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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This 3rd edition has been expanded and updated to account for recent developments, while new illustrative examples as well as an enlarged reference list have also been added. It naturally retains the successful concept of its predecessors in presenting a unified perspective on molecular charge and energy transfer processes, thus bridging the regimes of coherent and dissipative dynamics, and establishing a connection between classic rate theories and modern treatments of ultrafast phenomena. Among the new topics are: - Time-dependent density functional theory - Heterogeneous electron transfer, e.g. between molecules and metal or semiconductor surfaces - Current flows through a single molecule. While serving as an introduction for graduate students and researchers, this is equally must-have reading for theoreticians and experimentalists, as well as an aid to interpreting experimental data and accessing the original literature.
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Seitenzahl: 964
Veröffentlichungsjahr: 2011
Ähnliche
Contents
Cover
Related Titles
Title page
Copyright page
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Chapter 1: Introduction
Chapter 2: Electronic and Vibrational Molecular States
2.1 Introduction
2.2 Molecular Schrödinger Equation
2.3 Born–Oppenheimer Separation
2.4 Electronic Structure Methods
2.5 Condensed Phase Approaches
2.6 Potential Energy Surfaces
2.7 Diabatic versus Adiabatic Representation of the Molecular Hamiltonian
2.8 Supplement
References
Further Reading
Chapter 3: Dynamics of Isolated and Open Quantum Systems
3.1 Introduction
3.2 Time-Dependent Schrödinger Equation
3.3 The Golden Rule of Quantum Mechanics
3.4 The Nonequilibrium Statistical Operator and the Density Matrix
3.5 The Reduced Density Operator and the Reduced Density Matrix
3.6 The Reservoir Correlation Function
3.7 Quantum Master Equation
3.8 Reduced Density Matrix in Energy Representation
3.9 Generalized Rate Equations: The Liouville Space Approach
3.10 The Path Integral Representation of the Density Matrix
3.11 Quantum-Classical Hybrid Methods
3.12 Supplement
References
Further Reading
Chapter 4: Interaction of Molecular Systems with Radiation Fields
4.1 Introduction
4.2 Absorption and Emission of Light
4.3 Nonlinear Optical Response
4.4 Laser Control of Molecular Dynamics
References
Further Reading
Chapter 5: Vibrational Dynamics: Energy Redistribution, Relaxation, and Dephasing
5.1 Introduction
5.2 Intramolecular Vibrational Energy Redistribution
5.3 Intermolecular Vibrational Energy Relaxation
5.4 Polyatomic Molecules in Solution
5.5 Quantum-Classical Approaches to Relaxation and Dephasing
5.6 Supplement
References
Further Reading
Chapter 6: Intramolecular Electronic Transitions
6.1 Introduction
6.2 The Optical Absorption Coefficient
6.3 Absorption Coefficient and Dipole–Dipole Correlation Function
6.4 The Emission Spectrum
6.5 Optical Preparation of an Excited Electronic State
6.6 Pump–Probe Spectroscopy
6.7 Internal Conversion Dynamics
6.8 Supplement
References
Further Reading
Chapter 7: Electron Transfer
7.1 Classification of Electron Transfer Reactions
7.2 Theoretical Models for Electron Transfer Systems
7.3 Regimes of Electron Transfer
7.4 Nonadiabatic Electron Transfer in a Donor–Acceptor Complex
7.5 Nonadiabatic Electron Transfer in Polar Solvents
7.6 Bridge-Mediated Electron Transfer
7.7 Nonequilibrium Quantum Statistical Description of Electron Transfer
7.8 Heterogeneous Electron Transfer
7.9 Charge Transmission through Single Molecules
7.10 Photoinduced Ultrafast Electron Transfer
7.11 Controlling Photoinduced Electron Transfer
7.12 Supplement
References
Further Reading
Chapter 8: Proton Transfer
8.1 Introduction
8.2 Proton Transfer Hamiltonian
8.3 Adiabatic Proton Transfer
8.4 Nonadiabatic Proton Transfer
8.5 The Intermediate Regime: From Quantum to Quantum-Classical Hybrid Methods
8.6 Infrared Laser–Pulse Control of Proton Transfer
References
Further Reading
Chapter 9: Excitation Energy Transfer
9.1 Introduction
9.2 The Aggregate Hamiltonian
9.3 Exciton–Vibrational Interaction
9.4 Regimes of Excitation Energy Transfer
9.5 Transfer Dynamics in the Case of Weak Excitonic Coupling: Förster Theory
9.6 Transfer Dynamics in the Case of Strong Excitonic Coupling
9.7 The Aggregate Absorption Coefficient
9.8 Excitation Energy Transfer Including Charge Transfer States
9.9 Exciton–Exciton Annihilation
9.10 Supplement
References
Further Reading
Index
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The Author
Dr. Volkhard MayHumboldt-Universität zu BerlinInstitut für PhysikNewtonstraße 1512489 Berlinmay@physik.hu-berlin.de
Prof. Dr. Oliver KühnUniversität RostockInstitut für PhysikUniversitätsplatz 318055 Rostock
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Cover Design Adam Design, Weinheim
ISBN 978-3-527-40732-3
Preface to the Third Edition
The continued interest in our book since its first publication in 2000 and its second edition in 2004 triggered the idea to prepare a third edition in order to account for more recent developments in the field of molecular charge and energy transfer research.
Following the concept of the previous editions, we start by providing some general background on gas and condensed phase interaction potentials and Hamiltonians, now including a discussion of quantum mechanics/molecular mechanics hybrid methods for the explicit treatment of condensed phase environments. The methodologically oriented Chapter 8 on the dynamics of quantum systems has been extended in several respects. Most notable is an exposition on the calculation of transfer rates within the Liouville space approach. Here, special emphasis is put on the fourth-order rates that are crucial for an understanding of the more involved electron and excitation energy transfer processes. Further, we give an account on the multiconfiguration time-dependent Hartree method that in recent years has been proven to be a versatile tool for the numerically exact treatment of the quantum dynamics of thousands of degrees of freedom.
The introduction to basic theoretical concepts has been expanded by a new Chapter 4 devoted to some general aspects of the interaction between light and molecular systems. This comprises a derivation of the interaction Hamiltonian in dipole approximation, an introduction to field quantization for the description of emissions, and an outline of the basics of linear and nonlinear spectroscopy. In the second edition, laser control was covered in a separate chapter. For the present edition we have incorporated a discussion of the topic into the text, which reflects the development of laser control into an almost routine tool for the investigation of molecular dynamics phenomena. The theoretical foundations and, in particular, optimal control theory are now part of Chapter 4, while the various applications are covered in Chapters 7–9.
The applications start with Chapter 5, where the discussion of vibrational dynamics has been expanded to include quantum-classical approaches to the calculation of pure dephasing induced line broadening. Chapter 6, focusing on intramolecular electronic transitions, now contains a section on pump–probe spectroscopy and its role in interrogating molecular dynamics in the condensed phase. Major changes in Chapter 7 on electron transfer include the incorporation of heterogenous electron transfer at surfaces as well as of single-molecule electron transfer in the context of molecular electronics. The quantum dynamics treatment of proton transfer reactions has flourished recently due to the development of time-dependent multiconfiguration approaches, as mentioned above; an example is discussed in Chapter 8. Finally, Chapter 9, on excitation energy (Frenkel exciton) transfer, has been substantially rewritten. Topics that have been added include Dexter transfer and two-electron-assisted as well as photon-mediated exciton transfer.
The “Suggested Reading” section of the previous editions, which served to give a systematic starting point to explore the original literature, has been merged into the main text to become a list of “Further Reading” suggestions at the end of each chapter. As before, we would like to emphasize that these lists are by no means exhaustive, that is, it is not the purpose of this book to review all relevant literature on the title subject.
While working on the manuscript of this third edition we enjoyed the inspiring atmosphere of the Berlin Collaborative Research Center (Sfb450) “Analysis and Control of Ultrafast Photoinduced Reactions” and the Rostock Sfb652 “Strong Correlations and Collective Effects in Radiation Fields.”
Finally, we wish to thank E. Petrov (Bogolyubov Institute for Theoretical Physics, Kiev) and L. Wang (University of Science and Technology, Beijing) for reading parts of the new manuscript and K. Mishima (University of Tokyo) for drawing our attention to corrections required for the second edition.
Berlin and Rostock, October 2010
Volkhard May and Oliver Kühn
Preface to the Second Edition
The positive response to the First Edition of this text has encouraged us to prepare the present Revised and Enlarged Second Edition. All chapters have been expanded to include new examples and figures, but also to cover more recent developments in the field. The reader of the First Edition will notice that many of the topics which were addressed in its “Concluding Remarks” section have now been integrated into the different chapters.
The introduction to dissipative quantum dynamics in Chapter 3 now gives a broader view on the subject. Particularly, we elaborated on the discussion of hybrid quantum-classical techniques which promise to be able to incorporate microscopic information about the interaction of some quantum system with a classical bath beyond the weak coupling lirnit. In Chapter 4 we give a brief account on the state-space approach to intramolecular vibrational energy and the models for treating the intermediate time scale dynamics, where the decay of the survival probability is nonexponential. Chapter 5 now compares different methodologies to compute the linear absorption spectrum of a molecule in a condensed phase environment. Furthermore, basic aspects of nonlinear optical spectroscopy have been included to characterize a primary tool for the experimental investigation of molecular transfer processes. Bridge-mediated electron transfer is now described in detail in Chapter 6 including also a number of new examples. Chapter 7 on proton transfer has been supplemented by a discussion of the tunneling splitting and its modification due to the strong coupling between the proton transfer coordinate and other intramolecular vibrational modes. Chapter 8 dealing with exciton dynamics has been considerably rearranged and includes now a discussion of two-exciton states.
Finally, we have added a new Chapter 9 which introduces some of the fundamental concepts of laser field control of transfer processes. This is a rapidly developing field which is stimulated mostly by the possibility to generate ultrafast laser pulse of almost any shape and spectral content. Although there are only few studies on molecular transfer processes so far, this research field has an enormous potential not only for a more detailed investigation of the dynamics but also with respect to applications, for instance, in molecular based electronics.
Following the lines of the First Edition we avoided to make extensive use of abbreviations. Nevertheless, the following abbreviations are occasionally used: DOF (degrees of freedom), ET (electron transfer), IVR (intramolecular vibrational redistribution), PES (potential energy surface), PT (proton transfer), QME (quantum master equation), RDM (reduced density matrix), RDO (reduced density operator), VER (vibrational energy relaxation) and XT (exciton transfer).
We have also expanded the “Suggested Reading” section which should give a systematic starting point to explore the original literature, but also to become familiar with alternative views on the topics. Additionally, at the end of each chapter, the reader will find a brief list of references. Here, we included the information about the sources of the given examples and refer to the origin of those fundamental concepts and theoretical approaches which have been directly integrated into the text. We would like to emphasize, however, that these lists are by no means exhaustive. In fact, given the broad scope of this text, a complete list of references would have expanded the book’s volume enormously, without necessarily serving its envisaged purpose.
It is our pleasure to express sincere thanks to the colleagues and students N. Boeijenga, B. Brüggemann, A. Kaiser, J. Manz, E. Petrov, and B. Schmidt, which read different parts of the manuscript and made various suggestions for an improvement. While working on the manuscript of this Second Edition we enjoyed the inspiring atmosphere, many seminars, and colloquia held within the framework of the Berlin Collaborative Research Center (Sfb450) “Analysis and Control of Ultrafast Photoinduced Reactions”. This contributed essentially to our understanding of charge and energy transfer phenomena in molecular Systems. Finally, we would like to acknowledge financial support from the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie (O.K.).
Berlin, September 2003
Volkhard May and Oliver Kühn
Preface to the First Edition
The investigation of the stationary and dynamical properties of molecular systems has a long history extending over the whole century. Considering the last decade only, one observes two tendencies: First, it became possible to study molecules on their natural scales, that is, with a spatial resolution of some Ångström (10−10 m) and on a time scale down to some femtoseconds (10−15 s). And second, one is able to detect and to manipulate the properties of single molecules. This Progress Comes along with a steadily growing number of theoretical and experimental efforts crossing the traditional borderlines between chemistry, biology, and physics. In particular the study of molecular transfer processes involving the motion of electrons, Protons, small molecules, and intramolecular excitation energy, resulted in a deeper understanding of such diverse phenomena as the photoinduced dynamics in large molecules showing vibrational energy redistribution or conformational changes, the catalysis at surfaces, and the microscopic mechanisms of charge and energy transfer in biological systems. The latter are of considerable importance for unraveling the functionality of proteins and all related processes like the primary steps of photosynthesis, the enzymatic activity, or the details of the repair mechanisms in DNA strands, to mention just a few exarnples. In a more general context also molecular electronics, that is, the Storage and processing of information in molecular structures on a nanometer length scale, has triggered enormous efforts. Finally, with the increasing sophistication of laser sources, first steps towards the control of chemical reaction dynamics have been taken.
The ever growing precision of the experiments requires on the theoretical side to have microscopic models for simulating the measured data. For example, the interpretation of optical spectroscopies in a time region of some tenths of femtoseconds, demands for an appropriate simulation of the molecular dynamics for the considered System. Or, understanding the characteristics of the current flowing through a single molecule in the context of scanning tunneling microscopy, needs detailed knowledge of the electronic level smcture of the molecule as well as of the role of its vibrational degrees of freedom. These few example already demonstrate, that advanced theoretical concepts and numerical simulation techniques are required, which are the combination of methods known from general quantum mechanics, quantum chemistry, molecular reaction dynamics, solid state theory, nonlinear optics, and nonequilibrium statistical physics.
Such a broad approach is usually beyond the theoretical education of chemists and biologists. On the other hand, quantum chemistry and chemical reaction dynamics are quite often not on the curriculum of physics students. We believe that this discrepancy quite naturally does not facilitate communication between scientists having different backgrounds. Therefore it is one of the main intentions of the present book to provide a common language for bridging this gap.
The book starts with an introduction and general overview about different concepts in Chapter 1. The essentials of theoretical chemical physics are then covered in Chapter 2. For the chemistry student this will be mostly a repetition of quantum chemistry and in particular the theory of electronic and vibrational spectra. It is by no means a complete introduction into this subject, but intended to provide some background mainly for physics students. The prerequisites from theoretical physics for the description of dynamical phenomena in molecular systems are presented in Chapter 3. Here we give a detailed discussion of some general aspects of the dynamics in Open and closed quantum systems, focusing on transfer processes in the condensed phase.
The combination of qualitative arguments, simple rate equations, and the powerful formalism of the reduced statistical Operator constitutes the backbone of the second part of the book. We start in Chapter 4 with a discussion of intramolecular transfer of vibrational energy which takes place in a given adiabatic electronic state. Here we Cover the limits of isolated large polyatomic molecules, small molecules in a matrix environment, up to polyatomics in solution. In Chapter 5 we then turn to processes which involve a transition between different electronic states. Special emphasis is put on the discussion of optical absorption, which is considered to be a reference example for more involved electron-vibrational transfer phenomena such as internal conversion which is also presented in this chapter. Chapter 6 then outlines the theoretical frame of electron transfer reactions focusing mainly on intramolecular processes. Here, we will develop the well-known Marcus theory of electron transfer, describe nuclear tunneling and superexchange electron transfer, and discuss the influence of polar solvents. In Chapter 7 it will be shown that, even though Proton transfer has many unique aspects, it can be described by adapting various concepts from electron transfer theory. The intermolecular excitation energy transfer in molecular aggregates is considered in Chapter 8. In particular the motion of Frenkel excitons coupled to vibrational modes of the aggregate will be discussed. In the limit of ordinary rate equations this leads us to the well-known Förster expression for the transfer rate in terms of emission and absorption characteristics of the donor and acceptor molecules, respectively.
By presenting a variety of theoretical models which exist for different types of transfer processes on a common formal background, we hope that the underlying fundamental concepts are becoming visible. This insight may prepare the reader to take up one of the many challenging problems provided by this fascinating field of research. Some personal reflections on current and possible future developments are given in Chapter 9.
The idea for writing this book emerged from lectures given by the authors at the Humboldt University Berlin, the Free University Berlin, and at the Johannes Gutenberg University Mainz during the last decade. These Courses have been addressed to theoretically and experimentally oriented undergraduate and graduate students of Molecular Physics, Theoretical Chemistry, Physical Chemistry, and Biophysics, being interested in the fast developing field of transfer phenomena. The book is self-contained and includes detailed derivations of the most important results. However, the reader is expected to be familiar with basic quantum mechanics. Most of the chapters contain a supplementary part where more involved derivations as well as special topics are presented. At the end of the main text we also give some comments on selected literature which should complement the study of this book.
Of course this book would not have been possible without the help, the critical comments, and the fruitful discussions with many students and colleagues. In this respect it is a pleasure for us to thank I. Barvik, N.P. Ernsting, W. Gans, L. González, O. Linden, H. Naundorf, J. Manz, S. Mukamel, A.E. Orel, T. Pullerits, R. Scheller, and D. Schirrmeister. We also are grateful for continuous financial Support which has been provided by the Deutsche Forschungsgemeinschaft, in particular through the Sonderforschungsbereich 450 “Analysis and Control of Ultrafast Photoinduced Reactions”.
Berlin, September 1999
Volkhard May and Oliver Kühn
Chapter 1
Introduction
The understanding of transfer phenomena in molecular systems calls for a unified theoretical treatment that should have its foundation in a microscopic definition of the constituent parts and their interactions. There are three questions that need to be answered in this regard. First, what is the appropriate theoretical description of the molecular system? Second, what is the form of the dynamical equations that describe the transfer process? And third, how can the computed results be related to experimental observations?
In what follows the term “molecular system” shall cover single molecules and simple molecular aggregates, but also larger arrangements of molecules like supramolecular complexes. In particular, molecules embedded in different types of environments will be of interest. The definition even encompasses biological macromolecules such as membrane-bound protein complexes. The common link between these molecular systems is that they show transfer processes. By “transfer process” we understand the flow of vibrational energy, the dynamics of electrons, protons, and electronic excitation energy.
From a general point of view, quantum mechanics gives the framework for all phenomena occurring in molecular systems. Given the broad scope of transfer processes to be discussed, it is clear that an exact quantum mechanical treatment is impossible if we go beyond the level of simple model systems. Therefore, it is a particular challenge for theory to develop versatile models that provide answers to the initially raised three questions.
Chapter 2 addresses the first question discussing the steps that lead us from the formally exact to some approximate molecular Hamilton operator. Given a molecule in gas phase (vacuum) as shown in the upper part of Figure 1.1, the Born–Oppenheimer separation of nuclear and electronic motions can be performed. Here, the molecular wave function is split up into an electronic and a nuclear part, a procedure that is justified by the large mass difference between both types of particles. This results in a Schrödinger equation for the electronic wave function alone, for given fixed positions of the nuclei. Calculating the electronic energy spectrum for different positions of the nuclei one obtains potential energy surfaces that govern the motion of the nuclei. These potential energy surfaces are at the heart of the understanding of stationary molecular spectra and molecular dynamics. If nuclear motion and electronic motion are adiabatically separable, that is, if the coupling between different electronic states is negligible, one can carry out the Born–Oppenheimer approximation. Under certain conditions, however, so-called nonadiabatic transitions between different electronic states as a consequence of the nuclear motions are to be expected.
Figure 1.1 The problem of the interaction between electrons and nuclei is transformed to some tractable level by employing the Born–Oppenheimer separation of their motions. Upper panel: Three-atomic molecule with nuclei labeled A, B, and C. The electronic wave function is indicated by a gray area. Middle panel: The bond length between atom A and B (left) as well as atom B and C (right) is increased accompanied by an instantaneous deformation of the electronic wave function. As a result, a potential energy curve is formed determining the dynamics of the bond length coordinate. Lower panel: If the molecule is taken from the gas into the condensed phase, its stationary properties cannot be calculated without invoking further approximations.
If we move from the gas to the condensed phase, for example, by considering a molecule in solution as shown in the lower part of Figure 1.1, the effect of the molecule–environment interaction has to be taken into account. The simplest way to do this is to add an additional external potential to the molecular Hamiltonian. Often the environment can be described as a macroscopic dielectric and its influence can be judged from its dielectric properties.
Having discussed the stationary molecular properties we turn in Chapter 3 to the second question related to molecular dynamics. Here, the reader will become familiar with concepts ranging from incoherent to coherent transfer events. The connection between these limits is provided by the relevant time scales; of particular importance is the relation between intramolecular relaxation and intermolecular transfer times. In view of experimental advances in ultrafast optical spectroscopy, our treatment reflects the historical evolution of knowledge about molecular dynamics from simple transfer rates to quantum mechanical wave packet dynamics.
An important ingredient for the theoretical modeling is the concept of an open molecular systemS interacting with its environment (reservoir) R by collision processes or via other means of energy exchange. A schematic illustration of this situation is given in Figure 1.2. The relevant system S may represent any type of molecule, but it may also comprise selected so-called active degrees of freedom of a particular molecule.
Figure 1.2 Open molecular system S interacting with its environment (reservoir) R. In addition, the system may be influenced by external fields (wiggly line).
The most general description of the total system, S plus R, is given by the quantum statistical operator , as indicated in the left-hand part of Figure 1.3. This operator is based on the concept of a mixed quantum state formed by S and its macroscopic environment. However, the operator contains much more information than will ever be needed, for instance, to simulate a particular experiment. Indeed, it is the relevant system S we are interested in. Making use of a reduction procedure we obtain a reduced statistical operator that contains the information on the dynamics of S only, but including the influence of the environment R (right-hand part of Figure 1.3). When deriving equations of motion for the reduced statistical operator, the so-called quantum master equations, a number of approximations have to be invoked. Most fundamental in this respect will be the assumption of a weak interaction between the system S and the reservoir R, which in practice requires a proper separation into relevant and environmental coordinates for the molecular system at hand. If there is no interaction at all, the quantum master equation would be equivalent to the time-dependent Schrödinger equation. This is the regime of coherent dynamics. If the interaction is not negligible, however, the system dynamics gradually changes with increasing coupling strength from a partially coherent one to an incoherent one. The incoherent motion of a quantum system is commonly described using ordinary rate equations that are based on the Golden Rule rate expression of quantum mechanics.
Figure 1.3 The total system S + R is completely described by the quantum-statistical operator . By means of a reduction procedure one can focus on the relevant system using the reduced statistical operator .
The concept of the statistical operator provides a quantum-statistical description of S and R. However, in many situations it is sufficient to describe R by means of classical mechanics. Then, S can be characterized by a wave function Ψ, and the dynamics of the environmental degrees of freedom is governed by Newton’s equations. Often the dynamics is split up in such a way that the classical particles move in the mean field of the quantum particle. This situation is visualized in Figure 1.4.
Figure 1.4 Mixed quantum-classical description of condensed phase dynamics. The classical particles move in the mean field generated by the quantum particle described by the wave function Ψ.
The overwhelming amount of data on transfer processes in molecular systems is obtained by spectroscopic techniques working in the infrared, the visible, and, more recently, also in the ultraviolet region and beyond. Therefore, we will discuss the third question related to experimental observation mostly in the context of optical spectroscopy. As a means of preparation, Chapter 4 gives a brief account on general theoretical concepts of the interaction of molecular systems with the electromagnetic radiation field. A successful analysis of molecular transfer processes triggers the desire to take active control of the dynamics. For example, it would be intriguing to have a means for depositing energy into specific bonds or reaction coordinates such as to dissociate a polyatomic molecule into desired products. Theoretical approaches to the active control of transfer processes will be discussed in the second part of Chapter 4.
The general concepts presented in Chapters 2–4 are then applied to describe different transfer phenomena. In principle, transfer processes can be classified according to the type of transferred particle. In addition, one can distinguish between intramolecular and intermolecular particle transfer. The common frame is provided by the molecular Schrödinger equation together with the Born–Oppenheimer separation of electronic and nuclear motions as mentioned above.
The coupled nuclear dynamics in polyatomic molecules that might be immersed in some condensed phase environment is treated in Chapter 5. We will show how an initially prepared vibrational state decays while its excitation energy is distributed over all possible environmental modes, as illustrated in the left-hand part of Figure 1.5. For small polyatomic molecules the energy flow out of the initial state is called intramolecular vibrational energy redistribution. For condensed phase situations the dissipation of energy into the environment is called vibrational energy relaxation. In both cases the transferred objects are the quanta of vibrational energy.
Figure 1.5 Scheme of molecular potential energy surfaces including the levels of the quantized motion of some reaction coordinate. After optical preparation of an electronically and vibrationally excited initial state (absorption), different transfer processes can occur. If the electronic state is not changed, but there is a coupling to some manifold of vibrational states, intramolecular vibrational energy redistribution (IVR) or vibrational energy relaxation (VER) can be observed. If there is some coupling to another electronic state, intramolecular internal conversion (IC) or electron transfer (ET) takes place. At the same time, one has VER as indicated by the wiggly lines. In addition the system may return to the ground state by emitting a photon.
The preparation of the initial state can be due to an optical transition between two electronic states as a consequence of the interaction between the molecular system and an external electromagnetic field (cf. Figure 1.5). In Chapter 6 we will discuss the processes of photon absorption and emission sketched in Figure 1.5. It will be shown that the coupled electron–vibrational dynamics responsible for the absorption line shape can be described by a combined density of states that is the Fourier transform of some correlation function. This theoretical result will turn out to be quite general. In particular we will show that different types of transfer processes can be accommodated in such a framework. For example, the internal conversion dynamics of nonadiabatically coupled electronic states (right-hand part of Figure 1.5) can, in the incoherent limit, be described by a combined density of states.
The external field interaction, on the other hand, provides the means for preparing nonequilibrium initial states that can act as a donor in a photoinduced electron transfer reaction, which is discussed in Chapter 7. The concerted electron–vibrational dynamics accompanying electron transfer reactions can often be modeled in the so-called curve-crossing picture of two coupled potential energy surfaces representing two electronic states along a reaction coordinate (right-hand part of Figure 1.5). Generalizations of this picture to larger molecular systems and to the case where the molecule is in contact with metal electrodes and a voltage is applied will also be discussed.
In contrast, the proton or hydrogen atom transfer investigated in Chapter 8 usually does not involve electronic transitions. In Figure 1.6 we have sketched a typical situation for intramolecular proton transfer that is realized as an isomerization reaction in the adiabatic electronic ground state. Since the proton has a rather small mass, tunneling processes may play an important role for proton transfer. The small mass ratio between the proton and the other heavy atoms provides the background for the introduction of a second Born–Oppenheimer separation. This will enable us to adapt most of the concepts of electron transfer theory to the case of proton transfer.
Figure 1.6 Hydrogen bonding, which governs the proton transfer (PT) dynamics, often leads to a double minimum potential along a reaction coordinate. The interaction between the proton and some environment may cause vibrational relaxation (wiggly lines).
In Chapter 9 we discuss excitation energy transfer or so-called exciton transfer in molecular aggregates as another example of coupled electron–vibrational motion. In Figure 1.7 the mechanism of excitation energy transfer in the limit of localized excitations is shown. The donor (left) is initially excited, for example, by an external field. As a consequence of the Coulomb interaction between the excited molecule and surrounding molecules, excitation energy is transferred to some acceptor (right). Due to the large spatial separation, donor and acceptor are usually described by different sets of nuclear (reaction) coordinates. The process can formally be understood in a picture where the donor emits radiation energy that is in turn absorbed by the acceptor. If the Coulomb interaction between different molecules becomes large enough, then excitation energy transfer has to be discussed by introducing quantum mechanical superposition states of all excited molecules, the so-called Frenkel excitons. Their introduction gives a new view on excitation energy transfer via the motion of spatially delocalized states.
Figure 1.7 Excitation energy transfer (EET), which occurs after optical preparation of an electronically and vibrationally excited initial state (donor, left). The Coulomb interaction is responsible for deexcitation of the donor and excitation of the acceptor (right). The nuclear dynamics may be subject to relaxation processes (wiggly lines). Often two independent nuclear (reaction) coordinates are used for the donor and the acceptor site.
Chapter 2
Electronic and Vibrational Molecular States
This chapter provides the background material for the subsequent development of a microscopic description of charge and energy transfer processes in the condensed phase. After introducing the molecular Hamiltonian operator we discuss the Born–Oppenheimer separation of electronic and nuclear motions as the key to the solution of the molecular Schrödinger equation. The Hartree–Fock method, which is a simple yet very successful approach to the determination of the ground state electronic structure, is explained next. It enables us to obtain, for instance, the potential energy surface for nuclear motions. To prepare for the treatment of condensed phase situations, we further introduce the dielectric continuum model as a means for incorporating static solvent polarization effects into the electronic structure calculations.
The topology of the potential energy surface can be explored by calculating the first and second derivatives with respect to the nuclear coordinates. Of particular interest are the stationary points on a potential energy surface that may correspond to stable conformations of the molecule. In the vicinity of a local minimum, it is often possible to analyze nuclear motions in terms of small-amplitude normal-mode vibrations. If one wants to model chemical reaction dynamics, however, the shape of the potential energy surface away from the stationary points is required as an input. We present two different approaches in this respect: the minimum energy reaction path and the Cartesian reaction surface model. Particularly the latter will provide the microscopic justification for the generic Hamiltonians used later on to simulate small molecular systems embedded in some environment. Finally, we discuss the diabatic and adiabatic representations of the molecular Hamiltonian.
2.1Introduction
The development of quantum theory in the 1920s was to a considerable extent triggered by the desire to understand the properties of atoms and molecules. It was soon appreciated that the Schrödinger equation, together with the probabilistic interpretation of its solutions, provided a powerful tool for tackling a variety of questions in physics and chemistry. The mathematical description of the hydrogen atom’s spectral lines could be given and developed to a textbook example of the success of quantum mechanics. Stepping into the molecular realm one faces a complicated many-body problem involving the coordinates of all electrons and all nuclei of the considered molecule. Its solution can be approached using the fact that nuclei and electrons have quite different masses allowing their motion to be adiabatically separated. This concept was first introduced by Born and Oppenheimer in 1927. Within the Born–Oppenheimer adiabatic approximation the simplest molecule, the hydrogen molecule ion, , can be treated.
From an electronic point of view the appearance of one more electron, for instance, in H2, necessitates the incorporation of the repulsive electronic interaction. Moreover, since one deals with two identical electrons, care has to be taken that the wave function has the proper symmetry with respect to an exchange of any two particle labels. In a straightforward way this is accomplished by the self-consistent field method according to Hartree, Fock, and Slater. Despite its deficiencies, Hartree–Fock theory has played an enormous role in the process of exploring the electronic structure of molecules during the last decades. It still serves as the basis for many of the more advanced approaches used nowadays.
However, it is not only the electronic structure at the equilibrium configuration of the nuclei that is of interest. The form of the potential energy hypersurfaces obtained upon varying the positions of the nuclei proves crucial for the understanding of the vibrational structure of molecular spectra. Moreover, it provides the key to chemical reaction dynamics. While the adiabatic Born–Oppenheimer ansatz is an excellent approximation in the vicinity of the ground state equilibrium configuration, nonadiabatic couplings leading to transitions between electronic states become a ubiquitous phenomenon if the nuclei explore their potential surface in processes such as photodissociation and electron transfer reactions, for example.
This chapter introduces the concepts behind the keywords given so far and sets up the stage for the following chapters. Having this intention it is obvious that we present a rather selective discussion of a broad field. We first introduce the molecular Hamiltonian and the respective solutions of the stationary Schrödinger equation in Section 2.2. This leads us directly to the Born–Oppenheimer separation of electronic and nuclear motions in Section 2.3. A brief account of electronic structure theory for polyatomic molecules is given next (Section 2.4). This is followed by a short summary of the dielectric continuum model in Section 2.5.1 and the atomistic quantum-classical approach in Section 2.5.2, both allow for incorporation of solvent effects into electronic structure calculations. On this basis we move on in Section 2.6 to discuss potential energy surfaces and the related concepts of harmonic vibrations and reaction paths. In Section 2.7 we focus our attention on the problem of nonadiabatic couplings, which are neglected in the Born–Oppenheimer adiabatic approximation. Finally, the issue of diabatic versus adiabatic pictures that emerges from this discussion is explained and alternative representations of the molecular Hamiltonian are given.
2.2Molecular Schrödinger Equation
In what follows we will be interested in situations where atoms made of pointlike nuclei and electrons are spatially close such that their mutual interaction leads to the formation of stable molecules. Let us consider such a molecule composed of Nnuc atoms having atomic numbers . The Cartesian coordinates and conjugate momenta for the Nel electrons are denoted rj and pj, respectively. For the Nnuc nuclei we use Rn and Pn. The Hamiltonian operator of the molecule has the general form
Here the kinetic energy of the electrons is given by (mel is the electron mass)
and for the nuclei it is
with Mn being the mass of the nth nucleus. Since both kinds of particles are charged, they interact via Coulomb forces. The repulsive Coulomb pair interaction between electrons is
and for the nuclei we have
(Note that the factor 1/2 compensates for double counting.) The attractive interaction between electrons and nuclei is given by
Since there are Nel electrons and Nnuc nuclei, the molecule has 3(Nel+Nnuc) spatial degrees of freedom (DOF). Each electron is assigned an additional quantum number σj to account for its spin. The purely quantum mechanical concept of electron spin was introduced to explain the fine structure of certain atomic spectra by Uhlenbeck and Goudsmit in 1925. Later its theoretical foundation was laid in the relativistic extension of quantum mechanics developed by Dirac in 1928. When using the nonrelativistic Hamiltonian equation (2.1) we have no means to rigorously introduce spin operators and to derive the interaction potential between coordinate and spin variables (spin–orbit coupling). Therefore, the existence of spin operators is usually postulated and their action on spin functions defined. We will not consider relativistic effects in this text and therefore carry the spin variable along with the electron coordinate only in the formal considerations of Section 2.4.
All quantum mechanical information about the stationary properties of the molecular system defined so far is contained in the solutions of the time-independent nonrelativistic Schrödinger equation
Here and in what follows we will combine the set of electronic Cartesian coordinates in the multi-index coordinate . A similar notation is introduced for the nuclear Cartesian coordinates, . In addition we will frequently use the more convenient notation . Momenta and masses of the nuclei will be written in the same way. (In this notation M1M2M3 is the mass of nucleus number one, etc.) For the spin we use the notation .
As it stands Eq. (2.7) does not tell much about what we are aiming at, namely, electronic excitation spectra, equilibrium geometries, and so on. However, some general points can be made immediately: first, the solution of Eq. (2.7) will provide us with an energy spectrum λ and corresponding eigenfunctions, Ψλ(r,σ;R). The energetically lowest state 0 is called the ground state. If λ is negative, the molecule is in a stable bound state. Note that in what follows we will also make use of the more formal notation where the eigenstates of the molecular Hamiltonian are denoted by the state vector . The wave function is obtained by switching to the (r,σ;R) representation: .
Second, owing to the Pauli principle, which states that the wave function of a system of electrons has to be antisymmetric with respect to the interchange of any two electronic indices, Ψ(r,σ;R) will be antisymmetric in electronic Cartesian plus spin coordinates. The fact that there can be identical nuclei as well is frequently neglected when setting up the exchange symmetry of the total wave function. This is justified since the nuclear wave function is usually much more localized as compared with the electronic wave function, and the indistinguishability is not an issue. Exceptions may occur in systems containing, for example, several hydrogen atoms.
Third, the probability distribution, |Ψλ(r,σ;R)|2, contains the information on the distribution of electrons as well as on the arrangement of the nuclei. Having this quantity at hand one can calculate, for example, the charge density distribution ρλ(x) for a particular molecular state at some spatial point x. The classical expression
is quantized by replacing the coordinates by the respective operators. Taking the matrix elements of the resulting charge density operator with respect to the state Ψλ(r,σ;R) we get
Finally, since the Hamiltonian does not depend on spin, the solution of Eq. (2.7) can be separated according to
Here, ζ(σ) is the electronic spin function, which is obtained by projecting the molecule’s spin state vector onto the spin states of the individual electrons, . The individual spin states, |, describe electrons whose spin is parallel (spin up) or antiparallel (spin down) with respect to some direction in coordinate space.
2.3Born–Oppenheimer Separation
The practical solution of Eq. (2.7) makes use of the fact that, due to the large mass difference (mel/Mn<10−3), on average electrons can be expected to move much faster than nuclei. Therefore, in many situations the electronic degrees of freedom can be considered to respond instantaneously to any changes in the nuclear configuration, that is, their wave function corresponds always to a stationary state. In other words, the interaction between nuclei and electrons, Vel–nuc, is modified due to the motion of the nuclei only adiabatically and does not cause transitions between different stationary electronic states. Thus, it is reasonable to define an electronic Hamiltonian that carries a parametric dependence on the nuclear coordinates:
As a consequence the solutions of the time-independent electronic Schrödinger equation describing the state of the electrons in the electrostatic field of the stationary nuclei (leaving aside the electron’s spin)
will parametrically depend on the set of nuclear coordinates as well. Here, the index a labels the different electronic states. The adiabatic electronic wave functions define a complete basis in the electronic Hilbert space. Hence, given the solutions to Eq. (2.12) the molecular wave function can be expanded in this basis set as follows:
The expansion coefficients in Eq. (2.13), χa(R), depend on the configuration of the nuclei. It is possible to derive an equation for their determination after inserting Eq. (2.13) into Eq. (2.7). One obtains
Multiplication of Eq. (2.14) by from the left and integration over all electronic coordinates yields the following equation for the expansion coefficients χa(R) (using the orthogonality of the adiabatic basis):
Since the electronic wave functions depend on the nuclear coordinates we have using and the product rule for differentiation
The last term is simply the kinetic energy operator acting on χa(R). The other terms can be combined into the so-called nonadiabaticity operator
Thus, we obtain from Eq. (2.15) an equation for the coefficients χa(R) that reads
This result can be interpreted as the stationary Schrödinger equation for the motion of nuclei, with the χa(R) being the respective wave functions. The solution to Eq. (2.18), which is still exact, requires knowledge of the electronic spectrum for all configurations of the nuclei that are covered during their motion. Transitions between individual adiabatic electronic states become possible due to the electronic nonadiabatic coupling, Θab. This is a consequence of the motion of the nuclei as expressed by the fact that their momentum enters Eq. (2.17). The diagonal part of the nonadiabaticity operator, Θaa, is usually only a small perturbation to the nuclear dynamics in a given electronic state.
Looking at Eq. (2.18) we realize that it will be convenient to introduce the following effective potential for nuclear motion if the electronic system is in its adiabatic state :
This function defines a hypersurface in the space of nuclear coordinates, the potential energy surface (PES), which will be discussed in more detail in Section 2.6. Its exceptional importance for a microscopic understanding of molecular transfer phenomena will become evident in Chapters 5–9.
The solution to Eq. (2.18) is given by . The index M denotes the (set of) vibrational quantum numbers. The molecular wave function is
By virtue of the expansion (2.20) it is clear that the vibrational quantum number M in general is related to the total electronic spectrum and not to an individual electronic state.
2.3.1Born–Oppenheimer Approximation
Solving the coupled equations (2.18) for the expansion coefficients in Eq. (2.20) appears to be a formidable task. However, in practice it is often possible to neglect the nonadiabatic couplings altogether or take into account the couplings between certain adiabatic electronic states only. In order to investigate this possibility let us consider Figure 2.1. Here we have plotted different adiabatic electronic states for a diatomic molecule as a function of the bond distance. Without going further into the details of the different states we realize that there is one state, the electronic ground state , which, particularly close to its minimum, is well separated from the other states . Intuitively we would expect the nonadiabatic couplings, , to be rather small in this region. In such situations it might be well justified to neglect the nonadiabatic couplings, that is, we can safely set in Eq. (2.18). The nuclear Schrödinger equation then simplifies considerably. For we have
Figure 2.1 Potential energy curves for different adiabatic electronic states along the bond distance R of a diatomic molecule (ground and valence states of I2).
where Ha(R) defines the nuclear Hamiltonian for the state . Thus, the nuclei can be considered to move in an effective potential Ua(R) generated by their mutual Coulomb interaction and the interaction with the electronic charge distribution corresponding to the actual configuration R. The solutions of Eq. (2.21) are again labeled M, but this quantum number is now related to the individual adiabatic electronic states. The total adiabatic wave function becomes
The neglect of the nonadiabatic couplings leading to the wave function (2.22) is called the Born–Oppenheimer approximation.
Going back to Figure 2.1 it is clear, however, that in particular for excited electronic states one might encounter situations where different potential curves are very close to each other. If does not vanish for symmetry reasons, it can no longer be neglected. The physical picture is that electronic and nuclear motions are no longer adiabatically separable, that is, the change of the nuclear configuration from R to some R+ΔR causes an electronic transition.
In order to estimate the magnitude of this effect we consider a perturbation expansion of the energy with respect to the nonadiabaticity operator. The second-order correction to the adiabatic energies is obtained as
where the are the Born–Oppenheimer nuclear wave functions. Apparently, the matrix elements have to be small compared to the energy difference in order to validate the adiabatic Born–Oppenheimer approximation. Looking at the definition of it is clear that this operator will be a small perturbation whenever the character of the electronic wave function does not change appreciably with R. On the other hand, the denominator in Eq. (2.23) will become small if two electronic states approach each other. Thus, knowledge about the adiabatic states is necessary to estimate the effect of nonadiabatic couplings. The actual calculation of is a rather complicated issue, and an alternative representation of the Hamiltonian will be discussed in Section 2.7.
2.3.2Some Estimates
We complete our qualitative discussion by considering the dynamical aspect of the problem. For simplicity let us take a diatomic molecule in the vicinity of the potential minimum where the potential is harmonic, that is, Ua(R)=κR2/2. Here κ is the harmonic “spring” constant that is calculated from the second derivative of the potential with respect to R (see below). The frequency of harmonic vibration is obtained from , with Mnuc being the reduced mass of the vibration. If and denote the average velocity and deviation from the minimum configuration, respectively, the virial theorem tells us that . According to quantum mechanics this will also be proportional to . Now consider the electrons: let us assume that the most important contribution to the potential energy comes from the electrostatic electronic interaction. If del is some typical length scale of the electronic system, for example, the radius of the electron cloud, its potential energy will be proportional to e2/del. Further, the average electronic velocity is . Applying a reasoning similar to the virial theorem gives for the electronic subsystem. Equation (2.21) tells us that the average electronic energy is of the order of the potential energy for nuclear motion . This gives for the spring constant . Using this result we obtain the relations
and
Since the nuclei move on average much slower than the electrons and explore a smaller region of the configuration space.
With del and at hand we can estimate the period for the bound electronic motion as . Thus, the average energy gap between electronic states is of the order of . Comparing this result with the vibrational frequency given above we obtain
Thus, it is the large mass difference that makes the gap for vibrational transitions much smaller than for electronic transitions in the vicinity of a potential minimum. Therefore, the denominator in Eq. (2.23) is likely to be rather large and the second-order correction to the adiabatic energy becomes negligible in this case.
2.4Electronic Structure Methods
Our knowledge about the microscopic origin of spectral properties of molecules, their stable configurations, and their ability to break and make chemical bonds derives to a large extent from the progress made in electronic structure theory in recent decades. Nowadays modern quantum chemical methods routinely achieve almost quantitative agreement with experimental data, for example, for transition energies between the lowest electronic states of small and medium-size molecules. With an increasing number of electrons the computational resources limit the applicability of the so-called ab initio (that is, based on fundamental principles and not on experimental data) methods and alternatives have to be exploited. Semiempirical methods, such as the Hückel or the Pariser–Parr–Pople method, simplify the exact ab initio procedure in a way that gives results consistent with experimental data. On the other hand, ongoing developments in density functional theory shift the attention to this more accurate method. Switching to situations of molecules in the condensed phase, for example, in solution, requires more approximate methods as given, for example, by the reduction of the solvent to a dielectric continuum surrounding the solute1) (Section 2.5.1).
In what follows we will outline a tool for the practical solution of the electronic Schrödinger equation (2.12) for fixed nuclei. For simplicity our discussion will mostly be restricted to the electronic ground state E0(R). Specifically, we will discuss the Hartree–Fock self-consistent field procedure in some detail. It is the working horse of most more advanced ab initio methods that also include the effect of electronic correlations missing in the Hartree–Fock approach. Whereas these methods are based on the electronic wave function, density functional theory (discussed afterwards) builds on the electron density function. We note in caution that this section by no means presents a complete treatment of the field of electronic structure theory. The intention is rather to provide a background for the following discussions. The reader interested in a more comprehensive overview of the state of the art is referred to the literature quoted at the end of this chapter.
Let us start with the situation in which the Coulomb interaction between electrons is switched off. Then the electronic Hamiltonian equation (2.11) becomes a sum of single-particle Hamiltonians, , containing the kinetic energy of the jth electron and the Coulomb energy due to its interaction with the static nuclei. Note that in the following discussion we will drop the parametric dependence on the nuclear coordinates. The stationary Schrödinger equation for hel(ri) is solved by the single-particle wave function ,
Here the index αi runs over all possible single-particle states (including spin) of the Nel-electron system, which have the energy . The single-particle functions are called spin orbitals.
There are several points to make concerning the solutions of Eq. (2.27). First, since we are dealing with identical particles, the single-particle spectrum is the same for all electrons. Second, for the spin-independent Hamiltonian we use here, the spin function can be separated from the spatial orbital in the single-particle wave function according to . As mentioned above the orthogonal spin functions describe spin-up or spin-down electrons. Therefore, for Nel spatial orbitals there will be 2Nel possible spin orbitals . Thus, given Nel electrons, the electronic ground state would correspond to the situation where we fill in electrons in the different spin orbitals starting from the one with the lowest energy. Of course, we must be mindful of the Pauli principle, that is, each electron must have a distinct set of quantum numbers. In the present case this implies that each spatial orbital may be occupied by two electrons having spin up and spin down, respectively. The result of the distribution of electrons over the available spin orbitals is referred to as an electronic configuration.
Depending on whether there is an even number of electrons in the ground state (closed shell configuration) or an odd number (open shell configuration) all electrons will be paired or not, respectively. For simplicity we will focus in what follows on the electronic ground state of closed shell systems only. Here Nel spin orbitals are occupied. One can further require the spatial orbitals to be identical for spin-up and spin-down electrons so that there will be Nel/2 doubly occupied spatial orbitals in the ground state. Needless to say, the total spin of this many-electron system is zero. A closed shell situation is shown for the water molecule in Figure 2.2.
Figure 2.2 Orbital diagram for water calculated using Hartree–Fock theory (the energy is given in atomic units . There are Nel/2=5 doubly occupied orbitals; the empty orbitals are called virtual. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are assigned and shown on the right (different colors correspond to positive and negative values). Notice that Hartree–Fock theory predicts the LUMO energies to be positive, implying that additional electrons cannot bind and the negative ion does not exist.
The Pauli principle, which we invoked above, can be traced back to a fundamental property of the total wave function of a many-electron system. First, we observe that, in contrast to classical mechanics, in quantum mechanics the electrons described by a wave function are not distinguishable. This means that the total probability distribution, , should be invariant with respect to the exchange of any two particle indices. The permutation of the particle indices is conveniently written using a permutation operator which, when acting on a many-particle wave function, exchanges the indices of any two particles. After the application of the wave function can change at most by a constant factor ξ (of modulus 1). Therefore, applying twice one should recover the original wave function, that is, we have ξ2=1 or ξ=±1. For spin 1/2 particles like electrons it turns out that ξ=−1 and therefore the total wave function has to be antisymmetric with respect to the exchange of any two electron indices.
If we go back to the single-particle spin orbitals defined by Eq. (2.27), it is clear now that even in the absence of the electron interaction, the so-called Hartree product ansatz
cannot be correct since it does not have the required antisymmetry ({αj} comprises the set of quantum numbers αj). However, Eq. (2.28) can be used to generate an antisymmetric wave function. To this end we make use of the permutation operator . Keeping track of the number of permutations, p, that have been performed one obtains an antisymmetric wave function by the prescription
Here the summation is carried out over all Nel! possible permutations of the electron indices (rj,σj) (j=1,…,Nel) in the Hartree product. Alternatively, Eq. (2.29) can be written in the form of a determinant, the so-called Slater determinant, where the rows contain the single-particle spin orbitals for a given state and all possible electron coordinates, and the different electronic states for a given coordinate are recorded in the columns. The elementary properties of determinants then guarantee the antisymmetry of the ansatz for the total electronic wave function.
2.4.1The Hartree–Fock Equations
So far we have not considered the effect of the Coulomb interaction between electrons. Within Hartree–Fock theory this is usually done by starting from the correct antisymmetric ansatz (2.29) for the wave function. Then the goal is to optimize the single-particle spin orbitals such that the total energy is minimized. This can be achieved by invoking the calculus of variation. Consider a Slater determinant φ(r,σ), which shall be a function of some parameters. In practice the spatial orbitals are expanded in terms of some fixed basis set and the expansion coefficients then take the role of the parameters. The basis set is usually chosen to consist of functions that are centered at the different atoms in the molecule (linear combination of atomic orbitals, LCAO).
The expectation value of the energy is then given by
The first term denotes the single-particle Hamiltonian including the electron–nuclei Coulomb interaction, Eq. (2.27), and the second term describes the electron–electron repulsion, Eq. (2.4). In Section 2.8.1 it is shown that variational optimization of Eq. (2.30) leads to the following so-called Hartree–Fock integrodifferential equations for determination of the optimal orbitals for a closed shell configuration:
Here, εa is the energy associated with the spatial orbital φa(x). Further, the operator on the left-hand side is called the Fock operator; it is an effective one-electron operator.
Without the electron–electron interaction and wave function antisymmetrization the Fock operator reduces to the single-electron Hamiltonian, hel(x). Different spatial orbitals are coupled by means of the Coulomb operator Jb(x) (see Eq. (2.149)) and the exchange operator Kb(x) (see Eq. (2.150)). The Coulomb operator represents the average local potential of an electron in orbital φb(x) felt by the electron in φa(x). Thus, the exact two-particle Coulomb interaction is replaced by an effective one-electron potential. The fact that each electron only sees the mean field generated by all other electrons is a basic characteristic of the Hartree–Fock approach. Of course, in this way the interaction between electrons becomes blurred and correlations between their individual motions are lost.
It has been discussed above that for electrons having parallel spins there is a particular correlation introduced by the antisymmetric ansatz for the wave function. This effect is contained in the exchange operator. However, the action of Kb(x) on the orbital φa(x) obviously cannot be viewed in terms of a local potential for the electron in φa(x). In fact it is the exchange operator that makes the Fock operator nonlocal in space.
The Hartree–Fock equations are nonlinear since the Fock operator itself depends on the orbitals φa(x). Hence the solution can only be obtained by iteration. Starting from some trial orbitals one first constructs the Fock operator and then uses it to obtain improved orbitals that are the input for a new Fock operator. This iterative procedure is continued until the potentials Ja(x) and Ka(x) are consistent with the solutions for the orbitals. Therefore, the approach is usually termed the Hartree–Fock self-consistent field method.
Given the solution of the Hartree–Fock equations one has at hand the ground state energy as well as the ground state adiabatic electronic wave function, which follows from a single
