Theor Chem Acc (2015) 134:113
1 3
where E
(2)
l
is the incremental contribution to the secondorder energy E (2) [and also to its correlation component
E (2)
corr for l = 0 ] arising from l (ω; r 1 , r 2 ) . Application of
well-known algebraic techniques [ 29 ] to these expressions,
which follow for l = 0 from straightforward arguments
based upon perturbation theory [ 30 ], produces
and
In Eqs. ( 24 ), ( 28 ), and ( 29 ), L
l+1/2
n
(t) , Y m
l (θ, ϕ) ,
and 3 F 2
a 1 , a 2 , a 3
b 1 , b 2
t
are the pertinent generalized
Laguerre polynomial, spherical harmonic, and generalized
hypergeometric function, respectively. Equation ( 29 )
yields the leading asymptotics {˜ ν l (ω)} of the collective
(28)
E
(2)
l = −
4 −l
π (2l + 1) l
3 F 2
l, l +
1
2 , l +
1
2
l +
3
2 , 2l + 2
1
(29)
˜
ν l (ω) =
4 −l
2 π l 2 3 F 2
l, l, l +
1
2
l + 1, 2l + 2
1
+ E
(2)
l
ω
−1 .
occupancies at the limit of ω → ∞ that, in excellent agreement with the previously published
results
of
numerical
calculations
[ 14 ],
equal
127−48π+36 ln 2
24π
ω −1 ≈ 1.534 321 462 · 10 −2 ω −1 for l = 1
and
−2053+720π−300 ln 2
360π
ω −1 ≈ 8.864 544 969 · 10 −4 ω −1
for l = 2 .
2.3 The strong-correlation limit
At the strong-correlation limit of ω → 0 , the wavefunction
( 3 ) is given by its asymptotic expression [ 13 , 14 ]
where r 0 = (
2
ω 2 ) 1/3 . The corresponding leading asymptotics {˜ ν l (ω)} of the collective occupancies is given by
(30)
(ω; ; r 1 , r 2 ) ≈ ˜
(ω; ; r 1 , r 2 )
=
3 1/8
2 5/6 π 3/2 ω
5/3 exp
−
ω
4
( r 1 + + r 2 )
2
× exp
−
√
3
4
ω (|| r 1 − − r 2 | − r 0 )
2
,
Table 1 The collective occupancies of NOs at the four largest values of ω that correspond to polynomial correlation factors
l
ν l (ω K )
K = 1
K = 2
K = 3
K = 4
0
9.755557 × 10
−1
9.146744 × 10
−1
8.392861 × 10
−1
7.637280 × 10
−1
1
2.375865 × 10
−2
8.478905 × 10
−2
1.606437 × 10
−1
2.344602 × 10
−1
2
5.910989 × 10
−4
4.626236 × 10
−4
5.577958 × 10
−5
1.809789 × 10
−3
3
7.141400 × 10
−5
5.589217 × 10
−5
9.702106 × 10
−6
1.412707 × 10
−6
4
1.539745 × 10
−5
1.205081 × 10
−5
2.993862 × 10
−6
2.857688 × 10
−7
5
4.574478 × 10
−6
3.580215 × 10
−6
1.016854 × 10
−6
1.383782 × 10
−7
6
1.671905 × 10
−6
1.308516 × 10
−6
3.973200 × 10
−7
6.519386 × 10
−8
7
7.070209 × 10
−7
5.533500 × 10
−7
1.746623 × 10
−7
3.183796 × 10
−8
8
3.332811 × 10
−7
2.608425 × 10
−7
8.439335 × 10
−8
1.641902 × 10
−8
9
1.708878 × 10
−7
1.337453 × 10
−7
4.400492 × 10
−8
8.940612 × 10
−9
10
9.370289 × 10
−8
7.333657 × 10
−8
2.441966 × 10
−8
5.114637 × 10
−9
20
1.690606 × 10
−9
1.323153 × 10
−9
4.578093 × 10
−10
1.054168 × 10
−10
50
7.564718 × 10
−12
5.920527 × 10
−12
2.071120 × 10
−12
4.896421 × 10
−13
100
1.217710 × 10
−13
9.530410 × 10
−14
3.339285 × 10
−14
7.924784 × 10
−15
(31)
˜
ν l (ω) =
16 π 2
2l + 1
3 1/4
2 5/3 π 3 ω
10/3
∞
0
∞
0
⎛
⎝ 2l + 1
2
π
0
exp
−
ω
4
(r
2
1 + r
2
2 + 2 r 1 r 2 cos θ 12 )
× exp
−
√
3
4
ω
r 2
1 + r 2
2 − 2 r 1 r 2 cos θ 12 − r 0
2
P l (cos θ 12 ) sin θ 12 dθ 12 )
2 r
2
1 r
2
2 dr 1 dr 2 .
153
Reprinted from the journal
1 3
where E
(2)
l
is the incremental contribution to the secondorder energy E (2) [and also to its correlation component
E (2)
corr for l = 0 ] arising from l (ω; r 1 , r 2 ) . Application of
well-known algebraic techniques [ 29 ] to these expressions,
which follow for l = 0 from straightforward arguments
based upon perturbation theory [ 30 ], produces
and
In Eqs. ( 24 ), ( 28 ), and ( 29 ), L
l+1/2
n
(t) , Y m
l (θ, ϕ) ,
and 3 F 2
a 1 , a 2 , a 3
b 1 , b 2
t
are the pertinent generalized
Laguerre polynomial, spherical harmonic, and generalized
hypergeometric function, respectively. Equation ( 29 )
yields the leading asymptotics {˜ ν l (ω)} of the collective
(28)
E
(2)
l = −
4 −l
π (2l + 1) l
3 F 2
l, l +
1
2 , l +
1
2
l +
3
2 , 2l + 2
1
(29)
˜
ν l (ω) =
4 −l
2 π l 2 3 F 2
l, l, l +
1
2
l + 1, 2l + 2
1
+ E
(2)
l
ω
−1 .
occupancies at the limit of ω → ∞ that, in excellent agreement with the previously published
results
of
numerical
calculations
[ 14 ],
equal
127−48π+36 ln 2
24π
ω −1 ≈ 1.534 321 462 · 10 −2 ω −1 for l = 1
and
−2053+720π−300 ln 2
360π
ω −1 ≈ 8.864 544 969 · 10 −4 ω −1
for l = 2 .
2.3 The strong-correlation limit
At the strong-correlation limit of ω → 0 , the wavefunction
( 3 ) is given by its asymptotic expression [ 13 , 14 ]
where r 0 = (
2
ω 2 ) 1/3 . The corresponding leading asymptotics {˜ ν l (ω)} of the collective occupancies is given by
(30)
(ω; ; r 1 , r 2 ) ≈ ˜
(ω; ; r 1 , r 2 )
=
3 1/8
2 5/6 π 3/2 ω
5/3 exp
−
ω
4
( r 1 + + r 2 )
2
× exp
−
√
3
4
ω (|| r 1 − − r 2 | − r 0 )
2
,
Table 1 The collective occupancies of NOs at the four largest values of ω that correspond to polynomial correlation factors
l
ν l (ω K )
K = 1
K = 2
K = 3
K = 4
0
9.755557 × 10
−1
9.146744 × 10
−1
8.392861 × 10
−1
7.637280 × 10
−1
1
2.375865 × 10
−2
8.478905 × 10
−2
1.606437 × 10
−1
2.344602 × 10
−1
2
5.910989 × 10
−4
4.626236 × 10
−4
5.577958 × 10
−5
1.809789 × 10
−3
3
7.141400 × 10
−5
5.589217 × 10
−5
9.702106 × 10
−6
1.412707 × 10
−6
4
1.539745 × 10
−5
1.205081 × 10
−5
2.993862 × 10
−6
2.857688 × 10
−7
5
4.574478 × 10
−6
3.580215 × 10
−6
1.016854 × 10
−6
1.383782 × 10
−7
6
1.671905 × 10
−6
1.308516 × 10
−6
3.973200 × 10
−7
6.519386 × 10
−8
7
7.070209 × 10
−7
5.533500 × 10
−7
1.746623 × 10
−7
3.183796 × 10
−8
8
3.332811 × 10
−7
2.608425 × 10
−7
8.439335 × 10
−8
1.641902 × 10
−8
9
1.708878 × 10
−7
1.337453 × 10
−7
4.400492 × 10
−8
8.940612 × 10
−9
10
9.370289 × 10
−8
7.333657 × 10
−8
2.441966 × 10
−8
5.114637 × 10
−9
20
1.690606 × 10
−9
1.323153 × 10
−9
4.578093 × 10
−10
1.054168 × 10
−10
50
7.564718 × 10
−12
5.920527 × 10
−12
2.071120 × 10
−12
4.896421 × 10
−13
100
1.217710 × 10
−13
9.530410 × 10
−14
3.339285 × 10
−14
7.924784 × 10
−15
(31)
˜
ν l (ω) =
16 π 2
2l + 1
3 1/4
2 5/3 π 3 ω
10/3
∞
0
∞
0
⎛
⎝ 2l + 1
2
π
0
exp
−
ω
4
(r
2
1 + r
2
2 + 2 r 1 r 2 cos θ 12 )
× exp
−
√
3
4
ω
r 2
1 + r 2
2 − 2 r 1 r 2 cos θ 12 − r 0
2
P l (cos θ 12 ) sin θ 12 dθ 12 )
2 r
2
1 r
2
2 dr 1 dr 2 .
153
Reprinted from the journal