Local density aroximation The local density aroximation (LDA), Treat a molecular density as a collection of tiny bits of uniform electron gases (UEGs)
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1 Density-functional theory using finite uniform electron gases Pierre-François Loos 1,2 1 Laboratoire de Chimie et Physique Quantiques, UMR 5626, Université Paul Sabatier, Toulouse, France 2 Research School of Chemistry, Australian National University, Canberra, Australia 3rd May 2017
2 Local density aroximation The local density aroximation (LDA), Treat a molecular density as a collection of tiny bits of uniform electron gases (UEGs), The LDA is an ab initio model with no adjustable arameters, This is an attractive aroach to molecular electronic structure, It also forms a foundation for more accurate aroximations / Not very accurate for molecules: underestimate the exchange by roughly 10% overestimate the correlation by roughly 100%
3 Local density aroximation LDA exchange The LDA exchange energy (Dirac formula) is Z Z Ex LDA = (r)ex LDA ( )dr = C x (r) 4/3 dr where C x = 3 2 e LDA x ( ) = C x 1/3 1/3 3 = has been obtained based on the infinite uniform electron gas (IUEG) Dirac, Proc Cam Phil Soc 26 (1930) 376
4 Infinite uniform electron gas The uniform electron gas in Flatland The infinite uniform electron gas (IUEG), One of the most oular models in condensed matter hysics, Characterized by one arameter: Seitz radius r s / 1/D / Clearly suitable for metals. Less clearly suitable for molecules The jellium recie 1 Put n electrons into a D-dimensional cube of volume V 2 Add a background of ositive jelly to achieve neutrality 3 Increase both n and V so that = n/v remains constant 4 In the limit as n!1and V!1,oneobtainsaninfinite UEG Parr & Yang, DFT for atoms and molecules (1989) Loos & Gill, WIREs Comut Mol Sci 6 (2016) 410
5 Finite uniform electron gases The uniform electron gas in Shereland Finite UEGs (FUEGs) One can also construct UEGs using a finite number of electrons The recie: 1 Put n electrons onto a D-dimensional shere 2 Add a background ositive charge to achieve neutrality 3 That s all, For n!1,wegettheinfinite UEG!! Loos & Gill, JCP 135 (2011) Gill & Loos, TCA 131 (2012) 1069
6 L-sherium and L-glomium Uniform electron gases in Shereland We fill each (hyer)sherical harmonic Y`m(n) u to ` = L with one sin-u and one sin-down electron L-Sherium L-Glomium `X Y`m (, ) 2 = m= ` = 2` (L +1)2 4 R 2 = 1 r 2 s `X m=0 n= m = mx Y`mn (,, ) 2 = (` +1) (L +1)(L +2)(2L +3)/3 4 2 R 3 = 3 4 r 3 s Loos & Gill, JCP 135 (2011)
7 L-sherium and L-glomium Uniform electron gases in Shereland We fill each (hyer)sherical harmonic Y`m(n) u to ` = L with one sin-u and one sin-down electron L-Sherium L-Glomium `X Y`m (, ) 2 = m= ` = 2` (L +1)2 4 R 2 = 1 r 2 s `X m=0 n= m = mx Y`mn (,, ) 2 = (` +1) (L +1)(L +2)(2L +3)/3 4 2 R 3 = 3 4 r 3 s Loos & Gill, JCP 135 (2011)
8 L-sherium and L-glomium Uniform electron gases in Shereland We fill each (hyer)sherical harmonic Y`m(n) u to ` = L with one sin-u and one sin-down electron L-Sherium L-Glomium `X Y`m (, ) 2 = m= ` = 2` (L +1)2 4 R 2 = 1 r 2 s `X m=0 n= m = mx Y`mn (,, ) 2 = (` +1) (L +1)(L +2)(2L +3)/3 4 2 R 3 = 3 4 r 3 s Loos & Gill, JCP 135 (2011)
9 Jellium-based correlation functionals alied to sherium/glomium Non-uniqueness of the uniform electron gas Are jellium-based functionals accurate for finite UEGs? Exact Jellium-based Kohn-Sham DFT Error 2R E T Eee E T S E V E J E X E jell c E KS E KS E 0-sh / / glo / / Why? We are missing some two-electron information Loos & Gill, PRL 103 (2009) Gill & Loos, TCA 131 (2012) 1069
10 Jellium-based correlation functionals alied to sherium/glomium Curvature of the Fermi hole The curvature of the Fermi hole* is(0 ale <1): = W = x 2 C F = 3 IUEG IUEG 4C F 5 (6 2 ) 2/3 Xocc = r i 2 is the kinetic energy density W = r 2 4 i IUEG = C F 5/3 is the von Weizsäcker kinetic energy density is the kinetic energy density of the IUEG Becke & Edgecombe, JCP 92 (1990) 5397 Loos, Ball & Gill, JCP 140 (2014) 18A524 *Remember ELF!? ELF = (1 + 2 ) 1
11 L-sherium vs 2D jellium High-density (r s! 0) limit: L-sherium vs 2D jellium ejellium(r 2D " s)= 2 + " 1 +(" rs 2 0,J + " 0,K )+ 1r s ln r s + O(r s) r s " 2 = " 1 = " 0,J = " 0,K = 1 = Loos & Gill, PRB 83 (2011) ; ibid 84 (2011)
12 L-sherium vs 2D jellium High-density (r s! 0) limit: L-sherium vs 2D jellium L(L +2) " 2 = + 2(L +1) 2 ale 1 " 1 = F 2 " 0,J = 2 n " 0,K = 1 n ejellium(r 2D " s)= 2 + " 1 +(" rs 2 0,J + " 0,K )+ 1r s ln r s + O(r s) r s Xocc virt ij Xocc virt ij X ab X ab 1 = (resummation) 1 L, L +2, 2, L 2, L + 3 2, 2 hij abi 2 ale a + ale b ale i ale j hij abihba iji ale a + ale b ale i ale j Loos & Gill, PRB 83 (2011) ; ibid 84 (2011)
13 L-sherium vs 2D jellium High-density (r s! 0) limit: L-sherium vs 2D jellium ejellium(r 2D " s)= 2 + " 1 +(" rs 2 0,J + " 0,K )+ 1r s ln r s + O(r s) r s L(L +2) " 2 = +! 2(L +1) L!1 2 ale 1 1 L, L +2, " 1 = F 2, L 2, L + 3 2, 2! L!1 3 " 0,J = 2 n " 0,K = 1 n Xocc virt ij Xocc virt ij X ab X ab hij abi 2 ale a + ale b ale i ale j! L!1 ln 2 1 hij abihba iji! ale a + ale b ale i ale G 8 (4) j L!1 2 1 = (resummation)! L! Loos & Gill, PRB 83 (2011) ; ibid 84 (2011)
14 L-sherium vs 2D jellium High-density (r s! 0) limit: L-sherium vs 2D jellium ejellium(r 2D " s)= 2 + " 1 +(" rs 2 0,J + " 0,K )+ 1r s ln r s + O(r s) r s L(L +2) " 2 = +! 2(L +1) L!1 2 ale 1 1 L, L +2, " 1 = F 2, L 2, L + 3 2, 2! L!1 3 " 0,J = 2 n " 0,K = 1 n Xocc virt ij Xocc virt ij X ab X ab hij abi 2 ale a + ale b ale i ale j! L!1 ln 2 1 hij abihba iji! ale a + ale b ale i ale G 8 (4) j L!1 2 1 = (resummation)! L!1 Loos & Gill, PRB 83 (2011) ; ibid 84 (2011)
15 L-glomium vs 3D jellium High-density (r s! 0) limit: L-glomium vs 3D jellium ejellium(r 3D " s)= 2 + " 1 + rs 2 0 ln r s +(" 0,J + " 0,K )+O(r s ln r s) r s " 2 " 1 0 " 0,J " 0,K
16 L-glomium vs 3D jellium High-density (r s! 0) limit: L-glomium vs 3D jellium ejellium(r 3D " s)= 2 + " 1 + rs 2 0 ln r s +(" 0,J + " 0,K )+O(r s ln r s) r s " 2! L!1 " 1! L!1 0 resum.! L!1 " 0,J resum.! L!1 " 0,K! L! ln /3 1/3 1 ln (3)
17 L-glomium vs 3D jellium High-density (r s! 0) limit: L-glomium vs 3D jellium ejellium(r 3D " s)= 2 + " 1 + rs 2 0 ln r s +(" 0,J + " 0,K )+O(r s ln r s) r s " 2! L!1 " 1! L!1 0 resum.! L!1 " 0,J resum.! L!1 " 0,K! L! ln /3 1/3 1 ln (3)
18 Our conjecture and corrolary Our conjecture e(r s)= " 2 r 2 s + " 1 r s + 1X [ ` ln r s + "`]r `s [... ] the high-density exansions are identical to all order `=0, short-sightedness of electronic matter Kohn PRL 76 (1996) 3168 Loos & Gill, JCP 135 (2011)
19 Uniform electron gases Flatland vs Shereland Exchange functionals Acknowledgements Wigner crystallization Low-density (rs! 1) limit of L-sherium 2D ejellium (rs ) = 3/ / rs rs rs Thomson roblem determine the minimum energy configuration of n electrons on the surface of a shere that reel each other with a force given by Coulomb s law esh. (rs ) rs (large-n limit) Note: identical to the Wigner crystal hase of 2D jellium (hexagonal lattice) Bowick et al. PRL 89 (2002) Agboola, Knol, Gill & Loos, JCP 143 (2015) Pierre-Franc ois Loos (loos@irsamc.us-tlse.fr)
20 DFT with finite UEGs DFT with finite UEGs
21 Generalized local density aroximation Jacob s ladder of DFT revisited Rung 2: GGA (, x) Chemical accuracy Rung 4: HGGA (, x,, E HF x ) Rung 3: MGGA (, x, ) Rung 1: LDA ( ) Hartree Rung 2 0 : GLDA (, ) Rung 2.9: FMGGA (, x, ) Loos, JCP 146 (2017) Rung 2 0 :GeneralizedLDAs(GLDA) with e GLDA x (, ) =e LDA x lim F x GLDA ( ) =1!1 ( )Fx GLDA ( ) Rung 2.9: FactorizableMGGAs(FMGGA) with e FMGGA x F FMGGA x (, x, )=ex LDA (x, )=F GGA x ( )Fx FMGGA (x, ) (x)fx GLDA ( ) lim F x FMGGA (x, )=Fx GLDA ( ) x!0 lim F x FMGGA (x, )=Fx GGA (x)!1
22 Finite UEGs How to create finite UEGs (FUEGs)? We confine n electrons on the surface of a 3-shere (or a glome) For magic numbers of electrons (full shell), the density is uniform over the shere = n (L +2)(L +3/2)(L +1) = V 6 2 R 3 The curvature of the Fermi hole is L(L +3) = [(L +1)(L +3/2)(L +2)] 2/3 The exchange energy is C x(l) = C LDA x which yields lim =0 lim =1 n!1 n!1 1 2 L L Z E x(l) =C x(l) h 1 2 H 2L /3 dr i +ln2 + L (L +1) L (L +2) 4/3 2 L 2 +3L
23 GLDA with FUEGs How to use FUEGs in DFT? The GX functional (which is a GLDA) is defined as F GX x ( ) = ( Fx gx ( ), 0 ale ale 1 1+(1 1) 1 1+ >1 with Fx gx ( ) = C ale x GLDA (0) Cx GLDA (1) + c 0 + c 1 Cx GLDA (0) 1 1+(c 0 + c 1 1) Cx GLDA (1) where c 0 = , c 1 = are fitted on the exchange energy of FUEGs. For =1,werecover the LDA: Cx GLDA (1) = Cx LDA = 3 1/ For =0,weget: Cx GLDA (0) = 4 1/3 2 3
24 GLDA with FUEGs Plots of the GX functional Problem of GLDAs: cannot discriminate between homogeneous and inhomogeneous one-electron systems ( =0)
25 FMGGA with finite UEGs The PBE-GX functional The PBE-GX functional (which is a FMGGA) is defined as where F PBE-GX x (x, )=F PBE x is a PBE-like GGA enhancement factor How do we set the free arameters? Fx PBE 1 (x) = 1+µ x 2 (x)fx GX ( ) µ = to get the exact exchange energy of the hydrogen atom 1 = to obtain good exchange energies for neutral atoms Unlike GX, PBE-GX is accurate for both the (inhomogeneous) hydrogen-like ions and the (homogeneous) one-electron FUEGs
26 FMGGA with finite UEGs Plot of the PBE-GX functional
27 Concluding Remarks Students, Postdocs, Collaborators and Funding Collaborator: Peter Gill Honours students: Anneke Knol & Fergus Rogers PhD students: Caleb Ball Postdocs: Davids Agboola Centre National de la Recherche Scientifique (CNRS) Research School of Chemistry & Australian National University Australian Research Council (DECRA13 & DP14)
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