Зайцевский А.В. Релятивистская теория электронного строения молекул
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s
+
s
−
l
[η, p
η
′
] = iδ
ηη
′
η, η
′
= x, y, z
[l
z
, α
x
p
x
] = [xp
y
, α
x
p
x
] − [yp
x
, α
x
p
x
]
| {z }
=0
= iα
x
p
y
[l
z
, α
y
p
y
] = −iα
y
p
x
[l
z
, α
z
p
z
] = 0,
[l
z
, (α, p)] = i(α
x
p
y
− α
y
p
x
).
[s
z
, α
x
p
x
] =
1
2
σ
z
0
0 σ
z
,
0 σ
x
σ
x
0
p
x
=
1
2
0 [σ
z
, σ
x
]
[σ
z
, σ
x
] 0
p
x
= i
0 σ
y
σ
y
0
p
x
= iα
y
p
x
[s
z
, α
y
p
y
] = −iα
x
p
y
[s
z
, α
z
p
z
] = 0,
[s
z
, (α, p)] = −i(α
x
p
y
− α
y
p
x
) = −[l
z
, (α, p)] .
l s
(α, p )
j = (j
x
, j
y
, j
z
) = l + s,
j
x
, j
y
, j
z
l
s
[j
x
, j
y
] = ij
z
, [j
z
, j
x
] = ij
y
, [j
y
, j
z
] = ij
x
j
2
j
z
j
2
Ψ
jm
= j(j + 1)Ψ
jm
, j
z
Ψ
jm
= mΨ
jm
m = −j, −j + 1, . . . , j j
x
j
y
,
2j + 1
j
{Ψ
jm
} j
2j +1
θ
n
g
n,θ
= e
iθ(n,j)
n z g
n, θ
= e
iθj
z
2j +1 Ψ
jm
m = −j, −j +1, . . . , j
Ψ
jm
T (g)Ψ
jm
=
X
m
′
Ψ
jm
′
D
j
m
′
m
(g)
D
j
m
′
m
(g) n, θ
g z
D
j
m
′
m
(g
z, θ
) = δ
m
′
m
e
−imθ
.
3 × 3
SO( 3, R)
2×2
SU(2)
SU(2)
j
κ
j
z m
ψ
κjm
L
= f
κjm
↑
1
0
+ f
κjm
↓
0
1
,
f
↑
f
↓
f
κjm
↑
• Y
l=j−1/2, m
l
=m−1/2
• Y
l=j+1/2, m
l
=m−1/2
l
κ l
j
m l
s | s =
1
/
2
, m
s
=
1
/
2
E
≡
1
0
| s =
1
/
2
, m
s
= −
1
/
2
E
≡
0
1
Y
ljm
=
X
m
s
, m
l
:
m
s
+m
l
=m
*
l s
m
l
m
s
j
m
+
Y
lm
l
| s m
s
i
=
*
l
1
/
2
m−
1
/
2
1
/
2
j
m
+
Y
l (m−1/2)
1
0
+
*
l
1
/
2
m+
1
/
2
−
1
/
2
j
m
+
Y
l (m+1/2)
0
1
=
*
l
1
/
2
m−
1
/
2
1
/
2
j
m
+
Y
l (m−1/2)
*
l
1
/
2
m+
1
/
2
−
1
/
2
j
m
+
Y
l (m+1/2)
*
l s
m
l
m
s
j
m
+
Y
ljm
r f
ljm
↑
f
ljm
↓
*
l
1
/
2
m−
1
/
2
1
/
2
j
m
+
*
l
1
/
2
m+
1
/
2
−
1
/
2
j
m
+
ψ
ljm
L
= φ
lj
L
(r)Y
ljm
ψ
L
l = j + 1/2 ψ
S
j − 1/2
l = 2j − l
Ψ
ljm
=
φ
lj
L
(r)Y
ljm
φ
lj
S
(r)Y
ljm
=
φ
lj
L
(r)
*
l
1
/
2
m−
1
/
2
1
/
2
j
m
+
Y
l,m−1/2
φ
lj
L
(r)
*
l
1
/
2
m+
1
/
2
−
1
/
2
j
m
+
Y
l,m+1/2
φ
lj
S
(r)
*
l
1
/
2
m−
1
/
2
1
/
2
j
m
+
Y
l,m−1/2
φ
lj
S
(r)
*
l
1
/
2
m+
1
/
2
−
1
/
2
j
m
+
Y
l,m+1/2
1/2
j =l+
1
/
2
:
*
l
1
/
2
m−
1
/
2
1
/
2
j
m
+
=
v
u
u
t
j + m
2j
,
*
l
1
/
2
m+
1
/
2
−
1
/
2
j
m
+
=
v
u
u
t
j −m
2j
j =l−
1
/
2
:
*
l
1
/
2
m−
1
/
2
1
/
2
j
m
+
=−
v
u
u
t
j+1−m
2j+2
,
*
l
1
/
2
m+
1
/
2
−
1
/
2
j
m
+
=
v
u
u
t
j+1+m
2j+2
j = 3/2
l = 1 l = 2 m = 3/2
Ψ
l=1,j=3/2,m=3/2
=
φ
L
Y
1, 3/2, 3/ 2
φ
S
Y
2, 3/2, 3/2
=
φ
L
Y
11
0
−
q
1/5φ
S
Y
21
q
4/5φ
S
Y
22
G = {g
k
}
{T (g
k
)}
T (g
k
)T (g
l
) = ω(g
k
, g
l
)T (g
k
g
l
),
{ω(g
k
, g
l
)}
{ω(g
k
, g
l
)}
{T (g
k
)}
G
G
G
π z C
2
(z)
2
SO( 3, R)
T (C
2
(z))T (C
2
(z)) = −1 = (−1) · T (C
2
(z)C
2
(z)),
ω(C
2
(z), C
2
(z)) = −1
SO( 3, R) C
2
(z)
2
2π
C
2
(z)
4
= I
2 × 2
SU(2)
SO( 3, R) ⊗C
i
SO( 3, R) ⊗ C
i
= SU(2) ⊗ C
i
2π I
I SU(2)
C
2v
C
2v
I
C
2
(z) σ
v
= σ(yz) σ
v
′
= σ(xz)
C
2
2
=
I
C
2v
C
2v
• π
C
2
(x) C
2
(y)
P I
σ
v
2
= (C
2
(x)P)
2
=
I, σ
2
v
′
=
I,
•
−π 3π C
2
(z)
C
2
(z)
3
= C
2
(z)I
xz yz π
−π C
2
(z) σ
v
σ
v
′
σ
v
C
2
(z)σ
−1
v
= IC
2
(z), σ
v
′
C
2
(z)σ
−1
v
′
=
IC
2
(z)
C
2
(z) C
2
(z)I
C
2
(z) σ
v
σ
v
I σ
v
′
σ
v
′
I
C
2v
C
2v
G
•
G I
T (I) T (gI) = T (g) g G
C
2v
• T (I)
C
2v
E
C
2v
T I I
C
2
C
2
I
σ
v
σ
v
I
σ
v
′
σ
v
′
I
A
1
A
2
− −
B
1
− −
B
2
− −
E −
I
−1 n
T (I)Ψ = (−1)
n
Ψ
I
G T (I)Ψ = −Ψ
Ψ G
C
∞v
|m| j
z
C
∞v
|m|
C
∞v
. . . Σ
Π ∆
C
∞v
|m|
|m| |m| = 0
0
+
0
−
D
∞h
D
∞h
g u
0
+
g
0
+
u
0
−
g
0
−
u
1
g
1
u
2
g
2
u
1/2
g
1/2
u
3/2
g
3/2
u
. . .
Ψ
µ
Ψ
ν
T
µ
T
ν
T
∗
µ
× T
r
× T
ν
T
r
T
r
θ j
χ
j
(θ) =
sin(j + 1/2)θ
sin(θ/2)
, sin(θ/2) 6= 0
(2j + 1) cos(j + 1/2)θ
cos(θ/2)
, sin(θ/2) = 0
j = 1
T
r
C
2v
I I C
2
, C
2
I σ
v
, σ
v
I σ
v
′
, σ
v
′
I
T
r
A
1
B
1
B
2
C
2v
A
1
− A
2
B
1
− B
2
E
T
r
0
+
1 (C
∞v
), 0
+
u
1
u
(D
∞h
).