The quest for accurate exchange-correlation functionals has long remained a grand challenge in density functional theory (DFT), as it describes the many-electron quantum mechanical behavior through a computationally tractable quantity—the electron density—without resorting to multi-electron wave functions. The inverse DFT problem of mapping the ground-state density to its exchange-correlation potential is instrumental in aiding functional development in DFT. However, the lack of an accurate and systematically convergent approach has left the problem unresolved, heretofore. This work presents a numerically robust and accurate scheme to evaluate the exact exchange-correlation potentials from correlated ab-initio densities. We cast the inverse DFT problem as a constrained optimization problem and employ a finite-element basis—a systematically convergent and complete basis—to discretize the problem. We demonstrate the accuracy and efficacy of our approach for both weakly and strongly correlated molecular systems, including up to 58 electrons, showing relevance to realistic polyatomic molecules.
We present a computationally efficient approach to perform large-scale all-electron density functional theory calculations by enriching the classical finite element basis with compactly supported atom-centered numerical basis functions that are constructed from the solution of the Kohn-Sham (KS) problem for single atoms. We term these numerical basis functions as enrichment functions, and the resultant basis as the enriched finite element basis. The compact support for the enrichment functions is obtained by using smooth cutoff functions, which enhances the conditioning and maintains the locality of the enriched finite element basis. The integrals involved in the evaluation of the discrete KS Hamiltonian and overlap matrix in the enriched finite element basis are computed using an adaptive quadrature grid that is constructed based on the characteristics of enrichment functions. Further, we propose an efficient scheme to invert the overlap matrix by using a block-wise matrix inversion in conjunction with special reduced-order quadrature rules, which is required to transform the discrete Kohn-Sham problem to a standard eigenvalue problem. Finally, we solve the resulting standard eigenvalue problem, in each self-consistent field iteration, by using a Chebyshev polynomial based filtering technique to compute the relevant eigenspectrum. We demonstrate the accuracy, efficiency and parallel scalability of the proposed method on semiconducting and heavymetallic systems of various sizes, with the largest system containing 8694 electrons. We obtain accuracies in the ground-state energies that are ∼ 1 mHa with reference ground-state energies employing classical finite element as well as gaussian basis sets. Using the proposed formulation based on enriched finite element basis, for accuracies commensurate with chemical accuracy, we observe a staggering 50−300 fold reduction in the overall computational time when compared to classical finite element basis. Further, we find a significant outperformance by the enriched finite element basis when compared to the gaussian basis for the modest system sizes where we obtained convergence with gaussian basis. We also observe good parallel scalability of the numerical implementation up to 384 processors for a representative benchmark system comprising of 280-atom silicon nano-cluster.
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.
Accurate exchange−correlation (XC) potentials for three-dimensional systems via solution of the inverse density functional theory (DFT) problemare now available to test the quality of DFT approximations. Herein, the exact XC potential for seven molecules dihydrogen at four different bond-lengths, lithium hydride, water, and ortho-benzyneare computed from full configuration interaction reference densities. These are compared to model XC potentials from nonlocal (B3LYP, HSE06, SCAN0, and M08-HX) and semilocal/local (SCAN, PBE, and PW92) XC functionals. Whereas for most systems, relative errors in the ground-state densities are − − −
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.