Hylleraas and James-Coolidge wave functions

Already in 1929 Hylleraas found that the orbital expansion of the wave function of helium converges extremely slowly. The problem could be overcome by including terms in the wave function that depend explicitly on interelectronic coordinates [6, 7]. The proposed explicitly correlated wave function was of the form 

The singlet ground state (para-helium case) requires particular spatial symmetry of the function. Therefore, only even powers of t were considered here. Even very short expansions of this type [Eq. (4)] led to very good results reducing the discrepancy between theory and experiment from 0.12 to 0.01 eV in terms of the ionization potential of the helium atom. 

A natural idea was to extend the concept of Hylleraas to molecules. It was done in 1933 by James and Coolidge [8], and the generalized function was of the form 

Tai is the distance between the nucleus a and the i-th electron, R is the internuclear distance, p is the explicitly correlated term p = 2ri2/J , a is the nonlinear parameter to be optimized and m, rzj, ki, li, pi are non-negative integers. Similar to the Hylleraas case. 

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