the system, rather then to particles on the surface as a volume change in a

real experiment would do. This, of course, is also a choice that in¬‚uences the

instantaneous pressure, but not the ensemble-averaged pressure. Equation

(17.138) is obtained.26 It is interesting that this coupling to all particles

is equivalent to Irving and Kirkwood™s choice for the local pressure de¬ni-

tion. Explicit equations for use with Ewald summation have been given by

Nos´ and Klein (1983) and for use with the Particle Mesh Ewald method by

e

Essmann et al. (1995).

Pressure from center-of-mass attributes

Thus far we have considered detailed atomic motion and forces on atoms

to determine the pressure. However, pressure is a result of translational

motion and forces causing translational motion. For a system consisting of

molecules it is therefore possible to consider only the center-of-mass (c.o.m.)

velocities and the forces acting on the c.o.m. Equation (17.128) is equally

valid when F i are the forces acting on the c.o.m. and r i and v i denote

c.o.m coordinates and velocities. Somehow the extra “intramolecular” virial

should just cancel the intramolecular kinetic energy. Can we see why that

is so?27

Consider a system of molecules with c.o.m. position Ri , each consisting

of atoms with positions r i (see Fig. 17.10). A total force F i is acting on

k

k

this atom. Denoting the mass of the molecule as Mi = k mi , the c.o.m.

k

coordinate is given by

mi r i .

Mi R i = (17.141)

kk

k

Now we de¬ne intramolecular coordinates si of each atom with respect to

k

26 Pressure calculations on the basis of Hamiltonian derivatives and their use in constant pressure

algorithms have been pioneered by Andersen (1980) for isotropic pressure and by Parrinello

and Rahman (1980, 1981) for the pressure tensor. A good summary is to be found in Nos´ e

and Klein (1983) and an extensive treatment of pressure in systems with constraints has been

given by Ciccotti and Ryckaert (1986).

27 We roughly follow the arguments given in an appendix by Ciccotti and Ryckaert (1986).

490 Review of statistical mechanics

atom k

mass mki

ski

rki molecule j

mass Mj

Rj

Ri

molecule i

mass Mi

origin

Figure 17.10 De¬nition of atoms clustered in molecules for discussion of the c.o.m.

virial.

the c.o.m.:

def

si = r i ’ R i , (17.142)

k k

with the obvious relation

mi si = 0. (17.143)

kk

k

The total virial (see (17.130)) on an atomic basis can be split into a c.o.m.

and an intramolecular part:

1 1

Ξtot = ’ F i ri = ’ F i (Ri + si )

kk k k

2 2

i i

k k

1 1

=’ F i Ri ’ F i si = Ξcom + Ξintra . (17.144)

tot tot

kk

2 2

i k

The forces are the total forces acting on the atom. Likewise we can split the

kinetic energy:

1 1 ™ ™k ™

mi r i r i = mi (Ri + si )(Ri + si )

k ™k ™k ™k

Ekin = k

2 2

i i

k k

1 1

™™ mi si si

k ™k ™k

= Mi R i R i +

2 2

i i k

com intra

= Ekin + Ekin . (17.145)

If we can prove that Ξintra = Ekin , then we have proved that the pressure

intra

tot

computed from c.o.m. forces and velocities equals the atom-based pressure.

17.7 Pressure and virial 491

The proof is simple and rests on the fact that neither the internal coordinates

nor the internal velocities can grow to in¬nity with increasing time. First

realize that F i = mi r i ; then it follows that for every molecule (we drop

k ¨k

k

the superscripts i and use (17.143))

’ F k sk = ’ ¨

mk sk sk . (17.146)

k k

Ensemble-averaging can be replaced by time averaging:

T

1

’ = ’ lim ¨

mk sk sk dt

F k sk

T ’∞ T 0

k k

1

mk sk sk ’ lim [sk sk (T ) ’ sk sk (0)].(17.147)

™™ ™ ™

=

T ’∞ T

k

Since the last term is a ¬‚uctuating but bounded quantity divided by T , the

limit T ’ ∞ is zero and we are left with the equality of intramolecular virial

and kinetic energy, if averaged over an equilibrium ensemble. The reasoning

is equivalent to the proof of the virial theorem (17.132) (Hirschfelder et al.,

1954). Note that the “molecule” can be any cluster of particles that does

not fall apart in the course of time; there is no requirement that the cluster

should be a rigid body.

The subtlety of virial-kinetic energy compensation is nicely illustrated

by the simple example of an ideal gas of classical homonuclear diatomic

molecules (bond length d) with an internal quadratic potential with force

constant k. We can calculate the pressure from the total kinetic energy mi-

nus the internal atomic virial, but also from the kinetic energy of the c.o.m.

minus the molecular virial. So the virial of the internal forces, which can