7.3 Microscopic Formulas of Quadrupolar Susceptibilities
175
where g, m, n denote eigenstates of the molecule, ρ
(0)
g is a diagonal element of
the density matrix at thermal equilibrium, and mn = nm is the damping rate.
In this equation and hereafter, the symbol ˆ for an operator is omitted in a matrix
element. This Eq. (7.76) describes the second-order molecular hyperpolarizability,
and is equivalent to Eq. (3.31) in Chap. 3.
Equation (7.76) can be decomposed into the vibrationally resonant (α D0,res ) and
non-resonant (α D0,nonres ) terms, α D0 = α D0,res + α D0,nonres , where
α
D0,res
pqr ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
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)
α
D0,nonres
pqr
((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|μ p |nn|μ r |mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ r |nn|μ p |mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω mn + ii mn )
.
(7.78)
The former includes (ω 2 − ω mg + ii mg ) in the denominator, indicating possible
resonance with the infrared frequency ω 2 , whereas the latter does not.
Then we employ the extended perturbation Hamiltonian including electric
quadrupole and magnetic dipole interactions in Eq. (7.74) and carry out the perturbation expansion up to the second order after Sect. 3.2. Since the derivation is in
parallel with that in Sect. 3.2, we omit here to repeat the derivation procedure. The
outcome of the second-order perturbation gives rise to additional hyperpolarizability
terms, α D1 , α D2 and α Q , due to the extra term − ˆ
q : ∇E of Eq. (7.74). The sumover-state expressions of these terms are as follows.
α D1
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
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 + ii mg )(( − ω ng + ii ng )
−
g|μ r |nn|μ p |mm|q sq (ω 1 )|g
(ω 1 − ω mg + ii mg )(( − ω mn + ii mn )
+
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 )
,
(7.79)
175
where g, m, n denote eigenstates of the molecule, ρ
(0)
g is a diagonal element of
the density matrix at thermal equilibrium, and mn = nm is the damping rate.
In this equation and hereafter, the symbol ˆ for an operator is omitted in a matrix
element. This Eq. (7.76) describes the second-order molecular hyperpolarizability,
and is equivalent to Eq. (3.31) in Chap. 3.
Equation (7.76) can be decomposed into the vibrationally resonant (α D0,res ) and
non-resonant (α D0,nonres ) terms, α D0 = α D0,res + α D0,nonres , where
α
D0,res
pqr ((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
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)
α
D0,nonres
pqr
((, ω 1 , ω 2 ) =
1
¯
h 2
states
g,m,n
(ρ
(0)
g − ρ
(0)
m )
g|μ p |nn|μ r |mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω ng + ii ng )
−
g|μ r |nn|μ p |mm|μ q |g
(ω 1 − ω mg + ii mg )(( − ω mn + ii mn )
.
(7.78)
The former includes (ω 2 − ω mg + ii mg ) in the denominator, indicating possible
resonance with the infrared frequency ω 2 , whereas the latter does not.
Then we employ the extended perturbation Hamiltonian including electric
quadrupole and magnetic dipole interactions in Eq. (7.74) and carry out the perturbation expansion up to the second order after Sect. 3.2. Since the derivation is in
parallel with that in Sect. 3.2, we omit here to repeat the derivation procedure. The
outcome of the second-order perturbation gives rise to additional hyperpolarizability
terms, α D1 , α D2 and α Q , due to the extra term − ˆ
q : ∇E of Eq. (7.74). The sumover-state expressions of these terms are as follows.
α D1
pqrs ((, ω 1 , ω 2 ) =
1
¯
h 2
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 + ii mg )(( − ω ng + ii ng )
−
g|μ r |nn|μ p |mm|q sq (ω 1 )|g
(ω 1 − ω mg + ii mg )(( − ω mn + ii mn )
+
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 )
,
(7.79)