Breakdown of the Independent Electron Approximation

Both the LCAO and NFE methods are complementary approaches to one-electron band theory, in which electrons are allowed to move independently of one another, through an averaged potential generated by all the other electrons. The true Hamiltonian is a function of the position of all the electrons in the solid and contains terms for all the interactions between these electrons, that is, all of the electron-electron Coulombic repulsions. Electronic motion is correlated the electrons tend to stay away from one another because of Coulombic repulsion. 

Consider a J-electron system, such as a transition metal compound. The valence d atomic orbitals do not range far from the nucleus, so COs comprised of Bloch sums of d orbitals and, say, O 2p orbitals, tend to be narrow. As the interatomic distance increases, the bandwidth of the CO decreases because of poorer overlap between the d and p Bloch SUMS. In general, when the interatomic distance is greater than a critical value, the bandwidth is so small that the electron transfer energy becomes prohibitively large. Thus, the condition for metallic behavior is not met insulating behavior is observed. 

The archetypal examples are the 3d transition metal monoxides NiO, CoO, FeO, MnO, VO, and TiO. AU of these oxides possess the rock-salt stmcture (which makes both cation-cation and cation-anion-cation overlap important). One-electron band theory correctly predicts the metallic behavior observed in TiO, which is expected of a 

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Centrifugal barriers have a profound effect on the physics of many-electron atoms, especially as regards subvalence and inner shell spectra. One aspect not discussed above is how energy degeneracies arising from orbital collapse can lead to breakdown of the independent electron approximation and the appearance of multiply excited states. Similarly, we have not discussed multiple ionisation (the ejection of several electrons by a single photon) enhanced by a giant resonance. Both issues will be considered in chapter 7. 

As a consequence of the breakdown of the independent particle approximation, it then emerged that the quantisation of individual electrons was not completely reliable. This was referred to in the classic texts on the theory of atomic spectra [309] as a breakdown in the I characterisation, and it manifests itself in the appearance of extra lines, which could not be classified within the independent electron scheme. The proper solution would, of course, be to revisit the initial theory and correct its inadequacies by a proper understanding of the dynamics of the many-electron problem, including where necessary new quantum numbers to describe the behaviour of correlated groups of electrons. Unfortunately, this plan of action cannot be followed through it would require a deeper understanding of the many-body problem than exists at present (see, e.g., chapter 10 for some of the difficulties). 

As long as n and remain good quantum numbers, the independent particle model and the central field approximation both apply, and quantum chaos does not arise. We can thus identify two situations where chaos could emerge the first is a complete breakdown in the independent electron approximation (due, for example, to strong correlations) and the second is a distortion of the central field approximation (due, for example, to a strong external field). 

Breakdown of the Born-Oppenheimer Approximation. The B-O approximation is based on the independence of the motions of nuclei and electrons. This is generally a reasonable assumption, except at the crossing point of two electronic states where a minor nuclear displacement is linked to the transition between two electronic states (Figure 3.31). 

Various suggestions for the likely effects of anhar-monicity include very short lifetime of the excited vibrational state (uncertainty broadening) and breakdown of the Born-Oppenheimer approximation (i.e. motion of electrons and nuclei are no longer independent, which might be viewed as leading to a multiplicity of potential functions). Special factors may operate in the liquid state in water the combination of distribution of bond lengths and... 

The application of PMO theory to photo-pericyclic reactions has been discussed by Dougherty 1971), and he draws attention to the consequences of the breakdown in the Born-Oppenheimer (BO) approximation in certain excited state processes. The BO approximation assumes that nuclear motions are very slow compared with the speed of electronic transitions, so that effectively the nuclei behave as if they were fixed in space (cf. Franck-Condon principle). Since electronic motions are thus independent of nuclear motions, then the potential energy versus reaction co-ordinate curves for the ground state and first-excited state processes should roughly parallel one another. However, the relative energies of the bonding wells (i,e, the hollows in the P.E. surface corresponding to reactant and product) are reversed on the excited state surface so that the transition state on this surface moves from one side of the reaction co-ordinate to the other. 

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