Degenerate interaction

When the interaction between xi and xi is riot strong (5,2 is small), some very useful mathematical approximations may be used to simplify equation 2.7. Using the first two expressions in Table 2.1, results in equations 2.8 and 2.9  

For any realistic case, e is negative and normally (Hq — e°S 2) is negative too (i.e., H 2 e S 2 ). Hence, is stabilized by the presence of the second term in equation 2.8, but 2 is destabilized by the second term in equation 2.9. Both levels are destabilized by the third term in equations 2.8 and 2.9. These results are shown pictorially in 2.2. The important result is that with respect to the atomic orbital at an 

Putting electrons into these resultant molecular orbitals allows calculation of the total interaction energy, A , on bringing together the two atomic orbitals xi and xi-Two important cases are shown in 2.3 and 2.4, the two-orbital two-electron case and the two-orbital four-electron case, respectively. These orbital interaction diagrams 

Since for S 2 0 the term (Hq — e°Si2) is negative, the two-orbital-two-electron interaction is stabilizing (i.e., A 0) but the two-orbital-four electron interaction is destabilizing (i.e., 0). 

The arrangement shown in 2.3 is not the only way to put two electrons into these two molecular orbitals. An alternative pattern is shown in 2.5 with a total interaction energy of A V2. The pattern in 2.5 is called the high-spin case, to be contrasted with the low-spin arrangement of 2.3, where they are paired. We shall 

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The trans compound cannot be formed by a concerted reaction from the 33t-7t state because the predominant level perturbation is the almost degenerate interaction of K(jr) and 0(jr), that would give cis geometry of product. Since the trans adduct must be formed via biradical intermediate (the only other possibility), it has usually been convenient to suppose that the cis adduct is also formed from the same biradical intermediate. Another choice of mechanism is that cis compound is formed via a concerted reaction, and the trans compound arises from a biradical pathway. In this case, the spin prohibition could be outweighed by two factors, the favorable geometry and the stabilizing first-order perturbation. 

The regioselectivity of each one of the previously cited reactions, Eqs. 29—31, is well-correlated by the interaction diagram. The degenerate interaction of the bonding levels is controlling, and whether the reaction is concerted or biradical the major orientation should be as shown in 19. The olefin 1,1-dichloroethylene was taken as the model for 1,1-dimethoxy-ethylene. 

There is a doubly degenerate interaction between the e-type orbitals of the nitrogen atom and those of the ligand hydrogen atoms to give the... 

Conformation 30 is destabilized by two severe repulsions the lone pairs are degenerate and they overlap very strongly. That eclipsed lone pairs overlap well seems obvious, but the interactions are no smaller in the trans case, as you can convince yourself by factorizing the cis and trans hybrids into their components.24 Conformation 31 is destabilized less, as each repulsion involves a non-degenerate interaction between a p lone pair and the sp2 hybrid of the adjacent oxygen. Furthermore, these orbitals are staggered, so their overlaps are relatively small. 

Since in the ZDO approximation K,j vanishes, the S and D states are degenerate. Interactions with additional configurations are needed in order to stabilize D to such an extent that E < E. ... 

Figure 5.4 Phase space mapping for the harmonic model. Two pairs of exactly degenerate interacting levels are shown. Since V2 is less steep than Vi, 0J2 < wi and (u( —Vi)< (v 2 — U2).

We will now apply these ideas specifically to the orbital. situations depicted in 2.3, 2.4, 2.7, and 2.8. Initially for the degenerate interaction of 2.2 for the orbital occupation 2.3... 

An example will show the application of some of the ideas introduced above. Let us start with the simple two-ccntcr-two-orbital problem described exhaustively in Chapter 2. In the language of perturbation theory these two orbitals experience a degenerate interaction for the case of H2 where the energies of each atomic orbital are the same. The result is an in-phase (bonding) combination and an out-of-phase (antibonding) combination, between the centers A and B. A more complicated example arises when there are two orbitals on A and one on B as when the orbitals of linear H3 are constructed from those of II2 + H (3.10). This is shown in Figure 3.1, where the relative phases of the orbitals have been chosen so that Sij and are positive. 

The molecular orbitals of linear II4 (5.4) may be constructed by interacting the orbitals of two H2 fragments as shown in I igure 5.4 where the orbitals and 0/(/= 1, 2) are arranged sucli that <0/10 -> > 0. The orbitals 0/ and 0/ enter into degenerate interactions as showm in 5.5 where each of the atomic orbital coeffi-... 

FIGURE 5.5. Second-order interaction of the levels - 4 produced by degenerate interactions between two collinear H2 units (5.5). 

Fig U RE 2.1. Molecular orbital dlsgram showing details of the degenerate interaction between the two atomics orbitals, xi and xs. 

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