Numerical Study of Density Functional Theory of Multi-electronic Atoms: Case of Carbon and Helium

The ab-initio method based on density functional theory was used via local density approximation (LDA) to analyze the electronic structure of helium atoms and carbon. This involves calculating physical properties of these atoms such as effective potential, electron density and radial wave functions in both the ground state and the first three excited states. Thus, taking into account the structural complexity of the system, the calculations were carried out implicitly by the finite element method via the MATLAB software in deterministic mode. At the end of the study, the results obtained are consistent and revealing. They have shown that the effective potential of the helium atom decreases with position both in its ground state and in its first three excited states, so for the carbon atom, they have shown that these two states evolve in the same way. Regarding the electron density, the results revealed that at each radius of an atomic orbital, the electron density passes through a maximum for the helium atom while for carbon, the results show a succession of two and three peaks. Finally, with regard to radial wave functions, the results showed that the 1s orbital is closer to the nucleus than the 2s, 2p and 3s orbitals for the helium atom. For the carbon atom, the results confirm the information provided by the electron density. From the results it can be concluded that the effective potential of a multielectronic atom decreases with position and that the densest orbitals are those near the nucleus.