1549380323-Statistical Mechanics Theory and Molecular Simulation

Numerical evaluation 479

×exp

{

−β

∑P

k=1

[

p^2 k

2 m′k

+

1

2

mkω^2 Pu^2 k+

1

P

U(xk(u))

]}

. (12.6.9)

In eqn. (12.6.9), the parametersmkare defined to be

m 1 = 0, mk=

k

k− 1

m, k= 2,...,P, (12.6.10)

andm′ 1 =m,m′k=mk. The notationxk(u) indicates the inverse transformation in
eqn. (12.6.6) or (12.6.7). In order to evaluate eqn. (12.6.9), we canemploy a classical
Hamiltonian of the form

H ̃cl(u,p) =

∑P

k=1

[

p^2 k

2 m′k

+

1

2

mkω^2 Pu^2 k+

1

P

U(xk(u))

]

, (12.6.11)

which leads to the equations of motion

u ̇k=

pk

m′k

p ̇k=−mkω^2 Puk−

1

P

∂U

∂uk

. (12.6.12)

From the chain rule, the forces on the staging variables can be expressed recursively
as

1

P

∂U

∂u 1

=

1

P

∑P

l=1

∂U

∂xl

1

P

∂U

∂uk

=

1

P

[

∂U

∂xk

+

k− 2

k− 1

∂U

∂uk− 1

]

. (12.6.13)

The recursive staging force calculation is performed starting withk= 2 and using
the first expression for∂U/∂u 1. Eqns. (12.6.12) need to be thermostatted to ensure
that the canonical distribution is generated. The presence of thehigh-frequency force
on each staging variable combined with the 1/Pfactor that attenuates the potential-
energy derivatives leads to a weak coupling between these two forces. Therefore, it is
important to have as much thermalization as possible in order to achieve equiparti-
tioning of the energy. It is, therefore, strongly recommended (Tuckermanet al., 1993)
that path-integral molecular dynamics calculations be carried out using the “massive”
thermostatting mechanism described in Section 4.10. This protocolrequires that a
separate thermostat be attached to each Cartesian componentof every staging vari-
able. Thus, for the single-particle one-dimensional system described by eqns. (12.6.12),
if Nos ́e–Hoover chain thermostats of lengthMare employed, the actual equations of
motion would be