AO integral evaluation using Cartesian exponential ...

A 0 integral evaluation using Cartesian exponential type orbitals (CETOS)' RAMONCARBO? and EMILIBESALU Laborarori de Quitnicu Cot~1puraciotzu1,Estirdi de CiPncies, Esrudi Getleral de Girona, 17071 Girot~a,Spain Received July 15, 1991

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This paper is dedicccred ro Profe.rsor Sigeru Huzit~ccgccotl the occasiotl of his 65th birrhdccy RAMON CARBO and EMILIBESALU. Can J. Chem. 70, 353 (1992). CETO functions and their properties are defined and described, to provide a nieans of obtaining general expressions for many-center many-electron integral formulae. Compact integral expressions are written by means of nested summation symbols, a new concept developed in this paper. Integrals over CETO functions are computed by means of a set of several auxiliary integral forms. No transformations other than frame rotations are needed to compute the usual integral terms. The formulae obtained are imrncdiately programmable in any high lcvcl language and the parallclizable terms are obtained with a simple rule. Results can be considered an encouraging alternative way to solve the S T 0 integral problem. Key words: many-center A 0 integrals, molecular basis sets, ETO, STO, CETO. RAMON CARBOet EMILIBESALU. Can. J . Chem. 70, 353 (1992) On dtfinit et dtcrit des fonctions d'orbitales de type carttsien exponentiel (CETO) et leurs proprittks; ces fonctions pourraient ouvrir la voie pour obtenir des expressions gentrales pour des formules d'inttgrales plusieurs electrons et B plusieurs centres. On dtrive des expressions inttgrales compactes en utilisant des symboles desommation emboitts les uns dans les autres, un nouveau concept dtvelopfi dans ce travail. Pour calculer les inttgrales sur les CETO, on utilise un ensemble de quelques formes d'integrales auxiliaires. I1 n'est pas ntcessaire de faire appel i d'autres transformations que des rotations du repkre de rtftrence pour calculer les termes des inttgrales usuelles. Les formules qui sont obtenues peuvent immtdiatement &tre programmtes dans n'importe lequel language de haut niveau et on obtient des termes parallelisables avec une rkgle simple. On peut considtrer que les rtsultats sont une alternative encourageante pour la solution du problkme des integrales du type de Slater. Mors clPs : inttgrales d'orbitales atomiques a plusieurs centres, bases moltculaires, orbitales de type exponentiel, orbitales de type Slater, orbitales de type carttsien exponentiel. [Traduit par la redaction] I

1. CETO

Introduction

I

I

Although defined in 1960 by Boys and Cook ( I ) , Cartesian Exponential Type Orbital (CETO) functions did not attract notice until a GTO integral review made by Saunders (2), where they were cautiously manipulated as a subproduct. As integrals over Exponential Type Orbitals (ETO) are raising interest among various research groups (3-8), we tried to open anew the CETO functions study as a way to approach the ETO integral problem from another point of view. This paper approaches the computation of CETO integrals by using a notation where all the necessary elements, integrals plus coordinate transformations, are compactly expressed. We prove in this way how CETO integrals are expressible in terms of accessible concepts and auxiliary functions well known in the literature. Care has been taken to provide the simplest formulation possible, suited to easy translation to any high level programming language and well adapted to parallelization. To fulfill this primary goal, the present paper will be roughly divided into four parts. The first one will introduce the concepts of CETO functions and their transformation properties. The second discusses one- and two-center oneelectron integrals. The third is devoted to obtaining twocenter Coulomb repulsion integrals and in the fourth part many-center integrals are analyzed. 'A contribution of the Grup de Qimica Quantica del Institut d'Estudis Catalans. 2 ~ u t h oto r whom correspondence may be addressed. Printed in Canada

Definition I : CETofunctions

Cartesian Let US define a real Type Orbital (CETO) with the screening parameter a and centered at the point A as

where the following conditions hold: 1 a = ( a ,,a,,a3), a i integer 2 0 V i 2 tz integer 2 0 3 a real >O 4 rA = Ir - Al, r = (x,,x2,x3),A = (Al,A2,A3) and where one can distinguish the following terms: ( i ) T h e tzormalization factor

with a

=

Z , a , a n d p = IIi(2a; - I)!!.

( i i ) T h e angular part

where a is defined in eq. [1.2]. (iii) T h e radial part

In some cases condition 2 above can be relaxed, allowing negative values, and condition 3 may be extended with a =

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CAN. J . CHEM. VOL. 70. I992

1.2. Relationship of CETO with ST0 functions Slater Type Orbital (STO) (2, 9a) functions have been well known since their description by Slater ca. 60 years ago (10). S T 0 have been massively studied and used as atomic basis sets (1 1). In the convenient definition by Wahl et a / . (12), normalized S T 0 are constructed over spherical coordinates. After some elementary manipulations one can see that it is possible to write any S T 0 in terms of CETO functions (9, 13a). The interesting pattern here is the appearance of the structure of Cartesian S T 0 functions, expressible as finite linear combinations of CETO functions. This property can be also used in similar contexts where spherical harmonic functions are present. Take also as an example the atomic spinor functions, see ref. 13. It will be easy to design relativistic one-electron orbitals as formed by four-dimensional vectors whose components are CETO linear combinations. CETO functions can be considered the building blocks of Cartesian STO. In this sense, solving the integral evaluation problem over CETO functions will be the same as solving the integral S T 0 problem. 1.3. Practical details on CETO products As is well known, basis functions when used in a LCAO framework appear in product pairs at the moment of computing molecular integrals. Analysis of the structure of these CETO products will permit easy definition of canonical integral forms. 1.3.1 Orientation of CETOs over the line joining a pair of centers (i) To be useful for integral computation a product of two CETO functions must be well-oriented, in such a way that the x3-axis (z-axis) coincides with the line joining the two centers involved. (ii) Here we analyze how to perform an effective rotax ~ ) the Eulerian tion over the coordinates x0 = ( x ~ , x ~ ,using angles w = (cp,8,6) and the rotation matrix R(w) (14). The rotation R(w) is chosen in order to express the coordinates x0 in terms of the coordinate system x = (x,,x2,x3)having the same origin as xO,say A,,and which is well oriented with respect to another center, B . (iii) Thus, the rotations in the present paper will be used to orient well two CETO functions, centered at different sites. The rotation angles involved for this kind of manipulation will be constructed by the vector: w, = (cp,O,O). This is because of the cylindrical symmetry that the prolate spherical coordinates system (see Definition A1 in Appendix A) possesses with respect to a rotation around the axis line joining both centers. (iv) Cartesian coordinate power products such as those forming the CETO angular part, H:(A,~), are of special interest when dealing with this kind of A 0 functions. Using the result of the rotation matrix product over coordinate vectors, the following relationship can be found: [1.5]

H;(~,a)=C(i=o,a) Z ( j = 0,i) M(i,j,aY(i,j,a,cp,e> H,(A,g)

where a = C,g, = Ciai. Certain definitions should be taken into account in order to interpet eq. [ I S ] . They are written as follows: 1. We will call C(i = O,a), C(j = 0,i) nested summation symbols. A general description of these is given in Appendix A, Definition A2.

2. The M coefficients correspond to the binomial coefficient product:

3. The f function is defined as [ 1.71

f (i,j,a,cp,O)

=

(sin 8)"l (cos 8)" (sin cp)"' (cos cplb4

where the four-dimensional vector b is built up in terms of vectors a , i, and j elements using the rules: [1.8]

b=b(a,i,j)=(i,+i,+i,-(jl+j,), a,

+ j, + j,

-

i3, a ,

+ i2 - i , , a 2 + il - i,)

4. The three-dimensional vector g components are constructed by the rules: [1.9]

g

=

g(a,i,j) = (j, a,

+ j2 + i3,

+ a,

-

(i,

+ i,), a, + i, + i2 - g l )

5. In the present framework the index j3must be considered constant and equal to zero. 1.3.2 General CETO products arid their properties (i) Using CETO Definition 1 one can easily find that if two CETO functions share the same center, that is, if A = B and R = 0 , their product is but a new CETO within the same center, or:

=

x(A,a

+ a', n + n ' , a + a')

(ii) The product of two CETO functions placed at centers A and B respectively, separated by a distance R, giving when rotated, can be written as a charge distribution

a),,

where the f and M coefficients and the g and g' vectors arise from the rotations that put each CETO in a well-oriented form with respect to the other. The W functions are a particular form of CETO functions, which are defined below in the next section. 1.3.3. Well-oriented CETO form When observing eq. [1.5] it is easy to consider how the general CETO structure changes. The CETO functions, once transformed, become linear combinations of unnormalized Well-Oriented CETO (WO-CETO) functions, which can be defined as follows: Definition 2 : WO-CETO functions Using the conventions: 1. n = (n,,n,,n,> 2. m~{0,1) 3. a r 0 and real 4. R r 0 and real 5. s€{-l,O,+l)

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6 . Z ( S ) = sR/2 7 . r(s) = (sy + x; + (s,- ~(s))')"' an unnormalized WO-CETO function is written as

2.1 . Overlap integrals (i) One center The one-center overlap integrals can be easily obtained considering the structure of eq. [ l .lo]:

One can represent unnormalized WO-CETO functions as a product of the following terms, when the convention z(s) = (O,O,z(s))and the definitions [ 1.31 and [ 1.41 apply: (i) An atzgular part:

where a = a'

where tz = &ni. (ii) A rndinl part: [ 1.141

P,,,(z(s),a)= r-(s)"'exp{-ar(s)}

In this way r, = r(- l ) , so one can also use the symbol W ( A , n , m , a ) for W ( z ( - l ) , n , m , a ) ; moreover, one can write r, = r ( + l ) and, consequently, W(B,n,nz,a) = W ( z ( +l),n,rn,a).m A still simpler expression, having the form of some sort of elementary CETO building block, can be also described: Definition 3: E-CETO functions Taking into account the same conventions as in Definition 2 , an Elementary CETO (E-CETO) function is:

1.3.4. WO-CETO products. Cartesian form in terms of E-CETOS The product of two WO-CETO functions can be written in terms of the third axis components, so:

with

The two-dimensional vectorp is defined as (a,,b,) and

2. Integrals over one-electron operators Although already described by Saunders (2), overlap and kinetic energy integrals over a CETO framework will be briefly discussed for the sake of completeness. One- and twocenter nuclear attraction integrals will also be studied here. The notation presented previously in the above discussion is used in all integral cases, leading in this way to a compact description of every integral form, usually needed in quantum chemical problems. All one-electron integrals studied here can be written in terms of overlap integrals.

+ a", tz = n' + tz", a =

=

+ a", and

a'

6(A,(a,= 2 ) ) F , ( a , + a, + l,a,) F 2 ( n Z , ~ tz!I a-(l+l' )

-

with t = 2 + n + Cial. The integral [2.2]vanishes unless ai = 2 , V i , as signaled by the logical Kronecker delta function, described in Appendix A , Definition A3, appearing as a first factor. The functions F,, coming from the integration of the angular part of the CETO function, are defined in Appendix B , Definition B5. (ii) Two center Taking into account the rotations outlined in section 1.3.1, the two-center overlap integral can be expressed as

where it is necessary to compute overlap integrals over WO-CETO functions, as

F,(n2 + b l , a , + 6 , ) C ( r = 0 , t ) T(r,c)A,, (a)Br2( 7 ) with t = ( t , t ) ,t = 2 + m + tz + Ci(ai + b,), and having a = R(a + P)/2 and T = R(a - P)/2. The coefficients T(r,c)arise from the development of the prolate coordinates function P(c,u,v), with the definition of the six-dimensional vector c = (tz + 1, rn + 1, a,, b,, k , k ) , as explained in Appendix A, Definition A4. The Ak(x) and B,(x) integrals are defined in Appendix B , Definitions B 1 and B2. The sum a , + b , + a2 + 6 , is even for nonvanishing integrals, due to the properties of the F, functions, defined in Appendix B , Definition B5. Thus, the index k = ( a , + 6 , + a, + b 2 ) / 2 ,appearing in the expression of the vector c , has always an integer value. 2.2. Kinetic energy itztegrals Kinetic energy integrals may be computed using overlap integrals and the symmetric form of the associated kinetic energy operator derived from Green's first identity (15): 12.51 TAB = ( & I 1 / 2 V'IXB)= 1/2 ( V % I V X ) Furthermore, one obtains:

+ n W ( A , a + e,, n - 2 , a )

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CAN. J . CHEM. \

where {e,}are the rows of the (3 x 3) unit matrix. The fact must be noted here that if a, or n is zero in the original WO-CETO, the first or the second term on the right side of eq. [2.6] will vanish. Once the rotations described in paragraph 1.3.1 are performed over the CETO functions appearing in eq. [2.5], this integral can be expressed as a sum involving overlap integrals of type [2.4] as

One can see that the nuclear attraction integral is a linear combination of two-center overlap integrals:

+

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employing the auxiliary definition: [2.8]

K,(s,p) = b, (W(A, a

+ se,, n

-

p, a)

+ m(W(A, a + se,, n - p , -

P (W(A, a + se,, tz

+

+

where a = a' a", tz = n' nu, and a = a' a". The M and f coefficients and the vectors g = g(a,i,j ) , constructed as in eq. [I .9], arise from the rotations transforming the CETO centered at A into a well-oriented form with respect to the CETO centered at B . Also the term r j ' = x(B,O,-1,O) = W(B,O,-1,O) can be expressed either as a CETO or a WO-CETO, relaxing conditions 2 and 3 of CETO Definition 1. The overlap-like integrals appearing in eq. [2.13], which involve the operator W(B,0,- 1,0), can be taken as a particular case of the two-center overlap integrals described in eq. [2.4]. (iii) Two-center integrals of the form (AB:A) or (AB:B) In this case, one has

a)

- p, a )

+

, an integer, one obwhere SE{- 1, 1) is a sign and p ~ ( 01,2) tains:

In one-center kinetic energy integrals, the term [2.9] can be written using the following combination:

then:

where the symbol (U,,) stands for an integral like the onecenter overlap including all the CETO functions present in eq. [2.10].

2.3. Nuclear attraction integrals (i) One-center integrals of the form (AA:A) This type of integral can be computed by integrating within a spherical coordinates framework. They can be associated with a one-center overlap integral [2.2] as follows: [2.12]

(AA1:A) = J" x(A,a,n,a) r i ' x(A,a1,n',a') dV = ( x ( A , a + a ' , n n' - 1 , a a ' ) )

+

+

(ii) Two-center integrals of the form (AA:B) This type of integral, and the following case, can be obtained by integrating in a prolate spherical coordinates framework.

Thus, a finite linear combination of WO-CETO overlap integrals suffices to compute the two-center (AB:A) integrals.

3. Coulomb repulsion integrals Coulomb repulsion integrals over CETO functions, conserving the important role already obtained in GTO and S T 0 choices, are discussed here. The one- and two-center Coulomb integral computation is set as was done in earlier classical work, see for example refs. 12 and 16. We use the structure of the nuclear attraction monoelectronic integrals to solve the integration over the coordinates of the first electron for the one-center problem, in the same way as Wahl et a/. (12) proposed when dealing with S T 0 functions in general. However, we reverse this integration scheme for the two-center case, as did 0-ohata and Ruedenberg, also within the S T 0 functions framework, in an interesting paper published in 1966 (16). This twofold integration scheme was adopted after realizing that, when considering the two-center case, the potential first integration drives towards a second integration step with plenty of difficulties, if general analytical formulae are sought. In fact, Wahl et al. (12) proposed to perform the second integration numerically. The integral analytical evaluation over the coordinates of the second electron, in the two-center CETO function case, corresponds to a calculation of integrals, involving in the worst situation an infinite series if the 0-ohata and Ruedenberg (16) scheme is used. It must be said now, taking into account the property outlined in eq. [ l .lo], that all the Coulomb repulsion integrals will transform into integral forms, which we have called reduced Coulomb integrals. For example, a Coulomb integral will ) be rewritten schematically in the form ( A A l r z l ~ ~ ) . this kind of integral is discussed duced to ( ~ l r z l ~Only in the present paper, due to the fact that the coordinate rotations and the use of property [1.10] will transform

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Coulomb integrals into this last form, I A ) and IB) being WO-CETO functions. Moreover, as will be shown in Sect. 4 , when following the approach outlined there, the manycenter integrals require only reduced Coulomb integrals in order to be evaluated. Two kinds of reduced Coulomb integrals over WO-CETO functions will be studied here: oneand two-center. 3.1. Potetztial ,functions The general integration structure adopted in order to obtain analytical formulae in the one-center case is based on the first integration, associated with a nuclear attraction integral like those appearing in eq. [2.141. In fact, writing the following general reduced Coulomb integral:

integration with respect to the electron ( 2 ) is equivalent to computing a nuclear attraction integral such as:

which in turn is a function of electron ( 1 ) coordinates. Thus, in general, a reduced repulsion integral may be written:

P.31 R,,, = S [W(A,a,tz,a){l)l dV, The integration over the coordinates of the electron number ( 2 ) must be done over a prolate spherical coordinate system {rB,,rI2,r,,)defined in Appendix A, Definition A l , and the obtained analytic expression of the nuclear attraction integral shown in eq. [3.2].The intercenter distance, r,,, will act here as a variable and it has to be integrated when considering the electron ( 1 ) coordinates. To have electron (2) in a well-oriented system with respect to electron ( 1 1 , it is necessary to perform a rotation, for each position of electron (11, from the original system to the prolate spherical system. Let the WO-CETO attached to electron (2) coordinates be symbolized by W(B,b,m,P). Electrons ( 1 ) and ( 2 ) are described by means of typical spherical coordinate systems {rA,,O,,cpl)and (rB2,02,cp2), respectively. After performing the needed rotations outlined in Sect. 1.3.1, one can express the 6(1)potential function [3.2]as

where C(i = 0,b) and C ( j = 0,i) are nested summation symbols and the coefficients M(i,j,b) andf(i, j,b,cpl,O,,) are defined in Sect. 1.3.1. Note that the angle O,, is measured from the center B. The W({1),0,- 1,O) notation stands for a WO-CETO centered on the position of the electron (1). The vector g , according to eq. [1.9], is defined here as g(b,i,j ) . Using eq. [2.4],eq. [3.4]can be written:

where: (i) T , = m

+ C,b,

(ii) The function N is defined by

(iii) The coefficients A are constructed by means of where -

r appears as in eq. [2.4]with t = ( p , , p Z )The . A coefficient is defined as A ( p ) = ( p , ! p l ! ) / ( p 3 ! p ,and ! ) s = p, + p2

+3 PA). (iv) The six components of the vector h are (

~

with k = ( g , + g2)/2. ( v ) Finally,

Although the previous scheme seems very promising for both one- and two-center cases, we faced the same difficulties as Wahl et al. (12) have surely met, when integration over the second electron was to be performed. In consequence, we use this approach only for the one-center integrals, discussed next. 3.2. One-center reduced Coulomb integrals Let be W ( l ) = W(A,a,n,a)and W ( 2 )= W(B,b,m,P)two WO-CETOs centered at the origin, that is, A = B = 0 . One can define the following quantities: ( i ) T, = n + Ciai and T, = m + Cib,as the total order of each WO-CETO. (ii) T = T, + T, as the total order of the integral. The integral to be calculated is

which can be written using the potential integral described in eq. [3.2].

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CAN. J . CHEM. VOL. 70. 1992

One can obtain the expression for 6(1)using eq. [3.5] where the prolate spherical system needed is configured by means of the parameters {r2,rI2,r1). Integration over the electron (1) is carried out in a spherical coordinates framework {r,,0, ,cp,) and leads to the final result: [3.11]

RAA= S(A,(a,+ b, = 2)) N(T, - 1,P) C(i = 0,b) X( j

=

O,i)C(i,a,b)C ( p = 0,q) A(p,h,P) D(p,n,a,P)

where the following definitions hold. ( i )The C coefficients are defined as

with the B coefficients being defined in turn by means of

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[3.13]

+ b2 - ( i , + i2),jl+ j2 + i3)F,(a, + b,

B(i,a,b) = M(i,j,b) F,(b,

-

i,

+ & , a , + b~ + il - i,)

where M(i,j,b) are the transformation coefficients constructed as in eq. [ 1.61. (ii) N and A are the same as the coefficients defined in eqs. [3.6] and [3.7] respectively, and the functions F, are described in Appendix B, Definition BS. (iii) The six components of the vector h are the same as these defined in eq. [3.8], with the same definition of index g,. (iv) The vector q has its components constructed as: (T, + 1 , T, + 1, p,, p,). ( v ) The coefficients D are defined as: [3.14]

D(p,n,a,P) = (t - l)! {(- 1)'' a-'

-

(a

+ P)-')

+ +

with t = 3 + n - ( p , p,) + p3 + p4. (vi) From the parity restrictjons associated with the F, functions, it is easy to determine that the following conditions necessarily hold: {A;(a; b; = 2)), in order to have non-vanishing integrals. This generates the logical Kronecker delta appearing in eq. [3.11]. (vii) T o avoid integration divergence when obtaining 6 ( 1 ) , before carrying out any calculation as described in this section, it is necessary to ensure that T, 2 T,. 3.3 Two-center reduced Coulomb integrals Let W,, W, be two WO-CETO functions placed at the centers A and B respectively, with A f B and R being the intercenter distance. Due to the problems discussed at the end of Sect. 3.1, the integration in the two-center case has been done using the methodology outlined by 0-ohata and Ruedenberg (16). The integral to be computed now is a reduced Coulomb integral written as

[3.15]

RAE= J J WA(rI- r,,) Ir,

-

r , l - ' ~ , ( r , - r,) dr, dr,

Making a change of variable:

one must evaluate integral [3.15] in the form:

T o obtain this final result an overlap integral has to be calculated between the CETO functions h,xc, resulting from the translation of W,, W, to the new frame defined in eq. [3.16]: [3.18]

S(rc - r,,)

=

J xA(rI- r,) xc(r, - rC)dr, = C(i = 0,a) C ( j = 0,i) C(il = 0,b) C ( j '

=

0,i')

- M(i,j,a> M(i1,j',b)f(i,j,a,cp,,0,)f(i',f,b,cpA,0~)(W(~,g,n,a) with g = g(a,i,j ) and g' = g(b,il,j ' ) following the convention outlined in eq. [1.9]. Using the expression [3.18] in eq. [3.17] and the Laplace expansion of IrC - r,l-I, one arrives at the final result: [3.19]

R,,

=

+

&(a, b ,

=

2 A a,

+ b, = 2 ) ~ (' ~ / 2 ) ' +C(i ' = 0,a) C ( j = 0 , i ) C(il = 0,b) C ( f

where: ( i )The c and c' vectors are defined by means of the general vector structure: [3.20]

c'

=

c=c(a,i,j)=(a,-i,+iz,a2+il-iz,i,+i2+i3-(jl+j2),a3+jl+j~-i3)

c(b,il,j ' ) being defined in a similar manner.

= O,il)

of both functions RAEwill be expanded, using a more general structure than the one proposed by Ruedenberg ( 1 8 ) , consisting of the sum of two bilinear forms:

( i i ) The function P is

P,(cos 0)(sin 0)"' (cos 0)" d0

[ 3 . 2 1 ] P(k,m,n) = =

2-,

V-1)'

(2(k - p ) ) !

P ! (k

p ) ! (k - 2p)!

,=o

-

F,(m,k

+ n - 2p)

where q = [ k / 2 ] is the largest integer in the quotient k / 2 , the function F , is defined in Appendix B , Definition B5, and P,(x) are Legendre polynomials of order k.

[ 3 . 2 2 ] (iii) J(u,u,T) = K(u,u,T) + L(u,u,T) Can. J. Chem. Downloaded from www.nrcresearchpress.com by 118.99.76.178 on 06/03/13 For personal use only.

where u = ( r , s , t ) ,r = ( r ,,r2), and r"

One can use a product A,(x)B,(y) expansion form avoiding negative powers of x in the expression of L(u,u,T).In this manner, the K and L integrals lead to computing the C , ( x ) and D,(x) auxiliary functions, defined in Appendix B , Definitions B3 and B4. ( i v ) In this formulation t = ( t , t ) , t = 2 m n C(a, b , ) , the vector d = ( n + 1 , m + 1 , g,, g;, h , h) with h = ( g , + g; + g, + g;)/2, as well as the parameters u and T , which are defined as in eq. [ 2 . 4 ] . (vi) The parity restrictions associated with the F, functions have their origin in the logical Kronecker delta appearing in eq. [ 3 . 1 9 ] .

+ + +

+

4. Many-center integrals T o compute many-center integrals, two choices can be proposed in a first instance. The first is based on optimally expressing a two-center CETO product in terms of a linear combination of CETOs, centered at each of the sites belonging to the product functions. This approach will be discussed here and then used to describe in the following sections how to compute three- and four-center integrals. More details will be given elsewhere. The second option can be based on trivially expressing the 1s exponential part of the CETO functions in terms of a linear combination of primitive GTO, as did FernAndez Rico ( 4 e ) when dealing with STO. This approach, which possesses an extreme simplicity and is well adapted to the same form as the Cartesian angular part of the CETO and GTO functions, seems very promising because the exponential part of the CETO functions only has to be expanded. In fact, when using a Gaussian expansion, a unique fixed primitive GTO linear combination is needed, as a consequence of the scaling theorem demonstrated by 0-ohata, Taketa, and Huzinaga ( 1 7 ) when dealing with a similar problem. In this easy numerical environment, integrals over CETO functions become transformations of primitive cartesian GTO integrals. This second approach will be studied in depth elsewhere. 4.1. Two-center expansion of a CETO function product Let there be two CETO functions centered at points { A , B ) at a distance R: {xA,xB).In the present approach the product

But, as has been shown before in eq. [ l . 101, the CETO products i2,,. and Rep.can be substituted by unique CETO functions X , and x,, respectively. Thus the product RABcan also be expressed in terms of WO-CETO functions { Y t , Y i )centered separately on A and B . Connecting the present approach with earlier ideas of Harris and Rein (19a), Newton ( 1 9 b ) , and Okninski ( 1 9 c ) , one can also write eq. [ 4 . 1 ] as

In fact, after obtaining this simplified result, it is crucial to prove the product RA8can be expanded at the centers A and B with sufficient accuracy. A general theoretical discussion on the way these linear combinations can be calculated follows. Optimal linear coefficients can be computed using the compact matrix form of eq. [ 4 . 2 ] ,supposing both sides equal:

[4.3]

RAE= q A c A+ q B c B

then { c A , c B ) the , column vectors of the expansion coefficients, can be obtained by multiplying equation [ 4 . 3 ]on the and left by the WO-CETO basis set column vectors (qA)+ (qB)+ where both basis sets are stored respectively, and integrating. The following linear system can be built up:

[4.4]

sA = Sm c A + SABc B s,

=

S , c A + SBBc B

where s , = (*"IRA,) and S,, = (qAIqA) is the metric matrix of the WO-CETO basis set centered on A . The same definitions apply but with respect to center B for sB and SBB. Collecting {sA,sB)and the other matrices in [ 4 . 4 ] in a hypermatrix system one can easily write:

with obvious definitions for the organization of the matrices in eq. [ 4 . 4 ] as hypermatrices in eq. [ 4 . 5 ] .As S is the metric and consematrix of the union of the basis sets {qA,qB), quently S > 0 , then an inverse will always exist: S - ' , which can be used to solve eq. [ 4 . 5 ] . After some tests, we recommend Cholesky's decomposition ( 2 0 ) in order to compute S - I , due to the numerical stability of the method and to the possibility of performing iterative computations when new functions are added to the linear combination [ 4 . 3 ] .When adding functions to the expansion [4.3] it is mandatory to choose them so as to achieve a good fitting between the expansion [ 4 . 3 ] and RAE.The parameter taken as an indicator of the degree of fitting is the norm:

[4.6]

q("

=

IRA, - ( q A c A+ qBcB)12

Supposing all the involved functions are normalized, then the measure:

[4.7]

5 = S RAE( q A c A+ q B c B )dV

360

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-

is nothing more than the degree of fitting of the linear comas bination [ 4 . 3 ]of the WO-CETO charge distribution E; 1, IR,is better covered by the two-center expansion and q'2' + 0 . Numerical experience shows how the measure E; tends to one with sufficient accuracy ( - l o - ' ) using a reasonable number of terms ( ~ 2 0in) the combination [ 4 . 2 ] .

center and many-center integrals; ( b ) easy buildup of a program structure in any high level language for any integral; ( c ) parallelization of the terms lying inside nested summation symbols.

4 . 2 . Reduction of three- and four-center integrals to tnorzo and bicentric terms

This work has been supported by the Spanish "Ministerio d e Industria y Energia" under the "Programa Nacional d e Investigatibn y Desarrollo FarmacCuticos" through Grant No. FAR 88-0617. O n e of us (E. B.) benefits by a grant of the "Departament d'Ensenyament d e la Generalitat d e Catalunya." Comments made by the referees are gratefully acknowledged: they have generated considerable improvement in the paper.

a,,:

Using the C E T O expansion method outlined in the preceding section, it can be seen that molecular integrals may be constructed as linear combinations of other integrals within a reduced number of centers. Three important cases can be considered: ( i ) Three-center nuclear attraction integrals (AB:C) will reduce to an expression involving two-center integrals of the (AA:C) and (BB:C) type. Here, the expansion expressed in eq. [ 4 . 2 ] is used in order to compute the (AB:C) integral, and one can write:

( i i ) In general, it can be considered as the evaluation of a two-electron four-center integral of any kind: (ABJTJCD). Expanding both charge distributions involved as described in expression [ 4 . 2 ] ,the integral will be a sum of four bilinear terms that will depend on the four kinds of integral {(AITIc), (AITID), (BITIC), (BITID)}, reduced to bicentric forms:

Thus, three- and four-center Coulomb integrals will be expressed in terms of the reduced Coulomb integrals as defined in eq. [ 3 .l l . (iii) An interesting situation, which can easily be solved within the present approach, corresponds to two-center integrals of the "exchange" type: (ABITIAB), which have not been previously discussed. It is well known that in the S T 0 framework and with the Coulomb operator substituting the operator T , they are not so trivially computed as the twocenter "Coulomb" type integrals: (AAJTIBB),as the papers by Ruedenberg (21) or Wahl et al. (12) prove. But, using eq. [ 4 . 9 ] , they are simply expressed in terms of one-center (AITb'), (BITIB'), and two-center (AITIB) integrals. The same can be said for the "hybrid" type integrals (AAITIAB).

Conclusions A general but simple framework has been described in order to compute integrals over C E T O functions. T h e proposed formulation has been written in such a way as to permit: ( a ) construction of a procedure to compute basic two-

Acknowledgements

1. S. F. Boys and G. B. Cook. Rev. Mod. Phys. 32, 285 (1960). 2. V. R. Saunders. In Computational techniques in quantum chemistry and molecular physics. Edited by G. H. F. Diercksen et al. D. Reidel, Dordrecht. 1975. p. 347. 3. C. A. Weatherford and H. W. Jones (editors). ETO multicenter molecular integrals. D. Reidel, Dordrecht. 1982. 4. (a) R. Lopez, G. Ramirez, J. M. Garcia de la Vega, and J. Fernandez Rico. J. Chim. Phys. 84, 695 (1987); (b) J. Fernindez Rico, R. Lopez, and G. Ramirez. J. Comput. Chem. 9 , 790 (1988); (c) J. Comput. Chem. 10, 869 (1989); (d) Int. J. Quantum Chem. 37,69 (1989); (e) J. Chem. Phys. 91,4204 (1989); 91,4213 (1989); ( f ) J. Chem. Phys. 94, 5032 (1991). 5. (a) I. I. Guseinov. J. Chem. Phys. 72, 2198 (1980); (b) Phys. Rev. A , 22 369 (1980); (c) It1 ETO multicenter molecular integrals. Edited by C. A. Weatherford and H. W. Jones. D. Reidel, Dordrecht. 1982. p. 171; (d) J. Chem. Phys. 78, 2801 (1983); (e) Phys. Rev. A , 31 2851 (1985); ( f ) Int. J. Quantum Chem. (Q. C. Symp.), 19, 149 (1986); (g) In Quantum chemistry: basic aspects, actual trends. Edited by R. Carb6. Stud. Phys. Theor. Chem. 62, 37 (1989). 6. (a) H. W. Jones. Int. J. Quantum Chem. 20, 1217 (1981); (b) Int. J. Quantum Chem. 21, 1079 (1982); (c) Phys. Rev. A, 30 1 (1984); (d) Int. J . Quantum Chem. 29, 177 (1986); (e) H. W. Jones, B. Bussery, and C. A. Weatherford. Int. J. Quantum Chem. (Q. C. Symp.), 21,693 (1987); ( f ) H. W. Jones. Phys. Rev. A , 38 1065 (1988); (g) H. W. Jones and B. Etemadi. Int. J. Quantum Chem. (Q. C. Symp.), 24, 405 ( 1990). 7. (a) E. 0 . Steinborn. In ETO multicenter molecular integrals. Edited by C. A. Weatherford and H. W. Jones. D. Reidel, Dordrecht. 1975. p. 7, and references therein; (0) E. 0 . Steinborn. In Methods in computational molecular physics. Edited by G. H. F. Diercksen er al. D. Reidel, Dordrecht. 1983. p. 37; (c) E. 0 . Steinborn and H. H. H. Homeier. Int. J. Quantum Chem. (Q. C. Symp.), 24,349 (1990); (d) E. 0. Steinborn and E. J. Weniger. J. Mol. Struc. (Theochem), 210, 71 (1990); (e) H. H. H. Homeier and E. 0 . Steinborn. Int. J. Quantum Chem. 39, 625 (1991). 8. (a) H. Tai. NASA Technical Memorandum 101545. Harnpton, Va. (1989); (b) Phys. Rev. A, 40, 6681 (1989); (c) Chin. J. Phys. 28,481 (1990). 9. (a) P. W. Atkins. Quanta. Oxford Chemistry Series. Clarendon Press, Oxford. 1974; (0) L. C. Biedenharn and J. D. Louck. Angular momentum in quantum physics. Addison-Wesley, Reading, Mass. 1981. 10. J. C. Slater. Phys. Rev. 36, 51 (1930). 11. (a) S. Huzinaga. Approximate atomic functions. Technical reports I, I1 (197 l ) , and I11 (1973). Division on Theoretical Chemistry. Department of Chemistry. University of Alberta; (b) E. Clementi and C. Roetti. At. Nucl. Data Tables, 14, 177 (1974). 12. A. C. Wahl, P. E. Cade, and C. C. J. Roothaan. J. Chem. Phys. 41, 2578 (1964). 13. (a) E. Clementi (Editor). Modem techniques in ~0mpUta-

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tional chemistry 1990. IBM Corporation, Kingston, 1990; (b) A. D. McLean and Y. S. Lee. 111 Current aspects of quantum chemistry 1981. Edited by R. Carb6. Stud. Phys. Theor. Chem. 21, 219 (1982). (a) C. G . Gray and K. E. Gubbins. Theory of molecular fluids. Vol. 1. Appendix A. Clarendon Press, Oxford. (1984); (b) R. N. Zare. Angular momentum. John Wiley & Sons, New York. 1988. M. R. Spiegel. Mathematical handbook of formulas and tables. McGraw-Hill, New York. 1968. K . 0-ohata and K. Ruedenberg. J. Math. Phys. 7, 547 (1966). K. 0-ohata, H. Tateka, and S. Huzinaga. J. Phys. Soc. Jpn. 21, 2313 (1966). K . Ruedenberg. J. Chem. Phys. 19, 1433 (1951). (a) E. Harris and R. Rein. Theort. Chim. Acta, 6, 73 (1966); (6) M. D. Newton. J. Chem. Phys. 51, 3917 (1969); ( c ) A. Okninski. J. Chem. Phys. 60, 4098 (1974). ( a ) J. H. Wilkinson. The algebraic eigenvalue problem. Clarendon Press. Oxford. 1965; (b) E. Durand. Solutions numCriques des equations algebriques. Masson et Cie., Paris. 1961; ( c ) R. Carb6 and L1. Domingo. Algebra matricial y lineal. Serie Schaum. McGraw Hill, Madrid. 1987. K . Ruedenberg. J . Chem. Phys. 19, 1459 (1951). E. Besald and R. Carb6. Comput. Chem. Submitted. R. Carb6 and J. A. Hernindez. Chem. Phys. Lett. 47, 85 (1977). M. Abramowitz and I. A. Stegun. Tables of mathematical functions. Dover, New York. 1972. W. Grobner and H. Hofreiter. Integraltafel. Springer-Verlag, Wien. 1966.

Appendix A: Definitions Definition A l : Prolate spherical system When we refer to the prolate spherical parameter- system {rA,rB,R)we mean to use the coordinates:

Definition A2: Nested sumtnation symbols Let us use a tzested summation symbol as a general expression like C ( k = r,s) describing a sequence of summation symbols. The meaning of such a convention corresponds to performing all the sums involved in the generation of all possible forms of vector k , whose elements are defined by the limits k, = r,,s,;Vi. Nested summation symbolism is very convenient because successive generation of k vector elements can be programmed in a general way under any high level language, using a unique do or for loop irrespective of the number of sums involved. These symbols are well suited for programming in a parallel hardware environment. Inside a nested sum, all operations attached to each vector k can be run into a separate CPU. More details about nested summation symbology can be found in ref. 22. Definition A3: Logical Krotzecker delta A logical Kronecker delta, S({L)),is a symbolic factor that is unity when the logical expression { L ) is true and zero otherwise.. An initial and crude form of this symbol was given some years ago by one of us within the context of the development of a MCSCF ( 2 3 ) . More details concerning Logical Kronecker and related topics can be found in ref. 22. Definition A4: P(a,u,v)function and T(r,a) coeflcients For some of the integrals appearing in this paper, it is helpful to obtain an expression for the expansion of the binomial products, which can be written as [A.2]

P(a,u,v) = ( u

+ 71)"' ( u - v)"' ( 1 + uv)""

One can easily deduce the equivalent compact form: the volume element being dV = ( R / ~ ) ~ (u 'v7)du dv dq and l U E { - 1, + I ) , ( P E { ~ , ~ T T } . . the integration limits: u ~ {,x), [A.4]

T(r,a) = C ( k = 0 , a ) S(k, + k,

+

+

where p = ( t 2a5, t 2a6)with t = a , + a? + a3 + a,. The coefficients in eq. [A.3] are defined as

+ k3 + k, + 2k5 = r,)S(a, + a, - ( k , + k Z )+ k3 + k4 + 2k6 = r r )C ( k , a )

where

Definition B4: D,(x) integrals 6

/

rl

\

with the index definition: t = a,

+ a, + k, + k, + k5 + k6.

Appendix B: Auxiliary integrals Definition B l : A,(x) integrals

The evaluation and extensive computational properties of these integrals can be found in the handbook of Abramowitz and Stegun (24). Definition B.5: F,(m,n) functions

[B.1]

[B .5]

A,(x)=

(sin x)"' (cos x)" dx

F,(m,n) =

Definition B2: Bk(x)integrals [B .2]

B,(x) =

dt..

Definition B3: C,(X)integrals rm

[B .3]

Ck(x)=

J,

dt.. t-ke-x'

+

+

with k ~ { 1 , 2 )p, = ( m 1)/2, q = ( n 1)/2, and T(x) is the gamma function.. Partial information on this definition can be found in ref. 25.