Optimization of Trial Wave Function in Determining the Ground State Energy of Helium Atom

Abstract

The atomic ground state properties of helium, especially with regard to the ground state energy, can be studied by using various methods. One such method is the variational method. In this study, the atomic ground state energy for helium was calculated by using the trial wave function with four various parameters. The optimization process was performed by using Nelder-Mead method implied in script-code of Fortran. From calculation results, the symmetry new wave function, which meets both Pauli exclusion principle and Hund’s rule, was obtained. This obtained wave function can be written as $\Psi=0.3891exp(-1.9240r_1-0.9301r_2)+0.3891exp(-0.9301r_1-1.9240r_2)$. The atomic ground state energy obtained by this trial wave function is in good agreement with experimental energy, i.e.  eV (with discrepancy of  from experimental result). It can be indicated that the obtained trial wave function is quite good in explaining the atomic ground state properties of helium. In addition, the Nelder-Mead method used in this study is quite good in optimizing the equation with four variation parameters. The atomic ground state properties of helium, especially with regard to the ground state energy, can be studied by using various methods. Such method is the variational method. In this study, the atomic ground state energy for helium was calculated by using the trial wave function with four various parameters. The optimization process was performed by using Nelder-Mead method implied in script-code of Fortran. From calculation results, the symmetry new wave function, which meets both Pauli exclusion principle and Hund’s rule, was obtained. This obtained wave function can be written as Iˆ=0.3891 exp - 1.9240 r 1 - 0.9301 r 2 +0.3891 exp - 0.9301 r 1 - 1.9240 r 2 . The atomic ground state energy obtained by this trial wave function is - 79.004623  eV (with discrepancy of 0.000671%  from the experimental energy), where this result is in good agreement with the experimental energy. It can be inferred that the obtained trial wave function is quite good in explaining the ground state properties of helium atom. In addition, the Nelder-Mead method used in this study has good capability in optimizing the equation with four variation parameters.