1549380323-Statistical Mechanics Theory and Molecular Simulation

Numerical evaluation 483

with the conditionr(iP+1)=r(1)i. An important point to make about eqn. (12.6.24) is

that the potentialU(r( 1 k),...,r(Nk)) only acts between beads with the same imaginary
time indexk. This means that all beads with imaginary-time index 1 interact with each
other, but these do not interact with beads having imaginary-time indices 2,3,....,P
and so forth. This is illustrated for the case of two particles in Fig. 12.11.

1

2

3

4

5

6

7

8

1

2

(^43)

6

7 8

5

Fig. 12.11Interaction pattern between two quantum particles represented as cyclic polymer
chains within the discrete path-integral framework. The cyclic chains obey the rule that only
“beads” with the same imaginary-time index on different chains interact with each other.

The construction of a path-integral molecular dynamics algorithm forN Boltz-
mann particles inddimensions proceeds in the same manner as for a single particle in
one dimension. First, a transformation from primitive to staging or normal-mode vari-
ables is performed for each quantum particle’s cyclic path. In staging or normal-mode
variables, the classical Hamiltonian from which the equations of motion are derived is

H=

∑P

k=1

[N

∑

i=1

p(k)

2

i

2 m(k)

′

i

+

∑N

i=1

1

2

m(ik)ω^2 Pu(k)

2

i +

1

P

U

(

r( 1 k)(u 1 ),...,r(Nk)(uN)

)

]

.(12.6.25)

Here, each primitive variabler(ik)depends on the staging or normal-mode variables
with the same particle indexi. In deriving the equations of motion from eqn. (12.6.25),
the forces on the mode variables are obtained using eqns. (12.6.13)or (12.6.20). Im-
portantly, if the equations of motion are coupled to Nos ́e–Hooverchains, it is critical
to follow the protocol of coupling each component of each staging or normal-mode
variable to its own thermostat, for a total ofdNP thermostats. At first sight, this
might seem like overkill because it addsdNMP additional degrees of freedom to a
system, whereMis the length of each Nos ́e–Hoover chains. However, if we think back
to Fig. 4.12 and note that a path integral, according to eqn. (12.6.25), is a collection
of weakly coupled harmonic oscillators, then this protocol makes sense. Generally, the
computational overhead ofdNMPthermostats is small compared to that associated
with the calculation of the forces.