Orbital Energies and Interaction Matrix Elements

At this point, we have completed the presentation of the key equations which will be crucial to the development of a predictive theory of molecular structure. These equations will form the basis for determining the relative stability of isomers, the relative stabilization of a cationic, radical or anionic center by substituents, etc. On the other hand, the differential expressions (9) to (12) will form the basis for determining how substitution affects the relative stability of isomers, the relative stabilization of cationic, radical and anionic centers, etc. It is then obvious that a working knowledge of Eqs. (1) to (6) presupposes a great familiarity with the key quantities involved in these equations, namely, orbital energies and interaction matrix elements. 

We shall first consider the effect of atomic replacement on sigma or pi orbital energies. 

The HOMO-LUMO gap calculated by an ah initio method using an ST0-3G basis set seems to support these ideas. A more definitive test will be possible after the nature of the lowest excited state in carbon unsaturated sterns is understood  

We shall next consider the effect of substitution on sigma and pi orbital energies. 

The equation describing the energy change of an orbital due to its interaction with a degenerate orbital is given below  

---

The extension of independent electron treatments—e.g. of the type proposed by Huckel for n systems —to sigma systems, and in particular to hydrocarbons, has a long and well-known history. The early treatments used an orthonormal basis of atomic or bond orbitals with parametrized coulomb energies and interaction matrix elements restricted to nearest neighbours only. The most attractive approximation of this kind, proposed by Hall and Lennard-Jones in 1951, is the equivalent bond orbital (EBO) model, which has been used extensively since, with variations due mainly to Lorquet, Brailsford and Ford, Herndon, Murrell and Schmidt, and Gimarc . The conceptual consequences of such a treatment, in particular the phenomenon of -conjugation in saturated hydrocarbons, have been discussed in detail by Dewar ... 

Table 5.2. The NBOs nB and oah and associated orbital energies (en and ea ) of binary B - HAH-bondedcomplexes (cf. Fig. 5.1), with Fna = (nB. F cTAH ) interaction matrix element (note that two oxygen lone pairs contribute to Id-bonding in H2CO- H3N)...

Consider the two systems CH2F—SH and CH2F—OH. According to our approach both are predicted to exist in a preferred gauche conformation. However, the extent to which the nx-o F interaction obtains in the two molecules may be subject to matrix element control simply because ns is a better donor than no but yields a smaller interaction matrix element with a F- The variation of these two effects may conceivably be comparable and subject to matrix element control due to the fact that the n—o orbital interaction involves well separated energy levels. Hence, one... 

H—F. In the crudest approximation, one may say that the orbitals of C, N, O, and F are all approximately the same size and therefore the interaction matrix element hab will be approximately the same size for any A- pair. The dominant factor determining the heterolysis energy therefore is the difference in orbital energies in the denominator, and one has directly the prediction (Figure 4.2) that ease of heterolytic cleavage for C—X is in the order C > N > > F. The C—C bond is least likely to dissociate heterolytically and the C—F bond the most likely. In an absolute sense, of course, heterolytic cleavage is not a likely process for any of these bonds in the absence of other factors, as discussed below. 

The acceptor ability may be improved in two ways, by lowering the energy of the a orbital and by polarizing the orbital toward one end. The first improves the interaction between it and a potential electron donor orbital by reducing the energy difference, i A — eB, the second by increasing the possibility of overlap and therefore increasing the value of the intrinsic interaction matrix element, hAB. 

In SHMO, the core energies of heteroatoms, X, are specified in terms of a and / , and the interaction matrix elements for p orbitals overlapping in fashion on any pair of atoms, X and Y, are specified in terms of / . Thus,... 

Answer to 6(a). We consider the radical to be derived by addition of an electron to N-chloropyridinium cation. The character of the radical is in doubt because it is uncertain whether the LUMO of the A-chloropyridinium cation will be one of the n orbitals, or orbitals involved in the a-type interaction, n and 3pcl, are both low in energy to start with and (b) the fact that a 3p rather than a 2p orbital is involved reduces the magnitude of the intrinsic interaction matrix element as explained in Chapter 4. The experimental evidence of Abu-Raqabah and Symons is consistent with a cr-type radical as shown in Figure 11.2. Occupancy of the o cl orbital would lead to a substantial reduction in the N—Cl bond order. The consequent lengthening of the N—Cl bond is accompanied by reduction of the a-a gap, which is seen as a red shift of the 0-0 UV absorption. 

---