This leads to the fact that the summation over I in (4.6.27) does not include all
possible I values and depends on the symmetry of the rovibronic state. The sign of
the rotational components of the 0
þ
g , 0
À
g , 0
þ
u , 0
À
u , 1 g and 1 u and so on states,
Hund’s case (c)), e, is determined by the number (–1)
J (4.6.5) for the gerade
(g) states, and that of the (–1)
J+1 (4.6.6) for the ungerade (u) states.
The assignment of quantum numbers, indices (labels) and parities to the
molecular states in (4.6.27) for the 0
þ
g , 0
À
g , 0
þ
u , 0
À
u , 1 g and 1 u states are clarified in
Table 4.4.
The hyperfine terms of both states mixed by hyperfine interaction must have the
same parity: + $ + , − $ −. Selection rules for hyperfine interaction matrix
elements are:
DK
j j 1ð DX
j j 1Þ; DJ
j j 1
ð4:6:30Þ
for the nuclear magnetic-dipole interaction with the electrons (MD interaction) and
DK
j j 2ð DX
j j 2Þ; DJ
j j 2
ð4:6:31Þ
for the nuclear electric-quadrupole interaction with the electrons (MQ interaction)
(see Table 4.1). The hyperfine Hamiltonian is of even parity, and therefore conserves sign (p) but not electronic parity e. Since p is conserving and taking into
account (4.6.28), one sees that:
The u $ g hyperfine coupling implies an odd DI;
ð4:6:32Þ
whereas
The u $ u and g $ g hyperfine couplings imply an even DI:
ð4:6:33Þ
One can use Table 4.4 to understand what hyperfine sublevels can interact. There
is no precise selection rule for vibrational quantum number m. For a good
approximation, propensity rule is defined by the Franck–Condon factors (FCFs)
(see [7], p.60 and [39, 49, 50]).
4.6.1.5 Two-State Perturbation Model
It is convenient to use the two-level model to describe interactions of close-lying
rovibrational states, hyperfine interaction (HFI), in particular. Let two close-lying
rovibrational states W = XvJn
j
iand W
0 = X
0
v
0
J
0
n
0
j
iare coupled by the interaction.
According to the two-state perturbation model [7], p.61, [51, 52], the mixed 1
j i and
2
j i states are the following:
120
4 Photolysis of Free Molecules
possible I values and depends on the symmetry of the rovibronic state. The sign of
the rotational components of the 0
þ
g , 0
À
g , 0
þ
u , 0
À
u , 1 g and 1 u and so on states,
Hund’s case (c)), e, is determined by the number (–1)
J (4.6.5) for the gerade
(g) states, and that of the (–1)
J+1 (4.6.6) for the ungerade (u) states.
The assignment of quantum numbers, indices (labels) and parities to the
molecular states in (4.6.27) for the 0
þ
g , 0
À
g , 0
þ
u , 0
À
u , 1 g and 1 u states are clarified in
Table 4.4.
The hyperfine terms of both states mixed by hyperfine interaction must have the
same parity: + $ + , − $ −. Selection rules for hyperfine interaction matrix
elements are:
DK
j j 1ð DX
j j 1Þ; DJ
j j 1
ð4:6:30Þ
for the nuclear magnetic-dipole interaction with the electrons (MD interaction) and
DK
j j 2ð DX
j j 2Þ; DJ
j j 2
ð4:6:31Þ
for the nuclear electric-quadrupole interaction with the electrons (MQ interaction)
(see Table 4.1). The hyperfine Hamiltonian is of even parity, and therefore conserves sign (p) but not electronic parity e. Since p is conserving and taking into
account (4.6.28), one sees that:
The u $ g hyperfine coupling implies an odd DI;
ð4:6:32Þ
whereas
The u $ u and g $ g hyperfine couplings imply an even DI:
ð4:6:33Þ
One can use Table 4.4 to understand what hyperfine sublevels can interact. There
is no precise selection rule for vibrational quantum number m. For a good
approximation, propensity rule is defined by the Franck–Condon factors (FCFs)
(see [7], p.60 and [39, 49, 50]).
4.6.1.5 Two-State Perturbation Model
It is convenient to use the two-level model to describe interactions of close-lying
rovibrational states, hyperfine interaction (HFI), in particular. Let two close-lying
rovibrational states W = XvJn
j
iand W
0 = X
0
v
0
J
0
n
0
j
iare coupled by the interaction.
According to the two-state perturbation model [7], p.61, [51, 52], the mixed 1
j i and
2
j i states are the following:
120
4 Photolysis of Free Molecules