Selçuk J. Appl. Math. Selçuk Journal of Vol. 6. No. 1. pp. 13-20, 2005 Applied Mathematics
A Study of One-Electron Orbital-Zeeman Integral over Cartesian Gaussian Functions
Ercan Türkkan1, Ömer Dereli1, Ayhan Özmen2, Hüseyin Yüksel2 1Department of Physics, Faculty of Education, Selçuk University, 42090, Konya, Turkey 2Department of Physics, Faculty of Arts and Sciences, Selçuk University, 42079,
Konya, Turkey;
e-mail:eturkan@ selcuk.edu.tr, o dereli@ selcuk.edu.tr, aozm en@ selcuk.edu.tr, hyuksel@ selcuk.edu.tr
Received: April 1, 2005
Summary. A formalizm is devoloped for cartesian Gaussian one-electron in-tegral of the orbital-Zeeman operator r1c p1 appearing in Breit-Pauli Hamil-tonian. In this study Kx(l; m; n) Ky(l; m; n) and Kz(l; m; n) integrals are de-…ned, and using these integrals, expressions for the Orbital-Zeeman integrals are derived.
Keywords. Orbital-Zeeman operator, orbital-Zeeman integrals, cartesian Gaussian functions, molecular integrals.
1. Introduction
The e¢ cient evaluation of molecular integrals is the …rst step of an electronic structure calculation. Cartesian Gaussian functions of the form
xlAyAmzAnexp ArA2
were …rst proposed as basis functions by Boys [1]. The obvious advantage of Gaussian functions over Slater-type orbitals (STO’s) is the ease with which a product of Gaussians on two di¤erent centers can be written as a simple function on a common center [2]. Until the early 1960’s when calculations on diatomic molecular systems over STO’s were already very common, however the intractability of the four –center integral appeared to present a major block to polyatomic calculations [3]. Gaussians began to enjoy increased popularity when it was found of them a …xed linear combination ( so-called “contracted Gaussian”) could be used to approximate an atomic orbital to good accuracy. Initially the trend was to use combinations of simple Gaussian “lobes” (n =
l = m = 0) [4]. The alternative to contracted Gaussian lobes is contracted Cartesian Gaussians (n,l,m are positive integers).
Overlap, kinetic energy, nuclear attraction and electronic repulsion integrals with higher angular quantum numbers (l + m + n = 0; 1; 2; :::) can be obtained by expressing them as analytical derivatives of s-type (l = m = n = 0) Gaussians [5], or by employing gaussian lobe functions, as …rst suggested by Preuss [6] and further devoloped by Whitten[4]. The choice of lobe separations in the latter approach has been discussed by Pekte et. al. [7] and Shih et. al. [8] for p-type functions (l + m + n = 1) and by Driessler and Ahlrichs [9] for d-and f-type functions (l + m + n = 2) and (l + m + n = 3). In addition, expression for overlap, kinetic energy, nuclear attraction and electron repulsion integrals over Hermite Gaussian Functions are given by Zivocovic and Maksic [10]. Finally Taketa, Huzinaga and Ohata [11] have developed an alternative formulation of these integrals over Cartesian Gaussians.
With the availability of high quality wave functions, there is renewed interest in the study of molecular …ne structure and related properties. Such quanti-ties include spin-orbit splittings in electronic spectra, electronic and magnetic properties such as magnetic susceptibilities, shieldings, orbital-Zeeman, polar-izabilities,ESR hyper…ne cuopling, g-tensor, NMR spin-spin coupling and elec-tric quadrupole coupling to name but a few. Kern and Karplus [12] employed Gaussian transform methods for such integrals, but use of their expressions for one –electron spin-orbit integrals is rather cumbersome. The one-elctron inte-grals in their work are obtained as the limit of two-electron inteinte-grals (e. g., a three-center one-electron spin-orbit integral is obtained from a two-electron four-center integral.) and are obtained directly only for s-type functions; ex-pressions for integrals involving orbitals of higher angular quantum number can be obtained only by di¤erentiating the expressions for s-type orbitals[13]. For these integrals have employed Gaussians as lobe functions, a large number of theoretical calculations of spin-orbit interactions in polyatomic molecules have employed this type of formulation [14]. More recently McMurchie and David-son [15] have discussed a computer algorithm for dealing with this problem for Cartesian Gaussians of large angular momentum. At the other hand other expressions for these integrals over speci…c operators have been derived using Cartesian Gaussian functions [16,17]. These integrals over Breit-Pauli opera-tors have been developed using general Cartesian Gaussian function for both one-electron and two-electrons[18].
In this study, integrals over Cartesian Gaussian functions for orbital-Zeeman operator which occur in the Breit-Pauli Hamiltonian [19,20] are systematicaly treated.
2. Cartesian Gaussian Integral for the Orbital-Zeeman operator r1c p1
Following the formalism of Taketa et. al. [11] a cartesian Gaussian with orbital exponent a located at center A(Ax; Ay; Az)is expressed as
a(1) = Na(x1 Ax)la(y1 Ay)ma(z1 Az)na (1) exph a n (x1 Ax)2+ (y1 Ay)2+ (z1 Az)2 oi = Naxlaayamazanaexp ar2a where the normalization costant Na is given by
(2) Na = " 2 a 3 2 (4 a)la+ma+na (2la 1)!! + (2ma 1)!! + (2na 1)!! #1 2 :
Here la; ma and na are positive integers greater than or equal to zero, and l + m + n = 0; 1; 2; :::
corresponds to s-,p-,d-,f-,. . . types of Gaussians. The product of two Gaussians a(1) and b(1) can be written as a linear combination of Gaussians at a point P, whose coordinates are
(3) P = aA + bB:
and the product a(1) b(1) is
a(1) b(1) = Gab laX+lb L=0 fL la; lb; P Ax; P Bx xLp (4) maX+mb M =0 fM ma; mb; P Ay; P By y naX+nb N =0 fN na; nb; P Az; P Bz zpN exp rp2 ; where (5) = a+ b; (6) Gab= NaNbexp a b a+ b AB2 ;
(7) fL la; lb; P Ax; P Bx = la X i=0; i+j=L lb X j=0 la i lb j P Ax la i P Bx lb j :
It is obvious that L6 la+ lb in (4) and (7). In addition to these relationships the well-known de…nite integral result
(8) 1 Z 1 x2nexp x2 dx = 1 2(2n 1)!! (2 )n will also be used.
The overlap integral is given by
Ovlp = ZZZ a(1) b(1)dxdydz (9) = GabIxIyIz where (10) Ix= (la+lb) 2 X L=0 f2L(la; lb; P Ax; P Bx) (2L 1)!! (2 )L 1 2
with analogous expressions for Iy and Iz.
For a charge distribution of the type a(1)@x@ b(1), it is convenient to de…ne a new function, gL(la; lb;P Ax;P Bx) = l bfL(la; lb 1;P Ax;P Bx) (11) 2 bfL(la; lb+1;P Ax;P Bx); in terms of which a(1) @ @x b(1) = Gab la+lXb+1 L=0 gL la; lb; P Ax; P Bx xLp (12) maX+mb M =0 fM ma; mb; P Ay; P By yMp naX+nb N =0 fN na; nb; P Az; P Bz zpNexp r2p :
Finally in this section we will discuss the operator r1c p1, where p1is the linear momentum of electron 1 and r1cis vector de…ning the position of electron 1 with respect to the position of nucleus c.
(13) r1c= rc r1
r1c= r1p+ P C r1c p1operator may be expressed in the form
r1c p1= i h yc@z@1 zc@y@1 ^{ + xc@z@1 + zc@x@1 | + x^ c@y@1 yc@x@1 ^k i (14) = ih(r1c p1)x^{ + (r1c p1)y^| + (r1c p1)z^k i :
given by McMurchie and Davidson [15]. Cartesian Gaussian integrals for the r1c p1 operator can be dealt with in a systematic manner by considering integrals of the forms
(15) Kx(l; m; n) =R a(1)xlcymc zcn@x@1 b(1)d Ky(l; m; n) =R a(1)xlcycmzcn@y@1 b(1)d Kz(l; m; n) =R a(1)xlcymc znc@z@1 b(1)d
Kx(l; m; n)integral can be expressed by substituting (4),(8),(10),(11) and (12) in (15) as (16) Kx(l; m; n) = G ab R la+lPb+1 L=0 gL la; lb; P Ax; P Bx xLp maP+mb M =0 fM ma; mb; P Ay; P By ypM naP+nb N =0 fN na; nb; P Az; P Bz zpN exp ( r 2 p)x l cycmzcnd where xc and xlc are given by
(17) xc= xp+ P Cx and (18) xl c = xp+ P Cx l = l X =0 xp P Cx l l p :
Substituting (17) and (18) in (16) we …nd, Kx(l; m; n) = GabR la+lPb+1 L=0 gL la; lb; P Ax; P Bx xLp maP+mb M =0 fM ma; mb; P Ay; P By yMp naX+nb N =0 fN na; nb; P Az; P Bz zpN l X =0 xp P Cx l l p m X =0 yp P Cx m m n X =0 zp P Cx n n d = Gab Z la+lPb+1 L=0 gL la; lb; P Ax; P Bx l X =0 P Cx l l p (19) maP+mb M =0 fM ma; mb; P Ay; P By m X =0 P Cx m m naP+nb N =0 fN na; nb; P Az; P Bz n P =0 P Cx n n xL+ p ypM + zpN + exp r2p d By using (8),(18) this expression can be simpli…ed to give
Kx(l; m; n) =Gab (la+lb+1) 2 P L=0;L0=L+ gL la; lb; P Ax; P Bx l P =0 P Cx l l p (20) (ma+mb) 2 P M =0;M0=M + fM ma; mb; P Ay; P By m X =0 P Cx m m (na+nb) 2 X N =0;N0=N + fN na; nb; P Az; P Bz n X =0 P Cx n n 3 2 (2L0 1)!! (2M0 1)!! (2N0 1)!! (2 )L0+M0+N0 ; where L0= L + , M0= M + , N0= N + and g2L(la; lb; P Ax; P Bx) = lbf2L(la; lb 1; P Ax; P Bx) (21) 2 bf2L(la; lb+ 1; P Ax; P Bx):
Analogous expressions can be obtained for Ky(l; m; n) and Kz(l; m; n). In terms of these quantities,
Z a(1) h (r1c p1)y i b(1)d = i [Kz(0; 0; 1) Ky(1; 0; 0)] (22) Z a(1) h (r1c p1)y i b(1)d = i [Kx(0; 0; 1) Kz(1; 0; 0)] Z a(1) [(r1c p1)z] b(1)d = i [Ky(0; 0; 1) Ky(1; 0; 0)] 3. Conclusion
In this study a formalism has been derived for the evaluation of Cartesian Gaussian integrals for quantum mechanical orbital-Zeeman operator. Kx(l; m; n); Ky(l; m; n) and Kz(l; m; n) integrals were de…ned for calculating orbital-Zeeman integrals. Linear combination of these integrals gives orbital-Zeeman integral components. Direct calculation of the integral of orbital-Zeeman operator is very di¢ cult and it does not have a general form. However the formalism pro-posed here overcomes the calculation di¢ culties. The new formalism which we derived in this study has a systematic form. It is very easy to implement this formalism in computational applications.
4. References
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