Poisson Green's function method for increased computational efficiency in numerical calculations of Coulomb coupling elements

摘要

Often, the calculation of Coulomb coupling elements for quantum dynamical treatments, e.g., in cluster or correlation expansion schemes, requires the evaluation of a six dimensional spatial integral. Therefore, it represents a significant limiting factor in quantum mechanical calculations. If the size or the complexity of the investigated system increases, many coupling elements need to be determined. The resulting computational constraints require an efficient method for a fast numerical calculation of the Coulomb coupling. We present a computational method to reduce the numerical complexity by decreasing the number of spatial integrals for arbitrary geometries. We use a Green's function formulation of the Coulomb coupling and introduce a generalized scalar potential as solution of a generalized Poisson equation with a generalized charge density as the inhomogeneity. That enables a fast calculation of Coulomb coupling elements and, additionally, a straightforward inclusion of boundary conditions and arbitrarily spatially dependent dielectrics through the Coulomb Green's function. Particularly, if many coupling elements are included, the presented method, which is not restricted to specific symmetries of the model, presents a promising approach for increasing the efficiency of numerical calculations of the Coulomb interaction. To demonstrate the wide range of applications, we calculate internanostructure couplings, such as the Forster coupling, and illustrate the inclusion of symmetry considerations in the method for the Coulomb coupling between bound quantum dot states and unbound continuum states.