78
3 Microscopic Expressions of Nonlinear Polarization
Fig. 3.4 Raman process in
off-resonant condition
g g
g
g
g
g
m
g
g
m
m
n
n
n
n
n
e
e
e
v
v
v
E -
- E
e
e
n
g
E
v
g
E
v
n
∼ =
−
E e
n − E e
g
¯
h
+ ii ng
−1
=
1
− ω e
ng + ii ng
,
(3.61)
∗
1
+ ω mn + ii mn
∼ =
1
+ ω e
nm + ii nm
=
1
+ ω e
ng + ii ng
.
(3.62)
Using these approximations of Eqs. (3.61) and (3.62), 6 the Raman tensor element of
Eq. (3.37) is represented by
g|α pq (()|m
∼ = −
n e
n v
g v |
g e |μ p |n e |n v n v |
n e |μ q |m e |m v
− ω e
ng + ii ng
−
g v |
g e |μ q |n e |n v n v |
n e |μ p |m e |m v
+ ω e
ng + ii ng
=
g
v
−
n e
g e |μ p |n e
n e |μ q |g e
− ω e
ng + ii ng
−
g e |μ q |n e
n e |μ p |g e
+ ω e
ng + ii ng
m
v
=
g
v (R)|α pq ((, R)|m
v (R)
,
(3.63)
where the completeness condition
n v |n v n v | = 1 has been adopted. The
final expression of α pq ((, R) is no longer an explicit function of the electronic
coordinate r, as r is already integrated out in Eq. (3.63). α pq ((, R) means the
polarizability of the electronic ground state at the frequency and given nuclear
6 Equations (3.61) and (3.62) also involve the approximation of the dampling factor for electronic
states. Actually the damping factor is insignificant here in the electronically off-resonant
conditions, and often neglected in the present discussion.