Free, electron molecular orbital theory energy

V-UV Application First Excited State of Linear Polyenes. The first electronic absorption band of perfect linear aromatic polyenes (CH)X, or perfect polyacetylene shifts to the red (to lower energies) as the molecule becomes longer, and the bond length alternation (BLA) would be zero. This was discussed as the free-electron molecular orbital theory (FEMO) in Section 3.3. If this particle-in-a-box analysis were correct, then as x > oo, the energy-level difference between ground and first excited state would go to zero. This does not happen, however first, because BLA V 0, next, because these linear polyenes do not remain linear, but are distorted from planarity and linearity for x > 6. 

The simplest semiempirical w-electron theory is the free-electron molecular-orbital (FE MO) method, developed about 1950. Here the interelectronic repulsions l/r,y are ignored, and the effect of the cr electrons is represented by a particle-in-a-box potential-energy function V" = 0 in a certain region, while V = oo outside this region. With the interelectronic repulsions omitted, in (16.1) becomes the sum of Hamiltonians for each electron hence (Section 6.2)... 

The electron-sea model is a simple depiction of a metal as an array of positive ions surrounded by delocalized valence electrons. Molecular orbital theory gives a more detailed picture of the bonding in metals. Because the energy levels in a metal crowd together into bands, this picture of metal bonding is often referred to as band theory. According to band theory, the electrons in a crystal become free to move when they are excited to the unoccupied orbitals of a band. In a metal, this re-... 

The first indication of the existence of a captodative substituent effect by Dewar (1952) was based on 7t-molecular orbital theory. The combined action of the n-electrons of a donor and a captor substituent on the total Jt-electron energy of a free radical was derived by perturbation theory. Besides the formulation of this special stabilizing situation and the quotation of a literature example [5] (Goldschmidt, 1920, 1929) as experimental evidence, the elaboration of the phenomenon was not pursued further, neither theoretically nor experimentally. 

The Fermi surface plays an important role in the theory of metals. It is defined by the reciprocal-space wavevectors of the electrons with largest kinetic energy, and is the highest occupied molecular orbital (HOMO) in molecular orbital theory. For a free electron gas, the Fermi surface is spherical, that is, the kinetic energy of the electrons is only dependent on the magnitude, not on the direction of the wavevector. In a free electron gas the electrons are completely delocalized and will not contribute to the intensity of the Bragg reflections. As a result, an accurate scale factor may not be obtainable from a least-squares refinement with neutral atom scattering factors. 

Like benzenoid hydrocarbons, pyridine-like heterocycles give well-developed two-electron waves on reduction at the dropping mercury electrode. The latter are polarographically much more reducible than the former. This can be explained easily in terms of the HMO theory It is assumed (cf. ref. 3) that the value of the half-wave potential is determined essentially by the energy of the lowest free 7r-molecular orbital (LFMO) of the compound to be reduced, and for models of hetero analogues this quantity is always lower than that for the parent hydrocarbons. Introduction of an additional heteroatom into the molecule leads to a further enhancement of the ease of polarographic reducibility.95 On the other hand, anodic oxidation of the heterocyclic compounds is so much more difficult in comparison with benzenoid hydrocarbons that they are not oxidizable under the usual polarographic conditions. An explanation in terms of the HMO theory is obvious. 

Molecular orbital theory predicts that if the total energy of the electrons in a molecule is lower (more negative) than the total electron energy of its constituent atoms, then that molecule will be stable with respect to the free atoms. 

Problem 8.28 (a) Apply the MO theory to the allyl system (cf. Problem 8.26). Indicate the relative energies of the molecular orbitals and state if they are bonding, nonbonding, or antibonding, (b) Insert the electrons for the carbocation C,H, the free radical C,H, and the carbanion CjH, and compare the relative energies of these three species. 

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