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

Abstract

Abstract Non-relativistic Helium-like ions (−e, −e, Ze) with static nucleus in a d−dimensional space (d > 1) are considered. Assuming r−1Coulomb interactions, a 2-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 whilst 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 E0(Z, d) in the form of compact analytical expressions. We call them generalized Majorana solutions. They reproduce the first leading terms in the celebrated 1Z expansion, and serve as generating functions for certain correlation-dependent properties. The (first) critical charge Zc vs d and the Shannon entropy S(d)r 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 3-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+, H+2 and H− are indicated explicitly. Our variational analytical results are in excellent agreement with the exact numerical values reported in the literature.