9 Quantum molecular dynamics
9.1 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 nuc-
lei, we can study the dependence of the total energy on these positions. The energy
E(R 1 ,R 2 ,...,RN)as a function of the nuclear positionsRiis called thepotential
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. The only relevant parameter describing the relative positions of
the two nuclei is their separationX. The force between the nuclei is then given
byF=−∂E(X)/∂X(seeFigure 9.1). These forces are usually parametrised and
the parameters are fixed by comparison with quantum mechanical calculations for
a few configurations, or by comparison with experimental results. This paramet-
rised form can then be used to calculate the motion of the nuclei on a classical
level, for example to find the equilibrium conformation of the molecule, which is
the configuration of nuclei that minimises the total energy. This is called the method
offorce fields; it is often used by chemists.
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