The Hubbard intra-atomic energy

Much of this book is concerned with the properties of narrow bands to which the tight-binding approximation is appropriate. In this case, if the band is half full or nearly so, the short-range repulsion between the electrons may have very important effects on the properties of the electrons in the bands, producing magnetic moments and non-conducting properties. These are a major theme of this book. At this point we introduce the Hubbard intra-atomic energy  

More generally, for many-electron atoms we define U as the energy required to transfer an electron from one atom to another, so that 

For an ion such as Mn2+ in the state 3d5, where the electron must be transferred to a state with opposite spin, clearly 

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The distance between two electrons at a given site is given as ri2. The electron wave function for one of the electrons is given as (p(ri) and the wave function for the second electron, with antiparallel spin, is Hubbard intra-atomic energy and it is not accounted for in conventional band theory, in which the independent electron approximation is invoked. Finally, it should also be noted that the Coulomb repulsion interaction had been introduced earlier in the Anderson model describing a magnetic impurity coupled to a conduction band (Anderson, 1961). In fact, it has been shown that the Hubbard Hamiltonian reduces to the Anderson model in the limit of infinite-dimensional (Hilbert) space (Izyumov, 1995). Hence, Eq. 7.3 is sometimes referred to as the Anderson-Hubbard repulsion term. 

To put it on a more quantitative basis, Johansson uses the expression for the critical point of the Mott transition as reformulated by Hubbard in terms of the bandwidth W[ and the polar state formation energy Uh (or effective intra-atomic correlation) (Eq. (36)). 

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