7.3 Microscopic Formulas of Quadrupolar Susceptibilities
177
formulas are derived with the phenomenological Liouville equation including
(see Chap. 3). Accordingly, the states g, m, n in the above equations could refer to
a partial system embedded in bath. Equivalent expressions are possible in principle
when we define these states g, m, n for the whole system (partial system + bath).
Then the corresponding equations do not include the damping parameters;
α
D0
pqr ((, ω 1 , ω 2 ) =
1
¯
h 2
whole
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|μ p |nn|μ r |mm|μ q |g
(ω 1 − ω mg )(( − ω ng )
−
g|μ r |nn|μ p |mm|μ q |g
(ω 1 − ω mg )(( − ω mn )
+
g|μ p |nn|μ q |mm|μ r |g
(ω 2 − ω mg )(( − ω ng )
−
g|μ q |nn|μ p |mm|μ r |g
(ω 2 − ω mg )(( − ω mn )
,
(7.84)
α
D1
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
whole
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|μ p |nn|μ r |mm|q sq (ω 1 )|g
(ω 1 − ω mg )(( − ω ng )
−
g|μ r |nn|μ p |mm|q sq (ω 1 )|g
(ω 1 − ω mg )(( − ω mn )
+
g|μ p |nn|q sq (ω 1 )|mm|μ r |g
(ω 2 − ω mg )(( − ω ng )
−
g|q sq (ω 1 )|nn|μ p |mm|μ r |g
(ω 2 − ω mg )(( − ω mn )
,
(7.85)
α
D2
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
whole
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|μ p |nn|q sr (ω 2 )|mm|μ q |g
(ω 1 − ω mg )(( − ω ng )
−
g|q sr (ω 2 )|nn|μ p |mm|μ q |g
(ω 1 − ω mg )(( − ω mn )
+
g|μ p |nn|μ q |mm|q sr (ω 2 )|g
(ω 2 − ω mg )(( − ω ng )
−
g|μ q |nn|μ p |mm|q sr (ω 2 )|g
(ω 2 − ω mg )(( − ω mn )
,
(7.86)
α
Q
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
whole
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )