Analytical estimate of effective charge and ground-state energies of two to five electron sequences up to atomic number 20 utilizing the variational method

Abstract Background The variational method, a quantum mechanical approach, estimates effective charge distributions and ground-state energy by minimizing the Hamiltonian's expectation value using trial wave functions with adjustable parameters. This method provides valuable insights into system behavior and is widely used in theoretical chemistry and physics. This paper aims to investigate ground-state energies and isoelectronic sequences using the variational method, introducing a novel approach for analyzing multi-electron systems. This technique allows for determining effective charge values and ground-state energies for 2–5 electrons sequence up to Z ≤ 20. Hydrogenic wave functions are used as a trial wave function to calculate effective charge in 1 s, 2 s, and 2p states. Two varying parameters were used to calculate an approximate wave function for the system. These values are then used in non-relativistic Hamiltonian with electron–electron interaction terms to calculate the ground-state energy of an atom. Result The results align with the reported experimental values, showing a marginal 1% error. Conclusion A Python algorithm is established based on the variational principle. It was found that, based on a few selected parameters in scripting the program, a very promising result was obtained. Furthermore, adding more variational parameters can minimize the difference between experimental and theoretical values, and this technique can be extended to elements with higher atomic numbers.