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

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 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 – p. 2/25

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

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

Decomposition of the interaction 1 r = w lr ,µ ee ( r ) + w sr ,µ ee ( r ) Two decompositions tested: • erf interaction: • erfgau interaction: ee ( r ) = erf( µr ) ee ( r ) = erf( cµr ) − 2 cµ √ π e − 1 3 c 2 µ 2 r 2 w lr ,µ w lr ,µ r r 3 µ = 1 2.5 1 /r 2 1.5 1 /µ 1 0.5 0 0 0.5 1 1.5 2 2.5 3 r ( r ) = 1 Limits: w lr ,µ =0 ( r ) = 0 and w lr ,µ →∞ ee ee r – p. 5/25

Decomposition of the universal functional Two alternatives: • Choice 1: Ψ → n � Ψ | ˆ T + ˆ F lr ,µ [ n ] = min W lr ,µ ee | Ψ � F [ n ] = F lr ,µ [ n ] + ¯ F sr ,µ [ n ] F sr ,µ [ n ] = E sr ,µ ¯ [ n ] + ¯ [ n ] + E sr ,µ E sr ,µ [ n ] x c H • Choice 2: Ψ → n � Ψ | ˆ T + ˆ F sr ,µ [ n ] = min W sr ,µ ee | Ψ � F [ n ] = F sr ,µ [ n ] + ¯ F lr ,µ [ n ] F sr ,µ [ n ] = T s [ n ] + E sr ,µ [ n ] + E sr ,µ [ n ] + E sr ,µ [ n ] H x c Limits: [ n ] = E x [ n ] and ¯ For µ = 0 : E sr ,µ =0 E sr ,µ =0 [ n ] = E sr ,µ =0 [ n ] = E c [ n ] x c c [ n ] = ¯ For µ → ∞ : E sr ,µ →∞ E sr ,µ →∞ [ n ] = E sr ,µ →∞ [ n ] = 0 x c c – p. 6/25

Short-range exchange energy E sr ,µ x � Local density approximation (LDA): E sr ,µ n ( r ) ε sr ,µ x, LDA [ n ] = x, unif ( n ( r )) d r For the He atom: 0 � 0.2 erf � 0.4 sr, Μ E x � 0.6 exact LDA � 0.8 � 1 0 1 2 3 4 Μ – p. 7/25

Short-range exchange energy E sr ,µ x � Local density approximation (LDA): E sr ,µ n ( r ) ε sr ,µ x, LDA [ n ] = x, unif ( n ( r )) d r For the He atom: 0 � 0.2 erf � 0.4 erfgau sr, Μ E x � 0.6 exact LDA � 0.8 exact LDA � 1 0 1 2 3 4 Μ – p. 7/25

Short-range correlation energy ¯ (choice 1) E sr ,µ c � Local density approximation (LDA): ¯ E sr ,µ ε sr ,µ c, LDA [ n ] = n ( r )¯ c, unif ( n ( r )) d r For the He atom: 0 � 0.02 erf � 0.04 � 0.06 sr, Μ �� E c exact � 0.08 LDA � 0.1 � 0.12 0 1 2 3 4 5 6 Μ – p. 8/25

Short-range correlation energy ¯ (choice 1) E sr ,µ c � Local density approximation (LDA): ¯ E sr ,µ ε sr ,µ c, LDA [ n ] = n ( r )¯ c, unif ( n ( r )) d r For the He atom: 0 � 0.02 erf � 0.04 erfgau � 0.06 sr, Μ �� E c exact � 0.08 LDA � 0.1 exact LDA � 0.12 0 1 2 3 4 5 6 Μ – p. 8/25

Short-range correlation energies E sr ,µ (choice 2) c � Local density approximation (LDA): E sr ,µ n ( r ) ε sr ,µ c, LDA [ n ] = c, unif ( n ( r )) d r For the He atom: 0 erf � choice 2 � � 0.02 � 0.04 erf � choice 1 � � 0.06 sr, Μ E c exact � 0.08 LDA � 0.1 exact LDA � 0.12 0 1 2 3 4 5 6 Μ = ⇒ In choice 1, LDA can treat well a larger part of correlation energy – p. 9/25

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

Multi-determinantal DFT Ground-state energy � � � Ψ → n � Ψ | ˆ T + ˆ ee | Ψ � + ¯ E sr ,µ W lr ,µ E = min min H xc [ n ] + v ne ( r ) n ( r ) d r n � � � Ψ | ˆ T + ˆ ee + ˆ V ne | Ψ � + ¯ E sr ,µ W lr ,µ = min H xc [ n Ψ ] Ψ Euler-Lagrange equation � � Ψ µ = E µ Ψ µ T + ˆ ˆ + ˆ V ne + ˆ V sr ,µ W lr ,µ H xc [ n Ψ µ ] ee � �� � H µ H xc [ n ] /δn ( r ) . Ψ µ is a multi-determinantal wave function H xc [ n ]( r ) = δ ¯ with v sr ,µ E sr ,µ giving the exact density. Final expression of the energy E = � Ψ µ | ˆ T + ˆ + ˆ V ne | Ψ µ � + ¯ E sr ,µ W lr ,µ H xc [ n Ψ µ ] ee – p. 11/25

Ground-state energy of Be E = � Ψ µ | ˆ T + ˆ ee + ˆ [ n Ψ µ ] + ¯ V ne | Ψ µ � + E sr ,µ W lr ,µ E sr ,µ xc [ n Ψ µ ] H CI in limited configurational spaces LDA � 14.35 � 14.4 � 14.45 � 14.5 E � 14.55 erf 1s2s � 14.6 � 14.65 FCI 0 2 4 6 8 Μ – p. 12/25

Ground-state energy of Be E = � Ψ µ | ˆ T + ˆ ee + ˆ [ n Ψ µ ] + ¯ V ne | Ψ µ � + E sr ,µ W lr ,µ E sr ,µ xc [ n Ψ µ ] H CI in limited configurational spaces LDA � 14.35 � 14.4 � 14.45 � 14.5 E � 14.55 erf 1s2s � 14.6 1s2s2p � 14.65 FCI 0 2 4 6 8 Μ – p. 12/25

Ground-state energy of Be E = � Ψ µ | ˆ T + ˆ ee + ˆ [ n Ψ µ ] + ¯ V ne | Ψ µ � + E sr ,µ W lr ,µ E sr ,µ xc [ n Ψ µ ] H CI in limited configurational spaces LDA � 14.35 � 14.4 � 14.45 erfgau � 14.5 E � 14.55 erf 1s2s � 14.6 1s2s2p � 14.65 FCI 0 2 4 6 8 Μ – p. 12/25

Ground-state energy of Be E = � Ψ µ | ˆ T + ˆ ee + ˆ [ n Ψ µ ] + ¯ V ne | Ψ µ � + E sr ,µ W lr ,µ E sr ,µ xc [ n Ψ µ ] H CI in limited configurational spaces LDA � 14.35 � 14.4 � 14.45 erfgau � 14.5 E � 14.55 erf 1s2s � 14.6 1s2s2p � 14.65 FCI 0 2 4 6 8 Μ – p. 12/25

Ground-state energy of Be E = � Ψ µ | ˆ T + ˆ ee + ˆ [ n Ψ µ ] + ¯ V ne | Ψ µ � + E sr ,µ W lr ,µ E sr ,µ xc [ n Ψ µ ] H CI in limited configurational spaces LDA � 14.35 � 14.4 � 14.45 erfgau � 14.5 E � 14.55 erf 1s2s � 14.6 1s2s2p PBE � 14.65 FCI 0 2 4 6 8 Μ – p. 12/25

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

[ n ] for µ → ∞ Asymptotic expansion of E sr ,µ x For closed-shell systems: � |∇ n ( r ) | 2 � � � − A 0 n ( r ) 2 d r + A 2 E sr ,µ [ n ] = n ( r ) + 4 τ ( r ) d r + · · · x µ 2 µ 4 2 n ( r ) For the He atom: 0 � 0.2 � 0.4 sr, Μ E x � 0.6 exact � 0.8 erfgau LDA � 1 0 1 2 3 4 Μ – p. 14/25

Asymptotic expansion of ¯ [ n ] for µ → ∞ E sr ,µ c � � � � [ n ] = B 0 n 2 ,c ( r , r ) d r + B 1 n 2 ,c ( r , r ) − 1 ¯ E sr ,µ 2 n ( r ) 2 d r + · · · c µ 2 µ 3 For the He atom: 0 � 0.02 � 0.04 sr, Μ �� � 0.06 E c exact � 0.08 erfgau LDA � 0.1 5 0 1 2 3 4 6 Μ – p. 15/25

Local analysis of LDA ε sr ,µ Local short-range exchange-correlation energy per particle ¯ xc ( r ) (no unique definition!): � ¯ E sr ,µ ε sr ,µ xc [ n ] = d r n ( r ) ¯ xc ( r ) The exact formula (non-linear adiabatic connection) � ∞ �� xc ( r 1 , r 2 ) ∂w lr ,µ ′ xc [ n ] = 1 ee ( r 12 ) ¯ d r 1 d r 2 n ( r 1 ) n lr ,µ ′ E sr ,µ dµ ′ 2 ∂µ ′ µ suggests to define � ∞ � xc ( r 1 , r 2 ) ∂w lr ,µ ′ xc ( r 1 ) = 1 ee ( r 12 ) d r 2 n lr ,µ ′ ε sr ,µ dµ ′ ¯ 2 ∂µ ′ µ – p. 16/25

Local analysis of LDA Local short-range exchange energy per particle ε sr ,µ ( r ) for the Be atom: x 0 0 � 0.2 � 0.2 � 0.4 � 0.4 sr, Μ sr, Μ � x � x erfgau erfgau � 0.6 � 0.6 exact exact LDA LDA Μ � 0.00 Μ � 0.27 � 0.8 � 0.8 0 2 4 6 8 0 2 4 6 8 r r 0 0 � 0.2 � 0.2 � 0.4 � 0.4 sr, Μ sr, Μ � x � x erfgau erfgau � 0.6 � 0.6 exact exact LDA LDA Μ � 1.19 Μ � 2.96 � 0.8 � 0.8 0 2 4 6 8 0 2 4 6 8 r r – p. 17/25

Local analysis of LDA ε sr ,µ Local short-range correlation energy per particle ¯ ( r ) for the Be atom: c 0 0 � 0.02 � 0.02 � 0.04 � 0.04 �� sr, Μ �� sr, Μ � � erfgau erfgau c c � 0.06 � 0.06 Μ � 0.00 Μ � 0.27 � 0.08 � 0.08 exact exact LDA LDA � 0.1 � 0.1 0 2 4 6 8 0 2 4 6 8 r r 0 0 � 0.02 � 0.02 � 0.04 � 0.04 sr, Μ sr, Μ �� �� � � erfgau erfgau c c � 0.06 � 0.06 Μ � 1.19 Μ � 2.96 � 0.08 � 0.08 exact exact LDA LDA � 0.1 � 0.1 0 2 4 6 8 0 2 4 6 8 r r – p. 18/25

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

Gradient Expansion Approximation (GEA) GEA for the short-range exchange energy : � E sr ,µ x, GEA [ n ] = E sr ,µ d r n ( r ) C µ x ( n ) |∇ n | 2 x, LDA [ n ] + For the Be atom: 0 � 0.5 erfgau � 1 sr, Μ E x � 1.5 exact � 2 LDA GEA � 2.5 0 2 4 6 8 Μ – p. 20/25