Second Quantized Approach to Quantum Chemistry: An Elementary Introduction

ISBN-13: 9783642747571 / Angielski / Miękka / 2011 / 184 str.

Peter R. Surjan

Opis

The aim of this book is to give a simple, short, and elementary introduction to the second quantized formalism as applied to a many-electron system. It is intended for those, mainly chemists, who are familiar with traditional quantum chemistry but have not yet become acquainted with second quantization. The treatment is, in part, based on a series of seminars held by the author on the subject. It has been realized that many quantum chemists either interested in theory or in applications, being educated as chemi ts and not as physicists, have never devoted themselves to taking a course on the second quantized approach. Most available textbooks on this topic are not very easy to follow for those who are not trained in theory, or they are not detailed enough to offer a comprehensive treatment. At the same time there are several papers in quantum chemical literature which take advantage of using second quantization, and it would be worthwhile if those papers were accessible for a wider reading public. For this reason, it is intended in this survey to review the basic formalism of second quantization, and to treat some selected chapters of quantum chemistry in this language.Most derivations will be carried out in a detailed manner, so the reader need not accept gaps to understand the result.

Spis treści

1 Introduction.- 1.1 Importance of Second Quantization.- 1.2 The One-Electron Model.- 2 Concept of Creation and Annihilation Operators.- 2.1 The Vacuum State.- 2.2 Creating Electrons.- 2.3 Particle Number Representation.- 2.4 Annihilating Electrons.- 2.5 Commutator Relation between Creation and Annihilation Operators.- 2.6 The Adjoint Relation — Role of Orthogonality of One-Particle States.- 2.7 Summary of the Properties of Creation/Annihilation Operators.- 3 Particle Number Operators.- 4 Second Quantized Representation of Quantum Mechanical Operators.- 4.1 General.- 4.2 One-Electron Operators.- 4.3 Two-Electron Operators.- 4.4 Second Quantized Form of the Born-Oppenheimer Hamiltonian.- 4.5 Hermiticity of Second Quantized Operators.- 5 Evaluation of Matrix Elements.- 5.1 Basic Matrix Elements.- 5.2 Concept of the Fermi Vacuum.- 6 Advantages of Second Quantization — Illustrative Examples.- 6.1 General.- 6.2 Overlap of two Determinants.- 6.3 Hückel Energy Expression.- 6.4 Interaction of Two Electrons.- 7 Density Matrices.- 7.1 First-Order Density Matrix.- 7.2 Second-Order Density Matrix.- 7.3 Hartree-Fock Energy Expression.- 8 Connection to “Bra and Ket” Formalism.- 9 Using Spatial Orbitals.- 10 Some Model Hamiltonians in Second Quantized Form.- 10.1 ?-Electron Hamiltonians.- 10.1.1 Hückel Level.- 10.1.2 Hubbard Model.- 10.1.3 Pariser-Parr-Pople (PPP) Model.- 10.2 Particle-Hole Symmetry.- 10.3 All-Valence Electron Hamiltonians.- 10.4 The Hartree-Fock Hamiltonian.- 11 The Brillouin Theorem.- 12 Many-Body Perturbation Theory.- 13 Second Quantization for Nonorthogonal Orbitals.- 13.1 Anticommutation Rules.- 13.2 The Hamiltonian in Nonorthogonal Representations.- 13.3 Extended Hückel Theory.- 14 Second Quantization and Hellmann-Feynman Theorem.- 14.1 General.- 14.2 Variation of Energy-Orthogonal Basis Set.- 14.3 Variation of Energy-Nonorthogonal Basis Set.- 14.4 Special Case: The SCF Gradient Formula.- 15 Intermolecular Interactions.- 15.1 The Operator for Interaction.- 15.2 Symmetry-Adapted Perturbation Theory.- 16 Quasiparticle Transformations.- 16.1 One-Particle Transformations.- 16.2 Two-Particle Transformations.- 16.3 A Theory of the Local Chemical Bond.- 17 Miscellaneous Topics Related to Second Quantization.- 17.1 Spin Operators and Spin Hamiltonians.- 17.2 Unitary Group Approach.- 18 Problem Solutions.- 19 References.