KelbgLIP: Program implementation of the high-temperature Kelbg density matrix for path integral and molecular dynamics simulations with long-range Coulomb interaction - Demyanov, G.S.; Levashov, P.R. (Computer physics communications, 2024)
In this paper, we present the KelbgLIP code to implement the previously obtained analytical density matrix that includes Coulomb long-range interactions. The method is based on the work of G. Kelbg, who derived a high temperature density matrix for the Coulomb potential. To include all long-range interactions in the density matrix, we use the Ewald technique, specifically the angular-averaged Ewald potential (AAEP). The solution of the Blöch equation within the AAEP has a direct analytic form that can be easily implemented in classical and quantum Monte Carlo or molecular dynamics codes, including exchange effects. The potential part of the density matrix remains finite at small distances, preventing the collapse of a two-component system. Using KelbgLIP, one can calculate the diagonal Kelbg-AAE pseudopotential and the pair density matrix. In the case of a hydrogen plasma, the code is able to calculate action, kinetic and potential energy in the path integral representation. We validated our approach by simulating a nondegenerate weakly coupled hydrogen plasma and obtained the thermodynamic limit in agreement with the Debye-Hückel approximation. In addition, we observe the agreement with classical simulations using the unbounded from below AAEP, which is possible in the weakly-coupled regime. Program Title:KelbgLIP, version 1.0 CPC Library link to program files:https://doi.org/10.17632/p773jhdz69.1 Licensing provisions: GPLv3 Programming language: C++11 Nature of problem: The calculation of thermodynamic properties of two-component Coulomb systems relies on the formalism of the density matrix to avoid the collapse due to the unbounded Coulomb potential. The direct consideration of long-range Coulomb potential interactions with periodic boundary conditions requires the solution of the Blöch equation for the density matrix in the case of the Ewald potential. However, typically only the solution for the Coulomb potential is used, omitting all long-range effects, because the Ewald potential has a sufficiently complicated form. Solution method: To account for long-range effects in solving the Blöch equation, one can use the angular-averaged Ewald potential (AAEP). This potential, with its simple analytic form, effectively captures the long-range effects and has a finite interaction range. In order to solve the Blöch equation with the AAEP, we use the Kelbg approach [1], in which the solution was obtained for the Coulomb potential in a high-temperature limit. Hence, we derive the density matrix with the Kelbg-AAE pseudopotential (Kelbg-AAEPP) that is finite at small distances [2]. Notably, this density matrix has a direct analytic form, which avoids a summation over Coulomb energy levels or squaring method [3]. The Kelbg-AAE density matrix is then used in quantum simulations via path integral Monte Carlo and in classical simulations using the diagonal Kelbg-AAEPP with a finite interaction radius. In the case of Boltzmannons, the expressions for action, kinetic and potential energy are derived. Special attention is paid to the calculation involving the non-diagonal Kelbg-AAEPP. The code calculates the diagonal Kelbg-AAEPP with possible improvement at short distances [4], pair Kelbg-AAE density matrix, as well as action, kinetic energy, and potential energy. These routines can be easily integrated into other molecular dynamics or path integral Monte Carlo codes designed for fermions and bosons, which is particularly valuable for the simulations of two-component Coulomb systems. Additional comments including restrictions and unusual features: GSL library version 2.x is required for compilation. References[1]G. Kelbg, Theorie des Quanten-Plasmas, Ann. Phys. 467 (3–4) (1963) 219–224.[2]G.S. Demyanov, P.R. Levashov, Accounting for long–range interaction in the Kelbg pseudopotential, Contrib. Plasma Phys. 62 (10) (2022) e202200100.[3]B. Militzer, Computation of the high temperature Coulomb density matrix in periodic boundary conditions, Comput. Phys. Commun. 204 (2016) 88–96.[4]A.V. Filinov, V.O. Golubnychiy, M. Bonitz, W. Ebeling, J.W. Dufty, Temperature-dependent quantum pair potentials and their application to dense partially ionized hydrogen plasmas, Phys. Rev. E 70 (2004) 046411.