The accurate atomic potential determination is an essential task in the molecular simulations. The so-called ab-initio simulations using the quantum mechanics are of great interest in the computational physics and computational mechanics. Basically, the potential interactions can be obtained by means of the Schrödinger's equation. The main obstacle in the quantum mechanics is the solution of this equation whose application leads to the multi-dimensional PDEs with the Hamiltonian operator for every single electron. The Kohn-Sham's method which establishes the Density Functional Theory, is widely used in the quantum mechanics field. Basically, the 3D finite element method can be used to solve the Schrödinger's equation for the mono-electronic cases. In the present contribution, the solution of the Schrödinger's equation as well as KS model have been brought via the numerical implementations under 3D finite element method and eigenvalue problem solvers. Some conclusions and outlooks pertaining to the quantum mechanics and finite element method have been outlined herein.