Computational Physics

234 Molecular dynamics simulations

vary as the square root of theα-coefficients. In general, the bond length vibrations
are the most rapid, followed by the bending vibrations. For an MD integration to
be accurate, the time step should be chosen smaller than the fastest degree of free-
dom. But as this degree of freedom will vibrate with a small amplitude, because
of the strong potential, we are using most of the computer time for those parts of
the motion that are not expected to contribute strongly to the physical properties
of the system. Moreover, if there is a clear separation between the time scales of
the various degrees of freedom of the system, energy transfer between the fast and
slow modes is extremely slow, so that it is difficult, if not impossible, to reach
equilibrium within a reasonable amount of time. In such a case it is advisable to
‘freeze’ the fast modes by keeping them rigorously fixed in time. In practice this
means that lengths of chemical bonds can safely be kept fixed, and perhaps some
bending angles. In a more approximate description it is also possible to consider
entire molecules as being rigid. In the next subsections we shall describe how to
deal with rigid and partly rigid molecules.

8.6.2 Rigid molecules

We consider molecules which can be treated as rigid bodies whose motion consists
of translations of the centre of mass and rotations around this point. The forces acting
between two rigid molecules are usually composed of atomic pair interactions
between atoms belonging to the two different molecules.^8 The total force acting
on a molecule determines the translational motion and the torque determines the
rotational motion. In the next subsection, we shall describe a direct formulation of
the equations of motion of a simple rigid molecule – the nitrogen molecule. In the
following subsection we shall then describe a different approach in which rigidity
is enforced through constraints added to the Lagrangian.

Direct approach for the rigid nitrogen molecule

As a simple example consider the nitrogen molecule, N 2. This consists of two nitro-
gen atoms, each of massm≈14 atomic mass units (a.m.u.) and whose separation
dis kept fixed in the rigid approximation. The coordinates of the molecule are the
three coordinates of the centre of mass and the two coordinates defining its orient-
ation. The latter can be polar angles but here we shall characterise the orientation
of the molecule by a unit direction vectornˆ, pointing from atom 1 to atom 2 (see
Figure 8.5).
The motion of the centre of mass of the molecule is determined by the total force
Ftotacting on a particular molecule. This force is the sum of all the forces between

(^8) Sometimes, off-centre interactions (i.e. not centred on the atomic positions) are taken into account too but
we shall not consider these.