Solutions of multi-electron systems using a hyperspherical approach and a 1-D basis function method

With hyperspherical coordinates, the Schrödinger equation for some quantum mechanical many-body systems such as electrons in atoms like He and charged excitons in quantum wells can be solved in a similar way with the wave functions being expanded into orthonormal complete basis sets of the hyperspherical harmonics of hyperangles and generalized Laguerre polynomials of the hyperradius. Numerical results obtained explicitly by solving a simple secular equation show good agreement with those obtained through other computationally intensive methods. The eigenenergies and particle correlation in the low-lying states are investigated. Alternatively, the solutions of multi-electron systems can be obtained by using a linear combination of one-dimensional wave functions. Numerical results of small systems like He and H2 show that the one-dimensional bases along the x, y, z axes are good choices which facilitate easy numerical integration. The method developed is an effective numerical approach to the many-body problem and could be extended to larger atomic and molecular systems.