On the Computation of (2-2) Three- Center Slater-Type Orbital Integrals of 1/r 12 Using Fourier-Transform-Based Analytical Formulas

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1 On the Computation of (2-2) Three- Center Sater-Type Orbita Integras of /r 2 Using Fourier-Transform-Based Anaytica Formuas DANKO ANTOLOVIC, HARRIS J. SILVERSTONE 2 Pervasive Technoogy Labs, Indiana University, Boomington, IN Department of Chemistry, Johns Hopkins University, Batimore, MD 228 Received 6 February 2004; accepted 20 February 2004 Pubished onine 0 June 2004 in Wiey InterScience ( DOI 0.002/qua.2023 ABSTRACT: We describe a computationa scheme devised for using Fouriertransform-based anaytic formuas for three-center integras of r 2, wherein each eectron is described by a two-center product of Sater-type orbitas. The asymptotic behavior of the auxiiary functions, which are reated to modified spherica Besse functions and to exponentia integras, is investigated, and recursive computationa schemes are derived that are shown to be numericay stabe for high summation indices and arge internucear distances Wiey Periodicas, Inc. Int J Quantum Chem 00: 46 54, 2004 Key words: Sater-type orbita integras; exponentia orbita integras; muticenter integras; three-center integras Correspondence to: H. Siverstone; e-mai: hjsiverstone@ jhu.edu or D. Antoovic; e-mai: dantoov@indiana.edu Introduction There has been recent renewed interest in moecuar integras with exponentia-type orbitas [ 7], incuding the use of techniques to acceerate convergence. Accordingy, we present an anaysis of a cass of Sater-type orbita (STO) moecuar integras, the (2-2) three-center eectron eectron repusion integras that contain two bicentric eectron distributions that share one center in common. The (2-2) three-center integra is defined in the notation of Ref. [8] as I nc cm c c,n d dm d d;n a am a a,n b bm b b R, R 2 N a N b N c N d dv dv 2 r 2 * nc cm c c r 2 nd dm d d r 2 R 2 * * na am a a r nb bm b b r R () I cd;ab R, R 2 (2) Internationa Journa of Quantum Chemistry, Vo 00, (2004) 2004 Wiey Periodicas, Inc.

2 COMPUTATION OF (2-2) THREE-CENTER SLATER-TYPE ORBITAL INTEGRALS where the Sater orbita is defined by nm r Nr n expry m,. (3) The N is a normaization constant factored out of Eq. (); Y m (, ) is a spherica harmonic, and n. R and R 2 are internucear-distance vectors characterizing the two bicentric eectron distributions. The STOs a and c share a common center the origin. This work buids upon the methodoogy deveoped in Refs. [8 26], which we summarize here:. The six-dimensiona integra is converted into a three-dimensiona integra in Fourier transform space by using the convoution theorem. 2. To evauate the Fourier transform of the twocenter STO product that is required for Step (), expand the orbita at the dispaced center in an infinite series around the origin. The terms of this series invove spherica Besse functions, spherica harmonics, and vectorcouping coefficients. Carry out the anguar integration, and obtain a series of one-dimensiona Fourier transforms. The fina one-dimensiona Fourier integration can be carried out by means of the cacuus of residues and can be expressed in terms of exponentia-type integras. 3. The convoution integra is resoved first by carrying out the anguar integrations, which are easy, because each Fourier-transformed two-center STO product is a sum over spherica harmonics, and second by evauating the radia integration using contour integration techniques and the residue theorem. In this way, the moecuar integras are expressed as infinite series over anguar momentum numbers {Eq. (7) of Ref. [8]}: I cd;ab R, R 2 0 b b a c 20 2 d 2 2 d m min a, 2 c / 2 c b m b ; m max a, 2 c c 2 d m d ; 2, m m c m a c m ; a m a c 2, m m c m a ; c m c m Y bm m R, R Y dm cm am * 2 R2, R2 I ; 2 2; cd;ab R, R 2. (4) Each term of the series contains vector couping coefficients, spherica harmonics of the two internucear distance vectors, and a radia term. The series is a fivefod sum over anguar-momentum indices, 2,, 2, and that go to infinity together : the difference between any two summation indices is bounded by a b c d, because the vector-couping coefficients otherwise vanish. The radia term consists of modified spherica Besse functions and exponentia-type integras with internucear distances times STO exponents ( s) as arguments, and it decomposes into four terms {Eqs. (7) (20) of Ref. [8]}: I ; 2 2 cd;ab R, R 2 I I 2 I 3 I 4. (5) Our main task is to compute the radia term. is The Radia Term The simpest of the four parts of the radia term I 2 2 b d/d b nbb b d/d b b b b I b R b b d/d b b R n a d/d d ndd d d/d d d d d K 2 d R 2 d 2 d d/d d 2 d R 2 n c 2 R 2 2 E n a a b R ˆ nc 2 c d R 2 ˆ nc 2 c d R 2 },R R 2 (6) where the modified spherica Besse functions are given by I x x x d/dx x sinhx (7) K x x x d/dx x expx. (8) The (reative) transparency of the formua (6) is due to the use of derivative operators with respect to STO exponents, which commute with a of the Fourier-transform operations. As was the case for the (-2) three-center integra [25], before the formuas can be programmed for a computer, the de- INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 47

3 ANTOLOVIC AND SILVERSTONE rivatives appearing in them must be deat with expicity. We give the resuts here and sketch the use of stabe recursion formuas for the cacuation of the I (k). The reader is referred to [24] for a more compete account of the detais. The derivative operators in Eq. (6) for I (2) and in the companion formuas for the other I (k) appear in a common, four-eement structure: first, (d/d e ) n e e ( e d/d e ) e e e, where e specifies a STO; next, a modified spherica Besse function with argument e R i, where i is or 2; then, e i ( e d/d e ) i e ; then at the far right, a function of exponentia-integra cass. Our strategy [20] is to express (d/d e ) n e e in powers of ( e d/d e ), then absorb the atter either into the spherica Besse functions or into auxiiary functions based on the exponentia-integra functions. In such a manner we obtain for the I (k) : I 4R n an bn c2 R 2 n d nbnd n b n d u b0 u d0 ubud R /R 2 ud j b ( b )!! ( b 2s b )!! j bs b b I t b b n b b j b j b j d n b j b s b t b u b n d j d s b n d d j d j d s d s d t d u d d 2 2!! d 2 2 2s d!! j ds d 2d I 2t d 2d A u b na b, a E 2u d 2n c d, c E u b, 2u d,n a 2n c b, d, a, c I 2 4R n an b R 2 n cn d nbndd b R 2 /R ncud2 A 2u d nc 2 2d, 2c E u b n a b, a E 2u d, u b,n c 2 n a d, b, c, a (9) j bs b I t b b j d I 3 4R n an bn c2 R 2 n d nbndb n b n d u b0 u d0 n d d j d j d n b n d u b0 u d0 ub j b s d n b b j b j b s b n b j b s b t b u b b!! b 2s b!! n d j d d 2 2!! s d t d u d d 2 2 2s d!! j ds d 2d K 2t d 2d ud R /R 2 ud K tb b I 2td 2d E u b n a b, a  2u d nc 2 2d, 2c (0) E 2u d 2n c d, c  u b na b, a R 2 /R nancubud3 A 2u d nc 2 2d, 2c Ẽ u b n a 2b, 2a R /R 2 naub Ẽ u b n a b, a Ẽ 2u d, u b,n c 2 n a d, b, c, a R 2 /R nancubud3 Ẽ 2u d, u b,n c 2 n a 2d, 2b, 2c, 2a u b, 2u d,n a 2n c b, d, a, c R 2 /R nancubud3 u b, 2u d,n a 2n c 2b, 2d, 2a, 2c () I 4 4R n b R 2 n an bn cn d2 nbndbd n b n d u b0 u d0 R /R 2 ub K t b b K 2t d 2d  2u d nc 2 2d, 2c Ẽ u b n a 2b, 2a R /R 2 naub Ẽ u b n a b, a u b, 2u d,n a 2n c 2b, 2d, 2a, 2c 2u d, u b,n c 2 n a 2d, 2b, 2c, 2a. (2) We have introduced the notation: a R a b R b 2c R 2 c 2a R 2 a 2b R 2 b c R c 2d R 2 d d R d (3) n!! nn 2n or (4)!!2!! (5) 48 VOL. 00, NO. 2

4 COMPUTATION OF (2-2) THREE-CENTER SLATER-TYPE ORBITAL INTEGRALS and a number of auxiiary functions, the anaysis of which foows in the next section. 0 dtt n I xtexpyt (25) Auxiiary Functions In Eqs. (9) (2) for the I (k), there are three sets of functions: the modified spherica Besse functions I and K, whose properties are we known; and auxiiary functions that we sha ca - (one upper index) and 3- (three upper indices) functions, which are based on exponentia-type integras. The - functions have an integra representation invoving the spherica Besse functions, and the 3- functions have an integra representation invoving both the - and spherica Besse functions. These integra representations are usefu for deriving recursion reations and asymptotics. -L FUNCTIONS The four - functions are the descendants of, ˆ, E, and Ẽ, which were used in Refs. [8, 2 25] (see aso Ref. [26]): E n x t n e xt dt (6) E n x, y x x d/dx x E n x y (26) A n x, y (27) Ẽ n x, y x x d/dx x Ẽ n y x 2 / 2 (28) 2i where 0 dt ne i tt n I xtexpyt (29) ˆ n y x 2 ˆ n y x ˆ n y x. (30) 3-L FUNCTIONS The four 3- functions are simiary defined. E n 2 3 x, y, z, w x x d/dx x y 2 y d/dy 2 y d/dz 3 x z E n x y z w (3) n x (7) Ẽ n x E n x x n ogx n /n!, n 0 (8) E n x x n n!, n 0 (9) ˆ n x (20) n d/dnn n (2) A n x, y x x d/dx x n x y (22) dtt n K xtexpyt (23) Â n x, y x x d/dx x ˆ n y x 2 / 2 (24) Ẽ n 2 3 x, y, z, w 0 dtt n233 A 3 xt, ztk 2 ytexpwt x x d/dx x y 2 y d/dy 2 y (32) d/dz 3 x z 2 Ẽ n x y z w 2 n 2 3 x, y, z, w dtt n233 A 3 xt, zti 2 ytexpwt x x d/dx x y 2 y d/dy 2 y d/dz 3 2 z x Ẽ n x y z w (33) (34) Ẽ n y w 2 (35) INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 49

5 ANTOLOVIC AND SILVERSTONE n x, y, z, w dtt n233 Â 3 xt, ztk 2 ytexpwt x x d/dx x y 2 y d/dy 2 y d/dz 3 4 z x Ẽ n z x y w (36) Ẽ n w y 4 (37) Asymptotic formuas for a the - and 3- auxiiary functions can simiary be found [24]. In short, A n, E n, and E 2 3 n have K-ike asymptotics; Â n, Ẽ n, and 2 3 n are I-ike; Ẽ 2 3 n and 2 3 n have opposite asymptotics for arge indices and 2. Because the integra terms I ()... I (4) contain products of I-ike and K- ike functions, which have opposite asymptotic behavior, it is convenient to use a set of reduced functions in the numerica impementation, where the factors x [(2 )!!] and x (2 )!! have been extracted and expicity canceed out. 0 dtt n233 Â 3 xt, zti 2 ytexpwt In Eq. (38), we use the notation, z x Ẽ n z x y w Ẽ n w y 4 z x Ẽ n z x y w 2 Ẽ n w y 2 z x (38) Ẽ n z x y w 2 Ẽ n w y 2. (39) ASYMPTOTIC BEHAVIOR OF THE AUXILIARY FUNCTIONS There are two asymptotic cases of interest: arge indices (for the summation of the infinite series) and arge R i (the physica case of arge internucear distances). In both cases, the asymptotic forms for the - and 3- functions foow from the formuas for the modified spherica Besse functions: For instance, A n x, y I x x 2!!, as 3, (40) K x x 2!! I x e x /2x, as x 3. (4) K x e /x x 2!! x n y, as 3, (42) exy A n x, y, as x 3. (43) x x y Recursion Reations There are a arge number of recursion reations for the auxiiary functions. On the whoe, they foow a few simpe patterns. For each - function, there is a pair of three-term recursions in the n pane one homogeneous, the other inhomogeneous. Each 3- function has two anaogous pairs of recursions, one pair in the n 2 pane, and the other in the 3 pane. Within each pane, new recursions can be derived as inear combinations of the basic pair. To iustrate the use of recursion formuas, and especiay the question of numerica stabiity, we take as an exampe the function A n. The basic pair can be derived from the integra representation (23) by appying a recursion formua for the modified spherica Besse functions, or by integration by parts: A n x, y 2 x A n x, y A 2 n2 x, y A n x, y xa n2 x, y ya n x, y (44) e y K x/n. (45) These recursions can be represented by diagrams show in Figures and 2, in which we note the reative power of by which the term in the recursion foows as 3. One can see from the diagrams that recursive computation of ow- terms from high- terms woud necessariy invove numerica canceation, but that upwards recursion shoud be safe. In the imit of arge R i (x and y), we have aso annotated the diagrams with the reative powers of x and y that the term foows as x 3 and y 3. Because of numerica canceation, it woud be unsafe to cacuate the midde term in Figure or the 50 VOL. 00, NO. 2

6 COMPUTATION OF (2-2) THREE-CENTER SLATER-TYPE ORBITAL INTEGRALS diagrams can be transated reative to one another in the index pane a diagram does not change if the reative -, x-, or y-dependency is increased uniformy for every term if two diagrams overap, one and ony one pair of overapping points may be eiminated, provided that the same -, x-, and y-dependencies are assigned to both points if the overapping points are not eiminated, the resuting point acquires the higher -, x-, and y-dependencies of the two points By foowing these rues, we construct the (inhomogeneous) recursion FIGURE. Diagram for the homogeneous A n (x, y) recursion formua (44) that shows reative dependence for arge and reative x dependence for arge x. The term marked by an open circe can be safey computed (with the east numerica canceation). bottom term in Figure 2 when x and y are arge. In both Figures and 2, the safe term is indicated by an open circe. The foowing simpe rues describe how to generate new recursions from the existing ones: xa n x, y 2 n A n x, y ya n x, y e y K x (46) with diagram shown in Figure 3 and predict that ony the term marked by an open circe may be safey computed. The same principes appy to diagrams for the other auxiiary functions: the basic pairs of recursions aways ead to the same shape diagrams within each cass (- and 3-), but with different FIGURE 2. Diagram for the inhomogeneous A n (x, y) recursion formua (45) that shows reative dependence for arge and reative x dependence for arge x and y. The term marked by an open circe can be safey computed (with the east numerica canceation). FIGURE 3. Diagram for A n (x, y) recursion formua (46) that shows reative x- and y-dependence for arge x and reative -dependence for arge. The term marked by an open circe may be safey computed (with the east numerica canceation). INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 5

7 ANTOLOVIC AND SILVERSTONE FIGURE 4. Domain of indices of the A n (x, y) functions. reative asymptotics, because of the different asymptotics of the various auxiiary functions. Computation of Auxiiary Functions We iustrate the strategy of cacuating auxiiary functions with A n (x, y). The ranges of the indices n and are determined by the STO indices, how many terms are to be cacuated in the series (4), and the fact that the vector-couping coefficients vanish uness the three quantum numbers have even sum and satisfy the trianguar inequaity. For the function A n (x, y), the domain in the n pane is a paraeogram, as shown in Figure 4. Before any recursion reation can be used, initia vaues must be cacuated. We use the formua A n x, y mx m nm x y (47) m0 to seed the top row and the eftmost edge, as indicated by soid dots in Figure 4. A the remaining positions are fied by recursion. First, the inhomogeneous recursion (45) is used to compute the santing ine adjacent to the eft edge and marked by open circes, and in the direction indicated by the open circe in Figure 2, for which it is asymptoticay stabe for arge and R i (x and y). Second, the second row of Figure 4, marked by s, is competed using the recursion (46) with diagram Figure 3. Third, the rest of Figure 4 is fied in using recursion (44) in its asymptoticay stabe direction as indicated in Figure. We remark that it is possibe to construct seemingy stabe recursions, such as to connect A n (x, y), A n (x, y), and A n 2 (x, y), but which contain the factor x 2 y 2 in the inhomogeneous term, and which are unstabe for cose vaues of the two arguments. For this reason we have avoided the use of such recursions. Computationa schemes for the other - auxiiary functions foow aong the ines simiar to the above exampe, except that the n domain of E-functions is somewhat different. The ranges for the 3- auxiiary functions fi a four-dimensiona directproduct of two two-dimensiona ranges, one in the n 2 pane, the other in 3. The reader is referred to Chapter 3 of Ref. [24] for the detais. Canceations in I (3) and in I (4) When the Internucear Distances Are Equa: G Functions There are terms in Eqs. () and (2) for I (3) and I (4) that contain differences of - Ẽ functions and powers of the ratio of radii, r R /R 2, and simiary for 3- Ẽ functions. These differences van- 52 VOL. 00, NO. 2

8 COMPUTATION OF (2-2) THREE-CENTER SLATER-TYPE ORBITAL INTEGRALS ish ineary as r 3. For sma R and R 2 there are no significant numerica probems, but for arge R and R 2 the two Ẽ functions can be exponentiay arger than their difference, with a significant oss of accuracy. To iustrate the point, we abstract the simpest of the difference terms in I (4), which can be shown to have the integra representation, Ẽ n x, y r n2 Ẽ n rx, ry r dtt n I xte yt (48) I xe y r, as r 3. (49) The first-order canceation for sufficienty sma r is obvious. Note that the integrand in Eq. (48) is stricty positive. Now consider the vaue of the difference for arge x and y: Ẽ n x, y r n2 Ẽ n rx, ry exy r, 2x r 3, x and y 3. (50) If x y, the factor mutipying (r ) is exponentiay arge; if x y, it is exponentiay sma. On the other hand, the asymptotic behavior of Ẽ n (x, y) is quite different when x y. Take, for instance, Ẽ 2 x, y x y 2x 2 n y x, y x 3, (5) y x c R, (52) where c is a constant (note that x/y is a ratio of orbita exponents). The exponentiay sma asymptotics of Eq. (50) is therefore the resut of canceation of the dominant nonexponentia terms exempified by Eq. (52). Such a canceation can be avoided computationay by introducing a new auxiiary function G n (x, y) to use instead of Ẽ n (x, y): G n x, y dtt n I xte yt (53) E n x, y E n x, y. (54) 2 We use G n (x, y) ony when x y 0. G n (x, y) has the asymptotic formua, exy G x, y n 2x x y (55) and recasts Eq. (50) as a difference of auxiiary functions which have the same asymptotic form as their difference itsef: Ẽ n x, y r n2 Ẽ n rx, ry G n x, y r n2 G n rx, ry (56) exy e r xy rn 2xx y 2xx y (57) exy r Or O/x y. (58) 2x Anaogous considerations appy to the other difference terms. Recursion formuas for the G functions are identica to their Ẽ counterparts, and the numerica stabiity patterns of the recursions are identica as we. Vector-Couping Coefficients The same vector-couping coefficients recur in a the STO integras, but their number is arge, and a standard array on the indices woud be sparse. One strategy to avoid such a arge array woud be to cacuate the coefficients efficienty as needed, and not to store and reuse them. We have opted for an aternative strategy, tabuation in a packed array: the coefficients were computed once by means of Racah s formua, then stored via a packing scheme that eaves no empty spaces in the storage array. Summary We have shown how to cast Fourier-transformbased anaytic formuas for (2-2)-type STO threecenter integras of r 2 in a computationay convenient form. The expicit derivatives are repaced by summations over auxiiary functions, for which, in the ight of the asymptotics of the auxiiary func- INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 53

9 ANTOLOVIC AND SILVERSTONE tions for arge indices and for arge R i, stabe recursion schemes have been derived. References. Safouhi, H.; Hoggan, P. E. J Phys A 998, 3, Safouhi, H.; Hoggan, P. E. J Phys A 999, 32, Safouhi, H.; Hoggan, P. E. J Math Chem 999, 25, Safouhi, H.; Hoggan, P. E. J Comput Phys 999, 55, Vieie, L.; Beru, L.; Combourieu, B.; Hoggan, P. J Theor Comput Chem 2002, (2), Beru, L.; Hoggan, P. J Theor Comput Chem 2003, 2(2), Safouhi, H.; Hoggan, P. Mo Phys 2003, 0(), Siverstone, H. J.; Kay, K. G. J Chem Phys 968, 48, Siverstone, H. J. J Chem Phys 966, 45, Siverstone, H. J. Proc Nat Acad Sci USA 967, 58, Siverstone, H. J. J Chem Phys 967, 47, Siverstone, H. J. J Chem Phys 968, 48, Siverstone, H. J. J Chem Phys 968, 48, Siverstone, H. J.; Kay, K. G. J Chem Phys 969, 50, Kay, K. G.; Siverstone, H. J. J Chem Phys 969, 5, Kay, K. G.; Todd, H. D.; Siverstone, H. J. J Chem Phys 969, 5, Kay, K. G.; Todd, H. D.; Siverstone, H. J. J Chem Phys 969, 5, Kay, K. G.; Siverstone, H. J. J Chem Phys 969, 5, Siverstone, H. J.; Todd, H. D. Int J Quantum Chem 97, 4, Todd, H. D.; Kay, K. G.; Siverstone, H. J. J Chem Phys 970, 53, Kay, K. G.; Siverstone, H. J.; Keiman, B. J Chem Phys 970, 53, Kay, K. G.; Siverstone, H. J. J Chem Phys 970, 53, Antoovic, D.; Dehae, J. Phys Rev A 980, 2, Antoovic, D. On the anaytica and computationa aspects of the (2-2)-type three center integras over Sater type atomic orbitas. Ph.D. Dissertation, Johns Hopkins University, Batimore, MD, 983. [Avaiabe from University Microfims Internationa, Ann Arbor, MI, U.M.I. No ]. 25. Todd, H. D.; Daiker, K. C.; Coney, R. D.; Moats, R. K.; Siverstone, H. J. In ETO Muticenter Integras; Weatherford, C. A.; Jones, H. W., Eds.; Reide: New York, 982; p Handbook of Mathematica Functions; Abramowitz, M.; Stegun, I. A., Eds.; App Math Ser No. 55, Nationa Bureau of Standards, Washington, DC, VOL. 00, NO. 2