Density-functional theory at noninteger electron numbers

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1 Density-functional theory at noninteger electron numbers Tim Gould 1, Julien Toulouse 2 1 Griffith University, Brisbane, Australia 2 Université Pierre & Marie Curie and CRS, Paris, France July 215

2 Introduction Study of DFT with fractional electron numbers is useful: Pioneer work of Perdew, Parr, Levy, Balduz 1982: energy is piecewise linear wrt and discontinuity of the KS potential as passes through an integer = relevant for fundamental gap, molecular dissociation, charge transfer,... Revived interest since 26: Delocalization error of semilocal DFAs explained as deviation from piecewise linearity (Yang, Perdew,...) Construction of improved DFAs (e.g.: Kraisler & Kronik 213) Only a few numerical studies of exact DFT at fractional, e.g.: Savin, Colonna, Pollet 23: KS potentials for 1 2 for H atom Sagvolden & Perdew 28: discontinuity at = 1 for H atom Gori-Giorgi & Savin 28: T s and E xc wrt for 1 2 for Hooke and He atoms Gori-Giorgi & Savin 29: discontinuity at = 1 for Hooke atom = Here we extend these studies to larger atoms (Li, C, F) 2/12

3 Quantum mechanics for fractional electron numbers 3/12 For a fixed = M +f where M is an integer and f 1 Ground-state energy = minimum over density matrices ˆΓ yielding )] E = min Tr[ˆΓ (ˆT + ˆV ext +Ŵ ee ˆΓ Assuming convexity of energy, E M (E M 1 +E M+1 )/2, the minimizing ˆΓ is linear wrt f: Thus, the energy is linear in f: and also the density: ˆΓ = (1 f) ˆΓ M +f ˆΓ M+1 E = (1 f) E M +f E M+1 n (r) = (1 f) n M (r)+f n M+1 (r)

4 Kohn-Sham DFT for fractional electron number Explicit e-e interaction operator Ŵ ee replaced by a density functional E Hxc [n] { } E = min Tr[ˆΓ (ˆT + ˆV ext )]+E Hxc [nˆγ ] ˆΓ The minimizing non-interacting ˆΓ s is not linear wrt f: ˆΓ M,f s ˆΓ M+1,f s ˆΓ s = (1 f) ˆΓ M,f s +f ˆΓ M+1,f s and are density matrices of single-determinant wave functions made from a common set of KS orbitals {φ i (r)} [ vs (r) ] φ i (r) = ε i φ i (r) with the KS potential v s (r) = v ext (r)+δe Hxc [n ]/δn(r) The density is n (r) = f i φ i (r) 2 i where f i = 2 for inner orbitals, f i = for unoccupied orbitals, and f h 2 for (degenerate) HOMOs so that i f i = 4/12

5 Fundamental gap and derivative discontinuity Ground-state total energy for fractional : E = T s +V ext [n ]+E H [n ]+E xc [n ] The fundamental gap of the M-electron system is ( ) ( ) ( ) E Eg M = I M A M E δe = δn(r) M + M = For the non-interacting Kohn-Sham system: ( ) δt Eg KS,M = ε M LUMO ε M HOMO = s δn(r) Difference between E M g and E KS,M g M + must come from a derivative discontinuity of E xc : E M g = E KS,M g where M = M + ( ) δt s Eg M δn(r) M + M ( ) δe δn(r) ( ) ( ) δexc δexc = vxc M+ (r) vxc M (r) δn(r) M δn(r) + M M 5/12

6 Accurate densities from quantum Monte Carlo (QMC) For Li, C, and F: calculations of n Z 1 (r), n Z (r), and n Z+1 (r) by QMC Jastrow multideterminant full-valence CAS wave functions fully optimized in VMC (Toulouse & Umrigar 28) densities calculated in DMC using an improved statistical estimator (Toulouse, Assaraf, Umrigar 27) n(r) (a.u.) usual estimator improved estimator neutral C atom r (a.u.) 6/12

7 Kohn-Sham potentials at fractional 7/12 Fit of QMC densities to analytical expressions Densities for fractional obtained by interpolation: n (r) = (1 f) n M (r)+f n M+1 (r) Kohn-Sham potentials for fractional calculated by a modification of the iterative method of Wang & Parr 1992: v,m+1 s (r) = vs,m (r)+q n (r) n,m (r) G,m (r) where G,m (r) i f i φ,m i (r) 2 /ε,m i parameter and Q > is a convergence Starting potential: linear EXX (Gould & Dobson 213) At convergence: n (r) n,m (r) dr < 1 6

8 Kohn-Sham potential for Li atom 8/12 Total Kohn-Sham potential v s (r) as a function of r and : v s (r) (a.u.) r (a.u.) 8 1 = spatially constant discontinuity at = 3

9 Kohn-Sham potential for C atom 9/12 Total Kohn-Sham potential v s (r) as a function of r and : v s (r) (a.u.) r (a.u.) 8 1 = discontinuity at = 6 but decreases at large r because of the finite value = 6.1 used for right-hand limit

10 Kohn-Sham potential for F atom 1/12 Total Kohn-Sham potential v s (r) as a function of r and : v s (r) (a.u.) r (a.u.) 8 1 = discontinuity at = 9 but decreases at large r because of the finite value = 9.1 used for right-hand limit

11 Kohn-Sham potential for F atom 11/12 v s (r) v Z s (r) as a function of : v s (r) - vs Z - (r) (a.u.) r=.1 r=1 r=2 r=4 r= =.519 = same discontinuity at = 9 for all r but v s (r) is increasingly curved near = 9 + for large r ( exact = I A =.516)

12 Summary and perspectives 12/12 Summary: accurate KS data for Li, C, F at fractional from QMC densities numerical reproduction of the discontinuity in v s (r) T. Gould & J. Toulouse, Physical Review A 9, 552(R), 214 Similar work for Li, Be, B published a few weeks after our paper: R. Morrison, JCP 142, 1411, 215 Perspectives: calculations on larger atoms and molecules analysis of density-functional approximations use in the construction of new approximations

Fractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation.

Fractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation. Bastien Mussard, Julien Toulouse Laboratoire de Chimie Théorique Institut du Calcul