230 Isobaric ensembles
a
bc
α
βγFig. 5.2A general parallelepiped showing the convention for the cell vectors and angles.In terms of the cell matrix, the volumeVis easily seen to be
V= det(h). (5.6.3)On the other hand, a little reflection shows that, in fact, only six numbers are needed
to specify the cell: the lengths of the edgesa=|a|,b=|b|, andc=|c|and the angles
α,β, andγbetween them. By convention, these three angles are defined such thatα
is the angle between vectorsbandc,βis the angle between vectorsaandc, andγ
is the angle between vectorsaandb. It is clear, therefore, that the full cell matrix
contains redundant information—in addition to providing informationabout the cell
lengths and angles, it also describes overall rotations of the cell in space, as specified
by the three Euler angles (see Section 1.11), which accounts for the three extra degrees
of freedom.
In order to separate isotropic from anisotropic cell fluctuations,we introduce a
unit box matrixh 0 related tohbyh=V^1 /^3 h 0 such that det(h 0 ) = 1. Focusing
on the isothermal-isobaric ensemble, the changing cell shape underthe influence of an
isotropic applied pressurePcan be incorporated into the partition function by writing
∆(N,P,T) as
∆(N,P,T) =
1
V 0
∫∞
0dV∫
dh 0 e−βPVQ(N,V,h 0 ,T)δ(det(h 0 )−1), (5.6.4)where
∫
dh 0 is an integral over all nine components ofh 0 and theδ-function restricts
the integration to unit box matrices satisfying det(h 0 ) = 1. In eqn. (5.6.4), the explicit
dependence of the canonical partition functionQon both the volumeVand the shape
of the cell described byh 0 is shown.
Rather than integrate overV andh 0 with the constraint of det(h 0 ) = 1, it is
preferable to perform an unconstrained integration overh. This can be accomplished
by a change of variables fromh 0 toh. Since each element ofh 0 is multiplied byV^1 /^3 to
obtainh, the integration measure, which is a nine-dimensional integration, transforms