A quantitative study of the scaling properties of the Hartree–Fock method

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Abstract
Although it is usually stated that the Hartree–Fock method formally scales as N4, where N is the number of basis functions employed in the calculation, it is also well known that mathematical bounds computed with the Schwarz inequality can be used to screen and eliminate four-center two-electron integrals smaller than a certain threshold. In this work, quantitative data is presented to illustrate the effects of this integral screening on the scaling properties of the Hartree–Fock (HF) method. Calculations are performed on a range of carbon–hydrogen model systems, two-dimensional graphitic sheets, and three-dimensional diamond pieces, to determine the effective scaling exponent α of the computational expense. The data obtained in this paper for calculations including over 250 carbon atoms and 1500 basis functions shows two significant trends: (1) in the asymptotic limit of large molecules, α is found to be approximately 2.2–2.3, and (2) for molecules of modest size, α is still very much less than 4. Therefore, integral screening is quantitatively shown to substantially reduce the Hartree–Fock scaling from its formal value of N4.