1549380323-Statistical Mechanics Theory and Molecular Simulation

248 Isobaric ensembles

Ri=

∑n

∑α=1mi,αri,α

n

α=1mi,α

. (5.11.2)

We saw in Section 1.11 that the center-of-mass motion of each molecule can be sep-
arated from internal motion relative to a body-fixed frame. Thus,if the derivation
leading up to eqn. (4.6.57) is repeated using the transformation in eqn. (5.11.1), the
following pressure estimator is obtained:

Pmol(P,R) =

1

dV

∑N

i=1

[

P^2 i

Mi

+Ri·Fi

]

, (5.11.3)

whereMiis the mass of theith molecule, andPiis the momentum of its center of
mass:

Pi=

∑n

α=1

pi,α, (5.11.4)

andFiis the force on the center of mass

Fi=

∑n

α=1

Fi,α. (5.11.5)

The virial term appearing in eqn. (5.11.3)

∑N

i=1

Ri·Fi

is known as themolecular virial.
Given the molecular virial, it is straightforward to derive a molecular dynamics
algorithm for the isoenthalpic-isobaric ensemble that uses a molecular virial. The key
feature of this algorithm is that the barostat coupling acts only on the center-of-mass
positions and momenta. Assuming three spatial dimensions and no constraints between
the molecules, the equations of motion take the form

r ̇i,α=

pi,α

mi,α

+

pǫ

W

Ri

p ̇i,α=Fi,α−

(

1 +

1

N

)

pǫ

W

mi,α

Mi

Pi

V ̇=dV pǫ

W

p ̇ǫ=dV(Pmol−P) +

1

N

∑N

i=1

P^2 i

Mi

. (5.11.6)

These equations have the conserved energy

H′=

∑

i,α

p^2 i,α

2 mi,α

+U(r) +

p^2 ǫ

2 W

+PV, (5.11.7)

where therinU(r) denotes the full set of atomic positions. The proof that these
equations generate the correct isobaric-isoenthalpic ensemble is left as an exercise