conical intersections electronic structure

Advanced Serles In Physlcal Chernlstry 15

CONICAL INTERSECTIONS Electronic Structure, Dynamics & Spectroscopy

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Advanced Series in Physical Chemistry 15

CONICAL INTERSECTIONS E I ec t ro n ic S t ru c t u re, Dynamics & Spectroscopy

Editors

Wolfgang Domcke

David R Yarkony

Horst Koppel

Technicul University of Munich/ Germuny

Johns Hopkins UniversiM USA

University of Heidelber~~ Germuny

\b World Scientific N E W JERSEY LONDON SINGAPORE BElJ lNG SHANGHAI HONG KONG TAIPEI C H E N N A I

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

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CONICAL INTERSECTI ONS: ELECTRONIC STRUCTURE,DYNAMICS & SPECTROSCOPY

May 22, 2004 9:5 Conical Intersections: Electronic Structure, Dynamics and Spectroscopy contents

ADVANCED SERIES IN PHYSICAL CHEMISTRY

INTRODUCTION

Many of us who are involved in teaching a special-topic graduate course mayhave the experience that it is dicult to nd suitable references, especiallyreference materials put together in a suitable text format. Presently, severalexcellent book series exist and they have served the scientic communitywell in reviewing new developments in physical chemistry and chemicalphysics. However, these existing series publish mostly monographs consist-ing of review chapters of unrelated subjects. The modern development oftheoretical and experimental research has become highly specialized. Evenin a small subeld, experimental or theoretical, few reviewers are capable ofgiving an in-depth review with good balance in various new developments.A thorough and more useful review should consist of chapters written byspecialists covering all aspects of the eld. This book series is establishedwith these needs in mind. That is, the goal of this series is to publishselected graduate texts and stand-alone review monographs with specicthemes, focusing on modern topics and new developments in experimen-tal and theoretical physical chemistry. In review chapters, the authors areencouraged to provide a section on future developments and needs. Wehope that the texts and review monographs of this series will be more use-ful to new researchers about to enter the eld. In order to serve a widergraduate student body, the publisher is committed to making available themonographs of the series in a paperbound version as well as the normalhardcover copy.

Cheuk-Yiu Ng


PREFACE

The BornOppenheimer adiabatic approximation represents one of the cor-nerstones of molecular physics and chemistry. The concept of adiabaticpotential-energy surfaces, dened by the BornOppenheimer approxima-tion, is fundamental to our thinking about molecular spectroscopy andchemical reaction dynamics. Many chemical processes can be rational-ized in terms of the dynamics of the atomic nuclei on a single BornOppenheimer potential-energy surface. Nonadiabatic processes, that is,chemical processes which involve nuclear dynamics on at least two coupledpotential-energy surfaces and thus cannot be rationalized within the BornOppenheimer approximation, are nevertheless ubiquitous in chemistry,most notably in photochemistry and photobiology. Typical phenomenaassociated with a violation of the BornOppenheimer approximation arethe radiationless relaxation of excited electronic states, photoinduced uni-molecular decay and isomerization processes of polyatomic molecules.

During the last few decades, we have witnessed a change of paradigmsin nonadiabatic chemistry. First, the remarkable advances achieved in fem-tosecond laser technology and time-resolved spectroscopy have revealedthat the radiationless decay of excited electronic states may take placemuch faster than previously thought. The traditional theory of radia-tionless decay processes, developed in the sixties and seventies of thelast century, cannot explain electronic decay occurring on a time scaleof few tens of femtoseconds. Second, the development and widespreadapplication of multi-reference electronic structure methods for the cal-culation of excited-state potential-energy surfaces have shown that so-called CONICAL INTERSECTIONS of these multidimensional surfaces,predicted by von Neumann and Wigner in 1929, are the rule rather than

vii


viii Preface

the exception in polyatomic molecules. As a result, the concept of coni-cal intersections has become widely known in recent years. That conicalintersections may be responsible for ultrafast radiationless processes hasbeen surmised as early as 1937 by Teller. Nowadays, it is increasingly rec-ognized that conical intersections play a key mechanistic role in molecularspectroscopy and chemical reaction dynamics.

There are a number of intricate issues associated with conical inter-sections, concerning both the electronic structure, the nuclear dynamics,and the interaction of the molecule with the radiation eld. Few of us arefamiliar with all these aspects of the problem, which have been analyzedin research papers scattered throughout the chemical physics literature. Itis the intention of the present book to make this knowledge available tointerested graduate students and researchers in the thriving eld of femto-chemistry.

The chapters of this book are all theoretical in character. This reectsthe fact that the conical intersection is a theoretical concept, and as suchis not directly accessible to experimental observation. Nevertheless, theconcepts, techniques and results discussed in this book are crucial for theinterpretation of the observations in time-resolved spectroscopy and chem-ical kinetics on femtosecond time scales. It is hoped, therefore, that thisbook is of value not only for the theoretician, but also for the practitioneerin molecular spectroscopy, photochemistry, and collision-induced reactiondynamics.

Wolfgang Domcke, MunichDavid R. Yarkony, BaltimoreHorst Koppel, Heidelberg


HISTORICAL INTRODUCTION

Josef Michl

Department of Chemistry and Biochemistry,University of Colorado,Boulder, CO 80309-0215

In the last half century, our understanding of the qualitative features oforganic photochemistry has undergone a spectacular transformation. In themid-1900s, the community of chemists investigating light-induced reactionsof polyatomic organic molecules was not even sure that the theoretical con-cepts already well established in the ground electronic state, such as a singlepotential energy surface dictating nuclear motions, would be applicable toelectronically excited molecules. How useful would the BornOppenheimerapproximation be, given the energetic proximity of other electronic states?Would there be time for vibrational equilibration and could transition statetheory be applied? Kashas and Vavilovs early formulation of their impor-tant empirical rule provided a ray of hope that at least the lowest excitedstate of each multiplicity could be thought of in such familiar terms. Ofcourse, although the existence of metastable excited states was known,their recognition as triplet states and the appreciation of their importancein organic photochemistry were yet to come.

It was clear that in large organic molecules internal conversion fromupper to lower excited states occurs on a subpicosecond scale. In the manyorganic molecules that exhibit no emission, such ultrafast conversion evi-dently occurred all the way to the ground electronic state, in some casesaccompanied by a chemical transformation, in others not. When a chemi-cal transformation did occur, it seemed to be almost always accompanied

ix


x Conical Intersections: Electronic Structure, Dynamics and Spectroscopy

by electronic relaxation all the way to the ground state. Only in very fewexceptional cases, mostly simple proton transfer reactions, was the chemicalproduct produced in an electronically excited state.

The initial versions of the theory of internal conversion seemed towork ne for the weak coupling cases in which this process was rela-tively slow and competed with uorescence or with intersystem crossingto a triplet state. However, this theory did not cast light on the nature ofthe ultrafast process. Although theoreticians such as Teller1 in the thir-ties, Kauzman2 in the fties, and Forster3 in the early seventies seemedto have no doubt that ultrafast internal conversion proceeded by passagethrough conical intersections, in which potential energy surfaces touch,4

they did not attempt to provide any guidance as to the nature of moleculargeometries at which these points of rapid return occurred. This was rstprovided by Zimmerman5 in his reformulation of the WoodwardHomanrules. He pointed out that in the Huckel approximation, the ground andexcited potential energy surfaces touch at a point located on a symmetry-forbidden path of a pericyclic reaction, providing a natural explanation ofthe opposite nature of the rules for thermal and photochemical reactions ofthis type. A general analysis was also provided by Dougherty.6 Oosterho7

introduced electron repulsion into the theoretical treatment, which causedthe exact degeneracy to be avoided along the symmetric path, and hisemphasis on the exclusive importance of the resulting minimum in thedoubly excited state for the interpretation of WoodwardHoman rulesled to a lively controversy with the rules authors, who preferred to empha-size the presence or absence of correlation-diagram-imposed barriers in thesingly excited state at the start of the reaction path.8 Michls experimentsled him to conclude that both are important: with a barrier along the way,the pericyclic minimum will not be reached even if it is present, and theallowed product will not form unless extra energy is provided (e.g. by asecond photon) to pass above the barrier.9 His 1974 review10 summarizedthe state of understanding of the physical nature of processes involved inorganic photochemistry at the time and of the qualitative MO argumentsthat can be used for the rationalization of specic reaction paths. Thereview referred to conical intersections as funnels and emphasized theirtendency to occur at biradicaloid geometries, making these the likely start-ing points for vibrational relaxation after return to the ground state surface,thus permitting the prediction of likely products. It stated that true surface


Conical Intersections: Electronic Structure, Dynamics and Spectroscopy xi

touching is a relatively uncommon occurrence and along most paths suchcrossings, even if intended, are more or less strongly avoided. At the time,general searching for geometries of true touching was not computationallyfeasible, and did not seem to be particularly important: given the higherdimensionality of the regions of near touching, it seemed that most of theinternal conversion to the ground state would go through them. The qual-itative MO arguments only provided a very approximate notion of whatthe funnel geometries were likely to be. However, some of the intriguingcharacteristics of conical intersections, such as dynamic eects, when a fateof a molecule emerging on a ground state surface after passing through afunnel may depend on the direction from which it entered the funnel on theupper surface, were clearly recognized.

In the next phase of development, numerical computations started toplay a signicant role. Several groups of authors explored conical intersec-tions that are imposed by symmetry and therefore are easy to nd. Quitea few were known in very small systems, and Evleth and coauthors appar-ently were the rst to locate them in a large molecule.11 Salems extendedwork on photoreactions of carbonyl compounds12 advanced Zimmermansearlier more qualitative notions,13 and the results of Michl and collabora-tors for H4, taken as a model pericyclic entity, rationalized the nature ofbonding14 and cross-bonding15 that occurs in pericyclic reactions, and therole of excimers in them. Bonacic-Koutecky and Michl later explored severalother symmetry-imposed conical intersections related to cis-trans isomer-ization processes16 and jointly with Koutecky came up with a generalizedversion17 of the 3-state model of biradical structure18,19 that providedthe rst qualitative guide for searching for the geometry of a conical inter-section starting from a symmetrical geometry, with an application to theprimary process in vision.20

The modern era in work with conical intersections is characterized by theremoval of the condition that a symmetry element be present. It was ush-ered in by the groups of Yarkony21 and Ruedenberg,22 working with smallmolecules, and the international team of Robb, Bernardi and Olivucci,23

working with larger organic molecules. Armed with new algorithms andincreased computer power, these groups showed that conical intersectionsare far from relatively uncommon, as once thought, but instead are verycommon, and it now appears likely that it will be hard to nd an avoidedtouching of surfaces that does not have a true touching in its vicinity. Of


xii Conical Intersections: Electronic Structure, Dynamics and Spectroscopy

course, it continues to be true that the dimensionality of the space of neartouching is higher than that of true touching.

Robb, Bernardi, and Olivucci have been particularly prolic in identify-ing conical intersections along the paths of a vast number of organic reac-tions and rening the detailed understanding of organic reaction paths.24

They developed a 2-state VB model for the visualization of the originof the intersections25 which complements nicely the 3-state MO model17

mentioned earlier. Additional groups are publishing conical intersectiongeometries,26 and although quantitative improvements are needed and canbe expected in the future, the computational problem seems to be undercontrol. Discussions of conical intersections have become standard in pho-tochemistry textbooks.27

The opening sentence of this introduction stated that our understand-ing of the qualitative features of organic photochemistry has undergonea spectacular transformation in the last half century, and the above textdocuments it clearly. How about the quantitative features? We are nowherenear the end of the path. Unlike thermal reactions, whose absolute rates arenow fairly predictable from a combination of potential surface calculationsand transition state theory and/or modular dynamics, quantitative aspectsof organic photochemical reactions, such as quantum yields, still cannotbe predicted any better now than half a century ago. Although work onimproving our ability to obtain reliable potential energy surfaces clearly hasto continue, we need much more before this goal is achieved. We require pro-cedures for the calculation of molecular dynamics on excited state surfacesof polyatomic organic molecules and the requisite nonadiabatic couplingmatrix elements, particularly, in the regions of conical intersections. With-out such information, we cannot even tell whether all the computationaleort that has gone into identifying the lowest energy points in the conicalintersection subspace in meaningful one might guess that the internalconversion is likely to occur as soon as subspace is entered, or even slightlybefore, at a geometry at which the crossing is still weakly avoided, and thatthe geometries of the energy minima within the subspace are hardly everreached.

In a fully quantum mechanical treatment of the issue, the geometryof the point of return to the ground electronic state is not even sharplydened. In short, we may not be doing so vastly better than we did at thetime of the 1974 review as we would like to think, and perspectives for thefuture employment of imaginative theoretical photochemists are bright. At


Conical Intersections: Electronic Structure, Dynamics and Spectroscopy xiii

least, we know much more about what to ask next, and how to go aboutgetting the answers. These activities represent the subject of this book.

References

1. E. Teller, J. Phys. Chem. 41, 109 (1937).2. W. Kauzmann, Quantum Chemistry (Academic Press: New York, 1957)

p. 696.3. Th. Forster, Pure Appl. Chem. 24, 443 (1970).4. G. Herzberg and H. C. Longuet-Higgins, Disc. Faraday Soc. 35, 77 (1963).5. H. E. Zimmerman, J. Am. Chem. Soc. 88, 1566 (1966).6. R. C. Dougherty, J. Am. Chem. Soc. 93, 7187 (1971).7. W. Th. A. M. van der Lugt and L. J. Oosterho, J. Am. Chem. Soc. 91,

6042 (1969).8. R. B. Woodward and R. Homann, Angew. Chem. Int. Ed. Engl. 8, 781

(1969).9. J. Michl, J. Am. Chem. Soc. 93, 523 (1971).

10. J. Michl, Top. Curr. Chem. 46, 1 (1974).11. R. J. Cox, P. Bushnell and E. M. Evleth, Tetrahedron Lett. 207 (1990).12. L. Salem, J. Am. Chem. Soc. 96, 3486 (1974).13. H. E. Zimmerman, Adv. Photochem. 1, 183 (1963).14. W. Gerhartz, R. D. Poshusta and J. Michl, J. Am. Chem. Soc. 98, 6427

(1976).15. W. Gerhartz, R. D. Poshusta and J. Michl, J. Am. Chem. Soc. 99, 4263

(1977).16. V. Bonacic-Koutecky and J. Michl, J. Theor. Chim. Acta 68, 45 (1985).17. V. Bonacic-Koutecky, J. Koutecky and J. Michl, J. Angew. Chem. Int. Ed.

Engl. 26, 170 (1987).18. L. Salem and C. Rowland, Angew. Chem. Int. Ed. Engl. 11, 92 (1972).19. J. Michl, Mol. Photochem. 4, 257 (1972).20. V. Bonacic-Koutecky, K. Schoel and J. Michl, J. Theor. Chim. Acta 72,

459 (1987).21. D. R. Yarkony, J. Chem. Phys. 92, 2457 (1990).22. G. J. Atchity, S. S. Xantheas and K. Ruedenberg, J. Chem. Phys. 95, 1862

(1991) and reference therein.23. F. Bernardi, S. De, M. Olivucci and M. A. Robb, J. Am. Chem. Soc. 112,

1737 (1990).24. For a list of references, see F. Bernardi, M. Olivucci, J. Michl and M. A.

Robb, The Spectrum 9, 1 (1996).25. F. Bernardi, M. Olivucci and M. A. Robb, Accounts Chem. Res. 23, 405

(1990).26. J. Bentzien and M. Klessinger, J. Org. Chem. 59, 4887 (1994).27. M. Klessinger and J. Michl, Excited States and Photochemistry of Organic

Molecules (VCH Publishers: New York, 1995).


CONTENTS

Introduction v

Preface vii

Historical Introduction ixJ. Michl

Part I. Fundamental Concepts and ElectronicStructure Theory

1. BornOppenheimer Approximation andBeyond

3

L.S. Cederbaum

2. Conical Intersections: Their Descriptionand Consequences

41

D.R. Yarkony

3. Determination of Potential EnergySurface Intersections and DerivativeCouplings in the AdiabaticRepresentation

129

D.R. Yarkony

4. Diabatic Representation: Methods forthe Construction of Diabatic ElectronicStates

175

H. Koppel

xv


xvi Contents

5. Modeling and Interpolation of GlobalMulti-Sheeted Potential EnergySurfaces

205

A.J.C. Varandas

6. Conical Intersections and OrganicReaction Mechanisms

271

A. Migani and M. Olivucci

Part II. Conical Intersections in Photoinduced andCollisional Dynamics

7. The Multi-Mode Vibronic-CouplingApproach

323

H. Koppel, W. Domcke and L.S. Cederbaum

8. Model Studies of the Dynamics atConical Intersections

369

A. Lami and G. Villani

9. Generic Aspects of the Dynamics atConical Intersections: InternalConversion, Vibrational Relaxation andPhotoisomerization

395

W. Domcke

10. JahnTeller and Pseudo-JahnTellerIntersections: Spectroscopy andVibronic Dynamics

429

H. Koppel

11. Quantum Mechanical Studies ofPhotodissociation Dynamics UsingAccurate Global Potential EnergySurfaces

473

R. Schinke


Contents xvii

12. Geometric Phase Eects in ChemicalReaction Dynamics

521

B.K. Kendrick

13. Quantum Reaction Dynamics onCoupled Multi-Sheeted PotentialEnergy Surfaces

555

S. Mahapatra

14. Multidimensional Dynamics Involving aConical Intersection: WavepacketCalculations Using the MCTDHMethod

583

G.A. Worth, H.-D. Meyer and L.S. Cederbaum

15. Mixed Quantum-Classical Description ofthe Dynamics at Conical Intersections

619

G. Stock and M. Thoss

Part III. Detection and Control of Chemical Dynamicsat Conical Intersections

16. Absorption, Emission, andPhotoelectron Continuous-Wave Spectra

699

A. Lami, C. Petrongolo and F. Santoro

17. Femtosecond Time-ResolvedSpectroscopy of the Dynamics atConical Intersections

739

G. Stock and W. Domcke

18. Nonadiabatic Quantum Dynamics andControl Strategies

803

R. de Vivie-Riedle and A. Hofmann

Index 829

April 27, 2004 9:36 Conical Intersections: Electronic Structure, Dynamics and Spectroscopy chap01

PART I

Fundamental Concepts and ElectronicStructure Theory


CHAPTER 1

BORNOPPENHEIMER APPROXIMATION AND BEYOND

L.S. Cederbaum

Theoretische Chemie, Universitat Heidelberg, Im Neuenheimer Feld 229,D-69120 Heidelberg, Germany

Contents

1. Introduction 42. The BornOppenheimer Expansion 53. The BornOppenheimer Approximation 10

3.1. Separation of Electronic and Nuclear Motion 103.2. Group-BornOppenheimer Approximation 11

4. Gauge Potentials and Quasidiabatic States 144.1. Gauge Invariance of Group-BornOppenheimer

Approximation 144.2. Strictly Diabatic and Quasidiabatic Electronic States 16

5. Analysis in Powers of (1/M)1/4 235.1. General 235.2. Traditional Approach for Quasi-Rigid Molecules 27

6. Miscellaneous 306.1. BornOppenheimer Approximation in Magnetic Fields 306.2. Conical Intersections and Molecular Rotations 326.3. BornOppenheimer Approximation for Continuum

Electronic States? 35References 37

3


4 Conical Intersections: Electronic Structure, Dynamics and Spectroscopy

1. Introduction

The purpose of this review article is to present a comprehensive accountof what is generally known as the BornOppenheimer approximation, itsmeaning, its implications, its properties, and very importantly also its lim-itations and how to cure them. This approximation and the underlyingidea have been a milestone in the theory of molecules and actually alsoof electronic matter in general. Still today this approximation is basic toall molecular quantum mechanics and even in those cases where it fails,it remains the reference to which we compare and in terms of which wediscuss this failure.

The large mass of a nucleus compared to that of an electron permits anapproximate separation of the electronic and nuclear motion. Molecules arecomplicated quantum objects and this separation greatly simplies theirquantum mechanical treatment. It also allows us to visualize the dynamicsof molecules and provides the essential link between quantum mechanicsand traditional chemistry. We should be aware that even the notion ofmolecular electronic states is a consequence of the approximate separa-bility of the electronic and nuclear motion. Without this separability theintroduction of molecular electronic states would be relatively useless andjust a mathematical construction.

Most of our knowledge on the quality of the BornOppenheimer approx-imation stems from studies of the nuclear motion in the ground electronicstate of diatomic and polyatomic molecules and in excited electronic statesof diatomics. In diatomics, there is only a single vibrational degree of free-dom and the BornOppenheimer approximation is accurate in most cases.In polyatomics, the electronic ground state is typically well separated ener-getically from other electronic states, and this supports the a priori valid-ity of the BornOppenheimer approximation in these cases. Historically,the picture has thus emerged that this approximation is generally accuratewith the exception of some more or less exotic cases. In the last two decadesor so, there have been considerable advances in the development of experi-mental and theoretical techniques and a growing number of polyatomics intheir excited electronic states have been investigated. These studies reveal afast growing number of cases where the BornOppenheimer approximationfails. A particularly impressive failure is encountered in situations whereso-called conical intersections of the electronic energies exist. For such an


BornOppenheimer Approximation and Beyond 5

intersection to appear, at least two nuclear coordinates are necessary. Forpolyatomics with their dense electronic states and many nuclear degrees offreedom, conical intersections of the electronic energies are generic featuresand their absence is rather unusual. We, therefore, expect to see a growingrelevance of conical intersections in the future and shall concentrate in thisarticle on cases where the standard BornOppenheimer approximation mayfail and discuss its systematic improvement.

2. The BornOppenheimer Expansion

The objective of the present section is to provide a general ansatz to solvethe Schrodinger equation for problems with several degrees of freedom. Theansatz is particularly ecient if there are slow and fast degrees of freedom.To be specic, we consider molecules where the nuclei and electrons are theslow and fast degrees of freedom, respectively, but shall keep the generalityof the approach in mind.

The common molecular Hamiltonian is

H = Tn + Te + U(r,R) (1)

where Tn and Te are the kinetic energy operators of the nuclei and electrons,respectively, and U(r,R) is the total potential energy of the nuclei and theelectrons. The vector r denotes the set of electronic coordinates and R theset of nuclear coordinates. At the moment, we do not have to be specicabout the choice of these coordinates and we will address the matter atseveral places of this review.

By setting the kinetic energy of the nuclei equal to zero, one denes thefamiliar electronic Hamiltonian:

He = Te + U(r,R). (2)

Obviously, He is an operator in the electronic space that depends paramet-rically on R. Its eigenvalues Vi(R) and eigenfunctions i(r,R) fulll

Hei(r,R) = Vi(R)i(r,R). (3)

The set of eigenfunctions {i(r,R)} form a complete basis in the electronicspace at every value of R

i

i (r,R)i(r,R) = (r r) (4a)



and we write the orthonormality condition in the formi (r,R)j(r,R)dr i(R)|j(R) = ij . (4b)

In the bra and ket notation, only the indices i and j are retained, and thedependence of the electronic wavefunctions on the nuclear coordinates isindicated.

Of course, we are interested in solving the Schrodinger equation for thetotal Hamiltonian in Eq. (1) describing the electronic plus nuclear motionin our system. To this end, we expand the total wavefunction in theelectronic eigenfunctions of He:

(r,R) =

i

i(r,R)i(R). (5)

This expansion is known as the BornOppenheimer expansion.1 For-mally, Eq. (5) is exact, since the set {i(r,R)} is complete. It is onlywhen the expansion is truncated that approximations are introduced. TheBornOppenheimer expansion certainly provides a perfectly valid ansatz if(r,R) describes a bound state solution of the full Schrodinger equation

H(r,R) = E(r,R). (6)

It has been argued, see, for instance,2 that the ansatz (5) may not bejustied for describing continuum states. There are suciently many argu-ments which allow one to take the pragmatic stand point that the BornOppenheimer expansion is valid also for continuum states, see, for instance,Ref. 3.

From the Schrodinger Eq. (6) one can easily determine the coupledequations for the expansion coecients i(R) in the ansatz (5). InsertingEq. (5) into (6), multiplying from the left by j (r,R) and integrating overthe electronic coordinates leads to

[Tn + Vj(R)]i(R)

i

jii(R) = Ej(R) (7a)

where the so-called nonadiabatic couplings ji describe the dynamical inter-action between the electronic and nuclear motion. They are given by

ji = jiTn j(R)|Tn|i(R) (7b)and are obviously operators in R-space.



To proceed, we need a specic form of the nuclear kinetic energy oper-ator. Using atomic units and conveniently scaled rectangular nuclear coor-dinates, we may express Tn as follows

Tn = 12M = 1

2M2 (8)

where the gradient is a vector in nuclear space and the dot denotes theusual scalar product. M is an averaged nuclear mass, see Sec. 5.2, wherean explicit choice of nuclear coordinates is discussed. Of course, in certaincases one may wish to choose more general nuclear coordinates and expressthe Laplacian in these coordinates. When better adapted to the geometricstructure at hand, these coordinates will be curvilinear and the Laplacian(see, e.g. Ref. 4) and the nonadiabatic couplings (see Ref. 5) will have amore involved appearance.

With the denition (8), we readily obtain the following explicit form forthe nonadiabatic couplings:

ji =1

2M[2Fji + Gji]. (9a)

These elements split into a derivative operator and a c-number. The non-adiabatic derivative couplings are given by

Fji(R) = j(R)|i(R) (9b)and are obviously vectors in the nuclear coordinate space. The nonadiabaticscalar couplings Gji take on the following appearance

Gji(R) = j(R)|2i(R). (9c)Note that some authors dene the scalar couplings with a dierent signthan done here.

It is illuminating to express the coupled equations of nuclear motion inmatrix notation. To this end, we dene a diagonal matrix V of adiabaticpotential energy with diagonal elements Vj(R), which are the electronicenergies in Eq. (3), a matrix G with matrix elements Gji, and a matrix Fwith matrix element Fji. We recall that the latter quantities are vectorsin nuclear space. We may thus view F as a matrix in electronic space,where each element is a vector in nuclear space, or, equivalently, as a vectorin nuclear space with components which are matrices in electronic space.F may be called a vector matrix. Furthermore, denotes a column vectorwith components j , and in our matrix notation is also a vector matrix,



though a trivial one. It is the usual gradient in nuclear space, multipliedby the unit matrix in electronic space. Using the above straightforwardnotation, it is possible to rewrite the equations of nuclear notion (7) in thefollowing appealing form:5

[ 1

2M( + F)2 +V E

] = 0. (10)

This equation provides a natural theoretical formulation of the nuclearmotion. Showing some analogy to gauge theories,6,7 it is a helpful start-ing point for further investigations (see also Sec. 4).

The formulation (10) demonstrates that the coupled motion of elec-trons and nuclei can be reduced to the study of nuclear motion in thematrix potential V. The impact of the coupling on the electronic motionhas been transferred to the dressed kinetic energy operator, which nowreads (1/2M)( + F)2, instead of (1/2M)2 for the bare nuclei.

For completeness, we would like to list below the relevant relations ofthe nonadiabatic couplings. These follow immediately from the denitions(9) and the orthonormalization (4b) of the electronic states:

F = F (11a)G = ( F) + F F (11b)( F) = ( F) (11c)

F = ( F) + F (11d)(2F +G) = 2F +G (11e)

RF

RF + [F,F ] = 0. (11f)

Here, a few comments are in order. The matrix of derivative couplingsF is antihermitian. The matrix of scalar couplings G is composed of anhermitian as well as an antihermitian part. Of course, the dressed kineticenergy operator (1/2M)( + F)2 in our basic Eq. (10) is hermitian, asis also the case for the nonadiabatic couplings in Eq. (9a). The latterfollows immediately from the relation (11e). The notation ( F) is selfevident from Eq. (11d). Since F is a vector matrix, it can be written asF = (F1,F2, . . . ,FNn), where the matrices F are simply dened by their



matrix elements

(F)ji =j(R)

R

i(R)

(12)

and R = (R1, R2, . . . , RNn) are the Nn nuclear coordinates of the systemunder consideration. The relation (11f) plays an important role in the con-struction of diabatic states,5,8 see also Sec. 4.

The equation of motion of the nuclei (10) and the relations (11) are validfor both real and complex choices of the electronic wavefunctions i(r,R)and expansion quantities i(R). If these quantities are chosen to be real,the derivative coupling matrix F vanishes along its diagonal because of theantisymmetric nature of this matrix. In contrast, the scalar coupling matrixG contributes to the diagonal part of the dressed kinetic energy operatorvia its symmetric component F F, see Eq. (11b). If, on the other hand, theelectronic wavefunctions are chosen to be complex functions, the deriva-tive couplings do contribute to the diagonal of the dressed kinetic energyoperator. In some cases, for instance, in the presence of an external mag-netic eld, the electronic wavefunctions are in general necessarily complexfunctions,9,10 see also Sec. 6.1. In other cases we may choose them to becomplex for convenience.

To this end, let us introduce a phase factor for each electronic state

j(r,R) = eij(R)i(r,R) (13)

thus dening new wavefunctions j . These new wavefunctions are alsoeigenfunctions of the electronic Hamiltonian He and fulll all the equationsabove. In particular, the equation of nuclear motion (10) and the relations(11) are valid when the new functions are used instead of the original ones.It is interesting to express the equation of nuclear motion in terms of theoriginal derivative couplings F and the phases j(R). The result reads[

12M

[ + F+ 2i()]2 +V E] = 0 (14)

where () is a diagonal vector matrix with elements (i) ij . Sincethe phases j(R) are at our disposal, we may choose them to simplify orimprove the calculations, for instance, to make the j single valued in caseswhere the j are not.18



3. The BornOppenheimer Approximation

3.1. Separation of Electronic and Nuclear Motion

If the Hamiltonian were truly separable in the degrees of freedom denedby r and R, the solution of the Schrodinger equation would, of course, bea product of a function of r and a function of R. The separability one hasin mind when discussing electronic systems is of a dierent origin, however.It relies on the dierent masses of the underlying particles and hence ontheir dierent velocities. The fast electrons can instantly adjust to the slownuclei and, consequently, their wavefunction depends parametrically on thecoordinates of the nuclei. The total wavefunction thus reads:

(r,R) = (r,R)(R). (15)

Inserting this adiabatic ansatz into the Schrodinger Eq. (6), followed bymultiplication from the left with and integration over the electroniccoordinates, immediately leads to the following equation for the nuclearmotion

[Tn + V (R)](R) = E(R) (16)in the electronic state under consideration. For simplicity, we have droppedthe index of the electronic state. Clearly, this equation also follows fromthe general Eq. (7a), when all nondiagonal elements of the nonadiabaticcouplings ji, j = i, in (7b) are put to zero. If the electronic wave-function is taken to be real, the derivative coupling F vanishes, and = G/2M , as is seen from Eqs. (9). The resulting equation is calledBornOppenheimer approximation.11 If, on the other hand, the electronicwavefunction is complex, either by its nature or by choice (see precedingsubsection), the element in Eq. (16) contains derivative couplings. Whenthe electronic wavefunction is deliberately chosen complex, it is appropriateto call Eq. (16) the complex BornOppenheimer approximation, to distin-guish it from the common, i.e. real equation.

In which cases is the adiabatic ansatz (15) expected to be accurate?As shall become clear in the next subsection and from the discussion inSec. 5, the adiabatic ansatz is accurate for an electronic state which is wellseparated energetically from all other electronic states. Under the sameelectronic conditions, the ansatz improves with increasing nuclear mass. Onthe other hand, we can expect that the larger the energetical distance of



the other electronic states is from the state under investigation, the weakerthe dependence of (r,R) on the nuclear coordinates. This immediatelyimplies that the nonadiabatic term is the smaller, the better the ansatz(15) is. Neglecting in Eq. (16) leads to

[Tn + V (R)](R) = E(R) (17)

which is commonly called the BornOppenheimer adiabatic approxi-mation11 or briey the adiabatic approximation. The above discussion alsoimplies that as long as is small, its inclusion does provide an improvementto the adiabatic approximation. However, once is sizeable, the adiabaticansatz (15) itself and hence also (16) become doubtful. A sizeable indi-cates in turn sizeable derivative couplings [see Eq. (11b)] and, consequently,other electronic states may be intruding (see next subsection).

3.2. Group-BornOppenheimer Approximation

The nonadiabatic couplings enter the nuclear equation of motion scaled by1/M see Eqs. (7) and (9) and one is tempted to assume their impactto be small. A more precise analysis on their scaling with the nuclear masswill be given in Sec. 5. Indeed, if we assume the Fji to be of the order ofunity, the impact on the nuclear motion would be small and the adiabaticapproximation discussed in the preceding subsection would be highly accu-rate. This is, unfortunately, not the case. By applying the gradient to theelectronic Schrodinger equation (3), one readily obtains from the denition(9b) the following useful expression for the derivative couplings (i = j):

Fji(R) =j(R)|(He)|i(R)

Vi(R) Vj(R) . (18)

Apart from special cases, e.g. where the numerator vanishes due to spatialsymmetries of the involved electronic states and nuclear coordinates, thereis no reason to assume that it takes on particularly small or large valuesin general. However, the denominator does inform us on the situations inwhich the derivative couplings become large. In the vicinity of a degeneracy,the derivative couplings can be substantial and the adiabatic approxima-tion for the involved electronic states can be expected to break down. Inparticular, in the presence of conical intersections of the potential surfacesVi(R) and Vj(R), the nonadiabatic derivative couplings diverge and theadiabatic approximation becomes meaningless12 (see also Chapter 2).



From the above we may expect the adiabatic approximation to be appro-priate for electronic states well separated energetically from other electronicstates to which they may couple. Equations (10) and (18) also provide uswith straightforward extensions of this approximation. Let us assume thata group of electronic states are well separated energetically from all otherstates to which they may couple. We denote the subspace of the full elec-tronic space spanned by this group by g. Clearly, the energy gap betweeng and its complement has only to be large in the relevant range of nuclearcoordinates. In analogy to the BornOppenheimer approximation, wherethe matrix nuclear Schrodinger equation (10) has been limited to a singleelectronic state, we obtain a truncated version of this equation for the groupg of states under consideration:5[

12M

{( + F)2}(g) +V(g) E](g) = 0. (19)

The superscript g denotes the truncated quantities; both matrices and vec-tors now refer only to the group g of electronic states. Due to the aboveconsiderations, we call this useful result the group-BornOppenheimerapproximation. It should be clear that truncating Eq. (10) is equivalentto truncating Eq. (7a).

It is worthwhile to have a closer look at Eq. (19). Because of thequadratic structure of the dressed kinetic energy operator, (1/2M)(+F)2, its block in the space spanned by the states belonging to g cannotbe written as a single quadratic term, but rather as a sum of two quadraticterms

{( + F)2}(g) = ( + F(g))2 F(c) F(c). (20)F(g) is that part of the full vector matrix F = {j(R)|i(R)} where |iand j| belong to our group g. The coupling between the group g and therest of the electronic Hilbert space is given by F(c) = {j(R)|k(R)},where j| belongs to g and |k to the complement of g. We may now makeuse of Eq. (20) to rewrite Eq. (19) and obtain[

12M

( + F(g))2 +W(g) E](g) = 0 (21a)

where the matrix Hamiltonian is now in g space only, i.e. its elements i, jrefer to states in our set g, and we have dropped the superscript for brevity



from and E. The quantities

W(g)ji (R) = V

(g)i (R)ij +

12M

k/g

j(R)|k(R)k(R)|i(R) (21b)

are the elements of the dressed potential matrix W(g). In deriving (21b),we have used the antihermitian property (11a) of derivative couplings.

To compute the dressed potential surface, it is convenient to intro-duce the projector P (g) which projects on the subset of electronic states g.Introducing

P (g) =ig

|i(R)i(R)| (22a)

leads to

W(g)ji (R) = V

(g)i (R)ij +

12M

j(R)|(1 P (g))|i(R) (22b)which can be evaluated within the group g of states without summingover the formally innite set of states |k which do not belong to g.Equation (22b) is often even simpler than it looks. For instance, if ourgroup of coupled states contains two members and this is a widespreadsituation then the nondiagonal term reduces to W (g)ji = 1/2Mj|ibecause of the orthogonality relation (4b) (real electronic wavefunctions areassumed).

Let us return to the group-BornOppenheimer approximation (21a) andpay further attention to the corrections in (21b) due to the rest of theelectronic states. From the denition of the derivative coupling (9b) and theexpression (18), we may conclude that these corrections have only a weakimpact on the nuclear dynamics in the g-manifold of states. The derivativecouplings in (21b) connect states within this manifold with states outsidethe manifold, separated by a substantial energy gap. Furthermore, thesecouplings are weighted by 1/2M in Eq. (21b). The situation is actuallyequivalent to that discussed in the preceding subsection. By including asingle state in our group g of states, Eq. (21a) boils down to Eq. (16),which in particular for real electronic wavefunctions is the well-knownBornOppenheimer approximation. This can be easily seen by using therelations (9a) and (11b).

In analogy to Sec. 3.1, we may neglect in many cases to a good approxi-mation the dierence between the naked and the dressed potential matrices



V(g)(R) and W(g)(R). This leads to the following simplied equation ofmotion for the nuclei in the manifold g of coupled electronic states[

12M

( + F(g))2 +V(g) E](g) = 0. (23)

Note that the potential matrix V(g)(R) is a diagonal matrix by denition,in contrast to W(g)(R). Again, in analogy to the common treatment inSec. 3.1, we call (23) the group-BornOppenheimer adiabatic approximationor briey the group-adiabatic approximation. This approximation assumesthat the states within the manifold g are much stronger coupled to eachother e.g. via the presence of a conical intersection of the potentialsurfaces than to the rest of the electronic space.

Additional properties of the nonadiabatic couplings are discussed inSec. 4.

4. Gauge Potentials and Quasidiabatic States

4.1. Gauge Invariance of Group-BornOppenheimerApproximation

The full Eq. (10) and its group-BornOppenheimer approximation (21a)are natural gauge theoretical formulations of the nuclear motion.5 Gaugetheories are widely used in other elds, e.g. elementary particle physics,7,13

the simplest one in electrodynamics, where the vector potential appears.4

We mention that there are also some other attempts in molecular physicsand quantum chemistry6,1416 and that an intimate relation exists betweengauge theory and the geometric phase in adiabatic transport in quan-tum systems.1719 In the presence of a conical intersection, this geomet-ric phase leads to the well-known sign change of the adiabatic electronicwavefunction,20 which has stimulated the rst investigation of gauge poten-tials in the context of the BornOppenheimer approximation.21

It is beyond the scope of this work to enter the subject of gauge the-ory deeply. We shall only illustrate some results relevant to the group-BornOppenheimer approximation. For more details we refer to Ref. 5 andreferences therein.

There is always a freedom of choice in using basis sets. In the precedingsections we have used as basis set the eigenfunctions i of the electronicHamiltonian He, see Eq. (3). If we transform these basis functions to a new



basis set i by a unitary transformation U(g), U(g)U(g) = 1, this impliesthat we have transformed the nuclear wavefunction according to

(g) = U(g)(g) (24)

because of the invariance of the total wavefunction (r,R) in Eq. (5). Thequestion immediately arises whether such a transformation leaves invariantthe form of the matrix Schrodinger equation (21a) in our subspace g ofinteracting electronic states. If the answer is positive, one says that thisequation or equivalently the group-BornOppenheimer approximation isgauge invariant under the gauge transformation U(g). Of particular interestis, of course, the case where this transformation depends parametrically onthe nuclear coordinates R:

U(g)(R)U(g)(R) = 1. (25)

It has been shown in Ref. 5 that the electronic basis set transformationleads to the following matrix Schrodinger equation:[

12M

( + F(g))2 + W(g) E](g) = 0. (26)

In other words, the group-BornOppenheimer is a gauge invariant approx-imation. In the above equation, the dressed potential transforms as

W(g) = U(g)W(g)U(g); (27a)

in addition, the relevant relation

F(g) = U(g)F(g)U(g) +U(g)(U(g)) (27b)

holds for the nonadiabatic derivative couplings in the new electronic basis.Since we did not specify the set g of interacting states, the above also holdsfor the full matrix Schrodinger equation (10); we just have to drop thesuperscript g. Equation (10) is thus also gauge invariant. It is importantthat the full equation and the truncated one share the basic property ofgauge invariance.

To facilitate the above, let us consider explicitly the common case inwhich g contains two electronic states 1 and 2 which are taken to be



real. The matrix of derivative couplings then reads

F(g) =(

0 0

)(28)

where (R) = i|2. The most general gauge transformation is

U(g)(R) =(

cos(R) sin(R) sin(R) cos(R)

). (29)

It is easy to show that

= + . (30a)Here, = 1|2 is just the derivative coupling in the transformedelectronic basis. Hence, Eq. (27b) now takes on the explicit appearance

F(g) =(

0 0

)=(

0 0

)+(

0 11 0

) . (30b)

A general remark on the transformed dressed potential in Eq. (27a) isalso in order. According to Eq. (22b), the potential matrix W(g) consists ofthe diagonal matrix V(g) of the electronic energies and a, in general non-diagonal, correction matrix. By transforming to a new electronic basis, thematrix V(g), of course, becomes a nondiagonal matrix of potential energieswhich we denote by V(g). What happens to the correction matrix? Becauseof the projection operator in Eq. (22b), this matrix retains its form by thegauge transformation. We obtain the general relation:

W(g)ji = V

(g)ji +

12M

j(R)|(1 P (g)|i(R) (31)which obviously also holds in our above explicit example of two states.

4.2. Strictly Diabatic and Quasidiabatic Electronic States

It is often cumbersome to solve the group-BornOppenheimerequation (21a) because of the terms F(g) . Furthermore, these termsdescribe the coupling between electronic states in our g manifold via themomenta of the nuclei, and we commonly have more experience in under-standing the impact of couplings via potentials than via momenta. It has,therefore, been popular and desirable, starting already many decades ago, toformulate the nuclear equations of motion in a so called diabatic electronicbasis instead of the adiabatic one, which we have used above in Secs. 2



and 3.2226 Thereby these diabatic electronic states have been a prioriassumed to depend so weakly on the nuclear coordinates that the deriva-tive couplings can be considered small or even put exactly to zero. The basicquestion arises whether we can indeed eliminate the derivative couplings bychoosing an appropriate basis of electronic states. Following previous work,we call a basis strictly diabatic if in that basis the derivative couplingsvanish8,27,28 and quasidiabatic, if in that basis these couplings are small,but not zero.5

The gauge invariance of the group-BornOppenheimer approximationprovides a good starting point to discuss diabatic states. In contrast toEq. (21a), where this approximation is formulated in the adiabatic elec-tronic basis, Eq. (26) is expressed in an arbitrary basis. Elimination of thederivative couplings appearing in the latter equation amounts to setting tozero the left hand side of Eq. (27b):

F(g)U(g) + (U(g)) = 0. (32)Note that F(g) is the vector matrix of derivative couplings in the adiabaticelectronic basis and the gauge transformation U(g)(R) is the unitary trans-formation matrix connecting the adiabatic and diabatic basis sets. In theabove example of two real electronic states, Eq. (32) is identical to Eq. (30a)where is set to zero:

+ = 0. (33)Does Eq. (32) have a solution or, equivalently, do strictly diabatic states

exist? There are two limiting cases where the answer to this question is obvi-ously positive. If our set g of interacting electronic states contains all elec-tronic states, then we can transform the adiabatic electronic states i(r,R)to a set i(r) which does not depend parametrically on the nuclear coordi-nates. In this complete basis set of so called crude adiabatic functions,29 thederivative couplings denitely vanish. The other limiting case is obtainedby restricting ourselves to a single nuclear coordinate, say R, keeping allother coordinates frozen at arbitrary values. Along this path R we can justintegrate Eq. (33) over R, determine = (R) and obtain via Eq. (29), astrictly diabatic basis along the path. Diatomics provide natural examplesfor this limiting case.

In general, no strictly diabatic basis exists in a nite set g of interactingelectronic states.8,27,30 To see this, let us return to Eqs. (27b) and (32) and



introduce the quantities

A(g) =F(g)R

F(g)

R+ [F(g) ,F

(g) ] (34)

which are the , components of the so called gauge eld tensor.4,7,13 A nec-essary and sucient condition for the existence of a solution of Eq. (32),and hence of a strictly diabatic basis, is the equality 2U(g)/RR =2U(g)/RR. Using Eq. (32), this immediately implies that all the com-ponents of the gauge eld tensor have to vanish: A(g) = 0. This conditionis, in general, only fullled in the complete electronic space and not in asubspace g; see the relation (11f) for derivative couplings. From the latterrelation, it is easy to show that

A(g) = F(c) F

(c) F(c) F(c) (35)

where F(c) comprises the derivative couplings between g and the comple-mentary space and has been introduced in Eq. (20).

We briey reformulate the above necessary and sucient condition forthe existence of a strictly diabatic basis in the context of gauge theory.In contrast to the gauge potential F(g), which transforms in a relativelycomplicated manner via Eq. (27b), the gauge eld tensor A(g) simply trans-forms as

A(g) = U(g)A(g)U(g). (36)

This can be shown using Eqs. (34) and (27b), i.e. the gauge eld tensoris gauge invariant. Single valuedness of the electronic basis requires thatA(g) = 05,13,16 and because of Eq. (36), the gauge eld tensor vanishesfor all unitary transformations. It can generally be shown that the gaugepotential can be eliminated by a gauge transformation, once the gauge eldtensor is zero.13

Owing to Eq. (35), there is no reason to expect that a strictly diabaticbasis exists. Nevertheless, one can construct quasidiabatic states whichare extremely useful in solving and understanding many relevant prob-lems abundantly discussed in the literature. With their help it is possi-ble to remove a substantial part of the derivative couplings and make thegroup-BornOppenheimer Eq. (26) more transparent and better amenableto explicit numerical calculations. That part of the derivative couplingswhich can be removed by an unitary transformation U(g)(R) is called



the removable part, and the remaining part the nonremovable part. In theimportant two-state case, these parts of the derivative couplings have beenanalyzed in some detail.5,8 In general, we may not expect to have elimi-nated even the removable part exactly, and it is convenient to denote thesum of all the remaining parts the residual couplings. Of course, as long aswe insert the residual couplings F(g) into Eq. (26), our solution is equiva-lent to that of the original Eq. (21a). It is only if we neglect the residualcouplings that we have to check whether they are large or small.

In this context, let us briey return to the two-state case. As discussedabove, strictly diabatic states exist in one dimension, e.g. for a diatomicmolecule with interatomic distance R. In a typical situation, the adiabaticpotentials exhibit an avoided crossing and the diabatic potentials cross eachother once a function of R. Assuming that the derivative couplings vanishat innite internuclear distance, and using Eq. (30) then gives

(R)dR =

d

dRdR = (37)

and no residual coupling is found. In dierent situations, similarly sim-ple results are obtained. With the aid of the same equation and d =

/R dR, we obtain

F(g) dR =

(0 11 0

)+

F(g) dR (38)

which provides an idea about the overall size of the residual couplings ifab initio data on the couplings F(g) are available. For such data see, forinstance, Ref. 31.

To arrive at a quantitative criterion for quasidiabaticity, it has beenproposed5 to minimize the integral of F(g)2, which is the Euclidean normfor matrices, over the nuclear coordinates. This natural requirement leadsto the simple equation5

F(g) = 0 (39)for the diabatic states, which in the language of gauge theory can be calledLorentz gauge. In the two-state case, it has been shown that this gauge isthe best possible gauge, i.e. transforming to the Lorentz gauge eliminatesfrom the derivative couplings what can be eliminated. The removable partis removed, and one is left only with the nonremovable part. We call thestates which result from the above criterion optimal quasidiabatic states.



The expression (18) has been useful in assessing the relevance of thederivative couplings in the adiabatic electronic basis. What to expectfor the residual couplings? By dierentiating both sides of the identityV

(g)ji = j|He |i with respect to the nuclear coordinates, we obtain the

following expression (i = j):

F(g)ji (R) =j|(He)|i V (g)ji

V(g)jj V (g)ii

. (40)

Here, the V (g)ji (R) are the elements of the potential energy matrix inthe quasidiabatic basis [see also Eq. (31)]. In contrast to Eq. (18) wherethe numerator cannot be expected to be small or even to vanish in gen-eral, the numerator in Eq. (40) can be made small by appropriately choos-ing the electronic basis. This happens when this basis depends only weaklyon the nuclear coordinates. In the extreme case in which the basis is inde-pendent on the nuclear coordinates, the numerator in Eq. (40) vanishes bydenition. We have seen above that in the two-state case a strictly dia-batic basis can be determined along a single coordinate (e.g. for a diatomicmolecule). Having only a single coordinate at our disposal, there is no reasonto expect the denominator in Eq. (40) to vanish (avoided crossing situa-tion), but the numerator can be made zero.

What happens in conical intersection situations? In the adiabatic elec-tronic basis the derivative couplings become singular as one approaches theconical intersection12 (see also Chapter 2). Starting from the Hamiltonianin a quasidiabatic basis with no singularities in the residual derivative cou-plings, we readily obtain singular couplings in conical intersection situationswhen transforming the Hamiltonian to the adiabatic electronic basis. Obvi-ously, in these cases the part that becomes singular as one approaches theconical intersection is removable. We may transform back from the adiabaticbasis to the initial quasidiabatic basis where no singularities are present.

Can the nonremovable part also exhibit singularities as one approachesa conical intersection? The nonremovable part of the derivative couplingsoriginates from the fact that the gauge eld tensor A(g) does not vanish [seeEqs. (34) and (35) and text]. Since by construction this tensor assumes onlynite values, there is no reason to expect the nonremovable part to exhibitsingularities at conical intersections, see also the discussion in Ref. 32 andexplicit examples in Ref. 5. Let us briey consult Eq. (40) in a conical



intersection situation and consider for simplicity two states i and j ofdierent spatial symmetry, and two coordinates, a totally symmetric one Rand a nontotally symmetric one R , which can couple the two electronicstates. Expanding all quantities in Eq. (40) around the geometry of theconical intersection R = R = 0, we obtain by symmetry arguments:12

Vjj Vii = aR + . . . ,Vji = cR + cRR + . . .

j|(He)|i = (bR + . . . , b + bR + . . .). (41a)

a, b and c are real numbers and, in particular, a, c and b are nonzero.Insertion into Eq. (40) gives in the vicinity of the conical intersection

F(g)ji =[(b c)R , (b c) + (b c)R]

aR(41b)

The singularity at the conical intersection can be made to vanish if only twoconditions are fullled by the quasidiabatic basis: b = c and b = c.Since we have two independent coordinates R and R , these conditionscan be met. In general, the quasidiabatic basis can perform more e-ciently than just removing the singularity as one approaches the conicalintersection.

To conclude this section, we briey comment on the practical approachesto compute quasidiabatic states. A general account of the subject has beenreviewed in Ref. 5 and additional recent developments are discussed inRef. 41 and in this book.33 Basically, we can divide the various methodsproposed into two classes, depending on whether the derivative couplingsin the adiabatic electronic basis (or some other well dened basis) are usedor not. The method proposed by Baer34,35 is the most prominent proto-type scheme belonging to the rst class. Baer gives a general method, inwhich use is made in a propagative manner of Eq. (32) to nd a transfor-mation U(g) from the adiabatic to a diabatic basis. In those cases in whichstrictly diabatic states do exist, the method provides them by eliminatingthe derivative couplings. In most cases, however, a strictly diabatic basisdoes not exist as discussed above, and it has been argued that the methodsleads to nearly diabatic states.36 The analysis made in Ref. 5 demonstratesthat Baers method does lead to states which do fulll the Lorentz gaugelocally, i.e. in a small neighborhood of the reference nuclear coordinates at



which the propagative computation has started. In this neighborhood, thestates obtained are thus optimal quasidiabatic states.

The second class of methods to construct diabatic states does not makeuse of the availability of derivative couplings. Such approaches, if success-ful, have the advantage that derivative couplings need not be computed.Although these couplings can now be computed by ab initio methods (see,for instance, Ref. 31), their determination is still a formidable task, in par-ticular for several potential energy surfaces and not too small moleculeswith several nuclear degrees of freedom. The method of block diagonaliza-tion of the electronic Hamiltonian,37 discussed and analyzed in detail inRef. 5, provides a promising proposal of broad applicability. The electronicHamiltonian He is usually represented as a matrix He in some initial elec-tronic basis. Instead of diagonalizing this matrix to obtain the adiabaticpotential surfaces as eigenvalues, we may block diagonalize it, i.e. bring Heinto a block diagonal form. One block denoted g-block comprises thoseelectronic states which belong to the group g of states chosen to participatein the group-BornOppenheimer equation (26). The other block containsthe remaining states and is not of our concern. Obviously, the eigenvaluesof the g-block coincide with eigenvalues of the full Hamiltonian matrix He,i.e. they are those adiabatic potential surfaces of the states participating inthe group-BornOppenheimer equation for the nuclear dynamics.

There are innitely many transformations which bring a Hermitianmatrix into block diagonal form. It is intriguing and satisfactory thata single elementary condition is sucient to determine uniquely (up tophase factors) the block diagonalizing transformation.38 It is the simpleand appealing requirement that this transformation does not do anythingbut block diagonalize, i.e. it is as close as possible to the unit matrix.The resulting transformation can be given explicitly and requires only theknowledge of those eigenvectors of He which belong to the g-block. Start-ing at some nuclear conguration, one can show that this block diago-nalization leads locally to the Lorentz gauge, i.e. to optimal quasidiabaticstates in the neighborhood of this conguration.5 By using these quasidi-abatic states obtained at a given nuclear conguration as the initial basisfor block diagonalization at a nearby nuclear conguration, one denesa propagative block diagonalization procedure. This procedure has beenshown5 to yield identical results of those of Baers method.34,35 As dis-cussed above, the latter method requires the derivative couplings, while the



block diagonalization approach does not. We remind that if the nuclearcoordinate space is one dimensional, both methods give strictly diabaticstates. Variants of the block diagonalization approach have been formu-lated and successfully used to generate quasidiabatic states in ab initiocomputations.5,39,40

5. Analysis in Powers of (1/M)1/4

5.1. General

The large mass of a nucleus compared to that of an electron permits anapproximate separation of electronic and nuclear motion, which is the basisfor the treatments discussed in the preceding sections. Historically, therewere several attempts to expand the solutions of the Schrodinger equation,i.e. the energies and the wavefunctions, in powers of a small quantity relatedto the ratio of electronic and nuclear masses. The breakthrough has beenachieved by Born and Oppenheimer in their classic paper in which theyhave chosen = (1/M)1/4 as the small quantity.42 As done in the precedingsections, M is some average nuclear mass and we work in atomic units wherethe electronic mass is 1.

When studying the expansion in powers of , traditionally one rst sepa-rates o the center of mass motion and expresses the remaining Hamiltonianfor relative motion of nuclei and electrons in suitable nuclear coordinatesdescribing molecular rotations and vibrations. The resulting expressions arerather complex and depend on the particular choice of coordinates made,see, for instance, Refs. 4345. On the other hand, one also arrives at the cor-rect analysis in powers of without rst separating o the center of massmotion. This has been demonstrated by many authors, see, for instance,Refs. 3, 29 and 4648.

The physics behind the expansion in powers of is most straightfor-wardly understood by resorting to the nuclear kinetic energy operator Tnin its simplest form, given in Eq. (8). There is no need to specify the under-lying mass-scaled nuclear coordinates. It suces to divide these coordinatesinto various classes according to their physical nature. For transparency, wedivide the nuclear coordinates in this work into two extreme classes only;all other classes lead to results in between the results we shall obtain forthese two classes below. Our rst class comprises all quasi-rigid vibrationsof the system. All other nuclear coordinates, describing soft or even free,



e.g. translational, nuclear motion, are collected in the second class. Therespective nuclear modes of motion will be briey denoted as rigid and soft.To be particularly simple, we initially assume that the electronic energyVj(R) under consideration depends only on the rigid coordinates.

Obviously, the energy Vj , which is the potential energy surface for thenuclear motion, has a minimum value Vej at some equilibrium congurationRe, where the set comprises the rigid coordinates. In the vicinity ofthe minimum, the potential Vj is essentially quadratic, giving rise to amultidimensional harmonic oscillator. According to the standard theory,the frequencies of the harmonic oscillator scale as 2 and the root-mean-square displacements from the Re scale as . Born and Oppenheimer haveexploited the latter property by introducing new coordinates Q for thedisplacement from equilibrium

R = Re + Q. (42)

For the harmonic oscillator, the root-mean square values of Q scale as 0

and can also be expected to be of order unity for all quasi-rigid motionsbeyond the harmonic approximation.

Expressed in the new coordinates Q, we may now rewrite the nuclearkinetic energy Tn in Eq. (8) and obtain the convenient result

Tn = 2

22Q

4

22S (43)

where Q is the gradient with respect to the coordinates Q and S is thegradient with respect to the remaining coordinates, i.e. the soft coordinates.The kinetic energies for the quasi-rigid and soft motions scale dierentlywith the nuclear mass. To proceed, we expand the potential energy in apower series in Q about the minimum and readily obtain

Vj(Q) = 0Vej+122,

VjQQ+163

,,

VjQQQ+ . (44)

where the linear term in vanishes.To complete the expansion of the general matrix Schrodinger equa-

tion (7) in powers of , we need to analyze the nonadiabatic coupling jias well. Inspection of Eq. (9) shows that these couplings split into a termQji, arising from the quasi-rigid motion, and a term

Sji, corresponding to



the soft motion

ji = Qji +

Sji (45a)

which scale dierently with . Remembering that = Q + S , whereby denition Q = 1Q, we can easily see from Eq. (9a) that

Qji = 3FQji Q +

124GQji (45b)

and

Sji = 4(FSji S +

12GSji

). (45c)

The derivative couplings FQji and FSji and the scalar couplings G

Qji and

GSji are dened as usual, just replace in the denitions (9) by Q andS , respectively. These couplings are, in general, functions of the nuclearcoordinates and should be expanded in powers of as well. Fortunately, theleading terms of these quantities do not depend on , i.e. they scale as 0.

Having completed the analysis of all the operators appearing in thematrix Schrodinger equation

[Tn + Vj E]j =

i

jii (46)

we may now turn to analyze the equation itself. To lowest order in ,one immediately obtains E = Vej , which is the electronic energy at theequilibrium conguration. The nuclei are frozen in space, does not appear,and only the electrons move. In the next order, 2, we have:

2

2

2Q +

,

VjQQ

j(Q) = (E Vej)j(Q). (47)

Now the nuclear motion is harmonic, but the nuclei move only along therigid coordinates. The soft motion, e.g. translation, is frozen. What happensin the order 3? At rst sight, one may be tempted to assume that thecubic anharmonicity of Vj(Q) in Eq. (44) contributes to this order. From aperturbational point of view, this is not the case, however. The harmonicwavefunctions are even functions of the coordinates, and the expectationvalues of all odd anharmonic terms vanish. These terms will contribute tothe energy in second order perturbation theory, i.e. earliest in 4 (note



that when adding cubic anharmonicity to Eq. (47), it appears within thebrackets as a linear term in ).

Interestingly, the impact of the nonadiabatic couplings begins at thesame order of as that of the anharmonicity. As can be seen fromEqs. (45b,c), the scalar nonadiabatic couplings enter the Schrodinger equa-tion (46) already at 4. When real electronic states are used, the diagonalelement Fjj of the derivative couplings vanish. In the presence of a mag-netic eld,10 or when phase factors are attached to the electronic states,as discussed at the end of Sec. 2, the Fjj do not vanish in general. Conse-quently, the derivative couplings lead to eects already in the order 3 [seeEq. (45b)]. Does this result imply that nonadiabatic couplings are morerelevant than anharmonicities? The answer depends on the values of thenonadiabatic couplings in the actual system under investigation.

At what order of do other electronic states mix into the descriptionof the dynamics? As can be seen from Eq. (46), this mixing depends onthe nonadiabatic couplings ji between our electronic state j and theother states i, i = j. For the rigid coordinates, these couplings scale as3, but their contribution to the energy starts at second order perturbationtheory and hence as 6. This mixing implies that the total wavefunctionceases to be merely a product of an electronic and a nuclear wavefunction.The adiabatic ansatz (15) looses its validity and one has to resort to theexpansion (5) of the total wavefunction.

Summarizing, if perturbation theory is applicable throughout and realelectronic wavefunctions are used, the lowest order contributions to theenergy are all proportional to even powers of the small parameter . In grow-ing powers of , these contributions are:

0: Electronic energy.2: Harmonic energy of quasi-rigid motion.4: Contribution of soft motion, anharmonic contribution of quasi-rigid

motion; contribution of scalar nonadiabtic coupling.6: Contribution of coupling between electronic states (derivative cou-

plings via quasi-rigid motion); contribution of coupling between softand quasi-rigid motion.

If perturbation theory is applicable, the smallness of not only justiesthe separation of electronic and nuclear motion, but also the separation ofquasi-rigid and soft motion. We know, however, that perturbation theory



breaks down in many cases. A prototype case is encountered in the pres-ence of a conical intersection between potential energy surfaces. The deriva-tive couplings diverge at such intersections, and the participating electronicstates must be included on the same footing in the calculations. This leadsto the group-BornOppenheimer approximation discussed in Sec. 3.2. Inanalogy to the above analysis, it is evident that the energetic contributionof the derivative couplings between the electronic states within our group gand the remaining states begins at 6. Since the values of these derivativecouplings are of the order of unity, the smallness of justies the group-BornOppenheimer approximation. There is a price to pay, however. Thediverging derivative couplings between the electronic states of our group gmust be correctly included in the calculations of the nuclear motion, andalthough they appear in the equations with a prefactor 3, see Eq. (45b),one cannot resort to the smallness of .

5.2. Traditional Approach for Quasi-Rigid Molecules

In the traditional approach, the center of mass motion of the moleculeis explicitly separated o, and nuclear coordinates are introduced whichsuitably describe molecular vibrations and rotations. The procedure is notunique and the resulting kinetic energy operators depend on the choice ofcoordinates made. We follow here the transparent analysis of Mead45 whichis briey sketched below. The center of mass coordinate is dened as usual,and the nuclear relative coordinates are introduced one at a time. Startingwith an arbitrary chosen nucleus, the new coordinate for each nucleus isdened relative to the center of mass of the nuclei already introduced. Thisdenition of relative nuclear coordinates, which we denote by R, leads to aparticularly simple form of the kinetic energy operators (Jacobi coordinates;for a discussion of suitable nuclear coordinates, see Ref. 51). The relativeelectronic coordinates ri are dened, as usual, relative to the center of massof the nuclei.

Now, an average nuclear mass M is introduced and coecients b =/M are dened, relating the reduced masses of the above coordinatesR to the average mass. Dening mass-scaled coordinates, R = b

1/2 R,

leads to the following kinetic energy operator

T = Te 12M2 + Tmp (48a)



where the term Tmp is called mass polarization. With the above choice ofrelative nuclear coordinates, the mass polarization takes on a particularlysimple appearance

Tmp =4

2b

(i

pi

)2. (48b)

Here, b is a quantity dened by Mn = bM , where Mn is the total mass ofthe nuclei, and the pi are the canonical electronic momenta. Other choicesof nuclear coordinates typically lead to more intricate expressions for Tmpin which also the nuclear momenta appear. The electronic operator Te isnot aected by the choice of nuclear coordinates.

To separate rotational degrees of freedom from the vibrational ones, amolecule-xed coordinate system is introduced with three angles giving theorientation of this frame relative to the laboratory. Variables conjugate tothese angles are provided by the vector J of angular momentum, whichis conveniently dened relative to the molecular-xed frame. The eect ofa component of J is to rotate the nuclei about the respective axis, whileholding the electrons xed, see e.g. Ref. 52. The remaining internal nucleardegrees of freedom describe the vibrations of our quasi-rigid molecule. Thereare various conventional methods to introduce them, depending on the pur-pose at hand, for instance, analyzing vibrational spectra. For our presentpurpose, i.e. analysis in powers of the small parameter , the particularchoice of these vibrational coordinates is irrelevant. For simplicity, we shallnot introduce a new notation for these coordinates and denote them by R,as done in the preceding subsection. The nuclear kinetic energy now takeson the following appearance in matrix notation:

Tn =1

2M[JA(R)J RB(R)R] (49)

where the matrix A(R) is the inverse of the nuclear inertia tensor and thematrix B(R) depends on the particular choice of vibrational coordinatesmade.

We are now in the position to make direct contact with the generalanalysis in Sec. 5.1 by identifying these vibrational coordinates with therigid coordinates in Eq. (42). These, in turn, have lead to the introductionof new coordinates Q for the displacement from equilibrium. In analogy toEqs. (42) and (43), the nuclear kinetic energy in Eq. (49) is easily rewritten



to give

Tn = 2

2QBQ +

4

2JAJ. (50)

As in Eq. (43), the kinetic energy for the quasi-rigid motion scales with 2,while that for the soft motion scales with 4. Since the translational motionhas been separated o exactly and a quasi-rigid molecule is considered here,we are left only with the rotational motion as soft motion.

The dierence to the analysis made in the preceding subsection is thatmatrices A and B appear in Eq. (50) which depend on the vibrationalnuclear coordinates. These can be expanded in a power series in the Q inthe vicinity of the equilibrium geometry Re, giving

A = Ae +

AQ +2

2

,

AQQ + . (51)

and an analogous expression for B. The interpretation and analysis of thenonadiabatic couplings presented in the preceding subsection is also validhere. Just for completeness, we mention that the precise expressions forthese couplings follow from Eq. (45): replace there Q and S by BQ andAJ, respectively, The derivative couplings with respect to the vibrationalcoordinates are dened, as usual, as the electronic matrix elements of R,and those with respect to rotation as matrix elements of J. The scalarcouplings, which are, according to Eq. (9), electronic matrix elements ofthe nuclear kinetic energy operator multiplied by 2M , follow immediatelyfrom the terms of this operator in Eq. (49).

All the analysis and discussion of the preceding subsection can now becarried over to the present situation. If perturbation theory is valid andreal electronic wavefunctions are used, the lowest order contributions tothe energy in growing powers of listed in Sec. 5.1 apply also here. One,of course, has to identify the quasi-rigid motion and the soft motion inSec. 5.1 with vibrational and rotational motion, respectively. Then, thediscussion in Sec. 5.1 for cases in which perturbation theory breaks down,in particular in the presence of conical intersections, also remains valid.Where are the dierences between the general analysis in Sec. 5.1 and thepresent one for quasi-rigid molecules? First, mass polarization, see Eq. (48),contributes here in the order of 4. This contribution is obviously missing inSec. 5.1, where the translational motion has not been separated o a priori.However, as discussed there, the translational motion starts to contribute



also at 4. Second, the leading contribution due to the coupling of vibrationsand rotations scales as 6. In the present case it arises from the secondorder contribution of the terms linear in in the expansion (51) of thematrix A. Third, and most important, the main dierences between Sec.5.1 and the present traditional analysis are due to the matrices A and B.Their expansions (51) give rise to additional contributions in higher ordersof and to a more general denition of anharmonicity.

6. Miscellaneous

The aim of the present section is to supplement the discussions of the BornOppenheimer approximation and of conical intersections presented in thisbook. In particular, we would like to briey discuss three subjects: thepresence of external elds, nonstationary electronic states, and the role ofmolecular rotations in conical intersection situations. Each of these subjectspresents by itself a large area of research and would deserve a detailedaccount. Here, we shall only touch upon these subjects without going intoany details.

The basic idea underlying the BornOppenheimer approximation is theseparation of the electronic and nuclear motions in terms of the smallquality = (1/M)1/4 as elaborated in Sec. 5. The discussion in Sec. 3.2has made clear that the existence of the small parameter is, however,insucient to solve the problem. The coecients in the expansion of theSchrodinger equation in powers of also play a relevant role. Indeed, someof these coecients can be very large and in conical intersection situa-tions even diverge and hinder the separation of electronic and nuclearmotion, inspite of the smallness of . This has led to the introduction ofthe group-BornOppenheimer approximation. We would like to stress thatthe failure or even breakdown of the BornOppenheimer approximation isa consequence of the forces prevailing in the system. For instance, if all theforces among the light and heavy particles in the system were harmonicforces, no conical intersections or similar diabolic features would appear,and the BornOppenheimer approximation would turn out to be an excel-lent approximation for all states throughout.53,54

6.1. BornOppenheimer Approximation in Magnetic Fields

Following the above line of thought, we may ask the question whetherthe quality of the BornOppenheimer approximation is inuenced by the



presence of an external eld.10 We shall discuss here very briey the impactof an external magnetic eld. Details can be found in Ref. 49. Inspite of thediculties encountered in separating the center of mass motion in the pres-ence of a magnetic eld,9 it is possible to formulate a BornOppenheimerexpansion as done in Sec. 2 and to arrive at an equation analogous tothe basic Eq. (7a).49 The kinetic energy operator of the nuclei, Tn, how-ever, must be replaced in the magnetic eld by an eective operator. For ahomonuclear neutral diatomic molecule this operator takes on the followinginteresting appearance:49

Tn(B) = 1M

[ + iZ

4 [B R]

]2. (52)

Z and M are the charge and mass of one of the nuclei, respectively. B isthe magnetic eld and R denotes the relative position vector of the twonuclei.

The expression for the nonadiabatic couplings in Eq. (7b) is still valid inthe presence of the magnetic eld B. One has just to use the operator Tn(B)in Eq. (52) instead of Tn = Tn(B = O) in the eld-free case. One peculiarityis that the diagonal element of the derivative couplings cannot be made tovanish in the magnetic eld case. Another peculiarity is the appearanceof [B R] and ([B R])2 in the expression for the nuclear kineticenergy. These quantities, of course, diverge as the molecule dissociates,i.e. |R| , for any eld strength. The consequences are far reaching.In particular, the BornOppenheimer adiabatic approximation (17) doesnot provide a meaningful description of the nuclear dynamics. This can bebest seen in the dissociative limit, where one expects two separate atomsmoving in a magnetic eld. Instead, this approximation treats the nucleiwith respect to their relative motion as naked charges in a magnetic eld.

What is missing in the widely used BornOppenheimer adiabaticapproximation when a magnetic eld is present? It does not contain theeect of the screening of the nuclear charges through the electrons againstthe magnetic eld. As shown in Refs. 10 and 49, the screening is pro-vided by the nonadiabatic couplings. These couplings can diverge at largeinteratomic distances to compensate for the above mentioned misbehaviorencountered in the BornOppenheimer adiabatic approximation. How toremedy the situation? By recognizing that in a magnetic eld, the operatorR = i[R, H] and not the canonical momentum i controls adiabatic-ity, it has been possible to rearrange the BornOppenheimer expansion.

conical intersections electronic structure

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