam27
1.2, since the “inversion” of a nitrile group can
be explained as in Scheme II shown below.27
The authors later showed27
that the
cyanohydrin 1.11 and 1.12 can be converted to the corresponding hydroxyl lactam 1.17
upon treatment with aqueous NaOH, providing experimental proof for their assumption
above.
The explanation27
for the conversion of 1.12 to 1.2 is discussed in Scheme II below.
Under basic reaction conditions, ring cleavage of the cyclic ether 1.12 forms the
intermediate 1.13 which is converted to intermediate 1.14. The electron deficient nitrile
carbon atom of the endo-orientated cyano group in 1.13 is in an extremely favourable
position to suffer attack from the nearby negatively charged oxygen atom. Intermediate
1.14 rearranges to form intermediate 1.15. Cyclisation of the intermediate 1.15 results
in the formation of lactim 1.16 and by implication the lactam 1.2.27
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H2O
1.151.161.2
C
OH
OH
ON
1.14
C
O
OH
O
N
H
OH
C
OO
N
H
1.13
OH
C
NH
OH
OH
O
1.12
C
N
OH
OH
OH
CN
OH
O
H2O
--
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Scheme II: Conversion of the cyclic ether
The above explanation does not account for the formation of a compound such as the
cyano hydroxylactam 1.2b (Scheme III).23,26
The nucleophilic attack of the hydroxide
on the cyclic cyanohydrin 1.12 is shown is Scheme III. This is necessary to “invert” the
cyanide group and is expected to be combined with the loss of the hydroxide group in
the cyclic cyanohydrin 1.12, whereby the cyanohydrin 1.19 should be formed. The
endo cyano group in 1.20 has an electron deficient carbon atom which is in an
extremely favourable position to suffer attack from the nearby negatively charged
oxygen atom thus producing the intermediate 1.21 and subsequently 1.22. The
rearrangement of 1.22 to the -cyano cation 1.23 is a postulated rearrangement and is
not a controversial one. This is so since -cyano cations of the general formula shown
in 1.24 are significantly stabilized by charge delocalisation through resonance structures
such as 1.25, even though this requires a portion of the charge to reside on a divalent
nitrogen.29,30
When attached to very unstable cations30
then only is the -donor effect
of cyano substituents manifested. The rearrangement proposed in Scheme III is
promoted since the negative charge on the nitrogen atom of 1.23 is sufficiently
stabilized by the adjacent carbonyl group to facilitate the rearrangement.
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+
+
+
-
--
-OH
CN
-
HOC O
CN
N
OOHNC
O
HOCN
HOC
O
NH C
N
CNC O
HO
HN
CNC O
HO
N
C C N
R
R
C C N
R
R
1.12 1.19 1.20
H2O
1.22 1.211.23
HOC
O
NHCN
1.24 1.25
1.2b
Scheme III: Nucleophilic attack of hydroxide to form the cyano hydroxy
lactam23,27
The formation of 1.2 shown in Scheme IV below is also based on the explanation of the
cyano group stabilization as postulated in Scheme III. The hydroxy group in 1.26
should similarly stabilize the cation and result in the formation of 1.2. Attack of
hydroxy anions instead of cyanide ions on the carbonyl carbon atom of the cyanohydrin
1.19 also results in the conversion of dione 1.1 to the dihydroxylactam.27
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1.2
HOC
O
NHOH
1.28
1.261.19
HOC
O
NH OH
O
HOC
N
HOC O
OH
N
HOC O
OH
HN
H2O
1.27
-
OH-
-
+
Scheme IV: Nucleophilic attack of hydroxide to form the
dihydroxylactam23,25,26,27
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CHAPTER 2
2. THEORETICAL TOOLS FOR MOLECULAR ORBITAL CALCULATIONS
2.1 Molecular Orbital Theory
Molecular orbital theory is an approach to molecular quantum mechanics which uses
one-electron functions or orbitals to approximate the full wavefunction. A molecular
orbital, (x, y, z), is a function of the cartesian coordinates of a single electron. Its
square, 2(or square modulus | |
2 if is complex) is interpreted as the probability
distribution of the electron in space. To describe the distribution of an electron
completely, the dependence on the spin coordinates , also has to be included. This
coordinate takes on one of two possible values (½) and measures the spin angular
momentum component along the z-axis in units of h/2 .31
2.2 Molecular Mechanics
Molecular mechanics (MM) simulations use the laws of classical physics to predict the
structures and properties of molecules. There are many different molecular mechanics
methods. Each one is characterized by its particular force-field. A force-field
comprises a set of equations defining how the potential energy of a molecule varies with
the locations of its component atoms and a series of atom types, defining the
characteristics of an element within a specific chemical context.32,33
The atom types
describe different characteristics and behaviour for an element depending upon its
environment. For example, a carbon atom in a carbonyl is treated differently than one
bonded to three hydrogens. The atom type depends on hybridization, charge and the
types of the other atoms to which it is bonded. Molecular mechanics calculations don‟t
explicitly treat the electrons in a molecular system. Instead, they perform computations
based upon the interactions among the nuclei. Electronic effects are implicitly included
in force-fields through parameterization. This approximation makes molecular
mechanics computations quite inexpensive computationally, and allows them to be used
for very large systems containing many thousands of atoms. However, it also carries
several limitations as well. The most important is that each force-field achieves only
good results for a limited class of molecules, related to those for which it was
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parameterized. No force-field can generally be used for all molecular systems of
interest. Neglect of electrons means that molecular mechanics methods cannot treat
chemical problems where electronic effects predominate. For example, they cannot
describe processes which involve bond formation or bond breaking. Molecular
properties which depend on subtle electronic details are also not reproducible by
molecular mechanics methods.34
2.3 Electronic Structure Methods
These methods use the laws of quantum mechanics rather than classical physics as the
basis for their computations. According to quantum mechanics, the energy and other
related properties of a molecule may be obtained by solving the Schrödinger equation:
H = E (2.1)
where, H = Hamiltonian, a differential operator which like the energy in classical
mechanics, is representative of the kinetic and potential energy of the
molecule,
E = numerical energy of the state, and
= corresponding wavefunction for molecular state.
The Hamiltonian used in the Schrödinger equation is that for nuclear motions,
describing the vibrational, rotational and translational states of the nuclei.35
Schrödinger‟s equation for molecular systems can only be solved approximately. Exact
solutions of the Schrödinger equation may only be obtained for the very simplest
molecules (e.g., H2) because of the inter-electronic repulsion terms in the equation,
where the motion of each electron depends on the motion of the other electrons, so
approximate methods have to be used for larger molecules, for example the variation
method.36
The two main classes of electronic structure methods are semi-empirical
methods and ab initio methods.33
Semi-empirical methods use parameters derived from
experimental data or high level ab initio calculations to simplify the computation. An
approximate form of the Schrödinger equation is solved which depends on having
appropriate parameters available for the type of chemical system in question.
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2.3.1 The Ab Initio Method
The term ab initio is given to computations which are derived directly from theoretical
principles with no inclusion of experimental data. The approximations are usually
mathematical approximations, such as using a simpler functional form for a function or
getting an approximate solution to a differential equation. The square of the
wavefunction 2 is interpreted as the probability density for the electrons within the
system. The first step in simplifying the general molecular problem in quantum mechanics
is in the separation of the nuclear and electronic motions. This is possible because the
nuclear masses are much greater than those of the electrons and, therefore, nuclei move
much more slowly. This separation of the general problem into two parts is called the
adiabatic or Born-Oppenheimer Approximation.37
Thus, the electron distribution within a
molecular system depends on the positions of the nuclei, and not on their velocities.
The advantage of ab initio methods is that they eventually converge to the exact
solution, once all of the approximations are made sufficiently small in magnitude.
However, this convergence is not monotonic. Sometimes, the smallest calculation gives
the best result for a given property.
The disadvantage of ab initio methods is that they are expensive. These methods often
take enormous amounts of computer CPU time, memory and disk space. In practice,
extremely accurate solutions are only obtainable when the molecule contains about half
a dozen electrons or less.
Restricted Hartree Fock (RHF) or Unrestricted Hartree Fock are the two forms of the
wave function that are used in quantum mechanic calculations. The RHF wave function
is used for singlet electronic states, for example, the ground states of stable organic
molecules. The UHF wave function is most often used for multiplicities greater than
singlets.38
The Mller-Plesset second order perturbation theory (MP2) specifies the
calculation of electron correlation energy. The MP2 option can only be applied to
Single Point calculations.39
In general, ab initio calculations give very good qualitative results and can give
increasingly accurate quantitative results as the molecules in question become smaller.31
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2.3.2 Semi-Empirical
Semi-empirical (SE) calculations are set up with the same general structure as a Hartree
Fock (HF) calculation. Within this framework, certain pieces of information such as
two electron integrals are approximated or completely omitted. In order to correct for
the errors introduced by omitting part of the calculation, the method is parameterized,
by curve fitting a few parameters or numbers in order to give the best possible
agreement with experimental data.40
The advantage of semi-empirical calculations is that they are much faster than the ab
initio calculations. The disadvantage is that the results can be erratic. If the molecule
being computed is similar to molecules in the database used to parameterize the method,
then the results may be very good. If the molecule being computed is significantly
different from anything in the parameterization set, the answers may be very poor.40
Semi-empirical calculations have been very successful in the description of organic
chemistry where there are only a few elements used extensively and the molecules are
of moderate size. However, semi-empirical methods have been devised specifically for
the description of inorganic chemistry as well.40
2.3.3 Density Functional Theory (DFT) Methods40
Density Functional Theory (DFT) is the third class of electronic structure methods that
have recently come into wide use. These methods are similar to the ab initio methods in
many ways. DFT calculations require approximately the same amount of resources as
the Hartree-Fock theory, but they produce results approaching the quality of the MP2
level of theory.
DFT methods include the effects of electron correlation, where electrons in a molecular
system react with each other's motion and attempt to keep out of each other's way. This
is what makes DFT methods more attractive than the expensive ab initio methods.40
DFT methods are based on the theory developed by Hohenberg and Kohn41
in which
they demonstrated that the ground state energy of any molecule can be described in
terms of the total electron density, in other words, each molecule has a unique
functional form which exactly determines the ground state energy and electron density
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(i.e. geometry) of the molecule.40,42
This system is different to the wave function
approach of ab initio techniques, where the complexity of the wave function increases
by a factor of 3N for an N-electron system (no spin included). For the DFT system, the
complexity of the function is less dependent to the system size since the electron density
has the same number of variables. The aim of DFT methods, therefore, is to design
functions which connect the electron density with energy.43
Kohn and Sham44
were
responsible for the introduction of orbitals which formed the basis for the use of DFT
calculations in computational chemistry.
The advantage of DFT is that only the total density is to be considered and to calculate
the kinetic energy with accuracy, orbitals need to be re-introduced. The disadvantage of
DFT is the derivation of suitable formulae for the exchange-correlation term. DFT
methods, however, have the ability to produce accurate results.42
2.3.4 Basis Sets45
A basis set is a mathematical description of the orbitals within a system used to perform
the theoretical calculation. The wavefunction , can be expanded in terms of a set of
atomic orbitals, in the linear combination of atomic orbitals (LCAO) method, to
give45
:
= c (2.4)
where c = molecular orbital expansion coefficient, and
= basis function of atomic orbital.
The coefficient c is varied to obtain the wavefunction , which will give the lowest
energy in the Schrödinger equation. The more vibrational parameters used to describe
an individual orbital, the lower the energy. However, a situation is reached when the
energy is no longer decreased when the number of vibrational parameters is increased
and then the best single determinant wavefunction is obtained. When this occurs,
changing the wavefunction , by an infinitesimal amount will not alter the energy. The
number and quality of the atomic orbitals determine the quality of the molecular
orbital . If there are many electrons in a molecule then the number of atomic integrals
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required increases rapidly and can be as many as several million for quite small
molecules. For this reason a fast computer which has a large storage capacity is
essential. The two types of atomic basis functions are Slater-type atomic orbitals (STOs)
and Gaussian-type atomic orbitals (GTOs). The former is not well suited to numerical
work, and their use in practical molecular orbital calculations has been limited. Almost all
modern ab initio calculations employ GTO basis sets. These basis sets, in which each
orbital is made up of a number of Gaussian probability functions, has considerable
advantages over STOs. The Gaussian series of programs deals, as the name implies,
almost exclusively with Gaussian-type orbitals.
2.3.4.1 Minimal Basis Sets45
These contain the minimum number of basis functions needed for each atom. Minimal
basis sets are characterized by fixed-size atomic-type orbitals. Each orbital is
characterized by its coefficient and its exponent. They therefore do not have the
capability to expand or contract because the exponent is fixed.
(i) The STO-3G Basis Set45
The series of minimal basis sets consists of expansions of Slater-type orbitals (STOs).
The STO-3G basis set yields properties that are reasonably close to limiting values and
in view of the relative computational times of the various expansions, it is this level that
has been selected as an optimum compromise for widespread application. Another
possible exception to the use of the three-gaussian expansion as the standard minimal
basis level occurs in the consideration of the properties of weakly-bound complexes
where long-range forces are important.
2.3.4.2 Split Valence Basis Sets45
The way that a basis set can be made larger is to increase the number of basis functions
per atom. Inclusion of two sets of isotropic p-functions in the representation, one tightly
held to the nucleus and the other relatively diffuse, will allow independent variation of
the radial parts of the two sets of p-functions, thus producing more contracted or more
diffuse functions that would be suitable for the descriptions of say, s- and p-systems,
respectively. A basis set formed by doubling all functions in a minimal representation
-
15
is known as a double-zeta basis, while one in which only the basis functions for the
outer valence shells are doubled, is known as a split-valence basis set.
(i) The 3-21G Basis Set45
The 3-21G basis set defined through the fourth row of periodic table of elements, typify
representations in which two basis functions, instead of one have been allocated to
describe each valence atomic orbital. Except for hydrogen, the 3-21G basis sets are
employed as is, that is, without rescaling of the valence functions to account better for
changes that might occur as a result of molecule formation.
2.3.4.3 Polarised Basis Sets45
While split valence basis sets allow orbitals to change their size, but not their shape,
polarised basis sets remove this limitation by adding orbitals with angular momentum
beyond that which is required for the ground state description of each atom. For
example, polarised basis sets add p-functions to hydrogen atoms, d-functions to the
main groups and f-functions to transition metals.
(i) The 6-31G* Basis Set45
The 6-31G* basis set was originally proposed for first-row atoms and later extended to
second-row elements. The 6-31G* basis set is constructed by the addition of a set of six
second order (d-type) gaussian primitives to the split valence 6-31G basis set
description of each heavy (non-hydrogen) atom.
(ii) The 6-31G** Basis Set45
The basis set described above does not allow for any polarisation of the s orbitals of
either hydrogen or helium atoms. The 6-31G** basis set is identical to 6-31G* except
for the addition of a single set of gaussian p-type functions to each hydrogen and helium
atom.
(iii) The 3-21G* Basis Set45
The 3-21G* basis set for second-row elements are constructed directly from the
corresponding 3-21G representations by the addition of a complete set of six second-
order gaussian primitives. Although the resulting representations contain the same
number of atomic basis functions per second-row atom as the 6-31G* polarisation basis
-
16
set previously described, these are made up of significantly fewer gaussians (three
instead of six for each inner-shell atomic orbital, and two gaussians instead of three for
the inner part of the valence description).
2.3.4.4 Basis Sets Incorporating Diffuse Functions45
The basis sets that have been discussed thus far are more suitable for molecules in
which electrons are tightly held to the nuclear centers than they are for species with
significant electron density far removed from those centers. Calculations involving
anions pose special problems. Since the electron affinities of the corresponding neutral
molecules are typically quite low, the extra electron in the anion is only weakly bound.
One way to overcome problems associated with anion calculations is to include in the
basis representation one or more sets of highly diffuse functions. These are then able to
describe properly the long-range behavior of molecular orbitals with energies close to
the ionization limit.
(i) The 3-21+G and 3-21+G* Basis Set45
The 3-21+G basis set for the first-row elements and the 3-21+G* basis set for the
second-row are constructed from the underlying 3-21G and 3-21G* representations by
the addition of a single set of diffuse gaussian s- and p-type functions. For first-row
elements with lone pairs, the effects of diffuse and polarisation functions are
complementary to some extent. Hence, the energies of processes involving changes in
the number of lone pairs, for example, protonation, hydrogen bonding, or other
interactions, are improved at diffuse-orbital-augmented levels even when large basis
sets are used.
This basis set produces entry level results and is relatively inexpensive in terms of
computer resources and time. If the system under study involves only C, H, N and O
atoms, the system (with RHF) is ideal for student training. Upgrading to a better basis
set is trivial and one does not loose too much valuable time during your learning curve.
2.3.5 Hartree-Fock Theory46
The most common type of ab initio calculation is the Hartree Fock (HF) calculation in
which the primary approximation is the central field approximation. This means that
the Coulombic electron-electron repulsion is not specifically taken into account.
-
17
However, its net effect is included in the calculation. This is a variational calculation,
meaning that the approximate energies calculated are all equal to or greater than the
exact energy. Because of the central field approximation, the energies from HF
calculations are always greater than the exact energy and tend to a limiting value called
the Hartree Fock limit. This variational principle leads to the following equations
describing the molecular orbital expansion coefficients, derived by Roothaan and
Hall46
:
01
vi
N
v
viv cSF (2.2)
where, = 1, 2, …, N and cvi = molecular orbital expansion coefficient.
Equation (2.2) can be re-written in matrix form:
FC = SC (2.3)
where F = the Fock matrix,
S = the overlap matrix,
= the diagonal matrix, and
N = one-electron function or basis function.
Equation (2.3) is not linear and therefore must be solved iteratively. The procedure by
which this is carried out is called the self-consistent field (SCF) method.46
Slater and
Gaussian type orbitals are used in these equations. The Hartree Fock equations are
applicable no matter how many electrons there are in the molecule. However, this
theory does not include a full treatment of the effects of electron correlation, that is, the
energy contributions arising from electrons interacting with one another. It is
reasonably good at computing the structures and vibrational frequencies of stable
molecules and some transition-states. As such, it is a good base-level theory.46
The second approximation in HF calculations is that the wave function must be
described by some functional form, which is only known exactly for a few one electron
systems. The functions used most often are linear combinations of Slater-type orbitals
-
18
(STOs) or Gaussian-type orbitals (GTOs). The wave function is formed from linear
combinations of atomic orbitals or more often from linear combinations of basis
functions. Because of this approximation, most HF calculations give computed energy
greater than the Hartree Fock limit. The exact set of basis functions46
used is often
specified by an abbreviation, such as STO-3G or 6-31g**.
-
19
CHAPTER 3
3. TRANSITION STRUCTURE MODELING47
A transition structure (TS) is the molecular species that is represented by the top of the
potential energy graph in a simple one dimensional reaction coordinate shown in Figure
3.1. In order to determine the energy barrier to reaction rate, the energy of this
transition-state species is needed. The geometry and energy of a transition structure
include important pieces of information for describing reaction mechanisms.
En
erg
y
Reaction coordinate
Reactants
Products
Transition state
Precomplex
Precomplex
Figure 3.1 Points on a simple reaction coordinate48
A transition structure is defined mathematically as the geometry which has a zero
derivative of energy with respect to moving every one of the nuclear coordinates and
has a positive second derivative of energy of all but one geometric movement which has
a negative curvature.47
This description however, describes many structures other than
a reaction transition, for example an eclipsed conformation or the intermediate point in a
ring flip, a simple rotation of a methyl group or any structure with a higher symmetry
than the ground state of the compound.
It is difficult to predict what a transition structure will look like without the aid of
computer simulation. Such a prediction might be made based on a proposed
mechanism, which may be incorrect. The potential energy surface (PES) around the
transition structure is often much more flat than the surface around a stable geometry,
thus there may be large differences in the transition structure geometry between two
-
20
seemingly very similar reactions and with very small differences in energy.47
It has
however been possible computationally, to determine transition structures, although it is
not always easy. Experimentally, it has only become possible to examine reaction
mechanisms directly using femtosecond pulsed laser spectroscopy. It will be some time
before these techniques can be applied to all of the compounds that are accessible
computationally. Furthermore, these techniques yield indirect information such as
vibrational information rather than a likely geometry for the transition structure.47
Synthetic approaches to obtain information about transition-states are also limited to
very special cases, such as the static SN2 transition-state 3.1 shown below.49
OMeMeO C
OMeMeO
3.1
An X-ray structure of the above mentioned molecule was reported.49
3.1 Transition-state modeling with empirical force-fields50
The transition-states (TS) involved in a conformational equilibrium can be studied using
the ground-state parameters developed from geometries and heats of formation of stable
molecules. One of the earliest applications of empirical force-fields to organic
chemistry was Westheimer‟s study of rotational barriers in biphenyls, which begun in
the 1940‟s and was reviewed in the 1950‟s.51,52
In the subsequent half-century, there
have been many studies of conformational rate processes in organic systems. Ground-
state parameters are fully appropriate to such studies; transition-states have more torsion
and non-bonded strains than energy minima, but have the same type of bonds.
However, when bonds are being formed or broken, the parameters suitable for ground
states are no longer appropriate.50
Consequently, parameters must be developed to
model partial bonds in a quantitative way when using semi-empirical methods.
Theoretical studies on transition structures of several class of reactions have shown that
-
21
bond lengths and other geometrical parameters in transition structures have a relatively
narrow range of values. For example, in pericyclic reactions, forming C-C single bonds
generally have bond lengths from 1.95 to 2.28 Ǻ, even though some of these reactions
are very exothermic and others are thermoneutral.53
Radical additions to alkenes have
been studied for a variety of carbon- and oxygen- centred radicals, with constant angles
and bond lengths around 105 ± 3 and 2.25 ± 0.01 Ǻ, respectively.54
Hydroborations of alkenes and alkynes have been studied for a variety of alkylboranes
and substituted ethylenes. Even in the presence of high steric hindrance, the formation
and the breaking up of bond lengths are relatively constant.55
These examples support
one critical procedure often used in simple force-field transition-state modeling, namely
the breaking and forming bonds are either fixed at some lengths, or these values are
treated as energy minima. The energy is actually a maximum for the reaction
coordinate, but the simple expedient of calculating transition-states as minima has been
used in many cases. The TS is a saddle point along the free-energy surface. This saddle
point has a negative curvature in only one direction. The negative force constant
corresponds to motion along the reaction coordinate. All other vibrational motions have
positive force constants exactly like energy minima on potential energy surfaces.
However the negative force constant causes the transition structures to have unique
properties different from force minima.
As mentioned at the beginning of this chapter the energy surface at the TS is often very
flat. This normally causes problems for using SE methods to determine transition-
states.
Due to the structural features and characteristics in the highly strained moieties, the
outer limits of what can be prepared and studied regarding thermodynamic stability and
kinetic reactivity is investigated. Studies of such systems provide an excellent test for
existing chemical theory and thus perhaps furnish the best opportunity for advancing the
frontiers of our chemical knowledge.7 It is therefore likely that SE methods will give
poor results for transition-states involving the cage structure.
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22
3.2 Locating minima on the seams of intersecting semi-empirical PES56
Although the word "minima" is used above, it actually implies the lowest maximum on
the energy surface between the product and the reactant.
In recent years, the location of TS‟s has become relatively routine due to improvements
in the optimization algorithms. The TS can be refined to any desired accuracy, if the
energy-generating function is of the ab initio type. However, the practical consideration
usually limits both the size and the level of sophistication. Semi-empirical methods can
be used for somewhat larger systems, but in this case only comparison with experiments
or accurate ab initio calculations can be used to assess the quality of results.
The primary concern in many applications is not in the prediction of absolute values of
activation parameters, but rather on how they vary for closely related systems. There
are two or more reaction pathways that have activation energies which differ by only a
few kcal mol-1
and synthetic sequences are often dependent on these pathways. In such
cases, the desired reaction can often be favoured by a careful selection of substituents at
specific sites in the molecule being studied. According to the influence they have on a
reaction, substituents can be divided into two limiting cases; those of a “structural” or
“steric” nature and those which mainly exert an “electronic” influence. While the latter
requires an explicit description of the electrons in the system, the former can be
modeled by less rigorous theoretical methods, i.e. SE or MM. However, the above
classification of substituents will depend on the given reaction.56
The treatment of a TS as a minimum on the PES is a more fundamental problem with
Houk's approach.56
These directions can in general be written as a linear combination
of internal or cartesian coordinates. Three different strategies can be employed in
transferring the ab initio structure to the force-field model:
(i) The "fixed atom” procedure, where the atoms directly involved in the
reaction are frozen by fixing their cartesian coordinates;
(ii) The “fixed parameter” procedure, where certain internal coordinates are
constrained by assigning large force constraints to these variables; and
-
23
(iii) The “flexible parameter” procedure, where all atoms are allowed to
move.
As an alternative, the equivalent of a TS in a force-field environment can be defined as
the lowest energy structure linking the reactant and product. When different sets of
parameters are needed for describing the two end points, the TS equivalent is thus the
lowest energy structure on the seam of the intersecting PES‟s. The advantage of the
current strategy over Houk‟s TS modeling, is that only information regarding the two
minima (reactant and product) on the PES is needed, and such data are in principle
accessible by experiments. The disadvantage is that the functional form of the energy
must be reasonably accurate over a wider range of geometries than just near the
"minima".
SE methods were initially employed in this study for the determination of transition
structures. It was found that the method indeed produced poor quality results, and in
many cases the correct transition structures could not be found. It was therefore decided
to use ab initio techniques for the determination of the required transition-states.
3.3 Transition-structure modeling of a PCU Cage Lactam using ab initio
methods
The TS algorithm uses a combination of Rational Functional Optimization (RFO) and
linear search step to search for the lowest maximum on the energy surface between the
reactant and the product. A mathematical algorithm by default uses a crude semi-
empirical (SE) guess (INDO guess is used for the first-row systems, CNDO for the
second-row, and Hückel for the third-row and beyond)57
for the initial start structure to
the solution of the transition-state wave function. If the starting structure is too far from
the real maximum on the energy surface, the search algorithm would not find the correct
transition structure within the multitude of local maxima.
In order to obtain a better guess for the solution to the wave function, one could use
either of two methods:
(i) carry out a Single Point (SP) calculation of the starting structure at the
same level of theory using the same basis set, followed by an additional
-
24
step in the calculation, with the option to read the guess (guess = read)
from the calculation done (obtained from the checkpoint file) at the
required level and basis set or
(ii) use the option “CALCFC” where the force constant is calculated at the
required level/basis set and used to start the solution to the wave
function. The second option is considerably more expensive in terms of
resources and time, since the “CALCFC” option does the same
calculation as required for a frequency calculation. (see Chapter 4 for a
discussion on CALCFC).57
A third method is the QST2 and QST3 methods in Gaussian which has the facility for
automatically generating a starting structure for a transition-state optimization based
upon the reactants and products that the transition-state connects,58
known as the
Synchronous Transit-Guided Quasi Newton (STQN) method. This method uses a
quadratic synchronous transit approach to get closer to the quadratic region of the
transition-state, and uses a quasi-Newton or eigenvector-following algorithm to
complete the optimization.57
QST2 requires two molecule specifications, i.e., the
reactants and products. QST3 requires three molecule specifications, that is, the
reactants, the products and an initial structure for the transition-state, respectively. The
QST2 and QST3 methods were utilized in this study, but it did not yield favourable
results.
A fourth option is to use the “CALCALL” option in combination with the transition-
state optimization. During this calculation, the force constant will be calculated for
each optimization step. This option is very expensive and is only used as a last resort
when the methods above do not show positive results.
In this study, the TS for a PCU cage lactam were found by locating maxima on the
potential energy surfaces using Restricted Hartree-Fock theory and the 3-21+G* basis
set. For each TS (see Chapter 5 for the mechanistic pathway) a SCAN calculation (see
Chapter 4 for computational details) was performed to establish the maxima. The final
structure corresponds to a minimum on the potential energy surface, or saddle point. In
order to determine the nature of the stationary point found, a frequency calculation was
-
25
performed. The frequency output file has information that is critical in characterizing
the stationary point, namely, the number of imaginary frequencies and the normal mode
corresponding to the imaginary frequencies. Imaginary frequencies are listed in the
output file as negative numbers. By definition, a structure which has n imaginary
frequencies is an nth
order saddle point. Thus transition structures are usually
characterized by one imaginary frequency since they are first-order saddle points.59
The
movement of atoms associated with the imaginary frequency should follow the atoms
on the reaction coordinate between the reactant and product. Any reaction profile will
have only one transition-state. Although many intermediate transition structures on the
reaction profile might exist, only one of them will be the rate-determining step. That
transition structure is defined as the transition-state for the reaction.
-
26
CHAPTER 4
4. COMPUTATIONAL DETAILS EMPLOYED
This study was carried out on a series of related molecules using the GAUSSIAN 9860
program implemented on a DEC Alpha DS20 workstation with two CPU‟s and a
Pentium II desktop computer. A true 64 bit operating system was implemented on the
DEC Alpha workstation with a parallelized version of GAUSSIAN 98 program.
4.1 The GAUSSIAN 98 Program
GAUSSIAN 98 (G98) is one of the series of electronic structure computer programs
which began with GAUSSIAN 70, GAUSSIAN 92 and GAUSSIAN 94. GAUSSIAN
03 is the latest program which was released in April 2003.
The GAUSSIAN programs are general-purpose programs capable of performing ab-
initio Hartree-Fock (HF) molecular orbital calculations based on the linear combination
of atomic orbitals (LCAO) approach. As the name implies, the program deals mainly
with Gaussian-type orbitals, which has been described in Chapter 2. However new
methods have been added to G98 so as to improve optimization procedures for
transition-state calculations.
In addition, G98 can compute energies, molecular structures, vibrational frequencies
and numerous other molecular properties for systems in the gas phase and in solution,
including the ground state and excited states.
The input section to the GAUSSIAN programs consists of the molecular charge and the
multiplicity, the symbols of the constituent atoms and a definition of the molecular
structure, either in the form of cartesian coordinates or the Z-matrix notation, which
defines the molecular geometry in terms of bond lengths, bond angles and dihedral
angles. The task to be performed, i.e. whether a single-point calculation, geometry
optimization or frequency calculation, must also be specified, together with the
appropriate basis set and the level of theory.
The 3-D structures on the CD accompanying this thesis are written in the Gaussian input
file format (gjf). The geometries of the structures can be viewed by using the Gausview
-
27
program or the freeware (Molekel) program. The installation files for Molekel can be
downloaded from the site: www.cscs.ch/molekel
4.2 The GaussView Program61
GaussView is a Graphical User Interface (GUI) program designed to simplify and
extend the use of the Gaussian 98 program. In this study, Gaussview was used to build
and edit molecules, set up and submit Gaussian jobs, and to display and use the results
from the Gaussian jobs. However, GaussView is not directly integrated into the
Gaussian program system, but acts as a front-end/back-end processor to facilitate its use
on a desktop computer workstation. GaussView was also used to verify the animation
of atoms associated with the negative eigenvalue of the different transition-states.
4.3 The SCAN Calculation
The scan option was exclusively used as an aid to finding an approximate starting
structure for a normal transition-state optimization. A relaxed SCAN calculation
involved changing of bond length from reactants to products, in a step-wise manner.
The only constraint in this calculation is the required reaction coordinate (i.e. bond
length, angle or the dihedral angles). The rest of the molecule is then optimized to find
the lowest possible structure and energy, subject to the imposed constraints, after which
the reaction coordinate is modified by a prescribed value and in the next step the
procedure is repeated. In this study only relaxed SCAN calculations were used. The
energy of each step was plotted against the reaction coordinate. By inspecting the
different structures at each step of the scan job, one could follow the course of the
reaction. The approximate starting structure for a full (non-restrained) transition-state
optimization was obtained by manually extracting the coordinates of the structure
closest to the maxima on the energy vs. reaction coordinate plot. The TS was verified
by performing a frequency (FREQ) calculation. A frequency calculation produces only
one negative eigenvalue, which is usually associated with the movement of atoms
involved in either bond breaking or bond formation.
http://www.cscs.ch/molekel
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28
4.3.1 Commands used during a SCAN or a TS Search60
(i) GDIIS
Specified the use of the modified GDIIS algorithm, recommended for use with
larger systems, tight optimizations and molecules with flat potential energy
surfaces. It is the default for semi-empirical calculations. This command makes
use of a smaller step-size down the potential energy valley.60
(ii) MODREDUNDANT or ModRed
Used in geometry optimization, e.g. the bond length to be scanned. The specific
coordinates that are to be constraint (Modified Redundant coordinates = ModRed) is
specified below the Cartesian coordinates.60
(iii) TS
Used in a search to request the optimization to use a mathematical search algorithm
which aims to find a local maxima on the potential energy surface (i.e. a transition-
state) rather than a local minimum.60
(iv) NOEIGENTEST
The default transition-state search in G98 makes use of the EigenTest. If only one
imaginary frequency is found, the calculation continues to find the transition-state
associated with this negative eigenvalue. If more than one imaginary frequency
exists, the default routine is to terminate the calculations. Since it is practically
almost impossible to find a starting structure for a TS with one and only one
negative eigenvalue, the default TS calculation terminates very often. In order to
overcome this oversensitive search criterion, one uses the "NoEigenTest" option
which overrides the default search criteria in G98.60
(v) CALCFC
Specified the force constants be computed at the first point using the current method
(available for the HF, MP2, CASSCF, DFT, and semi-empirical methods only). By
default Gaussian uses a MNDO (semi-empirical) guess for the solution to the wave
function of the specified system. The optimization uses this guess as starting point
-
29
where, after the ab initio calculation is “built” on this starting point. Since the
MNDO guess is based on a rather crude or inaccurate method, the calculation could
sometimes follow a wrong solution for the wave function. One observes this by
inspecting the geometry of the structure produced by the optimization, there are
basically two choices: (a) start with a better structure, i.e., a structure that was
optimized at a lower level of theory or (b) if a better starting structure was already
used, use the CalcFc option.3,60
4.4 Calculation Details
In this study the procedure for all structures was the structures in HyperChem Version
5.62
In order to remove any van der Waals contacts or overlap, an energy minimization
was performed with ChemOffice,63
using a molecular mechanics force-field. The final
structures were saved as input files for GAUSSIAN 9860
(G98W). A low level (STO-
3G) ab initio (full geometry optimization) calculation was carried out on the
unconstrained molecule. The (OPT) keyword requests that a geometry optimization be
performed. The geometry is adjusted according to a mathematical algorithm to follow
the energy surface “down hill” until a stationary point on the potential energy surface is
found. Note that the algorithm by nature would not overcome local minima and special
techniques such as automatic or manual conformational searches or molecular
dynamics64,65,66
should be employed to overcome, mainly, rotational energy barriers.
This is more problematic in flexible molecules such as peptides.
For the Hartree-Fock, DFT and semi-empirical methods, the Berny algorithm is the
default algorithm for minimizations to a local minimum and optimizations to transition-
states and higher order saddle points.57
The purpose of a geometry optimization is to
locate the lowest energy of the molecular structure that is in close proximity to the
specified starting structure. The output structure of the optimization was submitted on a
DEC Alpha workstation using the RHF/3-21+G* level of theory. A SCAN calculation
was performed to locate the maxima on the PES. The starting structure for the
transition-state (TS) calculation was obtained by manually extracting the coordinates of
the structure that correspond to the maxima. Once a TS was obtained a frequency
calculation was carried out to verify that there is only one negative eigenvalue.
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30
CHAPTER 5
5. RESULTS AND DISCUSSION
5.1 Introduction
The purpose of this study is to utilize computational techniques in the determination of
the proposed mechanistic pathways, for the one-pot conversion of pentacyclo-undecane
(PCU) dione 5.1 to pentacyclo-undecane cage lactam 5.2.23,25
OO
NaCN
H2O
OH
C O
NHHO
5.1 5.2
This chapter focuses on the calculation results obtained for the proposed mechanistic
pathway as illustrated in Figure 5. 1 (see Page 31). The basis set incorporating diffuse
functions, that is, 3-21+G* was used in this study as discussed in Chapter 2. The
mechanism, which was one of two possible pathways that was proposed23,25,26,27
is
based on basic chemical principles and intuition and was discussed in Chapter 1. In this
study, the pathway proposed above was used as basis for the calculation of the reaction
profile. Based on the observations made during this investigation (i.e., a starting
structure optimized to a different intermediate or transition structure) the proposed
pathway was also modified. The energies for the calculated mechanism proposed in this
study are depicted in the form of a reaction profile (Figure 5.2 Page 32).
The reaction profile was investigated by first calculating the geometries and energies of
the minima on the energy surface (Figure 5.2). Since the cage is very rigid, very few
problems with conformational isomers of higher energies, were experienced.
The structures and energies of the corresponding TS's were then calculated. The
procedure for locating the TS's is described in Chapter 3, Section 3.3.
-
31
5.1
NaCN
H2O
5.3.1
H2O
5.4
-OH
+ NaOH
5.9
5.2
5.5 5.6 5.7.1
5.10 5.11
N
OH
OH
C
5.13
O
O O
O-
CN
O
CN
OH
O
HO
CN
O
HO
CN
O
HO
HO
OH
CN
OH
HO
HN
OH
C
OHO
HN
O
OH
C
HOHN
O
OH
C
HO
5.12
HN
O
OH
C
HO
CN
OH
Figure 5. 1 Modified23,25,26,27
mechanism for the conversion of the dione 5.1 to
the lactam 5.2.
-
32
5.1
5.2
5.3.1
5.4
5.5
5.7.1
5.9
5.12
5.13
-100
-50
0
50
100
150
200
250
300
Reaction Co-ordinate
5.105.65.11
O
OO
CN
OH
CN
O
HO
CN
O
HO N
OH
OH
C
HO
OH
CN
OH
HO
HN
OH
C
OHO
HN
O
OH
C
HO
HN
O
OH
C
HO
HN
O
OH
C
HO
CN
OH
OHO
O-
CN
O
Rel
ati
ve
ener
gy
(k
cal
mo
l-1)
Figure 5.2 Calculated reaction profile for the proposed mechanism.
The cartesian coordinates of all the 3D structures presented in this Chapter are included on the CD accompanying this Thesis.
-
33
5.2 Local minima on the energy profile
Structures 5.1, 5.4, 5.6, 5.9, 5.11, and 5.2 shown below are stationary points classified
as local minima on the energy profile of the reaction.
5.1
O
O
5.4
O
CN
OH
5.6
CN
O
HO
5.9
N
OH
OH
C
HO
5.11 5.2
HN
OH
C
OHOHN
O
OH
C
HO
An energy minimization was performed for each of the above structures to remove
any Van der Waals contacts or bond overlap. The calculated energies for the above
structures are presented in the Table 5.1. Note that the only rotational flexibility in
these structures are the C-OH bonds illustrated above. The corresponding bonds were
rotated at angles of 30º intervals and re-optimized to ensure the lowest possible
isomer was obtained.
Table 5.1 Calculated energies of the local minima.
Relative energiesa
Structure numberb
STO-3G/Hartrees 3-21+G*/Hartrees 3-21+G*/kcal mol-1
5.1 -565.0296 -568.9543 0
5.4 -656.7586 -661.3513 27.66
5.6 -656.8147 -661.3657 18.66
5.9 -731.7744 -731.9821 19.79
5.11 -731.8214 -737.0049 6.19
5.2 -771.8263 -737.0314 -10.46
aRelative energies are expressed in Hartrees and kcal mol
-1, performed at the HF level using the STO-
3G basis set, followed by a higher 3-21+G* level of theory.
bStructure number as per proposed reaction mechanism shown in Figure 5.1.
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34
From the results presented in Table 5.1, it is evident that the higher level basis set
produces a significantly lower energy value. (Note that energies obtained with
different basis sets cannot be directly compared). In addition, when the reaction
profile as shown in Figure 5.2, was plotted, it was also evident that the energy values
(heats of formation) confirm that structures 5.1, 5.4, 5.6, 5.9, 5.11, and 5.2 are indeed
minima.
5.3 The Transition Structures (TS)
Structures 5.3.1, 5.5, 5.7.1, 5.10, 5.12 and 5.13 shown in Figure 5.1 are characterized
as the following transition-state structures.
5.3.1
O-
CN
O
5.5
CN
O
HO
5.7.1
CN
OH
O
HO
5.10
OH
CN
OH
HO
5.12
HN
O
OH
C
HO
5.13
HN
O
OH
C
HO
The cartesian coordinates of all the 3-D structures presented in this Chapter are
contained on the CD accompanying this Thesis. Please refer to Chapter 3 for a
discussion on the techniques used to determine the transition-states below.
An explanation of the different types of SCAN calculations used in the route section
for the location of the various transition structures is given in Chapter 4. The
calculated energies for the above structures are presented in the Table 5.2.
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35
Table 5.2 Calculated energies of the transition structures.
Relative energiesa
Structure numberb
3-21+G*/Hartrees 3-21+G*/kcal mol-1
5.3.1 -660.7663 127.39
5.3.2 -736.9142 63.11
5.5 -661.2569 86.90
5.7.1 -736.3609 188.43
5.7.2 -812.5102 77.74
5.10 -736.8741 88.27
5.12 -736.8873 79.95
5.13 -736.8880 79.55
aRelative energies are expressed in Hartrees and kcal mol
-1, performed at the HF level using the 3-
21+G* level of theory.
bStructure number as per proposed reaction mechanism shown in Figure 5.1.
In the discussion of the results for each of the transition-states, the three dimensional
(3-D) structures have been referred to as Scan start or Scan end. Scan start implies
the starting structure that was submitted for the calculation. Scan end refers to the
corresponding structure that was obtained at the end of the scan calculation. By
inspecting the different structures at each step of the scan job, one could follow the
reaction profile and by plotting the corresponding energies vs. the reaction coordinate,
one can obtain an approximate indication of the transition structure involved. This
approximate starting structure for a full (non-restrained) transition-state search was
obtained from the SCAN calculation by manual isolation of the coordinates of the
structure closest to the maxima on the reaction profile (see Figure 5.3).
The same basic computational methods used for the calculation of the local minima
energies were utilized in locating the geometries of the transition-state structures. The
only difference is that the last part of the procedure made use of a transition-state
optimization during a scan close to the maxima associated with the corresponding TS.
The transition structure that was obtained is referred to TS end. A summary of the
results obtained for each of the transition-state structures follows.
Note that the Gaussian 98 frequency output files of the different transition-states are
available on the CD attached to this thesis. Molekel (freeware) at the site:
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36
www.cscs.ch/molekel can also be used to view the frequency output files, in
particular the vibrations associated with the negative eigenvalue.
5.3.1 Transition Structure 5.3.1
A constrained optimized structure (C9-C13, fixed at 1.7 Å) was used as input for a
relaxed SCAN calculation. The three dimensional (3-D) input structure for the
relaxed SCAN calculation is represented below.
5.3.1 (Scan start)
It is clear from the geometry of the structure above that the structure would be close to
the maxima on the energy profile or perhaps closer to the formation of the
intermediates 5.4 or 5.6 (see Figure 5.2). Note the carbonyl carbon (C5) is bending
out of plane, starting to become sp3 hybridised. The reaction coordinate C9 and C13
was scanned from 1.7 Å to 2.2 Å since the TS involved the formation of the bond
between atoms C9 and C13. The output of the Scan optimization is shown below as
5.3.1 (Scan end).
http://www.cscs.ch/molekel
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37
5.3.1 (Scan end)
On closer inspection of the Scan end file, it is clear that, as the reaction coordinate
between atoms C9 and C13 increased, the CN group moved away from the cage, and
the distance between atoms C5 and O15 increased accordingly. The carbonyl carbon
(C5) is again sp2 hybridised. The reaction profile of the step-wise increase in bond
length between atoms C9 and C13 is graphically presented in Figure 5.3.
0.0
1.0
2.0
3.0
4.0
5.0
6.0
1.6 1.8 2 2.2
Reaction Coordinate/Å
Rel
ati
ve
En
erg
y/k
cal
mo
l-1
Figure 5.3 Graphical representation of energy vs reaction coordinate for
structure 5.3.1.
Figure 5.3 shows that a minimum energy value occurs at 1.9 Å and a maximum
energy at 2.1 Å. Closer examination of the structure at the maxima of 1.8 Å and the
structure at the minima of 1.9 Å suggests that it is due to the rotation of the cyano CN
group (C13-N14), that is, the cyano group moving from the front of the cage to the
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38
back, as shown in the 3-D structures below. This phenomenon is therefore not likely
to be associated with the required transition structure - one would rather expect bond
formation/dissociation between C9 and C13.
The coordinates of the structure closest to the second maxima (2.1 Å) was manually
extracted from the Scan output file. This geometry of the structure was found to be
close to the expected TS. Thus a TS search was carried out using the structure at 2.1
Å as approximate starting structure for a full transition-state optimization. The 3-D
TS optimization is shown below.
5.3.1 TS end
The TS calculation was verified by performing a frequency (FREQ) calculation
resulting in ONE negative eigenvalue only. The FREQ calculation was viewed using
the GaussView61
program, enabling the vibration mode, associated with the negative
eigenvalue, so that the bond formation between the cyano group (C13-N14) and C9 is
clearly visualized. The vibration associated with the imaginary frequency shows the
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39
cyano group (C13) moving towards the cage to form a bond with C9. It is also clear
that the carbonyl carbon (C9) is converted to a sp3 carbon while O15 is moving
towards C5 as the nucleophile C13 is approaching the carbonyl carbon C9.
5.3.2 Transition Structure 5.5
Finding transition structure 5.5, (see Figure 5.2), involved monitoring the
intramolecular transfer of the hydrogen atom (H26) between the two oxygen atoms
(O12 and O15). Shown below is the 3-D structure (5.5-Scan start) which was used
as the starting structure for the relaxed scan.
5.5 (Scan start)
The bond was scanned from 2.2 Å to 1.4 Å. The output 3-D structure 5.5 (Scan end)
is shown below.
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40
5.5 (Scan end)
It is clear that the hydrogen (H26) is transferred during the Scan calculation. The
structure 5.5 (Scan end), above, when compared to the 5.5 (Scan start) shows that
the atoms C9-C13-N14 keep a linear bond as expected. Atoms C5 and O15 forms a
bond as the distance between atoms O12 and H26 decreases as a result of the H-
transfer. Atom H26 orientates itself between atoms O12 and O15. The scan is
graphically depicted below.
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
1.2 1.4 1.6 1.8 2 2.2
Reaction Coordinate/Å
Rel
ati
ve
En
ergy/k
cal
mol
-1
Figure 5.4 Graphical representation of energy vs. reaction coordinate for
structure 5.5.
The corresponding structure closest to the maximum in Figure 5.4 is at 1.6 Å. The
structure at this bond length was manually extracted from the scan output file and
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41
submitted as input structure for a TS optimization. The resultant TS structure that
was obtained at the end of the TS optimization is represented below.
5.5 TS end
The TS structure was verified to be correct since the FREQ calculation produced only
one negative eigenvalue and the movement of atoms associated with the negative
eigenvalue correspond to the expected movement of atoms on the reaction profile.
This movement includes transfer of atom H26 between atoms O15 and O12 and bond
formation/dissociation between C5-O15 as discussed above.
5.3.3 Transition Structure 5.7.1
The transition structure depicted as structure 5.7.1 was complex and difficult to find
as will be described next.
In structure 5.7.1 the calculations involved monitoring the progress of the hydroxyl
group (O27-H28) attaching to the cage and the breaking of the C9-O15 (ether/acetal)
bond.
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42
5.7.1 (Scan start)
The graphical presentation of the relaxed scan, that is, the decrease of the reaction
coordinate as the hydroxyl group (O27-H28) attaches to C9 from 2.4 Å to 1.3 Å is
shown below.
-20.0
-10.0
0.0
10.0
20.0
30.0
40.0
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
Reaction Coordinate/Å
Rel
ati
ve
En
erg
y/k
cal
mo
l-1
Figure 5.5 Graphical representation of energy vs. reaction coordinate for
structure 5.7.1.
This transition structure is also an example of a case for which the graph exhibits a
maximum, but is not necessarily indicative of the required transition-state. The
explanation of how the "transition-state" was found follows. The maximum in Figure
5.5 is at 1.8 Å. Thus the transition-state optimization for structure 5.7.1 was carried
out using the structure closest to this maximum bond length (1.8 Å). The 3-D input
structure (5.7.1 TS1 start) which was used for the transition structure optimization
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43
and the corresponding 3-D non-restricted output structure (5.7.1 TS1 end) that was
obtained are shown below.
The input structure, 5.7.1 TS1 start, had the hydro