1549380323-Statistical Mechanics Theory and Molecular Simulation

Virials 247

Z=

∫

dNpdNrdhdpgdη 1 dηcdξ 1 dξcdMpηdMpξ[det(h)]^1 −dedNη^1 +ηced

(^2) ξ 1 +ξc
×δ



H(r,p) +

∑M

j=1

[

p^2 ηj

2 Qj

+

p^2 ξj

2 Q′j

]

+NfkTη 1 +d^2 kTξ 1 +kT[ηc+ξc]

+

Tr

[

pTgpg

]

2 Wg

+Pdet[h]−H

)

. (5.10.12)

If we now integrate overη 1 using theδ-function, we find

Z∝

∫

dh[det(h)]^1 −de−βPdet(h)dNpdNre−βH(r,p), (5.10.13)

where the constant of proportionality includes uncoupled integrations over the re-
maining thermostat/barostat variables. Thus, the correct isothermal-isobaric partition
function for fully flexible cells is recovered.

5.11 Atomic and molecular virials

The isotropic pressure estimator in eqn. (5.7.28) and pressure tensor estimator in eqn.
(5.7.1) were derived assuming a scaling or matrix multiplication of all atomic positions.
The resulting virial term in the estimator

∑N

i=1

ri·Fi

is, therefore, known as anatomic virial. Although mathematically correct and physi-
cally sensible for purely atomic systems, the atomic virial might seem to be an overkill
for molecular systems. In a collection of molecules, assuming no constraints, the force
Fiappearing in the atomic virial contains both intramolecular and intermolecular
components. If the size of the molecule is small compared to its container, it is more
intuitive to think of the coordinate scaling (or multiplication by the cellmatrix) as
acting only on the centers of mass of the molecules rather than on each atom indi-
vidually. That is, the scaling should only affect the relative positions ofthe molecules
rather than the bond lengths and angles within each molecule. In fact, an alternative
pressure estimator can be derived by scaling only the positions of the molecular centers
of mass rather than individual atomic positions.
Consider a system ofN molecules with centers of mass at positionsR 1 ,...,RN.
For isotropic volume fluctuations, we would define the scaled coordinatesS 1 ,...,SNof
the centers of mass by
Si=V−^1 /dRi. (5.11.1)

If each molecule hasnatoms with massesmi, 1 ,...,mi,nand atomic positionsri, 1 ,...,ri,n,
then the center-of-mass position is