Efficient “Middle” Thermostat Scheme …
277
×
e
L x δt/2 e
L N δt/2 e
L O δt e
L N δt/2 e
L x δt/2 e
L
( f )
p δt
n−2
× e
L x δt/2 e
L N δt/2 e
L O δt e
L N δt/2 e
L x δt/2 e
L
( f )
p δt+L
(s)
p
(71)
[denoted “LF-Middle-SIN(R)”]. Eqs. (70) and (71) yield the same efficiency
and accuracy for sampling the configuration space. These versions in the “middle”
scheme are all easily extended with more than two time steps for SIN(R).
We have simulated a system of liquid water [216 H2O molecules in a box with
periodic boundary conditions at the state point T = 298.15 K(temperature) and 0.997
g·cm
−3 (density)] to compare the original version of SIN(R) [Eq. (69)] and its new
version with the “middle” scheme (“VV-Middle-SIN(R)”) [Eq. (70)]. POLI2VS-a
polarizable and flexible force field [57] for liquid water is used. In MTS, one may
treat the force contributed by the intramolecular interactions as the fast part and
consider the force derived from the noncovalent interactions as the slow part. Fig. 6
demonstrates the results with the inner time interval δt varied from 0.1 to 2fs while
the outer time interval are fixed at 2 or 4fs. As the inner time step δt increases,
“VV-Middle-SIN(R)” performs more efficiently than conventional SIN(R).
All versions in the “middle” scheme for MD in Sect. 4.1 (with holonomic constraints) and Sect. 4.2 (with nonholonomic constraints as well as resonance-free
multiple time-step techniques) can easily be extended to PIMD. We have already
applied some of them in PIMD simulations for liquid water [10].
-1520
-1510
-1500
-1490
-1480
-1470
0
0 . 5
1
1 . 5
2
SIN(R)
VVMiddle-SIN(R)
VVMiddle-MTS
converged result
Potential energy per atom (K)
Inner time step δt (fs)
Liquid water (POLI2VS model)
Outer time step Δt=2 fs
(a)
-1520
-1500
-1480
-1460
-1440
-1420
-1400
-1380
0
0.5
1
1.5
2
SIN(R)
VVMiddle-SIN(R)
VVMiddle-MTS
converged result
Potential energy per atom (K)
Inner time step δt (fs)
Liquid water (POLI2VS model)
Outer time step Δt=4 fs
(b)
Fig. 6 Average potential energy per atom U (x)/(N atom k B ) (unit: Kelvin) of liquid water at
T = 298.15 K as a function of the inner time step δt with a fixed outer time step. a = 2 fs b
= 4 fs (Reproduced with permission from Ref. [12])
277
×
e
L x δt/2 e
L N δt/2 e
L O δt e
L N δt/2 e
L x δt/2 e
L
( f )
p δt
n−2
× e
L x δt/2 e
L N δt/2 e
L O δt e
L N δt/2 e
L x δt/2 e
L
( f )
p δt+L
(s)
p
(71)
[denoted “LF-Middle-SIN(R)”]. Eqs. (70) and (71) yield the same efficiency
and accuracy for sampling the configuration space. These versions in the “middle”
scheme are all easily extended with more than two time steps for SIN(R).
We have simulated a system of liquid water [216 H2O molecules in a box with
periodic boundary conditions at the state point T = 298.15 K(temperature) and 0.997
g·cm
−3 (density)] to compare the original version of SIN(R) [Eq. (69)] and its new
version with the “middle” scheme (“VV-Middle-SIN(R)”) [Eq. (70)]. POLI2VS-a
polarizable and flexible force field [57] for liquid water is used. In MTS, one may
treat the force contributed by the intramolecular interactions as the fast part and
consider the force derived from the noncovalent interactions as the slow part. Fig. 6
demonstrates the results with the inner time interval δt varied from 0.1 to 2fs while
the outer time interval are fixed at 2 or 4fs. As the inner time step δt increases,
“VV-Middle-SIN(R)” performs more efficiently than conventional SIN(R).
All versions in the “middle” scheme for MD in Sect. 4.1 (with holonomic constraints) and Sect. 4.2 (with nonholonomic constraints as well as resonance-free
multiple time-step techniques) can easily be extended to PIMD. We have already
applied some of them in PIMD simulations for liquid water [10].
-1520
-1510
-1500
-1490
-1480
-1470
0
0 . 5
1
1 . 5
2
SIN(R)
VVMiddle-SIN(R)
VVMiddle-MTS
converged result
Potential energy per atom (K)
Inner time step δt (fs)
Liquid water (POLI2VS model)
Outer time step Δt=2 fs
(a)
-1520
-1500
-1480
-1460
-1440
-1420
-1400
-1380
0
0.5
1
1.5
2
SIN(R)
VVMiddle-SIN(R)
VVMiddle-MTS
converged result
Potential energy per atom (K)
Inner time step δt (fs)
Liquid water (POLI2VS model)
Outer time step Δt=4 fs
(b)
Fig. 6 Average potential energy per atom U (x)/(N atom k B ) (unit: Kelvin) of liquid water at
T = 298.15 K as a function of the inner time step δt with a fixed outer time step. a = 2 fs b
= 4 fs (Reproduced with permission from Ref. [12])