Ruedenberg approximation

Equations (3.20) and (3.21) represent an identity in Hartree-Fock theory. (The Hellmann-Feynman and virial theorems are satisfied by Hartree-Fock wavefunc-tions.) The particular interest offered by (3.21) lies in the fact that 7 = 1 appears to be the characteristic homogeneity of both Thomas-Fermi [62,75,76] and local density functional theory [77], in which case (3.20) gives the Ruedenberg approximation [78], E = v,e,-, while (3.21) gives the Politzer formula [79], E = Vne-... 

D. M. Silver, E. L. Mehler, and K. Ruedenberg, Electron correlation and separated pair approximation in diatomic molecules. 1. Theory. J. Chem. Phys. 52(3), 1174-1180 (1970). 

This approximation has been investigated by McWeeny (1955, 1956), LSwdin (1956), Lykos (1961), Ruedenberg (1961), and Fischer-Hjalmars (1965), among others. 

Clearly, this approach can also be used in the case of Slater basis sets and, moreover, in the case of universal Slater basis sets. Ruedenberg and co-workers104 105 have shown that, within the molecular orbital approximation, this systematic approach gives a series of energy values which smoothly approach the Hartree-Fock limit. Similarly smooth convergence is to be expected in the calculation of correlated wave functions and expectation values, and will be the subject of future studies in this area.106... 

Cusachs reported (4) that the repulsive terms in the W-H model which assumes that electron repulsion and nuclear repulsion cancel nuclear-electron attraction, consist of one-electron antibonding terms only. Cusachs noted Ruedenberg s observation that the two-center kinetic energy integral is proportional to the square of the overlap integral rather than the first power. Cusachs used this to develop the approximation ... 

There are several other schemes that have been proposed for approximating the integrals that arise in molecular structure theory. In general, the approximations are made on the two-electron integrals, the N step, but care must always be taken to ensure that the balance of one- and two-electron terms that arise in the Fock equations is maintained. We will certainly not review all of them here, but will try in this section to highlight some of the more successful ones, and, in particular, those that are based on Ruedenberg s generalization of the simple Mulliken idea. 

As with the smaller compounds, reliable computational descriptions of methyl phenyl sulfoxide excited states are not available. Ground state computations are easily accessible for molecules of this size. At the RHF/6-31G(d,p) level, the HOMO is 7t with regard to the SO bond but delocalized throughout the whole n-system. The next two descending orbitals are localized on the phenyl and SO, respectively. (The sulfur lone pair is the HOMO-2 when the valence bond orbitals are approximated by the Edmiston-Ruedenberg method.) While the LUMO is extensively delocalized, the LUMO-fl is entirely localized on the phenyl ring. 

In order to avoid the laborious evaluation of the above integrals we propose a rational approximation which consists of a multipolar expansion the elementary charge distributions around atomic centres. This goal can be attained in two steps. First a Ruedenberg-type approximation [169] is introduced to reduce all multicentre integrals to at most two-centre ones and then the G(r, r ) functions are developed in bipolar series with respect to atomic centres. 

An analogous integral approximation was proposed by Ruedenberg (1951) for two-electron integrals. 

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