3.3 Properties of χ (2)
59
the initial state g is a vibrationally excited state. Therefore, one could categorize the
terms in Eq. (3.32) into those including the above-mentioned resonant denominators
((ω 2 − ω mg + ii mg ), (ω 2 − ω ng + ii ng ), (ω 2 + ω mg + ii mg )) and the rest. The
former is called the vibrationally resonant terms and denoted by χ (2),res , while the
latter is called the nonresonant terms χ (2),nonres . Accordingly,
χ
(2)
pqr = χ
(2),res
pqr
+ χ
(2),nonres
pqr
,
(3.33)
where
χ
(2),res
pqr ((, ω 1 , ω 2 ) =
1
¯
h 2
g,n,m
ρ
(0)
g
g|μ p |nn|μ q |mm|μ r |g
−ω ng +ii ng
ω 2 −ω mg +ii mg
−
g|μ q |mm|μ p |nn|μ r |g
(−ω nm +ii nm )
ω 2 −ω ng +ii ng
+
g|μ r |mm|μ q |nn|μ p |g
+ω ng +ii ng
ω 2 +ω mg +ii mg
−
g|μ r |mm|μ p |nn|μ q |g
(−ω nm +ii nm )
ω 2 +ω mg +ii mg
(3.34)
and
χ
(2),nonres
pqr
((, ω 1 , ω 2 ) =
1
¯
h 2
g,n,m
ρ
(0)
g
g|μ p |nn|μ r |mm|μ q |g
−ω ng +ii ng
ω 1 − ω mg + ii mg
−
g|μ r |mm|μ p |nn|μ q |g
( − ω nm + ii nm )
ω 1 − ω ng + ii ng
+
g|μ q |mm|μ r |nn|μ p |g
+ω ng +ii ng
ω 1 +ω mg +ii mg
−
g|μ q |mm|μ p |nn|μ r |g
(−ω nm +ii nm )
ω 1 +ω mg +ii mg
.
(3.35)
χ (2),res in Eq. (3.34) is further modified to
χ
(2),res
pqr ((, ω 1 , ω 2 )
=
1
¯
h 2
g,n,m
ρ
(0)
g
g|μ p |nn|μ q |mm|μ r |g
− ω ng + ii ng
ω 2 − ω mg + ii mg
−
g|μ q |mm|μ p |nn|μ r |g
( − ω nm + ii nm )
ω 2 − ω ng + ii ng
59
the initial state g is a vibrationally excited state. Therefore, one could categorize the
terms in Eq. (3.32) into those including the above-mentioned resonant denominators
((ω 2 − ω mg + ii mg ), (ω 2 − ω ng + ii ng ), (ω 2 + ω mg + ii mg )) and the rest. The
former is called the vibrationally resonant terms and denoted by χ (2),res , while the
latter is called the nonresonant terms χ (2),nonres . Accordingly,
χ
(2)
pqr = χ
(2),res
pqr
+ χ
(2),nonres
pqr
,
(3.33)
where
χ
(2),res
pqr ((, ω 1 , ω 2 ) =
1
¯
h 2
g,n,m
ρ
(0)
g
g|μ p |nn|μ q |mm|μ r |g
−ω ng +ii ng
ω 2 −ω mg +ii mg
−
g|μ q |mm|μ p |nn|μ r |g
(−ω nm +ii nm )
ω 2 −ω ng +ii ng
+
g|μ r |mm|μ q |nn|μ p |g
+ω ng +ii ng
ω 2 +ω mg +ii mg
−
g|μ r |mm|μ p |nn|μ q |g
(−ω nm +ii nm )
ω 2 +ω mg +ii mg
(3.34)
and
χ
(2),nonres
pqr
((, ω 1 , ω 2 ) =
1
¯
h 2
g,n,m
ρ
(0)
g
g|μ p |nn|μ r |mm|μ q |g
−ω ng +ii ng
ω 1 − ω mg + ii mg
−
g|μ r |mm|μ p |nn|μ q |g
( − ω nm + ii nm )
ω 1 − ω ng + ii ng
+
g|μ q |mm|μ r |nn|μ p |g
+ω ng +ii ng
ω 1 +ω mg +ii mg
−
g|μ q |mm|μ p |nn|μ r |g
(−ω nm +ii nm )
ω 1 +ω mg +ii mg
.
(3.35)
χ (2),res in Eq. (3.34) is further modified to
χ
(2),res
pqr ((, ω 1 , ω 2 )
=
1
¯
h 2
g,n,m
ρ
(0)
g
g|μ p |nn|μ q |mm|μ r |g
− ω ng + ii ng
ω 2 − ω mg + ii mg
−
g|μ q |mm|μ p |nn|μ r |g
( − ω nm + ii nm )
ω 2 − ω ng + ii ng