Inverse Kohn-Sham Density Functional Theory: Progress and Challenges

Abstract

Inverse Kohn-Sham (iKS) problems are needed to fully understand the one-to-one mapping between densities and potentials on which Density Functional Theory is based. They are also important to advance computational schemes that rely on density-to-potential inversions such as the Optimized Effective Potential method and various techniques for density-based embedding. Unlike the forward Kohn-Sham problems, numerical iKS problems are ill-posed and can be unstable. We discuss some of the fundamental and practical difficulties of iKS problems with constrained-optimization methods on finite basis sets. Various factors that affect the performance are systematically compared and discussed, both analytically and numerically, with a focus on two of the most practical methods: the Wu-Yang method (WY) and partial-differential-equation constrained-optimization (PDE-CO). Our analysis of the WY and PDE-CO highlights the limitation of finite basis sets and the importance of regularization. We introduce two new ideas that will hopefully contribute to making iKS problems more tractable: (1) A correction to the WY method that utilizes the null space of the relevant Hessian matrices; and (2) A finite potential basis-set implementation of the PDE-CO method. We provide an overall strategy for performing numerical density-to-potential inversions that can be directly adopted in practice. We also provide an Appendix with several examples that can be used for benchmarking.

References in corpus (27)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• Perspective on density functional theory
• By-passing the Kohn-Sham equations with machine learning
• Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics
• Ab-initio Optimized Effective Potentials for Real Molecules in Optical Cavities: Photon Contributions to the Molecular Ground state
• Exact Kohn-Sham potential of strongly correlated finite systems
• Numerical Methods for the Inverse Problem of Density Functional Theory
• Reduction of Electronic Wavefunctions to Kohn-Sham Effective Potentials
• Neural-network Kohn-Sham exchange-correlation potential and its out-of-training transferability
• Kohn-Sham calculations with the exact functional
• How is the derivative discontinuity related to steps in the exact Kohn-Sham potential?
• Exact density-functional potentials for time-dependent quasiparticles
• Improved method for generating exchange-correlation potentials from electronic wave functions
• Molecular Binding Energies from Partition Density Functional Theory
• Large-scale all-electron density functional theory calculations using an enriched finite element basis
• Degeneracy in Density Functional Theory: Topology in v- and n-Space
• Universal nature of different methods of obtaining the exact Kohn-Sham exchange-correlation potential for a given density
• Nonanalyticity of the optimized effective potential with finite basis sets
• Density-inversion method for the Kohn-Sham potential: role of the screening density
• KS-pies: Kohn-Sham Inversion Toolkit
• Time-Dependent Density Functional Theory Beyond Kohn-Sham Slater Determinants
• Adiabatic and local approximations for the Kohn-Sham potential in time-dependent Hubbard chains
• Accurate effective potential for density amplitude and the corresponding Kohn-Sham exchange-correlation potential calculated from approximate wavefunctions
• Coarse-grained V-representability
• Improvement of functionals in density functional theory by the inverse Kohn--Sham method and density functional perturbation theory
• A first step in the nuclear inverse Kohn-Sham problem: from densities to potentials
• Nuclear energy density functionals from empirical ground-state densities

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• A rigorous formulation of Density Functional Theory for spinless fermions in one dimension
• Regularised density-potential inversion for periodic systems: application to exact exchange in one dimension
• Calculating potential energy surfaces with quantum computers by measuring only the density along adiabatic transitions
• Constrained Search in Imaginary Time
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