Real-space density functional theory and time dependent density functional theory using finite/infinite element methods

We present a numerical approach using the finite element method to discretize the equations that allow getting a first-principles description of multi-electronic systems within DFT and TD-DFT formalisms. A strictly local polynomial function basis set is used in order to represent the entire real-space domain. Infinite elements are introduced to model the infinite external boundaries in the case of Hartree's equation. The diagonal mass matrix is obtained using a close integration rule, reducing the generalized eigenvalue problem to a standard one. This framework of electronic structure calculation is embedded in a high performance computing environment with a very good parallel behavior.Fil: Soba, Alejandro. Barcelona Supercomputing Center - Centro Nacional de Supercomputacion; España. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Bea, Edgar Alejandro. Barcelona Supercomputing Center - Centro Nacional de Supercomputacion; España. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Houzeaux, Guillaume. Barcelona Supercomputing Center - Centro Nacional de Supercomputacion; EspañaFil: Calmet, Hadrien. Barcelona Supercomputing Center - Centro Nacional de Supercomputacion; EspañaFil: Cela, José María. Barcelona Supercomputing Center - Centro Nacional de Supercomputacion; Españ

Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status solidi (b) - basic solid state physics" devoted to the CECAM workshop "State of the art developments and perspectives of real-space electronic structure techniques in condensed matter and molecular physics". v2: Minor stylistic and typographical changes, partly inspired by referee comment

Gradient type optimization methods for electronic structure calculations

The density functional theory (DFT) in electronic structure calculations can be formulated as either a nonlinear eigenvalue or direct minimization problem. The most widely used approach for solving the former is the so-called self-consistent field (SCF) iteration. A common observation is that the convergence of SCF is not clear theoretically while approaches with convergence guarantee for solving the latter are often not competitive to SCF numerically. In this paper, we study gradient type methods for solving the direct minimization problem by constructing new iterations along the gradient on the Stiefel manifold. Global convergence (i.e., convergence to a stationary point from any initial solution) as well as local convergence rate follows from the standard theory for optimization on manifold directly. A major computational advantage is that the computation of linear eigenvalue problems is no longer needed. The main costs of our approaches arise from the assembling of the total energy functional and its gradient and the projection onto the manifold. These tasks are cheaper than eigenvalue computation and they are often more suitable for parallelization as long as the evaluation of the total energy functional and its gradient is efficient. Numerical results show that they can outperform SCF consistently on many practically large systems.Comment: 24 pages, 11 figures, 59 references, and 1 acknowledgement

Towards chemical accuracy using a multi-mesh adaptive finite element method in all-electron density functional theory

Chemical accuracy serves as an important metric for assessing the effectiveness of the numerical method in Kohn--Sham density functional theory. It is found that to achieve chemical accuracy, not only the Kohn--Sham wavefunctions but also the Hartree potential, should be approximated accurately. Under the adaptive finite element framework, this can be implemented by constructing the \emph{a posteriori} error indicator based on approximations of the aforementioned two quantities. However, this way results in a large amount of computational cost. To reduce the computational cost, we propose a novel multi-mesh adaptive method, in which the Kohn--Sham equation and the Poisson equation are solved in two different meshes on the same computational domain, respectively. With the proposed method, chemical accuracy can be achieved with less computational consumption compared with the adaptive method on a single mesh, as demonstrated in a number of numerical experiments.Comment: 19pages, 17 figure

Adaptive Finite Element Approximations for Kohn-Sham Models

The Kohn-Sham equation is a powerful, widely used approach for computation of ground state electronic energies and densities in chemistry, materials science, biology, and nanosciences. In this paper, we study the adaptive finite element approximations for the Kohn-Sham model. Based on the residual type a posteriori error estimators proposed in this paper, we introduce an adaptive finite element algorithm with a quite general marking strategy and prove the convergence of the adaptive finite element approximations. Using D{\" o}rfler's marking strategy, we then get the convergence rate and quasi-optimal complexity. We also carry out several typical numerical experiments that not only support our theory,but also show the robustness and efficiency of the adaptive finite element computations in electronic structure calculations.Comment: 38pages, 7figure