One-Center Two-Electron Integral

The one-center two-electron integrals in the MNDO method are derived from experimental data on isolated atoms. Most were obtained from Oleari s work [L. Oleari, L. DiSipio, and G. DeMich-ells. Mol. Phys., 10, 97(1977)], but a few were obtained by Dewar using fits to molecular properties. 

For each atom there are a maximum of five one-center two-electron integrals, that is (ssiss), (ssipp), (spisp), (ppipp), and (ppip p ), where p and p are two different p-type atomic orbitals. It has been shown that the extra one-center two-electron integral, (pp Ipp ), is related to two of other integrals by 

If the five independent one-center two-electron integrals are expressed by symbols such as Gss, Gsp, defined above, then the Fock matrix element contributions from the one-center two-electron integrals are  

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The algorithms in Z[ DO/S are almost the same as those in ZlNDO/1, except of the one-center two-electron integral, b . ZINDO/S uses em pirical value of in stead of ii sin g ah initio vaine in terms of the Slater orbitals. 

The INDO model extends the CNDO model by adding flexibility to tlie description of the one-center two-electron integrals. In INDO, however, there continues to be only a single two-center two-electron integral, which takes on the value /ab irrespective of which orbitals on atoms A and B are considered. As already noted, this can play havoc witli the accurate representation of lone pair interactions. 

Thiel and Voityuk (1992, 1996) described the first NDDO model with d orbitals included, called MNDO/d. For H, He, and the first-row atoms, the original MNDO parameters are kept unchanged. For second-row and heavier elements, d orbitals are included as a part of the basis set. Examination of Eqs. (5.12) to (5.14) indicates what is required parametrically to add d orbitals. In particular, one needs and /ij parameters for the one-electron integrals, additional one-center two-electron integrals analogous to those in Eq. (5.11) (there are... 

The general guidelines are, as stated above, that the expression reduces to that for one-center two-electron integrals at zero distance, and tends toward e2jR at large distances (see Fig. 1). 

The molecular jr-type orbitals are normally expanded in terms of p-type STO, one per atom. The corresponding orbital exponents are chosen on the basis of independent considerations regarding the values of the one-center, two-electron integrals (see below). 

The intermediate neglect of overlap model (INDO) contains all the terms that CNDO contains, and all one-center two-electron integrals as well. The Fock matrix for the closed-shell then becomes... 

PM3 differs from AMI in that the former treats the one-center, two-electron integrals as pure parameters, as opposed to being derived from atomic spectroscopy. In PM3 all quantities that enter the Fock matrix and the total energy expression have been treated as pure parameters. 

To accomplish this large task of optimizing parameters an automatic procedure was introduced, allowing a parameter search over many elements simultaneously. These now include H, C, N, O, F, Br, Cl, I, Si, P, S, Al, Be, Mg, Zn, Cd, Hg, Ga, In, Tl, Ge, Sn, Pb, As, Sb, Bi, Se, Te, Br, and I. Each atom is characterized through the 13-16 parameters that appear in AMI plus five parameters that define the one-center, two-electron integrals. The PM3 model is no doubt the most precisely parameterized semiempirical model to date, but, as in many multiminima problems, one still cannot be sure to have reached the limit of accuracy suggested by the MNDO model. 

Eq. (6.6)), as in the PPP and CNDO methods, in INDO ZDO is not applied to those one-center two-electron integrals, (rj / ), with r, (t>s, and 4>u all on the same atom obviously, these repulsion integrals should be the most important. Although more accurate than CNDO, INDO is nowadays used mostly only for calculating UV spectra, in specially parameterized versions called DSDO/S and ZMDO/S [19],... 

These specifications define choices (a) and (b) in the MNDO formalism and thus constitute the MNDO model. The original implementation of the model may be summarized as follows [19]. Conceptually the one-center terms are taken from atomic spectroscopic data, with the refinement that slight adjustments of the parameters are allowed in the optimization to account for possible differences between free atoms and atoms in a molecule. Any such adjustments should be minor to ensure that the one-center parameters remain close to their spectroscopic values and thus retain their physical significance. The one-center two-electron integrals derived from atomic spectroscopic data are considerably smaller than their analytically calculated values, which is (at least partly) attributed to an average incorporation of electron correlation effects. For reasons of internal consistency, these integrals provide the one-center limit (/ AB = 0) of the two-center two-electron integrals A o- ),... 

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