Levy-Perdew-Sahni equation and its application to perform atomic calculations

Levy-Perdew-Sahni (LPS) derived the connection between the asymptotic decay of density and the ionization potential of a many-electron system using the equation for the square root of density. For this, they employed an expression for the corresponding effective potential in terms of the wavefunction of the system. In this paper, we explore the possibility of solving the LPS equation in conjunction with approximate wavefunction. For this, we first perform the variational derivation of the equation and use this to set up a self-consistent cycle to get the ground state properties of two-electron systems using the modified form of the Le Sech wavefunction. Furthermore, using the observation that even the approximate wavefunctions give the accurate effective potential for the LPS equation, we show that accurate densities are obtained through using the LPS equation with these wavefunctions. We demonstrate our method by performing calculations for closed-shell atoms.