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Emily A. Carter - One of the best experts on this subject based on the ideXlab platform.
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Kinetic Energy Density of nearly free electrons i response functionals of the external potential
Physical Review B, 2019Co-Authors: William C Witt, Emily A. CarterAbstract:Free electrons have a uniform Kinetic Energy Density (KED), which evolves into a spatially varying quantity as the electrons respond to the gradual imposition of an external potential. In this paper and a companion paper, we examine two sets of functionals for describing the local, non-negative KED that emerges after such a perturbation. In this paper, we emphasize potential functionals, deriving the first- and second-order deviations from the free-electron KED as functionals of the perturbing potential, also reconsidering the analogous functionals for the local Density of states and the electron Density. (In the second paper, we use these results to re-express the KED response in terms of functionals of the induced electron Density.) We develop reciprocal-space formulations of the response kernels to complement previously known real-space forms. The first-order function is straightforward to obtain, but the second-order function requires considerable effort. To manage the derivations, we relate the KED response to that of the one-electron Green function, and then examine the latter in detail. Finally, we provide extensive validation of the derived response functions based on asymptotic analysis of an integral representation, numerical integration of the same generating integral, and application to the linear potential model.
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Kinetic Energy Density of nearly free electrons ii response functionals of the electron Density
Physical Review B, 2019Co-Authors: William C Witt, Emily A. CarterAbstract:We present electron-Density-based response functionals yielding the non-negative Kinetic Energy Density (KED) of nearly free electron systems. In a previous paper, for a canonical free-electron system perturbed by an external potential, we derived the first- and second-order corrections to the KED as functionals of the potential, providing the response functions in reciprocal space. Here, we formulate the KED response in terms of the electron Density by converting the potential-based functionals into Density functionals. We also determine the related response of the Pauli KED, which is the KED in excess of the von Weizs\"acker KED. We anticipate that the structure of these Density functionals will help guide the design of the more sophisticated Kinetic Energy functionals required for orbital-free Density functional theory simulations. We conclude by examining the performance of the first- and second-order Density functionals for the KED when applied to electron densities generated from local pseudopotential calculations for Li, Al, and Si crystals.
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upper bound to the gradient based Kinetic Energy Density of noninteracting electrons in an external potential
Journal of Chemical Physics, 2019Co-Authors: William C Witt, Kaili Jiang, Emily A. CarterAbstract:We examine a simple upper bound to the gradient-based Kinetic Energy Density (KED) of noninteracting electrons in an external potential: t(r) ≤ [μ−v(r)]n(r)+14∇2n(r), where t(r) is the gradient-based (non-negative) KED, μ is the Fermi Energy, v(r) is the external potential, and n(r) is the electron Density. The bound emerges naturally from a well-known expression for t(r), leading to an intuitive physical interpretation. For example, t(r) approaches the upper bound in regions where the electron Density consists mainly of contributions from states with energies close to the Fermi Energy. This upper bound complements the orbital-free lower bound provided by the gradient form of the von Weizsacker (vW) KED, which is also non-negative. Both bounds yield t(r) exactly for single-orbital systems, and accordingly, they merge in single-orbital regions of more general systems. We demonstrate the universality of the two bounds over a wide range of test systems, including model potentials, atoms, diatomic molecules, and pseudopotential approximations of crystals. We also show that the exact t(r) frequently exceeds the sum of the vW and Thomas-Fermi KEDs, rendering that sum unsuitable as a strict upper bound to the gradient-based KED.We examine a simple upper bound to the gradient-based Kinetic Energy Density (KED) of noninteracting electrons in an external potential: t(r) ≤ [μ−v(r)]n(r)+14∇2n(r), where t(r) is the gradient-based (non-negative) KED, μ is the Fermi Energy, v(r) is the external potential, and n(r) is the electron Density. The bound emerges naturally from a well-known expression for t(r), leading to an intuitive physical interpretation. For example, t(r) approaches the upper bound in regions where the electron Density consists mainly of contributions from states with energies close to the Fermi Energy. This upper bound complements the orbital-free lower bound provided by the gradient form of the von Weizsacker (vW) KED, which is also non-negative. Both bounds yield t(r) exactly for single-orbital systems, and accordingly, they merge in single-orbital regions of more general systems. We demonstrate the universality of the two bounds over a wide range of test systems, including model potentials, atoms, diatomic molecules, ...
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upper bound to the gradient based Kinetic Energy Density of noninteracting electrons in an external potential
Journal of Chemical Physics, 2019Co-Authors: William C Witt, Kaili Jiang, Emily A. CarterAbstract:We examine a simple upper bound to the gradient-based Kinetic Energy Density (KED) of noninteracting electrons in an external potential: t(r) ≤ [μ−v(r)]n(r)+14∇2n(r), where t(r) is the gradient-based (non-negative) KED, μ is the Fermi Energy, v(r) is the external potential, and n(r) is the electron Density. The bound emerges naturally from a well-known expression for t(r), leading to an intuitive physical interpretation. For example, t(r) approaches the upper bound in regions where the electron Density consists mainly of contributions from states with energies close to the Fermi Energy. This upper bound complements the orbital-free lower bound provided by the gradient form of the von Weizsacker (vW) KED, which is also non-negative. Both bounds yield t(r) exactly for single-orbital systems, and accordingly, they merge in single-orbital regions of more general systems. We demonstrate the universality of the two bounds over a wide range of test systems, including model potentials, atoms, diatomic molecules, and pseudopotential approximations of crystals. We also show that the exact t(r) frequently exceeds the sum of the vW and Thomas-Fermi KEDs, rendering that sum unsuitable as a strict upper bound to the gradient-based KED.
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reply to comment on single point Kinetic Energy Density functionals a pointwise Kinetic Energy Density analysis and numerical convergence investigation
Physical Review B, 2015Co-Authors: Junchao Xia, Emily A. CarterAbstract:We find that the multivalued character of the $G$ factor as a function of the reduced gradient ($s$) still exists after accounting for pseudopotential artifacts and the Kinetic Energy global upper bound. We also find that the VT84F functional indeed exhibits stable convergence and more reasonable results for self-consistent bulk properties compared to other generalized gradient approximation (GGA) Kinetic Energy Density functionals (KEDFs) that we tested earlier. However, VT84F generally yields overestimated equilibrium volumes, which may result from its inability (as with all GGAs) to reproduce the $G\ensuremath{-}s$ multivalued character. The analogous failure to predict the multivalued character of $G$ as a function of the reduced Density $(d)$ is also likely to be responsible for the inaccuracy of our vWGTF functionals reported earlier. Our multivaluedness analysis therefore does not impugn any particular GGA KEDF. Instead, it merely confirms the importance of pointwise analysis for improving KEDFs by emphasizing the need to resolve the multivaluedness of $G$ with respect to various Density variables.
Hiromi Nakai - One of the best experts on this subject based on the ideXlab platform.
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orbital free Density functional theory calculation applying semi local machine learned Kinetic Energy Density functional and Kinetic potential
Chemical Physics Letters, 2020Co-Authors: Mikito Fujinami, Junji Seino, Ryo Kageyama, Yasuhiro Ikabata, Hiromi NakaiAbstract:Abstract This letter proposes a scheme of orbital-free Density functional theory (OF-DFT) calculation for optimizing electron Density based on a semi-local machine-learned (ML) Kinetic Energy Density functional (KEDF). The electron Density, which is represented by the square of the linear combination of Gaussian functions, is optimized using derivatives of electronic Energy including ML Kinetic potential (KP). The numerical assessments confirmed the accuracy of optimized Density and total Energy for atoms and small molecules obtained by the present scheme based on ML-KEDF and ML-KP.
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restoring the iso orbital limit of the Kinetic Energy Density in relativistic Density functional theory
Journal of Chemical Physics, 2019Co-Authors: Toni M Maier, Yasuhiro Ikabata, Hiromi NakaiAbstract:In contrast to nonrelativistic Density functional theory, the ratio between the von Weizsacker and the Kohn-Sham Kinetic Energy Density, commonly used as iso-orbital indicator t within exchange-correlation functionals beyond the generalized-gradient level, violates the exact iso-orbital limit and the appropriate parameter range, 0 ≤ t ≤ 1, in relativistic Density functional theory. Based on the exact decoupling procedure within the infinite-order two-component method and the Cauchy-Schwarz inequality, we present corrections to the relativistic and the picture-change-transformed nonrelativistic Kinetic Energy Density that restores these exact constraints. We discuss the origin of the new correction terms and illustrate the effectiveness of the current approach for several representative cases. The proposed generalized iso-orbital indicator tλ is expected to be a useful ingredient for the development of relativistic exchange-correlation functionals.
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semi local machine learned Kinetic Energy Density functional demonstrating smooth potential Energy curves
Chemical Physics Letters, 2019Co-Authors: Junji Seino, Ryo Kageyama, Mikito Fujinami, Yasuhiro Ikabata, Hiromi NakaiAbstract:Abstract This letter investigates the accuracy of the semi-local machine-learned Kinetic Energy Density functional (KEDF) for potential Energy curves (PECs) in typical small molecules. The present functional is based on a previously developed functional adopting electron densities and their gradients up to the third order as descriptors (Seino et al., 2018). It further introduces new descriptors, namely, the distances between grid points and centers of nuclei, to describe the non-local nature of the KEDF. The numerical results show a reasonable performance of the present model in reproducing the PECs of small molecules with single, double, and triple bonds.
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semi local machine learned Kinetic Energy Density functional with third order gradients of electron Density
Journal of Chemical Physics, 2018Co-Authors: Junji Seino, Ryo Kageyama, Mikito Fujinami, Yasuhiro Ikabata, Hiromi NakaiAbstract:A semi-local Kinetic Energy Density functional (KEDF) was constructed based on machine learning (ML). The present scheme adopts electron densities and their gradients up to third-order as the explanatory variables for ML and the Kohn-Sham (KS) Kinetic Energy Density as the response variable in atoms and molecules. Numerical assessments of the present scheme were performed in atomic and molecular systems, including first- and second-period elements. The results of 37 conventional KEDFs with explicit formulae were also compared with those of the ML KEDF with an implicit formula. The inclusion of the higher order gradients reduces the deviation of the total Kinetic energies from the KS calculations in a stepwise manner. Furthermore, our scheme with the third-order gradient resulted in the closest Kinetic energies to the KS calculations out of the presented functionals.
J. E. Alvarellos - One of the best experts on this subject based on the ideXlab platform.
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generalized nonlocal Kinetic Energy Density functionals based on the von weizsacker functional
Physical Chemistry Chemical Physics, 2012Co-Authors: David Garciaaldea, J. E. AlvarellosAbstract:We generalize the ideas behind the procedure for the construction of Kinetic Energy Density functionals with a nonlocal term based on the structure of the von Weizsacker functional, and present several types of nonlocal terms. In all cases, the functionals are constructed such that they reproduce the linear response function of the homogeneous electron gas. These functionals are designed by rewriting the von Weizsacker functional with the help of a parameter β that determines the power of the electron Density in the expression, a strategy we have previously used in the generalization of Thomas–Fermi nonlocal functionals. Benchmark calculations in localized systems have been performed with these functionals to test both their relative errors and the quality of their local behavior. We have obtained competitive results when compared to semilocal and previous nonlocal functionals, the generalized nonlocal von Weizsacker functionals giving very good results for the total Kinetic energies and improving the local behavior of the Kinetic Energy Density. In addition, all the functionals discussed in this paper, when using an adequate reference Density, can be evaluated as a single integral in momentum space, resulting in a quasilinear scaling for the computational cost.
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fully nonlocal Kinetic Energy Density functionals a proposal and a general assessment for atomic systems
Journal of Chemical Physics, 2008Co-Authors: David Garciaaldea, J. E. AlvarellosAbstract:Following some recent ideas on the construction of Kinetic Energy Density functionals that reproduce the linear response function of the homogeneous electron gas, a family of them with a nonlocal term based on the von Weizsacker functional and with a dependence on the logarithm of the Density is presented. As localized systems are the most difficult to study with explicit Kinetic functionals, in this paper we apply to atomic systems a number of families of fully nonlocal Kinetic functionals. We have put our attention in both the total Kinetic Energy and the local behavior of the Kinetic Energy Density, and the results clearly show the quality of these fully nonlocal functionals. They make a good description of the local behavior of the Kinetic Energy Density and maintain good results for the total Kinetic energies. We must remark that almost all the functionals discussed in the paper, when using an adequate reference Density, can be evaluated as a single integral in momentum space, with a quasilinear scal...
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approach to Kinetic Energy Density functionals nonlocal terms with the structure of the von weizsacker functional
Physical Review A, 2008Co-Authors: David Garciaaldea, J. E. AlvarellosAbstract:We propose a Kinetic Energy Density functional scheme with nonlocal terms based on the von Weizs\"acker functional, instead of the more traditional approach where the nonlocal terms have the structure of the Thomas-Fermi functional. The proposed functionals recover the exact Kinetic Energy and reproduce the linear response function of homogeneous electron systems. In order to assess their quality, we have tested the total Kinetic energies as well as the Kinetic Energy Density for atoms. The results show that these nonlocal functionals give as good results as the most sophisticated functionals in the literature. The proposed scheme for constructing the functionals means a step ahead in the field of fully nonlocal Kinetic Energy functionals, because they are capable of giving better local behavior than the semilocal functionals, yielding at the same time accurate results for total Kinetic energies. Moreover, the functionals enjoy the possibility of being evaluated as a single integral in momentum space if an adequate reference Density is defined, and then quasilinear scaling for the computational cost can be achieved.
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Kinetic Energy Density Study of Some Representative Semilocal Kinetic Energy Functionals
The Journal of chemical physics, 2007Co-Authors: David García-aldea, J. E. AlvarellosAbstract:There is a number of explicit Kinetic Energy Density functionals for non-interacting electron systems that are obtained in terms of the electron Density and its derivatives. These semilocal functionals have been widely used in the literature. In this work we present a comparative study of the Kinetic Energy Density of these semilocal functionals, stressing the importance of the local behavior to assess the quality of the functionals. We propose a quality factor that measures the local differences between the usual orbital-based Kinetic Energy Density distributions and the approximated ones, allowing to ensure if the good results obtained for the total Kinetic energies with these semilocal functionals are due to their correct local performance or to error cancellations. We have also included contributions coming from the laplacian of the electron Density to work with an infinite set of Kinetic Energy densities. For all the functionals but one we have found that their success in the evaluation of the total Kinetic Energy are due to global error cancellations, whereas the local behavior of their Kinetic Energy Density becomes worse than that corresponding to the Thomas-Fermi functional.
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Kinetic Energy Density study of some representative semilocal Kinetic Energy functionals
Journal of Chemical Physics, 2007Co-Authors: David García-aldea, J. E. AlvarellosAbstract:There is a number of explicit Kinetic Energy Density functionals for noninteracting electron systems that are obtained in terms of the electron Density and its derivatives. These semilocal functionals have been widely used in the literature. In this work, we present a comparative study of the Kinetic Energy Density of these semilocal functionals, stressing the importance of the local behavior to assess the quality of the functionals. We propose a quality factor that measures the local differences between the usual orbital-based Kinetic Energy Density distributions and the approximated ones, allowing us to ensure if the good results obtained for the total Kinetic energies with these semilocal functionals are due to their correct local performance or to error cancellations. We have also included contributions coming from the Laplacian of the electron Density to work with an infinite set of Kinetic Energy densities. For all but one of the functionals, we have found that their success in the evaluation of the total Kinetic Energy is due to global error cancellations, whereas the local behavior of their Kinetic Energy Density becomes worse than that corresponding to the Thomas-Fermi functional.
Pavlo Golub - One of the best experts on this subject based on the ideXlab platform.
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data driven Kinetic Energy Density fitting for orbital free dft linear vs gaussian process regression
arXiv: Computational Physics, 2020Co-Authors: Sergei Manzhos, Pavlo GolubAbstract:We study the dependence of Kinetic Energy densities (KED) on Density-dependent variables that have been suggested in previous works on Kinetic Energy functionals (KEF) for orbital-free DFT (OF-DFT). We focus on the role of data distribution and on data and regressor selection. We compare unweighted and weighted linear and Gaussian process regressions of KED for light metals and a semiconductor. We find that good quality linear regression resulting in good Energy-volume dependence is possible over Density-dependent variables suggested in previous literature. This is achieved with weighted fitting based on KED histogram. With Gaussian process regressions, excellent KED fit quality well exceeding that of linear regressions is obtained as well as a good Energy-volume dependence which was somewhat better than that of best linear regressions. We find that while the use of the effective potential as a descriptor improves linear KED fitting, it does not improve the quality of the Energy-volume dependence with linear regressions but substantially improves it with Gaussian process regression. Gaussian process regression is also able to perform well without data weighting.
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Kinetic Energy densities based on the fourth order gradient expansion performance in different classes of materials and improvement via machine learning
Physical Chemistry Chemical Physics, 2019Co-Authors: Pavlo Golub, Sergei ManzhosAbstract:We study the performance of fourth-order gradient expansions of the Kinetic Energy Density (KED) in semi-local Kinetic Energy functionals depending on the Density-dependent variables. The formal fourth-order expansion is convergent for periodic systems and small molecules but does not improve over the second-order expansion (the Thomas-Fermi term plus one-ninth of the von Weizsacker term). Linear fitting of the expansion coefficients somewhat improves on the formal expansion. The tuning of the fourth order expansion coefficients allows for better reproducibility of the Kohn-Sham Kinetic Energy Density than the tuning of the second-order expansion coefficients alone. The possibility of a much more accurate match with the Kohn-Sham Kinetic Energy Density by using neural networks (NN) trained using the terms of the 4th order expansion as Density-dependent variables is demonstrated. We obtain ultra-low fitting errors without overfitting of NN parameters. Small single hidden layer neural networks can provide good accuracy in separate KED fits of each compound, while for joint fitting of KEDs of multiple compounds multiple hidden layers were required to achieve good fit quality. The critical issue of data distribution is highlighted. We also show the critical role of pseudopotentials in the performance of the expansion, where in the case of a too rapid decay of the valence Density at the nucleus with some pseudopotentials, numeric instabilities can arise.
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Kinetic Energy densities based on the fourth order gradient expansion performance in different classes of materials and improvement via machine learning
arXiv: Computational Physics, 2018Co-Authors: Pavlo Golub, Sergei ManzhosAbstract:We study the performance of fourth-order gradient expansions of the Kinetic Energy Density (KED) in semi-local Kinetic Energy functionals depending on the Density-dependent variables. The formal fourth-order expansion is convergent for periodic systems and small molecules but does not improve over the second-order expansion (Thomas-Fermi term plus one-ninth of von Weizsacker term). Linear fitting of the expansion coefficients somewhat improves on the formal expansion. The tuning of the fourth order expansion coefficients allows for better reproducibility of Kohn-Sham Kinetic Energy Density than the tuning of the second-order expansion coefficients alone. The possibility of a much more accurate match with the Kohn-Sham Kinetic Energy Density by using neural networks trained using the terms of the 4th order expansion as Density-dependent variables is demonstrated. We obtain ultra-low fitting errors without overfitting. Small single hidden layer neural networks can provide good accuracy in separate KED fits of each compound, while for joint fitting of KEDs of multiple compounds multiple hidden layers were required to achieve good fit quality. The critical issue of data distribution is highlighted. We also show the critical role of pseudopotentials in the performance of the expansion, where in the case of a too rapid decay of the valence Density at the nucleus with some pseudopotentials, numeric instabilities arise.
Junji Seino - One of the best experts on this subject based on the ideXlab platform.
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orbital free Density functional theory calculation applying semi local machine learned Kinetic Energy Density functional and Kinetic potential
Chemical Physics Letters, 2020Co-Authors: Mikito Fujinami, Junji Seino, Ryo Kageyama, Yasuhiro Ikabata, Hiromi NakaiAbstract:Abstract This letter proposes a scheme of orbital-free Density functional theory (OF-DFT) calculation for optimizing electron Density based on a semi-local machine-learned (ML) Kinetic Energy Density functional (KEDF). The electron Density, which is represented by the square of the linear combination of Gaussian functions, is optimized using derivatives of electronic Energy including ML Kinetic potential (KP). The numerical assessments confirmed the accuracy of optimized Density and total Energy for atoms and small molecules obtained by the present scheme based on ML-KEDF and ML-KP.
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semi local machine learned Kinetic Energy Density functional demonstrating smooth potential Energy curves
Chemical Physics Letters, 2019Co-Authors: Junji Seino, Ryo Kageyama, Mikito Fujinami, Yasuhiro Ikabata, Hiromi NakaiAbstract:Abstract This letter investigates the accuracy of the semi-local machine-learned Kinetic Energy Density functional (KEDF) for potential Energy curves (PECs) in typical small molecules. The present functional is based on a previously developed functional adopting electron densities and their gradients up to the third order as descriptors (Seino et al., 2018). It further introduces new descriptors, namely, the distances between grid points and centers of nuclei, to describe the non-local nature of the KEDF. The numerical results show a reasonable performance of the present model in reproducing the PECs of small molecules with single, double, and triple bonds.
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semi local machine learned Kinetic Energy Density functional with third order gradients of electron Density
Journal of Chemical Physics, 2018Co-Authors: Junji Seino, Ryo Kageyama, Mikito Fujinami, Yasuhiro Ikabata, Hiromi NakaiAbstract:A semi-local Kinetic Energy Density functional (KEDF) was constructed based on machine learning (ML). The present scheme adopts electron densities and their gradients up to third-order as the explanatory variables for ML and the Kohn-Sham (KS) Kinetic Energy Density as the response variable in atoms and molecules. Numerical assessments of the present scheme were performed in atomic and molecular systems, including first- and second-period elements. The results of 37 conventional KEDFs with explicit formulae were also compared with those of the ML KEDF with an implicit formula. The inclusion of the higher order gradients reduces the deviation of the total Kinetic energies from the KS calculations in a stepwise manner. Furthermore, our scheme with the third-order gradient resulted in the closest Kinetic energies to the KS calculations out of the presented functionals.
