Real-Time Time-Dependent Density Functional Theory using Higher Order Finite Element Methods
[Technical Report, Research Paper]
MICHIGAN UNIV ANN ARBOR
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We present a computationally efficient approach to solve the time-dependent Kohn-Sham equations in realtime using higher-order finite-element spatial discretization, applicable to both pseudopotential and allelectron calculations. To this end, we develop an a priori mesh adaption technique, based on the semidiscrete discrete in space but continuous in time error estimate on the time-dependent Kohn-Sham orbitals,to construct a close to optimal finite-element discretization. Subsequently, we obtain the full-discrete errorestimate to guide our choice of the time-step. We employ spectral finite-elements along with Gauss-LegendreLobatto quadrature to render the overlap matrix diagonal, thereby simplifying the inversion of the overlapmatrix that features in the evaluation of the discrete time-evolution operator. We use the second-orderMagnus operator as the time-evolution operator in all our calculations. Furthermore, the action of thediscrete Magnus operator, expressed as exponential of a matrix, on the Kohn-Sham orbitals is obtainedefficiently through an adaptive Lanczos iteration. We observe close to optimal rates of convergence of thedipole moment with respect to spatial and temporal discretization, for both pseudopotential and all-electroncalculations. We demonstrate a staggering 100-fold reduction in the computational time afforded by higherorder finite-elements over linear finite-elements, for both pseudopotential and all-electron calculations. Wepresent comparative studies, in terms of accuracy and efficiency, of our approach against finite-differencebased discretization for pseudopotential calculations, and demonstrate significant computational savingswhen compared to the finite-difference method. We also demonstrate the competence of higher-order finiteelements for all-electron benchmark systems. Lastly, we observe good parallel scalability of the proposedmethod on many hundreds of processors.
- Operations Research