Helium-like ions in d-dimensions: analyticity and generalized ground state Majorana solutions

Abstract

Non-relativistic helium-like ions (−e, −e, Ze) with static nucleus in a d-dimensional space (d > 1) are considered. Assuming r −1 Coulomb interactions, a two-parametric correlated Hylleraas-type trial function is used to calculate the ground state energy of the system in the domain Z ⩽ 10. For odd d = 3, 5, the variational energy is given by a rational algebraic function of the variational parameters while for even d = 2, 4 it is shown for the first time that it corresponds to a more complicated non-algebraic expression. This twofold analyticity will hold for any d. It allows us to construct reasonably accurate approximate solutions for the ground state energy E 0(Z, d) in the form of compact analytical expressions. We call them generalized Majorana solutions. They reproduce the first leading terms in the celebrated expansion, and serve as generating functions for certain correlation-dependent properties. The (first) critical charge Z c vs d and the Shannon entropy vs Z are also calculated within the present variational approach. In the light of these results, for the physically important case d = 3 a more general three-parametric correlated Hylleraas-type trial is used to compute the finite mass effects in the Majorana solution for a three-body Coulomb system with arbitrary charges and masses. It admits a straightforward generalization to any d as well. Concrete results for the systems e − e − e +, and H − are indicated explicitly. Comparison of our variational analytical results with corresponding highly accurate numerical values reported in the literature indicates a fair agreement, within the limits of accuracy imposed by the approximate character of our treatment.