Computational Physics

8.6 Molecular systems 235

ω

n^

Figure 8.5. The nitrogen molecule.nˆis a unit vector,ωωωis the rotation vector.

each of the two atoms in the molecule and the atoms of the remaining molecules. The
atomic forces can be modelled by a Lennard–Jones interaction with the appropriate
atomic nitrogen parametersσ =3.31 Å,ε/kB=37.3 K andd=0.3296σ[46].
The equation of motion for the centre of massRCMis then
R ̈CM=Ftot, (8.106)
which can be solved in exactly the same way as in an ordinary MD simulation.
The motion of the orientation vectornˆis determined by the torqueNwith respect
to the centre of the molecule, which is given in terms of the forcesF(^1 )andF(^2 )
acting on atoms 1 and 2 respectively:
N=(d/ 2 )nˆ×(F(^1 )−F(^2 )). (8.107)
The torque changes the angular momentumLof the molecule. This is equal to
Iωωω, whereIis the moment of inertiamd^2 /2 andωωωis the angular frequency vector
whose norm is the angular frequency and whose direction is the axis around which
the rotation takes place (seeFigure 8.5). Note thatNis not necessarily parallel to
ωωω. The equation of motion for the angular momentum isL ̇=Nor

Iωωω ̇=N. (8.108)

The angular frequencyωωωis in turn related to the time derivative of the direction

vectornˆ:

nˆ ̇=ωωω×nˆ. (8.109)

CombiningEqs. (8.108)and(8.109)leads to

n ̇ˆ=ωωω×(ωωω×nˆ)+N×nˆ/I=−ω^2 nˆ+N×nˆ/I. (8.110)

This equation of motion leaves the norm of the direction vectornˆinvariant, as

it should – this follows directly from (8.109). In a numerical integration of the