Robust and variational fitting: Removing the four-center integrals from center stage in quantum chemistry B.I. Dunlap * Code 6189, Theoretical Chemistry Section, US Naval Research Laboratory, Washington, DC 20375-5342, USA Received 3 December 1999; accepted 3 January 2000 Abstract We are interested in developing very fast and very accurate first-principles molecular dynamics. The approach that we are developing is robust and variational fitting. Fitting can be used effectively in quantum chemistry in many ways beyond optimizing and generating molecular orbitals, provided the fits are variational, i.e. compatible with the variational principle. Robust-variational fits, in addition to being variational, correct the target function to first order in the error made due to fitting. Thus robust and variational fits and fitting-basis sets can be generated in the same manner that molecular orbitals and atomic- basis sets are generated. Robust-variational resolution-of-the-identity methods are described as well as methods for analytically treating Kohn–Sham exchange and the case of canted spins in a general treatment of magnetism and with the spin–orbit interaction. Published by Elsevier Science B.V. Keywords: Density-functional theory; Variational fitting; Robust fitting; Gaussian basis sets; Resolution of the Identity With the incorporation of density-functional theory, quantum chemistry is undergoing rapid change. Density-functional methods are complex enough to appear to require numerical methods be added to the analytic methods of traditional quantum chemistry. Gaussian basis sets offer no advantages in numerical approaches to the molecular Schro ¨dinger equation. The strengths of the Gaussian basis set is that Gaussians enable the analytic evaluation of a vast array of matrix elements and the use of a large number of basis functions in chemically interesting regions of the molecule. The variational principle is important for traditional quantum chemistry and intrinsic accu- racy is not important at present. The intrinsic accuracy of an atomic Gaussian basis set is how well it solves the appropriate approximate Schro ¨dinger equation, usually the Hartree–Fock (HF) equation for its atom or ion. HF solutions for atoms can be obtained to arbitrary precision using numerical methods; thus we can compare the best Gaussian HF solutions with the exact result. That is done for hydro- gen through krypton in Fig. 1. The log of the relative error is the common logarithm of the ratio of the error in the Gaussian HF energy to the exact HF energy. This plot gives what is likely to be an upper bound on the precision with which these basis sets can be used to solve the molecular Schro ¨dinger equation. The upper line of triangular data points is the precision of the best basis sets from a 1985 compendium of Gaussian basis sets [1]. Partridge improved the preci- sion of Gaussian basis sets substantially in 1987 and 1989 [2,3]. His data for lithium to krypton are plotted as circles in the figure. Recently the Gaussian-basis- set precision for gallium to krypton has been further Journal of Molecular Structure (Theochem) 529 (2000) 37–40 0166-1280/00/$ - see front matter Published by Elsevier Science B.V. PII: S0166-1280(00)00528-5 www.elsevier.nl/locate/theochem * Tel.: +1-202-767-3250; fax: +1-202-767-1716. E-mail address: dunlap@alchemy.nrl.navy.mil (B.I. Dunlap).