Efficient “Middle” Thermostat Scheme …
261
That is, in the harmonic limit, either the “side” or “end” schemes with a finite
time interval leads to the exact momentum distribution but an approximate configurational distribution, and in contrast, the “middle” scheme produces the exact
configurational distribution but an approximate momentum distribution. Since evaluation of most properties in MD and that of all properties in PIMD depend on only the
configurational distribution, the “middle” scheme is a better choice for MD/PIMD
simulations.
2.1.2 “Middle” scheme with the Nosé-Hoover chain thermostat
Nosé-Hoover chain (NHC) [21] performs deterministic MD by adding extended
degrees of freedom to control the temperature of the system. The equations of motion
for NHC read
˙
x i =
p i
m i
˙
p i = −
∂U
∂ x i
−
p
η
(i)
1
Q 1
p i
˙
η
(i)
j =
p
η
(i)
j
Q j
˙
p η
(i)
1
=
p
2
i
m i
− k B T −
p
η
(i)
2
Q 2
p η
(i)
1
˙
p η
(i)
j
=
p
2
η
(i)
j−1
Q j−1
− k B T −
p
η
(i)
j+1
Q j+1
p η
(i)
j
j = 2, M NHC − 1
˙
p η
(i)
M NHC
=
p
2
η
(i)
M NHC −1
Q M NHC −1
− k B T
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
i = 1, 3N
(13)
where M NHC means the number of pairs for the additional variables
η
(i)
j , p η
(i)
j
j = 1, M NHC
coupled to each degree of freedom
i = 1, 3N
, the
parameters Q 1 , . . . , Q M NHC are the NHC thermal masses. For achieving the accuracy, one often uses the multiple time-scale techniques including the reference system propagator algorithm (RESPA) [23] and higher-order factorizations such as the
Suzuki-Yoshida decomposition framework [42–44]. When the phase space density
propagator for the thermostat part e
L T t is effectively accurate, the thermostat propagation keeps the Maxwell-Boltzmann distribution for the momentum effectively
unchanged, i.e.,
e
L T t exp
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
−β
⎡
⎢
⎢
⎣
1
2
p T M −1 p +
3N
i=1
M NHC
j=1
p 2
η
(i)
j
2Q j
⎤
⎥
⎥
⎦
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
= exp
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
−β
⎡
⎢
⎢
⎣
1
2
p T M −1 p +
3N
i=1
M NHC
j=1
p 2
η
(i)
j
2Q j
⎤
⎥
⎥
⎦
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(14)
The stationary state distribution of the variables
x, p, p η
for the harmonic system
propagated with “middle-NHC” scheme is
261
That is, in the harmonic limit, either the “side” or “end” schemes with a finite
time interval leads to the exact momentum distribution but an approximate configurational distribution, and in contrast, the “middle” scheme produces the exact
configurational distribution but an approximate momentum distribution. Since evaluation of most properties in MD and that of all properties in PIMD depend on only the
configurational distribution, the “middle” scheme is a better choice for MD/PIMD
simulations.
2.1.2 “Middle” scheme with the Nosé-Hoover chain thermostat
Nosé-Hoover chain (NHC) [21] performs deterministic MD by adding extended
degrees of freedom to control the temperature of the system. The equations of motion
for NHC read
˙
x i =
p i
m i
˙
p i = −
∂U
∂ x i
−
p
η
(i)
1
Q 1
p i
˙
η
(i)
j =
p
η
(i)
j
Q j
˙
p η
(i)
1
=
p
2
i
m i
− k B T −
p
η
(i)
2
Q 2
p η
(i)
1
˙
p η
(i)
j
=
p
2
η
(i)
j−1
Q j−1
− k B T −
p
η
(i)
j+1
Q j+1
p η
(i)
j
j = 2, M NHC − 1
˙
p η
(i)
M NHC
=
p
2
η
(i)
M NHC −1
Q M NHC −1
− k B T
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
i = 1, 3N
(13)
where M NHC means the number of pairs for the additional variables
η
(i)
j , p η
(i)
j
j = 1, M NHC
coupled to each degree of freedom
i = 1, 3N
, the
parameters Q 1 , . . . , Q M NHC are the NHC thermal masses. For achieving the accuracy, one often uses the multiple time-scale techniques including the reference system propagator algorithm (RESPA) [23] and higher-order factorizations such as the
Suzuki-Yoshida decomposition framework [42–44]. When the phase space density
propagator for the thermostat part e
L T t is effectively accurate, the thermostat propagation keeps the Maxwell-Boltzmann distribution for the momentum effectively
unchanged, i.e.,
e
L T t exp
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
−β
⎡
⎢
⎢
⎣
1
2
p T M −1 p +
3N
i=1
M NHC
j=1
p 2
η
(i)
j
2Q j
⎤
⎥
⎥
⎦
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
= exp
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
−β
⎡
⎢
⎢
⎣
1
2
p T M −1 p +
3N
i=1
M NHC
j=1
p 2
η
(i)
j
2Q j
⎤
⎥
⎥
⎦
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
(14)
The stationary state distribution of the variables
x, p, p η
for the harmonic system
propagated with “middle-NHC” scheme is