MATCH
Communications in Mathematical
and in Computer Chemistry
MATCH Commun. Math. Comput. Chem. 60 (2008) 385-394
ISSN 0340 - 6253
Combined Theory of Two-Electron Nonrelativistic and Quasirelativistic
Multicenter Integrals over Integer and Noninteger n Slater Type Orbitals
Using Auxiliary Functions Qnsq and G−qns
I. I. Guseinov
Department of Physics, Faculty of Arts and Sciences, Onsekiz Mart University,
Çanakkale, Turkey
(Received January 23, 2008)
Abstract:
Using complete orthonormal sets of ψ α − exponential type orbitals ( α = 1, 0, −1, −2,... ) a
large number of series expansion formulas for the two-electron nonrelativistic and
quasirelativistic multicenter integrals over integer and noninteger n Slater type orbitals is
obtained through the non- and quasi- relativistic basic functions. The non- and quasirelativistic basic functions are determined by the linear combinations of auxiliary functions
Qnsq and G−qns introduced by the author. The coefficients of series expansion relations are
expressed in terms of two-center overlap integrals over integer n Slater functions. The
relationships obtained are valid for the arbitrary location, quantum numbers and screening
constants of orbitals.
1. Introduction
One of the oldest mathematical and computational problems of molecular electron
structure theory is the efficient and reliable evaluation of the notoriously difficult molecular
multicenter integrals of exponentially decaying basis functions. Over the years, a large variety
of different approaches have been tried out, but in spite of the heroic efforts of numerous
researches, a completely satisfactory solution has not been found yet. A principal tool is
reduction of a multicenter integral to so-called auxiliary functions, which are essentially
special functions. The use of auxiliary functions was a particularly “hot” topic in the fifties of
the last century, as documented in Refs. [1-4]. However, even today, there is a lot of work
which all deal with auxiliary functions [5-8]. We have also worked quite extensively on the
use of auxiliary functions in connection with the evaluation of multicenter integrals of
exponentially decaying functions [9-20]. The aim of this work is to use auxiliary functions
- 386 ∞ 1
Qnsq ( p, pt ) = ∫ ∫ ( μν ) q ( μ +ν ) n ( μ −ν ) s e − pμ − ptν d μ dν
(1)
1 −1
and
∞ 1
n −1
[ pa ( μ +ν )]k − pμ − ptν
( μν ) q ( μ −ν ) s
− pa ( μ +ν )
(1
−
e
)e
d μ dν
∑
( μ +ν )n
k!
k =0
1 −1
G−qns ( pa ; p, pt ) = ∫ ∫
(2)
presented in previous papers [21-23] in evaluation of two-electron non- and quasi-relativistic
(NQR) multicenter integrals of integer and noninteger n Slater type orbitals (ISTO and
NISTO) arising in the study of electronic structure of atomic and molecular systems when the
nonrelativistic and quasirelativistic theories are employed [24]. Here, pa > 0 , p > 0 and
− p ≤ pt ≤ p . The indices n, s and q are all nonnegative integers. The auxiliary functions Qnsq
and G−qns were all calculated in Refs. [9, 14, 16, 22, 23] by the use of recurrence relations,
analytical expressions and series expansion formulas for all values of parameters.
2. Use of expansion formulae for NISTO charge densities
The two-electron NQR multicenter integrals examined in the present work have the
following form:
G G G
I p* p′* , p* p′* (ζ 1ζ 1′, ζ 2ζ 2′ ; Rca , Rdb , Rab )
1 1
2 2
G
G
G
G
= ∫ ∫ χ *p* (ζ 1 , ra1 ) χ p′* (ζ 1′, rc1 )O(r21 ) χ p* (ζ 2 , rb 2 ) χ *p′* (ζ 2′ , rd 2 )dv1dv2
(3)
G G G
I ip* p′* , p* p′* (ζ 1ζ 1′, ζ 2ζ 2′ ; Rca , Rdb , Rab )
1 1
2 2
G
G
G
G
G
= ∫ ∫ χ *p* (ζ 1 , ra1 ) χ p′* (ζ 1′, rc1 )O i (r21 ) χ p* (ζ 2 , rb 2 ) χ *p′* (ζ 2′ , rd 2 )dv1dv2
(4)
G G G
I ijp* p′* , p* p′* (ζ 1ζ 1′, ζ 2ζ 2′ ; Rca , Rdb , Rab )
1 1
2 2
G
G
G
G
G
= ∫ ∫ χ *p* (ζ 1 , ra1 ) χ p′* (ζ 1′, rc1 )Oij (r21 ) χ p* (ζ 2 , rb 2 ) χ *p′* (ζ 2′ , rd 2 )dv1dv2
(5)
G G G
L pij* p′* , p* p′* (ζ 1ζ 1′, ζ 2ζ 2′ ; Rca , Rdb , Rab )
1 1
2 2
,
G
G
G
G
G
= ∫ ∫ χ *p* (ζ 1 , ra1 ) χ p′* (ζ 1′, rc1 )R ij (r21 ) χ p* (ζ 2 , rb 2 ) χ *p′* (ζ 2′ , rd 2 )dv1dv2
(6)
1
1
1
1
1
2
2
1
1
where
2
2
2
1
2
2
G
G G
i, j = 0, ±1, r21 = r1 − r2 ; x11 = x1 , x1−1 = y1 , x10 = z1 and
Cartesian coordinates of electrons;
p ≡ n l mk and
*
k
*
k k
2
x12 = x2 , x2−1 = y2 , x20 = z2 are
p′ ≡ n′ l ′ mk′ ;
*
k
*
k k
*
k
*
k
n and n′
the
are the
noninteger and integer (for n = nk and n = nk′ ) principal quantum numbers; the normalized
*
k
*
k
complex or real NISTO are determined by
G
χ n lm (ζ , r ) = [Γ(2n* + 1)]−1/ 2 (2ζ )
*
n* +
1
2
*
r n −1e −ζ r Slm (θ , ϕ ) .
(7)
- 387 The operators containing in Eqs. (3)-(6) are defined as [24]
O (r21 ) =
1
r21
(8)
xi
G
O i (r21 ) = 21
r213
(9)
3 x i x j − δ r 2 4π
G
G
O ij (r21 ) = 21 21 5 ij 21 −
δ ijδ (r21 )
r21
3
(10)
xi x j
G
R i j (r21 ) = 21 3 21
r21
(11)
Here, the operators (8) and (11) correspond to the classical relativistic correction to the
interaction between the electrons. The correction is due to the retardation of the
electromagnetic field produced by an electron. The operators (9) and (10) describe the spinorbit and spin-spin magnetic moment interactions of the two-electrons, respectively.
In order to evaluate integrals (3)-(6), we use the expansion formulae of the electron charge
densities established in previous paper [25] in the following form:
G
G
χ p (ζ , ra ) χ *p′ (ζ ′, rc ) =
*
*
N μ −1 ν
G
1
G
lim ∑∑ ∑ Wpα* Np′*q (ζζ ′z; Rca , 0) χ q ( z, ra ) ,
4π N →∞ μ =1 ν =0 σ =−ν
(12)
where α = 1, 0, −1, −2,... , q ≡ μνσ and z = ζ + ζ ′ . Then, we derive the expressions in terms
of the charge density expansion coefficients and two-center basic integrals:
G G G
I p* p′* , p* p′* (ζ 1ζ 1′, ζ 2ζ 2′ ; Rca , Rdb , Rab ) = lim
1 1
2 2
N1 N 2 →∞
N1 μ1 −1
1 =1 1 = 0
N 2 μ2 −1
×∑ ∑
μ2 =1ν 2 = 0
G G G
I ip* p′* , p* p′* (ζ 1ζ 1′, ζ 2ζ 2′ ; Rca , Rdb , Rab ) = lim
1 1
2 2
N1 N 2 →∞
ν2
N1 N 2 →∞
1 =1 1 = 0
ν1
ν2
∑
N1
p1* p1′*q1
1 =− 1
α N2
Wp* p′*q
2 2
N1 μ1 −1
2
ν1
1 =1 1 = 0
ν2
∑
μ2 =1ν 2 = 0 σ 2 =−ν 2
2
(14)
G
(ζ 1ζ 1′z1 ; Rca , 0)
G
G
Wpα* Np′2*q (ζ 2ζ 2′ z2 ; Rdb , 0)K qij1q2 ( z1 z2 , Rab )
2 2
(13)
G
(ζ 1ζ 1′z1 ; Rca , 0)
G
G
(ζ 2ζ 2′ z2 ; Rdb , 0)K qi1q2 ( z1 z2 , Rab )
N1
p1* p1′*q1
1 =− 1
G
(ζ 1ζ 1′z1 ; Rca , 0)
G
G
(ζ 2ζ 2′ z2 ; Rdb , 0)K q1q2 ( z1 z2 , Rab )
Wα
∑
∑
∑
μ ν
σ ν
N 2 μ2 −1
×∑ ∑
N2
p2* p2′*q2
2 =− 2
μ2 =1ν 2 = 0 σ 2 =−ν 2
2 2
1 =− 1
∑ν W α
σ
N1 μ1 −1
×∑ ∑
1 1
N1
p1* p1′*q1
∑ ∑ σ ∑ν W α
μ ν
N 2 μ2 −1
G G G
I ijp* p′* , p* p′* (ζ 1ζ 1′, ζ 2ζ 2′ ; Rca , Rdb , Rab ) = lim
ν1
∑ ∑ σ ∑ν W α
μ ν
(15)
- 388 G G G
L pij* p′* , p* p′* (ζ 1ζ 1′, ζ 2ζ 2′ ; Rca , Rdb , Rab ) = lim
1 1
N1 N 2 →∞
2 2
N1 μ1 −1
1 =1 1 = 0
N 2 μ2 −1
×∑ ∑
ν1
∑ ∑ σ ∑ν W α
μ ν
ν2
∑
μ2 =1ν 2 = 0 σ 2 =−ν 2
N1
p1* p1′*q1
1 =− 1
G
(ζ 1ζ 1′z1 ; Rca , 0)
G
G
W pα* Np′2*q (ζ 2ζ 2′ z2 ; Rdb , 0)Lijq1q2 ( z1 z2 , Rab )
2 2
.
(16)
2
G
G
G
G
Here, K q1q2 ( z1 z2 , Rab ) , K qi1q2 ( z1 z2 , Rab ) , K qij1q2 ( z1 z2 , Rab ) and Lijq1q2 ( z1 z2 , Rab ) are the two-electron
basic integrals defined as
G
K q1q2 ( z1 z2 , Rab ) =
1
4π
∫J
q1
G
G
( z1 , ra 2 ) χ q2 ( z2 , rb 2 )dv2
(17)
G
K qi1q2 ( z1 z2 , Rab ) =
1
4π
∫J
i
q1
G
G
( z1 , ra 2 ) χ q2 ( z2 , rb 2 )dv2
(18)
G
K qij1q2 ( z1 z2 , Rab ) =
1
4π
∫J
ij
q1
G
G
( z1 , ra 2 ) χ q2 ( z2 , rb 2 )dv2
(19)
G
Lijq1q2 ( z1 z2 , Rab ) =
1
4π
∫Λ
ij
q1
G
G
( z1 , ra 2 ) χ q2 ( z2 , rb 2 )dv2 .
(20)
The quantities J q , J qi , J qij and Λ ijq occurring in these formulas are the one-electron basic
integrals defined as
G
J q ( z , ra 2 ) =
1
4π
∫χ
*
q
G
( z , ra1 )O(r21 )dv1
(21a)
G
= f μν00,νσ ( z , ra 2 )
G
J qi ( z , ra 2 ) =
1
4π
=
∑ν
=
G
G
( z , ra1 )O i (r21 )dv1
(22a)
⎛ xai 2 ⎞ 11
G
G
i
10
aνσ
⎟ f μν ,νσ ( z, ra 2 )
, m f μν ,ν −1m ( z , ra 2 ) − (2ν + 1) ⎜
−1)
⎝ ra 2 ⎠
1
4π
∫χ
ν −2
∑
m =− (ν − 2)
×
*
q
ν −1
m =− (
G
J qij ( z , ra 2 ) =
∫χ
(21b)
ν −1
G
G
( z , ra1 )O ij (r21 )dv1
(23a)
G
ij
20
aνσ
, m f μν ,ν − 2 m ( z , ra 2 ) − (2ν + 1)
⎡ i ⎛ xaj2 ⎞
⎛ xai 2
j
⎢ aνσ , m ⎜
⎟ + aνσ ,m ⎜
−1) ⎢
⎝ ra 2 ⎠
⎝ ra 2
⎣
∑ν
m =− (
*
q
(22b)
⎞ ⎤ 21
G
G
21
⎟ ⎥ f μν ,ν −1m ( z , ra 2 ) − (2ν + 1)δ ij f μν ,νσ ( z , ra 2 )
⎥
⎠⎦
⎛ xi ⎞ ⎛ x j ⎞
4π
G
G
*
δ ij χ μνσ
( z , ra 2 )
+ (2ν + 1)(2ν + 3) ⎜ a 2 ⎟ ⎜ a 2 ⎟ f μν22,νσ ( z , ra 2 ) −
3
r
r
a
a
2
2
⎝
⎠⎝
⎠
(23b)
- 389 G
Λ ijq ( z , ra 2 ) =
G
G
( z , ra1 )R ij (r21 )dv1 .
(24a)
1
∑ (2 L + 1)(2l + 1)1/ 2 C l|m| (1 j,νσ ) Amjσ
3 lm , LM
G
×C L|M | (1i, lm) AimM f μ00L , LM ( z , ra 2 )
(24b)
1
4π
∫χ
*
q
=
See Ref. [21] for the exact definition of generalized Gaunt coefficients C L|M |
and
coefficients AM . We notice that the charge densities expansion coefficients W α N occurring in
Eqs. (13)-(16) depend on overlap integrals for the calculation of which one can use the
computer programs presented in previous papers [26, 27].
The relations (21)-(23) have been established in previous paper [28]. The quantities
G
f μνtk ,κ m ( z , r ) are the one-electron basic functions which are determined by
1/ 2
G
⎛ 4π ⎞
f μνtk ,κ m ( z , r ) = f μνtk ( z , r ) ⎜
⎟
⎝ 2κ + 1 ⎠
f μνtk ( z , r ) = f μνtk ,00 ( z , r ) =
Sκ*m (θ , ϕ )
(25)
t
μ +ν + k
N μν
(2 z ) ⎛
⎞
1 − e − zr ∑ γ νs k ( μ )( zr ) s ⎟ ,
ν + t +! ⎜
( zr )
s =0
⎝
⎠
(26)
where t = 0,1, 2 , 0 ≤ k ≤ t , κ = ν ,ν − 1,ν − 2 and
⎡ (2 z ) 2t −1 ⎤
t
N μν
(2 z ) = 2 μ −t +1 ( μ + ν + 1)! ⎢
⎥
⎣ (2ν + 1)(2 μ )!⎦
γ νs 0 ( μ ) = γ νs ( μ ) =
1/ 2
(27)
1
( μ −ν )!
−
s ! ( μ + ν + 1)!( s − 2ν − 1)!
γ νs 1 ( μ ) =
1
⎡ (2ν + 1 − s )γ νs ( μ ) + γ νs −1 ( μ ) ⎤⎦
2ν + 1 ⎣
γ νs 2 ( μ ) =
1
⎡ (2ν + 1 − s )(2ν + 3 − s )γ νs ( μ )
(2ν + 1)(2ν + 3) ⎣
.
(28)
(29)
(30)
+ (4ν + 5 − 2 s )γ νs −1 ( μ ) + γ νs − 2 ( μ ) ⎤⎦
The terms with negative factorials in Eq. (28) should be equated to zero and γ νs ( μ ) = 0 for
s < 0 and s > μ +ν .
Now we introduce the two-electron basic functions defined as
G
Dtkμ1ν1 ,νσ ; μ2ν 2σ 2 ( z1 z2 , Rab ) =
1
4π
G
∫ f μ ν νσ ( z , r
tk
1 1,
1
a2
G
) χ μ2ν 2σ 2 ( z2 , rb 2 )dv2 ,
(31)
- 390 G
where the f μtk1ν1 ,νσ ( z1 , ra 2 ) are one-electron basic functions determined by Eq. (25).
G
G
In order to express the two-electron basic integrals K q1q2 ( z1 z2 , Rab ) , K qi1q2 ( z1 z2 , Rab ) ,
G
G
K qij1q2 ( z1 z2 , Rab ) and Lijq1q2 ( z1 z2 , Rab ) in terms of basic functions, Eq. (31), we take into account
Eqs. (21)-(24) in (17)-(20). Then, we obtain:
G
G
K q1q2 ( z1 z2 , Rab ) = Dμ001ν1 ,ν1σ1 ;μ2ν 2σ 2 ( z1 z2 , Rab )
(32)
G
K qi1q2 ( z1 z2 , Rab ) =
(33)
G
K qij1q2 ( z1 z2 , Rab ) =
ν 1 −1
∑ν
m1 =− ( 1 −1)
ν1 − 2
∑ν
m1 =− ( 1 − 2)
G
G
aνi 1σ1 ,m1 Dμ101ν1 ,ν1 −1m1 ;μ2ν 2σ 2 ( z1 z2 , Rab ) − (2ν 1 + 1) Dμ11,1νi1 ,ν1σ1 ;μ2ν 2σ 2 ( z1 z2 , Rab )
G
aνij1σ1 ,m1 Dμ201ν1 ,ν1 − 2 m1 ;μ2ν 2σ 2 ( z1 z2 , Rab )
G
G
⎡ aνi1σ1 ,m1 Dμ21,1ν1j,ν1 −1m1 ;μ2ν 2σ 2 ( z1 z2 , Rab ) + aνj1σ1 ,m1 Dμ21,1ν1i ,ν1 −1m1 ;μ2ν 2σ 2 ( z1 z2 , Rab ) ⎤
⎣
⎦
G
G
21
22,ij
−(2ν 1 + 1)δ ij Dμ1ν1 ,ν1σ1 ;μ2ν 2σ 2 ( z1 z2 , Rab ) + (2ν 1 + 1)(2ν 1 + 3) Dμ1ν1 ,ν1σ1 ;μ2ν 2σ 2 ( z1 z2 , Rab )
G
1
− δ ij S μ1ν1σ1 , μ2ν 2σ 2 ( z1 z2 , Rab )
3
−(2ν 1 + 1)
ν 1 −1
∑
m1 =− (ν 1 −1)
G
1
Lijq1q2 ( z1 z2 , Rab ) = ∑ (2 L + 1)(2l + 1)1/ 2 C l|m| (1 j ,νσ ) Amjσ
3 lm , LM
,
G
L| M |
M
00
×C (1i, lm) Aim Dμ L , LM ;μ2ν 2σ 2 ( z1 z2 , Rab )
(34)
(35)
where
G
G
G
S μ1ν1σ1 , μ2ν 2σ 2 ( z1 z2 , Rab ) = ∫ χ μ*1ν1σ1 ( z1 , ra 2 ) χ μ2ν 2σ 2 ( z2 , rb 2 )dv2
(36)
are the overlap integrals. It is easy to express the quantities D11,i , D 21,i and D 22,ij occurring in
G
Eqs. (33) and (34) in terms of two-electron basic functions Dtkμ1ν1 ,νσ ;μ2ν 2σ 2 ( z1 z2 , Rab ) :
G
1 ⎛ xai 2 ⎞ 11
G
G
Dμ11,1νi1 ,ν1σ1 ;μ2ν 2σ 2 ( z1 z2 , Rab ) =
∫ ⎜ ⎟ f μ ν ,ν σ ( z1 , ra 2 ) χ μ2ν 2σ 2 ( z2 , rb 2 )dv2
4π ⎝ ra 2 ⎠ 1 1 1 1
=∑
lm
G
(2l + 1)
C l |m| (1i,ν 1σ 1 ) Aimσ1 Dμ111ν1 ,lm; μ2ν 2σ 2 ( z1 z2 , Rab )
1/ 2
[3(2v1 + 1)]
G
1 ⎛ xai 2 ⎞ 21
G
G
Dμ21,1ν1i ,ν1 −1m1 ;μ2ν 2σ 2 ( z1 z2 , Rab ) =
⎜
⎟ f μ1ν1 ,ν1 −1m1 ( z1 , ra 2 ) χ μ2ν 2σ 2 ( z2 , rb 2 )dV2
∫
4π ⎝ ra 2 ⎠
=∑
lm
G
2l + 1
C l |m| (1i,ν 1 − 1m1 ) Aimm1 Dμ211ν1 ,lm;μ2ν 2σ 2 ( z1 z2 , Rab )
[3(2ν 1 − 1)]1/ 2
(37a)
(37b)
(38a)
(38b)
- 391 G
1 ⎛ xai 2 ⎞⎛ xaj2 ⎞ 22
G
G
ij
( z1 z2 , Rab ) =
Dμ22,
∫ ⎜ ⎟⎜ ⎟ f μ ν ,ν σ ( z1 , ra 2 )χ μ2ν 2σ 2 ( z2 , rb 2 )dv2
1ν 1 ,ν 1σ 1 ; μ2ν 2σ 2
4π ⎝ ra 2 ⎠⎝ ra 2 ⎠ 1 1 1 1
(39a)
1/ 2
=
⎛ 2l + 1 ⎞
1
∑ (2L + 1) ⎜ 2ν + 1 ⎟
3 lm , LM
⎝ 1 ⎠
×C
L| M |
C l|m| (1 j ,ν 1σ 1 ) Amjσ1
G
(1i, lm) A Dμ1ν1 , LM ;μ2ν 2σ 2 ( z1 z2 , Rab )
M
im
.
(39b)
22
3. Expressions for two-electron basic functions in terms of auxiliary functions Qnsq and
G−qns
G
Now we express the two-electron basic functions Dtkμ1ν1 ,νσ ;μ2ν 2σ 2 ( z1 z2 , Rab ) in terms of
auxiliary functions Qnsq and G−qns . For this purpose we use Eqs. (5) and (17) of Ref. [29] for the
rotation of two-center overlap integrals over arbitrary atomic orbitals in molecular coordinate
system:
min(ν ,ν 2 )
G
*
Dμtk1ν1 ,νσ ;μ2ν 2σ 2 ( z1 z2 , Rab ) = ∑ Tνσλ ,ν 2σ 2 (θ ab , ϕ ab ) Dμtk1ν1 ,νλ ;μ2ν 2λ ( z1 z2 , Rab ) .
(40)
λ =0
Here, the quantities Dμtk1ν1 ,νλ ;μ2ν 2λ ( z1 z2 , Rab ) are the two-center basic functions relative to lined-
up coordinate systems:
Dμtk1ν1 ,νλ ;μ2ν 2λ ( z1 z2 , Rab ) =
1
4π
G
∫ f μ ν νλ ( z , r
tk
1 1,
1
a2
G
) χ μ2ν 2λ ( z2 , rb 2 )dv2 .
(41)
Transforming (41) from Cartesian to elliptical coordinates and integrating over azimuthal
angle ϕ2 = ϕa 2 = ϕb 2 , we finally obtain for two-center basic functions in terms of auxiliary
functions Qnsq and G−qns the expressions
q
Dμtk1ν1 ,νλ ;μ2ν 2λ ( z1 z2 , R ) = Aμt 1ν1 ,ν ,μ2 ( p1 , p2 )∑ gαβ
(νλ ,ν 2λ )
αβ q
μ1 +ν1 + k
⎧
Q |νq1 +α +t|, μ2 − β (p 2 , −p 2 ) − ∑ γ νs 1k ( μ1 )p 1sQ sq+|ν1 +α +t|, μ2 − β (p 12 , p 12t 12 ) for ν 1 + α +t ≤ 0
⎪
⎪
s =0
×⎨
μ1 +ν1 + k
q
⎪G
(p ; p , p t )−
γ νs 1k ( μ1 ) p1s Qsq−(ν1 +α +t ), μ2 − β ( p12 , p12t12 ) for ν 1 + α + t > 0.
⎪⎩ − (ν1 +α +t ), μ2 − β 1 12 12 12 s =ν∑
1 +α + t
Here, R = Rab , p1 = 1 z1 R, p2 = 1 z2 R, η = z2 / z1 and
2
2
(42a )
(42b)
- 392 -
Aμt 1ν1 ,ν , μ2 ( p1 , p2 ) =
2μ1 + μ2 −t +3 ( μ1 +ν 1 + 1)! p2μ2 −t + 2
[(2ν 1 + 1)(2ν + 1)(2 μ1 )!(2 μ2 )!( p1 p2 )]1/ 2 p1ν1 +1 .
(43)
It is easy to derive for one-center basic functions the following relations:
Dμtk1ν1 ,νσ ;μ2ν 2σ 2 ( z1 z2 ) = Dμtk1ν1 ,νσ ;μ2ν 2σ 2 ( z1 z2 , 0) = δνν 2 δσσ 2 Bμt 1ν1 ,ν , μ2 ( z1 , z2 )
⎧ | ν 1 − μ2 + t |! μ1 +ν1 + k ν1k
(| ν 1 − μ2 + t | + s )!
for ν 1 − μ2 + t ≤ 0
⎪ |ν1 − μ2 +t|+1 − ∑ γ s ( μ1 )
(1 + η )|ν1 − μ2 + t|+ s +1
s =0
⎪ η
×⎨
μ1 +ν1 + k
[ s − (ν 1 − μ2 + t )]!
⎪G
γ νs 1k ( μ1 )
for ν 1 − μ2 + t > 0,
∑
− (ν 1 − μ 2 + t ) (η ) −
⎪⎩
(1 + η ) s −(ν1 − μ2 +t )+1
s =ν 1 − μ 2 + t
(44a)
(44b)
where
Bμt 1ν1 ,ν , μ2 ( z1 , z2 ) =
⎛ z2 ⎞
2 μ1 + μ2 +1 ( μ1 + ν 1 + 1)!
⎜ ⎟
[(2ν 1 + 1)(2ν + 1)(2 μ1 )!(2 μ 2 )!]1/ 2 ⎝ z1 ⎠
μ 2 +1/ 2
z1t − 2
(45)
∞
n −1 s
dx ⎛
x ⎞
1 − e− x ∑ ⎟ e−η x
n ⎜
x ⎝
s =0 s ! ⎠
0
G− n (η ) = ∫
for n ≥ 1 .
(46)
q
(νλ ,ν 2λ ) . The functions G− n (η ) can
See Ref. [21] for the exact definition of coefficients gαβ
be determined by the use of the following relations:
G−1 (η ) = ln
G− n (η ) =
1 +η
η
for n = 1
n −1
(−η )n −1 ⎡
(−1) s ⎤
⎢G−1 (η ) + ∑
s ⎥
(n − 1)! ⎣
s =1 sη
⎦
(47)
for n ≥ 2 .
(48)
Taking into account the formulae obtained for two-electron one- and two-center basic
functions all of the two-electron nonrelativistic and quasirelativistic multicenter integrals over
ISTO and NISTO, Eqs. (13)-(16), can be calculated with the help of auxiliary functions Qnsq
and G−qns . Thus, by the use of relationships for basic functions f μtkν ,κ m and Dtkμ1ν1 ,νσ ;μ2ν 2σ 2
obtained in Ref. [26] and in this study, respectively, all the one- and two-electron
nonrelativistic and quasirelativistic multicenter integrals over integer and noninteger n STO
can be evaluated using Cartesian coordinates of the nuclei and the arbitrary values of
parameters of Slater orbitals. It should be noted that the relationships obtained for oneelectron NQR multicenter integrals can also be used in evaluation of multicenter electronic
attraction, electric field and electric field gradient integrals of integer and noninteger n STO
arising in investigation of the electrostatic potential created by the electrons of a molecule and
its derivatives with respect to the coordinates of nuclei.
- 393 References
1. C. C. J. Roothaan, J. Chem. Phys. 24 (1956) 947.
2. K. Ruedenberg, C. C. J. Roothaan, W. Jaunzemis, J. Chem. Phys. 24 (1956) 201.
3. F. J. Corbato, J. Chem. Phys. 24 (1956) 452.
4. T. C. Chen, J. Chem. Phys. 24 (1956) 1268.
5. M. P. Barnett, Int. J. Quantum Chem. 76 (2000) 464.
6. F. E. Harris, Int. J. Quantum Chem. 100 (2004) 142.
7. J. Fernandez, R. Lopez, I. Ema, G. Ramirez, J. F. Rico, Int. J. Quantum Chem. 106
(2006) 1986.
8. I. Ema, R. Lopez, J. Fernandez, G. Ramirez, J. F. Rico, Int. J. Quantum Chem. 108
(2008) 25.
9. I. I. Guseinov, J. Chem. Phys. 78 (1983) 2801.
10. I. I. Guseinov, R. F. Yassen, J. Mol. Struc. (Theochem) 339 (1995) 243.
11. I. I. Guseinov, A. Özmen, U. Atav, H. Yüksel, Int. J. Quantum Chem. 67 (1998) 199.
12. I. I. Guseinov, B. A. Mamedov, J. Mol. Struc. (Thoechem) 528 (2000) 245.
13. I. I. Guseinov, B. A. Mamedov, M. Kara, M. Orbay, Pramana J. Phys. 56 (2001) 691.
14. I. I. Guseinov, B. A. Mamedov, Int. J. Quantum Chem. 81 (2001) 117.
15. I. I. Guseinov, B.A. Mamedov, J. Theor. Comput. Chem., 1 (2002) 17.
16. I. I. Guseinov, B. A. Mamedov, Int. J. Quantum Chem. 86 (2002) 440.
17. I. I. Guseinov, B. A. Mamedov, J. Mol. Model. 8 (2002) 272.
18. I. I. Guseinov, J. Math. Chem. 36 (2004) 83.
19. I. I. Guseinov, B. A. Mamedov, J. Math. Chem. 38 (2005) 21.
20. I. I. Guseinov, B. A. Mamedov, Chem. Phys. 312 (2005) 223.
21. I. I. Guseinov, J. Phys. B 3 (1970) 1399.
22. I. I. Guseinov, J. Chem. Phys. 67 (1977) 3837.
23. I. I. Guseinov, J. Chem. Phys. 69 (1978) 4990.
24. H. A. Bethe, E. E. Salpeter, Quantum Mechanics of One- and Two- Electron Atoms,
Springer-Verlag, Berlin, 1957.
- 394 25. I. I. Guseinov, J. Math. Chem. 42 (2007) 415.
26. I. I. Guseinov, B. A. Mamedov, MATCH Commun. Math. Comput. Chem. 52 (2004)
47.
27. I. I. Guseinov, B. A. Mamedov, Z. Naturforsch. 62a (2007) 467.
28. I. I. Guseinov, J. Phys. A: Math. Gen. 37 (2004) 957.
29. I. I. Guseinov, Phys. Rev. A 32 (1985) 1864.