Volume 2 17, number 1,2 CHEMICAL PHYSICS LETTERS 7 January 1994 A simple yet powerful upper bound for Coulomb integrals Peter M.W. Gill ’ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Department of Chemistry and Biochemistry. Massey University, Palmerston North, New Zealand Benny G. Johnson Department of Chemistry, Carnegie Mellon University, Pittsburgh, PA 15213, USA and John A. Pople Department of Chemistry Northwestern University, Evanston, IL 60208, USA Received 25 October 1993; in final form 9 November 1993 A new simple upper bound for Coulomb integrals is presented and shown to be significantly more powerful than the bound based on the Schwarz inequality. 1. Introduction The last decade has witnessed remarkable progress in the development and application of quantum chemistry [ 1,2 ] and readily available computer pro- grams can now be used to study chemical systems which, until a few years ago, would have been con- sidered prohibitively large. Moreover, there is every reason to believe that the next decade will be just as fruitful as the last. At present, the most computa- tionally demanding step in well-implemented Har- tree-Fock (HF) and density functional theory (DFT) calculations [ 31 #I is the treatment of the non-local electron-electron interactions which, within finite basis set methods [ 41, reduce to clas- sical Coulomb integrals (1) ’ E-mail address: p.m.gill@massey.ac.nz x1 CADPACS: The Cambridge Analytic Derivatives Package Is- sue 5, Cambridge, 1992. A suite of quantum chemistry programs developed by R.D. Amos with contributions from LL. alberts, J.S. Andrews, S.M. Colwell, N.C. Handy, D. Jayatilaka, P.J. Knowles, R. Kobayashi, N. Koga, K-E. Laidig, P.E. Maslen, C.W. Murray, J.E. Rice, J. Sanz, E.D. Simandiras, A.J. Stone and M- D. Su. between charge distributions P(r) and Q(r). It can not be over-emphasized, however, that HF and DFT calculations on very large systems are currently fea- sible only because the costs of such calculations do not obey the frequently cited Lo (N3) and Lo (N4) “laws”, where N is the size of the basis set employed. In fact, it is easy to show that, although the total number of Coulomb integrals (ERIs) which arise in large systems is 0 (N 3, or Co ( N4) (depending upon whether or not density-projection techniques [ 5 ] are used), the number of non-negligible ERIs is only O(N’). To take advantage of the fact that most of the ERIs in large systems are negligible, modem quantum chemistry programs use upper bound formulae to es- timate the magnitudes of ERIs in order to avoid computing and handling any that would be suffi- ciently small. To be maximially effective, such bounds must be both strong (i.e. not too conserva- tive) and simple (i.e. based only on information about P(r) and Q(r) individually). Although many sophisticated bounds have been proposed over the years, relatively few satisfy both of these criteria. In section 2 of this Letter, we develop three simple upper bounds on the ERI ( 1) and then propose a 0009-2614/94/S 07.00 0 1994 Elsevier Science B.V. All rights reserved. SSDZ OOOOS-2614(93)El340-M 65