Explicit formulae for integrals of s and p type GTFs

Transcription

1 ELECTRONIC JOURNAL OF THEORETICAL CHEMISTRY, VOL., (997) Expliit formulae for integrals of s and p type GTFs CAROL A. BAXTER AND DAVID B. COOK The Department of Chemistry, The University of Sheffield, Sheffield, S3 7HF, UK SUMMARY A set of expliit formulae for the familiar energy integrals arising from s and p GTFs are given, whih are orreted versions of some formulae whih have been in the literature for three deades. Reeived Otober 996; Aepted November 996 Eletron. J. Theor. Chem., Vol., (997) No. of Figures: 0 No. of Tables: 0 No. of Referenes: KEY WORDS Gaussian Funtions; integrals. INTRODUCTION There are many different approahes to the omputation of moleular energy integrals over a basis of Gaussian type funtion primitives (GTFs) based on different hoies of optimum proessing method. For some purposes it is very useful to have expliit expressions for individual integrals rather than, for example, methods whih are optimized for the mass alulation of (e.g.) integrals over shells of GTFs. One suh olletion of expliit formulae is to be found in the MOTECC series of publiations from Clementi s researh group [] whih are part of the program suite KGNMOL. Unfortunately, however, some of these formulae are inorret (among the Speial formulas for the matrix elements of s and p type ) and we assume that the original formulae in the 965 tehnial report have simply been transribed without hange and that the atual omputer implementation in IBMOL and, later, KGNMOL are orret. That is, we assume that these programs do not give erroneous results if the speial s and p odes are used. We have been involved in the generation of software whih, among other things, omputes integrals diretly over hybrid atomi orbitals [] (of type sp x ) and expliit formulae for the primitive integrals over the GTFs of s and p type have obvious appeal. We have therefore derived the orret formulae and present them below in a form whih is as lose as possible to the original format in MOTECC. We list all the formulae for ompleteness and onveniene and the differenes between our formulae and the originals are marked by hange bars in the usual way.. GENERAL FORMULAE The normalization fator for an unontrated Gaussian funtion, χ, on a entre A, where χ(a, α, l, m, n) =x l Ay m Az n Aexp( αr A) () Correspondene to: D. B. Cook. These formula go bak, in idential form, at least to the IBM tehnial reports desribing the IBMOL system; see, for example, IBMOL Version 4 by Alain Veillard (IBM, San Jose, 965). CCC /97/ $ by John Wiley & Sons, Ltd.

2 INTEGRALS OF s AND p TYPE GTFS 67 is [ ( ] π )3 (l )!!(m )!!(n )!! N α = α (l+m+n) α (l+m+n) In defining the integrals over Gaussian funtions, the inomplete Gamma funtion is used: () F ν (t) = 0 u ν e tu du (t >0,ν =0,,,...) (3) The point P whih appears in later formulae is the entre of the new Gaussian funtion, the produt of the Gaussians on A and B: Similarly the point Q is defined to be P i = α A i + α B i α + α (4) i = x, y, z Q i = α 3C i + α 4 D i α 3 + α 4 (5) i = x, y, z 3. SPECIAL FORMULAE FOR THE MATRIX ELEMENTS OF s AND p TYPE The subsripts i, j, k and l define the axis of a p type orbital funtion; eah is replaed with the value x, y or z as appropriate. A s type orbital funtion is defined by i, j, k and l being equal to zero. 3.. Overlap integrals Here δ ij is the Kroneker delta. ( )3 ( ) π S 00 α α (a, b) = S a S b = N N exp AB α +α α + α S i0 (a, b) = P ia S b = α (A i B i )S 00 (a, b) (6) α + α S 0j (a, b) = S a P jb = α (B j A j )S 00 (a, b) α + α S ij (a, b) = P ia P jb = (α + α ) δ ij + α α (α + α ) (A i B i )(B j A j ) S 00 (a, b) 3.. Kineti energy integrals The kinetienergyintegralsare givenin terms of the overlapintegrals (Equation (6)) and the following K funtions: K 00 (a, b) = 3α α α α α +α (α +α ) AB 997 by John Wiley & Sons, Ltd. Eletron. J. Theor. Chem., Vol., (997)

3 68 C. A. BAXTER AND D. B. COOK K i0 (a, b) = K 0j (a, b) = K ij (a, b) = α α (α +α ) (A i B i ) α α (α +α ) (B j A j ) (7) α α (α +α ) δ ij The kineti integrals are defined as T 00 (a, b) = K 00 (a, b)s 00 (a, b) T i0 (a, b) = K i0 (a, b)s 00 (a, b)+k 00 (a, b)s i0 (a, b) T 0j (a, b) = K 0j (a, b)s 00 (a, b)+k 00 (a, b)s 0j (a, b) (8) T ij (a, b) = K ij (a, b)s 00 (a, b)+k i0 (a, b)s 0j (a, b) +K 0j (a, b)s i0 (a, b)+k 00 (a, b)s ij (a, b) 3.3. Nulear attration integrals A set of intermediate funtions are defined; these use the inomplete Gamma funtion (Equation (3)). L 00 (; a, b) = F 0 (t) L i0 (;a, b) = (C i P i )F (t) L 0j (;a, b) = (C j P j )F (t) (9) L ij (; a, b) = (P i C i )(P j C j )F (t) (α + α ) δ ijf (t) In eah ase the argument, t, equals (α + α ) PC,whereP is defined as in Equation (4). The nulear attration integrals are now written in terms of the overlap integrals (Equation (6))and the intermediate L funtions (Equation (9)). V 00 (a, b) = V i0 (a, b) = V 0j (a, b) = V ij (a, b) = Z S a r S b = θ Z S 00 (a, b)l 00 (; a, b) (0) Z P ia r S b = θ Z S i0 (a, b)l 00 (; a, b)+s 00 (a, b)l i0 (; a, b) Z S a r P jb = θ Z S 0j (a, b)l 00 (; a, b)+s 00 (a, b)l 0j (; a, b) Z P ia r P jb = θ Z S ij (a, b)l 00 (; a, b)+s i0 (a, b)l 0j (; a, b) +S 0j (a, b)l i0 (; a, b)+s 00 (a, b)l ij (; a, b) where θ = (α π + α ) 997 by John Wiley & Sons, Ltd. Eletron. J. Theor. Chem., Vol., (997)

4 INTEGRALS OF s AND p TYPE GTFS Eletron repulsion integrals The formulae are simplified by defining a set of intermediate funtions. The entres P and Q are defined in Equations (4) and (5). The argument of the inomplete Gamma funtion is t = (α + α )(α 3 + α 4 ) PQ (α + α + α 3 + α 4 ) To abbreviate these formulae the following onventions are used: S = α + α : S = α 3 + α 4 : = S + S = α + α + α 3 + α 4 The intermediate funtions are G 0000 (t) =F 0 (t) G i000 (t) =G 0i00 (t) = S (P i Q i )F (t) G 00i0 (t) = G 000i (t) = S (Q i P i )F (t) G ij00 (t) = S G 00ij (t) = S S F (t) S (P i Q i )(P j Q j )F (t) δ ij S (Q i P i )(Q j P j )F (t) δ ij F (t) S () G i0j0 (t) = G 0ij0 (t) =G 0i0j (t)=g i00j (t) = S S (P i Q i )(Q j P j )F (t)+ δ ijf (t) G ijk0 (t) = G ij0k (t) = S S S (P i Q i )(P j Q j )(Q k P k )F 3 (t) + [δ ij(p k Q k )+δ ik (P j Q j )+δ jk (P i Q i )]F (t) G 0ijk (t) = G i0jk (t) = S S S (P i Q i )(Q j P j )(Q k P k )F 3 (t) [δ ij(p k Q k )+δ ik (P j Q j )+δ jk (P i Q i )]F (t) 997 by John Wiley & Sons, Ltd. Eletron. J. Theor. Chem., Vol., (997)

5 70 C. A. BAXTER AND D. B. COOK G ijkl (t) = S S (P i Q i )(P j Q j )(Q k P k )(Q l P l )F 4 (t) S S [δ ij (P k Q k )(P l Q l )+δ ik (P j Q j )(P l Q l ) +δ il (P j Q j )(P k Q k )+δ jk (P i Q i )(P l Q l ) +δ jl (P i Q i )(P k Q k )+δ kl (P i Q i )(P j Q j )]F 3 (t) + 4 [δ ijδ kl + δ ik δ jl + δ il δ jk ]F (t) The eletron repulsion integrals an now be defined in terms of the G funtions. The multipliative term Λ appears in eah formula: ( ) S S Λ= () π S a S b S S d = ΛSabS 00 dg 00 P ia S b S S d = ΛSab i0 d G Sab 00 d Gi000 P ia S b P k S d = ΛS i0 abs 00 dg 00k0 + S i0 abs k0 dg S 00 ab S00 d Gi0k0 + S 00 ab Sk0 d Gi000 (3) P ia P jb S S d = ΛS ij ab S00 d G Sab i0 d G0j00 +S 0j ab S00 dg i000 + SabS 00 dg 00 P ia P jb P k S d = ΛS ij ab [Sk0 d G Sd 00 ] +Sab i0 d G0j00 + Sd 00 ] +S 0j ab [Sk0 d Gi000 + Sd 00 ] +Sab 00 d Gij00 + Sd 00 ] P ia P jb P k P ld = ΛS ij ab [Skl dg SdG k0 + SdG 0l +SdG 00 00kl ]+Sab[S i0 dg kl 0j00 + SdG k0 0j0l +SdG 0l 0jk0 + SdG 00 0jkl ]+S 0j ab [Skl dg i000 +SdG k0 i00l + SdG 0l i0k0 + SdG 00 i0kl ] +Sab 00 [Skl d Gij00 + Sd k0 Gij0l + Sd 0l Gijk0 +SdG 00 ijkl ] REFERENCES. E. Clementi (ed.) MOTECC (Modern Tehniques in Computational Chemistry), ESCOM, Leiden, annually up to 99.. C. A. Baxter and D. B. Cook, Int. J. Quant. Chem. 6D, 73 (996). 997 by John Wiley & Sons, Ltd. Eletron. J. Theor. Chem., Vol., (997)