Real time time-dependent density functional theory using higher order finite-element methods

Abstract

We present a computationally efficient approach to solve the time-dependent Kohn-Sham equations in real-time using higher-order finite-element spatial discretization, applicable to both pseudopotential and all-electron calculations. To this end, we develop an a priori mesh adaption technique, based on the semi-discrete (discrete in space but continuous in time) error estimate on the time-dependent Kohn-Sham orbitals, to construct an efficient finite-element discretization. Subsequently, we obtain the full-discrete error estimate to guide our choice of the time-step. We employ spectral finite-elements along with special reduced order quadrature to render the overlap matrix diagonal, thereby simplifying the inversion of the overlap matrix that features in the evaluation of the discrete time-evolution operator. We use the second-order Magnus operator as the time-evolution operator, wherein the action of the discrete Magnus operator, expressed as exponential of a matrix, on the Kohn-Sham orbitals is obtained efficiently through an adaptive Lanczos iteration. We observe close to optimal rates of convergence of the dipole moment with respect to spatial and temporal discretization, for both pseudopotential and all-electron calculations. We demonstrate a staggering 100-fold reduction in the computational time afforded by higher-order finite-elements over linear finite-elements, for both pseudopotential and all-electron calculations. Further, for similar level of accuracy, we obtain significant computational savings by our approach as compared to state-of-theart finite-difference methods. We also demonstrate the competence of higher-order finite-elements for all-electron benchmark systems. Lastly, we observe good parallel scalability of the proposed method on many hundreds of processors.

DFT-FE 1.0: A massively parallel hybrid CPU-GPU density functional theory code using finite-element discretization

“…Also related to exchange-correlation functional development is the ongoing effort of implementing systematically convergent inverse DFT approaches [102] into DFT-FE to compute exact exchange-correlation potentials from many-body ground-state electron densities [103], and subsequently improve the exchange-correlation models from this data. Finally, implementation of real-time time-dependent DFT [27] and its GPU porting is also being pursued.…”

Fast and robust all-electron density functional theory calculations in solids using orthogonalized enriched finite elements

“…Thus, the proposed orthogonalized enriched FE basis offers a robust, efficient, systematically convergent, and scalable basis for all-electron DFT calculations, ap-plicable to metallic and non-metallic systems. The use of the orthogonalized enriched FE basis for all-electron time-dependent density functional theory (TDDFT) calculations 74,75 holds good promise, and is currently being investigated. Additionally, the orthogonalized enriched FE ideas, in conjunction with the incorporation of atomic forces 48 , offers a powerful tool for all-electron Born-Oppenheimer molecular dynamics as well as Ehrenfest dynamics, and constitute an active line of our research.…”

Ionic forces and stress tensor in all-electron DFT calculations using enriched finite element basis

“…[48][49][50][51][52][53][54][55][56][57][58] ), recent developments 59,60 have demonstrated the utility of FE basis for conducting fast and accurate large-scale pseudopotential DFT calculations involving many tens of thousands of electrons [59][60][61][62] . However, for all-electron electronic structure calculations, although prior works 48,49,57,58,[63][64][65][66][67][68][69][70][71][72][73] have demonstrated the systematic convergence of the FE basis, the computational cost remains high, given the large number of FE basis functions that are needed to accurately describe the all-electron wavefunctions. The EFE basis, which enriches the FE basis with compactly supported enrichment functions-such as, for e.g., those constructed from single-atom wavefunctions-can be used to significantly reduce the number of FE basis functions, and consequently improve the computational efficiency of all-electron calculations.…”