Hyperspherical three-body model calculation for the bound $^{1,3}$S-states of Coulombic systems

TL;DR

This paper applies a hyperspherical three-body model to calculate the energies of low-lying bound states in Coulombic systems like helium, using advanced mathematical techniques to improve accuracy and compare with existing literature.

Contribution

It introduces a simplified calculation of coupling potentials using Raynal-Revai Coefficients and employs the renormalized Numerov method for solving the coupled differential equations.

Findings

Calculated energies agree with literature values.

Method effectively handles Coulombic three-body systems.

Provides a systematic approach for low-lying state energies.

Abstract

In this paper, hyperspherical three-body model formalism has been applied for the calculation energies of the low-lying bound $^{1,3}$S (L=0)-states of neutral helium and helium like Coulombic three-body systems having nuclear charge (Z) in the range Z=2 to Z=92. The calculation of the coupling potential matrix elements of the two-body potentials has been simplified by the introduction of Raynal-Revai Coefficients (RRC). The three-body wave function in the Schr\H{o}dinger equation when expanded in terms of hyperpherical harmonics (HH), leads to an infinite set of coupled differential equation (CDE). For practical reason the infinite set of CDE is truncated to a finite set and are solved by an exact numerical method known as renormalized Numerov method (RNM) to get the energy solution (E). The calculated energy is compared with the ones of the literature.