Two-center Coulomb integrals

All non-zero integrals over atomic orbitals on the two centers are set equal, as in CNDO/INDO, to an averaged y. Thus, (s sj s s ) = (s s I PbPb) = (PaPa I PbPb) = The two-center Coulomb integrals, rather than being calculated from first principles using s orbitals as in CNDO/INDO, are approximated by an Ohno-Klop-man [K.Ohno, Theor. Chim. Acta, 2, 219 (1964) G. Klopman,... 

The sum A is over all the atoms in the quantum region and B is over all the atoms in the classical region. The two-electron and two-center Coulomb integral, Y b, is computed in MINDO/3 by... 

In the NDDO scheme, two-center contributions involve also the penetration effects defined in [56] as the difference between the potential induced on the AOs of the atom M by the nucleus of atom K and by the electron distribution around atom K. The corresponding integrals are taken to be proportional to two-center Coulomb integrals ... 

Clearly, the simplified form of is only valid for almost neutral molecules. In the case of strongly polar molecules and ions, the last term of Eq. (6.14) has to be taken into account, at least through its long-range components j%i. It may be remarked that if one introduces the two-center Coulomb integrals in their asymptotic form... 

In this expression, Kab (always positive) is the exchange integral, Jaa and Jab the one and two-center Coulomb integrals, respectively. 

As in atomic spectra, the form of Eq. (128) shows why the semi-empirical parameters of w-electron theory tend to include correlation. Zero differential overlap allows the Jif integrals in the H.F. part of Eq. (128) to be broken down into one- and two-center Coulomb integrals. The large difference between the non-and semi-empirical values of these is accounted for by introduc-ingi >i correlation into (2 J . This is justified because the "zero differential overlap can equally well be made in the e fs, reducing them to one- and two-center Coulomb correlation energies. 

Apart of specific counting of spin effects, other particularity of MINDO respecting the CNDO/INDO is that all the non-zero two-center Coulomb integrals are set equal and parameterized by the appropriate one center two electrons integrals Aa and As within the Ohno-Klopman expression (Ohno 1964 Klopman 1964)... 

So, in CNDO, all the TERIs are neglected except one- and two-center Coulomb integrals, (//// AA). Moreover, in order for this approximation to have rotational invariance (a calculation must give the same result if a molecule is rotated in the molecular coordinate system) it is necessary to make the further approximation that the two-electron integrals (pp XX) are independent of the specific orbitals x L or (pa cr) = 6xo- Iab-... 

Insertion of (6.11) and the analogous formula for the centers L,K in (6.8) expresses the four-center integral in terms of two-center Coulomb integrals and overlap integrals ... 

All parameters are either of a one-center nature or involve two-center integrals that fall off rapidly with distance (i.e., two-center Coulomb integrals are not parametrized). The reference ab initio data set was defined by STO-3G calculations, which quantitatively reproduce Mulliken charges, intemuclear distances, and relative energies derived from pure Slater orbital calculations. The error function minimized includes intemuclear distances, Mulliken charges, and energetics. Four parameters were defined for each atom, except for second major row atoms with d orbitals, which have a fifth parameter. Their effect on the integrals is as follows ... 

The Coulomb integrals, Eq. (10), can be calculated by an efficient numerical algorithm, such as the one adapted from the SMILES program for integrals with Slater-type orbitals, developed by Rico and collaborators. As a reasonable approximation, only the two-center Coulomb integrals need to be taken into account whereas the three and four-center integrals can be disregarded because they are much smaller. The matrix elements for the electron hamiltonian, due to its interaction with the hole, are written as... 

If only the two-center Coulomb integrals are taken into account, the matrices and F given by Eqs. (11) and (12), respectively, are block diagonal matrices. Furthermore, the diagonal elements of V-j and VV are... 

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