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. Previously published as a hardback edition, this comprehensive and technical work covers 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.

Preface xxi

Overview xxv

Programs used in the preparation of this book xxix

**1. Second Quantization 1**

1.1 The Fock space 1

1.2 Creation and annihilation operators 2

1.3 Number-conserving operators 6

1.4 The representation of one- and two-electron operators 9

1.5 Products of operators in second quantization 14

1.6 First- and second-quantization operators compared 18

1.7 Density matrices 19

1.8 Commutators and anticommutators 25

1.9 Nonorthogonal spin orbitals 27

**2. Spin in Second Quantization 34**

2.1 Spin functions 34

2.2 Operators in the orbital basis 35

2.3 Spin tensor operators 41

2.4 Spin properties of determinants 46

2.5 Configuration state functions 51

2.6 The genealogical coupling scheme 53

2.7 Density matrices 61

**3. Orbital Rotations 80**

3.1 Unitary transformations and matrix exponentials 80

3.2 Unitary spin-orbital transformations 86

3.3 Symmetry-restricted unitary transformations 89

3.4 The logarithmic matrix function 93

**4. Exact and Approximate Wave Functions 107**

4.1 Characteristics of the exact wave function 107

4.2 The variation principle 111

4.3 Size-extensivity 126

4.4 Symmetry constraints 135

**5. The Standard Models 142**

5.1 One- and N-electron expansions 143

5.2 A model system: the hydrogen molecule in a minimal basis 146

5.3 Exact wave functions in Fock space 162

5.4 The Hartree-Fock approximation 167

5.5 Multiconfigurational self-consistent field theory 176

5.6 Configuration-interaction theory 181

5.7 Coupled-cluster theory 186

5.8 Perturbation theory 192

**6. Atomic Basis Functions 201**

6.1 Requirements on one-electron basis functions 201

6.2 One- and many-centre expansions 203

6.3 The one-electron central-field system 204

6.4 The angular basis 207

6.5 Exponential radial functions 218

6.6 Gaussian radial functions 229

**7. Short-Range Interactions and Orbital Expansions 256**

7.1 The Coulomb hole 256

7.2 The Coulomb cusp 259

7.3 Approximate treatments of the ground-state helium atom 262

7.4 The partial-wave expansion of the ground-state helium atom 267

7.5 The principal expansion of the ground-state helium atom 273

7.6 Electron-correlation effects summarized 278

**8. Gaussian Basis Sets 287**

8.1 Gaussian basis functions 287

8.2 Gaussian basis sets for Hartree-Fock calculations 288

8.3 Gaussian basis sets for correlated calculations 300

8.4 Basis-set convergence 315

8.5 Basis-set superposition error 327

**9. Molecular Integral Evaluation 336**

9.1 Contracted spherical-harmonic Gaussians 336

9.2 Cartesian Gaussians 338

9.3 The Obara-Saika scheme for simple integrals 344

9.4 Hermite Gaussians 349

9.5 The McMurchie-Davidson scheme for simple integrals 352

9.6 Gaussian quadrature for simple integrals 357

9.7 Coulomb integra;s over spherical Gaussians 361

9.8 The Boys function 365

9.9 The McMurchie-Davidson scheme for Coulomb integrals 372

9.10 The Obara-Saika scheme for Coulomb integrals 381

9.11 Rys quadrature for Coulomb integrals 387

9.12 Scaling properties of the molecular integrals 398

9.13 The multipole method for Coulomb integrals 405

9.14 The multipole method for large systems 417

**10. Hartree-Fock Theory 433**

10.1 Parametrization of the wave function and the energy 433

10.2 The Hartree-Fock wave function 438

10.3 Canonical Hartree-Fock theory 443

10.4 The RHF total energy and orbital energies 450

10.5 Koopmans’ theorem 454

10.6 The Roothaan-Hall self-consistent field equations 458

10.7 Density-based Hartree-Fock theory 465

10.8 Second-order optimization 478

10.9 The SCF method as an approximate second-order method 490

10.10 Singlet and triplet instabilities in RHF theory 496

10.11 Multiple solutions in Hartree-Fock theory 504

**11. Configuration-Interaction Theory 523**

11.1 The CI model 523

11.2 Size-extensivity and the CI model 527

11.3 A CI model system for noninteracting hydrogen molecules 535

11.4 Parametrization of the CI model 540

11.5 Optimization of the CI wave function 543

11.6 Slater determinants as products of alpha and beta strings 550

11.7 The determinantal representation of the Hamiltonian operator 552

11.8 Direct CI methods 554

11.9 CI orbital transformations 569

11.10 Symmetry-broken CI solutions 573

**12. Multiconfigurational Self-Consistent Field Theory 498**

12.1 The MCSCF model 498

12.2 The MCSCF energy and wave function 600

12.3 The MCSCF Newton trust-region method 610

12.4 The Newton cigenvector method 616

12.5 Computational considerations 621

12.6 Exponential parametrization of the configuration space 630

12.7 MCSCF theory for several electronic states 637

12.8 Removal of RHF instabilities in MCSCF theory 640

**13. Coupled-Cluster Theory 648**

13.1 The coupled-cluster model 648

13.2 The coupled-cluster exponential ansatz 654

13.3 Size-extensivity in coupled-cluster theory 665

13.4 Coupled-cluster optimization techniques 670

13.5 The coupled-cluster variational Lagrangian 674

13.6 The equation-of-motion coupled-cluster method 677

13.7 The closed-shell CCSD model 685

13.8 Special treatments of coupled-cluster theory 698

13.9 High-spin open-shell coupled-cluster theory 704

**14. Perturbation Theory 724**

14.1 Rayleigh-Schrödinger perturbation theory 725

14.2 Møller-Plesset perturbation theory 739

14.3 Coupled-cluster perturbation theory 749

14.4 Møller-Plesset theory for closed-shell systems 759

14.5 Convergence in perturbation theory 769

14.6 Perturbative treatments of coupled-cluster wave functions 783

14.7 Multiconfigurational perturbation theory 796

**15. Calibration of the Electronic-Structure Models 817**

15.1 The sample molecules 817

15.2 Errors in quantum-chemical calculations 819

15.3 Molecular equilibrium structures: bond distances 821

15.4 Molecular equilibrium structures; bond angles 832

15.5 Molecular dipole moments 836

15.6 Molecular and atomic energies 840

15.7 Atomization energies 854

15.8 Reaction enthalpies 865

15.9 Conformational barriers 874

15.10 Conclusions 879

List of Acronyms 885

Index 887