Analytical estimate of effective charge and ground-state energy of beryllium atom utilizing variational method
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ORIGINAL PAPER
Analytical estimate of effective charge and ground-state energy of beryllium atom utilizing variational method Kamran-ul-Haq Khan1*, M I Aslam2, M Naeem1 and I A Siddiqui1 1
Department of Physics, University of Karachi, University Road, Karachi, Pakistan
2
Department of Electronic Engineering, NED University of Engineering and Technology, University Road, Karachi, Pakistan Received: 27 July 2019 / Accepted: 02 March 2020
Abstract: In this paper, we present an analytical procedure to determine the effective charge and ground-state energy of a four-electron beryllium atom. The procedure followed is based on a variant of variational technique using modified hydrogenic one-electron wave function to calculate effective charge in 1s and 2s states of beryllium atom. The effective charge in 1s state is 3.68609 and in 2s state is 1.82362. Further, we have used these values in non-relativistic Hamiltonian with interaction terms to calculate the ground-state energy of beryllium atom. The estimated value of ground-state energy (- 14.3423 au) is in good agreement with the experimental results and with other theoretically obtained values. The calculated results without using any computationally intensive technique are the essence of this research work. Keywords: Analytical technique; Beryllium atom; Ground-state energy; Quantum mechanics; Variational method PACS Nos.: 03.65.-w; 44.05.?e; 45.10.Db
1. Introduction The estimation of ground-state energy of a multi-electron system is beneficial in atomic and molecular physics to understand the distribution of electrons in an atom and its underlying dynamics. The quantum mechanical description of a many-body interacting system requires complete knowledge of wave function of each interacting particle and an equation of motion to describe their mutual evolution. The proper choice of a wave function is non-trivial and is subject to certain quantum mechanical restrictions fulfilling the rules of occupancy of states in multi-electron (He, Li, Be, etc.) and multi-nuclear (H2, CO, NH3, etc.) quantum systems. In addition to electron–nucleus potential interactions, electron–electron and electron–other-nucleus potential interactions are also present which affects the overall electronic charge distribution and corresponding resulting states of the system. These electron–electron interactions lead to many new effects in multi-electron systems which are either absent or not noticeable in a single-electron system. Owing to the complexity of the
system, almost none of the multi-electron systems are exactly solvable, or if approached analytically, it results in huge mathematical burden. Therefore, different approximations are used to roughly sketch the properties of multielectron systems [1]. Many researchers have taken beryllium atom as an example to understand the properties of multi-electron systems. In majority of these studies, either experimental approach is followed or computationally rigorous techniques are used to determine the properties of these multielectron systems.
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