Multi-configurational DFT by long-range/short-range separation of - PowerPoint PPT Presentation

Multi-configurational DFT by long-range/short-range separation of the electron-electron interaction Julien Toulouse, Franc ¸ois Colonna, Andreas Savin Laboratoire de Chimie Th´ eorique, Universit´ e Pierre et Marie Curie, Paris COST 2004 – p. 1/12

Introduction Electronic correlation can be treated by DFT Wave function methods COST 2004 – p. 2/12

Introduction Electronic correlation can be treated by DFT Wave function methods • “dynamical” correlation effects are • a long expansion of determinants is well treated needed to treat “dynamical” correla- tion COST 2004 – p. 2/12

Introduction Electronic correlation can be treated by DFT Wave function methods • “dynamical” correlation effects are • a long expansion of determinants is well treated needed to treat “dynamical” correla- tion • near-degeneracy is not explicitly • near-degeneracy is easily treated treated COST 2004 – p. 2/12

Introduction Electronic correlation can be treated by DFT Wave function methods • “dynamical” correlation effects • a long expansion of determinants is are well treated needed to treat “dynamical” correla- tion • near-degeneracy is not explicitly • near-degeneracy is easily treated treated = ⇒ Combination of the two approaches to take advantage of both of them. COST 2004 – p. 2/12

Adiabatic connection � ˆ T + ˆ � E KS V KS | Φ � = E KS | Φ � E COST 2004 – p. 3/12

Adiabatic connection � ˆ T + ˆ � E KS V KS | Φ � = E KS | Φ � � ˆ T + ˆ V ee + ˆ � V ne | Ψ � = E | Ψ � E COST 2004 – p. 3/12

Adiabatic connection Continuously switching on the electron-electron interaction ˆ V µ ee according to a parameter µ at constant density n � ˆ T + ˆ � E KS V KS | Φ � = E KS | Φ � � ˆ T + ˆ V ee + ˆ � V ne | Ψ � = E | Ψ � E µ → ∞ µ = 0 COST 2004 – p. 3/12

Adiabatic connection Continuously switching on the electron-electron interaction ˆ V µ ee according to a parameter µ at constant density n � ˆ T + ˆ � E KS V KS | Φ � = E KS | Φ � � ˆ T + ˆ ee + ˆ V µ V µ � | Ψ µ � = E µ | Ψ µ � E µ � ˆ T + ˆ V ee + ˆ � V ne | Ψ � = E | Ψ � E µ µ → ∞ µ = 0 COST 2004 – p. 3/12

Adiabatic connection Continuously switching on the electron-electron interaction ˆ V µ ee according to a parameter µ at constant density n � ˆ T + ˆ � E KS V KS | Φ � = E KS | Φ � V µ = � ˆ i v µ ( r i ) � ˆ T + ˆ ee + ˆ V µ V µ � | Ψ µ � = E µ | Ψ µ � E µ � ˆ T + ˆ V ee + ˆ � V ne | Ψ � = E | Ψ � E µ µ → ∞ µ = 0 COST 2004 – p. 3/12

Ground-state electronic energy • Mono-determinantal DFT � � � Φ | ˆ T + ˆ E = min V ne | Φ � + U [ n Φ ] + E xc [ n Φ ] Φ • Multi-determinantal DFT � � � Ψ | ˆ T + ˆ ee + ˆ V ne | Ψ � + ¯ U µ [ n Ψ ] + ¯ V µ E µ E = min xc [ n Ψ ] Ψ COST 2004 – p. 4/12

Ground-state electronic energy • Mono-determinantal DFT � � � Φ | ˆ T + ˆ E = min V ne | Φ � + U [ n Φ ] + E xc [ n Φ ] Φ • Multi-determinantal DFT � � � Ψ | ˆ T + ˆ ee + ˆ V ne | Ψ � + ¯ U µ [ n Ψ ] + ¯ V µ E µ E = min xc [ n Ψ ] Ψ E = � Ψ µ | ˆ T + ˆ ee + ˆ V ne | Ψ µ � + ¯ U µ [ n ] + ¯ V µ E µ xc [ n ] COST 2004 – p. 4/12

Ground-state electronic energy • Mono-determinantal DFT � � � Φ | ˆ T + ˆ E = min V ne | Φ � + U [ n Φ ] + E xc [ n Φ ] Φ • Multi-determinantal DFT � � � Ψ | ˆ T + ˆ ee + ˆ V ne | Ψ � + ¯ U µ [ n Ψ ] + ¯ V µ E µ E = min xc [ n Ψ ] Ψ E = � Ψ µ | ˆ T + ˆ ee + ˆ V ne | Ψ µ � + ¯ U µ [ n ] + ¯ V µ E µ xc [ n ] - Ψ µ is the multi-determinantal ground-state wave function of ˆ T + ˆ ee + ˆ V µ V µ COST 2004 – p. 4/12

Ground-state electronic energy • Mono-determinantal DFT � � � Φ | ˆ T + ˆ E = min V ne | Φ � + U [ n Φ ] + E xc [ n Φ ] Φ • Multi-determinantal DFT � � � Ψ | ˆ T + ˆ ee + ˆ V ne | Ψ � + ¯ U µ [ n Ψ ] + ¯ V µ E µ E = min xc [ n Ψ ] Ψ E = � Ψ µ | ˆ T + ˆ ee + ˆ V ne | Ψ µ � + ¯ U µ [ n ] + ¯ V µ E µ xc [ n ] - Ψ µ is the multi-determinantal ground-state wave function of ˆ T + ˆ ee + ˆ V µ V µ - ¯ U µ [ n ] = U [ n ] − U µ [ n ] is the complement Hartree energy - ¯ E µ xc [ n ] = E xc [ n ] − E µ xc [ n ] is the complement exchange-correlation energy COST 2004 – p. 4/12

Long-range/short-range separation ˆ � V µ v µ ee = ee ( r ij ) : long-range part of the Coulomb interaction i<j COST 2004 – p. 5/12

Long-range/short-range separation ˆ � V µ v µ ee = ee ( r ij ) : long-range part of the Coulomb interaction i<j E = � Ψ µ | ˆ T + ˆ ee + ˆ V ne | Ψ µ � + ¯ U µ [ n ] + ¯ V µ E µ xc [ n ] ¯ E µ • xc [ n ] : short-range interaction ⇒ well approximated by (semi-)local approximations COST 2004 – p. 5/12

Long-range/short-range separation ˆ � V µ v µ ee = ee ( r ij ) : long-range part of the Coulomb interaction i<j E = � Ψ µ | ˆ T + ˆ ee + ˆ V ne | Ψ µ � + ¯ U µ [ n ] + ¯ V µ E µ xc [ n ] ¯ E µ • xc [ n ] : short-range interaction ⇒ well approximated by (semi-)local approximations • Ψ µ : reduced interaction ⇒ well approximated by a few determinants corresponding to the near-degenerate states COST 2004 – p. 5/12

Possible long-range interactions • erf interaction: ee ( r ) = erf( µr ) v µ r 3 µ = 1 1 /r 2.5 2 1 /µ 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 r COST 2004 – p. 6/12

Possible long-range interactions • erf interaction: • erfgau interaction: ee ( r ) = erf( µr ) ee ( r ) = erf(˜ µr ) − 2˜ µ µ 2 r 2 √ π e − 1 v µ v ˜ µ 3 ˜ r r For comparison, ˜ µ ≈ 3 µ 3 µ = 1 1 /r 2.5 2 1 /µ 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 r COST 2004 – p. 6/12

The exchange-correlation functional ¯ xc [ n ] E µ • The short-range functional ¯ E µ xc [ n ] can be divided as ¯ xc [ n ] = ¯ x [ n ] + ¯ E µ E µ E µ c [ n ] COST 2004 – p. 7/12

The exchange-correlation functional ¯ xc [ n ] E µ • The short-range functional ¯ E µ xc [ n ] can be divided as ¯ xc [ n ] = ¯ x [ n ] + ¯ E µ E µ E µ c [ n ] • Local density approximation (LDA) for ¯ E µ xc [ n ] : � ¯ E µ ε unif ,µ xc [ n ] = n ( r )¯ ( n ( r )) d r xc ε unif ,µ ¯ ( n ) : short-range exchange-correlation energy per particle of a xc uniform electron gas with modified interaction COST 2004 – p. 7/12

The exchange-correlation functional ¯ xc [ n ] E µ Performance of the LDA for the Be atom: Short-range exchange energy ¯ E µ x [ n ] 0 erf � 0.5 � 1 �� Μ E � 1.5 x � 2 exact LDA � 2.5 0 2 4 6 8 Μ COST 2004 – p. 8/12

The exchange-correlation functional ¯ xc [ n ] E µ Performance of the LDA for the Be atom: Short-range exchange energy ¯ E µ x [ n ] 0 erf � 0.5 � 1 �� Μ E � 1.5 x erfgau � 2 exact LDA exact � 2.5 LDA 0 2 4 6 8 Μ COST 2004 – p. 8/12

The exchange-correlation functional ¯ xc [ n ] E µ Performance of the LDA for the Be atom: Short-range exchange energy ¯ Short-range correlation energy ¯ E µ E µ x [ n ] c [ n ] 0 0 erf erf � 0.5 � 0.05 � 1 � 0.1 �� �� Μ Μ E E � 1.5 x c erfgau � 0.15 exact � 2 exact LDA LDA � 0.2 exact � 2.5 LDA 0 2 4 6 8 0 2 4 6 8 Μ Μ COST 2004 – p. 8/12

The exchange-correlation functional ¯ xc [ n ] E µ Performance of the LDA for the Be atom: Short-range exchange energy ¯ Short-range correlation energy ¯ E µ E µ x [ n ] c [ n ] 0 0 erf erf � 0.5 � 0.05 � 1 � 0.1 �� �� Μ Μ erfgau E E � 1.5 x c erfgau � 0.15 exact � 2 exact LDA LDA � 0.2 exact exact � 2.5 LDA LDA 0 2 4 6 8 0 2 4 6 8 Μ Μ COST 2004 – p. 8/12

The exchange-correlation functional ¯ xc [ n ] E µ Performance of the LDA for the Be atom: Short-range exchange energy ¯ Short-range correlation energy ¯ E µ E µ x [ n ] c [ n ] 0 0 erf erf � 0.5 � 0.05 � 1 � 0.1 �� �� Μ Μ erfgau E E � 1.5 x c erfgau � 0.15 exact � 2 exact LDA LDA � 0.2 exact exact � 2.5 LDA LDA 0 2 4 6 8 0 2 4 6 8 Μ Μ Asymptotic expansions for µ → ∞ : x [ n ] ≈ − A c [ n ] ≈ B � � ¯ ¯ E µ n ( r ) 2 d r + · · · E µ n 2 ,c ( r , r ) d r + · · · µ 2 µ 2 COST 2004 – p. 8/12

Ground-state electronic energy of the Be atom E = � Ψ µ | ˆ T + ˆ ee + ˆ V ne | Ψ µ � + ¯ U µ [ n ] + ¯ V µ E µ xc [ n ] COST 2004 – p. 9/12

Ground-state electronic energy of the Be atom E = � Ψ µ | ˆ T + ˆ ee + ˆ V ne | Ψ µ � + ¯ U µ [ n ] + ¯ V µ E µ xc [ n ] CI in limited configurational spaces COST 2004 – p. 9/12

Ground-state electronic energy of the Be atom E = � Ψ µ | ˆ T + ˆ ee + ˆ V ne | Ψ µ � + ¯ U µ [ n ] + ¯ V µ E µ xc [ n ] CI in limited configurational spaces LDA COST 2004 – p. 9/12

Ground-state electronic energy of the Be atom E = � Ψ µ | ˆ T + ˆ ee + ˆ V ne | Ψ µ � + ¯ U µ [ n ] + ¯ V µ E µ xc [ n ] CI in limited configurational spaces LDA � 14.35 � 14.4 � 14.45 � 14.5 E � 14.55 erf 1s2s � 14.6 � 14.65 exact 0 2 4 6 8 Μ COST 2004 – p. 9/12