Hylleraas integrals

Schwartz, H. M., Phys. Rev. 103, 110, "Ground state solution of the non-relativistic equation for He." More rapid convergence in the Ritz variational method by inclusion of half-integral powers in the Hylleraas function. 

Starting the schematical historic outline now, one can suppose ETO integral calculation starts with the pioneering work of Hylleraas [3], Kemble and Zener [4], Bartlett [5], Rosen [6], Hirschfelder [7], Coulson [8] and Ldwdin [9], solidifying as a quantum chemical discipline with the publication of overlap formulae and tables by MuUiken et al. [10]. 

If the phase shifts are given as a function of the energy a unique solution can only be obtained if, in addition, the bound state energies and the normalization constants of the bound state wave function are known (Gelfand and Levitan, 1951). The explicit construction of the potential leads to a complicated solution of a Fredholm integral equation. Other procedures (Agranovich and Marchenko, 1963 Hylleraas, 1963) may be more convenient for a practical application (see Benn and Scharf, 1967, O Brien and Bernstein,... 

This contribution examines current approaches to Coulomb few-body problems mainly from a methodological perspective, in contrast to recent reviews which have focused on the results obtained for benchmark problems. The methods under discussion here employ wavefunctions which explicitly involve all the interparticle coordinates and use functional forms appropriate to nonadiabatic systems in which all the particles are of comparable mass. The use of such wavefunctions for states of arbitrary angular symmetry is reviewed, and the kinetic-energy operator, written in the interparticle coordinates, is presented in a convenient form. Evaluation of the resultant angular matrix elements is discussed in detail. For exponentially correlated wavefunctions, problems of integral evaluation are surveyed, the relatively new analytical procedures are summarized, and relations among matrix elements are presented. The current status of Gaussian-orbital and Hylleraas methods is also reviewed. 

For the 5 states of three-body systems, the Hylleraas-method correlation factor rj2 leads to integrals that can be easily evaluated in r,y coordinates without the need to introduce an expansion. A recent paper by Drake et al. [61 ]... 

The family of variational methods with explicitly correlated functions includes the Hylleraas method, the Hyller-aas Cl method, the James-Coolidge and the KcAos-Wolniewicz approaches, as well as a method with exponentially correlated Gaussians. The method of explicitly correlated functions is very successful for two-, three-, and four-electron systems. For larger systems, due to the excessive number of complicated integrals, variational calculations are not yet feasible. 

Therefore only two-electron integrals, as in the case of the Cl method, and triangle integrals have to be computed. This fact will be extremely helpful when extending the application of Hy-CI method to larger systems. In our code, the same computer memory is needed for Cl and Hy-CI calculations. Note, that in the Hylleraas-CI method for any electron number no higher order integrals than four-electron ones appear. 

In fact, it was known since the work of Hylleraas published in 1929, i.e. in the early days of quantum mechanics, that a much better convergence is obtained if the wavefunction ansatz explicitly includes the distance between two electrons. This knowledge, however, was difficult to apply for systems with more than two electrons as numerous and complicated integrals over three, four, etc. electrons occur. In the end it took more than fifty years until an idea emerged, how to turn the ideas behind explicit electron correlation... 

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