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(PIMD), which has become a powerful tool for estimating quantum thermodynamic
properties of realistic molecular systems [5, 7, 9, 38–40, 47].
3.1 Thermodynamic properties
One can write any thermodynamic property of the canonical ensemble in quantum
mechanics in the general form
ˆ
B
=
1
Z
Tr
e
−β ˆ
H ˆ
B
,
(16)
where Z = Tr
e
−β ˆ
H
is the partition function and ˆ
B represents a relevant operator
of our interest. Eq. (16) can be expressed in the configuration space x, i.e.,
ˆ
B
=
dxx|e
−β ˆ
H ˆ
B|x
dxx|e −β ˆ
H |x
.
(17)
Substituting the expression of the identity operator in the configuration space
ˆ
1 =
dx i |x i x i |, one can express the partition function as
Z =
x 1 ≡x
dxx|e
−β ˆ
H
|x
= lim
P→∞
dx 1
dx 2 . . .
dx P
P
2πβ 2
3N P/2 |M|
P/2
× exp
−
P
2β 2
P
i=1
(x i+1 − x i )
T M(x i+1 − x i )
−
β
P
P
i=1
U (x i )
,
(18)
where x P+1 ≡ x 1 and P is the total number of path integral beads. Similarly, the
numerator of Eq. (17) becomes
dxx|e −β ˆ
H ˆ
B|x
x 1 ≡x
= lim
P→∞
dx 1
dx 2 . . .
dx P
P
2πβ 2
3N P/2
|M| P/2
× exp
⎧
⎨
⎩
−
P
2β 2
P
i=1
x i+1 − x i
T M
x i+1 − x i
−
β
P
P
i=1
U (x i )
⎫
⎬
⎭
× ˜
B(x 1 , . . . , x P )
(19)
The estimator ˜
B(x 1 , . . . , x P ) for the operator ˆ
B
ˆ
x
is
˜
B(x 1 , . . . , x P ) =
1
P
P
j=1
B
x j
.
(20)