Overlap density exchange integral

Coulomb integrals Jij describe the coulombic interaction of one charge density (( )i2 above) with another charge density (c )j2 above) exchange integrals Kij describe the interaction of an overlap charge density (i.e., a density of the form ( )i( )j) with itself ((l)i(l)j with ( )i( )j in the above). 

Excitations similar to giant resonances are observed for much deeper-shell excitation, and are more aptly named shape resonances, because the spatial overlap between the initial and final states is less complete than for giant resonances. An example is 3d — f in Ba, which is similar in character, but not identical, to Ad —> / in the same element in particular, the smaller spatial overlap means that the exchange integral has a less dramatic influence on the spectrum, which is why, when studying solids (see chapter 11), deep shell excitation is preferred if the resonance is to be used as a probe of final state density in this way, one avoids most of the term dependence. 

The exchange integral K12 originates from the overlap density distribution 0102 and is extremely small in magnitude (e.g., 1 2 kcal/mol in perpendicular C2ll4), ... 

Here, g[ti is a well-known (after Janak 1977, Brooks and Johansson 1983) function of the density. Tar-sd thus depends upon the 4f-5d overlap densities and is always positive. As the 4f shell contracts, the region of overlap between the outer part of the 4f density and the inner part of the 5d density decreases. Then, a decrease of the 4f-5d exchange integrals across each series is expected. This variation of r4f 5d can be given as (Brooks et al. 1991a-c)... 

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