Degenerate NBMOs

Some rules have been put forward to design molecules with degenerate NBMO s. Following the proposal of Longuet-Higgins, an alternant hydrocarbon has at least N-2T singly occupied NBMO s, where N is the number of carbon atoms and T is the maximum number of double bonds occurring in any resonance structure [383, 396]. These rules have later been unproved on the... 

Fig. 5.14 Schematic representation showing that the original Dsh symmetry of TMM is reduced to a lower one of C2v due to J-T effect for TMM so as to split the doubly degenerate NBMOs of e" into and b orbitals, together with plots of the latter two orbiuils projected to the x-z plane. The computations of 2 and bi MOs were performed using the B3LYP method with 6-31+G(d,p)...

The x-band in malachite green arises from an NBMO—>n transition, so that 3- and 4-substituents affect the energy of the excited state only and bring about spectral shifts of the first absorption band which vary linearly with the appropriate Hammett substituent constants. Thus, electron-withdrawing groups cause bathochromic shifts of the x-band whereas donor substituents cause hypsochromic shifts (Table 6.6) [64,67]. The 3-band arises from a n—>n transition [68] so that substituent effects are less predictable. As the donor strength of the 4-substituent increases, however, the 3-band moves bathochromically and eventually coalesces with the x-band - at 589 nm in the case of crystal violet (6.164), which possesses two NBMOs that are necessarily degenerate [69]. 

In order to find the energy x, we use first order perturbation theory. The degenerate orbitals involved are the doubly occupied AO s of Cl- and I-, and the empty NBMO of ArCHa. The necessary matrix elements are found as before in terms of the CC1 and Cl resonance integrals fta, ft. Solving the resulting three-row secular equation, we find for the perturbed energies—... 

On the basis of the simple Hiickel theory, the two NMBOs of both 33 and 34 will be degenerate. Hund s rule therefore dictates a triplet state. However, Borden and Davidson argued that since the electrons in the two NBMOs of 33 are specially isolated and have reduced the electrostatic repulsion, the singlet state will not violate the Pauli s exclusion principle and will be energy competitive with the triplet... 

Section 4.2, any normalized linear combination of degenerate wavefunctions is an equally valid wavefunction. The linear combinations of the NBMOs with the smallest possible amount of local overlap for 3 and 17 are shown in Figure 5.14. Those of 3 are separated entirely in space to the starred and unstarred sets of atoms. Hence 3 is a disjoint biradical. The same cannot be achieved by any linear combination of the NBMOs of 17 it is classified as a nondisjoint biradical. 

In NH, two valence MOs, corresponding to the N-H bond (2a) and lone pair on nitrogen (3a), are occupied by electron pairs. Two more valence electrons must be distributed between two degenerate, nonbonding molecular orbitals (NBMOs), Tlx, Ky, which consist of the 2px and 2py AOs on... 

Closer consideration of the NBMOs of these systems explains the differing spin preferences. For a degenerate pair, any linear combination of the MOs is acceptable. For CBD, we can make a linear combination of the NBMOs such that they have no atoms in common (Figure 14.32 A). One NBMO is confined to atoms 1 and 3, while the other is confined to atoms 2 and 4. Such orbitals are said to be disjoint they occupy different sets of atoms. In contrast, it is not possible to find a linear combination of the NBMOs of TMM that are disjoint (Figure 14.32 B). No matter what you do, there will always be atoms in common, and the NBMOsare termed non-disjoint. 

When two odd AH radicals unite, their NBMOs are necessarily degenerate and contain between them just two electrons. Here then there is a first-order change in n energy due to union (Fig. 3.8). Since first-order perturbations are much larger than those of second order, the union of odd AHs can be treated in terms of first-order perturbation theory alone. To this approximation, the energy of union arises entirely from interaction of the NBMOs (Fig.3.8). 

The even AH R can be constructed by union of methyl with the odd AH R derived from R by removal of atom k (Fig. 3.23a). To a first approximation, this union involves only an interaction between the NBMO Oq of R and the 2p AO ij/ of methyl (Fig. 32.3b). Since these orbitals are degenerate, the interaction between them is analogous to that between two hydrogen atoms when they combine to H2 (Section 1.5 p. 13). The resulting bonding MO S in Fig. 3.23(b) will then be composed equally of the two interacting orbitals. 

We can form (22) from the cation ArCH2 and the ions X (23), the relevant AOs of X" (0i and < 2) being doubly occupied. Let us assume as a first approximation that X has the same Coulomb integral as carbon. In that case, there will be first-order perturbations between the AOs of X and the NBMO of ArCH2 since all three will be degenerate. 

The combination of radicals can therefore be treated, like union, in terms of an interaction between a pair of orbitals, either NBMOs or AOs. In the case of hydrocarbons, the orbitals will be degenerate in the case of compounds containing heteroatoms, they will not be. However, the same situation will still hold, the relationship being similar to that between the homopolar molecule H2 and the heteropolar ion HHe" (Sections 1.5 and 1.7). 

In the union of heptatrienate cation with CH3" (Fig. 7.9e), the NBMO of the cation and the 2p AO of CH3" are degenerate. The interaction between the orbitals is consequently large, so the HOMO of the product, octatetraene, lies well below the nonbonding level. In tropylium, on the other hand, the LUMO is antibonding (see Section 3.13). The interaction of this antibonding MO with the 2p AO of CH3 on union of the two (Fig. 7.9f) is consequently much less. The HOMO of heptafulvene therefore lies much closer to the nonbonding level than does the HOMO of octatetraene. The ionization potential of heptafulvene must therefore be much less than that of octatetraene. 

Also depicted in Figure 6 is the Ai state of 2,4,6-cyclohepatrienylidene, which is a possible transition state for racemization of the conjugated cyclic allene, 1,2,4,6-cycloheptetraene. In the Aj state of 2,4,6-cyclohepatrieny-lidene two electrons occupy the o NBMO, and there are two unfilled n MOs, 2a2 and 3bi that are nearly degenerate. Excitation of one electron from the at o NBMO can thus lead to two triplets, and fi, and two open-shell singlets, A2 and Bi. 

The two degenerate (e") NBMOs of Dj/, TMM are shown schematically in Figure II. It can be seen that the NBMOs have atoms in common. Because the e" NBMOs of TMM are nondisjoint. it can be predicted that the ground state of Dy/, TMM should be a triplet ( A ) in which one electron occupies each e" NBMO. Experiments by Dowd have, indeed, found the triplet to be the ground state of TMM. ... 

Group theory shows that not only the NBMOs but also the two lowest singlet states of Dy, TMM are degenerate ( ) The simplest wave functions for the two components of have the form. 

The violations of Hund s rule that ate predicted for CBD and for Dg COT are of particular interest, because, unlike the NBMOs of TMB, symmetry requires that the NBMOs of D h CBD and Dg COT be degenerate. Therefore, at these geometries CBD and COT are both predicted to violate the strictest version of Hund s rule. ... 

---