232 Molecular dynamics simulations
The equations of motion now read:
dr
dt=
∂H
∂pi=
pi
mV^2 /^3 s^2(8.101a)ds
dt=
∂H
∂ps=
ps
Q(8.101b)dpi
dt=−
∂H
∂ri
=−∇iU(V^1 /^3 R) (8.101c)dps
dt=−
∂H
∂s=
( ∑
ip
2
i
mV^2 /^3 s^2−gkBT)
/s (8.101d)dV
dt=
∂H
∂pV=
pV
W(8.101e)dpV
dt=−
∂H
∂V
=
( ∑
ip
2
i
mV^2 /^3 s^2−
∑
i∇iU(V^1 /^3 R)·ri)
/( 3 V)−p. (8.101f )It can be shown in the same way as in the thermostat case that the distribution of
configurations corresponds to that of the (NpT) ensemble:
ρ(P′,R′,V)=VNexp{−[H 0 (P′,R′)+pV]/kBT}. (8.102)Hoover[37]proposed similar equations of motion which conserve phase space, but
they differ from this distribution by an extra factorVin front of the exponent[44].
The method described is restricted to isotropic volume changes and can therefore
not be used for studying structural phase transitions in solids. A method which
allows for anisotropic volume changes while keeping the pressure constant was
developed by Parrinello and Rahman[45].
8.6 Molecular systems
8.6.1 Molecular degrees of freedomInteractions in molecular systems can be divided into intramolecular and inter-
molecular ones. The latter are often taken to be atom-pair interactions similar to
those considered in the previous sections. The intramolecular interactions (i.e. the
interactions between the atoms of one molecule) are determined by chemical bonds,
so not only are they strong compared with the intermolecular interactions (between
atoms of different molecules) but they also include orientational dependencies. We
now briefly describe the intramolecular degrees of freedom and interactions (see
alsoFigure 8.4).
First of all, the chemical bonds can stretch. The interaction associated with this
degree of freedom is usually described in the form of a harmonic potential for the