180
7 Quadrupole Contributions from Interface and Bulk
g|μ p |nn|μ q |mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ q |nn|μ p |mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω mn + ii mn )
(7.77)
= −
1
¯
h
states
g,m
(ρ
(0)
g − ρ
(0)
m )
g|α pq (()|mm|μ r |g
ω 2 − ω mg + ii mg
,
(7.97)
α
D1,res
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
+
g|μ p |nn|q sq (ω 1 )|mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|q sq (ω 1 )|nn|μ p |mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω mn + ii mn )
(7.82)
= −
1
¯
h
states
g,m
(ρ
(0)
g − ρ
(0)
m )
g|β
psq (ω 1 )|mm|μ r |g
ω 2 − ω mg + ii mg
,
(7.98)
α
D2,res
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|μ p |nn|μ q |mm|q sr (ω 2 )|g
(ω 2 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ q |nn|μ p |mm|q sr (ω 2 )|g
(ω 2 − ω mg + ii mg )(( − ω mn + ii mn )
= −
1
¯
h
states
g,m
(ρ
(0)
g − ρ
(0)
m )
g|α pq (()|mm|q sr (ω 2 )|g
ω 2 − ω mg + ii mg
,
(7.99)
α
Q,res
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|q sp (()|nn|μ q |mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ q |nn|q sp (()|mm|μ r |g
(ω 2 − ω mg + ii mg )(( − ω mn + ii mn )
= −
1
¯
h
states
g,m
(ρ
(0)
g − ρ
(0)
m )
g|β psq (()|mm|μ r |g
ω 2 − ω mg + ii mg
.
(7.100)
These formulas of Eqs. (7.97), (7.98), (7.99), (7.100) are converted to the time
correlation ones, via the same procedure as described in Chap. 3. After the derivation
of Eq. (4.17), the equivalent formulas are