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

Non-relativistic Helium-like ions (-e,-e,Ze) with static nucleus in a d-dimensional space mathbb{R}^d (d>1) are considered. Assuming r^-1 Coulomb interactions, a 2-parametric correlated Hylleraas-type trial function is used to calculate the ground state energy of the system in the domain Z leq 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 frac{1}{Z} expansion, and serve as generating functions for certain correlation-dependent properties. The (first) critical charge Z_{rm c} vs d and the Shannon entropy S_r^(d) 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^+, H₂^+ and H^- are indicated explicitly. Our variational analytical results are in excellent agreement with the exact numerical values reported in the literature.