Abstract
The solution of the Poisson equation is a crucial step in electronic structure calculations, yielding the electrostatic potential-a key component of the quantum mechanical Hamiltonian. In recent decades, theoretical advances and increases in computer performance have made it possible to simulate the electronic structure of extended systems in complex environments. This requires the solution of more complicated variants of the Poisson equation, featuring nonhomogeneous dielectric permittivities, ionic concentrations with nonlinear dependencies, and diverse boundary conditions. The analytic solutions generally used to solve the Poisson equation in vacuum (or with homogeneous permittivity) are not applicable in these circumstances, and numerical methods must be used. In this work, we present DL_MG, a flexible, scalable, and accurate solver library, developed specifically to tackle the challenges of solving the Poisson equation in modern large-scale electronic structure calculations on parallel computers. Our solver is based on the multigrid approach and uses an iterative high-order defect correction method to improve the accuracy of solutions. Using two chemically relevant model systems, we tested the accuracy and computational performance of DL_MG when solving the generalized Poisson and Poisson-Boltzmann equations, demonstrating excellent agreement with analytic solutions and efficient scaling to ∼109 unknowns and 100s of CPU cores. We also applied DL_MG in actual large-scale electronic structure calculations, using the ONETEP linear-scaling electronic structure package to study a 2615 atom protein-ligand complex with routinely available computational resources. In these calculations, the overall execution time with DL_MG was not significantly greater than the time required for calculations using a conventional FFT-based solver.
Highlights
What is the electrostatic potential corresponding to a given charge density? This deceptively simple question is key to modeling the electronic structure of atoms, molecules, and materials, where the classical electrostatic potential forms a foundation upon which quantum mechanical many-body effects can be modeled
Article computed using the solver to be validated against known analytic solutions to the SPE, generalized Poisson equation (GPE), and Poisson−Boltzmann equation (P-BE)
We have described the implementation of DL_MG, a generalpurpose Poisson solver library, and examined its numerical accuracy and computational performance when applied to Article chemically relevant model systems and in large-scale electronic structure calculations
Summary
What is the electrostatic potential corresponding to a given charge density? This deceptively simple question is key to modeling the electronic structure of atoms, molecules, and materials, where the classical electrostatic potential forms a foundation upon which quantum mechanical many-body effects can be modeled. Without the screening effect of a solvent, it is possible for systems to develop unphysical surface states and dipole moments.[7] This issue can be resolved by including solvent molecules in the electronic structure calculation This explicit approach is very costly, even using linear-scaling methods, because of the significant increase in the number of atoms that must be treated quantum mechanically and the need to statistically average over solvent configurations. The generalized Poisson and Poisson−Boltzmann equations (see section 2.1) offer a computationally inexpensive means of embedding a charge density in an electrostatic environment, avoiding the complexities of explicit modeling of the environment Solving these equations yields an electrostatic potential which includes an implicit representation of the electrostatic effects of the environment a solvent, for example. Since DL_MG is freely available under a permissive open source license, we provide some brief information for developers in Appendix B to aid interested readers who may want to test and possibly integrate the library in their own codes
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