Semirigorous LCAO—MO—SCF Methods for Three-Dimensional Molecular Calculations Including Electron Repulsion

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Abstract
A semirigorous LCAO—MO—SCF method including electron repulsion, for three‐dimensional closed‐shell molecular calculations has been derived. The method starts with the complete many‐electron Hamiltonian (in which interelectronic interactions are included explicitly) and the self‐consistent molecular orbital equations of Roothaan, then makes a series of systematic approximations for the integrals involved. There are several different levels of approximation which evolve depending on how restrictive one makes the conditions for neglecting a′(1)a″(1), where a′ and a″ are two orbitals on Atom A; and a′(1)b″(1), where a is an orbital on Atom A and b is an orbital on Atom B, A ≠ B, and for neglecting (a′c′′′| G | b″d′v), where a′ is an orbital on Atom A, b″ is an orbital on Atom B, c′′′ is an orbital on Atom C, d′v is an orbital on Atom D, and A, B, C, D are, in general, different atoms. The approximations for the core operators are defined and the core Hamiltonian matrix elements are presented in detail for several of the levels of approximation. The complete expressions for Fa′a′, Fa′a″, and Fa′b″ for closed‐shell systems in the two simplest of the approximative schemes are included in the body of this paper. The salient details of the more complicated schemes (corresponding to lesser and lesser neglect of the interaction of charge distributions) are indicated. These semirigorous LCAO—MO—SCF schemes outlined in this paper hold promise for less‐than‐rigorous yet better‐than‐empirical calculations for the whole gamut of three‐dimensional molecules from simple inorganics through complicated organics. An analogous treatment with the same approximations for the neglect of certain integrals can also be applied to an open‐shell LCAO—MO—SCF procedure.

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