38
F. E. Harris
K
EHCI
3
=
l d (l d + 1) − (n d + p)(n d − 1)
2
⟨
𝛷 abc
|
|
|
|
|
f 12 f 13
r
2
1
|
|
|
|
|
𝛷 def
⟩
+
(2n d + p)𝛼 d
2
⟨
𝛷 abc
|
|
|
|
f 12 f 13
r 1
|
|
|
|
𝛷 def
⟩
−
𝛼
2
d
+ 𝛽 2
2
⟨ 𝛷 abc
|
| f 12 f 13
|
| 𝛷 def
⟩
−
p(n d + p)
2
⟨
𝛷 abc
|
|
|
|
|
f 12 f 13
r
2
12
|
|
|
|
|
𝛷 def
⟩
+
𝛼 d p
2
⟨
𝛷 abc
|
|
|
|
|
f 12 f 13 r 1
r
2
12
−
f 12 f 13 r
2
2
r 1 r
2
12
|
|
|
|
|
𝛷 def
⟩
+
(n d − 1)p
2
⟨
𝛷 abc
|
|
|
|
|
f 12 f 13 r
2
2
r
2
1
r
2
12
|
|
|
|
|
𝛷 def
⟩
+
𝛽(n d + 2p + 1)
2
⟨
𝛷 abc
|
|
|
|
f 12 f 13
r 12
|
|
|
|
𝛷 def
⟩
−
𝛼 d 𝛽
2
⟨
𝛷 abc
|
|
|
|
|
f 12 f 13 r 12
r 1
+
f 12 f 13 r 1
r 12
−
f 12 f 13 r
2
2
r 1 r 12
|
|
|
|
|
𝛷 def
⟩
+
𝛽(n d − 1)
2
⟨
𝛷 abc
|
|
|
|
|
f 12 f 13 r 12
r
2
1
−
r
2
2
f 12 f 13
r
2
1
r 12
|
|
|
|
|
𝛷 def
⟩
+
√
4𝜋
3
∑
𝜆=±1
(
l d (l d +1)(2l d +1−𝜆)
2(2l d + 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
] [
p
⟨
𝛷 abc
|
|
|
|
|
f 12 f 13 r 2
r 1 r
2
12
|
|
|
|
|
d
m d −𝜈
l d +𝜆
e
m e +𝜈
l e +𝜆 ′ f
m f
l f
⟩
−𝛽
⟨
𝛷 abc
|
|
|
|
f 12 f 13 r 2
r 1 r 12
|
|
|
|
d
m d −𝜈
l d +𝜆
e
m e +𝜈
l e +𝜆 ′ f
m f
l f
⟩ ]
.
(30)
To keep the above formula more compact, we have defined f 12 = r
p
12
e
−𝛽r 12 and f 13 =
r
p ′
13
e
−𝛽 ′ r 13 .
Equation (30) confirms that the kinetic-energy matrix elements in E-Hy-CI reduce
to contiguous potential-energy integrals.
8 Numerical Verification
The formulas for kinetic-energy intergrals in Hy-CI developed here and (in more
detail) in [37] were confirmed by comparing integrals produced using them with
similar integrals computed in other ways by Ruiz [25, 27, 28] and by Sims and
Hagstrom [10]. After making some adjustments needed to achieve consistency (see
[37]), complete agreement with the results of those investigators was obtained.
The errors noted in various equations of [37] arose while transcribing the formulas from computer programs and therefore did not affect the numerical verification
process.