Computational Physics

8.5 Molecular dynamics methods for different ensembles 231
The disadvantage of the real-time equations is that they are not Hamiltonian, in
the sense that they cannot be derived from a Hamiltonian. Although this might not
seem to be a problem, we prefer Hamiltonian equations of motion as they allow
for stable (symplectic) integration methods as discussed in Section 8.4. Winkler
et al.[43]have formulated canonical equations of motion in real-time but these are
subject to severe numerical problems when integrating the equations of motion for
large systems.

8.5.2 Keeping the pressure constant

In experimental situations not only the temperature is kept under control but also
the pressure. The partition function for the(NpT)-ensemble is given as

Q(N,p,T)=

∫

dVe−βpV

1

N!

∫

dRdPe−βH(R,P)=

∫

dVe−βpVZc(N,V,T)
(8.97)
(seeChapter 7). We use a lower-casepfor the pressure in order to avoid confusion
with the total momentum coordinateP. We now describe the scheme which is
commonly adopted for keeping the pressure constant but do not go into too much
detail as the analysis follows the same lines as the Nosé–Hoover thermostat, and
refer to the literature for details [ 32 , 34 , 37 ].
Andersen first presented this scheme. He proposed incorporating the volume into
the equations of motion as a dynamical variable and scaled the spatial coordinates
back to a unit volume:
r′i=riV^1 /^3 , (8.98)

where again the prime denotes the real coordinate – unprimed coordinates are those
of the virtual system. Moreover

p′i=pi/(sV^1 /^3 ). (8.99)

The canonical Hamiltonian is extended bytwovariables, the volumeVand the
canonical momentumpVwhich can be thought of as the momentum of a piston
closing the system.^7 The Hamiltonian has an extra ‘potential energy’ termpVand
a ‘kinetic’ termp^2 V/ 2 W(Wis the ‘mass’ of the piston, andpVits momentum):

H(P,R,ps,s,pV,V)=

∑

i

p^2 i

2 mV^2 /^3 s^2

+

1

2

∑

ij,i
=j

U[V^1 /^3 R]

+

p^2 s

2 Q

+gkTln(s)+pV+p^2 V/ 2 W. (8.100)

(^7) Note that the system expands and contracts isotropically, so instead of a piston, the whole system boundary
moves.