Simple derivations of Talmi coefficients for solid spherical Gaussian-type functions1

Journal of Molecular Structure (Theochem) 451 (1998) 35±39

Simple derivations of Talmi coef®cients for solid spherical Gaussian-type functions 1 Osamu Matsuoka Department of Chemistry, Kyushu University, Ropponmatsu, Fukuoka 810, Japan Received 22 December 1997; accepted 21 January 1998

Abstract The Talmi coef®cient for complex solid spherical Gaussian-type functions (SSGTF) is derived using the addition theorem of homogeneous solid spherical harmonics. The corresponding coef®cient for real SSGTFs is derived by unitary transformations of those for complex SSGTFs. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Talmi transformation; Talmi coef®cient; Moshinsky±Smirnov coef®cient; Spherical Gaussian-type function

1. Introduction The well-known Boys' relation [1] states that a product of two s-type Gaussian-type functions (GTFs) of coordinates r1 and r2 is again a product of two GTFs of coordinates r3 ˆ r1 2 r2 (relative coordinate) and r4 ˆ (a 1r1 1 a 2r2)/(a 1 1 a 2) (center of mass coordinate): exp…2a1 r12 †exp…2a2 r22 † ˆ exp…2a3 r32 †exp…2a4 r42 † (1) where a 1 and a 2 are exponent parameters, a 3 ˆ a 1a 2/ (a 1 1 a 2), and a 4 ˆ a 1 1 a 2. The generalization of Eq. (1) also holds for general GTFs, x nlm(r):

xa1 N1 …r1 †xa2 N2 …r2 † X xa3 N3 …r3 †xa4 N4 …r4 †T…a3 N3 ; a4 N4 ua1 N1 ; a2 N2 † ˆ N3 N 4

(2) where N ˆ (n,l,m) is a set of quantum numbers. It 1 Dedicated to Professor Sigeru Huzinaga on the occasion of his 70th birthday.

0166-1280/98/$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S0 166-1280(98)001 57-2

should be noted that the summations over N3 and N4 in Eq. (2) are done over ®nite number of products of the two GTFs on the right-hand side. Eq. (2) is known as the Talmi transformation [2] and the coef®cient T is called the Talmi coef®cient. In nuclear physics the Talmi transformation for harmonic oscillators has been used extensively to elucidate nuclear structures. In quantum chemistry it has been adopted by several workers [3±9] and has played a central role in deriving formulas of molecular integrals over spherical GTFs. Various methods [10±17] were adopted for the derivation of explicit formulas of Talmi coef®cients. However, most of them used terminology of nuclear physics, which is rather unfamiliar in the ®eld of quantum chemistry. Furthermore, they were applied exclusively to the coef®cients for harmonic oscillators, although the coef®cients can be the same as those of other types of GTFs. In Section 2 of the present paper, a simple derivation is given for the Talmi coef®cient of complex solid spherical GTFs (SSGTF) using the addition theorem of homogeneous solid spherical harmonics (HSSH) [18]. Then unitary transformations are applied to the

36

O. Matsuoka / Journal of Molecular Structure (Theochem) 451 (1998) 35±39

obtained coef®cients to derive formulas of those for real SSGTFs. In Section 3 some remarks are given to conclude the paper. 2. Talmi coef®cients for solid spherical Gaussian-type functions 2.1. Complex solid spherical Gaussian-type functions A complex SSGTF is de®ned by

xaN …r† ˆ YN …r†exp…2ar2 †

(3)

where YN is a complex HSSH: Ynlm …r† ˆ r 2n1l Ylm …u; f†

(4)

and N ˆ (n,l,m) is a set of non-negative integers n and l and m ˆ 0, ^ 1,¼, ^ l. We adopt the de®nition of Condon±Shortley [19] of the complex spherical harmonic, Ylm. For an HSSH we have an addition theorem [18, 6]: YN …r1 0 1 r2 0 † X YN1 …r1 0 †YN2 …r2 0 † ˆ

We also have an identity [6] X YN1 …r†YN2 …r† ˆ YN …r†J…lmul1 m1 ; l2 m2 †

where the summation over N is done under the same restrictions as Eq. (6). It should be noted that in Eqs. (5), (11) there exist implicit restrictions among l1, l2, and l due to the Gaunt coef®cients J [20, 21]. In order to obtain the expression of the Talmi coef®cient for complex SSGTFs we note that r1 ˆ …a2 =a4 †r3 1 r4

(12)

and r2 ˆ …2a1 =a4 †r3 1 r4

(13)

where a 4 ˆ a 1 1 a 2, and use the addition theorem, Eq. (5), to obtain X YN3 0 …r3 †YN4 0 …r4 †X…n1 l1 un3 0 l3 0 ; n4 0 l4 0 † YN1 …r1 † ˆ N3 0 N 4 0

£ J…l1 m1 ul3 0 m3 0 ; l4 0 m4 0 †…a2 =a4 †2n3 0 1l3 0 (14)

N1 N2

 X…nlun1 l1 ; n2 l2 †J…lmul1 m1 ; l2 m2 †

(5)

and similarly X YN3 00 …r3 †YN4 00 …r4 †X…n2 l2 un3 00 l3 00 ; n4 00 l4 00 † YN2 …r2 † ˆ N3 00 N4 00

where the summations over N1 and N2 are done under restrictions: 2n1 1 l1 1 2n2 1 l2 ˆ 2n 1 l and m1 1 m2 ˆ m (6) In Eq. (5) X ˆ 4pDnl =Dn1 l1 Dn2 l2

(7)

with Dnl ˆ …2n†!!…2n 1 2l 1 1†!!

(8)

and J is the Gaunt coef®cient de®ned through 3j symbols [20] as ! l1 l2 l m ~ (9) J ˆ …21† U…l1 ; l2 ; l† m1 m2 2m where U~ ˆ ‰…2l1 1 1†…2l2 1 1†…2l 1 1†=4pŠ1=2

(11)

N

!

l1

l2

l

0

0

0 (10)

£ J…l2 m2 ul3 00 m3 00 ; l4 00 m4 00 †…2a1 =a4 †2n3 00 1l3 00 (15) where for constant c we have used Ynlm …cr† ˆ c2n1l Ynlm …r†

(16)

In Eqs. (14), (15) the summations over N3 0 , N4 0 , N3 00 , and N4 00 are done so that 2n3 0 1 l3 0 1 2n4 0 1 l4 0 ˆ 2n1 1 l1 2n3 00 1 l3 00 1 2n4 00 1 l4 00 ˆ 2n2 1 l2

(17)

Forming a product of Eqs. (14), (15) and using Eq. (11) for YN3 0 …r3 †YN3 00 …r3 † and YN4 0 …r4 †YN4 00 …r4 †, we have YN1 …r1 †YN2 …r2 † X ˆ YN3 …r3 †YN4 …r4 †T…a3 N3 ; a4 N4 ua1 N1 ; a2 N2 † N3 N4

(18)

O. Matsuoka / Journal of Molecular Structure (Theochem) 451 (1998) 35±39

for 3j symbols, Eq. (C.15b).) Thus we have

where Tˆ

X

X

N3 0 N4 0 N3 00 N4 00

X T ˆ Ml …n3 l3 ; n4 l4 ua1 n1 l1 ; a2 n2 l2 †

X…n1 l1 un3 0 l3 0 ; n4 0 l4 0 †

l

 X…n2 l2 un3 00 l3 00 ; n4 00 l4 00 †…a2 =a4 †2n3  …2a1 =a4 †

2n3 00 1l3 00

0

1l3 0



4

J

(19)

with J4 ˆ J…l1 m1 ul3 0 m3 0 ; l4 0 m4 0 †J…l2 m2 ul3 00 m3 00 ; l4 00 m4 00 † £ J…l3 m3 ul3 0 m3 0 ; l3 00 m3 00 †J…l4 m4 ul4 0 m4 0 ; l4 00 m4 00 † (20) The summations in Eqs. (18), (19) are done under Eq. (17) and the restrictions: 2n3 0 1 l3 0 1 2n3 00 1 l3 00 ˆ 2n3 1 l3

(21)

2n4 0 1 l4 0 1 2n4 00 1 l4 00 ˆ 2n4 1 l4 Thus, from Eqs. (17), (21), we have a relation: 2n1 1 l1 1 2n2 1 l2 ˆ 2n3 1 l3 1 2n4 1 l4

(22)

If we multiply Eq. (18) by Eq. (1) we get Eq. (2). Hence T in Eq. (19) is an expression for the Talmi coef®cient. We can rewrite Eq. (19), if we use a formula for 3j and 9j symbols: ! ! X j3 j4 J34 j1 j2 J12 m1

m1 m2 m3 m4



ˆ

37

m2

j1

j3

J13

m1

m3

M13

X

…2J 1 1†

JM

8 j1 > > <  j3 > > : J13

j2 j4 J24

M12 !

J12

m3 j2

j4

m2

m4

J34

M34 9 J12 > > = J34 > > ; J

M12

m4 J24

M34 !

M24 ! J13 J

J24

J

M

M24

M

M13

!

…23†

(This formula can be proved, for example, by using Eq. (C.40c) of Ref. [21] and the orthogonality relation

l1

l2

l

m1

m2

m

!

!

l3

l4

l

m3

m4

m (24)

where Ml is the Moshinsky±Smirnov (M±S) coef®cient [10, 11] written as X 0 0 00 00 Ml ˆ …2l 1 1† …a2 =a4 †2n3 1l3 …2a1 =a4 †2n3 1l3  X…n1 l1 un3 0 l3 0 ; n4 0 l4 0 †X…n2 l2 un3 00 l3 00 ; n4 00 l4 00 † ~ 3 l4 00 l2 †U…l ~ 3 0 l3 00 l3 †U…l ~ 4 0 l4 00 l4 † ~ 3 0 l4 0 l1 †U…l  U…l 8 0 9 l3 l3 00 l3 > > > > < = 0 00  l4 l4 l4 > > > > : ; l1 l2 l

…25†

In Eq. (25) the summations are done over all singly and doubly primed n and l under the restrictions of Eqs. (17), (21). This expression is the same as Eq. (2.16) in ref. [6] with corrections [8], since, using Eqs. (17), (21), (22), the factor involving the exponent parameters can be rewritten as …a2 =a4 †2n3

0

1l3 0

…2a1 =a4 †2n3

00

1l3 00

ˆ …a1 =a4 †2n3 1l3 22n1 2l1 …a2 =a4 †2n3 1l3 22n2 2l2 …21†l2  …a1 =a4 †2n4

0

1l4 0

…2a2 =a4 †2n4

00

1l4 00

(26)

Using Niukkanen coef®cients N [22, 6], the M±S coef®cient can be further reduced to X Nk …l; n3 l3 ; n4 l4 ; n1 l1 ; n2 l2 †dk12 (27) Ml ˆ where

d12 ˆ …a1 2 a2 †=…a1 1 a2 †

(28)

We can use Eq. (27) bene®cially to compute integrals over contracted GTFs [6]. 2.2. Real solid spherical Gaussian-type functions A set of real spherical harmonics {Slm} is formed by unitary transformation of complex spherical

38

O. Matsuoka / Journal of Molecular Structure (Theochem) 451 (1998) 35±39

Talmi coef®cient t for real SSGTFs [7]: X tˆ Ump 3 m3 Ump 4 m4

Table 1 2 v and w coef®cients in Eq. (35); 1 ˆ …21†l1 1l 1l m1 1 0 2

m2

v(m1,m2) m1 1m2

‰…21† p 1 1Š=2 …21†m1 = 2 ‰…21†m1 p1 1…21†m2 11 Ši=2 …21†m2 = 2 1 p i= 2 p m2 ‰…21† 1 1…21†m1 11 Ši= 2 p i= 2 ‰21 2 1…21†m1 1m2 Š=2

1 0 2 1 0 2 1 0 2

w(m1,m2) m2

m 1 m2 m 3 m 4

m1

‰…21† p 1 1…21† Š=2 1= 2 ‰1p1 …21†m1 1m2 11 Ši=2 1= 2 0 p 1…21†m2 11 i= 2 p m1 1m2 11 ‰…21† p 1 1Ši= 2 m1 11 …21† i= 2 ‰…21†m1 1 1…21†m2 Š=2

 T…a3 N3 ; a4 N4 ua1 N1 ; a2 N2 †Um1 m1 Um2 m2

Using Eq. (24) of the Talmi coef®cient for complex SSGTFs T, t in Eq. (35) can be rewritten as X tˆ Ml …n2 l3 ; n4 l4 ua1 n1 l1 ; a2 n2 l2 † l

£ harmonics {Ylm} as Slm ˆ

l X

mˆ2 l

Umm Ylm

(29)

Hence its inverse transformation is Yl m ˆ

l X mˆ2 1

Ump m Slm

(30)

Only the following elements of the unitary matrix U in Eqs. (29), (30) are nonvanishing [23]:

Slm …u; f†

m3

m4

m

!!p

l1

l2

l

m1

m2

m

!! (36)

where 3j symbol for real spherical harmonics is de®ned by ! !! X l1 l2 l l1 l2 l Um1 m1 Um2 m2 ; m1 m2 m m1 m2 m m 1 m2 ! ! l1 l2 l l1 l2 l ˆ v…m1 ; m2 ; l† 1 m1 m2 m 2m1 m2 m 0  w…m1 ; m2 †

(37)

3. Concluding remarks

where a real HSSH is written as Snlm …r† ˆ r

l

(32)

A real SSGTF is de®ned by

2n11

l4

(31)

p p U2umu;2umu ˆ i= 2; U2umu;umu ˆ i…21†m11 = 2

haN …r† ˆ SN …r†exp…2ar2 †

l3

Here v and w coef®cients are both real or pure imaginary for the given m1 and m2 values as shown in Table 1. The M±S coef®cient Ml in Eq. (36) is exactly the same as Eq. (25) for the complex SSGTFs. The Talmi coef®cients for the complex and the real SSGTFs differ only by the 3j symbols. It should be noted that the Talmi coef®cient t in Eq. (36) is indeed real.

U0;0 ˆ 1 p p Uumu;2umu ˆ 1= 2; Uumu;umu ˆ …21†m = 2

(35)

(33)

If we de®ne the Talmi transformation for real SSGTFs as

ha1 N1 …r1 †ha2 N2 …r2 † X ha3 N3 …r3 †ha4 N4 …r4 †t…a3 N3 ; a4 N4 ua1 N1 ; a2 N2 † ˆ N3 N4

(34) then using Eqs. (29), (30) in Eq. (34), we obtain the

The Talmi transformation has its origin in the ®eld of nuclear physics and the terminology adopted for the derivations [10±17] of the Talmi coef®cients is rather unfamiliar in the ®eld of quantum chemistry. In the present paper the language of quantum chemistry was used to derive the Talmi coef®cients for SSGTFs, which might help one to understand the signi®cance of the Talmi transformation and further to use the formulas [3±9] of molecular integrals over spherical GTFs. We note that the derived expressions of Talmi coef®cients T, Eq. (19), and t , Eq. (35), for the

O. Matsuoka / Journal of Molecular Structure (Theochem) 451 (1998) 35±39

SSGTFs are the same as those for the Laguerre GTFs [4, 6, 7] which can be de®ned by

xaN L …r† ˆ Y^ N …7†exp…2ar2 †

(38)

or

haN L …r† ˆ S^N …7†exp…2ar2 †

(39)

where Y^ N …7† and S^ N …7† are operators obtained by replacing x, y, and z in Y^ N (r) and S^N (r) by 2 /2 x, 2 /2 y and 2 /2 z. For these operators the same operator equations as Eq. (18) hold. Finally, it can be mentioned that the formulas of Talmi coef®cients for Cartesian and Hermite GTFs have been derived recently [8]. Acknowledgements The present study was initiated when the author was at the University of Alberta, Edmonton, Canada, 1983, and derived the formula of the ``Alberta coef®cient'' which turned out afterwards to be the same as the Moshinsky±Smirnov coef®cient. It is the author's great pleasure to dedicate this paper to Professor Sigeru Huzinaga who has always given warm encouragement.

39

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

S.F. Boys, Proc. R. Soc. London A 200 (1950) 542. I. Talmi, Helv. Phys. Acta 25 (1952) 185. G. Fieck, J. Phys. B 12 (1979) 1063. G. Fieck, Theor. Chim. Acta 54 (1980) 323. D.M. Maretis, J. Chem. Phys. 71 (1979) 917. O. Matsuoka, J. Chem. Phys. 92 (1990) 4364. O. Matsuoka, Can. J. Chem. 70 (1992) 388. O. Matsuoka, J. Chem. Phys. 108 (1998) 1063. N. Fujimura, O. Matsuoka, Int. J. Quantum Chem. 42 (1992) 751. M. Moshinsky, Nucl. Phys. 13 (1959) 104. Yu.F. Smirnov, Nucl. Phys. 27 (1961) 177. K. Kumar, J. Math. Phys. 7 (1966) 671. M. Baranger, K.T.R. Davies, Nucl. Phys. 79 (1966) 403. M.M. Bakri, Nucl. Phys. A 96 (1967) 115. A. Gal, Ann. Phys. 49 (1968) 341. J.D. Talman, Nucl. Phys. A 141 (1970) 273. L. Trlifaj, Phys. Rev. C 5 (1972) 1534. K.G. Kay, H.D. Todd, H.J. Silverstone, J. Chem. Phys. 51 (1969) 2359. E.U. Condon, G.H. Shortley, Theory of Atomic Spectra, Cambridge University Press, London, 1951. A.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, Princeton, 1957. A. Messiah, Quantum Mechanics, North-Holland, Amsterdam, 1962. A.W. Niukkanen, Chem. Phys. Lett. 69 (1980) 174. K. O-ohata, K. Ruedenberg, J. Math. Phys. 7 (1966) 547.