Energy levels annulenes

Figure 2.15. The real form of the four frontier MOs of [10]annulene. The doubleheaded arrows indicate the cross-links that produce naphthalene (black arrows) and azulene (white arrows), respectively. First-order shifts are shown as vertical arrows on the orbital energy levels in the center (by permission from Michl, 1984).

This type of energy-level pattern was evident in the MO energy-spectrum of the alternant hydrocarbon, butadiene, discussed in Chapter Two ( 2.7) and in the energy levels of the [n]-annulenes (n even) of Fig. 5-2. 

A Matrix Proof of the Annulene Energy-Level Formula... 

The simplest and most elegant proof of the energy-level formula for annulenes (equation (5-2)) exploits the properties of circulant matrices. A circulant is a matrix of the form... 

As an illustration of these ideas, and of the 4p + 2 rule, let us consider the energy-level patterns, given by equation (5-2), in the annulenes, C H , where n runs from 3 to 8. These are shown in Fig. 5-2 in which the energy-level... 

The Hiickel Hamiltonian-matrix for an n-annulene (Fig. 5-1) is the matrix in the determinant found in equation (5-1), with the quantities ( — x), along the diagonal, replaced by zero (i.e. a). As pointed out in Appendix A, this matrix is isomorphic with the adjacency matrix, A(C ), of the correspondingly-labelled molecular graph furthermore, the eigenvalues of the matrix, A(C ), are the annulene energy-levels we require (expressed in units of fi and... 

One of the most interesting and important of these theorems is known as the Pairing Theorem. In his study of the polyenes, and annulenes, Huckel demonstrated [149] that thdr energy levels... 

Frost s circle A simple, geometrical approach to approximating the HMO energy levels for planar annulenes. The lowest energy MO has an HMO energy of a+ip. 

FIGURE 12.7. Generation Of the tt energy levels of napthalene by linking together a pair of atoms of 10-annulene. The energies are given in units of and no attempt has been made to represent the actual AO coefficients. 

Table 3. Computed distortion energies with their n and cr components, QMRE values for a series of annulenes, and singlet-triplet gaps A ST of the corresponding 2e-2c n bonds. The A ° values arise from the direct a-n partition of Eq. (1) in the ground states of the annulenes. The total distortion energies AEtot are all calculated at the 6-31G/7t-CI level. All energies are in kcal/mol. For all species, the distortion keeps constant nuclear repulsion between the atoms of the ring. The geometries of hetero-annulenes are described in Ref. [4]...

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