Theor Chem Acc (2015) 134:113
1 3
Combining Eqs. ( 3 ), ( 4 ), and ( 5 ) with Eqs. ( 7 )–( 10 )
produces
where the contributions due to the terms with even and odd
powers of r in the expansion ( 4 ) read
and
respectively.
(11)
l (ω; r 1 , r 2 ) =
+
l (ω; r 1 , r 2 ) +
−
l (ω; r 1 , r 2 ),
(12)
+
l (ω; r 1 , r 2 ) = exp
−
ω
2
(r
2
1 + r
2
2 )
∞
j=l
C 2j (ω)
×
⎡
⎣
j−l
k=0
A jlk (r 1 r 2 )
l+2k
(r
2
1 + r
2
2 )
j−l−2k
⎤
⎦
(13)
−
l (ω; r 1 , r 2 ) = exp
−
ω
2
(r
2
1 + r
2
2 )
∞
j=1
C 2j−1 (ω)
×
⎡
⎣
min(j,l)
p=0
j
q=p
j−q
k=0
B jlpqk r
l+2k−2p+2q
<
× r
−l+2k+2p−1
>
(r
2
1 + r
2
2 )
j−q−2k
⎤
⎦ ,
When employed in conjunction with the identity
where (t) and 2 F 1
a 1 , a2
b 1
t
are the pertinent gamma
and hypergeometric functions, respectively, Eqs. ( 12 ) and
( 13 ) yield
where
(14)
∞
0
∞
0
r
α
< r
α
> (r
2
1 + r
2
2 )
β exp
−ω (r
2
1 + r
2
2 )
r
2
1 r
2
2 dr 1 dr 2
=
2 −(α+α +4)/2 ω −(α+α +2β+6)/2
α + 3
×
α + α + 2β + 6
2
2 F 1
1, −
α +1
2
α+5
2
− 1
(15)
ν l (ω) =
∞
j=l
∞
j =l
C 2j (ω) C 2j (ω)
ω j+j +3
D
++
ljj
+ 2
∞
j=1
∞
j =l
C 2j−1 (ω) C 2j (ω)
ω j+j +5/2
D
−+
ljj
+
∞
j=1
∞
j =1
C 2j−1 (ω) C 2j −1 (ω)
ω j+j +2
D
−−
ljj ,
and
(16)
D
++
ljj =
π 3
2l + 1
(j + j
+ 2)!
j−l
k=0
j −l
k =0
2
−4(l+k+k +1/2)
2l + 2k + 2k + 2
l + k + k + 1
A jlk A j lk ,
(17)
D
−+
ljj =
π 2
2l + 1
j + j
+
5
2
min(j,l)
p=0
j
q=p
j−q
k=0
j −l
k =0
2 −(l+2k+2k +q−5/2)
2l + 2k + 2k − 2p + 2q + 3
× 2 F 1
1, −(k + k + p)
l + k + k − p + q +
5
2
𚵿
− 1
A j lk B jlpqk ,
(18)
D
−−
ljj =
π 2
2l + 1
(j + j
+ 1)!
min(j,l)
p=0
j
q=p
j−q
k=0
min(j ,l)
p =0
j
q =p
j −q
k =0
2 −(2k+2k +q+q −3)
2(l + k + k − p − p + q + q ) + 3
× 2 F 1
1, l − (k + k + p + p ) +
1
2
l + k + k − p − p + q + q +
5
2
𵫿
− 1
B jlpqk B j lp q k .
151
Reprinted from the journal