7.3 Microscopic Formulas of Quadrupolar Susceptibilities
179
β pqr (ω) = −
1
¯
h
states
g,m
(ρ
(0)
g − ρ
(0)
m )
g|q pq (ω)|m
m|μ r |g
ω − ω mg + ii mg
=
1
¯
h
states
g,m
ρ
(0)
g
−
g|q pq (ω)|m
m|μ r |g
ω − ω mg + ii mg
+
g|μ r |m
m|q pq (ω)|g
ω + ω mg + ii mg
,
(7.92)
β
pqr (ω) = −
1
¯
h
states
g,m
(ρ
(0)
g − ρ
(0)
m )
g|μ p |m
m|q qr (ω)|g
ω − ω mg + ii mg
=
1
¯
h
states
g,m
ρ
(0)
g
−
g|μ p |m
m|q qr (ω)|g
ω − ω mg + ii mg
+
g|q qr (ω)|m
m|μ p |g
ω + ω mg + ii mg
.
(7.93)
Equation (7.91) is equivalent to Eqs. (3.25) and (3.26) in Chap. 3. α pq (ω) in
Eq. (7.91) is known to be a thermal average of the Raman tensor (see Appendix A.3),
α pq (ω) = Tr[ρ α(ω)],
where ρ is the density matrix and α(ω) denotes the Raman tensor. Its matrix element
is represented by
g|α pq (ω)|n =
1
¯
h
states
m
−
g|μ p |m
m|μ q |n
ω − ω mg + ii mg
+
g|μ q |m
m|μ p |n
ω + ω mn + ii mn
,
(7.94)
which is equivalent to Eq. (3.37). We can define the quadrupolar Raman tensors in
an analogous manner by
g|β pqr (ω)|n =
1
¯
h
states
m
−
g|q pq (ω)|m
m|μ r |n
ω − ω mg + ii mg
+
g|μ r |m
m|q pq (ω)|n
ω + ω mn + ii mn
,
(7.95)
g|β
pqr (ω)|n =
1
¯
h
states
m
−
g|μ p |m
m|q qr (ω)|n
ω − ω mg + ii mg
+
g|q qr (ω)|m
m|μ p |n
ω + ω mn + ii mn
.
(7.96)
Using the extended Raman tensors, the vibrationally resonant terms of the dipolar
and quadrupolar hyperpolarizabilities are represented as follows.
α
D0,res
pqr ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )