Computational Photochemistry

16 THEORETICAL AND COMPUTATIONAL CHEMISTRY

Computational Photochemistry

THEORETICAL AND COMPUTATIONAL CHEMISTRY

S ERIES E DITORS Professor P. Politzer Department of Chemistry University of New Orleans New Orleans, LA 70148, U.S.A.

Professor Z.B. Maksic Rudjer Boškovic Institute P.O. Box 1016, 10001 Zagreb, Croatia

VOLUME 1

VOLUME 9

Quantitative Treatments of Solute/Solvent Interactions P. Politzer and J.S. Murray (Editors)

Theoretical Biochemistry: Processes and Properties of Biological Systems L.A. Eriksson (Editor)

VOLUME 2

VOLUME 10

Modern Density Functional Theory: A Tool for Chemistry J.M. Seminario and P. Politzer (Editors)

Valence Bond Theory D.L. Cooper (Editor)

VOLUME 3 Molecular Electrostatic Potentials: Concepts and Applications J.S. Murray and K. Sen (Editors)

VOLUME 4 Recent Developments and Applications of Modern Density Functional Theory J.M. Seminario (Editor)

VOLUME 5 Theoretical Organic Chemistry C. Párkányi (Editor)

VOLUME 6 Pauling’s Legacy: Modern Modelling of the Chemical Bond Z.B. Maksic and W.J. Orville-Thomas (Editors)

VOLUME 7 Molecular Dynamics: From Classical to Quantum Methods P.B. Balbuena and J.M. Seminario (Editors)

VOLUME 8 Computational Molecular Biology J. Leszczynski (Editor)

VOLUME 11 Relativistic Electronic Structure Theory, Part 1. Fundamentals P. Schwerdtfeger (Editor)

VOLUME 12 Energetic Materials, Part 1. Decomposition, Crystal and Molecular Properties P. Politzer and J.S. Murray (Editors)

VOLUME 13 Energetic Materials, Part 2. Detonation, Combustion P. Politzer and J.S. Murray (Editors)

VOLUME 14 Relativistic Electronic Structure Theory, Part 2. Applications P. Schwerdtfeger (Editor)

VOLUME 15 Computational Materials Science J. Leszczynski (Editor)

VOLUME 16 Computational Photochemistry M. Olivucci (Editor)

16 THEORETICAL AND COMPUTATIONAL CHEMISTRY

Computational Photochemistry

Edited by M. Olivucci Dipartimento di Chimica dell’Universita di Siena Siena, Italy

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ISBN-13: 978-0-444-52110-1 ISBN-10: 0-444-52110-0 ISSN: 1380-7323 (Series) The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

This book is dedicated to my beloved parents Armando and Anna

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Contents Foreword by Josef Michl Preface I.

Computational Photochemistry Massimo Olivucci and Adalgisa Sinicropi

ix xiii 1

II. Ab initio Methods for Excited States Manuela Merchan and Luis Serrano-Andres

35

III. Density Functional Methods for Excited States: Equilibrium Structure and Electronic Spectra Filipp Furche and Dmitrij Rappoport

93

IV. Electronic and Vibronic Spectra of Molecular Systems: Models and Simulations based on Quantum Chemically Computed Molecular Parameters Fabrizia Negri and G. Orlandi

129

V. Semiclassical Nonadiabatic Trajectory Computations In Photochemistry: Is The Reaction Path Enough To Understand A Photochemical Reaction Mechanism? G. A. Worth, M. J. Bearpark and Michael A. Robb

171

VI. Computation of Photochemical Reaction Mechanisms in Organic Chemistry Marco Garavelli, Fernando Bernardi and A. Cembran

191

VII. Computation of Reaction Mechanisms and Dynamics in Photobiology Seth Olsen, Alessandro Toniolo, Chaehyuk Ko, Leslie Manohar, Kristina Lamothe, and Todd J. Martinez

225

VIII. Development of Theory with Computation Howard Zimmerman

255

IX. Calculations of Electronic Spectra of Transition Metal Complexes Kerstin Pierloot

279

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X. Perspectives in Calculations on Excited State in Molecular Systems Bjorn Roos

317

Index

349

IX

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

Anyone who has dealt with ground state mechanistic chemistry of organic or inorganic reactions appreciates how complex and demanding it can be. Nevertheless, from theoretical and computational standpoints, its complexity pales compared to mechanistic photochemistry, with its maze of paths that can be followed after the absorption of a photon of U V-visible light by a molecule. Even for the simplest photoreactions, it is a major feat to figure out the details of the routes followed by the molecule in a qualitative way and to rationalize the nature of the final ground state product. A reliable calculation of a quantity as fundamental to the experimental photochemist as the quantum yield of product formation remains a distant goal. There is competition between nonradiative and radiative processes, between spin-allowed and spinforbidden processes, between adiabatic and diabatic processes, between vibrational relaxation into one and another minimum after return to the ground state. There is the issue of possible violations of Kasha's rule, since a variation of the initial excitation energy is not always without consequences, even in solution photochemistry. Energy transfer and electron transfer possibilities often lurk in the background. Solvent effects are complex and manifold. No wonder a distinguished ground-state computational chemist friend who attended a theoretical photochemistry meeting with me a few years ago shook his head in disbelief after the first day of lectures, and asked something like "Isn't there an easier way to earn a living?". Photochemists thrive on complexity, theoreticians more than most. Although we cannot predict the quantum yield of a simple reaction any more accurately than we could when my interest in photochemistry was first piqued nearly half a century ago, great conceptual progress has been made. Then, it was not even very clear just what to calculate. Today, there is little doubt that we need dynamics on lowest potential energy surfaces. The concept of a potential energy surface guiding an excited molecule, with only occasional hops from one surface to another, was new to most experimental photochemists then. It has proven its heuristic and computational value since, and pervades the present book. Only for molecules with a very high

density of lowest-energy electronic states, such as those of saturated compounds, is it likely to be inadequate. The basic notions that are so familiar today were established in the sixties and seventies, and perusal of a 1974 review article[l] reveals the whole slew of the necessary concepts: excited state barriers and funnels for ultrafast return to the ground state, reactions with vibrationally equilibrated intermediates and direct reactions proceeding through state crossings, vibrationally equilibrated and "hot" excited and ground state reactions, internal and external heavy atom effects on intersystem crossing, etc. Back then, we used symmetry considerations, correlation diagrams, or calculations on simple models to estimate at what geometries barriers and funnels are likely to lie. Our first numerical computation of a funnel (conical intersection) relevant for a photochemical isomerization in an organic molecule was published only twenty years ago,[2] and it was made possible by the presence of symmetry at the state touching point. Today, advances in computer technology and in quantum chemical methodology, especially in multireference methods, many due to the authors of the chapters that follow, permit quite reliable calculations of these essential features at general geometries, and a thick book on conical intersections has just appeared. [3] Yes, difficulties remain, especially in evaluating the relative energies of covalent ("dotdot") and zwitterionic ("hole-pair") states with sufficient accuracy. Another problem is the proper treatment of reaction dynamics in all but the smallest molecules. After all, only those conical intersections are relevant that can be reached by the excited molecule in the short time available to it. And yes, in my opinion, too much emphasis has been put in recent years on the geometries of the lowest energy point of a conical intersection. This is an issue on which I have had a gentle disagreement with many. I would argue that these points are usually nearly irrelevant, because a molecule that has reached the seam of a conical intersection will fall to the lower surface right away and will not have time to ride the seam, looking for its lowest energy point. Thus, the effective funnel locations are those in which the seam is first reached, and not the lowest energy point in the intersection subspace. Unfortunately, the former are harder to calculate. In fact, much of the wave packet most likely seeps to the lower surface at geometries at which the state touching is still weakly avoided, simply because of their higher dimensionality, and in that sense the regions of weakly avoided crossings need not be as immaterial as they are sometimes made out to be. In spite of these minor quibbles, we all agree that the improvement from the level of mechanistic interpretations standard a quarter of a century ago to that common today is striking. Another interesting comparison is with a book on theoretical photochemistry that was published fifteen years ago.[4] It was a monograph rather than an edited multiauthor volume, and was organized differently in that it attempted a systematic treatment of all important classes of organic photoreactions, organized by Salem's concept of topicity. However, the ab initio calculations presented were hardly more than glorified correlation diagrams, involved no geometry

XI

optimization, and were pathetic by the present book's standards. The qualitative concepts are all that survives. Clearly, computational photochemistry has made tremendous strides in recent decades, and continues to do so. The present collection of ten outstanding contributions provides a fine illustration of this statement.

REFERENCES [1] Michl, J. "Physical Basis of Qualitative MO Arguments in Organic Photochemistry", Topics in Current Chemistry 1974, 46,1. [2] Bonacic-Koutecky, V.; Michl, J. "Photochemical Syn-Anti Isomerization of a Schiff Base: A Two-Dimensional Description of a Conical Intersection in Formaldimine", Theor. Chim. Ada 1985, 68, 45. [3] Domcke, W.; Yarkony, D. R.; Koppel, Editors, Conical Intersections: Electronic Structure, Dynamics & Spectroscopy, World Scientific Publishing Co., Singapore 2004. [4] Michl, J.; Bonacic-Koutecky, V. Electronic Aspects of Organic Photochemistry, John Wiley and Sons, Inc.: New York, 1990.

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Preface The chemical community have recently witnessed a growing interest in the application of computational methods to problems involving electronically excited molecules. This is mostly due to a change in the field of photochemical sciences. In fact, until two decades ago, these were dominated by the search for novel photochemical or photophysical properties. In other words, photochemists were reporting on the effects of light at the molecular-level. In contrast, contemporary photochemists look for ways to exploit light to drive various molecular-level actions such as pollutant scavenging and removal, mechanical motion, sensing and signalling, photocatalysis and others. While such raising technologies require the preparation of molecules capable of performing specific functions, the lack of knowledge on the molecular mechanisms of light energy exploitation and wastage constitutes a severe limitation to the design of such systems. In this book, a selected group of experts show how the development, implementation and application of quantum chemical methods in photochemistry and spectroscopy provide a way to tackle this problem. Until recently the computer-aided investigation of photochemical reactions (i.e. reactions that are initiated by light absorption rather then by heat) was unpractical if not impossible. Because of this the simulation of fundamental chemical and biological events such as bleaching, fluorescence, phosphorescence, photochromism, vision, photosynthesis, phototropism, and others could not be performed. Thus, despite the growing availability of computer power, there were neither computer tools nor a clear theoretical basis for the investigation of photoexcited molecules. One key point for the solution of this problem was to establish the nature of the spatial arrangement of the atoms that allows a photoexcited molecule to efficiently decay from the excited state to the ground state thus initiating product formation. Loosely, this critical molecular structure, often called "photochemical funnel", plays in photochemistry, the role played by the transition structure in a thermal process. Thus the description of a reaction pathway in photochemistry must involve the description of the path leading to and departing from the photochemical funnel. About fifteen years ago different research lines started to change this unfavorable situation. Few lines were merely related to the elucidation of the general mechanism of photochemical reactions, including the nature of the photochemical funnel, while others involved the development of software tools allowing for an accurate evaluation and mapping of the potential energy surface of photoexcited molecules. For instance, at the end of the 80's improved ab initio quantum chemical methodologies became available which were suitable for computing, in a balanced way, excited and ground state energy surfaces taking into account the complete set of the 3N-6 nuclear degrees of freedom of the reacting system (N is the number of atoms). Such progress made possible to provide clear evidence that the ideas of

XIV

Edward Teller, Lionel Salem, Howard Zimmerman and Josef Michl, stating that for singlet photochemical reactions the photochemical funnel corresponds to a conical intersection of the excited and ground state energy surfaces, were correct. (A book on conical intersections has recently appeared: "Conical Intersections: Electronic Structure, Dynamics and Spectroscopy"; Domcke, W., Yarkony, D. R., Koppel, H., Eds.; World Scientific: Singapore, 2004) The target of the present book is two-fold. The first, and most ambitious one, is to contribute to establish a branch of computational chemistry that deals with the properties and reactivity of photoexcited molecules (see Chapter 1). Accordingly the book should give not only an historical view of the "scientific adventure" (see the Foreword written by one of the original and major player) that led to the emergence of the field but also a survey of the work that characterizes it. In order to satisfy this requirement the book provides an overview of few general strategies currently employed to investigate photochemical processes. The second target of the book is to give an account of the status of knowledge in either the mechanistic (conceptual) and methodological research lines in computational photochemistry. In fact, during the last ten years the potential energy surfaces of several organic chromophores were mapped. The resulting "maps" reveal prototype photochemical reaction mechanisms and form a firm body of computational photochemistry results. Accordingly, three book chapters focus on instructive case-studies comprising: (i) organic chromophores (Chapter 1, 6 and 8), (ii) biologically related chromophores (Chapter 7), (ii) photochemical funnels and reactive intermediates (Chapter 8 and 9). Such (still ongoing) systematic investigation could not be carried out without the development of novel computational tools that, nowadays, constitute the computational photochemist toolbox. These tools belong to four classes that will be reviewed in the remaining book chapters: (i) tools for the accurate computation of the excited state potential energy (Chapters 2, 3 and 10), (ii) tools for the prediction of absorption, fluorescence and Resonance Raman spectra (Chapter 4) (iii) tools for the mapping of excited state potential energy surfaces (including locating photochemical funnels and excited state reaction paths, Chapters 6) and, finally, (iv) tools for the computation of "photochemical" semi-classical trajectories (i.e. trajectories that start on the excited state energy surface and continue along the ground state surface, Chapter 5 and 7). A final chapter (Chapter 10), written by one of the major experts of electronic structure theories, provides a review and a perspective on the technologies for the ab initio computation of excited state energy surfaces. All authors have made an effort to write the chapters in a plain and simple way. Thus "Computational Photochemistry" should be readable not only by computational and theoretical chemists but also by chemists (e.g. photochemists, photobiologists and material scientists) interested in using computer tools in their laboratories. I feel deeply indebted to all authors that, not only have readily accepted my invitation, but have felt that the book may have provided a first, probably still crude, picture of an expanding field of computational chemistry. However, it is important to stress that many other scientists

XV

have given important contributions to the field and, indirectly, to the material reported in this book. The Editor feels indebted to the many colleagues including spectroscopists, photochemists, organic chemists and theoreticians that through both discussions and criticism have stimulated the present editorial effort. There remains only the pleasant task of thanking those who have otherwise been of help in the preparation of the book. Prof. Professor Zvonimir B. Maksic for originally inviting me to plan and edit the book and Andrew Gent of Elsevier for the attention devoted to progress of our work. A very special thank goes to Dr. Adalgisa Sinicropi (one of the author of Chapter 1) for taking care of many of the practical problems related to the assembly and revision of the chapters, for the production of the index and for her willingness to share the many, sometime frustrating, decision that I had to take during the various stages of the manuscript handling. As this is my first editorial work I cannot fail to acknowledge the fundamental role played, in the development of my scientific personality and career, by the chemists that have not only cast my education but shared with me their enthusiasm for many wonderful years: Fernando Bernardi and Michael A. Robb. Finally I am deeply indebted to my wife Matilde and my children Paolo, Enrico and Lidia for never let me feel alone during the too many days that I spend away from them. Massimo Olivucci Professor of Organic Chemistry Universita di Siena

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M. Olivucci (Editor) Computational Photochemistry Theoretical and Computational Chemistry, Vol. 16 © 2005 Elsevier B.V. All rights reserved

I. Computational Photochemistry Massimo Olivucci and Adalgisa Sinicropi Dipartimento di Chimica, Universita di Siena, Italy 1. INTRODUCTION The study of photochemical problems by means of computer simulations using specialized software tools and strategies enable us to get an understanding at the microscopic level of what happens to a molecule after absorption of a photon. A detailed understanding of the properties of electronically excited state species and the knowledge of the molecular mechanisms which control the fate of the energy deposited on a molecule after absorption, increase our ability to design efficient photochemical reactions and artificial photosynthetic systems. Furthermore, this represents a fundamental requirement for the rational design of novel materials, molecular devices and molecular level machines. On a more general ground, the ability to simulate, using complementary computational strategies, photoinduced events often allows to explore areas of chemistry that experiment could touch only indirectly. Together with the mechanistic ideas discussed below these strategies define the field of "computational chemistry". The application of quantum mechanics to chemical problems goes back to the end of the 1950s when computers came into use and it was possible to handle very complicated mathematical equations describing such complex systems as molecules. Note that even if the "... fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry were completely known ...", as Dirac affirmed in 1929, accurate calculations of molecular properties and chemical reaction pathways were not possible at that time. In 1970 Pople published the first release of the GAUSSIAN program [1], making thus computational methods available to scientists [2]. The following growth of the speed of computers along with the development of more and more accurate quantum chemistry tools and methodologies implemented in commercially available program have made computational investigation of molecular structures and reactivity a standard practise. The way in which a chemical reaction step is investigated involves the computation of the transition state structure (TS) that connects a reactant to a product along with the associated energy barriers (Fig. 1). In particular, the energy of the transition state provides information on the time scale of the reaction while the geometrical structure of the transition state (TS) provides information on the stereochemistry of the reaction and sensitivity to different substituents. The progression of the molecular structure of the reactant toward the TS and the product constitutes the so called "reaction path" which can be mapped computing the

R Fig. 1. Schematic representation of the structure of the potential energy surface for a thermal chemical reaction. The dashed curve indicates the minimum energy path. R and P are local energy minima corresponding to reactants and products. TS is a saddle point corresponding to the transition structure.

minimum energy path (MEP) connecting the reactant (R) to a product (P) along the 3N-6 dimensional potential energy surface (N is the number of nuclei in the reacting system). One of the first computations in photochemistry/photophysics has to be ascribed to R. G. Parr and R. Pariser. In the early 1950s, they developed a semi-empirical method based on LCAOMO jt-electron theory to predict the electronic spectra and electronic structure of complex unsatured molecules, initially of benzene and N-heterocyclic analogues [3; 4]. Later on, using one of the first computers, an IBM 701, they were able to assign the electronic structure and electronic spectra of azulene and of the polyacenes [5-8]. The theoretical approach of Parr and Pariser together with the contribution of Pople formed the basis of the Pariser-Parr-Pople (PPP) theory that is one of the first semi-empirical method based on the ZDO (zerodifferential overlap) approximation. During '60s, '70s and even '80s many researchers and experts in the field of organic photochemistry shared their knowledge and published several papers [9-40] trying to understand the behaviour of electronically excited molecules and make a wide-ranging classification of photochemical reactions. The formulation of photoreduction mechanisms was mainly based on the construction of correlation diagrams. Although the interest for a unique theory of photochemical reactions was well recognized, the computational investigation of photochemical reaction mechanisms could not be easily implemented at the same level seen for the thermal chemistry. This frustrating status is somehow described in a 1990 paper [41], where N. J. Turro stated: "... the use of computational methods to elucidate reaction mechanisms has not really made a major impact on the way organic photochemists think about such mechanisms. The Woodward-Hoffmann rules and Salem diagrams of the 1960s and 1970s still serve as the basis for the day-to-day analysis of photoreactions...". On the other hand, other research groups in the field of organic photochemistry were tackling this

computational problem and realized that a detailed knowledge of the excited state molecular structure could lie at the basis of the "resolution" of the reaction mechanism. A detailed account of the development of mechanistic ideas and early results in the laboratory of H. E. Zimmerman will be given on Chapter 8. The major difficulty encountered by chemists in doing an exhaustive investigation of photochemical reactivity resided in the absence of robust computer tools able to map the MEP for excited state species. In fact, while a thermal reaction is governed by the topography of a single potential energy surface (starts and ends on the ground state of the reacting system), a photochemical reaction path evolves at least on two potential energy surfaces. Thus, in order to compute such path one needs to connect a reactant that is located on an excited state energy surface to products that are located on the ground state energy surface. This could only be done establishing the nature of the spatial arrangement of the atoms that allows a photoexcited molecule to efficiently decay from the excited state to the ground state thus initiating product formation. Loosely, this critical molecular structure, often called "photochemical funnel" plays, in photochemistry, the role of the transition state of a thermal process. As we will detail below, the characterization of the molecular structure and relative stability of the "photochemical funnel" in terms of conical intersections and singlet/triplet crossings is of central importance in mechanistic photochemistry. Nowadays, computational strategies are available for locating conical intersection and singlet/triplet crossing points and for constructing inter-state "photochemical" reaction pathways. These tools comprise methodologies for the optimisation of low-lying crossings between pair of potential energy surfaces and the computation of relaxation paths from a photoexcited reactant (e.g. from the Franck-Condon (FC) structure) to a deactivation channel. More in general, it is possible to compute the entire pathway connecting an excited state molecule to its ground state product. The major computational tools as well as few case-studies in the field of organic photochemistry will be revised by Cembran et al. in Chapter 6. The field of computational photochemistry is a relatively young field, especially when applied to the study of ultrafast reactions, but it is now established as a branch of computational chemistry and as a powerful, sometimes unique, way to simulate the molecular mechanism underlying fundamental chemical and biological events such as vision, primitive photosynthesis, phototropism, photochromism, bleaching, fluorescence, phosphorescence. Accordingly, the 3rd edition of the "Glossary of Terms Used in Photochemistry" (to be published in 2005) will contain new terms related to the use of computational tools in photochemistry (like Conical intersection, Photochemical Reaction Path, Minimum Energy Reaction Path).

2. PHOTOCHEMISTRY, PHOTOPHYSICS AND PHOTOBIOLOGY MEDIATED BY CONICAL INTERSECTION FUNNELS As mentioned above, in the past, correlation diagrams were, in many cases, the only practical tools available to the chemists to formulate reaction mechanism for thermal and photochemical reactions. For instance for pericyclic reactions Woodward-Hoffmann orbital correlation diagrams [42] and Longuet-Higgins and Abrahamson state correlation diagrams [43-45] were used. In the field of photochemical reactions the Van der Lugt-Oosteroff diagrams [40] were based on the hypothesis that avoided crossings provide the point of return of an excited state species to the ground state. At such an avoided crossing, if the energy gap is larger than few kcal mol"1, the excited state species will rapidly thermalize and the decay probability will be determined by the Fermi Golden Rule. However, within this model, the probability of decay should be small (unless the energy gap is small) and the radiationless decay process should occur on the same time scale of fluorescence (in ns [46; 47]). On the other hand, it is well known that many photochemical processes are extremely fast (well below one picosecond, i.e. on the timescale of a single molecular vibration) and associated with a complete lack of fluorescence. Furthermore, they are often stereospecific, implying a concerted mechanism. Indeed, femtosecond excited state lifetimes have been observed, for instance, for simple dienes [48], cyclohexadienes [48-50], hexatrienes [51], and in both free [52] and opsin-bound [53] retinal protonated Schiff bases. These observations suggest that a real crossing is accessible to the system. At such surface crossing the probability of decay is very high and the corresponding molecular identify the photochemical funnel (such name for the excited state decay channel suggests that the excited reactant must be "funnelled" through this point to initiate product formation). Thus, a photochemical funnel corresponds to a molecular structure that "lives" for only few femtoseconds (10" seconds). The history of conical intersection goes back to more than 60 years ago when, in 1937, the physicist Edward Teller giving a lecture at the Symposium on Molecular Structure [54] suggested that it was the electronic factors that may play the dominant role in the efficiency of radiationless decay. Teller made two general observations: in a polyatomic molecule the non-crossing rule, which is rigorously valid for diatomics, fails and two electronic states, even if they have the same symmetry, are allowed to cross at a conical intersection. radiationless decay from the upper to the lower intersecting state occurs within a single vibrational period when the system "travels" in the vicinity of such intersection points. On the basis of these observations, in 1969, at the Twentieth Farkas Memorial Symposium, Teller proposed that conical intersections may provide a common and very fast decay channel from the lowest excited states of polyatomics, which would explain the lack of fluorescence of the funnel[55].

In 1966, the organic chemist H. E. Zimmerman presented an alternative approach to the well known Woodward-Hoffmann method to predict the factors controlling ground and excited state reactions [9-11]. Zimmerman proposed an "MO Following" procedure that was capable of dealing with reactions lacking the symmetry to construct correlation diagrams. As an example he used the butadiene to cyclobutene closure and he found that along the reaction route a crossing (i.e., degeneracy) occurs. Thus, he concluded that such crossing point are significant in organic photochemistry and may provide a route for conversion of excited state reactant to ground state product. Michl [32; 33] proposed, independently, the same idea and documented such features in ab initio calculations on the H4 system [29; 30]. In 1970, Evleth and co-workers, in their work on the photolysis of aryldiazonium salts, interpreted their quantum yield measurements in terms of a complex energy surface crossing patterns. In the same years, Salem [35] proposed his state correlation diagrams that illustrated the occurrence of conical intersections at symmetric geometries in the photochemistry of carbonyl compounds. A continuous exchange of ideas between Salem, Turro e Dauben (as documented by Turro in a recent paper [56]) lead to the first complete classification of photochemical reactions [21; 36] using Salem's development of energy surface theory. Subsequently, geometries of few conical intersections were computed for Schiff base syn-anti isomerization by Bonacic-Koutecky and Michl [57], using ab initio procedures and the "3x3" model of biradicaloid electronic structure [58] was elaborated to permit qualitative prediction of geometries at which Si/So conical intersections take place [19; 34]. More recently, Yarkony [59; 60] and Ruedenberg [61] identified conical intersections geometries in small molecules. Despite the fact that the idea of Teller, Zimmerman, Michl and Salem represented an important refinement of the avoided crossing model, conical intersections were thought to be extremely rare or inaccessible (i.e. located too high in energy) in organic compounds and thus were disregarded. The main difficult has to be ascribed to the fact that, in practice, excited state quantum chemical computations require non-conventional methodologies and strategies based upon the use of multi-reference wavefunctions. (i.e., the so called post-SCF methods) rather than the standard single-reference SCF wavefunction. For this reason excited state computations were not routinely used by chemists. At the end of the 80's improved ab initio quantum chemical methodologies became available which were suitable for computing, in a balanced way, excited and ground state potential energy surfaces. In particular the ab initio Multiconfigurational Self-Consistent Field (MCSCF) method, developed by M. A. Robb in London, had an analytical gradient which could be employed for efficient geometry optimisation (the search for the structure corresponding to energy minima and transition states) taking into account the complete set of the 3N-6 nuclear degrees of freedom of the reacting system (N is the number of atoms). With this new methodology it was possible to overcome the limitations of the model proposed previously by Van der Lugt and Devaquet [31]. These limitations mainly regarded the computation of the excited state reaction path that was assumed to correspond to an interpolation between the reactant and the product geometrical structure. With the new tools one could determine real excited state reaction path where the reaction coordinates is not assumed but computed in a substantially unbiased way.

Conical Intersection

Interpolated Coordinate

Fig. 2. The relationship between the Van der Lugt - Oosterhoff model and the Conical Intersection (CI) model. The inset indicates the position of the Van der Lugt - Oosterhoff avoided crossing in the conical intersection region.

In Fig. 2 we show the relation between an avoided crossing and the double cone topology of a real conical intersection. In two dimensions, the Van der Lugt and Oosteroff model is refined by replacing the "avoided crossing" with an "unavoided crossing", i.e., a conical intersection (CI). In other words, the Van der Lugt and Oosteroff avoided crossing path R -> P is replaced by a path involving a real surface crossing R ->CI ->P. Bernardi, Olivucci, Robb and co-workers [62; 63] in 1990, reported a first application of the ab initio MCSCF method. This was a study of the photoinduced cycloaddition of two ethylene molecules and showed that: -

A conical intersection exists right at the bottom of the excited state energy surface. The molecular structure of the conical intersection is related to the observed photoproducts and stereochemistry of the reaction.

As discussed below, the original idea of Teller, Zimmerman Michl and Salem are now fully supported by the results of further computational work [64] which definitely demonstrates, when taken in conjunction with modern experimental results, that frequently radiationless deactivation occur via a conical intersection between excited and ground states. Radiationless decay at a conical intersection implies that: a) The internal conversion process may be 100% efficient (i.e. the Landau-Zener [65] decay probability will be unity)

b) Any observed retardation in the internal conversion or reaction rate (i.e. the competition with fluorescence) may reflect the presence of some excited state energy barrier which separates M* from the intersection structure and c) In the case where the decay leads to a chemical reaction, the molecular structure at the intersection must be related to the structure of the photoproducts. Points a-c provide the theoretical basis for the computational modelling of photochemical reactions. Between 1992 and 2002, a long-term computational project involving one of the authors has been carried out to prove the general validity of the hypothesis supporting the existence of low-lying conical intersections in organic molecules. The systematic search was performed on different classes of organic molecules with the intensive use of the MCSCF quantum chemical method. The application of powerful tools led to a detailed mapping of the potential energy surfaces of ca. 25 different organic chromophores, thus allowing the characterization of the conical intersections involved in the reaction mechanisms. The first result of such an extensive computational effort is that conical intersections may mediate all types of chemical events such as bond making, bond breaking, group exchange, intermolecular and intramolecular hydrogen transfer, charge transfer. The second outcome of the research is that conical intersections do not necessarily take part in a successful chemical reaction (i.e. reaction where the light energy is exploited to produce chemical species different from the reactants) but can also mediate light energy wastage mechanisms such as in quenching and internal conversion processes. Recently, an Si/So conical intersection has been characterized even in protein and solution environments using an hybrid Quantum Mechanics/Molecular Mechanics (QM/MM) strategy. Olivucci and co-workers [66; 67] located a 90°-twisted low-lying Si/So conical intersection in Rhodopsin and Bacteriorhodopsin using a CASPT2//CASSCF/AMBER level of theory. Toniolo et al [68] characterized the solution-phase conical intersections of the Green Fluorescent Protein (GFP) and Photoactive Yellow Protein (PYP) chromophores at semiempirical CAS/CI level. In conclusion, conical intersections could, contrary to common belief, be frequent (if not ubiquitous) in organic and bio-organic systems and, for many reaction, they constitute the photochemically relevant decay channel. The majority of the conical intersection structures documented for organic and bio-organic chromophores corresponds to low-lying conical intersections located at the bottom of the excited state relaxation path. As discussed in Chapter 6 of this book, these points could be only located using methods that allows for the computation of the so-called photochemical reaction path. A conical intersection is a point of crossing between two electronic states of the same spin multiplicity (most commonly singlet or triplet). Moreover, if we plot the energies of the two intersecting states against two specific internal coordinates Xi and X2 we obtain a typical double cone shape (see Fig. 3a). The Xi and X2 molecular modes define the so-called "branching" [61] or "g, h" [69] plane and the (n-2)-dimensional subspace of the n nuclear coordinates is called the intersection space, or seam of intersection [69], an hyperline

Ground Slalc PES ]

(a)

(b)

Fig. 3. (a) Representation of the typical double-cone topology for a conical intersection, (b) Relation between the "branching space" and the "intersection space".

consisting of an infinite number of conical intersection points (see Fig. 3b) and it is, locally, orthogonal to the two-dimensional branching plane. A molecular structure deformation along the branching plane lifts the Si/So energy degeneracy. Furthermore the ground and excited state wavefunctions undergo a dramatic change when the molecular structure is changed along a closed loop lying on the plane defined by the Xj and X2 modes and comprising the conical intersection. In particular, these wavefunctions exchange their character along the loop. The rest of this chapter contains some case studies involving the analysis of the branching plane structure and of the behaviour of the wavefunctions of the Si and So states in the region of the conical intersection. This will give a clear idea of the importance of such an analysis in providing information on the nature of the "reactive" process mediated by the such mechanistic entities. The results reported for each example have been produced using a common strategy based on the CASPT2//CASSCF level of theory. In this "mixed" computational method the full reaction coordinate is computed at the CASSCF level while the energy profile is computed at the CASPT2 level. This means that one applies multireference second-order perturbation correction to the CASSCF potential energy surface to incorporate dynamic correlation effects. For further details on these methods, the reader shall refer to Chapter 2 by Merchan and Serrano-Andres and, in part, to Chapter 10 by Roos. A discussion of the alternative and currently emerging Time-Dependent Density Functional Theory (TDDFT) methodology for computing the potential energy of electronically excited states will be given by Furche et al. in Chapter 3.

2.1 An example of a photoisomerization mediated by a conical intersection: the cis/trans isomerization in a rhodopsin chromophore model. The protonated Schiff base of retinal (PSB) is the chromophore of rhodopsin proteins. A light-induced cis->trans isomerization of the chromophore triggers the biological activity of rhodopsin, which, in turn, induces a conformational change in the protein. The detailed structure of the excited and ground state potential energy surfaces of the rhodopsin retinal chromophore model ?Z^-penta-3,5-dieniminium cation (CW-C5H6NH2) (1) and in particular the structure of the excited and ground state reaction path branches has been fully elucidated. Furthermore the reduced dimension of the model has allowed for the computations of abinitio CASSCF semi-classical trajectories and evaluation of the excited state lifetime and time scale of the photochemical isomerization. The results demonstrated that 1 provides a reasonable model for more realistic structures. In particular, the two-state two-mode nature of the reaction coordinate computed and observed (both in solution and in the protein) is maintained in the minimal model and the computed ultrafast excited state dynamics is still characterized by two different timescales corresponding to a very initial stretching relaxation (i.e. an inversion of the single bond/double bond positions) and to the following torsional deformation (about the central C2-C3 bond) respectively.

1 In Fig. 4 we plot the branching plane vectors (Xi and X2) at the conical intersection of 1. The conical intersection structure features one highly twisted double bond (about 92°) and involves two electronic configurations, an ionic and a covalent state, that differ for the transfer of one electron between the C5-C4-C3- and -C2-C1-N fragments.

10

Fig. 4. Branching (or g,h) plane vectors for the CA structure of 1. The Xi and X2 vectors correspond to the derivative coupling (or non-adiabatic coupling) and gradient difference vectors between the Si and So states

From the structure of the branching plane it is apparent that in this molecule Xi and X2 describe two types of processes. As shown in Scheme 1, motion along the Xi corresponds to a coupled pyramidalization (wagging) modes at the Ci and C4 centers of the it-chain. This motion allows for a widening of the C4-C3-C2-C1 dihedral angle leading to a it-bond breaking process. The X2 mode is characterized by a stretching deformation (a double bond expansion and single bond contraction mode) of the N=Ci-C2=C3-C4=C5 chain segment. Thus, motion along the X2 direction would ultimately yield two structures which may be represented by (resonance) formulas with inverted single and double bonds and with the positive charge shifted from the N-terminal to the Cs-terminal. These two 92° twisted structures will be less stable than the generated by motion along the wagging mode since the deformation along Xi allows for reconstitution of the central double bond providing strong coupling with the Z/E double bond isomerization coordinate. Thus, structural analysis of the branching plane suggests that upon decay from CI the molecule will generate the Z and E stereoisomers.

11

NHz

-•KES)

NH2

+{GS)

CI

+ Nhfe Scheme 1 The analysis of the wavefunction, taken together with the analysis of the branching plane, provides the basis for the rationalization of the electronic structure of the ground state energy surface comprising the reactant and product valleys (and, eventually, the transition structures connecting them). The result of such an analysis for chromophore 1 is shown in Scheme 2 where the wavefunction is analyzed in terms of point charges of the C5-C4-C3- and -C2-C1-N fragments along a loop centered on the CI and lying along the plane defined by the Xj and X2 modes. The charge distribution of the system demonstrates the existence of two different regions. The first region 0°< co < 30°, 200°< w < 360° is characterized by a structure where the charge is mainly localized on the N-terminal part of the molecule. The second region 30°< co < 200° is characterized by a structure where the positive charge is mainly located on the Cterminal part of the molecule. The border between the two regions corresponds to the electron transfer events between the two fragments. Notice that the wavefunction changes are associated with the two minima in the energy gap diagram.

12

/ degrees

Scheme 2 As we have previously underlined, the low-lying conical intersections could be only provided through the computation of the photochemical minimum energy path (MEP). However, some cases have been documented where excited state reaction path does not necessarily hit the lowest energy point belonging to the intersection space (IS) and the decay may not occur in this region. One of these cases regards the excited state relaxation path of the PSB chromophore 1. Indeed the mapping of the low-lying segments of the IS for this chromophore (see Fig. 5), by means of constrained MEP computations, demonstrated that it ends at a conical intersection with a ca. 70° (Cl7o°) twisted structure. The intersection space remains then coincident with the reaction path up to the lowest energy intersection (Cl92°) that has a 92° twisted structure [70]. Notice that in this situation the main locus of excited state (Si) decay is predicted to be Cl7o°. Semi-classical dynamics calculations on this model together with the calculations of the intersection seam connecting the So and Si potential energy surface spanning the entire range of twisting of the central double bond from 0° to 180° reveals the relationship between the excited-state reaction path and the intersection seam. The results show that motion along the reaction path is a good description of the photochemistry of the rhodopsin model. For a complete discussion about dynamical considerations on this example see Section 4 of Chapter 5 of this book. More in general, Chapter 5 and Chapter 7 provide an introduction to the use of non-adiabatic molecular dynamics to the investigation of photochemical reactions. Such studies bring the description of the reaction mechanism well beyond the photochemical reaction path picture discussed in this Chapter and in Chapter 6.

13

hv MaJorS,->S, daeay channel

Reaction Coordinate

Fig. 5. The excited state reaction path of the cation 1 intercepts the conical intersection point CI70" located ca 5 kcal mol"1 above the minimum energy conical intersection C I ^ . FC->Cl7o° The values of the relevant structural parameters are given in A and degrees. Redrawn with permission from reference [71] © 2004 World Scientific Publishing Co. Pte. Ltd.

The same model has been chosen to test the applicability of TDDFT method to excited state reactivity problems. Vertical excitation energies computed using the two different CASPT2//CASSCF and TDDFT//CASSCF treatments are in agreement. On the other hand, quantitative discrepancies are found along the reaction coordinate demonstrating that the quality of TDDFT must be further investigated especially with respect to the calculation of excited state reaction coordinate. In spite of these differences, TDDFT//CASSCF energy profile is found to describe the conical intersection region [72]. For a complete and detailed description of TDDFT performances in excitation energy and excited state structure computations see Chapter 3.

14 2.2 An example of photoaddition mediated by a conical intersection: a peryciclic reaction (ethylene + ethylene) The conical intersection seen in Fig. 4 for a protonated polyene Schiff base is an example of conical intersection between two electronic states, which are related by a charge transfer from a region of the molecule to another. Here, we show an example of conical intersection between two states, which does not differ for a different charge distribution but for a different spin distribution among the active orbitals. The Si/So conical intersection for the [2jts+2its] cycloaddition of two ethylene molecules is shown in Fig. 6a [62; 63]. The intersection is formed by two interacting olefinic fragment bound in a rigid rhomboidal structure (2.19 A interfragment distance and 110° rhomboidal distortion) with C2h symmetry. The branching plane vectors are shown in Fig. 6b. Motion along Xi lifts the degeneracy by changing the intermolecular C1-C4 interfragment distance whereas motion along X2 lifts the degeneracy by changing the angle of attack. Both motions are schematically illustrated in Scheme 3a where it is clear that the evolution along Xi leads to the reactant pair (-Xi) and cyclobutane (Xi). Alternatively, the evolution along X2 yields the tetramethylene biradical (-X2) or a rectangular saddle point (X2). The same prediction can be made analyzing the electronic structure of the conical intersection. Indeed, Si is described by a doubly excited state configuration, '(jt|2-jt32), involving excitation of two electrons from the ethylene-dimer HOMO (112) to the ethylenedimer LUMO (113) and can be identified as the combination of two ethylene molecules in their one-electron excited n-n* state. The electronic structure is tetraradical: the four unpaired electrons can recouple in different ways leading to the ethylene dimer reactant (the coupling is C1-C2, C3-C4), cyclobutane product (the coupling is C1-C3, C2-C4), and tetramethylene biradical (the coupling is C1-C4).

15

2 28 A

(a)

(b)

Fig. 6. (a) S|/So conical intersection for the [2jts+2res] cycloaddition of two ethylene molecules, (b) Branching (or g,h) plane vectors, X! and X2, for the CI structure in (a).

A combined Natural Bond Orbital (NBO) and wavefunction analysis along a small loop around the conical intersection shows that, in contrast to the conical intersection of a retinal model, there is no substantial variation in charge distributions. In this case, the analysis based on a representation in terms of localized 2-center spin orbitals shows that there is a change in the electronic distribution between aa-c3(C2-C4) and Jtci-c2(C3-C4) bonds. Note that even in this case the two dramatic wavefunction (i.e. electronic structure) changes correspond to the two minima in the Si-So energy difference diagram of Scheme 3b.

16

(a) Scheme 3 2.3 An example of photofragmentation mediated by a conical intersection: the photodenitrogenation of a bicyclic azoalkane. The photochemical denitrogenation of the 'n-it* 2,3-diazabicyclo[2.2.1]hept-2-ene (DBH) has been the subject of an intense investigation since the first report of Salomon[73] and coworkers in 1968 especially because of the unusual stereoselectivity observed during the nitrogen extrusion and formation of the housane. Upon thermal or photochemical excitation, in fact, DBH and its derivatives lose molecular nitrogen.

hv DBH

I

N

linear-axial Cf

exo-ax/a/ C,-/*e 1 DZ

17

Fig. 7. X[ and X2 vectors for the CI structure of DBH (showed in the inset) corresponding to two orthogonal bendings of the NNC angle.

The intersection structure that mediates the C-N a-cleavage is characterized by a linear-axial arrangement of the NNC fragment in which one of the two CN bond is still intact (1.48 A) [74; 75]. In Fig. 7 we plot the branching plane vectors Xi and X2 which correspond to two orthogonal bendings of the NNC angle. As illustrated in Scheme 4, after structural considerations, it is clear that the two CNN bending prompt the formation of ground state diazenyl diradical (*DZ in the Scheme above), either in the exo or endo or endo-exo forms.

18

•N A—1.18 A

(N.

exo

CI

N i/

N

endo-exo Scheme 4

The electronic structure of the intersecting states of the a-CN bond cleavage of DBH is shown in Scheme 5. The azoalkane Si state is described by a tetraradical configuration with one electron residing in each a-CN a-orbital, one electron residing in the excited nitrogen lone pair and one electron inside the p-orbital of the other nitrogen. The azoalkane So state is described by a biradical configuration in which the two a-CN a-orbitals are singly occupied.

Si1n-re* Scheme 5

19

Reaction path calculations indicate that the observed inversion of stereoselectivity has to be ascribed to the impulsive population of the vibrational mode that triggers an axial-toequatorial ring inversion. This idea is supported by classical trajectories calculation. In fact, as shown in Fig. 8, after a first oscillation (within 40 femtoseconds) in the direction of an unstable (transient) bicyclic intermediate, the molecule reaches a highly strained structure form which it can only relax following the initial direction of motion. After 60 femtoseconds the structures reaches the exo-axial 'DZ configuration and after 80 femtoseconds the axial to equatorial transition structure. The inverted configuration is then reached in 100 femtoseconds. Such ground state trajectory computation has been started from a point close to the conical intersection. Chapter 5 and 7, in this book, deals with the computations of photochemical (non-adiabatic) trajectories that start on the excited state energy surface and end on the ground state photoproduct minima.

El kcal mol

-25 120 Time (fs) Fig. 8. Triplet DFT energy profile along the trajectory computations started at a point closed to the CI of DBH. The structures document the molecular changes along the simulation. Redrawn with permission from reference [74] © 2003 American Chemical Society.

20

2.4 An example of charge transfer and an hydrogen transfer (intermolecular) processes mediated by a conical intersection: the fluorescence quenching of bicyclic azoalkanes. In contrast to DBH, there exist a different class of azoalkanes that are essentially inert to photochemical denitrogenation, the so-called "reluctant" azoalkanes. A representative system is the 23-diazabicyclo[2.2.2]oct-2-ene (DBO, see Scheme 6) which display exceedingly long singlet n,jt*-excited lifetimes (up to 1 |is).

hv

CHClj

Z5 or INEt,

Scheme 6 Using the 1,2-diazacyclopent-l-ene (pyrazoline) as a reduced model of DBO, allowing for the use of these accurate but expensive methods, it has been possible to demonstrate, in combination with the experimental evidence, that there are two basic mechanisms for the quenching of 'n,jt* states [76-79]. Indeed, DBO is efficiently quenched by hydrogen donors (such as non protic solvents like chloroform, methanol, benzene) via either a concerted or a stepwise process and by electron donors (such as amines like triethylamine) via a concerted process only. The computations have been carried out using CH2CI2 and methanol to model hydrogen donors, and trimethylamine and dimethyl ether as prototypes of strong and weak electron donor solvents, respectively. As shown in Fig. 9 both quenching routes involve bimolecular photochemical reactions (that is an electron or an hydrogen atom transfer photoreductions) and a full deactivation through a Si/So CI channel, which is located roughly halfway along the reaction coordinate and prompts a reaction path branching. The first branch (full arrows) is associated with an "aborted" chemical reaction. The second branch (light arrows) is associated to production of a very unstable transient species that may not accumulate but reverts to the original material by passage through a low-lying transition state located on a ground state reaction coordinate (dashed curve). Thus, for both routes a chemical transformation is initiated but it is not achieved.

21

Pyrawiline

•

CHjCl.orNfCHj),

quencher chromophorc Reacliun Cmirtlinale

Fig. 9. Potential energy diagram showing the interplay between ground (So) and excited (Si) state surfaces in the fluorescence quenching of n,jt* state chromophores due to an hydrogen donor or electron donor species. The full and light arrows describe the concerted and stepwise energy wastage route.

The conical intersection (see the ball and stick structure below) for the hydrogen abstraction mechanism (in the present of CH2G2) is characterized, with respect to the reactant pair, by a shortening of the H—N distance (from 2.22 to 1.02 A) and by a simultaneous expansion of the C—H distance (from 1.07 to 2.01 A).

The branching plane vectors are shown in Fig. 10. Motion along Xi lifts the degeneracy by stretching the intermolecular N-H distance whereas motion along X2 lifts the degeneracy by an out of plane pyrazoline ring distorsion. The evolution along both motions is schematically illustrated in Scheme 7. Distortion along Xi leads to production of a radical pair (RP) while distortion along -Xi leads to production of an unstable ion pair (IP). Evolution towards the X2 and -X2 directions leads to two equivalent ground state transition structures which feature a distorted pyrazoline ring.

22

Fig. 10. Xi and X2 vectors for the CI structure of pyrazoline + CH2C12 involved in the hydrogen abstraction mechanism of Fig. 9.

ci

}

O^ oi < 200°

0°< w c 100° * N

H

290°< w < 0°

CI

CI

IP

i

RP

-x,

/

I* Scheme 7

200°* w < 290'

23

The result of the wavefunction analysis indicate that the n,jt*-excited state correlates with a radical pair structure (derived from a complete hydrogen atom abstraction), and the ground state correlates with an ion pair (derived from proton abstraction) (see Fig. 1 la). Similarly to retinal PSB models, the configurations that describe the intersecting states are interchanged by a charge transfer (intermolecular instead of intramolecular). Indeed, the charge distribution of the system (see Fig. 1 lb) obtained computing fragment charges along a small loop lying along the plane defined by the Xi and X2 modes and centred around CI demonstrated that exist two different region. The first region (0°< co< 100° and 290°< co< 360°) is characterized by an ion pair structure where the charge on the pyrazoline is positive while on CHCI2 is negative. The second region 100°< co< 290° is characterized by a covalent structure. The border between the two regions corresponds to two sudden electron transfer events, one from the CHCI2 anion to the pyrazoline cation yielding the RP configuration (co = 90°) and the second in opposite direction (00 = 270°).

(a)

(b)

0

60

120

180

240

300

360

CO / degrees

Fig. 11. (a) Modified state correlation diagram of the n,it*-excited state (ES) of pyrazoline + CH2C12 correlating with the radical pair (RP) derived from hydrogen atom abstraction and the ground state (GS) correlating with the ion pair (IP) derived from proton abstraction, (b) So fragment charges [a.u.] along a loop centered around the CI (pyrazoline fragment, open triangles; hydrogen atom, open squares; CHCI2 fragment, open circles). Redrawn with permission from reference [77] © 2001 WileyVCH Verlag GmbH.

24

The geometrical structure of the conical intersection and the vectors of the branching plane for the charge transfer process (i.e. the present of trimethylamine) are given in Fig. 12. Notice that Xi is dominated by the out of plane deformation of the pyrazoline ring and X2 is dominated by the interfragment distance. The computed photochemical reaction path demonstrates that the excited state branch of the path is dominated by the decrease in distance between the pyrazoline and N(CH3)3 fragment. After a small excited state barrier, the progression along the path leads to the formation of an exciplex located in the close vicinity of a conical intersection. The intersection is accessed when the distance between the pyrazoline and amine nitrogen atoms is ca. 2 A. At the exciplex the computed amount of charge transfer from the trimethylamine lone pair to the excited state half-vacant nonbonding orbital of one pyrazoline nitrogen atom is 0.3 electrons.

Fig. 12. X[ and X2 vectors for the CI structure of pyrazoline + trimethylamine involved in the charge transfer process of Fig. 9.

25

Q

O

N CH

... (

3)3

.

N CH

( 3)3

CI

Interfragment Distance

Fig. 13. Modified correlation diagram for the interaction of the n,jt*-excited state of pyrazoline with an electron donor such as the trimethylamine reflecting the occurrence of an exciplexes and a conical intersection along the reaction pathway. Redrawn with permission from reference [76] © 2000 WileyVCH Verlag GmbH.

The exciplex state N...N two-orbital/three-electron bond can be viewed as a mixture of a covalent (N=N*---:NMe3) and an ionic (N=N*~---+*NMe3) electronic configuration. The steep rise of the ground state energy surface toward the conical intersection is due to a destabilizing two-orbital/four-electron repulsive interaction (N=N5--- :NMe3) (see Fig. 13).

2.5 An example of stereoselective photochemical reaction mediated by conical intersection: a Norrish Yang photocyclization. The hydrogen abstraction process mediated by a conical intersection, and documented above for pyrazoline, can also be found for structure 1 (see Scheme 8). Here the hydrogen atom transfer is intramolecular rather than intermolecular. In this structure a carbonyl function replaces the azo function (-N=N-) of the n^it* chromophore. Structure 1 models an alanine derivative, which undergoes, upon photoexcitation, a Norrish-Yang photocyclization reaction. Minimum energy path calculations demonstrated that this is an example of 'n,jt* photochemical reaction displaying a chiral memory effect [80] in agreement with experimental results [81].

hv

26 Scheme 8

Intermediate

Z5

.,«OH

r

Reaction Coordinate

Intermediate

Fig. 14. Potential energy diagram showing the ground (So) and excited (Si) state potential energy surfaces of the Norrish-Yang photocyclization of 1.

In fact, as illustrated in Fig. 14, the computed reaction coordinate leads to a CI displaying an incomplete hydrogen atom transfer to the 'n-jt* carbonyl oxygen. Once again, there are two ground state relaxation paths that develop from the intersection. Accordingly, while the first path leads back to the starting material (light arrows), the other path (full arrows) leads to a diradical species (intermediate). This is also demonstrated by plotting the branching plane vectors, which are dominated by the transfer of the H atom. Most important, no torsional components which can lead to a loss of stereochemistry are present (Fig. 15). However, in contrast to the pyrazoline (i.e. DBO) quenching, the intermediate structure does not easily revert to the starting material but is the precursor of the stereospecific product 2.

27

Fig. 15. Branching (or g,h) plane vectors (Xi and X2) for the CI structure of Structure 1. This model of an alanine derivative undergoes, upon photoexcitation, a Norrish-Yang photocyclization reaction.

2.6 Towards Computational Photobiology: the Rhodopsin Proteins The recent implementation of a QM/MM computational method based on the use of an ab initio CASPT2//CASSCF/6-31G* strategy (i.e. geometry optimization at the CASSCF level and energy evaluation at the CASPT2 level) coupled with a protein force field such as AMBER (or CHARMM) paved the way for excited state computations and conical intersection search in proteins (e.g. in rhodopsins, fluorescent proteins and others). This new appealing method has been applied to study the spectroscopy of two photoreactive proteins: the visual pigment Rhodopsin (Rh) and the Green Fluorescent Protein (GFP). The results demonstrate that the method is capable to provide a qualitatively correct description of the geometrical and electronic structure of the protein chromophores and their Si/So energy gap (the absorption and emission (for GFP) maxima) within a dim VSD

(44)

The coefficients (Cj, j = 1,..., M} are obtained from the system of linear equations >

i=l,...M

(45)

where E(o) =(o|H°|o) is the zeroth-order wave function and must be solved iteratively. In standard CASPT2, the zeroth-order Hamiltonian is expressed in terms of a generalized Fock operator, which can be written as a sum of a diagonal, FD, and non-diagonal, FN, contributions F T =F D +F N

(46)

The operator is defined in such way that for a closed-shell HF reference wave function is equivalent to the Moller-Plesset Hamiltonian. For multiconfigurational single-reference perturbation theory, the choice of the zeroth-order Hamiltonian is not unique and it has been

60 the subject of active research and discussions yielding a number of different successful variants [16]. Eq. (45) can then b e rewritten as '|o)

i = l,...M

(47)

In order to simplify the notation, the following matrices and vectors are introduced X = D, N J

V; = i H O

(48) (49) (50)

where i, j = 1, ...M. The column vector C contains the coefficients Cj of the expansion. The difficulties in the resolution of Eq. (47) depend on the choice of the one-particle operator. In the simplest case, using F D , it leads to [F D -E (O) S]C = -

(51)

and the second-order correction to the energy comes out as the product of V'C. In most cases M > dim VSD and the linear dependences have to be removed. It is done by diagonalizing the overlap matrix S and discarding the eigenvectors with eigenvalues equal (or close) to zero. The resulting vectors are then orthogonalized and a subsequent diagonalization of the Fock matrix written in the orthogonal basis takes place. The E(2) correction is easily evaluated as function of the transformed matrices. The process becomes much more elaborated when the full operator FT is used [10]. It is the recommended procedure. The normalized wave function corrected up to first order is given by = C0 o) + C,

(52)

with C^ +C 2 = 1. The weight of the reference function (C^) can be used as a simple and rapid criterion of quality for the perturbation treatment carried out. Ideally, in order to get a fast convergence in the perturbation series, the weight should be close to unity. Nevertheless, its value depends on the number of correlated electrons [28]. Thus, upon enlarging the molecular system the reference weight decreases. The electronic excited states considered should have a similar magnitude for the weight as compared to the ground state, employing the same active space. Sometimes intruder states appear in the second-order calculation, which are normally related to the occurrence of large coefficients in the first-order expansion, leading to a low

61 value for the reference weight. Analysis of the states with large coefficients (intruder sates) may give a hint about the type of reformulation in the perturbation partition necessary to overcome the problem. Thus, a new CASSCF calculation might be designed comprising in the active space the orbitals implied in the description of the previous intruder states. It is the proper action to be taken when intruder states are strongly interacting with the CASSCF reference wave function, with contribution to the second-order energy larger than 0.1 au, because it points out to obvious deficiencies in the choice of the active space. Intruder states are often present in the treatment of excited states of small organic compounds when the active space does not include the full n valence system. Thus, the low weight for the zerothorder wave function in such a case just tells us that the active space has to be enlarged in a way that previous intruder states would be treated variationally, that is, they should be moved to the CAS-CI space. It is also frequent to find calculations where the reference weight of the excited state is "somewhat low" compared to that of the ground state, but a particular state cannot be identified as intruder in the first-order wave function, which is instead characterized by a large number of low-energy minor contributions. It occurs often in the simultaneous computation of valence and Rydberg states, where the one-electron valence basis set has been augmented with Rydberg-type functions. We have to face then accidental near-degeneracy effects, implying weakly interacting intruder states, and the level-shift (LS) technique is especially useful in order to check the validity of the perturbation treatment performed. Many times one has to apply both strategies: enlargement of the active space to overcome the problem of severe intruder states, and, with the enlarged active space, the LS technique is applied in order to minimize the effect of weak interacting intruder states. The level-shift CASPT2 (LS-CASPT2) method removes efficiently weak intruder states by the addition of a shift parameter, s, to the zeroth-order Hamiltonian and a subsequent back correction of its effect to the second-order energy [16, 28, 74]. It can be shown that the corrected level-shift second-order energy, Ej2*!, is equal to the standard CASPT2 energy, E(2), in first order of s 1

(53)

CO

where E(2)and ffi (weight of the CASSCF wave function) were obtained by using the shifted Hamiltonian. The relationship (53) might not be valid when intruder states appear in the firstorder interacting space. It is highly recommended to make an analysis of the trends for the weights a, total, and excitation energies upon varying the values of s. For instance, results at s = 0.0 (standard CASPT2), 0.1, 0.2, 0.3, 0.4 au are sufficient to establish the proper behavior of the LS-CASPT2 results. It is extremely dangerous to rely on just one result, because the appearance of an accidental near degeneracy might lead to large errors in the excitation energies. In order to demonstrate the proper performance of the LS-CASPT2 technique, calibration calculations of that type always have to be carried out. The best choice for s is the lowest possible value capable of removing intruder states. In the absence of intruder states the ELs3pz)

6.21

6.17

0.0532

1 'B,(4b2->3a2)

8.02

6.36

forbidden

3'E(7C7t*)

10.28

6.40

1.1096

5a,-»3d

6.75-7.35

6.57-6.80

0.0014

"Observed by electron energy-loss spectroscopy (EELS) [102].

Table 4 Convergence pattern for the lowest vertical transition 1 'A|—»1 'A 2 (5ai^3a 2 ) upon improving the contraction scheme. Basis set*

7I-CASPT2 (eV)

Previous (eV)

C[3s2p]/H[2s]

4.45

4.37b

C[3s2pld]/H[2s]

3.92

4.00°

C[3s2pld]/H[2slp]

3.92

C[4s3pld]/H[2slp]

3.90

C[4s3pld]/H[2slp] + 2s2p2d

3.79d

C[5s4p2dlf]/H[3s2pld]

3.80

"Primitive sets: C(14s9p4d3f)/H(8p4p3d) ANO-type basis set b

MRCI result [106].

C d

TI-CASPT2/6-31G* result taken from Ref. [101].

From Table 3.

6.02 6.42

76 Coming back to Table 3, the following remarks are pertinent: •

The most intense feature is related to the 3'E(7ITI*) state, the plus state described above. It is placed at about 10 eV at the CASSCF level and the CASPT2 result, 6.40 eV, is in agreement with experiment. The effect of dynamic correlation is crucial for the accurate location of the state. The l'E(7r7i*) state corresponds to the minus state and is close to the lowest Rydberg state.

•

The 3'Ai(7i7i*) state has a prominent weight (33.2%) of the doubly excited configuration (HOMO->LUMO)2 in the CASSCF wave function. The 3'AI(TI71*) and l'B2(o7i*) states are degenerate. The most plausible assignment responsible of the observed shoulder at 6.02 eV is the Rydberg transition to the 3pz orbital, although the valence state an* might also contribute to this feature in the gas phase. The primary Rydberg character for the shoulder recorded in the gas phase is supported by the fact that it is not observed in the absorption spectrum of COT in hexane. It is well recognized that Rydberg states are usually perturbed in condensed phases and they collapse in solution. Many different earlier assignments for the observed shoulder can be found in the literature [105]. However, the issue is now clarified theoretically and it would be highly desirable that the assignment could be confirmed unambiguously by experimental research.

As we see both valence and Rydberg states coexist in the same energy region. It is more a rule than an exception for molecules of medium molecular size [15-18]. COT is employed for efficient laser operation of dye solutions because of the unique properties of its lowest triplet state. During the operation of laser dye solutions, the triplet excited levels of the dyes are populated along with the singlet states, which causes a detrimental effect in their operation. A fraction of the excited singlet state population responsible for the laser action becomes deactivated by the intersystem-crossing mechanism. These triplet dye molecules exhibit broad optical absorption triplets-triplet spectra with relatively high intensities, with the consequent loss of laser efficiency. Hence, it is essential to use a suitable triplet scavenger, able to remove the triplet dye molecules, without interfering in the laser efficiency. The acceptor COT fulfils the requirement and it is widely used for this purpose. As can be seen from Fig. 5, where the main CASPT2 results for the So—»Tj transition are depicted, the energy difference between the vertical and adiabatic excitation energies is large, about 2 eV. Simultaneous to the electronic excitation of COT, a progressive structural reorganization towards planarity takes place. Therefore, COT has a pronounced non-vertical behavior and covers a wide range of triplet donors, D*(Ti). The origin of the So^Ti transition, about 0.8 eV, can be considered as an estimate of the lower limit for the triplet energy of a donor that the acceptor COT could still react with [108]. On the other hand, the vertical phosphorescence is predicted in the infrared range.

77

vertical 1

i

V emission

adiabatic

' "}-0.22eV

~i U 0.78 eV

2.82 eV '

0

J

So"

J

C

Fig. 5. The lowest singlet—^triplet electronic transition. CASPT2 results for the vertical absorption, vertical emission (phosphorescence), and adiabatic excitation energies for COT. The triplet energy of a donor D is also represented.

Additional information about the lowest triplet state was obtained from the photoelectron (PE) spectrum of the radical anion, where photodetachment to two distinct electronic states of neutral COT was observed. Wenthold et al. [103] identified these electronic states as: •

The l'Aig (D4h) state at 1.1 eV, which corresponds to the transition state of COT ring inversion along the So hypersurface, and

•

The l3A2g (D8h) state; the lowest triplet state, at 1.62 eV.

Employing the ground state structure optimized for the COT radical anion, the PE spectrum was computed. At the CASPT2 level, the ground state of the neutral system is found to be at 1.11 eV and the lowest triplet state at 1.47 eV in reasonable accordance with the experimental data. As can be seen in Table 5, the results at the CASSCF level are poor. The minus sign in the CASSCF result (-0.86 eV) implies that the radical anion is not bound. Therefore, the CASPT2 approach is capable of recovering (qualitatively and quantitatively) the right relative position between the respective states of the anion and the neutral systems. It is really amazing, especially if one realizes that CASPT2 is just a second-order perturbation approach. In this type of difficult cases, involving large differential dynamic correlation contributions, the CASPT2 method certainly plays an outstanding role. Similar comments are valid for the computed electron affinity. The vertical EA is negative at both CASSCF and

78 CASPT2 levels of theory. Therefore, we can confidently conclude that the radical anion is not actually bound at the ground-state equilibrium geometry of COT (1 Ai (D2d))- In addition, the CASPT2 adiabatic EA, 0.56 eV, is in agreement with recent experimental determinations (0.57 eV) [109]. The agreement is entirely due to the inclusion of dynamic correlation because at the TI-CASSCF level the ground-state radical anion is above the ground state of the neutral system by nearly one and half eV (-1.43 eV in Table 5). Table 5 Computed photoelectron spectrum for the planar cyclooctatetraene radical anion and electron affinity of COT employing the ANO-type C[4s3pld]/H[2slp] + 2s2p2d (Rydberg functions) basis set. Energy differences in eV [105], State

TI-CASSCF

71-CASPT2

Experimental

Ground state of cyclooctatetraene radical anion: 1 B hl (D4i, symmetry) -0.86

1.11

1.10a

0.16

1.47

1.62a

vertical EA

-2.79

-0.49

adiabatic EA

-1.43

0.56

1'Aig 3

l A 2e

0.57b

"Taken from the recorded photoelectron spectrum [103]. b

See Ref. [109] and cited therein.

5.4. Up-to-date Theoretical Spectroscopy. We shall leave for the next section the purely photochemical problems, that is, those in which different photoproducts are generated or those in which non-adiabatic state transitions occur, and focus here on spectroscopy, understood as the assignment of absorption and emission band positions and intensities, radiative lifetimes, and environmental effects. Theoretical ab initio spectroscopy can provide nowadays extremely accurate data that help to interpret and rationalize the experimental recordings and to predict new findings. Not all systems can be equally computed at the same level of accuracy. Excitation energies and oscillator strengths for organic systems up to the size of, for instance, the free base porphin molecule (C20N4H14), have been studied using accurate methods and basis sets: CASPT2 [110], in which a novel interpretation of the spectrum was put forward, SAC-CI [111], and EOM-CCSD [112], although the single-configuration methods showed less accuracy. In order to compute the low-energy spectra of larger systems such as fused zinc porphyrin dimers (Zn2C4oNsH22) at the SAC-CI level [56], severe approximations, such as lack of polarization functions in the basis sets or partial removal of virtual orbitals, were performed, undoubtedly decreasing the accuracy of the results. As a rough estimation, error boundaries smaller than 0.3 eV are required in order to obtain reliable interpretations of many spectra. DFT approaches for excited states, TD-DFT theories basically, were expected to be able to deal, although at low level of accuracy, with large systems were ab initio methods become too expensive. Unfortunately, recent findings have proved that the TD-DFT methods fail

79 dramatically in too many situations: charge transfer states [113], multiconfigurational states [113], doubly or highly-excited states [61, 113], valence states of large it extended systems such as acenes, from naphthalene to octaacene [114, 115], polyacetylene fragments or oligoporphyrins [116], polyenes, from butadiene to decapentaene [117], and the list increases every day. In some cases the errors are larger than 5 eV [114]. Regarding inorganic electronic spectroscopy, only multireference perturbation theory, CASPT2 basically, has been able to obtain general and accurate results in systems so different as ionic transition metal (TM) molecules, covalent actinide complexes or organometallic metal-ligand compounds. Typical examples are chromium hexafluoride and hexachloride anions, iron porphyrins, tetra-, pentaand hexacarbonyl or cyano TM complexes, TM dihalides, cyclometalated compounds, blue copper protein chromophores, and lanthanide and actinide oxides [16, 118-121]. The virtual extension of the multiconfigurational approaches to systems with several transition metal atoms is complicated because the selection of the reference becomes challenging [16, 118]. In the other side of the scale, ab initio methods can yield extremely accurate information for small systems, provided that high-level approaches and large basis sets are used. In these cases, the required accuracy is not far from the usually recognized as chemical accuracy, 1-2 kcal/mol. Not only electronic data are needed, also detailed vibrational or rotational spectroscopic information, and, typically, also vibronic or spin couplings have to be included. In very small systems, the MRC1 method can be considered extremely accurate and general, provided that the problem of the size-extensivity is corrected or estimated [5, 7]. If the system fulfils certain requirements such as a closed-shell ground state well represented by a single reference and excited states of clear singly excited character, single-configuration coupledcluster approaches including triple excitations, EOM-CCSD(T) or CC3 for instance, may offer high accuracy. Those methods are size extensive and can in practice be extended further than the MRCI approaches. In any case, the only generally applicable methods are the multireference perturbation approaches, which means, CASPT2 and related. CASPT2 is a non size-extensive methodology but, in practice, it can be shown that, in the calculation of spectroscopic properties, the corresponding effect is negligible [16]. The expected accuracy, being simply a second-order perturbation theory, cannot be as large as more elaborated approaches, except for the fact that it does not present unexpected failures in difficult cases, assuming a computation free of intruder states. As an example of the required accuracy needed to solve spectroscopic problems, a CCSD(T) study of the ground state of the van der Waals Ar-CO complex required the use of a basis set composed by aug-cc-pVQZ plus midbond functions in order to get an accuracy close to 0.3-0.4 cm" and assign conflictive rovibrational bands [122]. In general, methods for electronic excited states cannot reach the same precision. Last decade has known tremendous breakthroughs in the field of quantum chemistry of the excited state. The number, size, and accuracy of the computed problems have grown up to the point of being comparable in certain cases with the experimental measurements, in particular for gas-phase spectroscopy. Solvent simulations in spectroscopy, basically by the Reaction Field (RF) or Quantum Mechanics/Molecular Mechanics (QM/MM) approaches, cannot be

80 considered quantitative so far, although they are helpful to elucidate spectroscopic assignments [2, 123, 124]. If we summarize a number of achievements made by modern quantum-chemical theories in the field of spectroscopy, it is worth remembering that, nowadays, all type of states can be computed accurately, whether valence, Rydberg or multipole-bound anionic states, optically one-photon allowed or forbidden (dark) states, and covalent, ionic, and zwitterionic states [18]. Band origins (Te or To transitions) [15-18, 42, 125] and vibrational profiles for electronic absorption and emission bands, involving ground and excited states geometry optimizations and knowledge of the states force fields are also computed with high accuracy for medium size systems such as benzene [90], pyrrole [126], and /i-benzosemiquinone radical anion [127] leading to straightforward comparisons with experiment. Even the effects of the anharmonicities in the vibrational bands positions and intensities can be computed at different levels for, at least, small systems. As an example, the low-lying absorption and emission spectra of the formyl radical obtained at the CASPT2 level, which required the calculations of quartic potentials built by computing hundreds of points in the hypersurfaces [128]. Methods to incorporate vibronic couplings at different levels and obtain refined effects on the intensity of the vibrational bands, become also available, although at high cost [128-130]. Examples of the inclusion of accurate calculation of vibronic couplings considering the interaction of several electronic states include the CASSCF/MRCI description of the S2(TI7I*) absorption band of pyrazine [130], the Green's function treatment of the photoelectron spectrum of benzene [130], and CASPT2 and EOMCCSD studies of pyridazine and pyrimidine [131, 132]. Other consequences of the breakdown of the Born-Oppenheimer approximation such as the Jahn-Teller and Renner-Teller couplings have been widely studied in small systems, where high accuracy is needed [4]. Finally, spinorbit couplings and relativistie effects computed at ab initio levels are becoming generally available for the excited states of systems including heavy atoms [4]. An example is the recent implementations of the combination of two-component relativistie formulations using a Douglas-Kroll Hamiltonian to incorporate the scalar effects and the use of multiconfigurational CASSCF/CASPT2 or shifted RASSCF methods with relativistie basis sets to solve the spin-orbit Hamiltonian. This approach proved to get errors in the relativistie effects negligible if compared with the accuracy of the methods to account for the correlation energy [133]. The whole previous discussion leads to one simple conclusion: within certain limitations related to the size of the systems, quantum-chemical methods applied to theoretical spectroscopy have reached the point where a real and constructive interplay can be established with experiment [134-137]. Both approaches, experimental and theoretical, will become more accurate in different cases. For instance, nowadays, none experimental determination can probably match the theoretical calculation of the ground-state structure for an isolated molecule, that is, modeling the system in the vapor phase. In other cases, such as hyperfine couplings at different levels or situations where the environment produce fine effects, the theoretical methods do not have enough accuracy as compared with recorded data. Apart from energies, excited states properties and transition probabilities are now routinely computed for many systems. In some cases, such as multipole moments in excited states, the

81 accuracy reached by the theoretical methods is also unmatched by the experimental measurements. Many representative examples can be given of this new era in the quantum chemistry of the excited state in which the ab initio methods, especially the multiconfigurational CASPT2 approach [15-18], have been the main protagonists. Considering the confirmed weaknesses of the TD-DFT theory to deal with excited states and the accuracy needed to solve spectroscopic problems, the ab initio methods will probably be the basic tool in the near future. Better implementations of the methods and development of efficient geometry optimizers will be required to proceed and they are, indeed, becoming available [84]. 6. EXCITED STATES AND PHOTOCHEMISTRY This section is devoted to the computation of excited states specifically involved in photochemical reactions, that is, reactions initiated by light. The borderline between spectroscopy and photochemistry is extremely dim and vague. We can jump from one area to the other without even notice it. For instance, if one is interested in the calculation of the vertical excitation energies of cytosine [138], the results produced are certainly in the area of theoretical spectroscopy. Now, let us assume one wants to give a step forward by computing the equilibrium structures of the main valence singlet excited states [139], namely '(TOT*), '(no?!*), and '(n^*), one immediately enters in the field of non-adiabatic photochemistry. The CASSCF geometry optimization of the '(riN7t*) state (where nN refers to the lone pair located on the nitrogen atom) leads directly to a conical intersection with the ground state. On the other hand, the CASSCF equilibrium structure for the '(JITT*) state is essentially coincident with a conical intersection involving the excited states '(TTTT*) and '(nojr*). There is no problem to reach the minimum for the '(noft*) state, which becomes the lowest excited state at the CASSCF level [139]. However, when dynamic electron correlation is taken into account the photochemical picture is somewhat different, becoming the '(7171*) state the most stable [140]. Incidentally, the computation of these spectroscopic properties of cytosine by employing the CIS method, with the purpose in mind of getting a rapid qualitative vision of the situation, becomes a nightmare, facing all sort of "convergence" problems (not surprisingly), leading to meaningless results where the '(no7t*) state is completely missed. Let us start from the very beginning. Considering the excited and ground state potential energy surfaces and the different reaction paths that a system might evolve through, the molecular processes can normally be identified as photophysics, adiabatic photochemistry, and non-adiabatic photochemistry [141]. Absorption and emission can be regarded as photophysical processes. From the theoretical viewpoint they involve calculations at similar molecular structures. In an adiabatic reaction path, once that the vertical absorption takes place, the system proceeds along the hypersurface of the excited state to reach a local (or absolute) minimum leading eventually to an emitting feature. For instance, the dual fluorescence observed for dimethylaminobenzonitriles [142] and 1-phenylpyrrole [143] in polar solvents can be explained in terms of a photoadiabatic reaction that takes place in the lowest excited state. In those cases, the polar environment decreases the reaction barriers and

82 favors the process. In a non-adiabatic photochemical reaction path, part of the reaction occurs on the excited state hypersurface and after a non-radiative jump at the surface crossing (or funnel) continues on the ground state hypersurface. When the two hypersurfaces have the same multiplicity (e.g. singlet/singlet) the radiationless jump is denoted as internal conversion (IC), and intersystem crossing (ISC) is reserved for cases of different multiplicity (e.g. singlet/triplet). Internal conversion may occur through an avoided crossing (AC) or a conical intersection (CI). Among several researchers, Robb, Olivucci, Bernardi, and co-workers have specifically shown during the last decade the important role that conical intersections play in organic photochemistry [141]. A large number of photochemical reactivity problems has been studied in the last years involving CIs, including photoisomerizations, photocycloadditions, photorearrangements, and photodecompositions. Depending on the nature of the CI [144], the corresponding radiationless transition can yield specific photoproducts or relax the energy towards the ground-state initial situation. Geometry determination of a conical intersection, as well as localization of minima and transition states, is usually performed at the CASSCF level. In a subsequent step, the energy differences are corrected by including dynamic correlation. If it is done at the CASPT2 level, the protocol is denoted as CASPT2//CASSCF, which stands for geometry optimization at the CASSCF level and CASPT2 calculation at the optimized CASSCF structure. Two main situations do actually occur. In cases where the PES computed at the CASSCF and CASPT2//CASSCF level behave approximately parallel (CASE A), the CASSCF optimized geometries will be in general correct, despite they have been computed at a lower level of theory. It means that dynamic correlation contributions are quite regular and similar in ample regions of the PES. The photochemistry of the protonated Schiff bases constitutes a nice CASE-A example, where the CASPT2//CASSCF computational strategy can be confidently applied. As can be seen in Fig. 3 of Ref. [145], the CASSCF minimum energy path runs parallel to that obtained at the CASPT2//CASSCF level. When dynamic correlation is markedly different for the states considered and varies significantly along the PES of interest, geometry optimization has to be carried out at the highest correlated level (CASE B). Otherwise, the uneven contributions of dynamic correlation may lead to unphysical crossings and interactions between the two electronic states. A clear representative study of CASE B corresponds to the characterization of the nature of the So/Si crossing responsible for the radiationless decay in singlet excited cytosine. The excited DNA bases have a lifetime so small that they relax to their ground state before a photochemical reaction may take place. In fact, the excited-state lifetimes of the nucleic acid molecules fall in the sub-picosecond time scale, suggesting the presence of an ultrafast internal conversion channel [146, 147]. It is an intrinsic molecular property because very short lifetimes have also been determined in the gas phase for the isolated purine and pyrimidine bases [148]. The CASPT2 results [140] suggest that the conical intersection between the ground state and the nn* state, denoted by (gs/7t7i*)ci, is responsible for the ultrafast decay of singlet excited cytosine, which is in contrast to the picture offered by the CASSCF method [139]. Moreover, the no7i* state is involved in a S2/Si crossing and it does not contribute directly to the ultrafast repopulation of the ground state [140]. As stated above, optimization of the singlet nNit* state leads directly to

83 a conical intersection with the ground state but it is not found to be the preferential path of the observed decay. Whether this is a general relaxation mechanism for all the excited nucleobases or not is the subject of current research. A situation like cytosine where the CASSCF and CASPT2 reaction paths do not run parallel (CASE B) manifests an urgent necessity of efficient algorithms for computing conical intersection with inclusion of dynamic correlation. The two main open routes available at present, through the MRCI [149] and the MS-CASPT2 methods [150], are limited to systems of small molecular size. However, in order to tackle general CASE-B problems, where the CASPT2//CASSCF (or MRCI//CASSCF) protocols are not valid, no methodology is available in practice. It is clear that most of the biomolecules of interest cannot be confidently treated today at the MRCI level because of the severe truncations that have to be performed. On the other hand, caution has to be exercised when applying the MS-CASPT2 method to locate conical intersections [151], which is next addressed by using as example the penta-2,4dieniminium cation. De Vico et al. [152] have recently reported the optimized structures for the Si /So conical intersection computed at the MS-CASPT2 and CASSCF levels, hereafter denoted as Geom. I and Geom. II, respectively. Table 6 shows the CASSCF, CASPT2, and MS-CASPT2 energy differences (AE) between Si and So that we have computed at those geometries. The 6-31G* basis set was used throughout. Incidentally, because photochemical studies are mainly related to the lowest valence states, basis sets smaller than those used in spectroscopic studies, are frequently employed, which should be alright as far as no competitive Rydberg states are placed around the studied region. When the energy difference is less than 2 kcal/mol, the minimum reached is considered technically as a conical intersection; otherwise (AE > 2 kcal/mol) we are facing an avoiding crossing. The MS-CASPT2(6MOs/6e) result, 3.89 kcal/mol, is similar to the CASPT2 finding employing Geom. I, and the off-diagonal matrix elements of the asymmetric effective Hamiltonian (Heff) are small (less than 2 kcal/mol). Everything seems to be quite consistent. Apparently an avoiding crossing has been found at the MS-CASPT2(6MOs/6e) level. Using Geom. II, the CASSCF(6MOs/6e) and CASPT2(6MOs/6e) results for AE are within 1 kcal/mol. It indicates that the optimal geometry determined for the conical intersection at both levels of theory is probably very similar. However, the states become separated by 7.57 kcal/mol when they are allowed to interact. As can be seen in Table 6, the off-diagonal elements of the Heff are very different, 6.12 and 1.38 kcal/mol. Because the states are nearly degenerate at the CASPT2 level, the result for the off-diagonal symmetric Heff just comes out from averaging: (6.12+1.38)/2. As a consequence, the CASPT2 states are pushed down and up by that amount, 3.75 kcal/mol. Such interaction is definitely unphysical! Enlarging the active space with two extra orbitals (8MOs/6e results), which allows for radial correlation of the electrons involved in the 90°twisted double bond, the Hi 2 and H21 asymmetric elements become small enough, which reflects that the corresponding zeroth-order Hamiltonians are capable of yielding a balanced description for both states. Accordingly, the CASPT2 and MS-CASPT2 splitting between the Si and So states becomes small. In summary, the computed geometry at the

84 CASSCF(6MOS/6e) level represents also a conical intersection at the MSCASPT2(8MOs/6e) level, which confirms that protonated Schiff bases behave as CASE A. Unfortunately, in larger molecular systems the active space cannot be extended to the extreme that the off-diagonal elements become less than 2 kcal/mol and the MS-CASPT2 method may be forced to yield avoiding crossings. It is certainly a circumstance to be prevented in future applications of the MS-CASPT2 method. Table 6 Energy difference between Si and So, AE, computed at the optimized structures of the penta-2,4-dieniminium cation Si/S0 conical intersection3 at the MSCASPT2 (Geom. I) and the CASSCF (Geom. II) levels. The 6-31G* basis set was used throughout. The off-diagonal elements of the MS-CASPT2 effective Hamiltonian (Heff) are also included. Method

Geom. I

Geom. II

(6MOs/6e)

(6MOs/6e)

Geom. II (8MOs/6e)

AE(S,-So) (kcal/mol) CASSCF

3.40

0.07

4.78

CASPT2

3.83

0.99

0.60

MS-CASPT2

3.89

7.57

0.64

eff

Off-diagonal elements of H

(kcal/mol)

H|2(asymmetric)

0.62

6.12

0.18

H2| (asymmetric)

0.09

1.38

0.04

Hi2=H2] (symmetric)

0.35

3.75

0.11

"Optimized geometrical parameters taken from De Vico et al. [152].

7. FINAL REMARKS It is clear that we are living in a new era where experimental and theoretical research can talk to each other on an equal footing. In this privileged situation we should be able to join efforts addressed to elucidate the big challenges our society faces at present in the realms of atmospheric chemistry, material science, photobiology, and nanotechnology. Experimental and theoretical research work shares at least one characteristic: the results produced have to be interpreted. The most cumbersome task is to compare experimental and theoretical derived data properly. In many cases recorded values do not directly yield the studied property, which has to be obtained by indirect procedures within a given scheme. On the other hand, theoretical results are usually obtained for simplified models. The resolution of the scientific problem certainly requires a constructive interplay between both viewpoints. We must be able to design a research strategy in computational chemistry (RESICC) leading to results with predictive character, independent of any experimental information. Fig. 6 shows a proposed RESICC algorithm. Basic steps include:

85 1. Define objectives. This is surely one of the most important parts of a research. The aim of the study has to be clearly defined, as precisely as possible: What is the purpose of the computation? 2. Literature reviewing: What is the scientific background on the topic? Analysis of previous information has to be critically reviewed with open mind, because it can be extremely helpful to design the computation. 3. Actual computation. According to the previous steps the actual computation takes place at a given level of theory. 4. Once the results have been carefully analyzed two key questions rise. Are the obtained conclusions stable with respect to further theoretical improvements? Do they fulfill the initial objectives? 5. A proper action has to be taken if the calculation does not guarantee the required levels of quality. Theory has to be pushed further until stable conclusions are achieved. It is worth noting that ab initio methods, because of their well-defined hierarchical structure that allows convergence of the results upon the increasing level or theory, are currently the only type of quantum-chemical tools able to fulfill the requirements implicit in the RESICC scheme. The decision is up to you!

f

START

Define Objectives Literature Reviewing/

Actual Computation Improved Level of Theory Stable \ ^ .Conclusions?,

No

Fig. 6. Research strategy in computational chemistry.

86 8. ACKNOWLEDGMENTS We thank our co-workers for their valuable contributions. MCYT of Spain, projects BQU2001-2926 and BQU2004-01739, and Generalitat Valenciana, project GV04B-228, have financed the research.

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III. Density Functional Methods for Excited States: Equilibrium Structure and Electronic Spectra Filipp Furche* and Dmitrij Rappoport Institut fiir Physikalische Chemie, Universitat Karlsruhe, Kaiserstrafie 12. 76128 Karlsruhe, Germany 1. INTRODUCTION Density functional theory (DFT) is nowadays one of the most popular methods for ground state electronic structure calculations in quantum chemistry and solid state physics. Compared to traditional ab initio and semi-empirical approaches, contemporary density functional methods show a favorable balance between accuracy and computational efficiency. A number of commercial programs is available, and DFT calculations of ground state energies, structures, and many other properties are routinely performed by nonexperts in (bio-)chemistry, physics, and materials sciences. Hohenberg-Kohn density functional theory is strictly limited to ground states [1], which excludes applications to photochemistry. This is a serious drawback, because photoexcited molecules are experimentally much more difficult to characterize than molecules in their ground states. Reliable theoretical predictions for excited states are thus especially valuable. Several routes have been followed to extend conventional DFT to excited states (see, e.g., Refs. [2-5]). In the present review, we focus on time-dependent density functional theory (TDDFT), which is presently the most popular method to treat excited states in a DFT framework. Extensive reviews on TDDFT exist [6-10]; most of them emphasize formal aspects of the theory. The aim of the present work is to survey the use of TDDFT in photochemistry. It is primarily written for non-experts with little background in DFT. The literature in this field is growing rapidly, and we cannot claim to be exhaustive; instead, we give a selective introduction to important concepts and recent developments from a rather personal perspective. Sec. 2 contains a brief introduction to the theory. We do not give any derivations and merely state the most important results and explain their meaning. An overview of popular density functionals is given in Sec. 2.3. Algorithms to compute spectra and excited state properties are reviewed in Sec. 3. We mostly describe the steps necessary in a TDDFT excited state calculation and give details only where necessary. Some timings for typical applications are presented in Sec. 3.4. Sec. 4 summarizes the performance of TDDFT excitation energies, transition moments, and excited state properties * Electronic address: [email protected].

94 in benchmark studies. This section is recommended to the reader interested in the accuracy of TDDFT in general. Situations where present functionals fail are discussed as well. Specific applications are surveyed in Sec. 5. Classes of compounds include aromatic systems and fullerenes, porphyrins and related compounds, transition metal compounds, metal and semiconductor clusters, organic polymers, and biologically relevant systems. We close with an outlook in Sec. 6.

2. THEORETICAL FOUNDATIONS 2.1. Time-dependent response theory approach to excited states Excited states are solutions of the time-independent stationary Schrodinger equation; time-dependent response theory is used as a trick to reduce electronic excitations to ground state properties. Consider a molecule in its electronic ground state subject to a periodic perturbation by a uniform electric field E oscillating at frequency ui. The distribution of the electronic charge and current density of the molecule is described by the one-particle density matrix 7(i). j(t) will perform driven oscillations about its ground state value 7 ^ . The amplitude of these oscillations is given by the Fourier transform of 7(t), denoted j(u>) for simplicity. As a result of elementary perturbation theory. 7(0;) has the following expansion in powers of the field E: 7H

=7(0) - £ (-^»- - ^f)

E + O(E?)

(1)

If the frequency ui approaches an excitation energy O()n of the system, there is a resonance catastrophe and the amplitude of the oscillation diverges. Keeping the analogy to a system of harmonic oscillators [11,12], the excitation energies Jlo,,. are the eigenfrequencies of the electrons in the molecule, and the transition density matrices -fOn are the corresponding collective modes. After inversion of the relation between 7(0;) and E, the excitation energies are obtained as eigenvalues of an electronic Hessian which may be imagined as the matrix of second derivatives of the electron energy with respect to the electronic degrees of freedom. In this way, any ground-state theory can be extended to excited states, provided the time-dependent response is well-defined. The thus obtained excitation energies and transition moments are in turn consistent with ground-state properties because sum over states (SOS) expressions as in Eq. (1) hold. Both is generally not true for state-based methods. On the other hand, the reliability of response theory based methods crucially depends on the stability of the ground state (see below). The formal basis for an extension of common ground-state density functional methods to time-dependent perturbations is TDDFT. Within the time-dependent Kohn-Sham (TDKS) framework [13], one considers a system of N non-interacting fermions whose density is constrained to the physical density p(t,x). This leads to the time-dependent Kohn-Sham equations i^:j(t,x)=H[p](t,x)cbJ(t,x).

(2)

The effective TDKS one-particle Hamiltonian H[p\(t,x) = n2(t, x)/2 + vs[p\(t, x) consists of a kinetic energy part and a local time-dependent external potential vs [p]. The latter is

95 a unique functional of p(t, x) (up to a gauge transformation) for a given initial state, as stated by Runge and Gross [13]. vs is usually decomposed according to vs[p}(t,x)=vext(t,x)+vc[p]{t,x)+vxc[p}{t,x)

(3)

into the external one-particle potential vtxt. the time-dependent Coulomb (or Hartree) potential vc[p](t,x) = J dx'p(t,x')/\r — r'|. and the time-dependent exchange-correlation potential vxt.[p](t,x). Equivalently, one may consider the TDKS one-particle density matrix 7(£), which is related to the TDKS orbitals via the spectral representation

Its time evolution is governed by the von-Neumann equation (5)

subject to the idempotency constraint f

>y(t,x,x') =

dx1-y(t,x,x1)>y(t,x1,x').

(6)

This density matrix based approach [12,14-16] is particularly convenient for response theory, because the equations determining the first and higher order response of 7 can be derived by straightforward differentiation of Eqs. (5) and (6) with respect to an external perturbation [15]. Complicated intermediates such as perturbed orbitals or response functions are avoided. Equations (5) and (6) describe a non-interacting system and are therefore computationally manageable, while the solution of the full interacting TV-electron problem is exponentially more complex. This is, somewhat simplified, the main cause of the computational advantage of density functional methods over conventional wave-function methods. The price for this improved efficiency is that the potential vxc[p](t, x) has to be approximated. The construction of accurate and inexpensive approximations to vxc [p] is a central problem of TDDFT and will be discussed in Sec. 2.3. Formally, the TDKS construction implies [6] that 7(4) yields the interacting density p and the interacting current density j according to p(t,x) = j(t,x,x) (7) This means that the frequency-dependent TDKS density matrix response must have an SOS expansion of the type (1). Therefore, the physical excitation energies are accessible from the TDKS response, and the corresponding eigenmodes yield physical transition moments.

96 2.2. Excited state properties

2.2.1. The Lagrangian of the excitation energy The time-dependent response theory approach outlined in the last section provides a route to excitation energies and transition moments. Excited state total energies are accessible by adding the corresponding ground state energy to the excitation energy. But how to compute other excited state properties such as dipole moments without an excited state wavefunction? - First it is important to remember that the wavefunction is only an intermediate that relates properties of a system such as energies or densities to a Hamiltonian, i.e., external potentials. Properties of a stationary state may be defined without reference to the wavefunction by the dependence of the energy on an applied external perturbation. For example, the dipole moment may be denned as the first derivative of the energy with respect to a constant electric field at zero field strength. More generally, the excited state density can be defined as the functional derivative of the excited state energy with respect to an external perturbing potential at zero coupling. It is therefore sufficient to know the dependence of the excited state energy on the external potential to compute static excited state properties. The energy of a stationary state is stable with respect to the wavefunction: this leads to the Hellmann-Feynman theorem for first-order properties and to the more general Wigner 2n + 1 rule. The latter states that the wavefunction through n-th order determines all properties through order 2n + 1. The Lagrangian method establishes an analogous variational principle for excited states in TDDFT. Here we present a summary only; for a detailed derivation, the reader is referred to Ref. [17]. The Lagrangian of the excitation energy is defined by L[X,Y,Q,C,Z,W] = (X,Y\A\X,Y) ~n({X,Y\A\X,Y) - 1)

F is the ground state Fock matrix, and S denotes the overlap matrix. L depends on the ground state Kohn-Sham (KS) molecular orbital (MO) coefficients C; the latter are related to the ground state KS MOs <j)pa via the LCAO (linear combination of atomic orbitals) jKr(r) = YlC^Xu(r),

(9)

where \v are atom-centered basis functions. Indices i,j,... are used for occupied, a, 6,... for virtual, and p,q.... for general MOs. We assume that the MOs are real and eigenfunctions of the z component of the total spin. X and Y parameterize the transition density matrix 70,. of the n-th excited state, ^ a f f (r)^ f f (r') + ^ ^ ( r ^ r ' ) ) ;

(10)

we shall always refer to the n-th state and omit state labels where possible. X and Y are conveniently gathered in the two component "transition vector" y)=\X,Y).

(11)

97 Q, Z. and W are Lagrange multipliers enforcing additional constraints, as discussed below. If L becomes stationary, the additional "penalty" terms introduced by fi, Z. and W vanish by construction. One is thus left with the term (X. Y\A\X, Y) representing the excitation energy. It may be considered an expectation value of the orbital rotation Hessian A evaluated for the transition vector \X,Y). A and A are 2x2 "super-operators".

where A and B have the matrix representation (.4 + B)lar,:jbrj,

= (eaa - ela)5i:j5ab5arj> + 2(iaa\jba')

+ 2f™a]ba>

- cxSa(7'[(jaa\ibcj) + (aba\ija)} (A-B)itmjba>

= (eaa - el(J)5i:j5ab5arji + cxSaaf[(jaa\iba)

(13a) - (aba\ija)}.

(13b)

(pqa\rsa') is a two-electron repulsion integral in Mulliken notation, and f^aTsa> represents a matrix element of the exchange-correlation kernel in the adiabatic approximation (AA),

where i?xc is the static exchange-correlation energy functional. The hybrid mixing parameter cx [18,19] is used to interpolate between the limits of "pure" density functionals (cx = 0) and time-dependent Hartree-Fock (TDHF) theory (cx = l , £ x c = 0). 2.2.2. Stationarity conditions for L The following stationarity conditions determine the excited state energy and first order properties. 1. The ground-state KS equations (in unitary invariant form), - ^ - = Flaa = 0,

(15)

implying that the occupied-virtual block of the ground-state Fock operator F is zero. The Lagrange multiplier W enforces orthonormality of the KS MOs, J T T ^ - = Spqa - 5pq = 0.

(16)

2. The TDKS eigenvalue problem (EVP) - ^ -

= (A-nA)|X,y>=0,

(17)

together with the non-standard normalization condition for the transition vectors (X.Y\A\X.Y)-l

= 0.,

(18)

which is enforced by f2. The form of Eqs. (17) and (18) is familiar from Hartree-Fock (HF) theory [20]. This analogy was first recognized by Zangwil and Soven [21] and later

98 generalized by Casida [14]. Other schemes, including density based methods [22] and Dyson-type procedures [23] are special cases of the density matrix based formalism. The eigenvalues Q of A are electronic excitation energies, and the corresponding transition vectors \X, Y) are collective eigenmodes of the TDKS density matrix. Vt and \X, Y) are the solutions of the TDKS EVP (17). The normalization condition (18) can be used to assign a state in terms of excitations from occupied to virtual KS MOs. The weight of a one-particle excitation from the occupied orbital i to the virtual orbital a is

The configuration mixing reflects the change in the Coulomb and exchange-correlation potentials upon excitation. More elaborate methods to analyze transition vectors use transition natural orbitals [24] or attachment and detachment densities [25]. Denoting the electronic dipole moment operator by /it, the oscillator strength for the transition n | 2 .

(20)

Similarly, the rotatory strength is ROn = Im((/x|X n , Yn) • (Xn, y n | m » ,

(21)

where m denotes the magnetic dipole moment operator, /j, can be expressed in various forms, e.g., the dipole-length or the dipole-velocity form [26] which are related by a gauge transformation. Since the TDKS formalism is gauge invariant, the different forms of /J, lead to the same result in the basis set limit [15]. As expected for a response theory based approach, the oscillator strength and the rotatory strength satisfy sum rules. For example, the isotropic polarizability of the the ground state at frequency LO has the SOS expansion

This is true independent of the basis set and functional. 3. The aZ vector" equation and the determining equations for W. They follow from the stationarity condition ^ = 0 .

(23)

The Z vector equation is a static perturbed KS equation of the form (A + B)Z = -R.

(24)

The expressions for R and W involve third order functional derivatives and are explicity given in Ref. [17]. The difference between the excited and ground state density matrices is given by P = T + Z,

(25)

99 where the "unrelaxed" part T contains products of the excitation vectors only. Z accounts for relaxation of the ground state orbitals; it can be of the same order of magnitude as T. The information contained in P is complementary to the information contained in the transition vector. The latter is related to matrix elements between the ground and excited state, while P is related to the difference of expectation values for the excited and the ground state. For example, tr(Pfi) is the change of the dipole moment upon excitation from the ground state; by adding the ground state density matrix to P, excited state properties can be computed in this way. Population analysis or graphical representation of P can give insight in the re-distribution of the electronic charge due to the excitation process. The remaining Lagrange multiplier W accounts for first-order changes in the energy due to changes in the overlap matrix. W is therefore an "energy weighted difference density matrix", and is needed for gradient calculations only. The total gradient of L with respect to a perturbation £ has the form [17]

fiver

/iva

E

fifnXaa'

+ £ ^ ( O i V + E 01xAX + Y)lwa(X + Y)KXar,

(26)

h is the sum the kinetic and potential energy one-particle operators and Vxc is the static exchange-correlation potential. F is an effective two-particle density matrix that separates into two-index quantities. £ may represent, e.g., a component of an external electric field, in which case all terms except the first are zero: or it may represent a nuclear coordinate. Parentheses indicate that derivatives need to be taken only with respect to basis functions; MO coefficient derivatives do not occur as a consequence of the 2n+ 1 rule. L^ has nearly the same form as the ground state energy gradient [27], the definitions of P, F, and W being the main difference. Total excited state properties are obtained by simply adding the ground state contributions. 2.3. Approximate exchange-correlation functionals There are different approaches to the construction of approximate functionals. Empirical functionals contain a large number of parameters fitted to a "training set" of accurate experimental or calculated data. Non-empirical functionals contain few or no fitted parameters and are designed to satisfy known constraints. Empirical functionals should be accurate for systems and properties contained in the training set, but they can fail for other systems. In contrast, non-empirical functional usually exhibit a more uniform accuracy [28], The accuracy of approximate exchange-correlation functionals is limited by their form, i.e., there is a certain maximum accuracy that can be expected for local, semi-local, etc. functionals. The "perfect agreement" with experiment reported in some density functional studies should therefore rather give rise to concern, especially if highly parameterized or exotic functionals are used. The most common and universally used approximation in TDDFT is the above-mentioned A A [29]. It replaces the time-dependent exchange-correlation potential by its static counterpart, evaluated at the time-dependent density. The resulting potential is instantaneous, in contrast to the exact one, which has a "memory" of all times t' < t. In response

100 theory, the AA makes the exchange-correlation kernel and all higher derivatives of the exchange-correlation potential independent of the frequency. The AA has been considered uncritical for a long time. Only recently it has been clarified that the lack of higher excited states in TDDFT excitation spectra is a consequence of the AA [30]. This may be related to the failure of the AA in dissociating H2, where doubly excited states are important [31]. 2.3.1. Local and semi-local functionals Semi-local functionals have the form Exc = /d A r /(p a (r),p0(r), Vp Q (r), V M r ) , . . . ) .

(27)

In the local spin density approximation (LSDA), / depends on the spin densities at r only. The LSDA is derived from the exchange-correlation energy per particle of a uniform electron gas. which has been accurately parameterized [32,33]. For functionals of the generalized gradient approximation (GGA), / also depends on the gradient of the spin densities. Popular GGA functionals with few empirical parameters are Becke's 1988 exchange functional [34] together with the correlation functional of Lee, Yang, and Parr (BLYP) [35], or Perdew's 1986 correlation functional (BP86) [36]. The GGA of Pewdew, Burke, and Ernzerhof [37] (PBE) is parameter free, while Hamprecht, Cohen, Tozer, and Handy (HCTH) have proposed an empirical GGA functional [38]. In meta-GGA functionals, / depends on additional local information such as the kinetic energy density or the Laplacian of the density. Examples are the 21 parameter meta-GGA of Van Voorhis and Scuseria (VS98) [39], or the non-empirical meta-GGA of Tao, Perdew, Staroverov, and Scuseria (TPSS) [28]. 2.3.2. Hybrid functionals Hybrid functionals interpolate between HF theory and semi-local functionals [18,19]; the fraction of HF exchange is controlled by the exchange mixing parameter cx. The exchange is treated as in HF theory, using non-local potentials. This interpolation leads to an error compensation for many properties. Popular hybrid functionals are, e.g., B3LYP [40], B3PW91 [19], or PBEO [41]. 2.3.3. Optimized effective potential (OEP) based functionals Exact exchange (EXX) as a functional of the KS density matrix has the same form as HF exchange. Differences arise in the variation of the energy. In HF theory, the energy is minimized with respect to the density matrix. The resulting exchange potential is the well-known non-local exchange operator in HF theory, while it is a local multiplicative potential in KS theory. For a fixed density, this potential can be determined by an energy optimization procedure, as first shown for atoms by Talman and Shadwick [42]. Computation of the local exchange potential in molecules is a non-trivial problem [43], but there has been recent progress in developing more efficient methods [44,45] and approximations [46-48]. Full OEP calculations of the frequency-dependent exchange kernel have been reported for solids, but not for molecules so far [49]; see Refs. [50, 51] for a review. In most TDDFT applications, KS orbitals and orbital energies from an OEP calculation are combined with adiabatic LSDA or GGA exchange-correlation kernels.

101

2.3.4- Asymptotic corrections The exchange-correlation potentials of semi-local functionals decay too fast in the asymptotic region outside a molecule. In most cases, the decay is exponential, instead of the correct — 1/r. As a result, diffuse excited states are often predicted too low in energy, and higher Rydberg excitations may be absent from the bound spectrum [52], Various correction schemes have been suggest to remedy this problem [53-55]. These corrected potentials are not the derivative of any exchange-correlation energy functional, however. This does not affect vertical excitation energies, but makes a consistent definition of excited state total energies and properties difficult. 2.3.5. Current-dependent functionals Some deficiencies of semi-local functionals can be cured by using the current density j instead of the density. Vignale and Kohn have shown that the time-dependent exchangecorrelation vector potential of weakly inhomogeneous systems possesses a gradient expansion as a functional of j but not of p [56,57]. Current dependent functionals capture macroscopic polarization effects in solids which are ultra-non-local in the density [58]. First applications to molecular excitation energies [59] show a somewhat mixed picture, however. 3. COMPUTATIONAL STRATEGIES 3.1. Basis set methods As explained in the last section, performing a T D D F T excited state calculation amounts to finding the stationary points of the Lagrangian L. Introduction of a finite basis set (usually atom-centered) generates a finite number of MOs through the LCAO expansion (9). If the basis set is suitably chosen, the excited state energy may be well approximated by optimizing L on the corresponding subspace. We thus arrive at a finite-dimensional optimization problem which can be solved by matrix algebra. The basis set incompleteness can be checked by using hierarchical basis sets of different size, compare Sec. 3.3. The steps necessary to compute excited state energy and gradients parallel the stationarity conditions for L discussed in Sec. 2.2.2. A summary is given in Table 1, including the scaling of the computational cost with the system size measured by N.

Table 1 Steps in an excited state energy and gradient calculation, formal and asymptotic scaling of computational cost. Scaling Formal Asymptotic Ground state energy and wavefunction TV4 N2 4 Excitation energy iV A"2 4 Relaxed density a n d gradient A" A2

T h e first step, solution of t h e g r o u n d - s t a t e KS equations in a finite basis set, is a s t a n d a r d procedure in q u a n t u m chemistry a n d needs no further discussion here. In t h e

102 second step, (approximate) excitation energies and transition vectors are calculated by solving the finite-dimensional TDKS EVP. Complete diagonalization of the electronic Hessian A scales as TV6 and is prohibitive for systems with more than 10 heavy atoms. In most applications, however, especially in larger systems, only the lowest excited states are of interest. By iterative methods, the lowest part of the spectrum of A can be calculated much more efficiently than by complete diagonalization. Iterative methods minimize L by expanding the excitation vector on a subspace whose dimension is small compared to the full problem. One usually starts from unit vectors, i.e., the KS one-particle excitations. In each iteration, the best approximation to the excitation energy is calculated by a small diagonalization on the current subspace (Ritz step). The error is controlled by the norm of the residual which corresponds to the gradient of L. If the error is small enough, the process terminates; otherwise, the subspace is extended in the direction of the (preconditioned) gradient and a new iteration starts. Similar ideas can be found in the early work of Lanczos [60] and Hestenes and Stiefel [61] already, but it was only the preconditioning introduced by Davidson [62, 63] that made these iterative algorithms useful for quantum chemistry. The extension to the special EVPs occurring in response theory goes back to Olsen, Jensen, and J0rgensen [64]; in the meantime, several modifications have been suggested [65-68]. If "pure" functionals are used, it is favorable to transform the TDKS EVP to a symmetric problem of half the original dimension [69]; the latter is amenable to standard algorithms for symmetric-positive EVPs. The time-determining step in all iterative methods is the computation of matrix-vectorproducts \U, V) = A\X, Y). where \X, Y) is a subspace basis vector. This is most efficiently performed as (U + V) = {A + B)(X + Y),

(28a)

(U-V)

(28b)

= (A-B)(X -Y),

because the symmetry of (A ± B) (as a super-operator) and of (X ± Y) can be fully exploited. The diagonal contribution to (A ± B) resulting from the orbital energy differences, cf. Eqs. (13), is trivial to compute. The multiplication by the the remaining four-index integrals is best performed by transforming the vectors to the AO basis, in the spirit of direct CI methods [70] in an AO formulation [71, 72]. Denoting the transformed vectors by Greek indices, we have

(X ± Y)lwa = l- J^iX ± y\u,,{ClllaCvan

± CtumCmrT).

(29)

ia

With respect to the AO indices, (X + Y) is a symmetric and (X — Y) a skew-symmetric square matrix. After that, one computes

(U + V)llva = ] KXU'

(30a) (U - V % - = ^ c x < W [ ( H ^ ) - (II\\VK)](X

- YW-

(30b)

103

Back-transformation finally yields the product vectors in the MO basis,

(U ± V)iaa -+l- J2(U ± V)IW(J(ClaiJC,J(m ± CtumCvin).

(31)

The part resulting from the two-electron integrals is fully equivalent to a ground-state Fock matrix construction for a complex density matrix [65]. This means that highly efficient direct SCF techniques available for ground states can be carried over to excited state calculations with minimal modifications. Thus, in each iteration, only O(N2) non-zero two-electron integrals (/iz/|«;A) are calculated "on the fly", i.e., they are completely or partly discarded after use and not stored. In contrast, an integral transformation would lead to an O(7V5) scaling of CPU-time and O(iV4) I/O, because (A ± B) is generally not sparse in the MO basis. The analogy to ground-state calculations also holds for the contribution arising from the exchange-correlation kernel. The four-index quantities f*'vaKxa> are never actually calculated; instead, the contributions to (U + V) are formed directly on the quadrature grid and integrated, which is virtually equivalent to setting up the matrix of the ground-state exchange-correlation potential [17, 69]. For semi-local functionals, prescreening leads to a scaling of O(N) for the exchange-correlation contribution to the matrix-vector-products (U + V). The vector transformation steps (29) and (31) have a formal O(N'i) scaling: however, if efficient linear algebra subroutines are used, the cost is negligible for systems with up to ca. 10000 basis functions. For simulating electronic excitation spectra of larger systems, block iteration methods lead to dramatic further savings of computation time [73, 74]. In these methods, a number of states is treated simultaneously. This means that the two-electron integrals need to be calculated only once for all vectors of a block. In addition, block methods often show favorable convergence compared to single-vector methods. Molecular point group symmetry can be exploited in the MO basis by Clebsch-Gordan reduction of MO products and in the AO basis by skeleton operator techniques [65, 74, 75]. This leads to an overall reduction of computational cost by approximately the order of the point group. Advantage can be taken of spin symmetry as well. For closed-shell singlet ground states, the TDKS EVP decomposes into two separate EVPs for singlet and triplet excitations. A restricted open shell scheme for high spin ground states has been proposed recently [76]. If first-order excited state properties are to be calculated, the Z vector equation (24) needs to be solved in the third step. This is best done iteratively again, using the techniques outlined above. Once the relaxed density matrices P and W have been obtained, excited state properties can be evaluated in almost the same manner as ground state properties. It is important that the thus obtained relaxed density matrices do not depend on the perturbation. The cost for computing analytical gradients of the excited state energy is therefore independent of the number of nuclear degrees of freedom. In contrast, numerical differentiation leads to a cost that increases linearly with the number of nuclei. The cost for computing excited state energies and first-order properties differs from the cost for the corresponding ground-state calculations by a constant factor only. In conclusion, excited state geometry optimizations within the TDDFT framework are hence not significantly more expensive than conventional DFT ground state optimizations [17]. The prerequisite is, however, that the 2n + 1 rule is used and full advantage is taken of

104

the similarity to efficient ground state algorithms. 3.2. Approximations and extensions 3.2.1. Efficient treatment of the Coulomb energy As explained in Sec. 3.1, computation of the two-electron integrals (JJLI>\K\) is the bottleneck in larger TDDFT response calculations. For non-hybrid functionals (cx = 0), these integrals contribute to the Coulomb part of the excitation energy only.

Ec\pan] = 2 Yl (X + ^W(H«*)(* + y W 2}

|r-r'|

The last expression is identical to the ground state Coulomb energy functional, evaluated at the spin-averaged transition density pOn(r) = \ ^ 7(w(r. r) = T ^ ( ^ + Y) ma\ ,(r)xi,(r)

(33)

\1V Q_

o

10

15

20

25

30

35

40

Fig. 1. CPU time for computing a single-point excited state energy plus gradient with and without the RI- J approximation as a function of the number of thiophene rings n. We used the BP86 functional and a TZVPP basis set. The calculations were performed on a 1.2 GHz Athlon PC.

109 The geometry optimization of the 2lA state of chlorophyll a may serve as an example for the efficiency of the RI-J approximation, as implemented in TuRBOMOLE. The BP86 functional and a SV(P) basis set were used, leading to a total of 1114 Cartesian basis functions. The overall calculation took 13 geometry cycles starting from the optimized ground state geometry and required 29:57 h of CPU time on a 2.4 GHz Pentium IV PC. Fig. 3.4 displays the scaling of computational cost for single-point excited state gradient calculations with and without the RI-J approximation. We consider a, a'-oligothiophenes with increasing chain length. Both methods show the expected N2 scaling, but with different pre-factors. For the larger members of the series, the RI-J approximation leads to a reduction in total computation times of a factor of 4-6. 4. VALIDATION 4.1. Vertical excitation and CD spectra Semi-local functionals predict low-lying valence excitation energies with errors in the range of 0.4 eV [69,91,98,120-127], There is a systematic underestimation [69] which may be due to the missing integer derivative discontinuity [128]. This underestimation is larger for singlet-triplet excitations [129,130]. Hybrid functionals yield smaller but less systematic errors, at somewhat higher cost. Contemporary TDDFT methods certainly cannot claim "chemical accuracy" (errors < 0.05 eV), but they are often accurate enough to make useful predictions. Calibration with accurate experimental or theoretical results for small systems is always recommendable. The domain of TDDFT are larger systems, where experimental inaccuracies may be comparable to the systematic errors of TDDFT, and correlated ah initio methods are (still) too expensive. With errors of 1-2 eV and more, traditional CIS and TDHF methods are considerably less accurate than TDDFT, despite similar or higher computational requirements. There are situations where semi-local functional tend to produce much larger errors, though. Lower-lying diffuse states are often too low in energy, and higher Rydberg states are spuriously unbound [52,131]. Similarly, the excitation energies of charge transfer (CT) and ionic states may be considerably underestimated [128,132.133]. In conjugated aromatic compounds [134] and polymers [135], the error in CT excitation energies increases with the chain length, and excitons may be erroneously unbound [136]. These failures may partly be traced to the self-interaction problem of semi-local functionals which has been known for a long time [137]. The classical Coulomb energy contains self-interaction which semi-local functionals do not cancel properly in strongly inhomogeneous systems. As a result, an electron "sees" the effective charge of N rather than N — I other electrons in the asymptotic tail of the density. The asymptotic correction schemes mentioned in Sec. 2.3.4 partly remedy this problem by imposing the correct — 1/r-behavior on the exchange-correlation potential. They do not improve the description of CT states, however. Correction schemes have been devised to estimate the missing derivative discontinuity in CT excitation energies from A SCF calculations [132,133]. At present, these approaches are mainly of diagnostic value because they depend on assumptions such as complete charge separation that may not be satisfied in many situations. The EXX methodology offers a more fundamental solution to the self-interaction problem. EXX potentials are self-interaction free and lead to a correct description of diffuse

110

states [138-140], and optical properties of conjugated polymers are improved [141]. Efficient methods to generate exact [44, 45] or approximate [46-48] EXX potentials for molecular systems are available. So far. they have been combined with adiabatic LSDA or GGA kernels; the EXX kernel is frequency-dependent and applications have been reported for solids only [49]. The dilemma of the EXX method is that, although it solves the Coulomb self-interaction problem, it does not improve consistently upon semi-local functionals for all systems and properties. For example, excitation energies of valence excited states are not better or even worse [138,140]. Unfortunately, the error cancellation between approximate exchange and correlation in semi-local functionals is lost when exact exchange is combined with semilocal correlation functionals. Hybrid functionals compromise between these extremes by using only a fraction of exact exchange. While this is not a general solution, it works often surprisingly well even for CT [142] and diffuse [143] states. In the long term, the development of correlation functionals compatible with exact exchange remains desirable. Oscillator strengths of well-separated states are usually predicted with errors in the 10% range [125]. They can be qualitatively wrong for strongly coupled states (as in most other methods). As the excitation energy approaches the KS ionization threshold, i.e., the negative HOMO energy, the density of states increases and a reliable assignment of individual transitions becomes impossible. This can be a major limitation in applications, especially to smaller systems and negative anions, because GGA potentials are too repulsive which results in too few bound states, as explained above. In other cases, one finds spurious intruder states which "steal" intensity from adjacent transitions of the same symmetry [144], Nevertheless, apart form the technical difficulties associated with continuum states, the overall shape of the computed spectra is often accurate [145]. This is also true if states with strong double excitation character are involved [30]. Pure double excitations are entirely missing in the TDDFT spectra [129], as a consequence of the AA. Trends observed for calculated rotatory strengths are generally similar to those observed for oscillator strengths [73,146]. Rotatory strengths of individual transitions may even have the wrong sign; but the overall CD spectra are often fairly accurate. The use of gauge origin invariant London orbitals does not seem to be necessary [147]. The simulation of CD spectra by TDDFT calculations is becoming increasingly popular as an inexpensive method to determine the absolute configuration; additional information is provided by optical rotations which can be calculated as well [148-151]. TDDFT works for inherently chiral chromophores [152] and transition metal complexes [153,154], but has problems with weakly disturbed, inherently achiral chromophores and systems with Rydberg-valence mixing [155]. 4.2. Excited state properties As analytical gradients of the excited state energy have become available only recently [17,156-158], the literature on excited state properties obtained with TDDFT is still limited. A comparison with accurate spectroscopic data for small systems shows that TDDFT excited state structures, dipole moments, and vibrational frequencies are of similar accuracy as the corresponding DFT ground state properties [17]. Case studies for other systems [159,160] and correlated ah initio results [161] corroborate this finding, which is somewhat unexpected in view of the relatively large errors in the excitation en-

Ill

ergies. Obviously, properties such as structures or dipole moments are less sensitive to deficiencies of current exchange-correlation functionals, e.g., self-interaction. The traditional CIS method, which has almost exclusively been used for geometry optimization of excited states in larger systems, is considerably less accurate at similar or even larger computational cost. Another significant advantage of TDDFT over HF-based methods for excited states is the enhanced stability of the KS reference compared to the HF reference, as discussed in Sec. 3.2.2. As a result, even excited state minima distant form the ground state minimum are mostly reasonable with TDDFT. Adiabatic excitation energies thus show basically the same error pattern as vertical excitation energies. Excited state vibrational frequencies can be used to identify the structure of excited states by comparison with, e.g., time-dependent infrared (TIR) or time-dependent resonance Raman (TRR) spectra from pump-probe experiments [162]. This is a promising combination, because TDDFT is applicable to fairly large systems and the information contained in the experimental spectra is difficult to interpret. In addition, the vibronic fine structure of UV spectra can be simulated within the Franck-Condon and HerzbergTeller approximations. Applications to aromatic hydrocarbons show a very encouraging agreement with experiments [163]. 4.3. Excited state dynamics Early work by Casida [164] and Domcke and coworkers [165] indicated that TDDFT can provide qualitatively correct excited state reaction paths. The validation is difficult and has to rely almost exclusively on accurate ab initio results. For the conical intersection in the retinal model Z-penta-2,4-dieniminium, TDDFT and CASPT2 (complete active space self-consistent field plus second order perturbation theory) single-point results are in agreement, while deviations have been reported for other systems [166]. A limitation most studies is that the calculated reaction paths do not correspond to minimum energy paths (MEPs), i.e., the internal degrees of freedom other than the reaction coordinate are not relaxed. The first full MEP calculations using TDDFT have been performed only recently [162]. For an adequate treatment of conical intersections and excited state dynamics, non-adiabatic coupling needs to be taken into account [167,168]. It seems unlikely that present functionals are accurate enough for predicting, e.g., barrier heights, but definite conclusions will have to await further studies. 5. APPLICATIONS 5.1. Aromatic compounds and fullerenes Aromatic compounds are among the most frequently investigated molecules in TDDFT studies. Several papers on singlet and triplet excitation energies of condensed polycyclic aromatic hydrocarbons (PAHs) [92,169-172] have appeared. In a recent study Grimme and Parac [134] have pointed out that the energy of the ionic La states [173] is significantly underestimated by common functionals. PAHs and their cations have also attracted interest due to their proposed occurrence the dark interstellar matter [127,174-177]. Chiroptical properties of a series of helicenes have been investigated in a joint experimental and theoretical study [73]. The simulated CD spectra are accurate enough to assign the absolute configuration and can even be used to distinguish derivatives with

112

substituents coupling to the aromatic TT system. CD spectra calculated with the DFT/SCI method have been used by Grimme and co-workers for structure elucidation of paracyclophanes [178]. Recently, the absolute configuration of enantiopure 9,9'-biathryls could be assigned by means of CD calculations [179]. For small aromatic heterocycles accurate excited state calculations with correlated ab initio methods are available. TDDFT studies focus on solvation effects [180,181], excited state dynamics [182-185], and larger systems [186-191]. Moreover, TDDFT calculations complement experimental investigations of newly synthesized ring systems like tetrathiafulvalene [192,193] and trithiapentalene [194]. Other recent TDDFT studies deal with indole derivatives related to tryptophane metabolism and melanin formation [195,196]. Laaksonen an co-workers [197] have investigated photochemical properties of urocanic acid, a human skin chromophore which plays a role in photo-immunosuppression and skin cancer. Mechanisms of photoisomerization of prototypical molecular switches azobenzene [198,199] and stilbene [200,201] have been the subject of other studies. Finally, the biological activity of the naturally occurring heterocycles luciferin [202] and flavins [203] has been investigated with TDDFT. Luciferin is responsible for the bioluminescence of fireflies, while flavins play a role in hydrogen transfer in cells. So far, the only practicable route to prepare pure fullerenes is based on soot extraction. Because of the extremely small yields, electronic absorption spectroscopy is, besides NMR measurements, the most important method for the characterization of fullerenes. Apart from a uniform red-shift, TDDFT using GGA functionals predicts the absorption spectra of large gap fullerenes with surprising accuracy [204], Small gap fullerenes are highly reactive and can presently only be studied theoretically. For example, of the seven isomers of Cgo obeying the isolated pentagon rule, only three have a large gap, and two of those have been observed [205]. Other studies focus on functionalized and substituted fullerenes [206, 207], carbon nanotubes [208] and sheets [209, 210]. Lower symmetric larger fullerenes frequently exhibit inherent chirality. In contrast to semi-empirical methods, TDDFT is well suited to determine the absolute configuration of chiral fullerenes, as has been shown for -D2-C84 [152] as well as C76 and C78 isomers [74]. TDDFT calculations on C ^ have been used to assign the photoelectron spectrum of stable Cs4 dianions [211]. 5.2. Porphyrins and related compounds Porphyrins, phthalocyanines, porphyrazines, and similar heterocyclic systems show a variety of optical and photochemical properties that are of interest from a biochemical as well as a technological point of view. The first rationale of the characteristic features observed in the absorption spectra of porphyrins was given by Gouterman [212, 213] in 1961. It is based on a simple perimeter model for [18]-annulene, the basic building unit of porphyrins. In Gouterman's scheme two energetically close pairs of orbitals, the two highest occupied molecular orbitals (HOMO and HOMO-1) and the two lowest virtual MOs (LUMO and LUMO+1), are involved in the lowest singlet transitions and are responsible for the so-called Q- and B-bands of porphyrins. For the free base porphin, the weaker pair of Q-bands (Q;,, and Qy) is found in the visible region whereas the substantially more intensive B-band (Soret band) is located in the near UV, see Fig. 5.2. The Q.,; and Qy bands were ascribed to the HOMO —> LUMO transition and the antisymmetric combination of HOMO —> LUMO+1 and HOMO-1

113 B

1 -

intens;ity (arb. unr

CO

A 1

0.8 0.6 -

I

0.4 0.2 -

M

L

J

ll

PL

0 200 250 300 350 400 450 500 550 600 650 700 wavelength (nm)

Fig. 2. The absorption spectrum of free base porphin. The experimental spectrum is from Ref. [214]. Calculated BP86/aug-SVP oscillator strengths [215] are indicated by sticks.

—> LUMO transitions, respectively. The symmetric combination of the latter two was considered as the origin of the Soret band. Porphyrin derivatives and analogues exhibit characteristic energy shifts and intensity patterns in the same energy range. The first TDDFT results on free base porphin were reported by Bauernschmitt and Ahlrichs [69] and later confirmed by Scuseria and co-workers [66]. Subsequent studies by van Gisbergen, Baerends and co-workers and by Sundholm addressed the validation of the four-orbital model of Gouterman for the free base porphin and the assignment of its UV/VIS spectrum [215-218]. Investigations by Parusel and co-workers employed the DFT/SCI [219] and DFT/MRCI methods [220] for the same purpose. While a correspondence to the Gouterman model can be established for the Q bands, the origin of the intense B band is still under discussion. It appears that lower occupied orbitals are significantly involved in these transitions [215, 217], and a non-negligible contribution from double excitations is suggested from DFT/MRCI results [220]; thus, the simple four-orbital model does not hold. A similar picture emerges for porphyrazine [217, 221], corrphycene [222] and corrin [223] molecules where Gouterman's model provides a rough description of low-lying electronic transitions. Positions of electronic excitations in porphyrins are further strongly affected by conformational flexibility of the macrocycle, deviations from planarity leading to red shifts of Q- and B-bands. The suggestion that nonplanarity of hemes in hemoproteins and photosynthetic proteins may influence their biological activity [224] stimulated much research on saddled and ruffled forms on porphyrins. Porphyrin diacids [225,226] and complexes bearing aromatic substituents

114 [227-230] have been investigated as well. Porphyrinoid systems have a, tendency to form chelate complexes with various metal cations. Two large groups of complexes can be distinguished by their spectral behavior, denoted regular and irregular porphyrins by Gouterman [213]. Main group and closedshell transition metal cations form regular complexes that largely resemble the parent macrocycles because the contribution of the metal to the frontier orbitals is small. This was shown by Nguyen, Baerends, and co-workers for Zn11 [231-235] and by Sundholm for Mg n complexes [236]. In irregular metal complexes incomplete d-shells of transition metal cations interact strongly with the 7r-system of the ligand; substantially different optical properties [237-243] result. The most important representatives of this class are iron and cobalt complexes which are closely related to heme [244,245] and vitamin B12 [246,247]. 5.3. Transition metal compounds For calculations of optical properties of transition metal complexes, TDDFT is often the method of choice. In most cases the accuracy of TDDFT is sufficient for an assignment of excitations in closed-shell oxide, carbonyl and cyclopentadienyl complexes [121,248250]: hybrid functionals do not always lead to an improvement for these systems. Ligand field d to d transitions appear at too high energies as a result of self-interaction error, as Autschbach and co-workers have shown for Co111 and Rh m complexes [154]. Difficulties are encountered for small open-shell molecules such as ScO or VO [251,252]. The diversity of photophysical and photochemical properties of transition metal complexes is reflected in TDDFT investigations on this class of compounds. Possible applications in photocatalysis and solar energy conversion have triggered research on complexes of copper [253,254], chromium [255], ruthenium [256,257], paladium [258], platinum [259] and rhenium [260] with aromatic heterocyclic ligands as 9,10-phenantroline (phen), a, a'-bipyridyl (bipy) or dipyrido[3,2-a:2',3'-c]phenazine (dppz). Dissociation and rearrangement dynamics upon photoexcitation has been discussed in connection with [Fe(CN)5(NO)]2~ [261, 262], [Cr(CO)5L] and [Fe(CO)4L] [263], as well as on [Ru(PH3)3(CO)(H2)] complexes [264]. The catalytical activity of titanium complexes for polymerization and oxidation reactions has motivated several studies on titanocenes [265, 266] and alkoxy complexes [267]. Optical properties and bond dissociation of alkylplatinum complexes are the subject of a recent study by van Slageren and co-workers [268]. TDDFT calculations for neutral dithiolene complexes of nickel, palladium, and platinum have explained the uncommon properties of these compounds, especially the presence of an exceedingly strong absorption in the near IR region [269]. Other studies investigate the photophysics and the luminescence behavior of cyclometalated complexes of rhodium [270] and indium [271]. 5.4. Metal and semiconductor clusters Metal clusters differ substantially in their properties from the bulk phase [272, 273] and have received much attention in connection with possible applications in nanotechnology and heterogenous catalysis. Experimental structure determination is a difficult task even for small clusters, and theoretical results are particularly helpful. Flexible structures, a large number of competing minima, and low-lying excited states are difficult challenges for all electronic structure methods. Most theoretical work therefore address the most simple class of metal cluster compounds, alkali metal clusters, for which reliable experimental data as well as accurate quantum chemical calculations [273, 274] exist. TDDFT

115 applications on alkali metal clusters range from simple jellium [275-279] to full TDDFT calculations employing GGA functionals [280-282], Comparison with available experimental data indicates a good accuracy of TDDFT results with typical errors of 0.1-0.2 eV or less in excitation energies [283-285]. For the dimers Li2, Na2 and K2, experimental vertical excitation energies are overestimated by TDDFT [281], in contrast to the usual behavior of the method. Photoabsorption spectra are reproduced satisfactorily as well [285-287]: finite temperature effects have been investigated by molecular dynamics simulations [288,289]. Similar studies have been performed for Al clusters [290,291], Coinage metal (Cu, Ag, Au) clusters are more complicated due to the presence of rather polarizable d-electrons. Very little direct structural information is available from experiment. Of particular interest is the transition from the planar structures that are the most stable isomers for small clusters to bulk-like three-dimensional aggregates [292]. While the simple jellium model does not perform very well in this case, the polarizable cluster core approximation [293-295] or full TDDFT calculations [296-298] provide better results for photoabsorption spectra. Nevertheless, transitions with s —> d character are notoriously in error, which is a consequence of self-interaction [137], In summary, TDDFT absorption spectra can give useful hints, but are presently not accurate enough for a unique determination of the geometric structure of most metal clusters. The band gap of semiconductor clusters can be altered over a wide range by varying the particle size; this makes them suitable materials for optoelectronic devices [299]. Recent TDDFT investigations have addressed optical properties of silicon [283, 287, 300305], gallium arsenide [287], as well as zinc sulfide, cadmium selenide, and related 12-16 clusters [306-311]. Most studies focus on the size dependence of the optical gap. With increasing cluster size the band gap is reduced as a result of quantum confinement, e.g. for hydrogenated Si clusters from 3.8 eV for Si47H107 to 2.5 eV for Si147H247 [300], Another important factor is the constitution of the cluster surface, with abstraction of hydrogen or oxidation leading to a substantial decrease of the absorption edge [299]. The definition of the optical gap is not straightforward, however, since the lowest electronic transitions are very weakly allowed in large clusters. Within these limitations both LSD A and gradient corrected functionals yield results in good agreement with experimental data. 5.5. Organic polymers Two different theoretical approaches have been used for polymers: solid state methods employing periodical boundary conditions, and oligomer methods considering discrete fragments of increasing size. For calculations of excitation energies of organic polymers, the latter seems to be more widespread, although a LCAO-crystalline orbital implementation of excitation energies of extended systems has been reported [312, 313]. For oligomer methods, the convergence of the calculated properties to the bulk limit and the quality of extrapolated properties are of primary interest. Several papers by Ratner, Zojer, and coworkers summarize computational results on different classes of polymers [314, 315], e.g., polyenes, polythiophenes, and polyphenylenes. From these results, the authors concluded [316] that extrapolation techniques are capable of providing correct band gaps for the polymers. However, empirical extrapolations with respect to 1/n, where n is the number of monomer units, may show significant systematic errors. Cai and co-workers [135] note a tendency to spurious metallic behavior and wrong ground state multiplicities in large

116 conjugated yr-systems. For polyenes the relative stability of l 1 ^ and 21A!j states (in C-ih symmetry), which is of importance for carotenoids of the light harvesting complex, has been extensively discussed [166,317-319]. Polythiophene [320-326] and polypyrrole [327, 328] polymers are important industrial materials for optoelectronic devices such as light emitting diodes (LEDs) have been the subject of numerous TDDFT studies. 5.6. Charge and proton transfer The geometric and electronic structure of a molecule can significantly change upon photoexcitation. Transfer of charge or protons are among the most simple photochemical reactions, and excitation energy transfer plays a fundamental role for the photosynthesis. In work of Parusel, Grimme. and others, intramolecular charge transfer (ICT) in donoracceptor substituted aromatic systems was investigated by TDDFT [329], DFT/SCI [330, 331], and DFT/MRCI [332-334] methods (see Ref. [329] for an overview). Most of the studies addressed 4-(N,N)-dimethylaminobenzonitrile (DMABN), a prototypical dual fluorescent compound showing a strong emission from the ICT state in polar solutions. In extensive studies by Jamorski and co-workers [142,335-337], the accuracy of TDDFT for exploration of intramolecular charge transfer phenomena has been assessed, and a classification for the emission properties of these compounds was presented [338,339]. A definite assignment of the structure of the two lowest singlet states has recently been given by means of TDDFT calculations [162] and confirmed by coupled cluster calculations [340]. Further investigations have dealt with solvent effects and photophysical properties of donor-substituted pyridine derivatives [341,342], Excited state proton transfer phenomena have been the subject of a number of TDDFT studies. So far, excited state proton transfer in salicylic acid and related aromatic compounds [165,343-346] as well as in 7-azonindole-water complexes [347] has been investigated. 5.7. Biologically relevant systems Most molecules of biological relevance are a challenge due to their size. Calculations of optical properties of chlorophylls and bacteriochlorophylls by Sundholm [144, 236, 348, 349] and Yamaguchi [350, 351] showed that good accuracy can be achieved with the BP86 and B3LYP functionals. Different aspects of the interaction between chlorophyll molecules and carotenoids and of the dynamics in the photosynthetic apparatus have been extensively studied by Dreuw, Fleming and co-workers [317, 352-354], Pullerits and co-workers have investigated the dependence of excitation energies of bacteriochlorphyll on the local environment represented by a uniform electric field [355]. The dissociation dynamics of CO-hemoglobin complexes has recently been studied by Head-Gordon and co-workers [244, 245] who showed that excitation into the & A" and the 3lA' states of the complex leads to repulsive interaction and dissociation of the CO molecule. Photochemistry of nucleic acid bases is relevant for an understanding of DNA damage by UV irradiation and cellular repair mechanisms. Absorption spectra, tautomeric equilibria, and excited state geometries of adenine [356] and cytosine [357] have been reported. A comprehensive study on absorption properties of DNA bases has appeared recently [358]. The thermochemistry of thymine dimer formation and photoinduced cycloreversion reactions occuring in DNA repair mechanisms have been investigated by Durbeej and Eriksson [359,360], TDDFT calculations on complexes of thymine with psoralene have

117 been performed to clarify the effect of psoralenes which are utilized in photochemotherapy [361]. TDDFT calculations allow to go beyond model compounds and investigate larger fragments of biological systems like the photoactive centers of green fluorescent protein [362] or photoactive yellow protein [363]. Future improvements of TDDFT such as a better description of solvation effects or QM/MM coupling may help to provide deeper insight into photochemical processes in living organisms. 6. OUTLOOK Many phenomena in photochemistry are still not well understood, even in small model systems. The enormous complexity of photochemical processes will require a combined effort of theory and experiment to extend the frontier of our knowledge to real systems of technical and biological interest. It is clear by now that TDDFT has the potential to play an important role in this development, besides more accurate methods and experimental techniques. Nevertheless, contemporary TDDFT is not a black box method, and every user of commercial TDDFT codes should be aware of its limitations. ACKNOWLEDGMENTS We would like to thank R. Ahlrichs for helpful comments. This work was supported by the Center for Functional Nanostructures (CFN) of the Deutsche Forschungsgemeinschaft (DFG) within project C2.1. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]

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M. Olivucci (Editor) Computational Photochemistry Theoretical and Computational Chemistry, Vol. 16 © 2005 Elsevier B.V. All rights reserved

129

IV. Electronic and Vibronic Spectra of Molecular Systems: Models and Simulations based on Quantum Chemically Computed Molecular Parameters. F. Negri and G. Orlandi Dipartimento di Chimica 'G. Ciamician', Via F. Selmi, 2, 40126 Bologna, and INSTM, UdR Bologna, Italy 1. INTRODUCTION As the title of this chapter suggests, out of the huge field of molecular spectroscopy, we restrict here to a relatively narrow group of spectroscopic techniques, namely absorption, emission (fluorescence, phosphorescence) and resonance Raman scattering, and we will be concerned, specifically, with the vibronic structure associated to these spectra and the information that can be extracted from it. The interaction of radiation with molecules that occurs in all the spectroscopic measurements might be thought to pertain more closely to the field of photophysics rather than to the photochemistry of molecules. Thus, the subject of this chapter, namely the simulation of electronic spectra such as the absorption, emission or resonance Raman and, more specifically, of their associated vibronic structures, might appear only marginally related with the photochemistry of excited states. However, as we will try to make clear, electronic spectra may contain a wealth of information on the early dynamics of molecules in their excited states, hidden in the vibronic structure associated with the electronic transitions. In this sense, the analysis and interpretation of the vibronic structure associated with an electronic spectrum can reveal important details on the initial photodynamical driving forces in the selected excited state. This happens because, as it will be seen, very often vibronic intensities are proportional to the changes that occur on potential energy surfaces (PES) upon electronic excitation. Specifically, changes occur along selected nuclear motions and the corresponding vibrational normal modes are activated. These nuclear motions, often skeleton stretching modes, but also angular or torsional modes, are associated with the initial step in excited state dynamics. In this sense, the identification of active vibrations in electronic spectra may provide the bridge between spectroscopy and photochemistry, since the vibronic activity of one specific mode in an electronic spectrum may indicate that a similar nuclear motion characterizes the initial evolution of the excited molecule. Vibronic structures can be associated with electronic spectra (absorption, fluorescence, phosphorescence), but often the vibronic information is hidden under the broad band shape of these spectra. In these cases the most interesting information is lost under the diffuse bands. An alternative spectroscopic technique that has been shown to be extremely versatile and powerful to study the structure of molecules and their primary evolution in excited states is resonance Raman [1, 2]. In this case the wavelength employed to excite the Raman scattering falls within an electronic absorption band and it causes selective enhancement of the vibrations of the absorbing species. The selective enhancement is particularly useful since it allows to obtain the vibrational spectrum of a chromophore

130 embedded, for instance in a protein, without interference from the vibrations of the amino acid environment [1]. A second reason for discussing the vibronic activity associated with electronic transitions is that while most standard ab initio or semi-empirical codes calculate routinely excited states and electronic transition dipole moments or oscillator strengths, thus, they provide almost routinely parameters connected with pure electronic absorption or emission properties of molecules, the simulation of the associated vibronic structures is not a problem solvable with one standard equation or approach. The reason is that the vibronic structure can be originated by different mechanisms[3], and hence it is difficult to generalise the problem and provide a solution with a single universal approach. For this reason, one scope of this chapter is to provide an overview on the models generally employed for the simulation of the vibronic spectra of molecules and on the level of theory necessary to obtain quantum-chemically reliable molecular parameters required for the simulations. Notice that these two aspects, models and simulations, represent two separate steps that lead to the correct simulation of spectroscopic properties. The choice of the model is important since different degrees of approximation must be selected in accord with the dominant mechanism that originates the spectral intensities. For instance electronic transitions can be dipole allowed or dipole forbidden or spin forbidden. In Section 2 we will go through the different types of electronic transitions and we will see that they cannot be dealt with the same model. Conversely, various degrees of approximation can be retained or relaxed in view of the dominant mechanism that leads to the observed intensities. Once the model is chosen appropriately, molecular parameters have to be predicted with quantum-chemical methods, and introduced in the model to generate simulated intensities. The choice of the more appropriate level of theory for this purpose must take into account the fact that ground and excited states must be treated at comparable levels of theory and that accurate vibrational frequencies are required. The correct model implemented with unreliable molecular parameters can lead to discrepancies between observed and computed intensities. Similarly, discrepancies can occur if the model is too approximate, even if molecular parameters are computed with state of the art level of theory. In addition, it should be kept in mind that the definition of the 'more appropriate level of theory' in this case may be different from that required to study other molecular properties such as, for instance, photo-excited reaction paths. In the following we will try to make more clear all the concepts briefly outlined above, by presenting, first, a theoretical introduction on the origin of vibronic structures in the electronic spectra of molecules. We will introduce approaches of increasing complexity starting with the simplest situation of an allowed transition and moving to more complex situations involving symmetry forbidden transitions and spin forbidden transitions. In the second part of this contribution we will review some applications covering several classes of molecules ranging from polyenes to fullerenes and strictly related to the models presented in the first part of this chapter. 2. MODELS AND COMPUTATIONAL DETAILS In this section we review the fundamental concepts concerning the interaction of radiation with molecules which leads to absorption, emission spectra or resonance Raman scattering. The starting point is the concept of molecular vibronic state, namely a molecular level corresponding to a given electronic and vibrational excitation, indicated, in the following as for the initial state and f) for the final state.

131

Excited state Final vibronic state | j - \

Ground state nuclear coordinate Fig.l. Schematic representation of a vibronic transition. The Born Oppenheimer approximation enables the factorization of the wavefunctions representing the vibronic states, as simple products of electronic and vibrational wavefunctions: i) = | g, m) = g)| m)

and similarly

| f) = | e, n) = | e)| n)

where |g) and |e) indicate pure electronic wavefunctions and m) and n) indicate pure vibrational wavefunctions. If a photon of energy E = Ef — E( is absorbed, the molecule will be subject to a vibronic transition and it will be excited to the final vibronic state f) (see Fig. 1). The probability of such a transition is related to the matrix element of the electrical transition dipole moment between the two states, Mj f : electrons

Miif=e(i| £ rj |f)

(1)

i

where r; represents the coordinate of the \th electron e. The vectorial quantity M if can be expressed through its three Cartesian components (M*f ,M*f ,M*f ) and it can be interpreted as a measure of the charge migration during the transition that brings the molecule to the final state f) from the initial state i). Beside the transition dipole moment, a strictly related quantity is the oscillator strength / often employed to quantify the intensity of a transition./is connected to the transition dipole moment by the following relation

132

'

8n2mC

v, f M?, f

(2)

2.1. Absorption and emission: the Franck-Condon principle for allowed electronic transitions With the basic definitions given above we can now consider in more detail the problem of calculating the vibronic transition dipole moments corresponding to the vibronic transitions in an absorption or emission spectrum. According to Eq. (2) the intensity in absorption or emission is related to the square of the transition dipole moment. In Section 2.3 we will see that intensities in Raman scattering are also related to these fundamental integrals, the transition dipole moments. The vibronic transition dipole moment M _>c n is defined as

Mg.,m

(3)

= {m \(g | e 5 > , | e)\ n) = (m \Mg,e (Q) n)

where \x, c (Q) represents the electronic transition dipole moment

MB,c (Q) = (gI©" & Ie) = (g IA| e)

(4)

and it depends on the nuclear variable Q owing to the fact that electronic wavefunctions depend, parametrically, from nuclear coordinates |g) = |g(r,Q)>;

|e) = |e(r,Q)>

Eq. (3) shows that, owing to the Q-dependence of the electronic wavefunctions, the correct procedure to calculate the full transition dipole moment would be to determine the Qdependence of the electronic transition dipole moment Hge(Q) and then to integrate over nuclear coordinates. The problem can be simplified by adopting the Condon approximation, namely by assuming that the electronic transition dipole moment is constant and equal to the value computed at a reference geometry, usually the equilibrium geometry Qo of the ground electronic state of the molecule under investigation:

Under the above condition Eq. (3) simplifies to: ,c(Q^n) = ^ c ( m | n ) = ^ c S m , n

(6)

where Sm,n represents the overlap integral between vibrational wavefunctions of the initial and final states. Thus, the intensity of a vibronic transition is given by

133

g.m—>e,n

L^g,e J

(7)

m,n

^ n is known as the Franck-Condon (FC) factor of the transition.

Excited state

yy=3 Vertical electronic transition

v'=0

Ground state v"=l v"=0

n

O

Vg

- e)| n) is related to the product of the FC factors of each normal vibration belonging to the molecule. From a computational point of view, the exact evaluation of the full FC factor can be very difficult and time consuming, however several simplifications, briefly described in the following, can be introduced. First of all, the use of symmetry can reduce the dimensionality of the problem. Vibrational normal coordinates can be classified according to the irreducible representations of the symmetry point group to which the molecule belongs. For instance the vibrational normal coordinates of benzene, which belongs to the D^ symmetry point group, can be classified into the following irreducible representations: Aig, A2g, B2g, Eig, ..., E2U, etc... To simplify the calculation in Eq. (10) an easier classification is, however, sufficient, namely the separation between totally symmetric (TS) vibrations and non-totally symmetric (NTS) vibrations. TS vibrations can be easily distinguished from NTS vibrations owing to the fact that the first do not break the symmetry of the molecules while the second do break it. To further clarify this point, in Fig. 3 we present as an example, a typical TS and NTS vibration of 1,3,5-frans-hexatriene.

135

Excited state

NTS coordinate

Fig.4. (left): profile of the ground and excited state PES along a typical TS normal coordinate; (right) profile of the ground and excited state PES along a typical NTS normal coordinate. An additional way to distinguish between TS and NTS vibrations is to consider the potential energy surface (PES) profile of the ground and excited state along the selected normal coordinate. If the normal coordinate is TS, the PES profile will show, generally, the minimum of the ground state Qg shifted with respect to the minimum of the excited state Qe (see Fig. 4, left). In other words, the two PESs are displaced along the TS coordinate. Conversely, the two PESs are not displaced along a NTS coordinate. Indeed a displacement of the two minima along a NTS coordinate can occur only if the excited state belongs to a symmetry point group different (lower) from that of the ground state. If the symmetry in the two electronic states is the same, a displacement cannot occur (see Fig. 4, right). According to the above classification, the multidimensional FC factor in Eq. (10) can be factorized in two terms:

A further simplification arises if the harmonic approximation is adopted. The harmonic approximation is generally acceptable for medium-high frequency modes, while it can be unacceptable for very low frequency modes whose PES profiles are highly anharmonic even for small displacements away from the equilibrium position. Thus, if the vibronic activity is dominated by medium-high frequency vibrations, the harmonic approximation can be adopted with confidence, otherwise the simulated results must be taken with care or more appropriate procedures to evaluate overlap integrals must be considered [4]. Under the harmonic approximation, and assuming equal vibrational frequencies (cof = co^) in both the ground and excited states, simple analytical expressions can be derived for the overlap integrals involving wavefunctions of TS or NTS vibrations[2, 5]. Notice that in typical conditions molecules are in their ground electronic state and ground vibrational state, thus, the FC factors to be calculated correspond to m) = 10}:

136 6-

-

NTS .

-

TS -

-

kJ =nkJnkJ

Each term in the product above, for instance the sequence [s^0J ; [s^, J ;[s,^ 2 J ...;[s^n represents a progression in the given normal mode k, and is described by a simple Poisson distribution:

where

and Bk is the dimensionless displacement parameter defined as [6]

rkE

(15)

here Q^>g represents the projection of the geometry change occurring upon g) —> e) electronic excitation, expressed in Cartesian coordinates, on the kth vibrational coordinate Qk, namely:

k

(16)

In Eq.(16) xgje is the 3N-dimensional vector of the Cartesian coordinates corresponding to the equilibrium geometry of the e and g electronic states. M is the diagonal 3NX3N matrix of the atomic masses and Lk is the 3N-dimensional vector of the normal coordinate Qk in terms of mass weighted Cartesian coordinates. It is worth noting at this point, that the extension of the vibronic progression of a given vibrational mode is governed by the magnitude of the displacement parameter, as it can be appreciated by inspecting Fig. 5, where three progressions corresponding to different magnitudes of the Bk parameter are compared. From the above definition of the dimensionless displacement parameter Bk and considering the characteristic PES profile behaviour typical for TS and NTS modes, it is easy to conclude that, being yk =0 for the latter, Eq. (13) reduces to =0 and

n>l

(17)

137

1000

2000 3000 Energy / cm-1

4000

5000

Fig. 5. Extension of the vibronic progression of a single mode of frequency co=1000 cm"1, as a function of the magnitude of the displacement parameter Bk. From bottom to top, Bk=1.0, 1.5, 2.0.

(18) It should now be clear why it is convenient to separate TS from NTS modes, since the FC factor of Eq.(12) simplifies to (19)

namely, Eq.(19) shows that the vibronic structure in the electronic spectra of polyatomic molecules, under the validity of the FC approximation, is governed predominantly by TS modes, through their displacement parameters Bi and hence their j \ parameters.

138

1000

2000

3000

4000

5000

Energy / cm-1

Fig.6. Vibronic structure due to the superposition of two active modes with the following parameters: co,=1000 cm'1, B,= 1.0 and co2=100 cm"1, B2=2.0.

Emission

Absorption

r" a "S

Energy Fig. 7. Schematic representation of the mirror symmetry between absorption and emission spectra.

139 Although in general only TS are active according to Eq. (19), it should be kept in mind that the number of TS modes can be quite large for polyatomic molecules. This can be reflected in quite complicated vibronic structures, due to the superposition of the progressions of modes characterized by different frequencies and different activities (displacement parameters). In Fig. 6 we show a simple example, namely the distribution of vibronic activities for a twomode system with considerably different vibrational frequencies and displacement parameters. If vibrational frequencies are substantially different in the ground and excited states, the constraint of equal vibrational frequencies must be relaxed and the conclusions above must be slightly modified since features corresponding to NTS modes can now appear in the spectra. Indeed, owing to the fact that {DgTS * roeNTS, FC factors such as [s^ s ] 2 and [soNf]2 can be significantly different from zero and less approximated relations for the FC factors must be employed. For instance, the more exact expression of [S^ s J reads:

(20) (D g +(D C

Notice that, if the approximation of equal frequencies in the ground and excited states is adopted, the equations so far discussed also imply a vibronic structure in emission specular to the vibronic structure observed in absorption (see Fig. 7). The mirror symmetry between absorption and emission spectra is indeed observed when frequency variations upon excitations are minor and when rotation of vibrational coordinates (Duschinsky effect [7]) can be neglected. Indeed, the change in the orbital nature of the electronic state upon excitation is usually reflected in a remarkable change of the PES. This change is usually associated with some degree of normal coordinate rotation. Normal coordinates belonging to the same irreducible representation can mix according to the following relation

where the Rkj coefficients are the elements of the Duschinsky rotation matrix. The main consequence of Duschinsky rotation is the redistribution of vibronic intensities between the absorption and emission spectra, namely the loss of mirror symmetry. Nevertheless, the Duschinsky effect can be reasonably taken into account in a simple way, namely without the direct inclusion of Duschinsky rotation matrices, by calculating the Bk displacement parameters (Eq. 15) using vibrational normal coordinates and frequencies of the ground state in the case of emission spectra and vibrational normal coordinates and frequencies of the excited state for the simulation of absorption spectra [8]. 2.2. Beyond the Condon approximation: forbidden electronic transitions In the previous section we discussed the vibronic structure that can be predicted according to the Condon approximation and we concluded that it will be governed by progressions of TS modes and, only marginally, by features due to NTS modes, if remarkable frequency changes occur upon electronic excitation.

140 The Condon approximation relies on Eq. (5) namely on the assumption that the electronic transition dipole moment can be considered independent of the nuclear coordinates. This is not so in general, and thus the usual procedure is to expand the electronic transition dipole moment as a Taylor series in the nuclear coordinates about the equilibrium nuclear configuration Qo [3]. (22)

The above expansion of the electronic transition dipole moment parallels the Herzberg-Teller (HT) expansion of adiabatic electronic wavefunctions in terms of crude adiabatic (CA) wavefunctions, i.e. electronic wavefunctions defined at a specific nuclear configuration Qo, so that the Q dependence is removed [3]. The HT expansion reads |r(q,Q)} = |r(q,Q0)) + Xa s , r (Q)|s(q,Q 0 ))

(23)

where r(q,Q0)) and |s(q,Q0)) are CA wavefunctions. The Q dependence in the above expansion is also known as the breakdown of the Condon approximation. The CA electronic wavefunctions satisfy the so called static Schrodinger equation, namely H cl (q,Q o Mq.Qo))= Er(Qo)|r(q,Qo))

(24)

The static electronic Hamiltonian Hei(q,Qo) in Eq. (24) is related to the complete, dynamic electronic Hamiltonian Hd(q,Q) through Eq. (25) H c ] (q,Q)=T(q)+U(q,Q)=T(q)+u(q,Q 0 )+AU(q,Q)=H c l (q,Q 0 )+AU(q,Q)

(25)

where T is the kinetic energy term and U is the potential energy term. Thus, the coefficients in Eq. (23), which depend on the nuclear coordinates, are given by the perturbation relation

a

-(Q)=

(S(q,Q0)|AU(q,QHq,Q,,)) Er(Q.)-E.(Q.)

+

-

The potential AU(q,Q) in Eq.(26) can be expanded as a Taylor series about Qo to get

(27)

Substitution of Eq.(27) into Eq.(26) leads to the following expression for the coupling constants as,r:

141

d

*Q)l

-(,,QJ (28)

E,(Q.)-E,(Q.)

or in a more compact form

' AE

(29)

where K k r is the adiabatic vibronic interaction between the s) and |r) electronic states, perturbation mediated through the Mi vibrational coordinate. Being a perturbation expansion, the HT approach can only be used if the interacting states in Eq. (28) are well separated in energy, namely for weak coupling regimes. If the energy separation is small, other approaches must be adopted, for instance perturbation approaches taking into account the vibrational contribution to the energy gaps which separate vibronically coupled states [9, 10] or exact diagonalization of the Hamiltonian matrix in the vibronic basis [11, 12]. We will not go into the details of these additional approaches, since they go beyond the scope of this chapter. Notice that Eq.(28) or (29) show the same Qk dependence of the electronic transition dipole moment in Eq.(22). Coming back to Eq.(22) we notice that the Condon approximation discussed in the previous section requires the neglect of all but the first term. In this limit we have seen that the transition intensities are proportional to the FC factors. In the breakdown of the Condon approximation, more generally called the HT vibronic coupling, the higher order terms in Eq.(22) are introduced. The most dramatic effects due to HT vibronic coupling are observed, generally, for dipole forbidden transitions. In this case the first term in Eq.(22) vanishes and the Condon approximation would imply absence of intensities in correspondence of a similar electronic transition. This conclusion is, however, in marked contrast with the experimental evidence. The simpler example is represented by the absorption spectrum of benzene, shown in Fig. 8, where, disregarding the lowest energy spin and symmetry forbidden S0(Alg ) —> T^B^J transition, appreciable intensities are observed not only for the symmetry allowed S0(A]g j—> S 3 (E 1 U ) transition but also for the S0(A,,,)—> S,(B 2 U ) transition whose transition dipole moment is zero by symmetry. The discrepancy between the experimental evidence and the prediction provided by the Condon approximation implies that the model is too approximated in this case, and one has to move to the more accurate HT vibronic coupling approach. In the HT vibronic coupling approach, the first term in Eq.(22) is zero, but higher terms are retained and they account for the intensities observed for symmetry forbidden transitions. To clarify this point we can consider an even simpler example, namely a | g) —> e) symmetry forbidden electronic transition of a molecule with a single NTS mode such that —

# 0 . Following Eq. (22), the intensity I gm ^ cn of the g)|m) —> |e)|n) vibronic

transition in this NTS mode is given by

142

150

X (r*m)

Fig.8. Low resolution spectrum of benzene from ref.[13].

=

m NTS

'NTS

.0

r-g.e n "N TS MTS

)+ m (30)

where the vibrational overlap multiplied by (a°c has been neglected in the last line of Eq.(30) because the electronic transition is assumed to be dipole forbidden. It is interesting to employ Eq. (30) to obtain the expressions for two specific vibronic transitions, namely the | g)| 0} -» | e)| 0) and the | g)| 0} -> | e)| l): =0

(31) (32)

The first of the two (Eq.(31)) is zero owing to the properties of the vibrational wavefunctions. This implies that the g}| 0} —> e)| 0} transition, namely the origin band, of a forbidden transition will never appear in the absorption spectrum. Eq.(32), in contrast, shows that the g)| 0) —> e)| l) band can appear in the region of a forbidden transition, owing to the properties of the vibrational wavefunctions which ensure that the integral

^J9SI

Ji2-15

3.39

SSii

05 A 2 9M 9 2 67 - flT

A

0SlSi 12.42 02.4

3.39

Fig. 3. M-O^ and M-O# distances in the asymmetric structures obtained from B3LYP-DFT optimization of MO6Si4Al2(OH)12 (M = Co, Cu)

thoroughly affect the bonding properties and corresponding spectroscopic (electronic and ESR) properties of both ions in different zeolites. In particular, our calculations have indicated how different distributions of aluminums may actually lead to two distinctly different ESR signals in the zeolites Y, ZK4 and mordenite [68, 69, 72]. As for the ligand field spectra of Cu(II) and Co(II) coordinated to zeolites A and Y, similar excitation energies were obtained from CASPT2 calculations on optimized models with different numbers and distributions of aluminums. The results obtained for the model with two aluminums distributed as in Fig. 3 are included in Table 3.1. These data clearly reveal a much stronger ligand field surrounding of both ions as compared to the trigonal oxygen environment in the models with structures frozen at the XRD values. The splitting of the 2D ground state of Cu(II) is doubled, and the calculated transition energies now nicely correspond to the position of the two band maxima at 10 400 cm" 1 and 15 000 cm" 1 in the experimental spectra. For Co(II), the splitting of the 4F ground state is again almost doubled, but still remains slightly below the maximum of band I in the experimental spectrum. It is shown in ref [67] that the introduction of spin-orbit coupling leads to a further broadening by 800 cm" 1 of the transition energies belonging to 4F. Band II at 16 000 cm" 1 is now found to correspond to two distinct transitions, split by about 600 cm" 1 . This is in accord with the experimental splitting observed for this band, although the calculated splitting is still too small. Finally, the appearance of band III at 25 000 cm" 1 is well reproduced by the CASPT2 results. The experimental splitting of this band may at least be partly explained by spin-orbit coupling [67]. On the whole, the results presented here have indicated how the coordination of TM ions in zeolites may be succesfully explained and even predicted by means of cluster models and using a combined CASPT2//D FT approach. The CASPT2 excitation energies are especially useful since, in confrontation with experimental ligand field spectra, they provide a sensitive probe of the geometric distortions predicted by the DFT optimizations. Such distortions are also indicated by other spectroscopic experiments, e.g. IR, ESEEM and 27A1 NMR [83-85], but their actual extent has so far been impossible to obtain directly from experiment.

291

4. CHARGE-TRANSFER STATES: ELECTRONIC SPECTRUM OF THE PERMANGANATE ION The electronic spectrum of permanganate presents one of the theoretically most intensively studied spectra of transition metal systems. Based on a semi-empirical MO treatment, a basic interpretation of the electronic transitions was provided already in 1952 by Wolfsberg and Helmholtz [86]. During the seventies, numerous ab initio studies were reported, using either Hartree-Fock and limited configuration-interaction [87-91] or one of the earlier Hartree-FockSlater versions of DFT: HFS-SW [92], HFS-DVM [93]. More recently, MnOJ has been given the status of a prototype transition metal system, used as a benchmark for testing the performance of several newly developed methods for the calculation of excited states: DFT in its ASCF [94, 95] and time-dependent [96, 97] form and cluster expansion methods such as SACCI [98] and EOM- and extended EOM-CC [99]. This prototype status is however somewhat undeserved, given that when it comes to obtaining accurate electronic excitation energies for transition metal complexes, permanganate is an exceptionally hard rather than a representative case. The problem is related to the high (+7) formal charge on manganese in this molecule, giving rise to substantial static correlation effects involving excitations from all twelve doubly occupied orbitals originating from 0(2-) 2p, either bonding or nonbonding, into the five antibonding orbitals corresponding to Mn(7+) 3d, which are empty in the ground state [18] (see also section 2) . Moreover, these correlation effects change appreciably in the excited states [100]. Consequently, any single reference molecular orbital description of the permanganate ground state is quite dubious, and its shortcomings doomed to be reflected in the calculation of excitation energies using methods working with these ground state orbitals. The presence of important static correlation effects in permanganate also affects the DFT description of the excitation energies, with values that substantially differ between the ASCF and TD-DFT approaches, that fluctuate by up to 3 000 cm" 1 between different functionals, and that are on the whole considerably less satisfactory than the results obtained with the same methods for other transition metal systems [96, 97]. Another consequence if this general malaise is that the assignments of the permanganate spectrum have undergone a lot of changes throughout the years, and in fact a general consensus still has not been reached. Here, we will only discuss the most recent theoretical results, obtained with either TD-DFT or cluster expansion methods (an overview of the older results can be found in ref. [98]). Furthermore, we also present the results obtained from a CASPT2 treatment [101], based on CASSCF reference wavefunctions built from an active space comprising all seventeen valence orbitals originating from manganese 3d and oxygen 2p: 6a i, (l-2)e, lti, (5-7)^. CASSCF calculations with such a large active space have only recently become within the limits of computer power, but are still very time consuming. We report here the results obtained for the excitation energies of the lowest spin- and dipole-allowed 1Ai—>XT2 transitions in this tetrahedral molecule. A more detailed description of the spectrum will be presented in a separate paper [101]. In Table 4 calculated excitation energies of the four lowest XT2 states with different methods

Is) Is)

Table 2 Calculated electronic allowed transition energies (in cm" 1 ) of MnOj state Excitation energy (cm x)

Composition(CASSCF)(%) l*i

lr

a T2 bxT2

a

ASCF" LDA 21880 34 070 32 400 45 950

6

TDDFT BP/ALDA 22 825 31536 38 310 47 182

SAC-Cr E 20 728 30 003 28 874 46 940

ref 95; b ref 96-values differing by up

I CASPT2 18 066 29 035 29 600 46 779

18 389 27 906 31455 43 433

C

I 2e 48 3

6*2

l*i

6ai

6*2

2e 3 30 9

7*2

7*2

7*2

1 8 23

3 4

5 5

Assignment

l*i->7* 2 6*2^7*2

Experimental band position^ (cm"1) 18 300 28 000 32 200 43 956

293

are compared to the experimental band positions, shown in the rightmost column. Based on symmetry considerations it can easily be shown that the O 2p23—Mn 3d1 manifold can give rise to five lT2 states corresponding to one of the following excitations, or to a mixture of them: lti—>2e, 6£2—>2e, lii—>7t2, 6ai—>7t2, 6t2—>7£2- The calculated results in Table 4 have been ordered such that their principal character corresponds to the assignment in the second column from the right. One should however keep in mind that, with exception of the ASCF method, all methods describe the different excited states as a mixture of different orbital replacements. For example, with TDDFT the state b : T 2 which is assigned as 6£2—>2e in table 4 in fact consists of 63 % 6t2—>2e and 36 % l*i —>7t2 character, whereas the c *T2 state, assigned as lii—>7i2, is composed of 50 % lii—>7t2, 17 % 6t2—>2e and 20 % 6t2^-7t2 character [96]. Table 4 also shows the contribution of the five singly excited configurations to the CASSCF reference wavefunctions correponding to the calculated CASPT2 excitation energies. For one thing, these numbers reflect the extreme importance of nondynamic correlation and corresponding multiconfigurational character of the different state functions in permanganate, with a total contribution of singly excited character amounting to 51 % for the first a x T 2 state and decreasing further to only 9 % for the d XT2 state. The contribution of the HF configuration in the corresponding CASSCF ground state wavefunction is 58 %. All methods collected in Table 4 agree upon the assignment of the first band in the experimental spectrum to an excitation with predominantly lti—>2e character. Furthermore, as far as concerns single orbital replacements, the CASPT2/CASSCF results also agree with the other methods [96, 98, 99] that the fourth band in fact corresponds to an (almost equal) mix of the 6t2—>7t2 and (M\—>7t2 orbital transitions. However, the assignments of the second and third bands have so far remained controversial. The SAC-CI and ASCF methods assign the second band to \t\-^t7t2 and the third band to 6t2^2e. This also corresponds to the original assignment by Ballhausen and Gray [103], based on indirect evidence from various sources. On the other hand, according to the TDDFT, EOM-CC and also to the present CASPT2 results these assignments should be reversed. In view of the fact that (a) CA SPT2//CA SSC F is the only method that properly accounts for the strong multiconfigurational character of this molecule, and (b) the close correspondence (to within 1000 cm" 1 ) between the CASPT2 excitation energies and the experimental band maxima of all four bands, we hold it most likely that the assignment based on the CASPT2//CASSCF method is correct, and that the discussion on the assignment of the second and third bands in the permanganate spectrum can now finally be closed.

5. SPECTROSCOPY OF REDOX PROTEINS TM atoms or ions coordinated in proteins are usually found in ligand field environments consisting of very specific, sometimes remarkably complex ligands. The role of this environment is of course to provide the metal with the appropriate electronic structure to play its part in the catalytic cycle. In the case of redox active metalloenzymes an important task for the ligand environment is to modulate the metal reduction potential, the latter determining the driving force of the electron transfer reaction. The reduction potential is dependent on the ionization energy

294

of the reduced site as well as on the so-called reorganization energy, i.e the energy required to reorganize the ligand environment (inner-sphere reorganization energy) and the solvent and the rest of the protein (outer-sphere reorganization energy). Obviously the number, position and specific character of the ligands directly bound to the metal plays a crucial part in all this. Thus, strongly covalent interactions between the metal and one or more ligands will facilitate oxidation by lowering the effective charge on the metal. Moreover, the more strongly antibonding the redox active orbital the higher its energy and the easier ionization from this orbital. Since the same factors also determine the spectroscopic characteristics of the metal-ligand combination, electronic spectroscopy provides an excellent means of obtaining information concerning the catalytic potential of the metal active site in redox proteins. In the remaining of this section we will discuss the electronic structure of the active site in two important groups of redox active metalloenzymes, based on the computation and interpretation of electronic absorption spectra of model complexes. 5.1. Relation between the structure and spectroscopy of blue copper proteins Blue copper proteins are electron transfer proteins. The active site of all these proteins contains a single copper ion (with oxidation state varying between I and II), four- or five-coordinated with a typical short bond to the thiolate S of a cysteine residue. Two of the other ligands are N s atoms of histidine residues, and the fourth is normally a methionine thioether group. (Azurins have a fifth, weakly bound carbonyl oxygen ligand). A typical example is plastocyanin, a small protein involved in photosynthesis. In the oxidized form, the Cu(II) ligand surrounding in this protein is trigonal, the two histidine-N and cysteine-S forming a (distorted) trigonal plane, with the methionine S bound in an axial position at a large distance. Such a trigonal surrounding is quite peculiar for Cu(II): four-coordinated inorganic complexes of this ion are almost invariably (more or less distorted) square planar [32] (e.g. the square planar Cu(II) coordination in zeolites, discussed in section 3.2) . Instead, the trigonal Cu(II) surrounding in plastocyanin is closer to the (distorted) tetrahedral surrounding found for the reduced Cu(I) form of the same protein [104]. The close resemblance between the two geometries accounts for a small reorganization energy accompanying reduction, and hence to a high rate of electron transfer in this class of proteins [105]. This fact originally led to the suggestion that the unusual trigonal surrounding of the oxidized site is not the one preferred by Cu(II) itself, but is instead imposed upon it by the tertiary protein structure, in order to facilitate electron-transfer, the induced-rack [106, 107] andentatic state [108, 109] theories. However, DFT structure calculations on different realistic models of the Cu(II) surrounding but lacking the rest of the protein (e.g. Cu(imidazole)2(SCH3)(S(CH3)2)+) [9, 110, 111] have indicated instead that a trigonal structure is indeed what is preferred by the specific choice of Cu(II) ligands in plastocyanin. The ligand which is crucial in this respect is the Scv« thiolate group [112]. By forming a 7r-bond between the two lobes of its 3pw orbital and two of the lobes of the Cu 3d orbital in the trigonal plane, the S cy« ligand in fact plays the role of two ligands in an "apparent" square-planar coordination [113]. Electronic structure calculations further indicated that the Cu-Scy., yr-bond is highly covalent. The antibonding Cu 3d-S 3pw combination

295 involved is the orbital which is singly occupied in the Cu(II) ground state. Therefore, this is also the redox active orbital and its high covalency activates the blue copper site for its biological function as electron transfer site [9]. The highly covalent Cu-S^j,., 7r-bond in plastocyanin is also responsible for the appearance in the electronic spectrum of the "blue" band, i.e. an intense (e,,,.aa:=3000-6000 M~1cm~1) absorption band around 600 nm. This has been shown in several theoretical studies of the electronic spectrum of plastocyanin, where the blue band was found to originate from a charge-transfer excitation from the bonding to the antibonding Cu-thiolate IT orbital, gaining its high intensity from the large overlap between the two orbitals involved [114117]. 5.7.7. The spectrum of plastocyanin The electronic spectra of several blue copper proteins have been recorded by visual and nearinfrared absorption spectroscopy, circular dichroism and magnetic circular dichroism [9,115, 118-120]. For plastocyanin in total nine different absorption bands have been reported [115]. They all correspond to excitations of an electron into the Cu 3d-S 3pn antibonding combination. Four of these are LF transitions, i.e. where the electron originates from one of the other molecular orbitals with (predominantly) Cu 3d character. The other five are LMCT excitations, with the excited electron coming from a molecular orbital which is mainly centered on one of the ligands. The spectrum of plastocyanin was calculated by performing CASPT2/CASSCF calculations on different model systems, ranging from [Cu(NH3)2(SH)(SH2)]+ to [Cu(imidazole)2 (SC2H5)(S(CH3)2)]+ [117]. In order to include the important 3d radial correlation effects in the Cu2+ ion the active space of these calculation necessarily had to include a second d-shell (i.e. the double-shell effect; see also section 2). Since the model complexes which are closest to the real protein necessarily are unsymmetric, the size of the active space that could be used was limited to twelve orbitals, i.e two Cu d-shells, and the Scys 'ipa and 3pn lone pair orbitals. This active space allowed for the description of the lowest energy part of the spectrum, up to 20 000 cm"1, containing the ligand field states and the two most intense charge-transfer states, both originating from Scy!i- Two additional charge-transfer transitions at higher energy could be calculated for model systems that were forced into Cs symmetry, and were shown to originate from either imidazole or SMC*- For a more detailed discussion of these transitions and the corresponding experimental band positions we refer to ref. [117, 121]. The most accurate CASPT2 results for the spectrum of plastocyanin were obtained for the [Cu(imidazole)2(SH)(SH2)]+ model by making use of the plastocyanin crystal structure [122] but with CASPT2 optimized Cu-S^s a n c ' CU-SMC* distances, and after including the effect of the protein environment by means of a point-charge model [123]. These results are shown in Table 3. The ground state and the different excited states may most easily be characterized by analyzing the singly occupied natural orbital in each state. These orbitals are shown in Fig. 4. By choosing a coordinate system with the z axis along the Cu-S Met bond and the Cu-Sc,ys bond located in the xz plane, the Cu 3d orbitals involved can (approximately) be labeled as shown in the left column of Table 3.

Is)

Table 3 Comparison between the calculated and experimental spectra of plastocyanin and nitrite reductase " State Character Plastocyanin Nitrite reductase exp. exp. calc. calc. bA LF (Cu-S)o-*^(Cu-S)7r* 4 363(.0001) 4 408(.0000) 5 000(.0000) 5 600(.0000) 11 645(.0000) 10 800(.0031) 11 900(.0026) 12 329(.0003) cA LF Cu3dz2 —>(CU-S)TT* dA LF Cu3dyz -^(CU-S)TT* 12 981(.0025) 12 800(.0114) 13 500(.0086) 12 872(.0004) 12 671 (.0002) 13 95O(.OO43) 14 900(.0101) 13 873(.0028) eA LF Cu3dxz -KCU-S)TT* fA CT ( C U - S ) T T ^ ( C U - S K 15 654(.1162) 16 700(.0496) 17 550(.0198) 15 789(.0325) CT (CU-S)CT^(CU-S)TT* 21 974(.0012) 21 39O(.OO35) 21 900(.0299) 22 461(.1192) gA

Character LF (CU-S)TT*->(CU-S)CT *(+7T*)

LF LF LF CT(CU-S)TT^(CU-S)CT* (+7T*) CT(CU-S)CT->(CU-S)CT* (+7T*)

"Calculations were performed on a [Cu(imidazole)2(SCH3)(S(CH3)2)]+ model at the crystal geometry but with the Cu-Scya optimized at the CASPT2 level

and Cu-Sj^ e t distances

297

^

XA

bA

eA H Fig. 4. Plot of the singly occupied (natural) orbital in the ground state and the lowest excited states in [Cu(imidazole)2(SH)(SH2)]+ at the plastocyanin structure

As one can see, the ground state singly occupied orbital is indeed strongly delocalized between Cu 3dxv and Scy., 3p^, thus confirming the highly covalent character of this interaction. The four lowest excited states in the spectrum may formally be characterized as ligand field states. However, from Fig. 4 it is clear that all four in fact also contain a significant amount of charge-transfer character. Even if the departing orbitals in the second to fourth transition are almost pure Cu 3d, the accepting orbital is partly Scys 3pn such that these transitions involve a significant movement of charge from the copper into Scy.,. More importantly, the lowest excited state has a single electron in an orbital which is again strongly delocalized over the CuScya—>Cu3d charge-transfer bands is characteristic of the electronic spectrum of all blue copper proteins, providing them with the label type 1 copper proteins (as opposed to type 2 copper proteins, the spectrum of which only contains a number of weak ligand field transitions in the same region). The band positions of the two transitions, around 600 nm (16 600 cm"1) and 450 nm (22 000 cm" 1 ), remain approximately the same for all proteins. However, the relative intensity of the two bands varies between different proteins [124, 125]. Thus axial type 1 proteins, like plastocyanin and azurin, show only little absorption in the 450 nm region, while the 450 nm band becomes much more prominent in rhombic type 1 proteins like pseudoazurin and cucumber basic protein (the classification of blue Cu proteins as axial or rhombic is based on their ESR characteristics). The increasing intensity of the 450 nm band in the latter proteins goes together with a decrease in intensity of the 600 nm band, so the sum of e^omn and eeoonm remains approximately constant [124]. A limiting case is nitrite reductase from Achromobacter cycloclastes. The intensity of the 600 nm absorption peak is in this enzyme reduced by a factor 3 compared to the classic proteins, and the 460 nm absorption has actually become the more intense, resulting in a green color of the enzyme. The gradual change in the spectral characteristics of different blue copper proteins goes together with a structure which is gradually more distorted with respect to the trigonal Cu(II) surrounding in plastocyanin. The distortion comes down to a flattening, and can most conveniently be quantified by considering the angle cj> between the planes formed by the N ffiil-CuNiH,, bonds and the S ^ - C U - S M C * bonds respectively [123]. In a strictly trigonal structure,

299 both planes are perpendicular ((/> = 90°). In practice, axial type 1 proteins have angles ranging between 77° and 89° (e.g. for the poplar plastocyanin described in the previous section =82°). On the other hand, in the rhombic type 1 proteins significantly smaller angles are found, ranging between 70° and 75°. Again, nitrite reductase is a limiting case: different crystal structures of this enzyme have cf> angles ranging between 56° and 65°. Together with the flattening of the structure subtle variations of two important bond distances are observed: the Cu-S Met bond is considerably shortened in nitrite reductase as compared to plastocyanin, from 2.88 A to 2.56 A, whereas the Cu-Scj,., bond distance is increased, from 2.11 A to 2.17 A. In order to (qualitatively) investigate the relation between the structural variations between different blue copper proteins on the one hand and the variations in the relative intensity of the 460 nm and 600 nm bands on the other hand, test calculations were first performed on a small model [Cu(NH3)2(SH)(SH2)]+. A series of (B3LYP-DFT + restricted CASPT2) geometry optimizations was performed on this model at fixed <j> angles, ranging between = 90° (i.e. a trigonal structure) and <j> = 0° (i.e. a square-planar structure). A CASPT2//CASSCF calculation of the electronic spectrum was performed at different points along the twisting path. From these small model calculations, several important points were noted, all of them highly relevant for the structure-spectroscopy relationship in the actual proteins. First of all, these calculations indicated that the energy curve along the <j> twisting angle is extremely flat. At the CASPT2 level two minima were found, one for =90 ° (i.e. close to the plastocyanin ) and one for — 50° (i.e. close to the nitrite reductase cj>). Yet, the barrier between these two minima is low: only 8 kJ/mol. This explains how different protein environments are in fact able to stabilize a continuum of Cu(II) surroundings [9], ranging from the trigonal structure in plastocyanin to a distorted tetragonal structure in nitrite reductase. A second point emerging from these calculations (see also Fig. 5) is that the optimized Cu-SH and CU-SH2 distances indeed show the trends also observed for the actual proteins: the Cu-SH2 distance is gradually decreased by more than 0.3 A as 0 is reduced from 90° to 50°, while the Cu-SH bond is simultaneously elongated by about 0.1 A. As for the electronic spectra obtained for these small models, the relative intensity of the two HS—>Cu transitions, also indicated in Fig. 5, indeed follows the trend observed for the proteins. This trend can now also be related to the electronic structure of the ground state at different 4> angles, as reflected by the nature of the singly occupied orbital. In the trigonal ( = 0°) structure this orbital displays a TT antibonding interaction between copper 3d and the SH ~ S 3p orbital, similar to plastocyanin (see also the previous section and Fig. 4). As such, an intense TT-to-Tr* transition is calculated for this structure, whereas at higher energy the second, a-to-vr* transition, is calculated with little intensity. However, as angle for the [Cu(NH3)2(SH)(SH2)]+ model.

state singly occupied orbital gaining more and more Cu-SH a character (at the expense of TT) as the structure becomes more flattened. As a consequence of this changing ground state electronic structure, the second charge-transfer transition in the electronic spectrum, originating from the Cu-SH a bonding orbital, gains intensity at the expense of the first transition, originating from Cu-SH IT. The "crossover", i.e. the point where the second transition becomes more intense than the first, is for [Cu(NH3)2(SH)(SH2)]+ predicted at cf> around 74°. As such, even these small model calculations predict nitrite reductase to be green. Armed with the above qualitative considerations, the analysis and assignment of the lowenergy part of the experimental spectrum of nitrite reductase becomes straightforward. CASPT2/CASSCF calculations of this spectrum were performed in a similar way as for plastocyanin, i.e. using the [Cu(imidazole)2(SH)(SH2)]+ at the nitrite reductase crystal structure of Achromobacter cycloclastes [126] but with CASPT2 optimized Cu-S>cys and Cu-S^d distances, and a charge model for the protein environment [123]. The results are compared to the experimental spectrum [127, 9] and the corresponding data for plastocyanin in Table 3. The singly occupied natural orbitals characterizing the ground and each of the excited states in nitrite reductase are shown in Fig. 6. As one can see, the ground state singly occupied orbital in the nitrite reductase model has indeed become more of a* than of TT* type, and vice versa for the corresponding orbital of the first excited state. Interestingly, the corresponding bonding orbitals, depopulated in the two charge-transfer states, do retain their pure a or TT character. As such, the intense bands

301

Fig. 6. Plot of the singly occupied (natural) orbital in the ground state and the lowest excited states in [Cu(imidazole)2(SH)(SH2)]+ at the nitrite reductase structure

in the nitrite reductase spectrum should be assigned as Scys—>Cu 3d 7r-to-(a'(d:l!j 48% ; a'(SOj,)^a'(da.z) 36 % 21 657(.0068) a'(SOI,)->a"(d,,J 58 % ; a(dx2_y2)^a"(dyz) 26 % 23 029(.0002) a"(S op )^a"(d,, z ) 57 % ; a"(SOJ,)^a"(d,,y) 20 % 23 131(.O257) a"(S O! ,)^a'(d ci ) 74 % ; a'(d3;2_y2)->a"(d.1/,) 5 % 23 276(.0023) a"(S,;j,)^a'(d:c2_,y2) 90 % 23 467(.0044) a'(S o} ,)^a"(d y J 71 % ; a'(dal2_!/2)^a"(da;!/) 6 % 23 985(.0431) a'(S OJ) )^a'(d,, 0 ) 64 % ; a'(d:i;2_,y2)^a'(d,,z) 21 % 25 655(.0038) a (Sop)^a" (dyz) 17 % ; a(Sop)^a"(dxy) 39 %

f 2 A' h 2 A"

26 274(.0006) 26 308(.0144)

State

a'(S,:j,)->a'(d3:2_v2) 89 % a"(S o ,,)^a"(d 3 , z ) 50 % ; a'(d c 2_,, 2 )^a"(d,, z ) 21 % ;

g 2 A' 27 030(.0111) a'(S OJ ,)^a'(d l;0 ) 71 % ; a'(d :l; 2_, y 2)^a'(d l;z ) 14 % h 2 A' 27 395(.0131) a"(S o ,,)^a"(d,,J 62 % ; a"(SOj,)-J-a"(d:,:iy) 18 % i 2 A" 33 270(.0146) a (So,,)^'(d^y) 72 % ; a'(da!2_v2)^a"(d;!1!y) 13% i 2 A' 34 121(.0043) a"(S o p )^a"(d : r ; / ) 66 % ; a " ( S q , ) - » a " ( g 13 % j 2 A" 36 715(.0011) a.'(Sop)^'(dvy) 45 % ; a'(da!2_!/2)^a"(d;I!y) 35% j 2 A' 37 284(.0016) a"(S o p )^a"(d : r ; / ) 63 % ; a"(S^^a (dyZ) 17 % "Calculations were performed on the B3LYP-DFT structure Excitation energies are given in cm ~ : r Only the main contributions (> 5%) are shown

experiment exc.energy h (osc.str.) 9 100(.0056) 13 100(.0033) 15 800 (-) 19 400(.0160)

22 100(.0170)

25 100(.0920)

306

a(d,.2_y2)

a {dxz)

a (SOJ))

a'(So,,)

a"(S, :p ) Fig. 10. Plot of the (natural) orbitals involved in the lowest electronic excitations in (Tp)MoO(bdt)

307

Fig. 11. Schematic plot of the three-center pseudo-a bonding and antibonding combinations formed by the Mo Ad.xi-yi and a'(S,;p) orbitals

excitations out of a (Sop) and a (Sop), the former being the lowest. The calculated transition energies, 10 912 cm" 1 and 14 326 cm" 1 are 1 200-1 800 cm" 1 too high, whereas the oscillator strength of both transitions is grossly overestimated by the calculations. Nevertheless, there can be little or no doubt concerning the character of the lowest two excited states in these spectra. The same is not true for the feature at 19 400 cm" 1 . Although this band was originally (for (Tp*)MoO(tdt)) assigned as the Mo 4dx2_y2->Adxy LF transition [134], all later studies [128, 130, 135] report the assignment of the 19 400 cm" 1 as the in-plane a'(Sj,,->Mo 4rfa.2_y2 LMCT transition. The latter is based on the relatively high intensity of this band, which supposedly originates from a large overlap between the donor and acceptor orbital involved. In order to explain such overlap it was suggested that both orbitals are involved in a highly covalent threecentered pseudo a bond, as shown schematically in Fig. 11. However, our calculations do not give any indication for the existence of such a three-centered bond. As can be seen from Fig. 10 the Mo 4rfiI2_y2 and Srp orbitals do not show any significant mixing. Correspondingly, the calculated oscillator strength of the a'(S,p)->Mo 4dx2_y2 excitation is very low, 6 x 10~4. This excitation is predicted to occur at 26 274 cm" 1 (state f 2A' in Table 4). The corresponding a"(S,;}))—>Mo 4dx2_y2 excitation is predicted at 23 276 cm" 1 with an oscillator strength of 2.3 x 10"3 (state e 2A" in Table 4). Both transitions should in the experimental spectrum be obscured by the occurrence in the same region of much more intense transitions originating from a (SO2,) and a"(Sop. The present results therefore throw a new light on the possible role of the pyranopterin dithiolate Sn, orbital in the catalytic activity of oxomolybdenum enzymes, either acting as an electron transfer conduit or modulating the molybdenum reduction potential. Facile electron transfer in general requires that there be a high degree of electronic communication between the donor and acceptor sites. In ref. [128] it was suggested that the highly covalent three-center a interaction between the a'(S,{)) orbital and the redox active Mo 4dx2_y2 orbital should be involved in and responsible for efficient pyranopterin mediated electron transfer. The in-plane nature of this interaction is convenient, since it may be expected to give a minimal contribution to the reorganization energy involving distortions along the apical M=O bond during electron transfer reactions. Furthermore, this effective in-plane covalency was also anticipated [128]

308 to play an important role in modulating the electrochemical potential of the active site, by destabilizing the energy of the redox active (predominantly) Mo Ad,r2_y2 orbital. The latter role was however put into perspective by a number of comparative studies of (Tp*)MoO(S-S) complexes with different (S-S) ligands, such as tdt, qdt or bdtCl2. The electronic absorption spectra of these complexes all contain the same band around 19 400 cm" 1 with a very similar transition energy and intensity. This would suggest (still assuming that this band indeed corresponds to a transition between the bonding and antibonding combinations of Mo d X2_y2 and a'(S,j,)) that the covalency of the three-center pseudo a bond and its destabilizing effect on the redox active orbital is indifferent to the character of the (S-S), in particular on the presence of electron withdrawing or donating groups. On the other hand, electrochemical and He(I) PES studies [130, 131, 135, 139] have indicated that both the ease of oxidation and reduction of the Mo(V) center is indeed strongly affected by the presence of such groups. In particular, these studies indicate that the redox active orbital is stabilized as the ene 1,2 dithiolate ligand becomes more electron withdrawing, thus making ionization more difficult and reduction easier. Based on these observations it was proposed that instead anisotropic covalency contributions involving only the Sop orbitals of the coordinated thiolate control the Mo reduction potential by modulating the effective nuclear charge of the metal. In refs 135, 140 it is suggested that a regulating factor of this interaction is the S-S fold angle of the ene dithiolate chelate ring, i.e. the angle between the S-C=C-S and S-Mo-S planes. Folding the S-C=C-S plane upwards in the direction of the M=O bond leads to an increased overlap between the the a (Sop) and the Mo idx2_y2 orbital, thus giving rise to a reduced effective charge on the metal. Such overlap is indeed manifested by the plot of the bonding a'(SOJ)) orbital in Fig. 10, although it is virtually absent in the corresponding antibonding orbital, i.e. the ground state singly occupied orbital a(d.j.2_y2). The role of the SOJ) orbitals and the effect of the S-S fold angle on the electronic structure of these (Tp*)MoO(S-S) complexes and pyranopterin Mo enzymes they stand model for certainly warrants further investigation. Returning to the spectroscopic data for (Tp)MoO(bdt) in Table 4 and to the band at 19 400 cm"1 in particular, our calculations indicate that this band should instead be assigned as the Mo dx2-y2—>dxz LF transition, which is however strongly mixed with a'(Sop)—>dxz LMCT character. This mixing may explain the intensity of the 19 400 cm" 1 band, although the calculated oscillator strength is significantly lower than the experimental value. A second ligand field transition, Mo dx2_y2—tdyz is predicted to occur at lower energy, 16 457 cm" 1 , again strongly mixed with CT character out of a'(Sop) into the same id orbital. The calculated oscillator strength of this transition is very low. This is in agreement with the appearance of a very weak feature (only discernable in low-temperature MCD) at 15 800 cm" 1 in the experimental spectrum, which is therefore assigned as such. The third LF exitation, Mo dx2_y2^dxy is situated at a considerably higher energy. The lowest excited state containing a significant (6 %) dx2_y2^dxy contribution is the f 2A" state at 23 467 cm" 1 . However, none of the states included in Table 4 can even grosso modo be identified as the transition to dxy, the largest contribution, 35 %, being found for the j 2 A" state at 36 715 cm" 1 . The high energy of the metal dxy orbital is of course related to the strong antibonding character (primarily with S,p but also with the equatorial N u-donor

309 ligands) of this orbital, as is obvious from the orbital plot in Fig. 10. Ranging from 21 000 to 27 400 cm" 1 a whole series of transitions is calculated, corresponding to excitations from the a'(SOJ,), a"(SOJ>) orbitals into Mo dxz, dyz. In the experimental spectrum, two intense bands are distinguished in this region: a shoulder at 22 100 cm" 1 and a very strong band at 25 100 cm" 1 . They are assigned to the two excited states for which the calculations predict the highest oscillator strength: d 2A", calculated at 23 131 cm" 1 and e 2A', calculated at 23 985 cm" 1 . Both states primarily correspond to an excitation into Mo dxz. Corresponding excitations to Mo dyz are calculated at slightly lower energies but with a much lower oscillator strength. Finally, excitations out of the a'(SOj,), a"(SO2,) orbitals into Mo dxy are predicted to occur between 33 000 and 37 200 cm" 1 , i.e. outside the range of the experimental spectrum. 6. CONCLUSION In this chapter we have described a few theoretical studies of electronic spectra of transition metal ions in different coordination environments. We believe that these studies have convincingly demonstrated the strength of accurate computational approaches: (i) in explaining and predicting the relative position and intensities of different types of excited states (LF or CT) in these spectra; (ii) in relating this information to the ground state electronic structure and bonding characteristics of different metal-ligand combinations; (iii) in relating to the ground state electronic structure and spectroscopic characteristics other important properties such as the geometrical structure of the coordination environment or the possibility for redox activity of the central metal ion. An illustrative example is presented by the calculations on the Cu(II) ion in two thoroughly different coordination environments: the ionic oxygen ligand surrounding of a zeolite surface on the one hand, leading to an almost strictly square planar Cu(II) surrounding and a typical ligand field spectrum, and the trigonal Cu(II) surrounding in blue copper proteins on the other hand, caused by the presence of a highly covalent Cu-S cVa "" bond, the latter also giving rise to a low-lying intense LMCT band, i.e. the "blue" band. The calculations on (Tp)MoO(bdt) have indicated that the Mo(V)-dithiolate bond in these model complexes is in fact highly ionic, as opposed to the earlier suggested presence of a covalent three-center S-Mo-S pseudo a bond. The question of course remains whether this complex is indeed a representative model for pterin-containing molybdenum enzymes and how to explain the redox activity of such enzymes. Finally, the calculations on MnO J have demonstrated how the CASPT2/CASSCF method, used throughout this work, is able to provide very accurate results also for the notoriously difficult-to-calculate spectrum of this molecule, provided that the active space is chosen large enough to include all (very important) nondynamic correlation effects.

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M. Olivucci (Editor) Computational Photochemistry Theoretical and Computational Chemistry, Vol. 16 © 2005 Elsevier B.V. All rights reserved

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X. Perspectives in Calculations on Excited State in Molecular Systems Bjorn O. Roos Department of Theoretical Chemistry, Chemical Center P.O.B. 124, S-221 00 Lund, Sweden 1. INTRODUCTION The electronic Schrodinger equation has for most molecular systems an infinite number of bound solutions. Each of them corresponds to an electronic state with a specific electronic structure. Such a state gets populated when the molecule absorbs one or more quanta of light (photons). When a molecule is brought into one of the excited states, one of several things can happen. The energy surface for the excited state is different from that of the ground state, because the electronic structure has changed. The excitation process is fast, so the atoms will not have time to move. As a result, the molecule will find itself in a non-equilibrium position and will start vibrating. Ultimately, it may end up in the equilibrium geometry of the excited state (which in many cases means dissociation of one or several chemical bonds). That is, if there is time enough because the lifetime of the excited state is usually short and a competing process is emission of a light quantum and return to the ground state energy surface directly or via lower lying electronic states. But the dynamics may be even more complex. We show an example in Fig.l (for simplicity, the energy is shown for one geometrical coordinate only). A photon excites the system from the lowest vibrational level on the ground state energy surface, X, vertically to the first excited state surface, B. This state is bound but with another geometry than the ground state. Another state, A, crosses state B at a larger value for the geometry parameter. The crossing point lies below the energy that the molecule has adsorbed from the photon. So, the molecule has several possibilities. It can stay on surface B and loose the extra energy by thermal dissipation (possibly involving other molecules). Ultimately, it will then loose its energy by emitting a photon of lower energy (fluorescence). Or it can cross over the surface A and dissociate. It can also come back to surface B via the crossing point. Today it is possible to follow such dynamical processes experimentally using time-resolved laser spectroscopy. How can we deal with such problems theoretically and computationally? It is clearly not straightforward and simple. We need to be able to compute with good accuracy the energy surfaces for a number of excited states and we then need to be able to solve the dynamical problem for the nuclear motion on these surfaces. In this chapter we shall deal with the first of these problems. Chapters 5 and 7 will deal with different aspects of the nuclear dynamics.

318

hv

— X

Geometry

Fig.l. Energy surfaces in a hypothetical molecular system. Four electronic states are shown. The energies are given as functions of one arbitrary geometrical variable.

2. A BRIEF HISTORICAL BACKGROUND The history of electronic spectroscopy of atoms and molecules is older than quantum mechanics. Actually much of the experimental background that inspired the development of quantum mechanics in the beginning of the previous century was found in spectroscopy. It was a grand and immediate success of the Schrodinger equation [1] that it could explain with high accuracy the electronic spectrum of the hydrogen atom. The exact solution of this problem gave us the concept of the atomic orbital. It was almost immediately extended to many-electron systems by the introduction of the self-consistent field model, were the electron-electron interaction is modeled using a mean-field approximation. The model was introduced for atoms by R. D. Hartree in 1928 [2]. A similar model was actually proposed for molecules already in 1927 by F. Hund [3]. The approach was immediately used by R. S. Mulliken and others for the interpretation of electronic spectra of small (diatomic) molecules [4]. Today we know this approach as the Hartree-Fock (HF) method. The basic concept is the molecular orbital (MO) and its energy, leading to a shell model for molecules similar (but more complex in structure) to that of the atoms. This theoretical model explains the excited states of a molecule as follows: in the ground state the electrons reside in the MOs of lowest energy (the aufbau principle). If one electron absorbs a photon of the right energy it will move to an MO of higher energy, which is empty or only singly occupied in the ground state. This is a very powerful approach and forms still today the basis for our analysis of excited states in molecular systems. We now know that one has to go beyond the self-consistent field model to obtain quantitative results for excited states but we can always (almost) label them as excitations by one or more electrons from one MO to another. The development discussed above concerned mainly atoms and small molecules. A simple approach that was to become very influential for the development of the theory of electronic spectra of planar conjugated organic molecules was suggested by E. Huckel [5]. Huckel

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Fig.2. The orbital energy levels for the 7t-electrons in the benzene molecule. simplified the problem of the many electrons by only considering the loosely bound electrons in the 7t-orbitals. The method that emerged was a simple semi-empirical scheme that could be used to compute the structure and energy of these 7i-orbitals. The use of the approach for the interpretation of electronic spectra was, however, limited because the important electronelectron repulsion terms were not accounted for. This problem was realized by M. GoeppertMayer and A. L. Sklar in their landmark study of the electronic spectrum of the benzene molecule in 1938 [6]. In order to understand the importance of the electron-electron interaction, let us take a closer look at the energy levels in benzene. Experimentally, four excited singlet states are known which originates from excitations within the valence 7i-orbitals with energies around 4.9, 6.2, 7.0, and 7.8 eV. The energy levels for the orbitals are shown in Fig. 2. Actually, we can form four excited configurations by moving one electron from an occupied to a virtual orbital and one might think that this corresponds to the four singly excited valence states seen in the electronic spectrum of benzene. This is, however, a too simple picture. Consider the singly excited configuration a ^ e ^ e ^ . We can form 16 Slater determinants for this configuration because of the double degeneracy of the e2U and eig orbitals in the D^ point group of the molecule. 12 of them correspond to triplet states and the four others to singlet states. A little bit of group theory shows that they are: 'B2U, 'BIU, and 'EH, . In a Hartree-Fock picture they have the same one-electron energy, but the electron-electron repulsion energies are different. Goeppert-Mayer and Sklar succeeded to calculate these energies, which was quite an achievement in 1938, even if they were forced to use crude approximations. We refer to their paper for details [6]. What about the fourth state at 7.8 eV? Today we know that this state has gerade symmetry. It can thus be formed from the single excitations aiu—>e2U or eig—»big . We know now that the wave function is a combination of these two configurations, but that also the double excitation efg —^e^ gives a sizable contribution. It is thus an oversimplification to assume that only single excitations are important for the excited states. We shall later see that double excitations are sometimes of crucial importance for the description of the electronic structure. It was to take another 15 years until further progress was made. A rather straightforward extension of the ideas of Goeppert-Mayer and Sklar would be the following: represent each of

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the Ti-orbitals in a planar organic molecule with one basis function. Determine the MOs by solving a Hartree-Fock like problem. Then, expand the wave function for the excited states, V, as a linear combination of singly (and possibly multiply) excited configurations, O^ starting from the closed shell HF wave function:

(1)

^=1 ^

Use the variational principle to determine the expansion coefficients Q,. 50 years ago it was an impossible task to carry out such a calculation from first principles. Drastic approximations had to be made. One key problem was the calculation of all electron-electron repulsion integrals, which occur in the matrix element of the Hamiltonian:

(pg\rs)= \\^p{rl)^p — ^r{r2)Ur2)dVldV2 r

(2)

12

where <j>p etc. are the molecular 7t-orbitals. Even if each atomic basis function was a single Slater orbital, it was an impossible task to compute these integrals in 1950. Robert Parr then introduced a drastic approximation: the zero differential overlap, ZDO approximation [7]. All products of basis function, which where located on different centra were assumed to be zero everywhere. The only remaining integrals were then of the type (pp|rr) and the calculation of the matrix elements were drastically simplified. The integrals could be computed using a single Slater type orbital to represent the AOs. The one-electron matrix elements were taken from Hiickel theory. However, very accurate excitation energies were still not obtained. Rudolph Pariser [8] suggested that the one-center integrals for carbon should also be determined empirically from the valence state disproponation reaction: 2C->C~ +C+

(3)

The corresponding energy is (pp|pp) = IP - EA, where IP is the ionization energy and EA the electron affinity. Using experimental values for these quantities gave the value 11 eV for the one-center integral (pp|pp), a reduction of 6 eV from the theoretical value. These are the key ingredients in the, so called, Pariser-Parr-Pople (PPP) method to compute electronic spectra of planar conjugated hydrocarbons. It was later extended also to hetero-atoms and actually also to transition metal complexes with some success. For a nice historical review of the development we refer to articles by Pariser, Parr, and Pople in Ref. [9]. The PPP method was an immediate success and it became possible to assign a large number electronic states in conjugated organic molecules. Why was that? What was the underlying reason for the large reduction of the one-center two-electron integral? Actually, this was well understood immediately and Pariser wrote: the effect of the a-electrons can be approximately taken into account by changing the value of primarily one Coulomb repulsion integral [8]. Today we call this effect dynamic polarization. It is maybe easiest demonstrated on the Vstate of the ethene molecule C2H4. While the ground state of this molecule has two electron in the bonding 71-orbital, the V state has the electronic configurations (7ur*)s , where S stands for singlet coupling. It is easy to see that the state can be described (using localized orbitals) as a resonance between the two ionic states C+C~ and C~C+. The a-electrons will react dynamically to this polarization. One can actually show explicitly that if the o-system is

321 replaced by a polarizable medium, the effective Hamiltonian will have reduced values of the electron repulsion integrals. The one-center Coulomb integral is reduced from 16.2 to 11.5 eV, which is exactly what Pariser predicted [10]. It would take a long time before modern ab initio quantum chemistry was able to challenge the results obtained with the PPP method. However, chemistry is not only organic. What about the excited states of inorganic compounds. Actually, there was much less interest in this area with two exceptions, small molecules (mainly diatomics) and transition metal complexes. The chemical bond between transition metal ions and their ligands could not be explained using the classical valence picture. A different theory was suggested by Becquerel [11] and formulated more exactly by Bethe [12]. It was acronymed Crystal Field Theory. The applications in chemistry were initiated by the work of Van Vleck [13]. The metal ion was in this method assumed to be perturbed by an electric field from the surrounding ligands. The spectroscopy therefore was atomic in nature. Transition metal complexes often have high symmetry and much of the success of the theory was based on the possibility to use group theory for the analysis of the excited states. For octahedral complexes it was, for example, enough to introduce one single empirical parameter (lODq) to explain the perturbation on the d-type atomic orbitals by the ligands. The crystal field theory was, however, only moderately successful. It was clear that many experimental facts could only be explained if one assumed a delocalization of the open shell electrons onto the ligands. From the crystal field theory emerged Ligand Field Theory. The ligand orbitals directly interacting with the metal ion d-orbitals were introduced and a set of MOs were constructed as linear combinations of the d-orbitals and ligand orbitals. The open shell electrons are then delocalized onto the ligands and a number of spectroscopic properties could thereby be explained. Ligand field theory shares many of the features of the PPP method of organic chemistry both in its formulation and the semi-empirical parameterization. The two approaches have even been combined into one semi-empirical theory, which was used with some success to explain the spectra features of metal porphyrins and similar metalorganic systems [14]. For a detailed treatment of crystal and ligand field theory we refer to the classical book by Carl Ballhausen [15]. This was the general situation at the middle of the 60ies, which is the time when modern ab initio quantum chemistry started its development. We shall therefore leave the history here and continue with a description of the different methods to deal with excited states in molecules and complexes that have emerged during the last 30 years. 3. WAVE FUNCTIONS FOR EXCITED STATES The semi-empirical methods developed for excited state calculations were moderately successful in describing certain features of the electronic spectra. But they had severe deficiencies. The PPP method was, for example, only able to describe %—>%* excitations within the valence shell. It was not clear how to treat the Rydberg transition or n—>p* transitions that occur from a lone-pair orbitals («) in systems with hetero atoms. Today we know that such excitations form an important part of the electronic spectrum of conjugated molecules. Sometimes we also see strong interaction between valence and Rydberg excited states. A typical example is the V state in ethene, which was mentioned above. In transition metal chemistry we know that important excitations are of the charge-transfer type, which can hardly be treated using ligand-field theory. There was clearly a need for an unbiased nonempirical theory, which does not a priori make any assumption concerning the nature of the excited states. This was the aim, when the development of the ab initio methods started in the

322 middle of the 60ies. The first efforts were naturally concentrated to the ground states and the development for excited states came a little bit later. But let us start with the general formulation of modern quantum chemistry. 3.1. Orbitals and wave functions The quantum chemistry for the excited state is always formulated using molecular orbital theory. The basic entity is a set of one-electron functions, the molecular orbitals (MOs). They are multiplied with a spin function to yield a set of spin-orbitals (SOs). Usually, the MOs are expanded in a basis set consisting of functions centered on the different atoms in the molecule, the atomic orbitals (AOs). If we select« such functions, /p(p= 1, «), we can form n orthonormal MOs as linear combinations of the AOs (the LCAO method):

6- = y C- v

(4)

From the n MOs we can generate In SOs. This is the one-electron basis we use to build the wave function. Our results will depend strongly on the quality of the MOs and features not present here cannot be compensated anywhere else. A proper choice of the AO basis set is therefore crucial for the accuracy of the results. This is particularly important for excited states. Once the MO basis has been chosen we need to build a basis for the /V-electron wave function. We can generate In N Slater determinants, d^, by occupying the 2« SOs with N electrons in all possible ways. The total wave function can be expanded in these Slater determinants:

^ = EC /; O /(

(5)

The expansion coefficients can be determined using the variational principle, which leads to the well known secular problem:

J] Hfn,-ES/lr)Cr=0

(6)

When the AO basis set becomes complete, this equation becomes equivalent to the Schrodinger equation. In practice this is of course not possible, but we can (in principle) approach closer and closer to the exact solution by increasing the size of the AO basis set. This is the key feature of ab initio quantum chemistry in contrast to other approaches, like semi-empirical schemes, density functional theory, etc. With a finite basis set the, method is called Full Cl (FCI). It is the best (in the context of the variational principle) approximate solution to the Schrodinger equation that we can obtain with the chosen basis set. The problem is only that we cannot solve equation 6. The dimension is simply too large for anything but very limited basis sets and few electrons.

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Therefore, a second approximation is needed. There are many of them. Actually, all of modern quantum chemistry (with the exception of density functional theory) attempts to approximate the FCI equation in one way or another. Many of the approximations concentrate on the ground state and treat excited states as a secondary problem. The simplest of these methods is obviously the Hartree-Fock method, which only keeps one determinant in the expansion 5. A more advanced approach is coupled cluster theory. The most important of the methods that treat ground and excited states in a balanced way, is the complete active space SCF method and the multi-reference CI methods. All these methods try to select the most important terms in the FCI expansion, using knowledge about the detailed structure of the electron-electron interaction in molecular systems. We shall take a look at some of the methods. There are essentially two different approaches used in excited state calculations. The first, attempts to determine wave functions and total energies for each state of interest. To this category belongs the configuration interaction based methods, which directly tries to approximate equation 6. The second approach starts with the ground state and considers the excitation process as a perturbation of the ground state wave function due to the interaction with the electromagnetic field. These are the linear response methods. Here we compute directly the excitation energies and the transition moments, but do not have access to a wave function for the excited states. To this category belongs time dependent density functional theory and coupled cluster linear response methods. Let us first take a look at the wave function based methods. 3.2. Hartree-Fock and configuration interaction The solution of the FCI equation 6 is independent of the form of the MOs because the FCI wave function is invariant to unitary transformations among them. It is no longer the case when we approximate the equation. We are then faced with two problems, to determine the most optimal form of both the MOs and the CI expansion coefficients. So, what MOs should we use? The most immediate answer to the question is to use the HF orbitals for the ground state (we shall assume that the reader is familiar with the HF method). The eigenfunctions of the Fock operator can be divided into one occupied and one empty (virtual) subspace of MOs. The occupied orbitals will, for a closed shell molecule, contain two electrons each (cf. Fig. 3.)

Ground State

Singly Excited States

Doubly Excited State

Fig.3. Occupied and empty orbitals in the HF model for excited states.

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The HF method is quite successful in describing the electronic structure and energetics of many molecules. The reason is that the mean field approximation of the electron-electron interaction is able to recover almost all of the repulsion energy. The error in the total energy is in well behaved cases less than 0.5%. This is not always the case though. We shall later discuss cases, where it is not possible to start from a HF ground state, but here we shall assume that the HF wave function gives a good representation of the ground state electronic structure. The error in the HF energy compared to the energy of the exact electronic wave function comes from the mean field approximation for the electron-electron repulsion. Each electron will in HF theory experience the average electrostatic field from all the other electrons. The error is called correlation energy because it arises from the inability of the HF method to describe the correlated motion of the electrons, in particular when they come close together. The Pauli exclusion principle tells us that two electrons of the same spin will not occupy the same position in space. This Fermi correlation is included in HF theory through the antisymmetry requirement on the wave function. Thus, the major part of the correlation error comes from electrons of opposite spin. Only two electrons can simultaneously be close together. Thus, correlation can with a good approximation be described using pair-theories. With such a ground state, how do we describe excited states? It is rather obvious: by moving electrons from the occupied to the virtual orbitals. This will generate a set of electronic configurations, which we can use to describe the excited states (cf. Fig. 3). Will the configurations in themselves give a good representation of the electronic structure? Usually not. The reason is that the density of states is often large in the excited manifold and one can expect strong interaction between different configurations (the + and -combinations that occur in alternant hydrocarbons is a good example). So, we have to write the wave function as a linear configuration of the configurations (or more properly configuration state functions (CSFs) because we have to take linear combinations of Slater determinants in order to obtain functions with the correct spin symmetry, singlets or triplets). Our excited state wave function thus has the form:

^ = C 0 O 0 + X Ciaia + Z Cia,jbOiaJb

+....

(7)

where ®o is the HF wave functions, ia are singly excited CSFs and ®iajb are doubly excited CSFs, etc. Where do we stop the expansion? If we do not stop it at all, we shall recover the FCI wave function. We can expect that the singly excited CSFs will be most important, but we cannot exclude that in some cases also doubly excited CSFs will give important contributions. Actually, it is more common that one might expect that excited states dominated by double excited CSFs appear at low energies in the electronic spectrum. But if we believe that this is not the case, can we terminate the expansion after the singly excited CSFs? Such an approximation is today labeled CIS (CI singles). Let us consider an example, the V state of the ethene molecule. The dominating CSFs are presented in Table 1. The numbers are from an extensive multi-reference CI calculation [16]. All orbitals in the table are of cr-type, except those labeled n. The ground state configuration is (ITT)2. The excitations that involve a-orbitals describe dynamic polarization, which is the most important correlation effect for the V state in ethene (see the discussion in the previous section). We see in the table that the CSFs are both singly and triply excited with respect to the HF ground state, but they are, as expected, double excitations with respect to the main configuration of the V state. We conclude that CIS does not give a balanced treatment of the

325

dynamic correlation effects in the excited state [17]. Ideally, we ought to include up to triple excitations in the expansion, but such an expansion becomes quite long and difficult to handle computationally. There is another solution to the problem: instead of starting from a single HF configuration we add a second configuration (ITI*) 2 : ct>0 = a1(l^") + a2i}7r)

(8)

and determine the two expansion coefficients aj and 02 variationally. We then include in the CI expansion for the excited states singly and doubly excited states with respect to both the starting configurations. These are the multi-reference methods. Table 1 The most important CI coefficients for the V state in the ethene moleculea. CSF

Coefficient

lTT^lTI*

0.934

3a g —> 3biu

-0.101

S(D)

( l 7 i ) 2 3 a g ^ (171* ) 3b,u

0.071

T(D)

Ib 3 a ^2b 3 U

-0.064

S(D)

(17I) 2 ^(27C)(2TI*)

0.064

D(D)

Ib3g^3b3u

-0.063

S(D)

2b|U^4ag

-0.052

S(D)

(l7i) 2 3a g ^(27i*) 2 3b lu

-0.043

T(D)

2

Type

a

FromRef. [16].

b

S = single, D = double, T = triple. Within (): with respect to the V state.

But there is yet another way to view the table. Suppose we use the V state configuration (7i7i*)s , instead of the ground state, as the reference in the expansion 7. The most important CSFs will then be the double excitations. It is well known that the most important contributions to the correlation energy comes from the configurations, which are doubly excited with respect to a given reference state. So, it seems that the most natural way to approximate the FCI equation for a given excited state is to first find a good reference function, which describes well the qualitative features of the wave function and then keep in the CI expansion the reference function plus all doubly excited CSFs. For open shell systems, one also needs to include singly excited CSFs. Exactly the same result would be obtained if we tried to estimate the importance of the excited CSFs using perturbation theory. Only the singly and doubly excited CSFs will interact directly with the reference function and therefore appear in the low order corrections. The analysis points to a theory that first computes

326 reference wave functions for each excited state and then adds correlation using singly and doubly excited CSFs. This is the multiconfigurational approach. 3.3. Active orbitals and the CASSCF method The two configuration wave function given in Eq. (8) describes the ground state of ethene better than HF theory. Actually, the coefficients of the second configuration is as large as -0.17. The reason is the rather weak 71-bond, which leads to a large occupations of the antibonding orbital. It is clear that when such occupations become too large, the HF model will break down and we cannot describe the electronic structure with a single configuration. Such situations are obtained when bonds are weak or even broken (dissociation, transition states) and quite frequently in excited states. We need then to go beyond the single configurational model to describe the excited state. This type of electron correlation is called static. It is long range in contrast to the dynamic correlation, which describes what happens when two electrons come close together. The idea behind the Complete Active Space (CAS) model [18] is to find a wave function that is based only on the orbital concept (as is HF), but which can handle also situations where a single configuration is not sufficient to describe the electronic structure. This is achieved by partitioning the MOs into three subspaces: inactive, active, and virtual. The inactive orbitals are assumed to be HF like and thus doubly occupied. To them we add a number of active orbitals but we do not assume anything about their occupation. Instead we construct a full CI wave function in this orbital space. The number of active electrons is the total number minus the number occupying the inactive orbitals. The wave function is thus completely determined by the choice of active orbitals and the condition that it should be an eigenfunction of spin and have a given space symmetry (in the molecular point group): *V

=^C O

(9)

where the sum goes over all CSFs O^ that can be generated by occupying the active orbitals in all possible ways consistent with an overall spin and space symmetry. The coefficients of the MOs and the CI expansion coefficients are obtained using the variational principle. In the ethene case we would choose as active the n and TI* orbitals with two active electrons, which gives the wave function 8 for the ground state. The same active space can also be used to describe the V state and the corresponding triplet. If we start to rotate the molecule around the double bond we would find that also the, so-called, Z state, the singlet state orthogonal to 8 becomes interesting (see below). However, the choice of active orbital becomes less straightforward when we study excited states. We need to assume something about the nature of the excited states and the orbitals which are important for a qualitatively correct description of the wave functions. Take ethene as an example again. The problem to compute an accurate wave function and energy for the V state is related to a near degeneracy between the valence excited state and an excitation to a 3d Rydberg orbital. There is a qualitative difference between Rydberg and valence excited states. A Rydberg state has one electron in a very diffuse orbital which interacts only weakly with the other electrons. Thus, the correlation energy decreases and resembles more that of the positive ion. For the ionic V state, the correlation energy is, on the other hand, increased because of the strong dynamic polarization effects, which are more prominent here than they are in the ground state where the electronic structure has a more covalent character. In order

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to balance the correlation effects, we therefore need very accurate wave functions where most of the correlation energy is included. In order to be able to describe both valence and Rydberg excited states we need to extend the active space with Rydberg orbitals. The first step is to extend the AO basis set with enough diffuse functions. How it is done will not be described in detail here. Instead we refer to a review that [19], where the choice of active orbitals for excited states of organic molecules are discussed in detail. Ideally, one would like to be able to solve Eq. (9) separately for each electronic state. It is in general not possible. Different electronic states can be close in energy and the corresponding energy surfaces may even cross. Separate optimization of two states, which are close in energy and of the same symmetry, could also yield wave functions, which are non-orthogonal. This is undesirable. The solution is state average optimization. Several roots are extracted from the secular equation. The average density matrices are constructed and a common set of orbitals is optimized. Such wave functions are per construction orthogonal to each other. The use of a common set of orbitals is normally not a problem. The wave functions will still be good reference functions, which describe the qualitative features of the electronic structure well. To summarize: the CASSCF method will determine a qualitatively correct wave function for each electronic state. This function can now be used as a reference function in the CI expansion 7. So, the next step would then be to solve this equation to obtain the final accurate description of the electronic states. 3.4. Dynamic correlation In the previous section we sketched a two step procedure for the calculation of wave functions for excited states. The first step consists of determining an optimal set of MOs together with a wave function, which describes the electronic structure of the excited state qualitatively correct. We saw that it could be achieved by extending the HF scheme to a full CI in a limited set of active orbitals, the CASSCF method. It should be noted that the method can, with a properly chosen active space, be used not only in the Frank-Condon region around the equilibrium geometry of the molecule, but also to map out full energy surfaces, describing chemical transformations, conical intersections and other curve crossings, etc. The HF method cannot be used in such situations. The second step consists of solving a large CI problem where the expansion comprises all CSFs that can be generated by single and double excitations in each of the reference configurations. We call this approach multi-reference CI (MRCI). It became feasible for large scale applications with the development of the direct CI methodology in the early 70ies [20,21]. It is probably the most accurate method available for general studies of excited states. Today it is possible to use more than 108 CSFs in the CI expansion. The most efficient implementation is probably the COLUMBUS code developed by H. Lischka and co-workers [22]. Even if modern MRCI programs can handle long CI expansions, it is easy to see that with a CASSCF reference function, the number of CSFs will easily reach 108, when the number of correlated electrons and the basis set increase. The calculations will then become quite timeconsuming and rather soon impossible. One way to reduce them is to use only a subset of the CAS configurations in the reference function, thereby introducing one more ambiguity. Another possibility, which has been used with some success, is to use the entire CASSCF wave function as the reference instead of the individual CSFs. This is the internally contracted form of the MRCI method [23,24].

328 The MRCI method has a deficiency: it is not a size-extensive method. To explain what this means, let us consider two non-interacting systems A and B with wave functions *PA and ^PB • The wave function for the total system is ^FAB = ^ A ^ B • Suppose now that the wave functions for the two sub-systems are of the MRCI type with singly and doubly excited CSFs from a given reference function (MR-SDCI). The total wave function VAB will then include up to quadruple excitations with respect to the product of the two reference functions. An MRSDCI calculation will not describe the total system with the same accuracy as the subsystems and the energy will not be the sum of the energies for A and B. It is not easy to correct for this deficiency of the MR-SDCI method. Several methods have been developed, which account approximately for the missing higher excitations, which are products of the double excitations, but none of them are completely satisfactory. The only completely efficient way out is to include the product terms in the formal expression for the wave function. This leads to the coupled cluster expansion, which however has so far only been developed into an effective computational method for a single determinant reference function. 3.5. Second order perturbation theory Another way to treat the dynamic correlation effects is to use perturbation theory. Such an approach has the virtue of being size-extensive and ought to be computationally more efficient than the MRCI approach. Moller-Plesset second order perturbation theory (MP2) has been used for a long time to treat electron correlation for ground states, where the reference function is a single determinant. It is known to give accurate results for structural and other properties of closed shell molecules. The approach was in the late 80ies extended to reference functions of the CAS type, the CASPT2 method [25,26]. The approach turned out to be very productive especially in molecular spectroscopy. A large number of applications has been made for organic molecules, transition metal complexes, and more recently also in heavy element chemistry [19,27,28]. A number of examples will be given in other chapters of this book. A CASPT2 calculation starts with a CASSCF calculation, which has been designed to include the electronic states of interest. Each of the CAS wave functions are then used as reference functions for a CASPT2 calculation. The first order wave function is generated as a sum over all single and double excitations with respect to the entire reference function. In this respect CASPT2 is similar to internally contracted CI, but the expansion coefficients, CM are only determined to first order and the energy to second order. They are determined by the equation: =-VOfl

(10)

Here, H{^ are matrix elements of a zeroth order Hamiltonian, which is chosen as a oneelectron operator in the spirit of MP2. Sf,v is an overlap matrix: the excited functions are not in general orthogonal to each other. Finally, VOu represents the interaction between the excited function and the CAS reference function. The difference between Eq. (10) and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth order Hamiltonian. In ordinary MP2 it is a simple sum of orbital energies. Here it is a complex expression involving matrix elements of a generalized Fock operator combined with up to fourth order density matrices of the CAS wave function. We do not give further details here but refer to the original papers If one assumes the generalized Fock matrix to be diagonal, one can formally write the second order energy in the same form as for MP2:

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(11) where K runs over a set of orthogonalized excited states and s are sums of generalized orbital energies. In ordinary MP2, the denominators are simply: £a+£b-s~£/, where a, b are virtual and i, j occupied orbitals. The CASPT2 denominators are more complex, but describes in principle the same energy difference. The CASPT2 approach has several advantages. It is computationally effective, as the many large scale applications have shown. It can use arbitrarily complex reference functions with maybe up to a million CSFs in the reference function. It is size extensive (a small deviation from size extensivity can occur with some choices of the active space, but the deviations are negligibly small). The generality of the reference functions makes it possible to treat, not only the Frank-Condon region but to follow excited states energy surfaces over energy barriers, through conical intersections, to dissociation of a chemical bond, etc. Other articles in the book will describe such situations in more detail. The accuracy of the excitation energies is high with errors generally smaller than about 0.2 eV. There are, however, some drawbacks with the method, which can sometimes cause problems. One is the limitation of the active space. It is today difficult to perform calculations with more than about one million CSFs in the reference function. This can in some cases make accurate calculations impossible because of requirements on the size of the active space that yields larger sizes of the CAS CI space. Another problem is caused by the, so called, intruder states. The denominator in ordinary MP2 will rarely become small because there is usually a large HOMO-LUMO gap in the orbital energies. It is not always the case in CASPT2 because the active orbitals have energies that are intermediate to the energies of the doubly occupied and virtual orbitals. A low order perturbation approach will become invalid when the denominators become small in comparison to the numerators in Eq. (11). The cause of such behavior is usually a too small active space, but it is not always possible to extend the space further. A level shifting technique has been developed that will remove the intruder state [29,30], but it should only be used for weak intruder states that will have only a minor effect on the computed correlation energy. The CASPT2 method also has a small systematic error, which is due to the definition of the zeroth order Hamiltonian. States with a closed shell character are favored in relation to open shell structures. This leads to too small excitation and dissociation energies in some cases. A modification of the zeroth order Hamiltonian that reduces the error considerably has recently been suggested [31]. The CASPT2 method relies on a well defined CAS reference function, which will not be strongly affected by the addition of dynamic correlation. Normally, this model is satisfactory, but it may happen in more complex cases that several states of the same symmetry appear close in energy. They may then interact strongly with large changes occurring in the reference functions. Typical situations are areas of avoided crossings, conical intersections, etc. A multi-state version of CASPT2 has been developed to handles such situations. This is multistate (MS) CASPT2 [32]. Here an effective Hamiltonian is constructed for a set of CAS reference functions that has the normal CASPT2 energies in the diagonal and includes interaction between the first order wave functions in the off-diagonal elements. A detailed

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0.0

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HCH Rotation angle Fig.4. Weights of the CSFs (n) (upper line) and (71*) for the N-state of ethene as a function of the rotation angle.

discussion of the possibilities and limitations of the MS-CASPT2 method is given by Merchan and Serrano in another chapter of this book [33]. 3.6. The ethene molecule As an example of the importance of a multiconfigurational treatment in photochemistry we consider a very simple but illustrative and important example: the rotation of one CH2 group around the double bond in the ethene molecule, C2H4. The calculations on which the results are based have been made with a small DZP basis set. Also, the only parameter that has been varied is the rotation angle. All other geometry parameters have been kept at the values of the equilibrium geometry, which is a serious approximation, in particular for the CC bond. We should therefore not pay much attention to the actual numbers, but merely to the qualitative features. We construct CASSCF wave functions for all states, which can be formed by occupying the two 71-orbitals, n and n* in all possible ways. This gives rise to four electronic states: The N-state, dominated by the CSF (nf The V-state, a singlet state with the configuration: (TC)(TT*) The T-state, a triplet state with the configuration: (TT)(7T*) The Z-state, dominated by the CSF (n*f

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N-state Z-state V-state T-state

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Fig. 5. The total energy of the N-, T-, V-, and Z-states as a function of the rotation angle

The two CSFs (TI)2 and (TT*)2 belong to the same symmetry and can thus mix, so both states will have some multiconfigurational character. The question is how it varies with the rotational angle. We show in Fig. 4 how the weights of the two CSFs vary with the rotation angle for the Nstate. The corresponding picture for the V-state is the mirror image in this simple model. For planar ethene there is not much mixing. The 7i*-orbital is high in energy, so the N-state will be quite HF like with one dominating CSF. But, when we start rotating one of the HCH groups around the double bond the energy drops and finally, at 90°, the two orbitals it and TI* become degenerate. The n bond is now completely dissociated. Thus, the two CSFs have the same energy and there will be complete mixing, which is typical of a dissociation process. The variation of the (CASPT2) energies with the angle is shown in Fig. 5. The large spread in energies at zero angle is decreased substantially at 90° and the four states become pairwise degenerate. The lower pair correspond to a singlet (S) and triplet (T) coupled biradical system. The upper pair of states are ionic with the two electrons resonating between the two centers. The description is oversimplified here. In reality, Rydberg like states in the same energy region interact strongly with the upper states and change the nature of the wave functions in the FC region. The two degenerate ionic states are split by a Jahn-Teller distortion. A pyramidalization of one HCH group will stabilize one of the states further. At the same time, the ground state is further destabilized and eventually a conical intersection is reached, which makes it possible to move from this surface back to the ground state, (a more detailed discussion of the subject can be found in Ref. [34]).

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One might think that the above example is rather special and of little interest in photochemistry. On the contrary. This simple model is the key for the understanding of a large number of important photochemical processes in chemistry and biochemistry. Rotation around a double bond is a much used process in nature. The most striking example is probably the cis-trans isomerization of the retinal molecule in rhodopsin, the key molecular process in vision. A number of other examples could be given, for which we refer to other chapters in this book. We emphasize that these typical problems in photochemistry can only be dealt with using a multiconfigurational quantum chemical methods like CASSCF/CASPT2 or CASSCF/MRC1. With this example we finish the overview of the state specific methods. We shall next briefly discuss the linear response methods, where the focus is on the ground state wave function and the excitations are obtained through the response of the system to an electromagnetic field. 4. LINEAR RESPONSE THEORY An alternative way to treat excitation processes is to study the response of the ground state electronic structure perturbed by an electromagnetic field. The quality of such a method depends crucially on the quality of the ground state wave function. The first attempts along this line was based on Hartree-Fock theory and was named the Random Phase Approximation (RPA) [35]. The method is not much used today. Recent developments use more accurate wave functions and the two most popular approaches are based on Density Functional Theory (DFT) or the Coupled Cluster (CC) method. Because both these approaches assume the ground state to be single configurational HF like, the main area of application will be studies of electronic spectra in the FC region. It is not advisable to use LR methods far away from equilibrium where the ground state may be strongly multiconfigurational and doubly excited states are low in energy. The second limitation inherent in the methodology is that only excited states, which are dominated by singly excited configurations are included. LR theory cannot be use to study rotation around a double bond as in the ethene example discussed above. LR theory starts with the ground state wave function. A perturbation in the form of a time dependent electric field is applied: E(/) = E^cosiat)

(12)

The first order response function is then computed and the excitation energies are found as poles of this function while the transition moments are obtained from the residues. We shall not attempt to give a full account of the theory here but refer to the original papers. The method was first developed in CC theory by Monkhorst [36] and was extended into an effective computational tool by the Aarhus school (see Ref. [37] and further references therein). The CC-LR method is capable of giving a very accurate account of the excited states that are dominated by single excitations. It is available today in an integral direct implementation, which makes calculations with many basis functions possible. For HF like ground states and excitations dominated by singly excited CSFs it is probably the most accurate method available today. Due do its limitations it is, however, not useful for studies of photochemical processes. Linear response is in DFT a simple extension of the RPA approach. For a full account of LR in DFT we refer to the review by M. Casida where more references may be found [38]. The approach is available in most programs and is very simple to use. It gives a quick overview of

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all singly excited states, provided that an adequate basis set is used. One needs, however, to remember the limitations. We shall illustrate the performance of both LR-DFT and LRCC in the next section. A detailed and critical evaluation of the approach is given in the chapter on Ab Initio methods by Merchan and Serrano-Andres [33]. 5. THE BENZENE MOLECULE Let us now for a while leave the theory and relax with an example. We choose the organic molecule, which has through the years been the prototype for testing methods for excited states, benzene. At the same time it will be a recapitulation of the history. In our historical review, we discussed the calculations of Goeppert-Mayer and Sklar in 1938 and the use of benzene for the testing of the PPP method in the 50ies. It is not surprising that these early successes would be a challenge for ab initio quantum chemistry when it became available for studies of excited states in the late 60ies. The first attempts were made using small CI expansions, involving only the Tt-orbitals. The first calculation was performed by R. Buenker et al. [39]. It was a (by today's standard) a small CI calculation involving only the six 7i-orbitals. The results are shown in the first column of Fig. 4 (BWP). The results were not very impressive for obvious reasons. Actually, the calculations resemble the PPP calculation by Pariser from 1952, but now without any empirical correction of the integrals. Thus there is no account for the dynamic polarization of the o-electrons, which is reflected in the errors, which are especially large for the more ionic states 'Biu and ' E i u . Some improvement was obtained in the calculation by Peyerimhoff and Buenker (PB in Fig. 7) by extending the basis set and increasing the number of 7i-orbitals from six to nine [40], but the errors are still large. Hay and Shavitt (HS) performed a calculation including 23 7t-orbitals in 1974 [41], which we might consider as limit of what can be obtained without explicit inclusion of the a-electrons. The errors are still larger than one eV for some of the states. One should remember that the largest error in the PPP calculation of Pariser was 0.5 eV for the singlet states and 0.9 eV for the triplets and later developments improved the results further.

Fig. 6. The benzene molecule.

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eV 11 -,

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8

•

BWP

PB

HS

R

MRM

Expt.

Fig. 7. Early CI results for the electronic spectrum of benzene. For the references, see the text. So, it was a hard task to improve on the semi-empirical methods. The first CI calculation that included the a-electrons was performed in 1976 by Rancurel et al. [42] in a minimal AO basis set (R in Fig. 7). The results did not improve, which is due to the too small basis set used, which did not allow for any orbital relaxation. A much larger CAS-CI calculation was performed by Matos et al. in 1987 [43]. It also included corrections for the size-consistency error. It was the largest Cl calculation performed on the benzene spectrum before 1990. As can be seen in Fig. 7, the errors in computed excitation energies are still sizable (more than 0,5 eV in some cases), but the quality is now at least comparable to that obtained with the PPP method 30 years earlier. The more recent developments of linear-response theory (both in DFT and CC theory) and the CASSCF/CASPT2 approaches have of course been tested on benzene. The first

335 CASSCF/CASPT2 calculation was performed in 1992 [44] and a more complete treatment of the vertical spectrum (also including the Rydberg states) was made in 1995 [45]. The results of these calculations are shown in column 2 of Fig. 5 (labeled CASPT2). There is now good agreement with experiments with errors of only 0.1-0.2 eV. One notices that most CASPT2 energies are slightly on the low side, reflecting the systematic error of the method. The CASPT2 results are in Fig. 8 surrounded by two results obtained using linear response theory. The first column shows the results of a TD-DFT calculation [46]. Again, we see improvement compared to the older CI results, but errors are as large as 0.5 eV for some states. Nevertheless, this very cheap method is able to recover most of the dynamic

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•1B. 3

E