ORIGINAL PAPER The ground state of the lithium atom in dense plasmas using variational Monte Carlo method S B Doma 1 *, H S El-Gendy 2 , M A Abdel-Khalek 1 and M M Hejazi 1 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 E HF of the Slater determinant (SlDet), jw HF 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 hw 0 jHjwi j j, where jw 0 i is a SlDet and H is the exact Hamiltonian of an atom. They showed that the sequence a n ¼hw nþ1 jHjw n i     is showing a convergence. They applied this proceeding for identifying the eigenstate energies of a few configurations of H 3 , 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 1 2 state of Li they found that E = - 7.43269 Hartrees, while the value obtained by Koga et al. [5] was - 7.43272 Hartrees. In all cases, the devia- tion from the CI calculations ranged from 10 -8 to 10 -5 Hartees. Therefore, the method had certain features with respect to the density functional theory. In Ref. [6], Doma et al. applied the VMC method to study the ground state and some excited states of the lithium atom Z ¼ 3 and lithium-like ions up to Z ¼ 10 undergoing an external magnetic field regime with field strength c ¼ 0100, (c = B B 0 is the magnetic field strength in atomic units, B 0 ¼ hc ea 2 0 ¼ 2:3505  10 5 T ). They concluded that the VMC method can be considered as an efficient tool to study the three-electron system under the effect of strong magnetic *Corresponding author, E-mail: sbdoma@alexu.edu.eg Indian J Phys https://doi.org/10.1007/s12648-020-01920-2 Ó 2020 IACS