Free electron molecular orbital model

In the free-electron molecular orbital model, the electrons actually move in three dimensions. For 1,3-butadiene represent the electrons as particles in a three-dimensional box with a length in the x direction equal to 694 pm, width in the y direction equal to 268 pm, and height in the z direction equal to the width. Find the wavelength of the photons absorbed in the longest-wavelength absorption due to changes in the quantum numbers Uy and n. Explain why the representation as a one-dimensional box can successfully be used to understand the near-ultraviolet spectrum. 

Compound 41 can be expected to be the strongest 77-acid of the compounds 40, 41, and 43. It has been found84 that 41 is considerably more easily reduced polarographically than 40, and it is worth noting that the energy of the LFMO (lowest free 77-molecular orbital) of the model of compound 41 is substantially lower than that of 40 (Fig. 5). Much attention has been paid61 to the electronic structure of 44, and models of 49 and 50 have been studied by the HMO method. In structural formulas 49 and 50, X represents (CH3)2N, CH3, CF8, and CN. 

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. 

We start with some biographical notes on Erich Huckel, in the context of which we also mention the merits of Otto Schmidt, the inventor of the free-electron model. The basic assumptions behind the HMO (Huckel Molecular Orbital) model are discussed, and those aspects of this model are reviewed that make it still a powerful tool in Theoretical Chemistry. We ask whether HMO should be regarded as semiempirical or parameter-free. We present closed solutions for special classes of molecules, review the important concept of alternant hydrocarbons and point out how useful perturbation theory within the HMO model is. We then come to bond alternation and the question whether the pi or the sigma bonds are responsible for bond delocalization in benzene and related molecules. Mobius hydrocarbons and diamagnetic ring currents are other topics. We come to optimistic conclusions as to the further role of the HMO model, not as an approximation for the solution of the Schrodinger equation, but as a way towards the understanding of some aspects of the Chemical Bond. 

METALLIC BONDING (SECTION 12.4) The properties of metals can be accounted for in a qualitative way by the electron-sea model, in which the electrons are visualized as being free to move throughout the metal. In the molecular-orbital model the valence atomic orbitals of the metal atoms interact to form energy bands that are incompletely filled by valence electrons. Consequently,... 

When there is extensive delocalisation, it is more appropriate to use free electron-based molecular orbital models. 

This molecular-orbital model of metallic bonding (or band theory, as it is also called) is not so different in some respects from the electron-sea model. In both models Ihe electrons are free to move about in the solid. The molecular-orbital model is more quantitative than the simple electron-sea model, however, so many properties of metals can be accounted for by quantum mechanical calculations using molecular-orbital theory. 

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 Kronig-Penney model, although rather crude, has been used extensively to generate a substantial amount of useful solid-state theory [73]. Simple free-electron models have likewise been used to provide logical descriptions of a variety of molecular systems, by a method known in modified form as the Hiickel Molecular Orbital (HMO) procedure [74]. 

H2, N2, or CO dissociates on a surface, we need to take two orbitals of the molecule into account, the highest occupied and the lowest unoccupied molecular orbital (the HOMO and LUMO of the so-called frontier orbital concept). Let us take a simple case to start with the molecule A2 with occupied bonding level a and unoccupied anti-bonding level a. We use jellium as the substrate metal and discuss the chemisorption of A2 in the resonant level model. What happens is that the two levels broaden because of the rather weak interaction with the free electron cloud of the metal. 

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. 

In the four-orbital model (1 ), low-lying ir-ir states of free-base porphyrins (symmetry D2h) are considered as resulting from single electron excitation from a pair of nondegenerate highest occupied molecular orbitals (bi, bo) to a pair of nondegenerate lowest unoccupied molecular orbitals (ci, cg). In the case of symmetry D2h mutually perpendicular electric transition dipoles X and Y are not equivalent and, therefore, in the visible absorption spectra of free-base porphyrins two different electronic bands Qx(0>0) and Qy(0,0) are observed (Table 1 and Fig. 10). 

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. 

Table XII summarizes HMO quantities referring to models of sulfur compounds total 7r-electronic energy (W7), energies of the two highest occupied molecular orbitals (k2 and ky kx is referred to as HOMO in the text), energies of the two lowest free molecular orbitals (k-x and k 2 k-i is referred to as LFMO in the text), and the energy...

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