The formalism and matrix elements of a complete potential-harmonic scheme for directly solving the Schrödinger equation of the helium atom Original Research Article

By analyzing the wavefunctions and the precision of the calculated energy of several potential-harmonic (PH) expansion schemes previously developed by us, a complete PH scheme has been proposed which incorporates the electron-nucleus cusp as well as electron-electron correlation. The PH method including the coupled hyperradial ordinary differential equations and the secular equation for the energy has been formulated. The involved matrix elements of the two-body potentials and correlation function are analytically expressed through a sum of the grand angular momentums or beta functions. The new PH scheme has been applied to the helium atom, and the eigenenergies of the ground state as well as the low-lying singlet excited states are obviously improved over the previous CFPHGLF method.