Free, electron molecular orbital theory

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. 

Space does not unfortunately permit more than a mention of the free-electron molecular orbital theory and its application to the spectra of unsaturated hydrocarbons and heteromolecules. The very recent calculations of Ham and Ruedenberg on unsaturated hydrocarbons cover a more extensive range than the LCAO calculations of Pariser, with which they agree very well. It seems fair, however, to say that in spite of brilliant exploratory work in this field the free-electron theory in its present form still lacks foundations as secure as those which have now been provided for the LCAO theory. A particular difficulty in the free-electron molecular orbital theory is the proper inclusion of electron repulsion various ways have been devised of introducing it into the theory but the validity of these expedients still rests on goodwill rather than on rigor. Nevertheless the free-electron theory, in the hands of Platt and Kuhn, has already pointed the way to a sound theory of the spectra of linear and branched systems, and there seems little doubt that the next few years will witness advances in the theory of d)re spectra - as important as those which have already occurred in the theory of simpler molecules. 

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-... 

There are other methods. For a discussion of the free-electron method, see Streitwieser Jr., A. Molecular Orbital Theory for Organic Chemists Wiley NY, 1961, p. 27. For the nonpairing method, in which benzene is represented as having three electrons between adjacent carbons, see Hirst, D.M. Linnett, J.W. J. Chem. Soc., 1962,1035 Firestone, R.A. J. Org. Chem., 1969, 34, 2621. 

The free-electron model is a simplified representation of metallic bonding. While it is helpful for visualizing metals at the atomic level, this model cannot sufficiently explain the properties of all metals. Quantum mechanics offers a more comprehensive model for metallic bonding. Go to the web site above, and click on Web Links. This will launch you into the world of molecular orbitals and band theory. Use a graphic organizer or write a brief report that compares the free-electron and band-theory models of metallic bonding. 

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. 

We know that not all solids conduct electricity, and the simple free electron model discussed previously does not explain this. To understand semiconductors and insulators, we turn to another description of solids, molecular orbital theory. In the molecular orbital approach to bonding in solids, we regard solids as a very large collection of atoms bonded together and try to solve the Schrodinger equation for a periodically repeating system. For chemists, this has the advantage that solids are not treated as very different species from small molecules. 

Chapter 2 introduces the band theory of solids. The main approach is via the tight binding model, seen as an extension of the molecular orbital theory familiar to chemists. Physicists more often develop the band model via the free electron theory, which is included here for completeness. This chapter also discusses electronic condnctivity in solids and in particular properties and applications of semiconductors. 

When the unpaired electron is delocalized over a number of atoms, molecular orbital theory must be applied to obtain a molecular description of the resulting magnetic species. In this situation there is less opportunity for substantial contributions from L, and in general the more delocalized the electron the more like a free electron it appears. In some cases, the electron is delocalized over only a few atoms, and in these cases modest contributions from L are expected, especially if one of the atoms is a transition metal. If more extensive delocalization is present, or if all the atoms involved are light, only small contributions (e.g., from 2fi orbitals) may be observed. 

It is essential to have tools that allow studies of the electronic structure of adsorbates in a molecular orbital picture. In the following, we will demonstrate how we can use X-ray and electron spectroscopies together with Density Functional Theory (DFT) calculations to obtain an understanding of the local electronic structure and chemical bonding of adsorbates on metal surfaces. The goal is to use molecular orbital theory and relate the chemical bond formation to perturbations of the orbital structure of the free molecule. This chapter is complementary to Chapter 4, which... 

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. 

The spectra of benzene and naphthalene have a strong family resemblance (see Table 9). From a theoretical viewpoint this is expressed most clearly in the free-electron form of molecular orbital theory (Ham and Rudenberg, 1956b Klevens and Platt, 1949) which... 

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