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

 Adrian M. Escobar-Ruiz,∗Horacio Olivares-Pil ́on,†Norberto Aquino,‡and Salvador A. Cruz§

 

 Abstract

 Non-relativistic Helium-like ions (−e,−e,Ze) with static nucleus in ad−dimensional spaceRd(d >1) are considered. Assumingr−1Coulomb interactions, a 2-parametric correlated Hylleraas-type trial function is used to calculate the ground state energy of the system in the domainZ≤10.For oddd= 3,5, the variational energy is given by a rational algebraic function of the variationalparameters whilst for evend= 2,4 it is shown for the first time that it corresponds to a morecomplicated non-algebraic expression. This twofold analyticity will hold for anyd. It allows usto construct reasonably accurate approximate solutions for the ground state energyE0(Z,d) inthe form of compact analytical expressions. We call them generalized Majorana solutions. Theyreproduce the first leading terms in the celebrated1Zexpansion, and serve as generating functionsfor certain correlation-dependent properties. The (first) critical chargeZcvsdand the ShannonentropyS(d)rvsZare also calculated within the present variational approach. In the light of theseresults, for the physically important cased= 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-bodyCoulomb system with arbitrary charges and masses. It admits a straightforward generalization toanydas well. Concrete results for the systemse−e−e+,H+2andH−are indicated explicitly. Ourvariational analytical results are in excellent agreement with the exact numerical values reportedin the literature.

 

https://arxiv.org/pdf/2108.09439.pdf