1549380323-Statistical Mechanics Theory and Molecular Simulation

Anisotropic cells 229

H=−

∂

∂β

ln ∆ = (N+ 1)kT+

3

2

NkT≈

5

2

NkT, (5.5.8)

from which the constant pressure heat capacity is given by

CP=

(

∂H

∂T

)

=

5

2

Nk. (5.5.9)

Eqns. (5.5.8) and (5.5.9) are usually first encountered in elementaryphysics and chem-
istry textbooks with no microscopic justification. This derivation shows the micro-
scopic origin of eqn. (5.5.9). Note that the difference between the constant volume and
constant pressure heat capacities is

CP=CV+Nk=CV+nR, (5.5.10)

where the productNkhas been replaced bynR, withnthe number of moles of gas
andRthe gas constant. (This relation is obtained by multiplying and dividing byN 0 ,
Avogadro’s number,Nk= (N/N 0 )N 0 k=nR.) Dividing eqn. (5.5.10) by the number
of moles leads to the familiar relation for the molar heat capacities:

cP=cV+R. (5.5.11)

5.6 Extending the isothermal-isobaric ensemble: Anisotropic cell

fluctuations

In this section, we will show how to account for anisotropic volume fluctuations within
the isothermal-isobaric ensemble. Anisotropic volume fluctuations can occur under a
wide variety of external conditions; however, we will limit ourselves to those that de-
velop under an applied isotropic external pressure. Other external conditions, such as
an applied pressure in two dimensions, would generate a constant surface tension en-
semble. The formalism developed in this chapter will provide the reader with the tools
to understand and develop computational approaches for different external conditions.
When the volume of a system can undergo anisotropic fluctuations,it is necessary
to allow the containing volume to change its basic shape. Consider a system con-
tained within a general parallelepiped. The parallelepiped representsthe most general
“box” shape and is appropriate for describing, for example, solids whose unit cells are
generally triclinic. As shown in Fig. 5.2, any parallelepiped can be specified by the
three vectorsa,b, andcthat lie along three edges originating from a vertex. Simple
geometry tells us that the volumeV of the parallelepiped is given by

V=a·b×c. (5.6.1)

Since each edge vector contains three components, nine numberscan be used to char-
acterize the parallelepiped; these are often collected in the columnsof a 3×3 matrix
hcalled thebox matrixorcell matrix:

h=





ax bx cx

ay by cy

az bz cz



. (5.6.2)