Book contents
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- 1 Introduction
- 2 Quantum scattering with a spherically symmetric potential
- 3 The variational method for the Schrödinger equation
- 4 The Hartree–Fock method
- 5 Density functional theory
- 6 Solving the Schrödinger equation in periodic solids
- 7 Classical equilibrium statistical mechanics
- 8 Molecular dynamics simulations
- 9 Quantum molecular dynamics
- 10 The Monte Carlo method
- 11 Transfer matrix and diagonalisation of spin chains
- 12 Quantum Monte Carlo methods
- 13 The finite element method for partial differential equations
- 14 The lattice Boltzmann method for fluid dynamics
- 15 Computational methods for lattice field theories
- 16 High performance computing and parallelism
- Appendix A Numerical methods
- Appendix B Random number generators
- Index
4 - The Hartree–Fock method
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- 1 Introduction
- 2 Quantum scattering with a spherically symmetric potential
- 3 The variational method for the Schrödinger equation
- 4 The Hartree–Fock method
- 5 Density functional theory
- 6 Solving the Schrödinger equation in periodic solids
- 7 Classical equilibrium statistical mechanics
- 8 Molecular dynamics simulations
- 9 Quantum molecular dynamics
- 10 The Monte Carlo method
- 11 Transfer matrix and diagonalisation of spin chains
- 12 Quantum Monte Carlo methods
- 13 The finite element method for partial differential equations
- 14 The lattice Boltzmann method for fluid dynamics
- 15 Computational methods for lattice field theories
- 16 High performance computing and parallelism
- Appendix A Numerical methods
- Appendix B Random number generators
- Index
Summary
Introduction
Here and in the following chapter we treat two different approaches to the many-electron problem: the Hartree–Fock theory and the density functional theory. Both theories are simplifications of the full problem of many electrons moving in a potential field. In fact, the physical systems we want to study, such as atoms, molecules and solids, consist not only of electrons but also of nuclei, and each of these particles moves in the field generated by the others.Afirst approximation is to consider the nuclei as being fixed, and to solve the Schrödinger equation for the electronic system in the field of the static nuclei. This approach, called the Born–Oppenheimer approximation, is justified by the nuclei being much heavier than the electrons so that they move at much slower speeds. It remains then to solve for the electronic structure.
The Hartree–Fock methodcan be viewed as a variational method in which the wave functions of the many-electron system have the form of an antisymmetrised product of one-electron wave functions (the antisymmetrisation is necessary because of the fermion character of the electrons). This restriction leads to an effective Schrödinger equation for the individual one-electron wave functions (called orbitals) with a potential determined by the orbitals occupied by the other electrons. This coupling between the orbitals via the potentials causes the resulting equations to become nonlinear in the orbitals, and the solution must be found iteratively in a self-consistency procedure.
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- Computational Physics , pp. 43 - 88Publisher: Cambridge University PressPrint publication year: 2007
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