256 Isobaric ensembles
0.6 0.65 0.7 0.75 0.8
r*0
5
10
15
20
25
30
P(
r*)Fig. 5.5Density distribution for the argon system atP= 0.5 kbar for two different values
ofτb(Tuckermanet al., 2006). The solid curve with filled circles represents the fitto the
Gaussian form in eqn. (5.12.29)
5.13 The isothermal-isobaric ensemble with constraints: The
ROLL algorithm
Incorporating holonomic constraints into molecular dynamics calculations in the iso-
baric ensembles introduces new technical difficulties. The forces in the virial contri-
butions to the pressure and pressure tensor estimators must also include the forces of
constraint. According to eqn. (3.9.5), the force on atomiin anN-particle system is
F ̃i=Fi+∑
kλkF
(k)
c,i, whereF(k)
c,i=∇iσk(r^1 ,...,rN), whereFi=−∂U/∂ri, and the
virial part of the pressure is
P(vir)=1
dV∑N
i=1[
ri·Fi+ri·∑
kλkF(c,ik)]
. (5.13.1)
The integration algorithm for eqns. (5.9.5) encoded in the factorization of eqn. (5.12.4)
generates a nonlinear dependence of the coordinates and velocities on the barostat
variablesvǫorvg, while these variables, in turn, depend linearly on the pressure
or pressure tensor. The consequence is that the coordinates and velocities acquire a
complicated dependence on the Lagrange multipliers, and solving formultipliers is
much less straightforward than in the constant-volume ensembles(see Section 3.9).