Three-body exponentially correlated wavefunctions

In contrast to the four-body problem, real exponential wavefunctions in the interparticle coordinates led to readily evaluated integrals for three-body systems, and that fact was exploited in the context of extensive configuration interaction for adiabatic systems as long ago as 1977 by Thakkar and Smith [12]. Starting in 1987, this method was also applied to nonadiabatic systems by Petelenz and Smith [33,34], and to many adiabatic and nonadiabatic systems by Frolov, both alone [35,36] and in collaboration with Thakkar [37] and with Smith [38]. 

We add here one technical note about the matrix element evaluation. Evaluation of the kinetic energy involves the quantities alcos 0plj8) as for the four-body problem, these matrix elements can be related to overlap and potential-energy integrals. The line of reasoning that led to equation (42) yields the following formula for the three-body problem  

Here i, j, are 1, 2, 3 in any order. The fact that and are equal enables equation (44) also to be a starting point for the development of additional relations connecting overlap and potential-energy integrals. 

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Bases of fully exponentially correlated wavefunctions [1, 2] provide more rapid convergence as a function of expansion length than any other type of basis thus far employed for quantum mechanical computations on Coulomb systems consisting of four particles or less. This feature makes it attractive to use such bases to construct ultra-compact expansions which exhibit reasonable accuracy while maintaining a practical capability to visualize the salient features of the wavefunction. For this purpose, exponentially correlated functions have advantages over related expansions of Hylleraas type [3], in which the individual-term explicit correlation is limited to pre-exponential powers of various interparticle distances (genetically denoted r,j). The general features of the exponentially correlated expansions are well illustrated for three-body systems by our work on He and its isoelectronic ions, for... 

There have now been several applications reported for fiilly exponentially correlated four-body wavefunctions [7-9], also limited to bases without pre-exponential r,j. While it was found that pre-exponential are relatively unimportant for three-body systems, they can be expected to contribute in a major way to the efficiency of expansions for three-electron systems such as the Li atom and its isoelectronic ions, as is obvious from the fact that the zero-order description of the ground states of such systems has electron configuration s 2s. 

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