Neglect of overlap

Here the indices a and b stand for the valence orbitals on the two atoms as before, n is a number operator, c+ and c are creation and annihilation operators, and cr is the spin index. The third and fourth terms in the parentheses effect electron exchange and are responsible for the bonding between the two atoms, while the last two terms stand for the Coulomb repulsion between electrons of opposite spin on the same orbital. As is common in tight binding theory, we assume that the two orbitals a and b are orthogonal we shall correct for this neglect of overlap later. The coupling Vab can be taken as real we set Vab = P < 0. 

When dealing with carbon atoms, the coulomb integral (// ) is represented as a and the exchange integral (H, ) is represented as o. It is also assumed that interaction between nonadjacent atoms can be ignored so if i — j 15 2 the H,j = 0. Finally, in simple HMO there is complete neglect of overlap so... 

In the previous sections, we saw that, in most cases, a nonbonded attractive or repulsive interaction is enforced by both four electron destabilization and two electron stabilization. Hence, in order to simplify subsequent discussions, we shall adopt the OEMO model with neglect of overlap. Consequently, in the remainder of this work... 

In order to perform calculations on larger molecules in a reasonable amount of time, approximations are made, which may involve the neglect of certain terms, or the inclusion of experimentally determined parameters. The best known and simplest example of this level of approximation are Hiickel Molecular Orbital (HMO) calculations, which treat only pi-electrons, in conjugated hydrocarbons, with neglect of overlap (1). While obviously limited in use, HMO methods are still used in certain research applications. 

The different behavior of tertiary and quaternary carbon atoms seems to be due to either the complete neglect of overlap in these calculations or to polarization effects of the carbon-nitrogen bonds, Similar results are obtained for a series of 5-halouracils by plotting the 13C NMR chemical shifs versus 7r-electron charge densities calculated by the extended Hiickel theory [756], Though for several nitrogen heterocycles a better correla-... 

The difference between the lengths of the central CC bond in the equilibrium geometries of the radical cation of [ 1.1.1 Jpropellane (a one-electron bond, nearly the same length as in the neutral molecule) and of its radical anion (a three-electron bond, much longer than in the neutral molecule), well reproduced by ab initio calculations, emphasizes the inadequacy of the usual simplistic concepts of bonding based on Hiickel theory and neglect of overlap. An explicit introduction of overlap, however, permits a qualitative rationalization of the difference. 

Table 30 collects the effective -charges on the principal heteroatoms of the individual bases for the same mono-, di-, and trinucleotides of Table 29. These tables (see p. 37) provide the numerical evidence for the comments made above. The electron distributions in the individual bases are almost constant, independent of whether they participate in a polynucleotide or not. Of course, this is a conclusion valid only within the framework of the approximations in these treatments, including the omision of the orthophosphoric acid and the sugar and the neglect of overlap.

Ihe algebra considerably and is sufficiently accurate for most purposes. With complete neglect of overlap, our molecular wave functions become... 

Fig. 5.7 Energy levels for the H2 molecule with neglect of overlap. The quantity A ,. represents the difference in energy between the energy levels of the separated atoms and the bonding molecular orbital. It is equal to 458 kJ mol. ...

FIGURE 9. An enlargement of the (very narrow) ground state band. Curve A, complete neglect of overlap B orthogonalized basis functions and, C normalized with the overlap in the first zone. 

This is a semiempirical all-valence electron quantum mechanical method, apart from the Tr-approximation and the neglect of overlap integrals, as those of Hiickel molecular orbital (HMO) theory. The method reproduces, relatively well, the shapes and the order of the energy levels of molecular orbitals. To consider the overlapping, it is possible to describe the net destabilization caused by the interaction of the two doubly occupied orbitals, the effect of which is not reproduced by HMO theory. 

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... 

From equation (2-14), in the neglect-of-overlap case, this implies... 

Yet, even the hnite dimensional standard VB approach runs into a number of difficulties, such as the nightmare of the inner shells [11], neglect of overlap integrals, and the so-called V catastrophe (see, e.g. Ref. [12]). For this very reason, sometime during the second World War, VB theory was eclipsed by the computationally much more amenable molecular orbital (MO) method, relying on the independent-particle model (IPM), which reduces the V-electron problem to effectively a one-electron, though highly non-linear, one. A very important conceptual advance was achieved by the exploitation of the variation principle, which led to the formulation of Hartree-Fock (HF) equations... 

MINDO calculations, which pathologically favor biradical as opposed to concerted transition states, predict an unsymmetrical transition state for the 4 + 2 cycloaddition however, an early ab initio calculation revealed the presence of a symmetrical transition state with a biradical path roughly 2 kcal/mol higher in energy.The neglect of overlap in the MINDO framework may be a source of difficulty as pointed out by Houk. ... 

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