Solving Euler’s equation for non-interacting model systems and atoms

Abstract

In this project, we develop a numerical approach to solve Euler’s equation for many-body atomic systems within the framework of density functional theory (DFT) to calculate the electron density that minimizes the total energy functional. The energy functional includes contributions from kinetic energy, external potential, Hartree potential and exchange- correlation energy. The Pauli potential is incorporated to account for the quantum mechanical effects of fermionic antisymmetry, improving the rep- resentation of kinetic energy. Using a self-consistent field (SCF) method, the electron density is iteratively updated, starting from an initial guess, with the Hartree, exchange-correlation, and Pauli potentials computed at each step. The density is normalized to maintain the total number of elec- trons, and convergence is achieved when the error in density or energy falls below a defined threshold. This implementation employs Numerov meth- ods for radial grid discretizations and numerical integration to evaluate potentials and energies. The results demonstrate the successful compu- tation of electron density for multi-electron atomic systems, providing a robust framework for extending the method to more complex systems or incorporating advanced functionals.