Methods Involving Interelectronic Distances

The necessity for going beyond the HF approximation is the fact that electrons are further apart than described by the product of their orbital densities, i.e. their motions are con-elated. This arises from the electron-electron repulsion operator, which is a sum of ten-ns of the type 

Without these terms the Schrodinger equation can be solved exactly, with the solution being a Slater determinant composed of orbitals. 

The electron-electron repulsion operator has a singularity for r 12 = 0 which results in the exact wave function having a cusp (discontinuous derivative).-  

In other words, the exact wave function behaves asymptotically as a constant 4- l/2ri2 when ri2 is small. It would therefore seem natural that the interelectronic distance would be a necessary variable for describing electron correlation. For two-electron systems, extremely accurate wave functions may be generated by taking a trial wave function consisting of an orbital product times an expansion in electron coordinates such as 

In order to achieve a high aceuraey, it would seem desirable to explicitly include terms in the wave functions which are linear in the intereleetronie distanee. This is the idea in the R12 methods developed by Kutzelnigg and co-workers. The first order correction to the HF wave funetion only involves doubly exeited determinants (eqs. (4.35) and (4.37)). In R12 methods additional terms are included which essentially are the HF determinant multiplied with faetors. 

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