Hyperspherical Three-Body Calculation for Exotic Atoms

Abstract

Ground state energies of atomic three-body systems like negatively charged hydrogen, normal helium, positively charged-lithium, beryllium, carbon, oxygen, neon and negatively charged exotic- muonium and positronium atoms have been calculated adopting hyperspherical harmonics expansion method. Calculation of matrix elements of two body interactions needed in the hyperspherical harmonics expansion method for a three body system is greatly simplified by expanding the bra- and ket-vector states in the hyperspherical harmonics (HH) basis states appropriate for the partition corresponding to the interacting pair. This involves the Raynal–Revai coefficients (RRC), which are the transformation coefficients between the HH bases corresponding to the two partitions. Use of RRC become particularly essential for the numerical solution of three-body Schrődinger equation where the two-body potentials are other than Coulomb or harmonic. However in the present work the technique is used for two electron atoms 1H−(p + e − e −), D−(d + e − e −), Mu−(μ + e − e −),4He(4 He 2+ e − e −),6Li(6 Li 3+ e − e −),10Be(10 Be 4+ e − e −),12C(12 C 6+ e − e −),16O(16 O 8+ e − e −) etc. and the exotic positronium negative ion Ps −(e + e − e −) where the interactions are purely Coulomb. The relative convergence in ground state binding energy with increasing K max for 20Ne has been demonstrated as a representative case. The calculated energies at K max = 28 using RRC’s have been compared with those obtained by a straight forward manner in some representative cases to demonstrate the appropriateness of the use of RRC. The extrapolated energies have also been compared with those found in the literature. The calculated binding energies agree within the computational error.