Adaptive-coordinate real-space electronic-structure calculations for atoms, molecules, and solids

We report the development of a real-space approach to electronic-structure calculations which utilizes adaptive curvilinear coordinates. A regular real-space mesh would be desirable from computational considerations because it produces a sparse, local, and highly structured Hamiltonian, which enables the effective use of iterative numerical methods and parallel-computer architectures. However, a regular real-space mesh has equal resolution everywhere. This results in an inefficient distribution of mesh points, since actual physical systems are inhomogeneous. To remedy this inherent inefficiency without losing the computational advantages of a regular mesh, we use a regular mesh in curvilinear coordinates, which is mapped by a change of coordinates to an adaptive mesh in Cartesian coordinates. We discuss in detail the choices involved in the implementation of the method, including the form and optimization of the coordinate transformation, the expression for the discretized Laplacian, the regularization of the ionic potential for all-electron calculations, the method of calculating the forces, and the algorithms used. Band-structure calculations have been implemented by adding a phase shift at periodic boundary conditions. We report all-electron calculations for atoms and molecules with 1s and 2p valence electrons, and pseudopotential calculations for molecules and solids.