36
F. E. Harris
and that the final term of Eq. (17) reduces to
2g d (1) ̂
𝐫 12 ⋅ ∇ 1 Y
m d
l d
(1) = −
2r 2 g d (1)
r 1 r 12
√
4𝜋
3
1
∑
𝜈=−1
∑
𝜆=±1
( l d (l d + 1)(2l d + 1 − 𝜆
2(2l d + 1)
) 1∕2
×
⟨
l d + 𝜆 1 l d
m d −𝜈 𝜈 m d
⟩
Y
m d −𝜈
l d +𝜆
(1)Y
𝜈
1
(2).
(23)
In these equations the array in angle brackets is a Clebsch-Gordan coefficient as
defined in the Appendix; the notation we are using for it is not standard but the
author hopes it will become more widely adopted.
The spherical harmonic Y
𝜈
1
(2) will in overall computations occur multiplied by
the harmonic contained in 𝜙(2); that product can be reduced to a sum of single harmonics. This reduction takes the form
Y
𝜈
1
(2)Y
m e
l e
(2) =
∑
𝜆 ′ =±1
[
l e 1 l e + 𝜆
′
m e 𝜈 −m e −𝜈
]
Y
m e +𝜈
l e +𝜆 ′ (2) ,
(24)
where the array in brackets is a Gaunt coefficient (also in a nonstandard notation
suggested by the author). There is unfortunately no single widely-accepted definition for the Gaunt coefficients; we use here that given in the Appendix, which has
the properties of being both analogous to the definition of the Wigner 3-j symbol
[39] and in agreement with the Gaunt-coefficient definition chosen by Pinchon and
Hoggan [40].
Combining Eqs. (17), (18), and (24), we obtain a final expression containing no
differential operators for −
1
2
∇
2
1
r 12 𝜙 d (1)𝜙 e (2); it is then straightforward to insert that
expression into Eqs. (7) and (8). We display here the formula for K 3 ; that for K 2 is
included in [37]:
K 3 =
l d (l d + 1) − n
2
d
+ 1
2
⟨
𝛷 abc
|
|
|
|
|
r 12 r 13
r
2
1
|
|
|
|
|
𝛷 def
⟩
−
𝛼
2
d
2
⟨ 𝛷 abc
|
| r 12 r 13
|
| 𝛷 def
⟩
+
(2n d + 1)𝛼 d
2
⟨
𝛷 abc
|
|
|
|
r 12 r 13
r 1
|
|
|
|
𝛷 def
⟩
−
n d + 1
2
⟨
𝛷 abc
|
|
|
|
r 13
r 12
|
|
|
|
𝛷 def
⟩
+
𝛼 d
2
⟨
𝛷 abc
|
|
|
|
|
r 1 r 13
r 12
−
r
2
2
r 13
r 1 r 12
|
|
|
|
|
𝛷 def
⟩
+
n d − 1
2
⟨
𝛷 abc
|
|
|
|
|
r
2
2
r 13
r
2
1
r 12
|
|
|
|
|
𝛷 def
⟩
+
√
4𝜋
3
∑
𝜆=±1
(
l d (l d +1)(2l d +1−𝜆)
2(2𝜆 + 1)
) 1∕2 1
∑
𝜈=−1
(−1)
m e +𝜈
⟨
l d + 𝜆 1 l d
m d −𝜈 𝜈 m d
⟩
×
∑
𝜆 ′ =±1
[
l e 1 l e + 𝜆 ′
m e 𝜈 −m e −𝜈
] ⟨
𝛷 abc
|
|
|
|
r 2 r 13
r 1 r 12
|
|
|
|
d
m d −𝜈
l d +𝜆
e
m e +𝜈
l e +𝜆 ′ f
m f
l f
⟩
.
(25)