94
4 Two Computational Schemes of χ (2)
G(t) =
1
β
β
0
dλ
exp (λH)
α pq (t) − α ◦
pq
exp (−λH) (μ r − μ ◦
r )
=
1
β
β
0
dλ Tr
ρ exp(λH) exp
iHt
¯
h
α pq − α ◦
pq
exp
−iHt
¯
h
exp(−λH)
μ r − μ ◦
r
=
1
β
β
0
dλ
m
n
exp(−βE m )
Q
exp(λE m ) exp
iE m t
¯
h
m|α pq |n
−
m|α ◦
pq |n
· exp
−
iE n t
¯
h
exp(−λE n )
n|μ r |m −
n|μ ◦
r |m
=
1
β
m
n( =m)
exp(−βE m )
Q
exp
iE m t
¯
h
m|α pq |n
exp
−iE n t
¯
h
n|μ r |m
·
exp (β(E m − E n )) − 1
E m − E n
=
1
Q
m
n( =m)
exp (−βE m ) − exp (−βE n )
−β (E m − E n )
exp
iE m t
¯
h
m|δα pq |n
exp
−iE n t
¯
h
n|δμ r |m ,
(4.26)
where E m , E n are the energy eigenvalues for the states m, n. Q =
m
exp (−βE m )
is the partition function for the entire system. During the derivation in Eq. (4.26),
the diagonal terms (m = n) of A − A ◦ (A = α pq or μ r ) vanish, and m|δA|n =
m|A − A|n
= m|A|n is employed for m = n.
Now Eq. (4.21) is expressed with G(t) instead of F(t) using Eqs. (4.25)
and (4.26) in the following way,
χ
(2),res
pqr ((, ω 1 , ω 2 ) =
∞
0
dt exp (iω 2 t) F(t)
(4.21)
=
i
¯
h
∞
0
dt exp(iω 2 t)
m
n
exp(−βE m )
Q
exp
iE m t
¯
h
m|δα pq |n
exp
−iE n t
¯
h
n|δμ r |m
− m|δμ r |n exp
iE n t
¯
h
n|δα pq |m
exp
−iE m t
¯
h
=
i
¯
h
∞
0
dt exp(iω 2 t)
m
n( =m)
exp(−βE m ) − exp(−βE n )
Q