Electronic Structure via Potential Functional Approximations

Cangi, Attila; Lee, Dong‐Hyung; Elliott, Peter; Burke, Kieron; Gross, E. K. U.

doi:10.1103/physrevlett.106.236404

Cited by 49 publications

(65 citation statements)

References 23 publications

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“…Recently, the search for an orbital-free version of DFT (OF-DFT) has rapidly gained attention. [11][12][13][14][15] To apply "orbital free" approaches, it is essential to find highly accurate kinetic energy density functionals (KEDF). The subject is not new and its origins date back to the early years of quantum mechanics.…”

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confidence: 99%

Kinetic energy density for orbital‐free density functional calculations by axiomatic approach

Alharbi

Kais

2017

Int J of Quantum Chemistry

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An axiomatic approach is herein used to determine the physically acceptable forms for general D‐dimensional kinetic energy density functionals (KEDF). The resulted expansion captures most of the known forms of one‐point KEDFs. By statistically training the KEDF forms on a model problem of noninteracting kinetic energy in 1D (six terms only), the mean relative accuracy for 1000 randomly generated potentials is found to be better than the standard KEDF by several orders of magnitudes. The accuracy improves with the number of occupied states and was found to be better than 10−4 for a system with four occupied states. Furthermore, we show that free fitting of the coefficients associated with known KEDFs approaches the exactly analytic values. The presented approach can open a new route to search for physically acceptable kinetic energy density functionals and provide an essential step toward more accurate large‐scale orbital free density functional theory calculations.

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Kinetic energy density for orbital‐free density functional calculations by axiomatic approach

Alharbi

Kais

2017

Int J of Quantum Chemistry

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“…At zero temperature, potential functional theory (PFT) is a promising approach to the electronic structure problem [20,21]. It is also orbital-free, but skirts the troublesome issue of separately approximating the KS kinetic energy.…”

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“…It is also orbital-free, but skirts the troublesome issue of separately approximating the KS kinetic energy. PFT's coupling-constant formalism automatically generates a highly accurate kinetic energy potential functional approximation (PFA) for any density PFA [20]. In this way, one needs only find a sufficiently accurate density approximation [22].…”

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“…In analogy to the zero-temperature case [20], we introduce a coupling constant λ in the onebody potential, v λ (r) = (1 − λ)v 0 (r) + λv(r), where v 0 is some reference potential. Via the Hellmann-Feynman theorem, we rewrite the grand potential,…”

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“…Solution of these equations at every time-step is the most costly step of DFT molecular dynamics. The KS potential is defined [20,21] …”

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Efficient formalism for warm dense matter simulations

Cangi

Pribram‐Jones

2015

Phys. Rev. B

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Understanding kernel ridge regression: Common behaviors from simple functions to density functionals

Snyder

et al. 2015

Int J of Quantum Chemistry

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Accurate approximations to density functionals have recently been obtained via machine learning (ML). By applying ML to a simple function of one variable without any random sampling, we extract the qualitative dependence of errors on hyperparameters. We find universal features of the behavior in extreme limits, including both very small and very large length scales, and the noise-free limit. We show how such features arise in ML models of density functionals. V C 2015 Wiley Periodicals, Inc.DOI: 10.1002/qua.24939 IntroductionMachine learning (ML) is a powerful data-driven method for learning patterns in high-dimensional spaces via induction. It provides a whole suite of tools for analyzing data, fitting highly nonlinear functions, and dimensionality reduction. [1] Given a set of training data, ML algorithms learn via induction to predict new data. ML methods have been developed within the areas of statistics and computer science, and have been applied to a huge variety of data, including neuroscience, image and text processing, and robotics. We are interested primarily in kernel ridge regression (KRR), which is one such standard method in ML. In general, the quality of the KRR learning model performance is highly dependent on the hyperparameters chosen and the size of the training data.ML has enjoyed widespread success in many fields, and has recently become popular as a tool in quantum chemistry and materials science, [2][3][4][5][6][7][8][9][10] as shown by the articles in this special issue. In many of these applications, many ab initio calculations are performed, and ML is applied to various properties of the results of these calculations. Our primary interest is in creating a much more intimate relation between ML and electronic structure calculations. Many such calculations use density functional theory (DFT), because of its favorable balance between accuracy and computational efficiency. But all such calculations rely on some approximation of an energy component as a functional of the electronic density. In particular, we wish to explore the applications of ML to the construction of density functionals, [11][12][13][14][15] which have focused so far on approximating the kinetic energy (KE) of noninteracting electrons. An accurate, general approximation to this could make orbital-free DFT a practical reality. However, the ML methods that have been developed are quite general and have not been tailored to account for specific details of the quantum problem. For example, it was found that KRR could yield excellent results for the KE functional, while never yielding accurate functional derivatives. [11] The development of methods for bypassing this difficulty has been important for ML in general. [14] In this context, ML provides a completely different way of thinking about electronic structure. The traditional ab initio approach [16] to electronic structure involves deriving carefully constructed approximations to solve the Schr€ odinger equation, based on physical intuition, exact conditions and asymptotic b...

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Electronic Structure via Potential Functional Approximations

Cited by 49 publications

References 23 publications

Kinetic energy density for orbital‐free density functional calculations by axiomatic approach

Kinetic energy density for orbital‐free density functional calculations by axiomatic approach

Efficient formalism for warm dense matter simulations

Understanding kernel ridge regression: Common behaviors from simple functions to density functionals

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