Unified open-boundary electrostatics in real-space density functional theory
We present an electrostatic formulation in real-space density functional theory that provides a systematic and unified treatment of the open-boundary electrostatics of isolated and partially periodic systems, including in the presence of an applied uniform electric field along the open (finite) directions. Specifically, we formulate a local electrostatic energy functional whose stationarity yields the Poisson equation for the electrostatic potential, subject to periodic and Dirichlet boundary conditions along the periodic and open directions, respectively. Using a Green's function approach, we derive analytical expressions for the Dirichlet values arising from the total charge density of the system. We also derive the expressions for the energy, atomic forces, and stress tensor. We implement the resulting expressions within the large-scale parallel real-space SPARC electronic structure code. Using representative examples, we verify the accuracy and efficiency of the framework, demonstrating exponential convergence of the computed quantities with vacuum size and excellent agreement with established plane-wave codes while requiring significantly less vacuum at comparable accuracy. Since no existing implementation provides the stresses for such systems, we instead verify them against numerical derivatives of the energy, finding close agreement. Finally, we apply the framework to compute static polarizabilities and piezoelectric coefficients, obtaining very good agreement with values reported in the literature.