On Rüdenberg's integral approximations and their unrestricted and combined use in molecular orbital theories of Hartree-Fock type

• Published in: International journal of quantum chemistry Vol. 76; no. 2; pp. 148 - 160
• Main Author: Koch, Wolfhard
• Format: Journal Article
• Language: English
• Published: New York John Wiley & Sons, Inc 15.01.2000
• Subjects: extended Hückel theory (EHT); integral approximations according to Mulliken and Rüdenberg; neglect of diatomic differential overlap (NDDO); unrestricted (and restricted) Hartree-Fock molecular orbitals; zero differential overlap (ZDO)
• ISSN: 0020-7608; 1097-461X
• DOI: 10.1002/(SICI)1097-461X(2000)76:2<148::AID-QUA5>3.0.CO;2-R

Rüdenberg's well‐known letter of 1951 contains two implications which still have not been employed so far in numerical computations: (1) Whenever all types of attraction and repulsion integrals are subject to the Rüdenberg approximations in its simplest form partially known already from Mulliken, the attractive, the Coulomb, as well as the exchange part of the restricted Hartree–Fock–Roothaan equation can be led back to the calculation of corresponding diagonal elements, only. Using Rüdenberg's more elaborate concepts, which are invariant with respect to the rotation of local coordinate axes, the complete Fock‐matrix representation can be constructed equivalently from only its own diagonal blocks, each belonging to one atom. Similar formulas are valid for the unrestricted Hartree–Fock theory of Pople and Nesbet. (2) If, however, one prefers to calculate all types of one‐ and two‐center integrals exactly as suggested in Rüdenberg's headline, the original simplicity of both representations is lost. Instead, one is led to more complicated expressions, which arise from the fact that Rüdenberg's integral formulas, when applied to certain kinds of three‐center repulsion integrals, imply considerable oversimplifications. In spite of this critical result, Rüdenberg's ideas offer an extension together with an interpretation of the semiempirical Wolfsberg and Helmholz recipe (better known from Hoffmann's “extended Hückel” theory), on the one hand, and of the “neglect of differential overlap” schemes ZDO and NDDO, on the other, from a common point of view. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 148–160, 2000