Computational Physics

9.3 An example: quantum molecular dynamics for the hydrogen molecule 273

5 10 15 20 25 30 35 40 45

Energy

–2.1

–2.05

–2

–1.95

–1.9

–1.85

–1.8

0 50

Time steps

Figure 9.2. Evolution of the energy in a Car–Parrinello simulation of the electronic

structure of the hydrogen molecule with separationX=1 between the nuclei, with

frictional forces included.

Now we must add an amount−λSrsCs(t)to this solution, whereλis determined
by the requirement that the normalisation condition(9.28)holds:
∑

rs

C ̃r(t+h)SrsC ̃s(t+h)− 2 λ

∑

rst

SrsC ̃r(t+h)SrtCs(t)

+λ^2

∑

rstu

SrsCs(t)SrtStuCu(t)=1. (9.32)

This is a quadratic equation inλ, of which the lowest positive root is needed. The
Verlet solution of the equation of motion is now fully defined.
Modifying the HF program ofChapter 4to calculate the electronic structure is
relatively easy, as the Fock matrix and the overlap matrix are calculated already in
this program.

programming exercise

Take the program of Problem 4.9 and replace the self-consistency iteration by

a molecular dynamics algorithm with friction, using the Verlet algorithm.

A frictional force −γC ̇r is included using the algorithm given in

Section 8.4.1in order to let the system evolve towards the ground state.

CheckA reasonable value for the time step is 0.1 (in atomic units) and for the

frictional constantγthe value 1 (in atomic units) is chosen. In Figure 9.2, the

energy as a function of the ‘time’ is shown. It is seen that for a nuclear separation