Long-range/short-range energy decomposition in density functional - - PowerPoint PPT Presentation

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Long-range/short-range energy decomposition in density functional - - PowerPoint PPT Presentation

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Long-range/short-range energy decomposition in density functional theory Julien Toulouse Franc ois Colonna, Andreas Savin Laboratoire de Chimie Th eorique, Universit e Pierre et Marie Curie, Paris p. 1/25 Introduction Some

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SLIDE 1

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

Julien Toulouse Franc ¸ois Colonna, Andreas Savin Laboratoire de Chimie Th´ eorique, Universit´ e Pierre et Marie Curie, Paris

– p. 1/25

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SLIDE 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 = Elr + Esr explicit many-body approximation density functional approximation

– p. 2/25

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SLIDE 3

Outline

  • Long-range/short-range decomposition
  • Multi-determinantal DFT
  • Analysis of short-range density functionals
  • Beyond LDA for short-range density functionals

– p. 3/25

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SLIDE 4

Outline

  • Long-range/short-range decomposition
  • Multi-determinantal DFT
  • Analysis of short-range density functionals
  • Beyond LDA for short-range density functionals

– p. 4/25

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SLIDE 5

Decomposition of the interaction

1 r = wlr,µ

ee (r) + wsr,µ ee (r)

Two decompositions tested:

  • erf interaction:

wlr,µ

ee (r) = erf(µr)

r

  • erfgau interaction:

wlr,µ

ee (r) = erf(cµr)

r − 2cµ √π e− 1

3 c2µ2r2

1/r 1/µ µ = 1

0.5 1 1.5 2 2.5 3 r 0.5 1 1.5 2 2.5 3

Limits: wlr,µ=0

ee

(r) = 0 and wlr,µ→∞

ee

(r) = 1 r

– p. 5/25

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SLIDE 6

Decomposition of the universal functional

Two alternatives:

  • Choice 1:

F lr,µ[n] = min

Ψ→nΨ| ˆ

T + ˆ W lr,µ

ee |Ψ

F[n] = F lr,µ[n] + ¯ F sr,µ[n] ¯ F sr,µ[n] = Esr,µ

H

[n] + Esr,µ

x

[n] + ¯ Esr,µ

c

[n]

  • Choice 2:

F sr,µ[n] = min

Ψ→nΨ| ˆ

T + ˆ W sr,µ

ee |Ψ

F[n] = F sr,µ[n] + ¯ F lr,µ[n] F sr,µ[n] = Ts[n] + Esr,µ

H

[n] + Esr,µ

x

[n] + Esr,µ

c

[n] Limits: For µ = 0: Esr,µ=0

x

[n] = Ex[n] and ¯ Esr,µ=0

c

[n] = Esr,µ=0

c

[n] = Ec[n] For µ → ∞: Esr,µ→∞

x

[n] = ¯ Esr,µ→∞

c

[n] = Esr,µ→∞

c

[n] = 0

– p. 6/25

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SLIDE 7

Short-range exchange energy Esr,µ

x

Local density approximation (LDA): Esr,µ

x,LDA[n] =

  • n(r)εsr,µ

x,unif(n(r))dr

For the He atom:

1 2 3 4 Μ 1 0.8 0.6 0.4 0.2 Ex

sr,Μ

exact LDA erf

– p. 7/25

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SLIDE 8

Short-range exchange energy Esr,µ

x

Local density approximation (LDA): Esr,µ

x,LDA[n] =

  • n(r)εsr,µ

x,unif(n(r))dr

For the He atom:

1 2 3 4 Μ 1 0.8 0.6 0.4 0.2 Ex

sr,Μ

exact LDA exact LDA erf erfgau

– p. 7/25

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SLIDE 9

Short-range correlation energy ¯

Esr,µ

c

(choice 1)

Local density approximation (LDA): ¯ Esr,µ

c,LDA[n] =

  • n(r)¯

εsr,µ

c,unif(n(r))dr

For the He atom:

1 2 3 4 5 6 Μ 0.12 0.1 0.08 0.06 0.04 0.02 E

  • c

sr,Μ

exact LDA erf

– p. 8/25

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SLIDE 10

Short-range correlation energy ¯

Esr,µ

c

(choice 1)

Local density approximation (LDA): ¯ Esr,µ

c,LDA[n] =

  • n(r)¯

εsr,µ

c,unif(n(r))dr

For the He atom:

1 2 3 4 5 6 Μ 0.12 0.1 0.08 0.06 0.04 0.02 E

  • c

sr,Μ

exact LDA exact LDA erf erfgau

– p. 8/25

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SLIDE 11

Short-range correlation energies Esr,µ

c

(choice 2)

Local density approximation (LDA): Esr,µ

c,LDA[n] =

  • n(r)εsr,µ

c,unif(n(r))dr

For the He atom:

1 2 3 4 5 6 Μ 0.12 0.1 0.08 0.06 0.04 0.02 Ec

sr,Μ

exact LDA exact LDA erf choice 1 erf choice 2

= ⇒ In choice 1, LDA can treat well a larger part of correlation energy

– p. 9/25

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SLIDE 12

Outline

  • Long-range/short-range decomposition
  • Multi-determinantal DFT
  • Analysis of short-range density functionals
  • Beyond LDA for short-range density functionals

– p. 10/25

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SLIDE 13

Multi-determinantal DFT

Ground-state energy E = min

n

  • min

Ψ→nΨ| ˆ

T + ˆ W lr,µ

ee |Ψ + ¯

Esr,µ

Hxc[n] +

  • vne(r)n(r)dr
  • =

min

Ψ

  • Ψ| ˆ

T + ˆ W lr,µ

ee + ˆ

Vne|Ψ + ¯ Esr,µ

Hxc[nΨ]

  • Euler-Lagrange equation
  • ˆ

T + ˆ W lr,µ

ee

+ ˆ Vne + ˆ V sr,µ

Hxc [nΨµ]

  • Hµ

Ψµ = EµΨµ with vsr,µ

Hxc[n](r) = δ ¯

Esr,µ

Hxc[n]/δn(r). Ψµ is a multi-determinantal wave function

giving the exact density. Final expression of the energy E = Ψµ| ˆ T + ˆ W lr,µ

ee

+ ˆ Vne|Ψµ + ¯ Esr,µ

Hxc[nΨµ]

– p. 11/25

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SLIDE 14

Ground-state energy of Be

E = Ψµ| ˆ T + ˆ W lr,µ

ee + ˆ

Vne|Ψµ + Esr,µ

H

[nΨµ] + ¯ Esr,µ

xc [nΨµ]

CI in limited configurational spaces LDA

2 4 6 8 Μ 14.65 14.6 14.55 14.5 14.45 14.4 14.35 E

FCI 1s2s erf

– p. 12/25

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SLIDE 15

Ground-state energy of Be

E = Ψµ| ˆ T + ˆ W lr,µ

ee + ˆ

Vne|Ψµ + Esr,µ

H

[nΨµ] + ¯ Esr,µ

xc [nΨµ]

CI in limited configurational spaces LDA

2 4 6 8 Μ 14.65 14.6 14.55 14.5 14.45 14.4 14.35 E

FCI 1s2s 1s2s2p erf

– p. 12/25

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SLIDE 16

Ground-state energy of Be

E = Ψµ| ˆ T + ˆ W lr,µ

ee + ˆ

Vne|Ψµ + Esr,µ

H

[nΨµ] + ¯ Esr,µ

xc [nΨµ]

CI in limited configurational spaces LDA

2 4 6 8 Μ 14.65 14.6 14.55 14.5 14.45 14.4 14.35 E

FCI 1s2s 1s2s2p erf erfgau

– p. 12/25

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SLIDE 17

Ground-state energy of Be

E = Ψµ| ˆ T + ˆ W lr,µ

ee + ˆ

Vne|Ψµ + Esr,µ

H

[nΨµ] + ¯ Esr,µ

xc [nΨµ]

CI in limited configurational spaces LDA

2 4 6 8 Μ 14.65 14.6 14.55 14.5 14.45 14.4 14.35 E

FCI 1s2s 1s2s2p erf erfgau

– p. 12/25

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SLIDE 18

Ground-state energy of Be

E = Ψµ| ˆ T + ˆ W lr,µ

ee + ˆ

Vne|Ψµ + Esr,µ

H

[nΨµ] + ¯ Esr,µ

xc [nΨµ]

CI in limited configurational spaces LDA

2 4 6 8 Μ 14.65 14.6 14.55 14.5 14.45 14.4 14.35 E

FCI PBE 1s2s 1s2s2p erf erfgau

– p. 12/25

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SLIDE 19

Outline

  • Long-range/short-range decomposition
  • Multi-determinantal DFT
  • Analysis of short-range density functionals
  • Beyond LDA for short-range density functionals

– p. 13/25

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SLIDE 20

Asymptotic expansion of Esr,µ

x

[n] for µ → ∞

For closed-shell systems: Esr,µ

x

[n] = −A0 µ2

  • n(r)2dr + A2

µ4

  • n(r)

|∇n(r)|2 2n(r) + 4τ(r)

  • dr + · · ·

For the He atom:

1 2 3 4 Μ 1 0.8 0.6 0.4 0.2 Ex

sr,Μ

exact LDA erfgau

– p. 14/25

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SLIDE 21

Asymptotic expansion of ¯

Esr,µ

c

[n] for µ → ∞

¯ Esr,µ

c

[n] = B0 µ2

  • n2,c(r, r)dr + B1

µ3 n2,c(r, r) − 1 2n(r)2

  • dr + · · ·

For the He atom:

1 2 3 4 5 6 Μ 0.1 0.08 0.06 0.04 0.02 E

  • c

sr,Μ

exact LDA erfgau

– p. 15/25

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SLIDE 22

Local analysis of LDA

Local short-range exchange-correlation energy per particle ¯ εsr,µ

xc (r) (no unique

definition!): ¯ Esr,µ

xc [n] =

  • dr n(r) ¯

εsr,µ

xc (r)

The exact formula (non-linear adiabatic connection) ¯ Esr,µ

xc [n] = 1

2 ∞

µ

dµ′

  • dr1dr2 n(r1) nlr,µ′

xc (r1, r2)∂wlr,µ′ ee (r12)

∂µ′ suggests to define ¯ εsr,µ

xc (r1) = 1

2 ∞

µ

dµ′

  • dr2nlr,µ′

xc (r1, r2)∂wlr,µ′ ee (r12)

∂µ′

– p. 16/25

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SLIDE 23

Local analysis of LDA

Local short-range exchange energy per particle εsr,µ

x

(r) for the Be atom:

2 4 6 8 r 0.8 0.6 0.4 0.2 x

sr,Μ

exact LDA erfgau

Μ 0.00

2 4 6 8 r 0.8 0.6 0.4 0.2 x

sr,Μ

exact LDA erfgau

Μ 0.27

2 4 6 8 r 0.8 0.6 0.4 0.2 x

sr,Μ

exact LDA erfgau

Μ 1.19

2 4 6 8 r 0.8 0.6 0.4 0.2 x

sr,Μ

exact LDA erfgau

Μ 2.96

– p. 17/25

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SLIDE 24

Local analysis of LDA

Local short-range correlation energy per particle ¯ εsr,µ

c

(r) for the Be atom:

2 4 6 8 r 0.1 0.08 0.06 0.04 0.02

  • c

sr,Μ

exact LDA erfgau

Μ 0.00

2 4 6 8 r 0.1 0.08 0.06 0.04 0.02

  • c

sr,Μ

exact LDA erfgau

Μ 0.27

2 4 6 8 r 0.1 0.08 0.06 0.04 0.02

  • c

sr,Μ

exact LDA erfgau

Μ 1.19

2 4 6 8 r 0.1 0.08 0.06 0.04 0.02

  • c

sr,Μ

exact LDA erfgau

Μ 2.96

– p. 18/25

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SLIDE 25

Outline

  • Long-range/short-range decomposition
  • Multi-determinantal DFT
  • Analysis of short-range density functionals
  • Beyond LDA for short-range density functionals

– p. 19/25

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SLIDE 26

Gradient Expansion Approximation (GEA)

GEA for the short-range exchange energy: Esr,µ

x,GEA[n] = Esr,µ x,LDA[n] +

  • dr n(r) Cµ

x(n)|∇n|2

For the Be atom:

2 4 6 8 Μ 2.5 2 1.5 1 0.5 Ex

sr,Μ

exact LDA GEA erfgau

– p. 20/25

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SLIDE 27

Gradient Expansion Approximation (GEA)

GEA for the short-range correlation energy: ¯ Esr,µ

c,GEA[n] = ¯

Esr,µ

c,LDA[n] +

  • dr n(r) Cµ

c (n)|∇n|2

with Cµ

c (n) ≈ −Cµ x (n)

For the Be atom:

2 4 6 8 Μ 0.2 0.15 0.1 0.05 E

  • c

sr,Μ

exact LDA GEA erfgau

– p. 21/25

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SLIDE 28

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 ¯ Eµ

x[n] for the Be atom:

2 4 6 8 Μ 2.5 2 1.5 1 0.5 Ex

sr,Μ

exact LDA GEA PBE erfgau

– p. 22/25

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SLIDE 29

Short-range PBE functional

Short-range correlation energy ¯ Esr,µ

c

[n] for the Be atom:

2 4 6 8 Μ 0.2 0.15 0.1 0.05 E

  • c

sr,Μ

exact LDA GEA PBE erfgau

– p. 23/25

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SLIDE 30

Short-range PBE functional

Local short-range correlation energy per particle ¯ εsr,µ

c

(r) for the Be atom:

2 4 6 8 r 0.1 0.08 0.06 0.04 0.02

  • c

sr,Μ

exact LDA PBE erfgau

Μ 0.00

2 4 6 8 r 0.1 0.08 0.06 0.04 0.02

  • c

sr,Μ

exact LDA PBE erfgau

Μ 0.27

2 4 6 8 r 0.1 0.08 0.06 0.04 0.02

  • c

sr,Μ

exact LDA PBE erfgau

Μ 1.19

2 4 6 8 r 0.1 0.08 0.06 0.04 0.02

  • c

sr,Μ

exact LDA PBE erfgau

Μ 2.96

– p. 24/25

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SLIDE 31

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
  • f 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?

– p. 25/25