Long-range/short-range energy decomposition in density functional theory

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1 p. 1/2 Long-range/short-range energy decomposition in density functional theory Julien Toulouse François Colonna, Andreas Savin Laboratoire de Chimie Théorique, Université Pierre et Marie Curie, Paris

2 p. 2/2 Introduction Some problems in Kohn-Sham DFT with present (semi)local density functional approximations: near-degeneracy long-range interactions Consensus: (local) density functional approximations work well for short-range interactions A possible approach: long-range/short-range decomposition of the energy E = E lr + E sr explicit many-body approximation density functional approximation

3 p. 3/2 Outline Long-range/short-range decomposition Multi-determinantal DFT Analysis of short-range density functionals Beyond for short-range density functionals

4 p. 4/2 Outline Long-range/short-range decomposition Multi-determinantal DFT Analysis of short-range density functionals Beyond for short-range density functionals

5 p. 5/2 Decomposition of the interaction Two decompositions tested: erf interaction: w lr,µ ee (r) = erf(µr) r 1 r = wlr,µ ee (r) + wee sr,µ (r) interaction: w lr,µ ee (r) = erf(cµr) r 2cµ π e 1 3 c2 µ 2 r /r µ = /µ r Limits: w lr,µ= ee (r) = and wee lr,µ (r) = 1 r

6 Decomposition of the universal functional Two alternatives: Choice 1: F lr,µ [n] = min Ψ ˆT + Ŵ ee lr,µ Ψ Ψ n F [n] = F lr,µ [n] + F sr,µ [n] F sr,µ [n] = E sr,µ H [n] + Esr,µ x [n] + Ēsr,µ c [n] Choice 2: F sr,µ [n] = min Ψ ˆT + Ŵ ee sr,µ Ψ Ψ n F [n] = F sr,µ [n] + F lr,µ [n] F sr,µ [n] = T s [n] + E sr,µ H [n] + Esr,µ x [n] + Ec sr,µ [n] Limits: For µ = : Ex sr,µ= For µ : E sr,µ x [n] = E x [n] and Ē sr,µ= c [n] = Ēsr,µ c [n] = Ec sr,µ= c [n] = [n] = E sr,µ [n] = E c [n] p. 6/2

7 p. 7/2 Short-range exchange energy E sr,µ x Local density approximation (): E sr,µ x, [n] = n(r)ε sr,µ x,unif (n(r))dr For the He atom:.2 erf E x

8 p. 7/2 Short-range exchange energy E sr,µ x Local density approximation (): E sr,µ x, [n] = n(r)ε sr,µ x,unif (n(r))dr For the He atom:.2 erf.4 E x

9 p. 8/2 Short-range correlation energy Ēsr,µ c (choice 1) Local density approximation (): Ēsr,µ c, [n] = n(r) ε sr,µ c,unif (n(r))dr For the He atom:.2.4 erf E c

10 p. 8/2 Short-range correlation energy Ēsr,µ c (choice 1) Local density approximation (): Ēsr,µ c, [n] = n(r) ε sr,µ c,unif (n(r))dr For the He atom:.2 erf E c

11 p. 9/2 hort-range correlation energies E sr,µ c (choice 2) Local density approximation (): E sr,µ c, [n] = n(r)ε sr,µ c,unif (n(r))dr For the He atom:.2 erf choice 2 E c.4.6 erf choice = In choice 1, can treat well a larger part of correlation energy

12 p. 1/2 Outline Long-range/short-range decomposition Multi-determinantal DFT Analysis of short-range density functionals Beyond for short-range density functionals

13 p. 11/2 Multi-determinantal DFT Ground-state energy E = min min Ψ ˆT + Ŵ ee lr,µ Ψ n { = min Ψ ˆT + Ŵee lr,µ Ψ n { Ψ + Ēsr,µ Hxc [n] + + ˆV ne Ψ + Ē sr,µ Hxc [n Ψ] } } v ne (r)n(r)dr Euler-Lagrange equation ( ) ˆT + Ŵee lr,µ + ˆV ne + ˆV sr,µ Hxc [n Ψ µ] Ψ µ = E µ Ψ µ }{{} H µ with v sr,µ Hxc [n](r) = δēsr,µ Hxc [n]/δn(r). Ψµ is a multi-determinantal wave function giving the density. Final expression of the energy E = Ψ µ ˆT + Ŵ lr,µ ee + ˆV ne Ψ µ + Ēsr,µ Hxc [n Ψ µ ]

14 p. 12/2 Ground-state energy of Be E = Ψ µ ˆT + Ŵ lr,µ ee + ˆV ne Ψ µ + E sr,µ H [n Ψ µ] + Ēsr,µ xc [n Ψ µ] CI in limited configurational spaces E erf 1s2s FCI

15 p. 12/2 Ground-state energy of Be E = Ψ µ ˆT + Ŵ lr,µ ee + ˆV ne Ψ µ + E sr,µ H [n Ψ µ] + Ēsr,µ xc [n Ψ µ] CI in limited configurational spaces E erf 1s2s 1s2s2p FCI

16 p. 12/2 Ground-state energy of Be E = Ψ µ ˆT + Ŵ lr,µ ee + ˆV ne Ψ µ + E sr,µ H [n Ψ µ] + Ēsr,µ xc [n Ψ µ] CI in limited configurational spaces E erf 1s2s 1s2s2p FCI

17 p. 12/2 Ground-state energy of Be E = Ψ µ ˆT + Ŵ lr,µ ee + ˆV ne Ψ µ + E sr,µ H [n Ψ µ] + Ēsr,µ xc [n Ψ µ] CI in limited configurational spaces E erf 1s2s 1s2s2p FCI

18 p. 12/2 Ground-state energy of Be E = Ψ µ ˆT + Ŵ lr,µ ee + ˆV ne Ψ µ + E sr,µ H [n Ψ µ] + Ēsr,µ xc [n Ψ µ] CI in limited configurational spaces E PBE erf 1s2s 1s2s2p FCI

19 p. 13/2 Outline Long-range/short-range decomposition Multi-determinantal DFT Analysis of short-range density functionals Beyond for short-range density functionals

20 p. 14/2 Asymptotic expansion of Ex sr,µ [n] for µ For closed-shell systems: E sr,µ x [n] = A µ 2 n(r) 2 dr + A 2 µ 4 ( n(r) 2 n(r) 2n(r) ) + 4τ(r) dr + For the He atom:.2.4 E x

21 p. 15/2 Asymptotic expansion of Ēsr,µ [n] for µ c Ē sr,µ c [n] = B µ 2 n 2,c (r, r)dr + B 1 µ 3 (n 2,c (r, r) 12 ) n(r)2 dr + For the He atom:.2.4 E c

22 p. 16/2 Local analysis of Local short-range exchange-correlation energy per particle ε sr,µ xc (r) (no unique definition!): Ē sr,µ xc [n] = dr n(r) ε sr,µ xc (r) The formula (non-linear adiabatic connection) Ē sr,µ xc [n] = 1 2 suggests to define µ dµ dr 1 dr 2 n(r 1 ) n lr,µ xc (r 1, r 2 ) wlr,µ ee (r 12 ) µ ε sr,µ xc (r 1 ) = 1 2 µ dµ dr 2 n lr,µ xc (r 1, r 2 ) wlr,µ ee (r 12 ) µ

23 p. 17/2 Local analysis of Local short-range exchange energy per particle ε sr,µ x (r) for the Be atom:.2.2 ε x ε x r r.2.2 ε x ε x r r

24 p. 18/2 Local analysis of Local short-range correlation energy per particle ε sr,µ c (r) for the Be atom:.2.2 ε c ε c r r ε c ε c r r

25 p. 19/2 Outline Long-range/short-range decomposition Multi-determinantal DFT Analysis of short-range density functionals Beyond for short-range density functionals

26 p. 2/2 Gradient Expansion Approximation (GEA) GEA for the short-range exchange energy: E sr,µ x,gea [n] = Esr,µ x, [n] + dr n(r) C µ x (n) n 2 For the Be atom:.5 1 E x GEA

27 p. 21/2 Gradient Expansion Approximation (GEA) GEA for the short-range correlation energy: Ē sr,µ c,gea [n] = Ēsr,µ c, [n] + dr n(r) C c µ (n) n 2 with C µ c (n) C µ x (n) For the Be atom:.5 E c GEA

28 p. 22/2 Short-range PBE functional Extension of the PBE exchange-correlation functional to modified interactions: same ansatz but with µ-dependent constants. Short-range exhange energy Ē µ x [n] for the Be atom:.5 1 E x GEA PBE

29 p. 23/2 Short-range PBE functional Short-range correlation energy Ēc sr,µ [n] for the Be atom:.5 E c GEA PBE

30 p. 24/2 Short-range PBE functional Local short-range correlation energy per particle ε sr,µ c (r) for the Be atom:.2.2 ε c PBE ε c PBE r r.2.2 ε c PBE ε c PBE r r

31 p. 25/2 Conclusions Long-range/short-range decomposition: a rigorous way to combine many-body methods and DFT an understanding of density functional approximations in term of the range of the interaction Other works/perspectives: better short-range functionals MCSCF+DFT (J. Pedersen and H.J. Jensen, Odense) MP2+DFT for van der Waals (with J. Ángyán and I. Gerber, Nancy) RPA+DFT (with J. Ángyán and I. Gerber, Nancy) extensions?

Théorie de la fonctionnnelle de la densité avec séparation de portée pour les forces de van der Waals

Théorie de la fonctionnnelle de la densité avec séparation de portée pour les forces de van der Waals Julien Toulouse 1 Iann Gerber 2, Georg Jansen 3, Andreas Savin 1, János Ángyán 4 1 Laboratoire de Chimie