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?
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