Computational Physics

9.3 An example: quantum molecular dynamics for the hydrogen molecule 277

1.35

1.36

1.37

1.38

1.39

1.4

1.41

1.42

0 50 100 150 200 250 300

X

Time

Figure 9.3. The change of the separationXbetween the nuclei of a hydrogen

molecule as a function of time. The number of nuclear integration steps is shown

along theX-axis. The nuclear integration step size is 4.3 (in atomic units). The

integration step for the electrons was 0.1. Twelve thousand electron integration

steps were carried out. The electrons experience a friction with damping constant

γ=1, and the nuclei are damped with a friction constant of 5.

Check 1Take the nuclear mass e.g. 1000 times larger than the electron mass.
The nuclei will move very slowly in comparison with the electrons because they
are so much heavier. If friction is included, the nuclei should end up with zero
velocity at their equilibrium spacing, which is atX=1.3881a 0 (within the HF
approximation and using exclusively s-basis functions). This is to be compared
with the experimental value of 1.401a 0. The behaviour ofXas a function of time
is shown in Figure 9.3.
Check 2If friction is not included, the nuclei will oscillate around their equilib-
rium separation. Use 1836.15 for the proton mass. The frequency for an initial
separation of 1.35 Bohr radii is found to be 13.5× 1013 Hz, to be compared with
the value 13.64× 1013 Hz obtained above from fitting a parabola to the bottom
of the effective potential well in Figure 9.1, and with the experimental value,
which is 12.48× 1013 Hz. The parabola was characterised by a ‘spring constant’
κ=0.385. The behaviour ofXas a function of time is shown inFigure 9.4.
Check that the results inFigure 9.4comply with this value (note that the time
step in thisfigure is 4.3in reduced units).
It is possible and advisable to use fewer integration steps for the nuclear equation
of motion than for the electronic one: the nuclei move much more slowly than the