N-density representability and the optimal transport limit of the Hohenberg-Kohn functional.

We derive and analyze a hierarchy of approximations to the strongly correlated limit of the Hohenberg-Kohn functional. These "density representability approximations" are obtained by first noting that in the strongly correlated limit, N-representability of the pair density reduces to the requirement that the pair density must come from a symmetric N-point density. One then relaxes this requirement to the existence of a representing symmetric k-point density with k < N. The approximate energy can be computed by simulating a fictitious k-electron system. We investigate the approximations by deriving analytically exact results for a 2-site model problem, and by incorporating them into a self-consistent Kohn-Sham calculation for small atoms. We find that the low order representability conditions already capture the main part of the correlations.

[1]  D. Freedman,et al.  Finite Exchangeable Sequences , 1980 .

[2]  P. Gori-Giorgi,et al.  Kohn-Sham density functional theory for quantum wires in arbitrary correlation regimes , 2013, 1301.7323.

[3]  Brendan Pass,et al.  On the local structure of optimal measures in the multi-marginal optimal transportation problem , 2010, 1005.2162.

[4]  Paola Gori-Giorgi,et al.  Strong correlation in Kohn-Sham density functional theory. , 2012, Physical review letters.

[5]  J. Perdew,et al.  Hellmann-Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms. , 1985, Physical review. A, General physics.

[6]  Codina Cotar,et al.  Density Functional Theory and Optimal Transportation with Coulomb Cost , 2011, 1104.0603.

[7]  E. Davidson,et al.  Necessary conditions for the N-representability of pair distribution functions , 2006 .

[8]  M. Seidl.  Strong-interaction limit of density-functional theory , 1999 .

[9]  Codina Cotar,et al.  Infinite-body optimal transport with Coulomb cost , 2013, Calculus of Variations and Partial Differential Equations.

[10]  T. Bally,et al.  INCORRECT DISSOCIATION BEHAVIOR OF RADICAL IONS IN DENSITY FUNCTIONAL CALCULATIONS , 1997 .

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

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

[13]  Bruno de Finetti.  Sulla proseguibilità di processi aleatori scambiabili , 1969 .

[14]  Lin Lin,et al.  Kantorovich dual solution for strictly correlated electrons in atoms and molecules , 2012, 1210.7117.

[15]  C. Villani.  Optimal Transport: Old and New , 2008 .

[16]  Remarks on the semi-classical Hohenberg-Kohn functional , 2012, 1211.2766.

[17]  D. Mazziotti.  Reduced-Density-Matrix Mechanics: With Application to Many-Electron Atoms and Molecules , 2007 .

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

[19]  École d'été de probabilités de Saint-Flour,et al.  École d'été de probabilités de Saint-Flour XIII - 1983 , 1985 .

[20]  P. Gori-Giorgi,et al.  Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities , 2007, cond-mat/0701025.

[21]  B. Judd,et al.  Reduced Density Matrices: Coulson's Challenge , 2000 .

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

[23]  D. Aldous.  Exchangeability and related topics , 1985 .

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

[25]  G. Buttazzo,et al.  Optimal-transport formulation of electronic density-functional theory , 2012, 1205.4514.

[26]  Kyuho Lee,et al.  Higher-accuracy van der Waals density functional , 2010, 1003.5255.

[27]  W. Gangbo,et al.  The geometry of optimal transportation , 1996 .

[28]  Gero Friesecke,et al.  Asymptotics-Based CI Models for Atoms: Properties, Exact Solution of a Minimal Model for Li to Ne, and Application to Atomic Spectra , 2009, Multiscale Model. Simul..

[29]  A. J. Coleman,et al.  Reduced Density Matrices , 2000 .

[30]  G. Friesecke.  Pair Correlations and Exchange Phenomena in the Free Electron Gas , 1997 .

[31]  Michael Seidl,et al.  Strictly correlated electrons in density-functional theory , 1999 .

[32]  Strictly correlated uniform electron droplets , 2011, 1101.6049.

[33]  E. Davidson.  N-representability of the electron pair density , 1995 .

[34]  K. Hirao,et al.  An investigation of density functionals: The first-row transition metal dimer calculations , 2000 .

[35]  Jan M. L. Martin,et al.  On the structure and vibrational frequencies of C24 , 1996 .