1549380323-Statistical Mechanics Theory and Molecular Simulation

Approximations 563

U 0 (xc) =−

1

β

ln

{(

2 πβ ̄h^2

m

) 1 / 2 ∮

Dx(τ)δ(x 0 [x(τ)]−xc)e−S[x(τ)]/ ̄h

}

, (14.6.18)

wherex 0 [x(τ)] = (1/β ̄h)

∫

dτ x(τ). In eqn. (14.6.18),S[x(τ)] is the Euclidean time
action, and theδ-function restricts the functional integration to cyclic paths whose
centroid position isxc. Note that eqn. (12.6.23), in the limitP→∞, is equivalent to
eqn. (14.6.18). Of course, in actual calculations, we use the discretized, finite-Pversion
in eqn. (12.6.23). The centroid force atxc,F 0 (xc) is derived from eqn. (14.6.18) simply
by spatial differentiation:

F 0 (xc) =−

∮

Dx(τ)δ(x 0 [x(τ)]−xc)

[

1

β ̄h

∫β ̄h

0 dτ

′U′(x(τ′))

]

e−S[x(τ)]/ ̄h

∮

Dx(τ)δ(x 0 [x(τ)]−xc) e−S[x(τ)]/ ̄h

. (14.6.19)

In a path-integral molecular dynamics or Monte Carlo calculation, the centroid force
would be computed simply from

F 0 (xc) =−

〈

1

P

∑P

k=1

∂U

∂xk

δ

(

1

P

∑P

k=1

xk−xc

)〉

f

. (14.6.20)

Although formally exact within the CMD framework, eqns. (14.6.18),(14.6.19), and
eqn. (14.6.20) are of limited practical use: Their evaluation entails a full path integral
calculation at each centroid configuration, which is computationally very demanding
for complex systems.
In order to alleviate this computational burden, an adiabatic approximation, simi-
lar to that of Section 8.10, can be employed (Cao and Voth, 1994b; Cao and Martyna,
1996). In this approach, an ordinary imaginary-time path-integral molecular dynam-
ics calculation in the normal-mode representation of eqn. (12.6.17) isperformed with
two small modifications. First, the noncentroid modes are assignedmasses that are
significantly lighter than the centroid mass(es) so as to effect an adiabatic decoupling
between the two sets of modes. According to the analysis of Section 8.10, this al-
lows the centroid potential of mean force to be generated “on thefly” as the CMD
simulation is carried out. The decoupling is achieved by introducing an adiabaticity
parameterγ^2 (0< γ^2 <1), which is used to scale the fictitious kinetic masses of
the internal modes according tom′k=γ^2 mλkand thereby accelerate their dynamics.
Second, while the path-integral molecular dynamics schemes of Section 12.6 employ
thermostats on every degree of freedom, in the adiabatic CMD approach, because we
require the actual dynamics of the centroids, only the noncentroid modes are coupled
to thermostats.
The key assumption of CMD is that the Kubo-transformed quantumtime correla-
tion functionKAB(t) of eqn. (14.6.11), for operatorsAˆandBˆthat are functions of ˆx
can be approximated by

KAB(t)≈

1

Q(β)

∫

dxcdpca(xc)b(xc(t;xc,pc)) exp

[

−β

(

p^2 c

2 m

+U 0 (xc)

)]

.

(14.6.21)

Here, the functionb(xc(t;xc,pc)) is evaluated using the time-evolved centroid variables
generated by eqns. (14.6.17), starting from{xc,pc}as initial conditions. An analogous