Semi-local machine-learned kinetic energy density functional with third-order gradients of electron density.

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.

[1]  C. Hodges.  Quantum Corrections to the Thomas–Fermi Approximation—The Kirzhnits Method , 1973 .

[2]  Weitao Yang,et al.  Challenges for density functional theory. , 2012, Chemical reviews.

[3]  Yasuo Tomishima,et al.  On the Weizsäcker Correction to the Thomas-Fermi Theory of the Atom , 1965 .

[4]  Juvencio Robles,et al.  On the atomic kinetic energy functionals with full Weizsacker correction , 1982 .

[5]  Matthias Rupp,et al.  Machine learning for quantum mechanics in a nutshell , 2015 .

[6]  F. Della Sala,et al.  Semiclassical neutral atom as a reference system in density functional theory. , 2011, Physical review letters.

[7]  D. L. Cooper,et al.  The gradient expansions of the kinetic energy and the mean momentum for light diatomic molecules , 1985 .

[8]  Burke,et al.  Generalized Gradient Approximation Made Simple. , 1996, Physical review letters.

[9]  Emily A. Carter,et al.  Linear-scaling parallel algorithms for the first principles treatment of metals ✩ , 2000 .

[10]  J. Perdew,et al.  Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation. , 1986, Physical review. B, Condensed matter.

[11]  D. Lacks,et al.  Pair interactions of rare-gas atoms as a test of exchange-energy-density functionals in regions of large density gradients. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[12]  Tomasz Adam Wesolowski,et al.  Link between the Kinetic- and Exchange-Energy Functionals in the Generalized Gradient Approximation , 2002 .

[13]  Ilgyou Shin,et al.  Enhanced von Weizsäcker Wang-Govind-Carter kinetic energy density functional for semiconductors. , 2014, The Journal of chemical physics.

[14]  Levente Vitos,et al.  KINETIC-ENERGY FUNCTIONALS STUDIED BY SURFACE CALCULATIONS , 1998 .

[15]  S. K. Pogrebnya,et al.  A simple approximation to the kinetic energy functional , 1988 .

[16]  F. Della Sala,et al.  Generalized Gradient Approximations of the Noninteracting Kinetic Energy from the Semiclassical Atom Theory: Rationalization of the Accuracy of the Frozen Density Embedding Theory for Nonbonded Interactions. , 2011, Journal of chemical theory and computation.

[17]  Wang,et al.  Kinetic-energy functional of the electron density. , 1992, Physical review. B, Condensed matter.

[18]  Mark S. Gordon,et al.  General atomic and molecular electronic structure system , 1993, J. Comput. Chem..

[19]  Axel D. Becke,et al.  On the large‐gradient behavior of the density functional exchange energy , 1986 .

[20]  K. Burke,et al.  Condition on the Kohn-Sham kinetic energy and modern parametrization of the Thomas-Fermi density. , 2008, The Journal of chemical physics.

[21]  Vincent L. Lignères,et al.  Improving the orbital-free density functional theory description of covalent materials. , 2005, The Journal of chemical physics.

[22]  Li Li,et al.  Understanding Machine-learned Density Functionals , 2014, ArXiv.

[23]  Emily A. Carter,et al.  Introducing PROFESS: A new program for orbital-free density functional theory calculations , 2008, Comput. Phys. Commun..

[24]  Axel D. Becke,et al.  Density Functional Calculations of Molecular Bond Energies , 1986 .

[25]  Thakkar.  Comparison of kinetic-energy density functionals. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[26]  D. R. Murphy.  Sixth-order term of the gradient expansion of the kinetic-energy density functional , 1981 .

[27]  A. Becke,et al.  Density-functional exchange-energy approximation with correct asymptotic behavior. , 1988, Physical review. A, General physics.

[28]  Li Li,et al.  Bypassing the Kohn-Sham equations with machine learning , 2016, Nature Communications.

[29]  K. Yonei.  Energy Levels for an Extended Thomas-Fermi-Dirac Potential , 1967 .

[30]  John C. Snyder,et al.  Orbital-free bond breaking via machine learning. , 2013, The Journal of chemical physics.

[31]  Tarazona,et al.  Nonlocal kinetic energy functional for nonhomogeneous electron systems. , 1985, Physical review. B, Condensed matter.

[32]  Approximating the kinetic energy functional Ts[ρ] : lessons from four-electron systems , 2005 .

[33]  Emily A. Carter,et al.  libKEDF: An accelerated library of kinetic energy density functionals , 2017, J. Comput. Chem..

[34]  Mel Levy,et al.  Approximate noninteracting kinetic energy functionals from a nonuniform scaling requirement , 1991 .

[35]  J. Perdew.  Generalized gradient approximation for the fermion kinetic energy as a functional of the density , 1992 .

[36]  Emily A. Carter,et al.  Describing metal surfaces and nanostructures with orbital-free density functional theory , 2007 .

[37]  C. Weizsäcker.  Zur Theorie der Kernmassen , 1935 .

[38]  Patricio Fuentealba,et al.  Further evidence of the conjoint correction to the local kinetic and exchange energy density functionals , 1995 .

[39]  E. Lieb.  Thomas-fermi and related theories of atoms and molecules , 1981 .

[40]  Jean-Louis Reymond,et al.  Enumeration of 166 Billion Organic Small Molecules in the Chemical Universe Database GDB-17 , 2012, J. Chem. Inf. Model..

[41]  P. Hohenberg,et al.  Inhomogeneous Electron Gas , 1964 .

[42]  Emily A. Carter,et al.  Nonlocal orbital-free kinetic energy density functional for semiconductors , 2010 .

[43]  Linda Hung,et al.  Accurate simulations of metals at the mesoscale: Explicit treatment of 1 million atoms with quantum mechanics , 2009 .

[44]  P. Acharya,et al.  An atomic kinetic energy functional with full Weizsacker correction. , 1980, Proceedings of the National Academy of Sciences of the United States of America.

[45]  Kun Yao,et al.  Kinetic Energy of Hydrocarbons as a Function of Electron Density and Convolutional Neural Networks. , 2015, Journal of chemical theory and computation.

[46]  Joel D. Kress,et al.  Rational function representation for accurate exchange energy functionals , 1987 .

[47]  L. H. Thomas.  The calculation of atomic fields , 1927, Mathematical Proceedings of the Cambridge Philosophical Society.

[48]  Kress,et al.  Kinetic-energy functionals via Padé approximations. , 1987, Physical review. A, General physics.

[49]  Jackson,et al.  Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. , 1992, Physical review. B, Condensed matter.

[50]  A. Thakkar,et al.  Local density functional approximations and conjectured bounds for momentum moments , 1990 .

[51]  Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy , 2006, cond-mat/0612430.

[52]  Alexandre Tkatchenko,et al.  Quantum-chemical insights from deep tensor neural networks , 2016, Nature Communications.

[53]  Ari Harju,et al.  Orbital-Free Density Functional Theory Implementation with the Projector Augmented-Wave Method , 2014, The Journal of chemical physics.

[54]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[55]  A. Lembarki,et al.  Obtaining a gradient-corrected kinetic-energy functional from the Perdew-Wang exchange functional. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[56]  Klaus-Robert Müller,et al.  Finding Density Functionals with Machine Learning , 2011, Physical review letters.

[57]  Samuel B. Trickey,et al.  Issues and challenges in orbital-free density functional calculations , 2011, Comput. Phys. Commun..

[58]  E. Carter,et al.  Density-decomposed orbital-free density functional theory for covalently bonded molecules and materials , 2012 .

[59]  Parr,et al.  Conjoint gradient correction to the Hartree-Fock kinetic- and exchange-energy density functionals. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[60]  S. Trickey,et al.  Nonempirical generalized gradient approximation free-energy functional for orbital-free simulations , 2013, 1308.2193.