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X. Liu et al.
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Liquid water (298.15 K)
middle-Andersen
middle-Langevin
middle-NHC
side-Andersen
side-Langevin
side-NHC
Absolute difference between the
primitive and virial estimators
ΔE
kin
(K)
Time interval (fs)
(a)
-500
-480
-460
-440
-420
-400
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Liquid water (298.15 K)
middle-Andersen
middle-Langevin
middle-NHC
side-Andersen
side-Langevin
side-NHC
Potential energy per atom (K)
Time interval (fs)
(b)
Fig. 3 PIMD results using different time intervals for liquid water at T = 298.15 K. a Absolute
difference between the primitive and virial estimators for averaged kinetic energy per atom (unit:
Kelvin). b The averaged potential energy per atom U (x)/(N atom k B ) (unit: Kelvin). Statistical
error bars are included. (Reproduced with permission from Ref. [9])
the Maxwell momentum distribution. This is verified by the simulation results for
PIMD in Ref. [9] when the Andersen thermostat and NHC are used.
We apply the “middle” and conventional “side” schemes with PIMD algorithms
for liquid water system to study the canonical ensemble at constant temperature
T = 298.15 K and ρ l = 0.997 g · cm
−3 . As presented in Fig. 3, in comparison
to the “side” scheme, the “middle” scheme reduces the error by about an order of
magnitude with the same time interval.
3.3 Normal-mode Path Integral Molecular Dynamics
Consider the normal mode transformation of path integral beads [48, 51, 52]
x = C
norm q
(38)
with q =
q 0 . . . q P−1
T and x =
x 1 . . . x P
T , where the element of the
orthogonal transformation matrix C
norm is