|
Quantum numbers
Type of angular momentum
Operator
Total
Z-axis
z-axis
total orbital angular momentum
L
^
=
∑
i
=
1
N
l
^
i
L
M
L
Λ
=
∑
i
=
1
N
λ
i
=
0
,
1
,
2
,
…
(
Σ
,
Π
,
Δ
,
…
)
total electron spin angular momentum
S
^
=
∑
i
=
1
N
s
^
i
S
M
S
Σ
=
∑
i
=
1
N
σ
i
total electronic angular momentum[a]
J
^
a
=
L
^
+
S
^
Ω
=
∑
i
=
1
N
ω
i
=|
Λ
+
Σ
|
rotational angular momentum
R
^
R
K
R
,
k
R
total angular momentum excluding
electron and nuclear spins
N
^
=
J
^
-
S
^
=
R
^
+
L
^
N
M
N
Λ
K
,
k
total angular momentum of the molecule
excluding nuclear spins
J
^
=
R
^
+
L
^
+
S
^
=
N
^
+
S
^
J
M
J
Ω
P
nuclear spin angular momentum
I
^
I
M
I
total angular momentum
F
^
=
R
^
+
L
^
+
S
^
+
I
^
F
M
F
total angular momentum excluding nuclear spins
and electron orbital angular momentum
[b]
O
^
=
J
^
-
L
^
=
R
^
+
S
^
Single-Particle Angular Momentum Operators
single-electron orbital angular momentum
l
^
i
l
i
m
li
λ
i
=
0
,
1
,
2
,
…
(
σ
,
π
,
δ
,
…
)
single-electron spin angular momentum
s
^
i
s
i
m
si
σ
i
single-electron total angular momentum
j
^
i
j
i
m
ji
ω
i
=
λ
i
+
σ
i
Other Miscellaneous Conventions
principle quantum number
n
,
n
i
Cartesian coordinates in a laboratory-fixed frame of reference
X
'
,
Y
'
,
Z
'
Cartesian coordinates in the center-of-mass frame of reference,
parallel to the laboratory-fixed reference frame,
X
'
,
Y
'
,
Z
'
X
,
Y
,
Z
Cartesian coordinates in the molecule-fixed frame of reference
x
,
y
,
z
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