Efficient “Middle” Thermostat Scheme …
265
When operator ˆ
B involves the momentum, the estimator ˜
B(x 1 , . . . , x P ) can also
be expressed. For instance, the primitive version of the estimator for the kinetic
energy operator ˆ
B =
1
2
ˆ
p
T M
−1
ˆ
p is
˜
B(x 1 , . . . , x P ) =
3N P
2β
−
P
j=1
P
2β 2 2
x j+1 − x j
T M
x j+1 − x j
,
(21)
and the virial version is
˜
B(x 1 , . . . , x P ) =
3N
2β
+
1
2P
P
j=1
x j − x
∗
T ∂U
x j
∂x j
,
(22)
where x
∗ can be selected as the centroid of the path integral beads [48]
x
∗
= x c ≡
1
P
P
j=1
x j
(23)
or any specific bead.
3.2 Staging Path Integral Molecular Dynamics
Consider the staging transformation [6, 38, 47, 49]
ξ 1 = x 1
ξ j = x j −
( j − 1)x j+1 + x 1
j
j = 2, P
.
(24)
Define
ω P =
1
β
.
(25)
Eq. (18) becomes
Z
ξ 1 ≡x 1
= lim
P→∞
P
2πβ 2
3N P/2
|M|
P/2
dξ 1
dξ 2 . . .
dξ P
× exp
⎧
⎨
⎩
−β
P
j=1
1
2
ω
2
P ξ
T
j M j ξ j +
1
P
U
x j
ξ 1 , . . . , ξ P
⎫
⎬
⎭
,
(26)
265
When operator ˆ
B involves the momentum, the estimator ˜
B(x 1 , . . . , x P ) can also
be expressed. For instance, the primitive version of the estimator for the kinetic
energy operator ˆ
B =
1
2
ˆ
p
T M
−1
ˆ
p is
˜
B(x 1 , . . . , x P ) =
3N P
2β
−
P
j=1
P
2β 2 2
x j+1 − x j
T M
x j+1 − x j
,
(21)
and the virial version is
˜
B(x 1 , . . . , x P ) =
3N
2β
+
1
2P
P
j=1
x j − x
∗
T ∂U
x j
∂x j
,
(22)
where x
∗ can be selected as the centroid of the path integral beads [48]
x
∗
= x c ≡
1
P
P
j=1
x j
(23)
or any specific bead.
3.2 Staging Path Integral Molecular Dynamics
Consider the staging transformation [6, 38, 47, 49]
ξ 1 = x 1
ξ j = x j −
( j − 1)x j+1 + x 1
j
j = 2, P
.
(24)
Define
ω P =
1
β
.
(25)
Eq. (18) becomes
Z
ξ 1 ≡x 1
= lim
P→∞
P
2πβ 2
3N P/2
|M|
P/2
dξ 1
dξ 2 . . .
dξ P
× exp
⎧
⎨
⎩
−β
P
j=1
1
2
ω
2
P ξ
T
j M j ξ j +
1
P
U
x j
ξ 1 , . . . , ξ P
⎫
⎬
⎭
,
(26)