Molecular Electronic-Structure Theory - Trygve Helgaker - ebook

Molecular Electronic-Structure Theory ebook

Trygve Helgaker

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Ab initio quantum chemistry has emerged as an important tool in chemical research and is appliced to a wide variety of problems in chemistry and molecular physics. Recent developments of computational methods have enabled previously intractable chemical problems to be solved using rigorous quantum-mechanical methods. This is the first comprehensive, up-to-date and technical work to cover all the important aspects of modern Molecular Electronic-Structure Theory. Topics covered in the book include: * Second quantization with spin adaptation * Gaussian basis sets and molecular-integral evaluation * Hartree-Fock theory * Configuration-interaction and multi-configurational self-consistent theory * Coupled-cluster theory for ground and excited states * Perturbation theory for single- and multi-configurational states * Linear-scaling techniques and the fast multipole method * Explicity correlated wave functions * Basis-set convergence and extrapolation * Calibration and benchmarking of computational methods, with applications to moelcular equilibrium structure, atomization energies and reaction enthalpies. Molecular Electronic-Structure Theory makes extensive use of numerical examples, designed to illustrate the strengths and weaknesses of each method treated. In addition, statements about the usefulness and deficiencies of the various methods are supported by actual examples, not just model calculations. Problems and exercises are provided at the end of each chapter, complete with hints and solutions. This book is a must for researchers in the field of quantum chemistry as well as for nonspecialists who wish to acquire a thorough understanding of ab initio Molecular Electronic-Structure Theory and its applications to problems in chemistry and physics. It is also highly recommended for the teaching of graduates and advanced undergraduates.

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CONTENTS

PREFACE

OVERVIEW

PROGRAMS USED IN THE PREPARATION OF THIS BOOK

1 SECOND QUANTIZATION

1.1 The Fock space

1.2 Creation and annihilation operators

1.3 Number-conserving operators

1.4 The representation of one- and two-electron operators

1.5 Products of operators in second quantization

1.6 First- and second-quantization operators compared

1.7 Density matrices

1.8 Commutators and anticommutators

1.9 Nonorthogonal spin orbitals

References

Further reading

Exercises

Solutions

2 SPIN IN SECOND QUANTIZATION

2.1 Spin functions

2.2 Operators in the orbital basis

2.3 Spin tensor operators

2.4 Spin properties of determinants

2.5 Configuration state functions

2.6 The genealogical coupling scheme

2.7 Density matrices

References

Further reading

Exercises

Solutions

3 ORBITAL ROTATIONS

3.1 Unitary transformations and matrix exponentials

3.2 Unitary spin-orbital transformations

3.3 Symmetry-restricted unitary transformations

3.4 The logarithmic matrix function

References

Further reading

Exercises

Solutions

4 EXACT AND APPROXIMATE WAVE FUNCTIONS

4.1 Characteristics of the exact wave function

4.2 The variation principle

4.3 Size-extensivity

4.4 Symmetry constraints

References

Further reading

Exercises

Solutions

5 THE STANDARD MODELS

5.1 One-and N-electron expansions

5.2 A model system: the hydrogen molecule in a minimal basis

5.3 Exact wave functions in Fock space

5.4 The Hartree–Fock approximation

5.5 Multiconfigurational self-consistent field theory

5.6 Configuration-interaction theory

5.7 Coupled-cluster theory

5.8 Perturbation theory

References

Further reading

Exercises

Solutions

6 ATOMIC BASIS FUNCTIONS

6.1 Requirements on one-electron basis functions

6.2 One- and many-centre expansions

6.3 The one-electron central-field system

6.4 The angular basis

6.5 Exponential radial functions

6.6 Gaussian radial functions

References

Further reading

Exercises

Solutions

7 SHORT-RANGE INTERACTIONS AND ORBITAL EXPANSIONS

7.1 The Coulomb hole

7.2 The Coulomb cusp

7.3 Approximate treatments of the ground-state helium atom

7.4 The partial-wave expansion of the ground-state helium atom

7.5 The principal expansion of the ground-state helium atom

7.6 Electron-correlation effects summarized

References

Further reading

Exercises

Solutions

8 GAUSSIAN BASIS SETS

8.1 Gaussian basis functions

8.2 Gaussian basis sets for Hartree–Fock calculations

8.3 Gaussian basis sets for correlated calculations

8.4 Basis-set convergence

8.5 Basis-set superposition error

References

Further reading

9 MOLECULAR INTEGRAL EVALUATION

9.1 Contracted spherical-harmonic Gaussians

9.2 Cartesian Gaussians

9.3 The Obara–Saika scheme for simple integrals

9.4 Hermite Gaussians

9.5 The McMurchie–Davidson scheme for simple integrals

9.6 Gaussian quadrature for simple integrals

9.7 Coulomb integrals over spherical Gaussians

9.8 The Boys function

9.9 The McMurchie–Davidson scheme for Coulomb integrals

9.10 The Obara–Saika scheme for Coulomb integrals

9.11 Rys quadrature for Coulomb integrals

9.12 Scaling properties of the molecular integrals

9.13 The multipole method for Coulomb integrals

9.14 The multipole method for large systems

References

Further reading

Exercises

Solutions

10 HARTREE–FOCK THEORY

10.1 Parametrization of the wave function and the energy

10.2 The Hartree–Fock wave function

10.3 Canonical Hartree–Fock theory

10.4 The RHF total energy and orbital energies

10.5 Koopmans’ theorem

10.6 The Roothaan–Hall self-consistent field equations

10.7 Density-based Hartree–Fock theory

10.8 Second-order optimization

10.9 The SCF method as an approximate second-order method

10.10 Singlet and triplet instabilities in RHF theory

10.11 Multiple solutions in Hartree–Fock theory

References

Further reading

Exercises

Solutions

11 CONFIGURATION-INTERACTION THEORY

11.1 The CI model

11.2 Size-extensivity and the CI model

11.3 A CI model system for noninteracting hydrogen molecules

11.4 Parametrization of the CI model

11.5 Optimization of the CI wave function

11.6 Slater determinants as products of alpha and beta strings

11.7 The determinantal representation of the Hamiltonian operator

11.8 Direct CI methods

11.9 CI orbital transformations

11.10 Symmetry-broken CI solutions

References

Further reading

Exercises

Solutions

12 MULTICONFIGURATIONAL SELF-CONSISTENT FIELD THEORY

12.1 The MCSCF model

12.2 The MCSCF energy and wave function

12.3 The MCSCF Newton trust-region method

12.4 The Newton eigenvector method

12.5 Computational considerations

12.6 Exponential parametrization of the configuration space

12.7 MCSCF theory for several electronic states

12.8 Removal of RHF instabilities in MCSCF theory

References

Further reading

Exercises

Solutions

13 COUPLED-CLUSTER THEORY

13.1 The coupled-cluster model

13.2 The coupled-cluster exponential ansatz

13.3 Size-extensivity in coupled-cluster theory

13.4 Coupled-cluster optimization techniques

13.5 The coupled-cluster variational Lagrangian

13.6 The equation-of-motion coupled-cluster method

13.7 The closed-shell CCSD model

13.8 Special treatments of coupled-cluster theory

13.9 High-spin open-shell coupled-cluster theory

References

Further reading

Exercises

Solutions

14 PERTURBATION THEORY

14.1 Rayleigh–Schrödinger perturbation theory

14.2 Møller–Plesset perturbation theory

14.3 Coupled-cluster perturbation theory

14.4 Møller–Plesset theory for closed-shell systems

14.5 Convergence in perturbation theory

14.6 Perturbative treatments of coupled-cluster wave functions

14.7 Multiconfigurational perturbation theory

References

Further reading

Exercises

Solutions

15 CALIBRATION OF THE ELECTRONIC-STRUCTURE MODELS

15.1 The sample molecules

15.2 Errors in quantum-chemical calculations

15.3 Molecular equilibrium structures: bond distances

15.4 Molecular equilibrium structures: bond angles

15.5 Molecular dipole moments

15.6 Molecular and atomic energies

15.7 Atomization energies

15.8 Reaction enthalpies

15.9 Conformational barriers

15.10 Conclusions

References

Further reading

Exercises

Solutions

LIST OF ACRONYMS

INDEX

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PREFACE

Quantum chemistry has emerged as an important tool for investigating a wide range of problems in chemistry and molecular physics. With the recent development of computational methods and more powerful computers, it has become possible to solve chemical problems that only a few years ago seemed for ever beyond the reach of a rigorous quantum-mechanical treatment. Today quantum-mechanical methods are routinely applied to problems related to molecular structure and reactivity, and spectroscopic parameters calculated quantum-mechanically are often useful in the interpretation of spectroscopic measurements. With the development and distribution of sophisticated program packages, advanced computational electronic-structure theory has become a practical tool for nonspecialists at universities and in industry.

In view of the increasing importance of computational electronic-structure theory in chemical research, it is somewhat surprising that no comprehensive, up-to-date, technical monograph is available on this subject. This book is an attempt to fill this gap. It covers all the important aspects of modern ab initio nonrelativistic wave function-based molecular electronic-structure theory – providing sufficient in-depth background material to enable the reader to appreciate the physical motivation behind the approximations made at the different levels of theory and also to understand the technical machinery needed for efficient implementations on modern computers.

To justify the adoption of the various approximations, this book makes extensive use of numerical examples, designed to illustrate the strengths and shortcomings of each method treated. Statements about the usefulness or deficiencies of the various methods are supported by actual examples and not merely model calculations, making the arguments more forceful. The accuracy and applicability of the various methods have been established from extensive calculations on a sample of molecules followed by a statistical analysis. The purpose of this book is thus not only to motivate approximations and to explain techniques, but also to assess the reliability and accuracy of the various computational methods in quantum chemistry.

The important working equations of computational electronic-structure theory are derived in detail. The level of detail attempted is such that, from the equations presented in this book, the reader should be able to write a computer program without too much difficulty. Thus, all the important aspects of computations are treated: the evaluation of molecular integrals, the parametrization and optimization of the wave function, and the analysis of the results. A detailed description of the contents of the book is given in the Overview.

Some areas of computational electronic-structure theory are not treated in this book. All methods discussed are strictly ab initio. Semi-empirical methods are not treated nor is density-functional theory discussed; all techniques discussed involve directly or indirectly the calculation of a wave function. Energy derivatives are not covered, even though these play a prominent role in the evaluation of molecular properties and in the optimization of geometries. Relativistic theory is likewise not treated. In short, the focus is on techniques for solving the nonrelativistic molecular Born–Oppenheimer electronic Schrödinger equation ab initio and on the usefulness and reliability of the solutions.

The presentation of the material relies heavily on the techniques of second quantization. However, no previous knowledge of second quantization is assumed. Indeed, Chapter 1 provides a self-contained introduction to this subject at a level appropriate for the remainder of the text. It is the authors’ belief that the beauty and usefulness of second quantization will become apparent as the reader tackles the remainder of the material presented in this book.

Except for the techniques of second quantization, the presentation in this book makes no use of techniques that should not be familiar to any student of quantum chemistry after one or two years of undergraduate study. At the end of each chapter, problems and exercises are provided, complete with hints and solutions. The reader is strongly urged to work through these exercises as a means of deepening and consolidating the understanding of the material presented in each chapter.

The material presented in this book has – in different stages of preparation – been used as teaching material at the Quantum Chemistry and Molecular Properties summer schools, organized by the authors biennially since 1990. In addition, it has been used by the authors for teaching one-year courses in computational quantum chemistry in their own departments. As such, it has been subjected to the repeated scrutiny of a large number of students. Their critical comments have been a great help in the preparation of the final manuscript.

Thousands of calculations constitute the background material for this book. We are, in particular, indebted to our colleagues Keld L. Bak, Jürgen Gauss, Asger Halkier, Wim Klopper and Henrik Koch for making these calculations possible. Most of the calculations were carried out with the DALTON program, while ACES II was used especially for spin-unrestricted calculations, for the CCSDT calculations and for geometry optimizations. Many of the DALTON and ACES II calculations were performed by Keld L. Bak and Asger Halkier. Wim Klopper performed all the explicitly correlated calculations, modifying his codes to fit our needs; Jürgen Gauss helped us with the ACES II program, making a number of modifications needed for our nonstandard and seemingly exorbitant calculations; and Henrik Koch made it possible to carry out many of the nonstandard coupled-cluster calculations with the DALTON program. We are also indebted to Jacek Kobus, Leif Laaksonen, Dage Sundholm and Pekka Pyykkö for providing us with their numerical diatomic Hartree–Fock program. The CASPT2 calculations were carried out with the MOLCAS program, the CI and higher-order perturbation calculations with the LUCIA program, while the numerical atomic calculations were carried out with the LUCAS program.

A large number of colleagues and students have provided helpful comments and suggestions for improvements. For this we thank Lars K. Andersen, Alexander Auer, Keld L. Bak, Vebjørn Bakken, Anders Bernhardsson, Thomas Bondo, Ove Christiansen, Attila G. Csaszar, Pål Dahle, Knut Fægri, Jürgen Gauss, Kasper Hald, Asger Halkier, Nicholas C. Handy, Christof Hättig, Hanne Heiberg, Alf C. Hennum, Morten Hohwy, Kristian S. Hvid, Michal Jaszunski, Hans Jørgen Aa. Jensen, Wim Klopper, Jacek Kobus, Henrik Koch, Helena Larsen, Jan Linderberg, Jon Lærdahl, Per-Åke Malmqvist, Kurt V. Mikkelsen, Chris Mohn, Thomas Nyman, Jens Oddershede, Tina D. Poulsen, Antonio Rizzo, Bjørn Roos, Torgeir Ruden, Kenneth Ruud, Trond Saue, Jack Simons, Dage Sundholm, Peter Szalay, Peter R. Taylor, Danny L. Yeager and Hans Ågren.

Most of this manuscript was prepared during numerous stays at the Marinebiological Laboratory of Aarhus University, located at Rønbjerg near Limfjorden in Denmark. We would like to thank the staff of the laboratory – Jens T. Christensen and Ingelise Mortensen – for providing us with an enjoyable and productive setting. Hanne Kirkegaard typed parts of the numerous drafts. We thank her for her competent typing and for never giving up hope in a seemingly never-ending flood of modifications and corrections. Finally, we are grateful to our wives – Barbara, Lise and Jette – for patiently letting us go on with our work. Without their love and support, we would not have finished this book, which is therefore dedicated to them.

Trygve Helgaker, Poul Jørgensen and Jeppe OlsenRønbjerg, March 1999

OVERVIEW

Chapters 1–3 introduce second quantization, emphasizing those aspects of the theory that are useful for molecular electronic-structure theory. In Chapter 1, second quantization is introduced in the spin-orbital basis, and we show how first-quantization operators and states are represented in the language of second quantization. Next, in Chapter 2, we make spin adaptations of such operators and states, introducing spin tensor operators and configuration state functions. Finally, in Chapter 3, we discuss unitary transformations and, in particular, their nonredundant formulation in terms of exponentials of matrices and operators. Of particular importance is the exponential parametrization of unitary orbital transformations, used in the subsequent chapters of the book.

In Chapters 4 and 5, we turn our attention away from the tools of second quantization towards wave functions. First, in Chapter 4, we discuss important general characteristics of the exact electronic wave function such as antisymmetry, size-extensivity, stationarity, and the cusps that arise from the singularities of the Hamiltonian. Ideally, we would like our approximate wave functions to inherit most of these properties; in practice, compromises must be made between what is desirable and what is practical as illustrated in Chapter 5, which introduces the standard models of electronic-structure theory, presenting the basic theory and employing numerical examples to illustrate the usefulness and shortcomings of the different methods.

Chapters 6–8 are concerned with one-electron basis functions and the expansion of many- electron wave functions in such functions. In Chapter 6, we discuss the analytical form of the one-electron basis functions – the universal angular part and the different exponential and Gaussian radial parts, applying these to one-electron systems in order to illustrate their merits and shortcomings. In Chapter 7, we consider the expansion of the two-electron helium system in products of one-electron functions. In particular, we study the slow convergence of such expansions, arising from the inability of orbital products to tackle head-on the problems associated with short-range interactions and singularities in the Hamiltonian. We here also develop techniques for the extrapolation of the energy to the basis-set limit, later used for many-electron systems. Next, having discussed one- and two-electron systems in Chapters 6 and 7, we proceed in Chapter 8 to the construction of basis sets for many-electron systems, giving a detailed account of many of the standard Gaussian basis sets of electronic-structure theory – in particular, the correlation-consistent basis sets extensively used in this book. Their convergence is examined by carrying out extrapolations to the basis-set limit, comparing with explicitly correlated calculations.

Chapter 9 discusses the evaluation of molecular integrals over Gaussian atomic functions. We here cover all integrals needed for the evaluation of molecular electronic energies, deriving the working equations in detail. Many techniques have been developed for the calculation of molecular integrals; the more important are treated here: the Obara–Saika scheme, the McMurchie–Davidson scheme, and the Rys-polynomial scheme. However, for large systems, it is better to use multipole expansions to describe the long-range interactions. In this chapter, we develop the multipole method for the individual two-electron integrals; then, we discuss how the multipole method can be organized for the calculation of Coulomb interactions in large systems at a cost that scales linearly with the size of the system.

The next five chapters are each devoted to the study of one particular computational model of ab initio electronic-structure theory; Chapter 10 is devoted to the Hartree–Fock model. Important topics discussed are: the parametrization of the wave function, stationary conditions, the calculation of the electronic gradient, first- and second-order methods of optimization, the self-consistent field method, direct (integral-driven) techniques, canonical orbitals, Koopmans’ theorem, and size-extensivity. Also discussed is the direct optimization of the one-electron density, in which the construction of molecular orbitals is avoided, as required for calculations whose cost scales linearly with the size of the system.

In Chapter 11, we treat configuration-interaction (CI) theory, concentrating on the full CI wave function and certain classes of truncated CI wave functions. The simplicity of the CI model allows for efficient methods of optimization, as discussed in this chapter. However, we also consider the chief shortcomings of the CI method – namely, the lack of compactness in the description and the loss of size-extensivity that occurs upon truncation of the CI expansion.

In Chapter 12, we study the related multiconfigurational self-consistent field (MCSCF) method, in which a simultaneous optimization of orbitals and CI coefficients is attempted. Although the MCSCF method is incapable of providing accurate energies and wave functions, it is a flexible model, well suited to the study of chemical reactions and excited states. This chapter concentrates on techniques of optimization, a difficult problem in MCSCF theory because of the simultaneous optimization of orbitals and CI coefficients.

Chapter 13 discusses coupled-cluster theory. Important concepts such as connected and disconnected clusters, the exponential ansatz, and size-extensivity are discussed; the linked and unlinked equations of coupled-cluster theory are compared and the optimization of the wave function is described. Brueckner theory and orbital-optimized coupled-cluster theory are also discussed, as are the coupled-cluster variational Lagrangian and the equation-of-motion coupled-cluster model. A large section is devoted to the coupled-cluster singles-and-doubles (CCSD) model, whose working equations are derived in detail. A discussion of a spin-restricted open-shell formalism concludes the chapter.

Chapter 14 presents a discussion of perturbation theory in electronic-structure theory. Rayleigh-Schrödinger theory is first treated, covering topics such as Wigner’s 2n + 1 rule and the Hylleraas functional. Next, we consider an important special case of Rayleigh-Schrödinger theory – Møller – Plesset theory – deriving explicit expressions for the energies and wave functions. A convenient Lagrangian formulation of Møller–Plesset theory, in which only size-extensive terms occur, is then developed. The question of convergence is also examined, in particular the conditions under which the Møller–Plesset series is expected to diverge. This chapter concludes with a discussion of hybrid methods: the CCSD(T) model, in which the CCSD energy is perturbatively corrected for the lack of connected triples; and multiconfigurational perturbation theory, in which MCSCF energies are corrected for the effects of dynamical correlation.

In Chapter 15, we conclude our exposition of molecular electronic structure theory by applying the standard models of quantum chemistry to the calculation of a number of molecular properties: equilibrium structures, dipole moments, electronic energies, atomization energies, reaction enthalpies, and conformational energies. The performance of each model is examined, comparing with the other models and with experimental measurements. The large number of molecular systems considered enables us to carry out a statistical analysis of the different methods, making the investigations more forceful and reliable than would otherwise be the case. These studies demonstrate that rigorous electronic-structure theory has now advanced to such a level of sophistication that an accuracy comparable to or surpassing that of experimental measurements is routinely obtained for molecules containing five to ten first-row atoms.

PROGRAMS USED IN THE PREPARATION OF THIS BOOK

DALTON:

T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H. Ågren, T. Andersen, K. L. Bak, V. Bakken, O. Christiansen, P. Dahle, E. K. Dalskov, T. Enevoldsen, B. Fernandez, H. Heiberg, H. Hettema, D. Jonsson, S. Kirpekar, R. Kobayashi, H. Koch, K. V. Mikkelsen, P. Norman, M. J. Packer, T. Saue, P. R. Taylor and O. Vahtras, DALTON, an

ab initio

electronic structure program, Release 1.0, 1997.

DIRCCR12:

J. Noga and W. Klopper, DIRCCR12, an explicitly correlated coupled-cluster program.

SORE:

W. Klopper, SORE, a second-order R12-energy program.

ACES II:

J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, R. J. Bartlett, ACES II, Quantum Theory Project, University of Florida, Gainesville, Florida, 1992. See also J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, R. J. Bartlett,

Int. J. Quantum Chem.

S26

, 879 (1992). ACES II uses the VMOL integral and VPROPS property integral programs written by J. Almlöf and P. R. Taylor, a modified version of the integral derivative program ABACUS written by T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, J. Olsen, and P. R. Taylor, and a geometry optimization and vibrational analysis package written by J. F. Stanton and D. E. Bernholdt.

LUCIA:

J. Olsen, LUCIA, a CI program.

LUCAS:

J. Olsen, LUCAS, an atomic-structure program.

2D:

J. Kobus, L. Laaksonen, D. Sundholm, 2D, a numerical Hartree–Fock program for diatomic molecules.

MOLCAS4:

K. Andersson, M. R. A. Blomberg, M. P. Fülscher, G. Karlström, R. Lindh, P.-Å. Malmqvist, P. Neogrady, J. Olsen, B. O. Roos, A. J. Sadlej, M. Schütz, L. Seijo, L. Serrano-Andrés, P. E. M. Siegbahn, P.-O. Widmark, MOLCAS Version 4, Lund University, Sweden (1997).

1

SECOND QUANTIZATION

In the standard formulation of quantum mechanics, observables are represented by operators and states by functions. In the language of second quantization, the wave functions are also expressed in terms of operators – the creation and annihilation operators working on the vacuum state. The antisymmetry of the electronic wave function follows from the algebra of these operators and requires no special attention. From the creation and annihilation operators, we construct also the standard operators of first-quantization quantum mechanics such as the Hamiltonian operator. This unified description of states and operators in terms of a single set of elementary creation and annihilation operators reduces much of the formal manipulation of quantum mechanics to algebra, allowing important relationships to be developed in an elegant manner. In this chapter, we develop the formalism of second quantization, laying the foundation for our subsequent treatment of molecular electronic structure.

1.1 The Fock space

Let {ϕP (x)} be a basis of M orthonormal spin orbitals, where the coordinates x represent collectively the spatial coordinates r and the spin coordinate σ of the electron. A Slater determinant is an antisymmetrized product of one or more spin orbitals. For example, a normalized Slater determinant for N electrons may be written as

(1.1.1)

We now introduce an abstract linear vector space – the Fock space – where each determinant is represented by an occupation-number (ON) vector |k〉,

(1.1.2)

Thus, the occupation number kP is 1 if ϕP is present in the determinant and 0 if it is absent. For an orthonormal set of spin orbitals, we define the inner product between two ON vectors |k〉 and |m〉 as

(1.1.3)

This definition is consistent with the overlap between two Slater determinants containing the same number of electrons. However, the extension of (1.1.3) to have a well-defined but zero overlap between states with different electron numbers is a special feature of the Fock-space formulation of quantum mechanics that allows for a unified description of systems with variable numbers of electrons.

In a given spin-orbital basis, there is a one-to-one mapping between the Slater determinants with spin orbitals in canonical order and the ON vectors in the Fock space. Much of the terminology for Slater determinants is therefore used for ON vectors as well. Still, the ON vectors are not Slater determinants – unlike the Slater determinants, the ON vectors have no spatial structure but are just basis vectors in an abstract vector space. This Fock space can be manipulated as an ordinary inner-product vector space. For example, for two general vectors or states in the Fock space

(1.1.4)

(1.1.5)

the inner product is given by

(1.1.6)

The resolution of the identity likewise may be written in the usual manner as

(1.1.7)

where the summation is over the full set of ON vectors for all numbers of electrons.

The ON vectors in (1.1.2) constitute an orthonormal basis in the 2M-dimensional Fock space F(M). This Fock space may be decomposed as a direct sum of subspaces F(M, N)

(1.1.8)

where F(M, N) contains all ON vectors obtained by distributing N electrons among the M spin orbitals – that is, all ON vectors for which the sum of the occupation numbers is N:

(1.1.9)

The subspace F(M, 0), which consists of ON vectors with no electrons, contains a single vector – the true vacuum state

(1.1.10)

which, according to (1.1.3), is normalized to unity:

(1.1.11)

Approximations to an exact N-electron wave function are expressed in terms of vectors in the Fock subspace F(M, N) of dimension equal to the binomial coefficient .

1.2 Creation and annihilation operators

In second quantization, all operators and states can be constructed from a set of elementary creation and annihilation operators. In this section we introduce these operators and explore their basic algebraic properties.

1.2.1 CREATION OPERATORS

The M elementary creation operators are defined by the relations

(1.2.1)

(1.2.2)

where

(1.2.3)

The spin orbitals that are unoccupied in an ON vector (1.1.2) may be identified from the specification of the occupied spin orbitals. The explicit reference to the unoccupied spin orbitals may be avoided altogether by expressing the ON vector as a string of creation operators in the canonical order (i.e. in the same order as in the ON vector) working on the vacuum state:

(1.2.4)

We shall later see that the phase factor (1.2.3) is automatically kept track of by the anticommutation relations of the creation operators in (1.2.4), making any reference to this factor unnecessary.

The properties of the creation operators can be deduced from the relations (1.2.1) and (1.2.2), which we here combine in a single defining equation:

(1.2.5)

Operating twice with on an ON vector, we obtain from (1.2.5)

(1.2.6)

Since the product gives zero when applied to any vector, it must be identical to the zero operator:

(1.2.7)

For P ≠ Q, the operators and may act on an ON vector in two ways. For Q > P, we obtain

(1.2.8)

where the phase factor for P is unaffected by the application of since P appears before Q in the ON vector. Reversing the order of the creation operators, we obtain

(1.2.9)

The factor arises since – if it does not vanish – contains one more electron before spin orbital Q than does |k〉. Adding together (1.2.8) and (1.2.9), we obtain

(1.2.10)

(1.2.11)

holds for any pair of creation operators.

1.2.2 ANNIHILATION OPERATORS

Having introduced the creation operators of second quantization, we now proceed to the study of their Hermitian adjoints aP. We shall see that the creation operators and their adjoints are distinct operators and consequently that these operators are not self-adjoint (Hermitian).

The properties of the adjoint or conjugate operators aP can be inferred from those of the creation operators. Thus, from (1.2.11) the adjoint operators are seen to satisfy the anticommutation relation

(1.2.12)

To determine the action of aP on an ON vector |k〉, we invoke the resolution of the identity (1.1.7):

(1.2.13)

The matrix element in this expression may be written as

(1.2.14)

(1.2.15)

Hence, only one term in (1.2.13) survives and we conclude

(1.2.16)

The operator aP reduces kp from 1 to 0 if spin orbital P is occupied and it gives 0 if the spin orbital is unoccupied. It is therefore called an electron annihilation operator. An interesting special

case of (1.2.16) is

(1.2.17)

which states that there are no electrons to be destroyed in the vacuum state.

1.2.3 ANTICOMMUTATION RELATIONS

We have seen that the creation operators anticommute among themselves (1.2.11) and that the same is true for the annihilation operators (1.2.12). We shall now establish the commutation relations between creation and annihilation operators. Combining (1.2.5) and (1.2.16), we obtain

(1.2.18)

(1.2.19)

The phase factors cancel since they appear twice. Adding these equations together, we arrive at the following expression:

(1.2.20)

for any ON vector |k〉. The operator is therefore equal to the identity operator

(1.2.21)

For P > Q, we obtain

(1.2.22)

(1.2.23)

where the minus sign arises since, in aQ|k〉, the number of occupied spin orbitals to the left of spin orbital P has been reduced by one. Adding these two equations together, we obtain

(1.2.24)

Since |k〉 is an arbitrary ON vector, we have the operator identity

(1.2.25)

The case P < Q is obtained by taking the conjugate of this equation and renaming the dummy indices. Combination of (1.2.21) and (1.2.25) shows that, for all P and Q,

(1.2.26)

The anticommutation relations (1.2.11), (1.2.12), and (1.2.26) constitute the fundamental properties of the creation and annihilation operators. In view of their importance, they are here collected and listed in full:

(1.2.27)

(1.2.28)

(1.2.29)

From these simple relations, all other algebraic properties of the second-quantization formalism follow.

1.3 Number-conserving operators

The creation and annihilation operators introduced in Section 1.2 change the number of particles in a state and therefore couple ON vectors belonging to different subspaces F(M, N). We now turn to operators that conserve the particle number and thus couple ON vectors in the same subspace.

1.3.1 OCCUPATION-NUMBER OPERATORS

We first introduce the occupation-number (ON) operators as

(1.3.1)

The ON operator counts the number of electrons in spin orbital P:

(1.3.2)

Here we have used (1.2.18). The ON operators are Hermitian

(1.3.3)

and commute among themselves

(1.3.4)

The ON vectors are thus the simultaneous eigenvectors of the commuting set of Hermitian operators . Moreover, the set of ON operators is complete in the sense that there is a one-to-one mapping between the ON vectors in the Fock space and the eigenvalues of the ON operators. The eigenvalues of the ON operators characterize the ON vectors completely, consistent with the introduction of the ON vectors as an orthonormal basis for the Fock space.

In the spin-orbital basis, the ON operators are projection operators since, in addition to being Hermitian (1.3.3), they are also idempotent:

(1.3.5)

Here we have used the anticommutators (1.2.29) and (1.2.28) in that order. Applied to a linear combination of ON vectors (1.1.4), the operator leaves unaffected vectors where ϕP is occupied and annihilates all others:

(1.3.6)

Note that this property of the ON operators holds only in the spin-orbital basis where the occupations are either zero or one.

Using the basic anticommutation relations of creation and annihilation operators, we obtain for the commutators of the ON operators with the creation operators

(1.3.7)

and therefore

(1.3.8)

Taking the conjugate of this equation, we obtain the corresponding commutator with the annihilation operator

(1.3.9)

From these commutators, we may also conclude that, for an arbitrary string of creation and annihilation operators such as

(1.3.10)

the commutators with the ON operators become

(1.3.11)

where is the number of times occurs in minus the number of times aP occurs in the same string. To arrive at relation (1.3.11), we have used the commutator expansion (1.8.5) of Section 1.8.

1.3.2 THE NUMBER OPERATOR

Adding together all ON operators in the Fock space, we obtain the Hermitian operator

(1.3.12)

which returns the number of electrons in an ON vector

(1.3.13)

and therefore is known as the particle-number operator or simply the number operator. From (1.3.11), we see that the commutator of the number operator with an arbitrary string of operators is given by

(1.3.14)

where NX is the excess of creation operators over annihilation operators in the string. In particular, we find that the number operator commutes with any string that contains an equal number of creation and annihilation operators. Such strings are called number-conserving, since they conserve the number of particles in any vector:

(1.3.15)

In general, the application of the string to a Fock-space vector increases the number of electrons by NX.

1.3.3 EXCITATION OPERATORS

Apart from the particle-number operators (1.3.12), the simplest number-conserving operators are the elementary excitation operators , for which we shall occasionally use the notation

(1.3.16)

Applied to an ON vector, these operators give (see Exercise 1.1)

(1.3.17)

where

(1.3.18)

and where the ket on the right-hand side of (1.3.17) is an ON vector with the same occupation numbers as |k〉 except as indicated for spin orbitals P and Q.Equation (1.3.17) shows that aQ excites an electron from spin orbital Q to spin orbital P, thus turning |k〉 into another ON vector in the same subspace F(M, N). In fact, each ON vector in F(M, N) can be obtained from any other ON vector in the same subspace by applying a sequence of excitation operators aQ. The application of a single such operator yields a single excitation, two operators give a double excitation, and so on. The ‘diagonal' excitation operators correspond to the occupation-number operators (1.3.1).

In Box 1.1, we summarize the fundamentals of the second-quantization formalism. In Section 1.4, we proceed to discuss the second-quantization representation of standard first-quantization operators such as the electronic Hamiltonian.

Box 1.1 The fundamentals of second quantization

1.4 The representation of one- and two-electron operators

Expectation values correspond to observables and should therefore be independent of the representation given to the operators and the states. Since expectation values may be expressed as sums of matrix elements of operators, we require the matrix element of a second-quantization operator between two ON vectors to be equal to its counterpart in first quantization. An operator in the Fock space can thus be constructed by requiring its matrix elements between ON vectors to be equal to the corresponding matrix elements between Slater determinants of the first-quantization operator.

Before proceeding to determine the form of the operators in second quantization, we recall that the matrix elements between Slater determinants depend on the spatial form of the spin orbitals. Since the ON vectors are independent of the spatial form of spin orbitals, we conclude that the second-quantization operators – in contrast to their first-quantization counterparts – must depend on the spatial form of the spin orbitals.

First-quantization operators conserve the number of electrons. Following the discussion in Section 1.3, such operators are in the Fock space represented by linear combinations of operators that contain an equal number of creation and annihilation operators. The explicit form of these number-conserving operators depends on whether the first-quantized operator is a one-electron operator or a two-electron operator. One-electron operators are discussed in Section 1.4.1 and two-electron operators in Section 1.4.2. Finally, in Section 1.4.3 we consider the second-quantization representation of the electronic Hamiltonian operator.

1.4.1 ONE-ELECTRON OPERATORS

In first quantization, one-electron operators are written as

(1.4.1)

where the summation is over the N electrons of the system. Superscript c indicates that we are working in the coordinate representation of first quantization. Since each term in the operator (1.4.1) involves a single electron, this operator gives a vanishing matrix element whenever the Slater determinants differ in more than one pair of spin orbitals. The second-quantization analogue of (1.4.1) therefore has the structure

(1.4.2)

since the excitation operators shift a single electron in an ON vector. The summation is over all pairs of spin orbitals to secure the highest possible flexibility in the description. The order of the creation and annihilation operators in each term ensures that the one-electron operator produces zero when it works on the vacuum state.

To determine the numerical parameters fPQ in (1.4.2), we evaluate the matrix elements of between two ON vectors and compare with the usual Slater–Condon rules for matrix elements between determinants [1]. For one-electron operators there are three distinct cases.

1. The ON vectors are identical:

(1.4.3)

2. The ON vectors differ in one pair of occupation numbers:

(1.4.4)

(1.4.5)

(1.4.6)

3. The ON vectors differ in more than one pair of occupation numbers:

(1.4.7)

In these expressions, we have used indices I and J for the spin orbitals with different occupations in the bra and ket vectors.

Let us see how the above results are obtained. For the diagonal element

(1.4.8)

we note from the orthogonality of ON vectors that nonzero contributions can only arise when

(1.4.9)

(1.4.10)

has nonvanishing contributions only if

(1.4.11)

Comparing with the Slater–Condon rules for one-electron operators, we note that the second-quantization matrix elements (1.4.3), (1.4.6) and (1.4.7) will agree with their first-quantization counterparts if we make the identification

(1.4.12)

The recipe for constructing a second-quantization representation of a one-electron operator is therefore to use (1.4.2) with the integrals (1.4.12). For real spin orbitals, the integrals exhibit the following permutational symmetry

(1.4.13)

In conclusion, we note that the phase factors (1.2.3) were necessary to reproduce the Slater–Condon rules for matrix elements between Slater determinants.

1.4.2 TWO-ELECTRON OPERATORS

In first quantization, two-electron operators such as the electronic-repulsion operator are given by the expression

(1.4.14)

Other examples of two-electron operators are the two-electron part of the spin–orbit operator and the mass-polarization operator. A two-electron operator gives nonvanishing matrix elements between Slater determinants if the determinants contain at least two electrons and if they differ in the occupations of at most two pairs of electrons. The second-quantization representation of a two-electron operator therefore has the structure

(1.4.15)

The annihilation operators appear to the right of the creation operators in order to ensure that gives zero when it works on an ON vector with less than two electrons. The factor of one-half in (1.4.15) is conventional. Anticommuting the creation and annihilation operators and renaming the dummy indices, we obtain

(1.4.16)

The parameters gPQRS may therefore be chosen in a symmetric fashion

(1.4.17)

The numerical values of the parameters gPQRS may be determined by evaluating the matrix element of between two ON vectors and setting the result equal to the matrix element between the corresponding Slater determinants. There are four cases.

1. The ON vectors are identical:

(1.4.18)

2. The ON vectors differ in one pair of occupation numbers:

(1.4.19)

(1.4.20)

(1.4.21)

3. The ON vectors differ in two pairs of occupation numbers:

(1.4.22)

(1.4.23)

where I < J and K < L

(1.4.24)

4. The ON vectors differ in more than two pairs of occupation numbers:

(1.4.25)

We have in these expressions used the indices I, J, K and L for the spin orbitals with different occupations in the bra and ket vectors.

Let us consider the derivation of these matrix elements in some detail. The diagonal element

(1.4.26)

has nonvanishing contributions if

(1.4.27)

This condition holds in two different cases

(1.4.28)

If both sets of relations are fulfilled, then the expectation value of the creation and annihilation operators vanishes. We may therefore write the diagonal matrix element in the form

(1.4.29)

From the definition of the ON operators (1.3.1) and the commutator (1.3.9), we obtain

(1.4.30)

The diagonal element (1.4.29) therefore becomes

(1.4.31)

Next, we consider the case where the ON vectors (1.4.19) and (1.4.20) differ in the occupation numbers of one pair of spin orbitals. The matrix element

(1.4.32)

has nonvanishing contributions if

(1.4.33)

This condition holds in four different cases:

(1.4.34)

Since the matrix element vanishes if several of the above sets of relations hold, we obtain

(1.4.35)

Invoking the permutational symmetry (1.4.17) and the elementary anticommutation relations, we arrive at the final expression (1.4.21)

(1.4.36)

It does not matter whether the occupation numbers in this expression refer to |k1〉 or |k2〉 since the contributions vanish whenever the occupations differ. The matrix element between ON vectors differing in two pairs of occupations (1.4.24) can be treated in the same way and is left as an exercise.

The two-electron second-quantization matrix elements (1.4.18), (1.4.21), (1.4.24) and (1.4.25) become identical to the corresponding first-quantization elements obtained from the Slater–Condon rules if we choose

(1.4.37)

The recipe for constructing a two-electron second-quantization operator is therefore given by expressions (1.4.15) and (1.4.37). For any interaction between identical particles, the operator gc(x1, x2) is symmetric in x1 and x2. The integrals (1.4.37) therefore automatically exhibit the permutational symmetry in (1.4.17). We also note the following useful permutational symmetries for real spin orbitals:

(1.4.38)

Thus, for real spin orbitals there are a total of eight permutational symmetries (1.4.17) and (1.4.38) present in the two-electron integrals, whereas for complex spin orbitals there is only one such symmetry (1.4.17).

1.4.3 THE MOLECULAR ELECTRONIC HAMILTONIAN

Combining the results of Sections 1.4.1 and 1.4.2, we may now construct the full second-quantization representation of the electronic Hamiltonian operator in the Bom–Oppenheimer approximation. Although not strictly needed for the development of the second-quantization theory in this chapter, we present the detailed form of this operator as an example of the construction of operators in second quantization. In the absence of external fields, the second-quantization nonrelativistic and spin-free molecular electronic Hamiltonian is given by

(1.4.39)

where in atomic units (which are always used in this book unless otherwise stated)

(1.4.40)

(1.4.41)

(1.4.42)

Here the ZI are the nuclear charges, rI the electron–nuclear separations, r12 the electron–electron separation and RIJ the internuclear separations. The summations are over all nuclei. The scalar term (1.4.42) represents the nuclear-repulsion energy – it is simply added to the Hamiltonian and makes the same contribution to matrix elements as in first quantization since the inner product of two ON vectors is identical to the overlap of the determinants. The molecular one- and two-electron integrals (1.4.40) and (1.4.41) may be calculated using the techniques described in Chapter 9.

The form of the second-quantization Hamiltonian (1.4.39) may be interpreted in the following way. Applied to an electronic state, the Hamiltonian produces a linear combination of the original state with states generated by single and double electron excitations from this state. With each such excitation, there is an associated ‘amplitude’ hPQ or gPQRS, which represents the probability of this event happening. These probability amplitudes are calculated from the spin orbitals and the one- and two-electron operators according to (1.4.40) and (1.4.41).

1.5 Products of operators in second quantization

1.5.1 OPERATOR PRODUCTS

Let Ac and Bc be two one-electron operators in first quantization

(1.5.1)

(1.5.2)

and let and be the corresponding second-quantization representations

(1.5.3)

(1.5.4)

From the construction of the second-quantization operators, it is clear that the first-quantization operator aAc + bBc, where a and b are numbers, is represented by a + b. The standard relations

(1.5.5)

(1.5.6)

for linear operators in a linear vector space are also valid.

We now consider the representation of operator products. The product of the two first-quantization operators AcBc can be separated into one- and two-electron parts

(1.5.7)

where

(1.5.8)

(1.5.9)

The two-electron operator is written so that it is symmetric with respect to permutation of the particle indices. This symmetrization is necessary since we know only how to generate the second-quantization representation of two-electron operators that are symmetric in the particles.

The second-quantization representation of Pc is the sum of the second-quantization representations of Oc and Tc:

(1.5.10)

where

(1.5.11)

(1.5.12)

Inserting this expression for the two-electron parameters in the two-electron part of , we obtain

(1.5.13)

by substitution of dummy indices. Using the anticommutation relations, we may rewrite this expression as

(1.5.14)

Inserting (1.5.14) in (1.5.10), we finally arrive at the expression

(1.5.15)

which shows that the second-quantization representation of AcBc is in general not equal to the product of the representations of Ac and Bc.

We shall now demonstrate that the last term in (1.5.15) vanishes for a complete basis. We use the Dirac delta junction δ(x – x′), defined by the relationship [2,3]

(1.5.16)

For a complete one-electron basis, the delta function may be written in the form

(1.5.17)

Assuming that (1.5.17) holds, we may now write

(1.5.18)

and the last term in (1.5.15) vanishes. We therefore have for a complete one-electron basis

(1.5.19)

but for finite basis sets this expression does not hold.

The second-quantization operators are projections of the exact operators onto a basis of spin orbitals. For an incomplete basis, the second-quantization representation of an operator product therefore depends on when the projection is made. For a complete basis, however, the representation is exact and independent of when the projection is made.

1.5.2 THE CANONICAL COMMUTATORS

The previous discussion suggests that commutation relations that hold for operators in first quantization, do not necessarily hold for their second-quantization counterparts in a finite basis. Consider the canonical commutators

(1.5.20)

where we have contributions from each of the N electrons

(1.5.21)

(1.5.22)

(1.5.23)

and Greek letters denote Cartesian directions. The relationship (1.5.20) holds exactly for first-quantization operators. Note carefully that the operator (1.5.23) depends on the number of electrons and is not the usual Kronecker delta. The second-quantization representations of the position and momentum operators are

(1.5.24)

(1.5.25)

and the commutator of these operators becomes

(1.5.26)

In these expressions, square brackets around a first-quantization operator represent the one-electron integral of this operator in the given basis. This somewhat cumbersome notation is adopted for this discussion to make the dependence of the integrals on the first-quantization operators explicit.

In Section 1.8, the commutator between the two excitation operators is shown to be

(1.5.27)

and the commutator (1.5.26) therefore reduces to

(1.5.28)

For a complete basis, we may use (1.5.18) and arrive at the following simplifications:

(1.5.29)

(1.5.30)

The second-quantization canonical commutator therefore becomes proportional to the number operator in the limit of a complete basis:

(1.5.31)

This expression should be compared with its first-quantization counterpart (1.5.20). For finite basis sets, the second-quantization canonical commutator turns into a general one-electron operator (1.5.28).

1.6 First- and second-quantization operators compared

In Box 1.2, we summarize some of the characteristics of operators in the first and second quantizations. The dependence on the spin-orbital basis is different in the two representations. In first quantization, the Slater determinants depend on the spin-orbital basis whereas the operators are independent of the spin orbitals. In the second-quantization formalism, the ON vectors are basis vectors in a linear vector space and contain no reference to the spin-orbital basis. Instead, the reference to the spin-orbital basis is made in the operators. We also note that, whereas the first-quantization operators depend explicitly on the number of electrons, no such dependence is found in the second-quantization operators.

Box 1.2 First- and second-quantization operators compared

The fact that the second-quantization operators are projections of the exact operators onto the spin-orbital basis means that a second-quantization operator times an ON vector is just another vector in the Fock space. By contrast, a first-quantization operator times a determinant cannot be expanded as a sum of Slater determinants in a finite basis. This fact often goes unnoticed since, in the first quantization, we usually work directly with matrix elements.

The projected nature of the second-quantization operators has many ramifications. For example, relations that hold for exact operators such as the canonical commutation properties of the coordinate and momentum operators do not necessarily hold for projected operators. Similarly, the projected coordinate operator does not commute with the projected Coulomb repulsion operator. It should be emphasized, however, that these problems are not peculiar to second quantization but arise whenever a finite basis is employed. They also arise in first quantization, but not until the matrix elements are evaluated.

Second quantization treats operators and wave functions in a unified way – they are all expressed in terms of the elementary creation and annihilation operators. This property of the second-quantization formalism can, for example, be exploited to express modifications to the wave function as changes in the operators. To illustrate the unified description of states and operators afforded by second quantization, we note that any ON vector may be written compactly as a string of creation operators working on the vacuum state (1.2.4)

(1.6.1)

Matrix elements may therefore be viewed as the vacuum expectation value of an operator

(1.6.2)

and expectation values become linear combinations of vacuum expectation values. The unified description of states and operators in terms of the elementary creation and annihilation operators enables us to carry out most of our manipulations algebraically based on the anticommutation relations of these operators. Thus, the antisymmetry of the electronic wave function follows automatically from the algebra of the elementary operators without the need to keep track of phase factors.

1.7 Density matrices

Having considered the representation of states and operators in second quantization, let us now turn our attention to expectation values. As in first quantization, the evaluation of expectation values is carried out by means of density matrices [4]. Consider a general one- and two-electron Hermitian operator in the spin-orbital basis

(1.7.1)

The expectation value of this operator with respect to a normalized reference state |0〉 written as a linear combination of ON vectors,

(1.7.2)

(1.7.3)

is given by the expression

(1.7.4)

where we have introduced the matrix elements

(1.7.5)

(1.7.6)

Clearly, all information that is required about the wave function (1.7.2) for the evaluation of expectation values (1.7.4) is embodied in the quantities (1.7.5) and (1.7.6) called the one- andtwo-electron density-matrix elements, respectively. Overbars are used for the spin-orbital densities to distinguish these from those that will be introduced in Chapter 2 for the orbital basis. Since the density elements play such an important role in electronic-structure theory, it is appropriate here to examine their properties within the framework of second quantization.

1.7.1 THE ONE-ELECTRON DENSITY MATRIX

The densities (1.7.5) constitute the elements of an M × M Hermitian matrix the one-electron spin-orbital density matrix – since the following relation is satisfied:

(1.7.7)

For real wave functions, the matrix is symmetric:

(1.7.8)

The one-electron density matrix is positive semidefinite since its elements are either trivially equal to zero or inner products of states in the subspace F(M, N – 1). The diagonal elements of the spin-orbital density matrix are the expectation values of the occupation-number operators (1.3.1) in F(M, N) and are referred to as the occupation numbers of the electronic state:

(1.7.9)

This terminology is appropriate since the diagonal elements of reduce to the usual occupation numbers kp in (1.3.2) whenever the reference state is an eigenfunction of the ON operators – that is, when the reference state is an ON vector:

(1.7.10)

Since the ON operators are projectors (1.3.5), we may write the occupation numbers in the form

(1.7.11)

where the projected electronic state is given by

(1.7.12)

The occupation numbers may now be written in the form

(1.7.13)

and interpreted as the squared norm of the part of the reference state where spin orbital ϕP is occupied in each ON vector. The occupation numbers thus serve as indicators of the importance of the spin orbitals in the electronic state.

The expansion coefficients satisfy the normalization condition

(1.7.14)

Recalling that the occupation numbers kP of an ON vector are zero or one, we conclude that the occupation numbers of an electronic state (1.7.13) are real numbers between zero and one – zero for spin orbitals that are unoccupied in all ON vectors, one for spin orbitals that are occupied in all ON vectors, and nonintegral for spin orbitals that are occupied in some but not all ON vectors:

(1.7.15)

We also note that the sum of the occupation numbers (i.e. the trace of the density matrix) is equal to the total number of electrons in the system:

(1.7.16)

Here we have used the definition of the particle-number operator (1.3.12).

For a state consisting of a single ON vector, the one-electron spin-orbital density matrix has a simple diagonal structure:

(1.7.17)

By contrast, for an electronic state containing several ON vectors, the density matrix is not diagonal. Applying the Schwarz inequality in the (N – 1)-electron space, we obtain

(1.7.18)

which gives us an upper bound to the magnitude of the elements of the spin-orbital density matrix equal to the geometric mean of the occupation numbers:

(1.7.19)

Of course, since is a Hermitian matrix, we may eliminate the off-diagonal elements (1.7.19) completely by diagonalization with a unitary matrix:

(1.7.20)

The eigenvalues are real numbers 0 ≤ ≤ 1, known as the natural-orbital occupation numbers. The sum of the natural-orbital occupation numbers is again equal to the number of electrons in the system. From the eigenvectors U of the density matrix, we obtain a new set of spin orbitals called the natural spin orbitals of the system. However, we defer the discussion of unitary orbital transformations to Chapter 3.

1.7.2 THE TWO-ELECTRON DENSITY MATRIX

We now turn our attention to the two-electron density matrix. We begin by noting that the two-electron density-matrix elements (1.7.6) are not all independent because of the anticommutation relations between the creation operators and between the annihilation operators:

(1.7.21)

The following elements are therefore zero in accordance with the Pauli principle:

(1.7.22)

To avoid these redundancies in our representation, we introduce the two-electron density matrix with elements given by

(1.7.23)

There are M(M – 1)/2 rows and columns in this matrix with composite indices PQ such that P > Q. The elements of constitute a subset of the two-electron density elements (1.7.6) and differ from these by a reordering of the middle indices:

(1.7.24)

The reason for introducing this reordering is that it allows us to examine the two-electron density matrix by analogy with the discussion of the one-electron density in Section 1.7.1. Thus, as in the one-electron case, we note that the two-electron density matrix is Hermitian

(1.7.25)

and therefore symmetric for real wave functions

(1.7.26)

Also, the two-electron density matrix is positive semidefinite since its elements are either trivially equal to zero or inner products of states in F(M, N – 2).

We recall that the diagonal elements of correspond to expectation values of ON operators (1.7.9) and are interpreted as the occupation numbers of the spin orbitals. We now examine the diagonal elements of the two-electron density matrix:

(1.7.27)

Since P > Q, we may anticommute aP from the fourth to the second position, introduce ON operators, and arrive at the following expression analogous to (1.7.9):

(1.7.28)

Since the ON operators are projectors, we may now interpret the diagonal elements as simultaneous occupations of pairs of spin orbitals (pair occupations), noting that represents the squared amplitude of the part of the wave function where spin orbitals ϕP and ϕQ are simultaneously occupied:

(1.7.29)

The norm of the wave function is successively reduced by the repeated application of ON projectors; compare (1.7.12) and (1.7.29). This observation agrees with the expectation that the simultaneous occupation of a given spin-orbital pair cannot exceed those of the individual spin orbitals:

(1.7.30)

The reader may wish to verify that a weaker upper bound to the pair occupations

(1.7.31)

is arrived at by the application of the Schwarz inequality to (1.7.28). From the trace of the two-electron density matrix

(1.7.32)

we find that the sum of all pair occupations is equal to the number of electron pairs in the system in the same way that the sum of all single occupations is equal to the number of electrons in the system (1.7.16). We may summarize these results by stating that the one-electron density matrix probes the individual occupancies of the spin orbitals and describes how the N electrons are distributed among the M spin orbitals, whereas the two-electron density matrix probes the simultaneous occupations of the spin orbitals and describes how the N(N – 1)/2 electron pairs are distributed among the M(M – 1)/2 spin-orbital pairs.

For a state containing a single ON vector, the two-electron density matrix has a particularly simple diagonal structure with the following elements

(1.7.33)

recalling the conditions P > Q and R > S. Indeed, for such electronic states, the two-electron density matrix may be constructed directly from the one-electron density matrix

(1.7.34)

and likewise the expectation value of any one- and two-electron operator may be obtained directly from the one-electron density matrix. This observation is consistent with our picture of ON vectors (i.e. determinants) as representing an uncorrelated description of the electronic system where the simultaneous occupations of pairs of spin orbitals are just the products of the individual occupations.

For a general electronic state, containing more than one ON vector and providing a correlated treatment of the electronic system, the two-electron density matrix is in general not diagonal and cannot be generated directly from the one-electron density elements. As in the one-electron case (1.7.19), we may invoke the Schwarz inequality to establish an upper bound to the magnitude of the off-diagonal elements

(1.7.35)

and we may in principle diagonalize to obtain a more compact representation of the two-electron density, but this is seldom done in practice.

1.7.3 DENSITY MATRICES IN SPIN-ORBITAL AND COORDINATE REPRESENTATIONS

The density matrices we have discussed so far in this section have all been given in the spin-orbital representation. We shall now see how these matrices are represented in the coordinate representation of quantum mechanics. Since we have not here developed a second-quantization formalism appropriate for the coordinate representation, we shall draw on the equivalence between matrix elements in the first and second quantizations to establish the relationship between density matrices in the spin-orbital and coordinate representations.

We recall that, in the coordinate representation of first quantization, we may write the expectation value of any one-electron operator in the following form

(1.7.36)

in terms of first-order reduced density matrix

(1.7.37)

The density matrix in the spin-orbital representation was introduced in second quantization for the evaluation of one-electron expectation values in the following form

(1.7.38)

where the integrals are those given in (1.4.12):

(1.7.39)

Combining (1.7.38) and (1.7.39), we obtain

(1.7.40)

and we are therefore able to make the identification

(1.7.41)

involving the spin-orbital density matrix and the first-order reduced density matrix γ1(x1, ). Since the spin orbitals are orthonormal, we obtain the following expression after multiplication by spin orbitals and integration

(1.7.42)

where we have used (1.7.5). This equation displays the relationship between the density matrices in the spin-orbital representation and in the coordinate representation. We may establish similar relationships for the two-electron densities. Thus, introducing the second-order reduced density matrix

(1.7.43)

we obtain the following relationships

(1.7.44)

(1.7.45)

analogous to (1.7.41) and (1.7.42). The simple algebraic definitions for the densities in second quantization (as expectation values of excitation operators) should be contrasted with the more complicated expression in terms of the reduced densities.

1.8 Commutators and anticommutators

In the manipulation of operators and matrix elements in second quantization, the commutator

(1.8.1)

and the anticommutator

(1.8.2)

of two operators are often encountered. The elementary creation and annihilation operators satisfy the anticommutation relations (1.2.27)–(1.2.29). Referring to these basic relations, it is usually possible to simplify the commutators and anticommutators between strings of elementary operators considerably. Since manipulations in second quantization frequently involve complicated operator strings, it is important to establish a good strategy for the evaluation of commutators and anticommutators of such strings.