302 Monte Carlo
T 1T 2T 3T 4TMFig. 7.3Schematic of the parallel-tempering replica exchange Monte Carlo.(r(K),r(K+1))→( ̃r(K), ̃r(K+1)) is made, where ̃r(K)=r(K+1) and ̃r(K+1)=r(K);
this move is simply an exchange of coordinates between the systems. Since the co-
ordinates are not actuallychanged(they are not displaced, rotated, etc.) but merely
exchanged, the probability distribution for such trial moves satisfies
T(
̃r(K), ̃r(K+1)|r(K),r(K+1))
=T
(
r(K),r(K+1)| ̃r(K), ̃r(K+1))
, (7.5.3)
so that the acceptance probability becomes
A(
̃r(K), ̃r(K+1)|r(K),r(K+1))
=A
(
r(K+1),r(K)|r(K),r(K+1))
= min[
1 ,
fK(
r(K+1))
fK+1(
r(K))
fK(
r(K))
fK+1(
r(K+1))
]
= min[
1 ,e−∆K,K+1]
, (7.5.4)
where
∆K,K+1= (βK−βK+1)
[
U
(
r(K))
−U
(
r(K+1))]
. (7.5.5)
The improvement in conformational sampling efficiency gained by employing a
parallel-tempering replica exchange Monte Carlo approach is illustrated using the sim-
ple example of a 50-mer alkane system C 50 H 102 in the gas phase using the CHARMM22