240 Isobaric ensembles
In most molecular dynamics calculations, the isoenthalpic-isobaric ensemble is em-
ployed only seldom: the most common experimental conditions are constant pressure
and temperature. Nevertheless, eqns. (5.8.6) provide the foundation for molecular
dynamics algorithms capable of generating an isothermal-isobaric ensemble, which
we discuss next.
5.9 Molecular dynamics in the isothermal-isobaric ensemble I:
Isotropic volume fluctuations
Since most condensed-phase experiments are carried out under the conditions of con-
stant temperature and pressure (e.g. thermochemistry), the majority of isobaric molec-
ular dynamics calculations are performed in the isothermal-isobaric ensemble. Because
N,P, andTare the control variables, we often refer to theNPTensemble for short.
Calculations in theNPTensemble require one of the canonical methods of Chapter 4
to be grafted onto an isoenthalpic method in order to induce fluctuations in the en-
thalpy. In this section, we will develop molecular dynamics techniquesfor isotropic
volume fluctuations under isothermal conditions. Following this, we will proceed to
generalize the method for anisotropic cell fluctuations.
Although several algorithms have been proposed in the literature for generating an
NPTensemble, they do not all give the correct ensemble distribution function (Mar-
tynaet al., 1994; Tuckermanet al., 2001). Therefore, we will restrict ourselves to the
approach of Martyna, Tobias, and Klein (1994) (MTK), which has been proved to
yield the correct volume distribution. The failure of other schemes isthe subject of
Problem 5.7.
The starting point for developing the MTK algorithm is eqns. (5.8.6). In order to
avoid having to writeV / ̇ 3 Vrepeatedly, we introduce, as a convenience, the variable
ǫ= (1/3) ln(V/V 0 ), whereV 0 is the reference volume appearing in the isothermal-
isobaric partition function of eqn. (5.3.22). A momentumpǫcorresponding toǫcan be
defined according to ̇ǫ=pǫ/W=V / ̇ 3 V. Note that inddimensions,ǫ= (1/d) ln(V/V 0 )
andpǫ=V /dV ̇. In terms of these variables, eqns. (5.8.6) become, inddimensions,
r ̇i=pi
mi+
pǫ
Wrip ̇i=−∂U
∂ri−
pǫ
WpiV ̇=dV pǫ
Wp ̇ǫ=dV(P(int)−P), (5.9.1)whereP(int)is the internal pressure estimator of eqn. (4.6.57) or eqn. (4.6.58). Although
eqns. (5.9.1) are isobaric, they still lack a proper isothermal coupling and therefore,
they do not generate anNPT ensemble. However, we know from Section 4.10 that
temperature control can be achieved by coupling eqns. (5.9.1) to athermostat.
Before we discuss the thermostat coupling, however, we need to analyze eqns.
(5.9.1) in greater detail, for in introducing the “convenient” variablesǫandpǫ, we have