The ground state of the lithium atom in dense plasmas using variational Monte Carlo method
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ORIGINAL PAPER
The ground state of the lithium atom in dense plasmas using variational Monte Carlo method S B Doma1*, H S El-Gendy2, M A Abdel-Khalek1 and M M Hejazi1 1
Faculty of Science, Alexandria University, Alexandria, Egypt
2
College of Science and Humanities, Shaqra University, Shaqra, Kingdom of Saudi Arabia Received: 06 October 2019 / Accepted: 04 September 2020
Abstract: In this paper, the variational quantum Monte Carlo method is applied to investigate the ground state of the lithium atom. Moreover, the energy eigenvalues of the lithium atom in dense plasma are also investigated by using the Debye–Hu¨ckel model and the exponential cosine screened Coulomb potential model. The calculations are carried out by using trial wave functions in the form of the Slater determinant wave function multiplied by a correlation function due to the interaction between the electrons. Three types of correlation functions are used—with two, three and four variational parameters—one of which satisfies the well-known cusp conditions. Interesting results are obtained in comparison with results obtained by using other trial wave functions. Keywords: Lithium atom; Dense plasma; Exponential cosine screened Coulomb potential; Variational Monte Carlo method; Trial wave functions
1. Introduction The variational Monte Carlo (VMC) method [1, 2] is a very powerful technique that estimates the energy and all the desired properties of a given atom, molecule and nucleus by a suitably chosen trial wave function. In quantum mechanics, the Monte Carlo method has been extensively employed to evaluate the multi-dimensional integrals which arise in the different formulations of the many-body problem. These calculations involve the evaluation of integrals, whose dimensionality is three times the number of particles, typically hundreds. Using the VMC algorithm, the expectation value of the energy for any trial wave function form can be estimated by averaging the local energy Hw/w during a random walk in the configuration space, using a Metropolis algorithm [3], for example. It was acknowledged since chemical physics has appeared that the energy EHF of the Slater determinant (SlDet), jwHF i, obtained by the single-particle Hartree–Fock (HF) equation, does not synchronize with the lower energy of the functional hwjHjwi where jwi is a SlDet and H is the many-particle Hamiltonian. In Ref. [4], Thanos et al.
started from a SlDet jwi with its spin orbitals calculated by the standard HF equation; they looked for the maximum of the functional jhw0 jHjwij, where jw0 i is a SlDet and H is the exact Hamiltonian of an atom. They showed that the sequence an ¼ hwnþ1 jHjwn i is showing a convergence. They applied this proceeding for identifying the eigenstate energies of a few configurations of H3, the lithium atom, LiH and Be. As a conclusion, the new single determinant approximation gives a precision of the eigenstate energies, compared to those of the configuration interaction (CI). Consequently, for the S ¼ 12 state of Li they found that E = - 7
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