Wolfsberg-Helmholtz approximation

H is the one-electron operator and i is the Slater basis set function (2s, 2p). The diagonal elements of Htj (Hit) are approximated as the valence state ionization potentials and the off-diagonal elements Htj are estimated using the Wolfsberg-Helmholtz approximation,... 

The exchange integrals, / are evaluated by representing them as functions of the Coulomb integrals, Hii and the overlap integrals. One such approximation is known as the Wolfsberg-Helmholtz approximation, which is written as... 

It is clear that such a Hamiltonian can only rationalize a Huckel-type Hamiltonian (eventually in a non-minimal basis set) and that it essentially leads to a pure Mulliken-type approximation for the bicentric integrals (which play the key role in the construction of the bond through electron delocalization). It is known that such approximations usually fail an empirical k parameter is used in the proportionality of the hopping integrals to the overlap and atomic energies (Wolfsberg-Helmholtz approximation). 

These results show that it is necessary to recalibrate the scale of I.S, in terms of total s-electton density proposed in ref. (1), Preliminary results were obtained by calculating the 4s orbital occupation using the principle of electtonegativity equalization and from the relation between N.E. and I.S. for some Fe(III) complexes. A scale was obtained with the values +0,10 cm/sec for the relative I.S. of F ion (instead of +0,055 cm/sec in the previous calibration) and +0,14 cm/sec for the Fe ion. Table II compares the values of 4s occupation given by the two calibrations with that calculated by molecular orbitals with the Wolfsberg-Helmholtz approximation for the tetrahedral [FeCl ]" complex (8). 

Although the energy of a chemical bond as a function of internuclear distance can be represented by the potential energy curve shown in Figure 3.2, neither the Wolfsberg-Helmholtz nor the Ballhausen-Gray approximation is a function that possesses a minimum. However, the approximation of Cusachs is a mathematical expression that does pass through a minimum. 

The extended Hiickel theory calculations, used in this work and discussed below, are based on the approaches of Hoffmann Although VSIP values given by Cusachs, Reynolds and Barnard were explored for use as the Coulomb integrals, the VSIP values obtained from a Hartree-Fock-Slater approximation by Herman and Skillman were consistently used in the present EHT calculations by this author. Both the geometric mean formula due to Mulliken and Cusachs formula ) were considered for the Hamiltonian construction, but the Mulliken-Wolfsberg-Helmholtz arithmetic mean formula was chosen for use. 

The more difficult resonance integrals to approximate are the off-diagonal ones. Wolfsberg and Helmholtz (1952) suggested the following convention... 

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