Some considerations on U eff values entering the hubbard hamiltonian

One of the simplest ways to take into account correlations between electrons in solid state physics, beyond the usual one electron approximation, is to use the Hubbard Hamiltonian\documentclass{article}\pagestyle{empty}\begin{document}$$\begin{array}{*{20}c} {H = \sum\limits_{ij} {\beta _{ij} C^\dag _{i\sigma } + U\sum\limits_i {n_{i\sigma } n(1 - \sigma )} } } & {n_i = C^\dag _i C_i }\\ \end{array} $$\end{document}Although it corresponds to very severe approximations, since intersite correlations are neglected, and since the whole problem is reduced to a single parameter U , it is already very difficult to solve, and exact solutions exist in only a few cases. It is thus not desired to introduce more complications in the Hamiltonian; but it could be interesting to understand, at least qualitatively, which processes influence the value taken by parameter U . The question arises in concrete cases, such as transition metals or organic molecules; comparing experiments and approximate solutions of the Hubbard Hamiltonian has shown that, in most cases, the value of U that yields the best fit is very different from what would be expected from atomic considerations. For instance, if one wants to study correlations on atoms of type M , a pure atomic point of view would affect to U the change in energy U at in the redox‐type equation\documentclass{article}\pagestyle{empty}\begin{document}$$ 2{\rm M} \mathbin{\lower.3ex\hbox{$\buildrel\textstyle\rightarrow\over{\smash{\leftarrow}\vphantom{_{\vbox to.5ex{\vss}}}}$}} {\rm M}^{\rm + } + {\rm M}^ - $$\end{document} . However it seems that solid state effects play an important role and we discuss, in this paper, three of them: (i) the finite time that one electron spends on a given atom due to the delocalization of the electrons in a band (effect of β ij which increases U relatively to U at ); (ii) the mixing of bands of different orbital character which allow two electrons not to be on the same orbital of an atom (and thus decreases U ); and (iii) the existence of Coulomb repulsion between electrons on neighboring sites which reduces their effective exclusion on the same site.