Orbital perimeter

Thus, l,6-methano[10]annulene (77) and its oxygen and nitrogen analogs 78 and 79 have been prepared and are stable compounds that undergo aromatic substitution and are diatropic. For example, the perimeter protons of 77 are found at 6.9-7.3 5, while the bridge protons are at —0.5 5. The crystal structure of 77 shows that the perimeter is nonplanar, but the bond distances are in the range 1.37-1.42A. It has therefore been amply demonstrated that a closed loop of 10 electrons is an aromatic system, although some molecules that could conceivably have such a system are too distorted from planarity to be aromatic. A small distortion from planarity (as in 77) does not prevent aromaticity, at least in part because the s orbitals so distort themselves as to maximize the favorable (parallel) overlap of p... 

The chemical behavior of an atom is determined by the electrons that are accessible to an approaching chemical reagent. Accessibility, in turn, has a spatial component and an energetic component. An electron is accessible spatially when it occupies one of the largest orbitals of the atom. Electrons on the perimeter of the atom, farthest from the nucleus, are the first ones encountered by an incoming chemical reagent. An electron is accessible energetically when it occupies one of the least stable occupied orbitals of the atom. Electrons in less stable orbitals are more chemically active than electrons in more stable orbitals. 

Both singlet and triplet states are generated by the orbital promotion of an electron, n- -it transitions are totally allowed. These energy values can also be calculated from HQckel molecular orbital (HMO) method. For benzene, the free electron perimeter model has been found to be useful. The energy levels and nodal properties of benzene molecule are given in Figure 2.19. 

In contrast, CIDNP results indicate that the radical cations of barbaralane (157, X = C = 0) and semibullvalene (157, X = —) correspond to the elusive structure type with a single minimum [391, 424]. The spin density resides primarily on the termini (C-2,4,6,8) of the twin allyl moieties, whereas the remaining (internal) carbons of the 5 jr-electron perimeter have negative spin density. This spin density distribution reflects the coefficients of orbital 158, the HOMO of a bis-homoaromatic structure (Fig. 32) [424], More recently, ESR results have confirmed this assignment [392, 393],... 

A stationary state of an electron, considered as a wave phenomenon, signifies a stationary wave motion, whereby the perimeter of the orbit must therefore be a whole number of waves long. Thus discrete orbits of length X, 2X, 3X etc. are obviously produced as a result of which whole numbers already occur. However X is not equal for the different energy states on account of the connection between the wave length of the electron and the momentum, and so X also depends on the energy. 

The Wittig [75] and the McMurry reaction [76] are standard methods to form C=C double bonds. For example, the bridged biphenylene 98 was prepared in diluted solution by Wittig reaction [75a]. Vogel synthesized the bridged (14]annu-lene 100 with its phenanthrene perimeter from dialdehyde 99 by McMurry reaction [76 c]. 7t-Spherand 101 could be prepared likewise. Because of its a-orbitals... 

Platt s original perimeter model was a free-electron (FEMO) model based on a one-dimensional circular potential along which the jt electrons can move freely. The orbitals of an electron confined to such a circular ring are given by... 

The model becomes much more meaningful if the perimeter orbitals are described as linear combinations of n AOs (Moffitt, 1954a). Due to the properties of the cyclic point group C these LCAO MOs are determined completely by symmetry for regular polygons. (Cf. Cotton, 1971.) They may be written as... 

In the ground configuration of a perimeter with AN+ 2 n electrons all orbitals up to and including doubly occupied. Thus, the degenerate MOs nd 0 jv are the HOMOs and the degenerate MOs -n- are the LUMOs of the Ji system. If only HOMO LUMO excitations are considered, four singly excited configurations... 

Figure 2.11. The perimeter model of an (4N+2)-electron [njannulene, geometry on the left, energies of the MOs on the right. The angular momentum quantum number is given for each MO. The sense and magnitude of electron circulation and the resulting orbital magnetic moment are shown schematically in a perspective view. Orbital occupancy in the ground configuration and the four one-electron HOMO->LUMO excitations are indicated (by permission from Michl, 1978).

If the perimeter is uncharged, n = 4N + 2 and the configurations d>j and I>4 belong to the same irreducible representation b = e ,2 — e ,2 of the point group C . They differ in two spin orbitals and therefore interact only through the electron repulsion part iU,j) of the Hamiltonian. Consequently, the one-electron model fails and first-order configuration interaction has to be taken into account. f Q = , f d>4> is the interaction matrix element between configurations ] and d>4, the Cl matrix reads... 

The subscript A on a perimeter orbital defined in Equation (2.2) can be viewed as a quantum number related to the z component of orbital angular momentum associated with an electron in the orbital The selection rule for one-electron transitions between perimeter orbitals is similar to that familiar from atoms the quantum number k is allowed to increase or decrease by I. Thus promotions from to, and to, are allowed,... 

Bridging units tend to introduce new orbitals between the HOMOs and the LUMOs of the perimeter. This is especially true in cases such as in 16 where the bridge contains an odd number of n centers and possesses nonbonding orbitals. Excitations to or from these additional orbitals cannot of course be classified in terms of the unperturbed perimeter transitions. 

The change 5c, in the orbital energy is proportional to the product ( -c or to the square of the LCAO coefficients, respectively. The effect of introducing a bond between different fragments R and S (bridging or substitution) on the energy e, of the /-th perimeter orbital is approximated by the second-order expression... 

Figure 2.14. Frontier orbitals of benzene and [8)annulene dianion. The nodal planes ( ) are obtained from the polygons with 2k vertices inscribed into the perimeter, and the magnitude of the LCAO coefficients (indicated by the size of the circles) may be estimated from the location of the nodal planes.

An example is shown in Figure 2.15 where an even perturbation produces the orbitals of azulene and an odd perturbation (/ j) those of naphthalene from the perimeter orbitals of cyclodecapentaene. (Cf. Example 2.5.]) The energies of 0, and 0,. are not affected to the first order by the odd perturbation producing naphthalene, whereas 0s is stabilized and 0,- destabilized. Therefore, 0, becomes the HOMO and 0. the LUMO, and the HOMO-LUMO splitting AEhomo-lumo he same as for cyclodecapentaene. The HOMO- LUMO transition is referred to as L according to Platt. In azulene, on the other hand, 0, is destabilized and becomes the HOMO, whereas 0s- is stabilized and becomes the LUMO. The HOMO-LUMO splitting is markedly smaller than for cyclodecapentaene, and the HOMO-LUMO transition is of the Lb type. 

Figure 2.19. Orbital energy scheme of a 4 -eIectron perimeter. The reference configuration o is shown together with the single excitations 0- between the levels of the frontier orbitals doubly occupied (HO), singly occupied (SO), and unoccupied (LU) in Oo.

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