Self-consistent numerical-basis-set linear-combination-of-atomic-orbitals model for the study of solids in the local density formalism

A new approach to the fully self-consistent solution of the one-particle equations in a periodic solid within the Hohenberg-Kohn-Sham local-density-functional formalism is presented. The method is based on systematic extensions of non-self-consistent real-space techniques of Ellis, Painter, and collaborators and the self-consistent reciprocal-space methodologies of Chaney, Lin, Lafon, and co-workers. Specifically, our approach combines a discrete variational treatment of all potential terms (Coulomb, exchange, and correlation) arising from the superposition of spherical atomiclike overlapping charge densities, with a rapidly convergent three-dimensional Fourier series representation of all the multicenter potential terms that are not expressible by a superposition model. The basis set consists of the exact numerical valence orbitals obtained from a direct solution of the local-density atomic one-particle equations and (for increased variational freedom) virtual numerical atomic orbitals, charge-transfer (ion-pair) orbitals, and "free" Slater one-site functions. The initial crystal potential consists of a non-muffin-tin superposition potential, including nongradient free-electron correlation terms calculated beyond the random-phase approximation. The usual multicenter integrations encountered in the linear-combination-of-atomic-orbitals tight-binding formalism are avoided by calculating all the Hamiltonian and other matrix elements between Bloch states by three-dimensional numerical Diophantine integration. In the first stage of self-consistency, the atomic superposition potential and the corresponding numerical basis orbitals are modified simultaneously and nonlinearly by varying (iteratively) the atomic occupation numbers (on the basis of computed Brillouin-zone averaged band populations) so as to minimize the deviation, $\Delta \rho \left(\overrightarrow{\mathrm{r}}\right)$, between the band charge density and the superposition charge density. This step produces the "best" atomic configuration within the superposition model for the crystal charge density and tends to remove all the sharp "localized" features in the function $\Delta \rho \left(\overrightarrow{\mathrm{r}}\right)$ by allowing for intra-atomic charge redistribution to take place. In the second stage, the three-dimensional multicenter Poisson equation associated with $\Delta \rho \left(\overrightarrow{\mathrm{r}}\right)$ through a Fourier series representation of $\Delta \rho \left(\overrightarrow{\mathrm{r}}\right)$ is solved and solutions of the band problem are found using a self-consistent criterion on the Fourier coefficients of $\Delta \rho \left(\overrightarrow{\mathrm{r}}\right)$. The calculated observables include the total crystal ground-state energy, equilibrium lattice constants, electronic pressure, x-ray scattering factors, and directional Compton profile. The efficiency and reliability of the method is illustrated by means of results obtained for some ground-state properties of diamond; comparisons are made with the predictions of other methods.