r · dS = pV,

Ξext = p (17.133)

2 2

S

the virial theorem immediately yields (17.131) (see Exercise 17.6).

Periodic boundary conditions

The virial expression and the pressure equations given above are valid for

a bounded system, but not for an in¬nite system with periodic boundary

conditions, as so often used in simulations. Under periodic boundary condi-

tions the force on every particle may be zero while the system has a non-zero

pressure: imagine the simple case of one particle in a cubic box, interacting

symmetrically with six images. Then i F i r i is obviously zero. For forces

that can be written as a sum of pair interactions this problem is easily

remedied by replacing i F i r i by a sum over pairs:25

F i ri = F ij r ij , (17.134)

i i<j

where F ij is the force on i due to j and r ij = r i ’r j . This sum can be taken

over all minimum-image pairs if no more than minimum images are involved

in the interaction. For interactions extending beyond minimum images, the

pressure tensor can be evaluated over the volume of a unit cell, according

to Irving and Kirkwood™s distribution over a straight line, using the fraction

of the line lying within the unit cell. Consider the simple case, mentioned

above, with a single particle in a cubic unit cell of size a — a — a.

24 See, e.g., Hirschfelder et al. (1954). A very extensive review on the virial theorem, including a

discussion on the quantum-mechanical form, has been published by Marc and McMillan (1985).

25 Erpenbeck and Wood (1977).

17.7 Pressure and virial 487

F

T1

u p a2

Assume that each of its six images

exert a force F on the particle, with

a zero vector sum. Each contribu-

ccccc

tion F a to the sum i F i · r i counts

F3

for 0.5 since the interaction line lies

' T F4

'u u

E u

E

for 50% in the unit cell, so the to-

F2 c

F4 F2

tal sum is 3F a, and the virial con-

F1

tribution to the pressure (17.129)

equals F a/V = F/a2 . This is cor-

rect, as we can see that this pres-

u

sure, if acting externally on one side

c3

F

with area a2 , just equals the force

acting “through” that plane.

Now consider the interaction between two particles i and j (Fig. 17.9).

In a lattice sum interactions between all image pairs of both i and j are

included; note that there are always sets of equivalent pairs, e.g., in the case

of a single shift with n = (010) (Fig. 17.9b):

r i ’ (rj + Tn) = (r i ’ Tn) ’ rj , (17.135)

where T is the transformation matrix from relative coordinates in the unit

cell to cartesian coordinates (see (6.3) on page 143), i.e., a matrix of which

the columns are the cartesian base vectors of the unit cell, and n ∈ Z3 .

Figure 17.9 shows three examples of image pairs, with one, two and three

equivalent pairs, respectively. If we add up the fractions of the interaction

lines that run through the unit cell, we obtain just one full interaction line,

F ij · r ij is given by

so the contribution of that set of pairs to the sum

F ijn · (r i ’ rj ’ Tn). Note that each set of equivalent pairs contributes only

once to the total energy, to the force on i, to the force on j and to the

virial contribution to the pressure. Replacing the dot product by a dyadic

product, the scalar contribution is generalized to a tensorial contribution.

Summarizing, for a lattice sum of isotropic pair interactions vij (r), the total

potential energy, the force on particle i and the instantaneous pressure tensor

(see (17.127)) are given by

1

Epot = vij (rijn ), (17.136)

2

i,j,n

dvij rijn

Fi = F ijn , F ijn = , (17.137)

dr |rijn |

j,n

488 Review of statistical mechanics

(a)

(b) (c)

Figure 17.9 Three examples of “equivalent image pairs” of two interacting particles

(open circle and open square in the shaded unit cell; images are ¬lled symbols). For

each image pair the fraction of the dashed interaction line lying in the unit cell,

which contributes to the pressure, is drawn as a solid line.

1

ΠV = F ijn r ijn + mi v i v i . (17.138)

2

i,j,n i

Here we use the notation

rijn = |r ijn |, (17.139)

r ijn = r i ’ r j ’ Tn. (17.140)

Note that the volume V is equal to det T. The sum is taken over all

particles i and all particles j in the unit cell, and over all sets of three

integers n : n ∈ Z3 . This includes i and its images; the prime in the sum

means that j = i is excluded when n = 0. The factor 1 prevents double

2

counting of pairs, but of course summation over i, j; i = j can be restricted

to all i with all j > i because i and j can be interchanged while replacing

n by ’n. The factor 1 must be maintained in the iin summation. The

2

summation over images may well be conditionally convergent, as is the case

for Coulomb interactions. This requires a speci¬ed summation order or

17.7 Pressure and virial 489

special long-range summation techniques, as discussed in Section 13.10 on

page 362.

The equation for the pressure (17.138) is usually derived for the canonical

ensemble from the equation p = ’kB T (‚ ln Q/‚V )T or a tensorial variant

that implies di¬erentiating to the components of the transformation tensor

T. The volume dependence in the partition function is then handled by

transforming to scaled coordinates ρ (r = Tρ) which concentrates the vol-