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
9 - Quantum molecular dynamics
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
In the previous chapter we considered systems of interacting particles. They were treated as classical particles for which the interaction potential is known. We had to solve the classical equations of motion to simulate the behaviour of such a system at some nonzero temperature. Had we added frictional forces, the system would have evolved towards the ground state. In this chapter we discuss methods for simulating interacting atoms and molecules using quantum mechanical calculations. In fact, we consider the nuclei on a classical level but use quantum mechanics for the electronic degrees of freedom. Again, we can use this approach either to simulate a system of interacting particles at a finite temperature, or to find the ground state (minimum energy) configurations of solids and of molecules.
In Chapters 4 to 6 we studied methods for solving the electronic structure of molecular and solid state systems with a static configuration of nuclei (Born–Oppenheimer approximation). Knowledge of the electronic structure includes knowledge of the total energy. Therefore, by varying the positions of the nuclei, we can study the dependence of the total energy on these positions. The energy E(R1, R2, …, RN) as a function of the nuclear positions Ri is called the potential surface. As a simple example, consider the hydrogen molecule. We assume that the molecule is not rotating, so that the nuclear motion is a vibration along the molecular axis.
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- Chapter
- Information
- Computational Physics , pp. 263 - 294Publisher: Cambridge University PressPrint publication year: 2007