The single-particle approximation

In the previous chapter we saw that except for the simplest solids, like those formed by noble elements or by purely ionic combinations which can be described essentially in classical terms, in all other cases we need to consider the behavior of the valence electrons. The following chapters deal with these valence electrons (we will also refer to them as simply the electrons in the solid) we will study how their behavior is influenced by, and in turn influences, the ions. 

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Now we see clear the problem while the new dot Hamiltonian (154) is very simple and exactly solvable, the new tunneling Hamiltonian (162) is complicated. Moreover, instead of one linear electron-vibron interaction term, the exponent in (162) produces all powers of vibronic operators. Actually, we simply remove the complexity from one place to the other. This approach works well, if the tunneling can be considered as a perturbation, we consider it in the next section. In the general case the problem is quite difficult, but in the single-particle approximation it can be solved exactly [98-101]. 

As already mentioned, the Weisskopf estimates are calculated in the single-particle approximation, supposing constant radial wave function. The observed transition rates sometimes strongly deviate from the estimated values. Nevertheless, the Weisskopf units (W. u.) are very useful for comparison. For example, enhanced transition rates (compared to the Weisskopf estimate) signal collectivity. 

Why is the single-particle approximation successful We can cite the following three general arguments ... 

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