A divide and conquer real space finite-element Hartree-Fock method

Since the seminal contribution of Roothaan, quantum chemistry methods are traditionally expressed using finite basis sets comprised of smooth and continuous functions (atom-centered Gaussians) to describe the electronic degrees of freedom. Although this approach proved quite powerful, it is not well suited for large basis sets because of linear dependence problems and ill conditioning of the required matrices. The finite element method (FEM), on the other hand, is a powerful numerical method whose convergence is also guaranteed by variational principles and can be achieved systematically by increasing the number of degrees of freedom and/or the polynomial order of the shape functions. Here we apply the real-space FEM to Hartree-Fock calculations in three dimensions. The method produces sparse, banded Hermitian matrices while allowing for variable spatial resolution. This local-basis approach to electronic structure theory allows for systematic convergence and promises to provide an accurate and efficient way toward the full ab initio analysis of materials at larger scales. We introduce a new acceleration technique for evaluating the exchange contribution within FEM and explore the accuracy and robustness of the method for some selected test atoms and molecules. Furthermore, we applied a divide-and-conquer (DC) method to the finite-element Hartree-Fock ab initio electronic-structure calculations in three dimensions. This DC approach leads to facile parallelization and should enable reduced scaling for large systems.