1549380323-Statistical Mechanics Theory and Molecular Simulation

Sampling 295

of a random vector and normalizing the vector by the length

l=

√

ζ^2 x+ζy^2 +ζz^2 (7.3.39)

to given= (ζx/l,ζy/l,ζz/l). One final random numberηis used to determine a random
rotation angleθ= 2πη, and the rotation formula (see eqn. (1.11.3))

r′i=ricosθ+n(n·ri)(1−cosθ) + (ri×n) sinθ (7.3.40)

is applied to each particle in the rigid body. Once the trial move is generated, eqn.
(7.3.36) is used to determine if the move is accepted or rejected. Such rotational
moves can also be generated using the three Euler angles or quaternions described in
Section 1.11.

Sampling the isothermal-isobaric distribution

The isothermal-isobaric partition function for a system ofN particles at constant
external pressureP and temperatureTis

∆(N,P,T) =

1

V 0

∫∞

0

dVe−βPVQ(N,V,T)

=

1

V 0 N!λ^3 N

∫∞

0

dVe−βPV

∫

D(V)

dr 1 ···drNe−βU(r^1 ,...,rN), (7.3.41)

where the coordinate integrations are limited to the spatial domainD(V) defined by
the containing volume.
Sampling the isothermal-isobaric distribution requires sampling both the particle
coordinates and the volumeV. The former can be done using the uniform sampling
schemes of the previous subsection. A trial volume move fromV toV′can also be
generated from a uniform distribution. A random numberξVis generated and the trial
volume move is given by
V′=V+ (ξV− 0 .5)δ, (7.3.42)

whereδdetermines the size of the volume displacement. Volume moves need to be
handled with some care because each time the volume changes, the particle coordi-
nates must be scaled byr′i= (V′/V)^1 /^3 riand the total potential energy recalculated
before the decision to accept or reject the move can be made. Also, the dependence
of the integration limits in eqn. (7.3.42) on the volume presents an additional compli-
cation. As we saw in eqn. (4.6.51), we can make the volume dependence explicit by
transforming the spatial integrals to scaled coordinatessi=ri/V^1 /^3 , which gives