Kohn-Sham calculations with the exact functional

We present here a model of the exchange-correlation hole for strongly correlated systems using a simple nonlocal generalization of the Wigner-Seitz radius. The model behaves similarly to the strictly correlated electron approach, which gives the infinitely correlated limit of density functional theory. Unlike the strictly correlated method, however, the energies and potentials of this model can be presently calculated for arbitrary geometries in three dimensions. We discuss how to evaluate the energies and potentials of the nonlocal model, and provide results for many systems where it is also possible to compare to the strictly correlated electron treatment.

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“…For non-self-consistent densities, T S [n] must be evaluated using some other approach, e.g., via inversions [60][61][62][63][64][65][66][67]. …”

Section: Introductionmentioning

confidence: 99%

Electron avoidance: A nonlocal radius for strong correlation

Wagner

Gori‐Giorgi

2014

Phys. Rev. A

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“…Other modifications are also possible such as the use of an approximate density-density response matrix to help guide the iterations. [38] This method produces smooth potentials by construction provided that the underlying eigenvalue solver produces smooth orbitals. The reliance on a prefactor, however, requires some manual intervention or heuristics to ensure convergence.…”

Section: This Iterative Inversion Scheme Of Van Leeuwen and Baerends mentioning

confidence: 99%

Numerical methods for the inverse problem of density functional theory

Jensen

Wasserman

2017

Int J of Quantum Chemistry

129

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The inverse problem of Kohn-Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to onedimensional finite and periodic model systems. K E Y W O R D Sdensity functional theory, inverse problems, PDE-constrained optimization | I N TR ODU C TI ONDensity functional theory (DFT) is a very popular and formally exact quantum many-body theory that has successfully been used to model both finite and periodic systems. [1,2] Much of this success stems from the availability of approximate exchange-correlation functionals used as input in the direct problem of DFT to produce the ground-state electronic density of a given system. Although far less common, this process can be reversed and the true exchange-correlation potential can be recovered from a given electronic density by solving the inverse problem of Kohn-Sham (KS) DFT. (Other flavors of DFT exist but we focus on the standard ground-state version of KS DFT and drop the KS prefix for brevity throughout this tutorial.) Although both the forward and inverse problems use the same equations, the difference in input and output variables leads to major differences in solution methods for the two problems. We refer to the inverse problem of DFT as a density-to-potential inversion and the numerical methods required for solving this problem are the main focus of this tutorial. In this section, we give a brief introduction to inverse problems and explain how density-to-potential inversions can be used within DFT.Inverse problems are common in science and have been central in quantum mechanics since its inception. Much of what we know about the structure of matter has come from scattering experiments, [3] which can be described mathematically as inverse problems. A variety of mathematical and numerical techniques exist for solving scattering problems as well as other inverse problems.[4] These methods are often very different from the methods used for solving the direct problems due to the differences in mathematical structure and input data.According to Hadamard, [5] a problem is well-posed if a solution exists, it is unique, and it depends continuously on the data. If any of the three properties listed above is violated then the problem is ill-posed. The forward or direct problem of DFT is widely considered to be well-posed thanks to several proofs including the Hohenberg-Kohn Theorems and the constrained-search formulation of Levy and Lieb. [1] The inverse problem of DFT can also be well-posed for dis...

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“…The wavefunction-to-density map has been fully realized for fixed particle number, i.e. for the twoelectron singlet case by [54,55] and for the more-electron case by [56,57]. In principle, a small system size of the model used in this work allows to graphically illustrate the complete density-to-wavefunction map in the full Fock space of the system, and can be used to show how features such as the intra-system steepening and the inter-system derivative discontinuity of the density-to-potential map appear in the density-to-wavefunction map.…”

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confidence: 99%

Exact maps in density functional theory for lattice models

et al. 2016

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In the present work, we employ exact diagonalization for model systems on a real-space lattice to explicitly construct the exact density-to-potential and graphically illustrate the complete exact density-to-wavefunction map that underly the Hohenberg-Kohn theorem in density functional theory. Having the explicit wavefunction-to-density map at hand, we are able to construct arbitrary observables as functionals of the ground-state density. We analyze the density-to-potential map as the distance between the fragments of a system increases and the correlation in the system grows. We observe a feature that gradually develops in the density-to-potential map as well as in the density-towavefunction map. This feature is inherited by arbitrary expectation values as functional of the ground-state density. We explicitly show the excited-state energies, the excited-state densities, and the correlation entropy as functionals of the ground-state density. All of them show this exact feature that sharpens as the coupling of the fragments decreases and the correlation grows. We denominate this feature as intra-system steepening and discuss how it relates to the well-known inter-system derivative discontinuity. The inter-system derivative discontinuity is an exact concept for coupled subsystems with degenerate ground state. However, the coupling between subsystems as in charge transfer processes can lift the degeneracy. An important conclusion is that for such systems with a neardegenerate ground state, the corresponding cut along the particle number N of the exact density functionals is differentiable with a well-defined gradient near integer particle number. IntroductionOver the last decades ground-state density-functional theory (DFT) has become a mature tool in material science and quantum chemistry [1][2][3][4][5]. Provided that the exact exchange-correlation (xc) functional is known, DFT is a formally exact framework of the quantum many-body problem. In practice, the accuracy of observables in DFT highly depends on the choice of the approximate xc-functional. From the local density approximation (LDA) [6], to the gradient expansions such as the generalized gradient approximations (GGAs), e.g. Perdew-Burke-Enzerhof [7] and the hybrid functionals such as B3LYP [8], to the orbital-functionals such as optimized effective potentials [9] and to the range-separated hybrids such as HSE06 [10], the last decades have seen great efforts and achievements in the development of functionals with more accurate and reliable prediction capability.Nonetheless, available approximate functionals such as the LDA, the GGA's and the hybrid functionals have known shortcomings to model gaps of semiconductors [11], molecular dissociation curves [12], barriers of chemical reactions [13], polarizabilities of molecular chains [14, 15], and charge-transfer excitation energies, particularly between open-shell molecules [16].Practical applications of density functional theory encounter two major problems: (i) while the Hohenberg-Kohn theorem tells us that a...

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Kohn-Sham calculations with the exact functional

Cited by 50 publications

References 84 publications

Electron avoidance: A nonlocal radius for strong correlation

Electron avoidance: A nonlocal radius for strong correlation

Numerical methods for the inverse problem of density functional theory

Exact maps in density functional theory for lattice models

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