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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. [ABSTRACT FROM AUTHOR]

We calculate the energy of ground state S0 and the first two excited states S1 and T1 of the hydrogen molecule. The corresponding wave function is expanded with the bonding and non-bonding molecular orbitals where each hydrogen atom is represented with one unoptimized Slater type basis function, then all integrations for the energy components evaluations are performed and estimated with the Monte-Carlo method. Unlike the standard method of the Hartree-Fock, the construction of the Fock matrix followed with its diagonalization is circumvented. For considerably large data samplings of 50, 000, the computational time is still cheap, costing just a split of second. At the STO-3G level of theory, the ground state energies show similar accuracy with the Hartree-Fock calculations while superior accuracy for the singlets and triplets reveal with ease the electron correlation effect. [ABSTRACT FROM AUTHOR]