Four-body exponentially correlated wavefunctions

The complications associated with integrations over all the interparticle distances of a four-hody system have caused that approach to be used only when all the interparticle distances occur in exponential functions. Integrals of that kind were thought to be intractable until Fromm and Hill [24] presented a closed analytical formula for the basic four-body integral 

Fromm and Hill s paper, while a sophistieated and almost miraculous application of complex variable theory, produced a formula that exhibited two problems from a practical viewpoint. It contained the dilogarithm function, Li2, and squares of logarithmic functions, in combinations that were multiple-valued with respect to both their real and imaginary parts, and no simple recipe was provided to indicate which branches of these functions should be used. Fromm and Hill s provisional solution was to start from a point in the parameter space where the proper branch was known from asymptotic considerations, and then move in steps to the required parameter values. This procedure was referred to as branch tracking . 

The second problem was that individual terms of the formula became singular for physically relevant parameter values, with the singularities in the various terms expected on general grounds to combine into a nonsingular result. But there was no analysis of the singularity cancellation, and numerical calculation near the termwise singularities became seriously unstable. 

In 1997 the present author addressed these problems [26], showing how a modified definition of the multiply valued functions could eliminate the need 

The most extensive use of the analytical formulas for four-hody wavefunctions has been by Rebane and associates in 1992 Rebane and Yusupov [27] presented a preliminary study on model problems there followed a detailed study of the positronium molecule Ps2 (e e e e ) hy Rebane et al. [28] and an application to a number of four-particle mesomolecules by Zotev and Rebane [29]. These authors then refined the branch-tracking procedure so as to make it applicable to complex parameter sets [30,31]. At this point, the use of multiconfiguration exponential wavefunctions has produced results of a quality similar to that from more extensive Gaussian expansions, hut with what appears to be a comparable amount of effort. There are at present insufficient data to indicate whether the exponential wavefunctions have significant superiority over the Gaussian functions for short-range (e.g., delta-function) properties. 

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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... 

Eully correlated exponential wavefunctions for four-body systems... 

Energy Computation for Exponentially Correlated Four-Body Wavefunctions... 

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. 

A practical reason that pre-exponential r,y have not been used with exponentially correlated four-body wavefunctions has been the difficulty of managing analytical formulas for the integrals that thereby result that difficulty has now been reduced in importance by the author s recent presentation[10] of a recursive procedure for the integral generation. 

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