Orthogonalized Enriched Finite Elements - A Robust, Accurate and Efficient Basis for All-Electron Density Functional Theory Calculations

Abstract

A computationally efficient approach to perform systematically convergent real-space all-electron Kohn-Sham density functional theory calculations using an enriched finite element (FE) basis is presented. The enriched FE basis is constructed by augmenting the classical FE basis with atom-centered numerical basis functions, comprising of atomic solutions to the Kohn-Sham problem. Notably, to improve the conditioning, the enrichment functions are orthogonalized with respect to the classical FE basis, without sacrificing the locality of the resultant basis. In addition to improved conditioning, this orthogonalization procedure also renders the overlap matrix block-diagonal, greatly simplifying its inversion. Furthermore, the treatment of electrostatics adopted in this work ensures that the basis is seamlessly applicable to both periodic and non-periodic calculations. A Chebyshev polynomial based filtering technique is used to efficiently compute the occupied eigenspace in each self-consistent field iteration. The basis offers excellent parallel scalability with 92% efficiency at 22 times speedup for a system with 620 electrons.

Additionally, a formulation to compute nuclear forces and the stress tensor needed to perform structural relaxation in this basis is also presented. This is done using the notion of configurational forces which arise from the variational derivative of the Kohn-Sham free energy functional with respect to the position of the material point. This approach, originally proposed in the context of classical FE by Motamarri and Gavini [Phys. Rev. B, 97 , 16 (2018)] , enables computation of variationally consistent nuclear forces and the stress tensor in a unified framework without having to explicitly account for Pulay contributions. In this work, we demonstrate the accuracy of the formulation by comparing the computed forces and stresses for various benchmark systems with those obtained from finite-differencing the ground-state energy. Moreover, our calculations are benchmarked against the Gaussian basis for isolated systems and LAPW+lo basis for periodic systems.

Lastly, the applicability of the basis is extended to solve the time-dependent Kohn-Sham equations in real time. The formulation is based on a recent work by Kanungo and Gavini [Phys. Rev. B, 100 , 11 (2019)], wherein classical finite elements were used for spatial discretization. The presence of enrichment functions in the basis accelerates all-electron TDDFT calculations. Moreover, the block-diagonal nature of the overlap matrix afforded by the orthogonalized enriched finite element basis enables fast and convenient inversion of the overlap matrix that features in the evaluation of the discrete time-evolution operator. The formulation utilizes the second-order Magnus operator, which contains an exponential of a large sparse matrix, as the time propagator. The action of this operator on an orbital is efficiently performed by using a Krylov-subspace projection method. Calculations show close to optimal rates of convergence of the dipole moment with respect to spatial and temporal discretization.

Overall, the work presents an efficient, scalable and accurate basis to perform all-electron DFT calculations. This would be instrumental in the study of defects in crystalline systems where calculations on large systems are inevitable. Moreover, the systematic improvability afforded by the basis makes it a suitable tool to conduct pseudopotential transferability studies.