C
1 U
0
À
Á Â C
3 U
0
m
À
Á Â C
3
r
0
À Á ¼ C 1 þ . . .
ð4:2:18Þ
All this is a consequence of the total symmetry of the b
H SO operator at symmetry
point groups of a molecule, including both spatial and spin coordinates.
The classification of electron motion in a molecule according to symmetry types
taking into account the spin–orbit interaction depends on the geometry of the
molecule, nuclear charge, etc. Therefore, for theoretical group analysis of (4.2.15–
4.2.18), it is necessary to understand the bond classification in Hund’s case (c).
These concepts for diatomic molecules are developed in the Mulliken’s works
[15–19] (see [20–22] and references, also).
As is known, if there is a spin–orbit interaction in one of the atoms of a diatomic
molecule, the axial electric field may not break the connection between the orbital
moment of the electron l and its spin moment s. In this case, the total electron
moment j = l + s precesses around a molecule axis Z, and only the quantum
number corresponding to the projection of the total moment J =
P
i j i on the axis is
‘true’. The projections of the total orbital moment L =
P
i l i , and the spin S =
P
i s i
of a molecule lose their meaning. The corresponding quantum numbers L, S remove
the meaning, also. This rough description of the coupling of moments in a diatomic
molecule in the presence of a strong spin–orbit interaction corresponds to the Hund
(c) case [23], p. 224.
‘Close nuclei’ case (c). As a result of the nuclei proximity, the axial component
of the electric field is small, the precession of the orbital momentum about the
Z axis is weak. In this case, L, S, J a (J for the atom) as well X are good quantum
numbers (although the first three are not quite so). This is a rarely implemented case
(see [15, 17] for details).
X-x coupling. If a molecule can be represented as a charged core characterized
by the quantum numbers K c , R c , and X c with a fairly distant electron (so that the
interaction is weak), the state of the molecule may be described by analyzing the
projections of the orbital and spin momenta of the core and this electron on the
internuclear axis. For example, the electron configuration of the R
0 I diatomic
molecules, R
0 = H(
2 S), Hal(
2 P), takes the form [16]:
R
0 I r
2
p
4
R 0 p
2
1
À
Á 2 P 3=2 2r
Ã
h
i
2;1
ð4:2:19Þ
R
0 I r
2
p
4
R 0 p
3
1
À
Á 2 P 1=2 r
Ã
h
i
0;1
:
ð4:2:20Þ
In this case, the iodine molecule electronic configuration is:
I 2 r
2
g p
3
u p
4
g
2
P 3=2 r
Ã
g
!
2;1u
ð4:2:21Þ
4.2 Radiative Electronic Transitions …
91
1 U
0
À
Á Â C
3 U
0
m
À
Á Â C
3
r
0
À Á ¼ C 1 þ . . .
ð4:2:18Þ
All this is a consequence of the total symmetry of the b
H SO operator at symmetry
point groups of a molecule, including both spatial and spin coordinates.
The classification of electron motion in a molecule according to symmetry types
taking into account the spin–orbit interaction depends on the geometry of the
molecule, nuclear charge, etc. Therefore, for theoretical group analysis of (4.2.15–
4.2.18), it is necessary to understand the bond classification in Hund’s case (c).
These concepts for diatomic molecules are developed in the Mulliken’s works
[15–19] (see [20–22] and references, also).
As is known, if there is a spin–orbit interaction in one of the atoms of a diatomic
molecule, the axial electric field may not break the connection between the orbital
moment of the electron l and its spin moment s. In this case, the total electron
moment j = l + s precesses around a molecule axis Z, and only the quantum
number corresponding to the projection of the total moment J =
P
i j i on the axis is
‘true’. The projections of the total orbital moment L =
P
i l i , and the spin S =
P
i s i
of a molecule lose their meaning. The corresponding quantum numbers L, S remove
the meaning, also. This rough description of the coupling of moments in a diatomic
molecule in the presence of a strong spin–orbit interaction corresponds to the Hund
(c) case [23], p. 224.
‘Close nuclei’ case (c). As a result of the nuclei proximity, the axial component
of the electric field is small, the precession of the orbital momentum about the
Z axis is weak. In this case, L, S, J a (J for the atom) as well X are good quantum
numbers (although the first three are not quite so). This is a rarely implemented case
(see [15, 17] for details).
X-x coupling. If a molecule can be represented as a charged core characterized
by the quantum numbers K c , R c , and X c with a fairly distant electron (so that the
interaction is weak), the state of the molecule may be described by analyzing the
projections of the orbital and spin momenta of the core and this electron on the
internuclear axis. For example, the electron configuration of the R
0 I diatomic
molecules, R
0 = H(
2 S), Hal(
2 P), takes the form [16]:
R
0 I r
2
p
4
R 0 p
2
1
À
Á 2 P 3=2 2r
Ã
h
i
2;1
ð4:2:19Þ
R
0 I r
2
p
4
R 0 p
3
1
À
Á 2 P 1=2 r
Ã
h
i
0;1
:
ð4:2:20Þ
In this case, the iodine molecule electronic configuration is:
I 2 r
2
g p
3
u p
4
g
2
P 3=2 r
Ã
g
!
2;1u
ð4:2:21Þ
4.2 Radiative Electronic Transitions …
91