Many-electron correlation problem variational approaches

Explicitly correlated wave function fheory [14] is anofher imporfanf approach in quantum chemistry. One introduces inter-electron distances together with the nuclear-electron distances and set up some presumably accurate wave function and applies the variation principle. The Hylleraas wave function reported in 1929 [15] was the first of this theory and gave accurate results for the helium atom. Many important studies have been published since then even when we limit ourselves to the helium atom [16-28]. They clarified the natures and important aspects of very accurate wave functions. However, the explicitly correlated wave function theory has not been very popularly used in the studies of chemical problems in comparison with the Hartree-Fock and electron correlation approach. One reason was that it was generally difficult to formulate very accurate wave functions of general molecules with intuitions alone and another reason was that this approach was rather computationally demanding. 

The technique for dealing with this problem is well known from nonrelativistic calculations on many-electron systems. One-particle basis sets are developed by considering the behavior of the single electron in the mean field of all the other electrons, and while this neglects a smaller part of the interaction energy, the electron correlation, it provides a suitable starting point for further variational or perturbational treatments to recover more of the electron-electron interaction. It is only natural to pursue the same approach for the relativistic case. Thus one may proceed to construct a mean-field method that can be used as a basis for the perturbation theory of QED. In particular, the inclusion of the Breit interaction in the mean-field calculations ensures that the terms of O(a ) are included to infinite order in QED. 

In an ab initio approach, the first step is to solve the Hartree-Fock problem using a suitable basis set. In the Hartree-Fock model, each electron experiences only the average potential created by the other electrons. In reality, the instantaneous position of each electron, however, depends on the instantaneous position of the other electrons but the Hartree-Fock model cannot account for this electron correlation. In order to obtain quantitative results, electron correlation (also referred to as dynamical correlation) should be included in the model and there are many methods available for accomplishing this task based on either variational or perturbation principles. The easiest method to understand conceptually is variational configuration interaction (CI). In this method, the electronic wavefunction is expanded in terms of configurations that are formed from excitations of electrons from the occupied orbitals in the Hartree-Fock wavefunction to the virtual orbitals. The expansion can be written as... 

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