Pure density functional for strong correlation and the thermodynamic limit from machine learning

We use the density-matrix renormalization group, applied to a one-dimensional model of continuum Hamiltonians, to accurately solve chains of hydrogen atoms of various separations and numbers of atoms. We train and test a machine-learned approximation to $F[n]$, the universal part of the electronic density functional, to within quantum chemical accuracy. We also develop a data-driven, atom-centered basis set for densities which greatly reduces the computational cost and accurately represents the physical information in the machine-learning calculation. Our calculation (a) bypasses the standard Kohn-Sham approach, avoiding the need to find orbitals, (b) includes the strong correlation of highly stretched bonds without any specific difficulty (unlike all standard DFT approximations), and (c) is so accurate that it can be used to find the energy in the thermodynamic limit to quantum chemical accuracy.

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

[2]  Xuan Zhao,et al.  Evaluating randomness in cyber attack textual artifacts , 2016, 2016 APWG Symposium on Electronic Crime Research (eCrime).

[3]  Demis Hassabis,et al.  Mastering the game of Go with deep neural networks and tree search , 2016, Nature.

[4]  Matthias Rupp,et al.  Special issue on machine learning and quantum mechanics , 2015 .

[5]  Klaus-Robert Müller,et al.  Nonlinear gradient denoising: Finding accurate extrema from inaccurate functional derivatives , 2015 .

[6]  T. E. Baker,et al.  One Dimensional Mimicking of Electronic Structure: The Case for Exponentials , 2015, 1504.05620.

[7]  Li Li,et al.  Understanding Kernel Ridge Regression: Common behaviors from simple functions to density functionals , 2015, ArXiv.

[8]  Kieron Burke,et al.  DFT: A Theory Full of Holes? , 2014, Annual review of physical chemistry.

[9]  D. Matthews,et al.  Ab initio determination of the crystalline benzene lattice energy to sub-kilojoule/mole accuracy , 2014, Science.

[10]  T. E. Baker,et al.  Kohn-Sham calculations with the exact functional , 2014, 1405.0864.

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

[12]  Yu-An Chen,et al.  Density matrix renormalization group , 2014 .

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

[14]  Kieron Burke,et al.  Guaranteed convergence of the Kohn-Sham equations. , 2013, Physical Review Letters.

[15]  Eunji Sim,et al.  Understanding and reducing errors in density functional calculations. , 2012, Physical review letters.

[16]  Kieron Burke,et al.  Reference electronic structure calculations in one dimension. , 2012, Physical chemistry chemical physics : PCCP.

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

[18]  K. Müller,et al.  Fast and accurate modeling of molecular atomization energies with machine learning. , 2011, Physical review letters.

[19]  Kieron Burke,et al.  One-dimensional continuum electronic structure with the density-matrix renormalization group and its implications for density-functional theory. , 2011, Physical review letters.

[20]  C. V. Jawahar,et al.  The truth about cats and dogs , 2011, 2011 International Conference on Computer Vision.

[21]  U. Schollwoeck.  The density-matrix renormalization group in the age of matrix product states , 2010, 1008.3477.

[22]  Ashutosh Kumar Singh,et al.  The Elements of Statistical Learning: Data Mining, Inference, and Prediction , 2010 .

[23]  Weitao Yang,et al.  Insights into Current Limitations of Density Functional Theory , 2008, Science.

[24]  I. McCulloch.  Infinite size density matrix renormalization group, revisited , 2008, 0804.2509.

[25]  M. Tuckerman,et al.  Ab initio molecular dynamics: concepts, recent developments, and future trends. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[26]  T. Wesołowski,et al.  Introduction of the explicit long-range nonlocality as an alternative to the gradient expansion approximation for the kinetic-energy functional , 2002 .

[27]  W. Krauth,et al.  Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions , 1996 .

[28]  J. Zaanen,et al.  Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators. , 1995, Physical review. B, Condensed matter.

[29]  Burke,et al.  Escaping the symmetry dilemma through a pair-density interpretation of spin-density functional theory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[30]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[31]  Sols.  Gauge-invariant formulation of electron linear transport. , 1991, Physical review letters.

[32]  M. Levy.  Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. , 1979, Proceedings of the National Academy of Sciences of the United States of America.

[33]  F. L. Hirshfeld.  Bonded-atom fragments for describing molecular charge densities , 1977 .

[34]  E. Lieb,et al.  The Thomas-Fermi theory of atoms, molecules and solids , 1977 .

[35]  E. Fermi.  A Statistical Method for the Determination of Some Atomic Properties and the Application of this Method to the Theory of the Periodic System of Elements , 1975 .

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

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

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