Efficient “Middle” Thermostat Scheme …
269
C
norm
jk
=
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
√
1/P,
k = 0
√
2/P cos(2π jk/P), 1 ≤ k ≤ P/2 − 1
√
1/P(−1)
j
,
k = P/2
√
2/P sin(2π jk/P), P/2 + 1 ≤ k ≤ P − 1
j = 1, P
,
(39)
for even P and
C
norm
jk
=
⎧
⎨
⎩
√
1/P,
k = 0
√
2/P cos(2π jk/P), 1 ≤ k ≤ (P − 1)/2
√
2/P sin(2π jk/P), (P + 1)/2 ≤ k ≤ P − 1
j = 1, P
, (40)
for odd P. When one employs the normal mode transformation, the partition function
becomes
Z = lim
P→∞
P
2πβ 2
N P/2
|M|
P/2
dq 0
dq 1 . . .
dq P−1
× exp
⎧
⎨
⎩
−β
⎡
⎣
P−1
k=0
1
2
ω
2
k q
T
k M
norm
k
q k +
1
P
P
j=1
V
x j (q 0 , . . . , q P−1 )
⎤
⎦
⎫
⎬
⎭
.
(41)
The mass matrices are defined as M
norm
0
= 0 and M
norm
k
= PM
k = 1, P − 1
,
the frequency for each mode is given by
ω k = 2ω P sin(kπ/P)
k = 0, P − 1
,
(42)
respectively. Eq. (28) then becomes
φ(q 0 , . . . , q P−1 ) =
1
P
P
j=1
V
x j (q 0 , . . . , q P−1 )
,
(43)
and the derivatives ∂φ/∂q k is obtained from
∂φ
∂q
=
∂x
∂q
T ∂φ
∂x
=
C
norm
T ∂φ
∂x
.
(44)
Employing fictitious momenta (p 0 , . . . , p P−1 ) into Eq. (41) produces
Z = lim
P→∞
P
4π 2 2
N P/2
|M|
P/2
P−1
k=0
˜
M
norm
k
−1 / 2 P−1
k=0
dq k dp k
× exp
−β H
norm
eff (q 0 , . . . , q P−1 ; p 0 , . . . , p P−1 )
,
(45)