1549380323-Statistical Mechanics Theory and Molecular Simulation

Molecular dynamics 237

P(int)αβ (r 1 ,...,rN,p 1 ,...,pN) =

1

det(h)

∑N

i=1

[

pi,αpi,β

mi

+Fi,αri,β

]

, (5.7.27)

which is equivalent to eqn. (5.7.1), thus completing the derivation. The isotropic pres-
sure estimator forP(int)in eqn. (4.6.57) can be obtained directly from the pressure
tensor estimator by tracing:

P(int)(r,p) =

1

3

∑

α

P(int)αα (r,p) =

1

3

Tr

[

P(int)(r,p)

]

, (5.7.28)

whereP(int)(r,p) is the tensorial representation of eqn. (5.7.27). Finally, note that if
the potential has an explicit dependence on the cell matrixh, then the estimator is
modified to read

P(int)αβ (r,p) =

1

det(h)

∑N

i=1

[

(pi·ˆeα)(pi·ˆeβ)

mi

+ (Fi·ˆeα)(ri·ˆeβ)

]

−

1

det(h)

∑^3

γ=1

∂U

∂hαγ

hγβ. (5.7.29)

5.8 Molecular dynamics in the isoenthalpic-isobaric ensemble

The derivation of the isobaric ensembles requires that the volume beallowed to vary in
order to keep the internal pressure equal, on average, to the applied external pressure.
This suggests that if we wish to develop a molecular dynamics technique for generat-
ing isobaric ensembles, we could introduce the volume as an independent dynamical
variable in the phase space. Indeed, the work-virial theorem of eqn. (5.4.8) strongly
supports such a notion, since it effectively assigns an energy ofkTto a “volume mode.”
The idea of incorporating the volume into the phase space as an additional dynamical
degree of freedom, together with its conjugate momentum, as a means of generating
an isobaric ensemble was first introduced by Andersen (1980) and later generalized
for anisotropic volume fluctuations by Parrinello and Rahman (1980). This idea in-
spired numerous other powerful techniques based on extended phase spaces, including
the canonical molecular dynamics methods from Chapter 4, the Car-Parrinello ap-
proach (Car and Parrinello, 1985; Tuckerman, 2002; Marx and Hutter, 2009) for per-
forming molecular dynamics with forces obtained from “on the fly” electronic structure
calculations, and schemes for including nuclear quantum effects in molecular dynam-
ics (see Chapter 12). In this section, we present Andersen’s original method for the
isoenthalpic-isobaric ensemble and then use this idea as the basis fora non-Hamiltonian
isothermal-isobaric molecular dynamics approach in Section 5.9.
Andersen’s method is based on the remarkably simple yet very elegant idea that
the scaling transformation used to derive the pressure

si=V−^1 /^3 ri, πi=V^1 /^3 pi (5.8.1)

is all we need to derive an isobaric molecular dynamics method. This transformation is
used not only to make the volume dependence of the coordinates and momenta explicit