Energy levels, alternant hydrocarbons

FIGURE 2.7 Energy levels in odd- and even-alternant hydrocarbons. The arrows represent electrons. The orbitals are shown as having different energies, but some may be degenerate. 

The energy levels of an alternant hydrocarbon (AH), equal in number to the number of conjugated carbon atoms, are arranged symmetrically about ao (Fig. 9)... 

Fig. 9. Change of energy levels AH of alternant hydrocarbon with limiting levels coincide with the levels RM of the residual molecule.

The energy levels in an alternant hydrocarbon are symmetrical with respect to a. Therefore, any orbital Fp having an energy a + xp / has a corresponding orbital WLp of energy a — xp ft. The coefficients of the starred atoms are identical in Fp and for the non-starred atoms, they are equal but opposite in sign. 

Figure 2.23a gives a schematic representation of the frontier orbital energy levels of an uncharged odd alternant hydrocarbon. It is seen that... 

Figure 2.23. Odd alternant hydrocarbon radicals a) Schematic representation of the frontier orbital energy levels and of the various configurations that are obtained by single excitations from the ground configuration o. (It should be remembered that spin eigenstates cannot be represented correctly in these diagrams.) b) Energies of these configurations and effect of first-order configuration interaction.

Figure 2.24. Orbital energy levels of alternant hydrocarbon ions a) anions and cations of odd-alternant systems, and b) radical anions and radical cations of even-alternant systems in the HMO approximation and c) in the PPP approximation.

Acyclic hydrocarbon Linear hydrocarbon Alternant hydrocarbon Non-alternant hydrocarbon Topological matrix Topological MO MO energy level... 

In an alternant hydrocarbon the MO energy-levels are symmetrically paired about an appropriate zero (a) such that if , = a -I- kp is a root of the secular equations then e /+, = a — kp is also a root there is thus a complementary character about the molecular-orbital energies . 

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. 

The obvious and logical extension to the discussion so far outlined in this chapter is to ask whether, in the case of, say, a general, non-alternant hydrocarbon, it is possible and legitimate to perform an iterative calculation in which both Coulomb integrals ( 7.2 and 7.3) and resonance integrals ( 7.4) are varied simultaneously. In such a scheme, calculations based on relations (7-6) and (7-8) would be carried out in an iterative fashion such that the one-electron Hamiltonian-matrix [ar, 0rJ furnished qr and p identical with those which had served to calculate the particular ar - and / elements in question such LCAO-MO coefficients and energy-levels as were derived from this process would then in principle be truly self-consistent , in the sense implied. This approach has been called1122 the self-consistent Huckel-method or the / a/co" method . 

We may also note in passing that the Coulson-Rushbrooke Pairing -Theorem for the energy levels of alternant hydrocarbons (much discussed in Chapter Six and in Appendix D) is nicely illustrated by Fig. B2 (c/also... 

By an odd-alternant hydrocarbon is usually meant a free radical, such as the benzyl radical (Fig. 6-3). In this example there are seven carbon-atoms, hence seven atomic-orbitals and seven molecular-orbitals which must accommodate the seven jt-electrons. Of the seven energy-levels, six of them will occur quite naturally in pairs, according to the Theorem we have just proved, as shown schematically in Fig. 6-4. How can the seventh be... 

My friend Leszek Stolarczyk remarked on still another kind of symmeby in chemistry (in an impublished paper). Namely, the alternant hydrocarbons, defined in the Hueckel theory, despite the fact that they often do not have any spatial sjfmmetry at aU, have a S)mimetric energy level pattern with respect to a reference energy. We meet the same feature in the Dirac theory, if the electronic and positmiic levels are considered. This suggests a underlying, not yet known, internal reason common to the alternant hydrocarbons as viewed in the Hueckel theory and the Dirac model, which seems to be related somehow to the notion of supersymmetry. 

Theorem % The eigenvalues of an alternant are symmetrically distributed about the zero energy level. The corresponding eigenfunctions also show a mirror relationship, except for a difference of sign (only) in every other atomic orbital coefficient. The total charge density at any carbon atom in the neutral alternant hydrocarbon equals unity. 

Alternant hydrocarbons exhibit certain symmetry properties which are reflected by the special arrangement of their orbital energy levels and the specific form of their molecular orbitals. These special properties are often referred to as the pair-theorem or, using the terminology of physics, it is called the particle-hole symmetry. It holds under the first-neighbor approximation in the 7c-electron system. In the name of this symmetry, the term particle refers simply to the electrons of the molecule. We say that the occupied levels are filled with particles. 

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