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

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