Matrix Elements for Explicitly-Correlated Atomic Wave Functions
35
̂
𝐫 1 ⋅ ̂
𝐫 12 =
r
2
1
+ r
2
12
− r
2
2
2r 1 r 12
,
(15)
0 = ∇ 1 g d (1) ⋅ ∇ 1 Y
m
l
(1) .
(16)
With these substitutions, Eq. (10) becomes
∇
2
1 [r 12 𝜙 d (1)] =
[
n
2
d
− 1 − l d (l d + 1)
r
2
1
−
(2n d + 1)𝛼 d
r 1
+ 𝛼
2
d
]
r 12 𝜙 d (1)
+
[
n d + 1 − 𝛼 d r 1 +
𝛼 d r
2
2
r 1
−
(n d − 1)r
2
2
r
2
1
]
𝜙 d (1)
r 12
+ 2g d (1) ̂
𝐫 12 ⋅ ∇ 1 Y
m d
l d
(1).
(17)
Equation (17) corrects a sign error that was present in the corresponding equation of
[37].
One further simplification can now be easily made to the final term of Eq. (17):
The orthogonality of ̂
𝐫 1 and ∇ 1 Y
m d
l d
(1) permit us to write
2g d (1) ̂
𝐫 12 ⋅ ∇ 1 Y
m d
l d
(1) = −
2r 2 g d (1)
r 12
̂
𝐫 2 ⋅ ∇ 1 Y
m d
l d
(1) .
(18)
As explained in more detail in [37], the properties of vector spherical harmonics
can now be used to complete the evaluation of Eq. (18).
Introducing the complex unit vectors
̂
𝐞 1 = −
̂
𝐱 + i ̂
𝐲
√
2
, ̂
𝐞 −1 =
̂
𝐱 − i ̂
𝐲
√
2
, ̂
𝐞 0 = ̂
𝐳 ,
(19)
with orthogonality relation
̂
𝐞 𝜇 ⋅ ̂
𝐞 −𝜈 = (−1)
𝜈
𝛿 𝜇𝜈 ,
(20)
it can be shown that
̂
𝐫 2 =
√
4𝜋
3
1
∑
𝜇=−1
(−1)
𝜇 Y
−𝜇
1
(2) ̂
𝐞 𝜇 ,
(21)
∇ 1 Y
m d
l d
(1) =
1
r 1
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) ̂
𝐞 𝜈 ,
(22)