Molecular orbitals physical significance

The analytical determination of the derivative dEtotldrir of the total energy Etot with respect to population n, of the r-th molecular orbital is a very complicated task in the case of methods like the BMV one for three reasons (a), those methods assume that the atomic orbital (AO) basis is non-orthogonal (b), they involve nonlinear expressions in the AO populations (c) the latter may have to be determined as Mulliken or Lbwdin population, if they must have a physical significance [6]. The rest of this paper is devoted to the presentation of that derivation on a scheme having the essential features of the BMV scheme, but simplified to keep control of the relation between the symbols introduced and their physical significance. Before devoting ourselves to that derivation, however, we with to mention the reason why the MO occupation should be treated in certain problems as a continuous variable. 

It appears from the description of radical ions in Sects. 1 and 3 that redox reactions can significantly change the chemical and physical properties of conjugated 7r-systems. Whether the extended jc-species are treated within molecular orbital theory or within band-structure theory, the inherent assumption in these concepts is that an electron transfer is reversible and does not promote subsequent chemical reactions. While inspection of cyclic voltammetric waves and the spectroscopic characterization of the redox species provide reliable criteria for the reversibility of an electron transfer and the maintenance of an intact (T-frame, it is generally accepted that electron transfer, depending on the nature of the substrate and on the experimental conditions, can also initiate chemical reactions under formation or cleavage of er-bonds [244, 245],... 

The actual sign ("phase") of the molecular orbital at any given point r of the 3D space has no direct physical significance in fact, any unitary transformation of the MO s of an LCAO (linear combination of atomic orbitals) wavefunction leads to an equivalent description. Consequently, in order to provide a valid basis for comparisons, additonal constraints and conventions are often used when comparing MO s. The orbitals are often selected according to some extremum condition, for example, by taking the most localized [256-260] or the most delocalized [259,260] orbitals. Localized orbitals are often used for the interpretation of local molecular properties and processes [256-260]. The shapes of contour surfaces of localized orbitals are often correlated with local molecular shape properties. On the other hand, the shapes of the contour surfaces of the most delocalized orbitals may provide information on reactivity and on various decomposition reaction channels of molecules [259,260]. 

A fundamental postulate of quantum mechanics is that atoms consist of a nucleus surrounded by electrons in discrete atomic orbitals. When atoms bond, their atomic orbitals combine to form molecular orbitals. The redistribution of electrons in the molecular orbitals determines the molecule s physical and chemical properties. QM methods do not employ atom or bond types but derive approximate solutions to the Schrodinger equation to optimize molecular structures and electronic properties. QM calculations demand significantly more computational resources than MM calculations for the same system. In part to address computer-resource constraints, QM calcu-... 

L. Schafer was the first to begin to use in the late 1970s the results from molecular mechanics and ab initio calculations to supplement GED structure analysis. He also coined the acronym MOCED (Molecular Orbital Constrained Electron Diffraction) for such a combination [211,212]. The major problem facing the direct incorporation of ab initio results (r structure) into electron-diffraction structure analysis is the same as it was for the (GED + MW) combination (see Sec. IV B) the different physical significance of the parameters involved. 

For molecules of chemical interest it is not possible to calculate an exact many-electron wave function. As a result, we have to make certain approximations. The most commonly made approximation is the molecular orbital approximation, which is outlined in the next section. Within such a framework, it is useful to define various levels of computational method, each of which can be applied to give a unique wave function and energy for any set of nuclear positions and number of electrons. If such a model is clearly specified and if it is sufficiently simple to apply repeatedly, it can be used to generate molecular potential energy surfaces, equilibrium geometries, and other physical properties. Each such theoretical model can then be explored and the results compared in detail with experiment. If there is sufficient consistent success, some confidence can then be acquired in its predictive power. Each such level of theory therefore should be thoroughly tested and characterized before the significance of its prediction is assessed. 

In Figures 11 and 12, are depicted some examples of molecular orbital formation from separate atomic orbitals. The illustrations are of surfaces like those of the atomic orbitals we drew in chapter 3 they are of greater physical significance than the actual orbitals themselves. Again we will stress the point that the boundary surfaces are functions of whereas the... 

The idea of revolutionary progress in certain periods as developed by Kuhn [9] and very recently by McAllister [10], has some bearing on what I will discuss. It will be argued that theoretical organic chemistry has known three periods of dramatic change. The first of these periods (1850-1875) witnessed the birth of the structural formula and its development from formal representation to a reflection of physical reality. The second (1910-1935) saw the advent of quantum mechanics and the concepts of the electron pair, resonance and mesomerism, and hybridisation. In the third one (1955-1980), already mentioned, it is perhaps the succesful application of molecular orbital theory to chemical reactions, made possible by a very fruitful interplay of calculations and concepts, which is most significant. 

For systems that contain only one electron there is no difference in the molecular-orbital and the total electronic wave function. For many-electron systems, however, there is a considerable difference. It should be noted that for many-electron systems it is only the symmetry of the total wave function which has physical (and chemical ) significance. This quantity is the only observable quantity. ... 

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