Exact and Model Exchange-Correlation Potentials for Open-Shell Systems

Abstract

The conventional approaches to the inverse density functional theory problem typically assume nondegeneracy of the Kohn−Sham (KS) eigenvalues, greatly hindering their use in open-shell systems. We present a generalization of the inverse density functional theory problem that can seamlessly admit degenerate KS eigenvalues. Additionally, we allow for fractional occupancy of the Kohn−Sham orbitals to also handle noninteracting ensemblev-representable densities, as opposed to just noninteracting pure-v-representable densities. We present the exact exchange-correlation (XC) potentials for six open-shell systems�four atoms (Li, C, N, and O) and two molecules (CN and CH 2 )�using accurate ground-state densities from configuration interaction calculations. We compare these exact XC potentials with model XC potentials obtained using nonlocal (B3LYP, SCAN0) and local/semilocal (SCAN, PBE, PW92) XC functionals. Although the relative errors in the densities obtained from these DFT functionals are of (10 −3 to 10 −2 ), the relative errors in the model XC potentials remain substantially large� (10 −1 to 10 0 ).

How Does HF-DFT Achieve Chemical Accuracy for Water Clusters?

“…The inverse Kohn–Sham calculations are conducted using the partial differential equation constrained optimization approach presented in refs − . Although the Hohenberg–Kohn theorem guarantees a unique Kohn–Sham potential for a given density, numerical inverse Kohn–Sham calculations are ill-conditioned.…”

“…These incorrect asymptotics in the Gaussian densities induce spurious oscillations in the resulting potential. − Our approach addresses these issues by using (i) a systematically convergent finite-element basis, which renders the discrete problem well-posed; (ii) a small correction to the Gaussian densities to ensure cusps at the nuclei; and (iii) appropriate far-field boundary conditions on the Kohn–Sham potential to ensure the expected −1/ r decay. We refer to refs − for details of the method, and the accuracy and efficacy of our approach in obtaining accurate potentials from a target density. In this work, for our inverse Kohn–Sham calculations on the water trimer (3UUD conformation from BEGDB), we use fifth-order finite elements (fifth-order three-dimensional Lagrange polynomials) to discretize the Kohn–Sham orbitals, similar to those used in our previous work .…”

How does HF-DFT achieve chemical accuracy for water clusters?

“…This is done by the accurate Kohn-Sham inversion method described and benchmarked in Refs. [44][45][46][47] that yields a local Kohn-Sham effective potential and corresponding orbitals for that density. the CCSD(T) total energies and binding energies converge slowly with respect to basis set, and may not be well-converged with respect to the basis set.…”

Tucker Tensor Approach for Accelerating Fock Exchange Computations in a Real-Space Finite-Element Discretization of Generalized Kohn–Sham Density Functional Theory

“…The FE basis is a piecewise polynomial basis with desirable features such as locality of the basis which is suitable for good parallel scalability, spatial adaptivity to efficiently handle nonperiodic systems and all-electron calculations, and its ability to handle general boundary conditions (periodic, nonperiodic, and semiperiodic). DFT-FE, , a massively parallel, open-source, real-space DFT code using adaptive higher-order FE discretization, has been shown to handle large-scale systems with scriptO ( 10 5 ) electrons, exhibiting good parallel scalability on many-core and hybrid CPU-GPU architectures, , and has recently been employed in studying a range of problems, including the electronic structure of DNA molecules, dislocations in crystalline materials, , phase transformations in doped nanofilms, computing spin Hamiltonian parameters of systems with defects, , and tackling the inverse DFT problem. − …”