Two-center terms

However, the CNDO method showed systematic weaknesses that were directly attributable to the approximations outlined above, so that it was superseded by the intermediate m lect of diatomic differential overlap (INDO) method, introduced by Pople, Beveridge, and Dobosh in 1967 [13]. The approximation outlined in Eq. (50) proved to be too severe and was replaced by individual values for the possible different types of interaction between two AOs. These individual values, often designated Cgg, Ggp, Gpp and in the literature, can be adjusted to give better agreement with experiment than was possible for CNDO. However, in INDO the two-center terms remain of the same type as those given in Eqs. (51) and (52) (again, there are many variations). This approximation leads to systematic weaknesses, for instance in treating interactions between lone pairs. 

The summation of Eq. (3.7) contains one- and two-center terms for which (/> and 0V are centered on the same, and on different nuclei, respectively. The two-center terms represent the overlap density in a bond they can only give a significant contribution to the density if and < v(r) have an appreciable value in the same region of space, and are therefore not important for distant atoms. 

Diamond (1966) has applied a filtering procedure in the refinement of protein structures, in which poorly determined linear combinations are not varied. In charge density analysis, the principal component analysis has been tested in a refinement of theoretical structure factors on diborane, B2H6, with a formalism including both one-center and two-center overlap terms (Jones et al. 1972). Not unexpectedly, it was found that the sum of the populations of the 2s and spherically averaged 2p shells on the boron atoms constitutes a well-determined eigenparameter, while the difference is very poorly determined. Correlation between one- and two-center terms was also evident in the analysis. 

To evaluate Eq. (1) in a way that is analytic, robust and variational, it is first necessary to divide the density among the atoms. That is easy in any LCAO approach, where the only problem is to how to assign centers to the cross (two-center) terms of the density. The computationally most efficient way is to multiply each atomic orbital by a /8, where a is the appropriate Xa scaling factor for that atom. Then the scaled, and thus partitioned, density may be written... 

The two-electron integrals pq kl] are < p(l)0fc(2) e2/ri2 0,(l)0j(2) > and may involve as many as four orbitals. The models of interest are restricted to one and two-center terms. Two electrons in the same orbital, [pp pp], is 7 in Pariser-Parr-Pople (PPP) theory[4] or U in Hubbard models[5], while pp qq are the two-center integrals kept in PPP. The zero-differential-overlap (ZDO) approximation[3] can be invoked to rationalize such simplification. In modern applications, however, and especially in the solid state, models are introduced phenomenologically. Particularly successful models are apt to be derived subsequently and their parameters computed separately. 

As a consequence of the NDDO integral approximation, the overlap integrals in the ab initio secular equations are replaced by Kronecker deltas, and the MNDO Fock matrix elements contain only one-center and two-center terms. They are defined as... 

The two-center term in these diagonal matrix elements contains the nuclear attraction and electron repulsion for aU the other atoms in the systems. In FH these were both set to the point-charge classical limit. Thus,... 

Figure 6 Left the two-center terms in the bond-order potential for different local coordinations, and right their influence on the effective pair potential, z is the local coordination.

Since P involves summation over atomic centers, this expression for <7s involves both one- and two-center terms. 

Fig. 3. The dihedral angle dependences of the two-center terms in absolute energy scale for the vicinal H H (a), H 0 (b) and 0 0 (c) interactions. .e. the interaction between the circled atoms in each figure). Note that the abscissa is cj) defined by the 0—C—C—0 bond sequence.

The gauche-oxygen effect in 1,2-DME can be described approximately by the one- and two-center terms for the OCH2—CH2O fragment. Although the electrostatic interaction, mainly of the 0---0 pair, disfavors the G-conformation. 

The dihedral angle dependence study of the two-center terms for the OCH2—CH2O fragment showed the unique nature of the 0 0 interaction in comparison with the H H and H 0 interactions predominant with small and exc in the former, predominant E with small 5 exc el the latter. 

It was soon obvious that ab initio methods would be impractical for the study of large polyatomic systems. Attempts were made to use empirically determined data to approximate the complicated integrals used in ab initio theory. All of the difficult three- and four-center integrals were ignored, and the one- and two-center terms were approximated using a mixture of functions based on atomic spectra and on formal theory. Procedures of this type, which have both experimental and theoretical components, are called semiempirical methods. 

The idea is to decompose the one-, two-, three- and four-center integrals into one- and two-center terms, put them into matrices, and read directly the numerically more Important (bonding) and less important (non-bonding) terms of interaction in molecules and molecular complexes. The method is convincingly Illustrated with water and methane plus water. Unfortunately the formulae are not free from errors. 

The three-center terras can be separated from the one- and two-center terms if the overlap between atomic orbitals is small.In the case of a minimal basis set, the one- and two-center terms of the matrix elements are equal to the DIM matrix elements obtained in the ZOAO approximation. Thus the three-center terms are those terms ignored by the DIM method.For example, in the simplest case, a symmetrical linear ACB system where each atom has a single valence electron of s S3nnmetry, the energy of the system is expressed as a sum of a DIM energy and a three-center term... 

---