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

SLIDE 1

Multi-configurational DFT by long-range/short-range separation

f 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

SLIDE 2

Introduction

Electronic correlation can be treated by DFT Wave function methods

COST 2004 – p. 2/12

SLIDE 3

Introduction

Electronic correlation can be treated by DFT Wave function methods

“dynamical” correlation effects are

well treated

a long expansion of determinants is

needed to treat “dynamical” correla- tion

COST 2004 – p. 2/12

SLIDE 4

Introduction

Electronic correlation can be treated by DFT Wave function methods

“dynamical” correlation effects are

well treated

a long expansion of determinants is

needed to treat “dynamical” correla- tion

near-degeneracy is not explicitly

treated

near-degeneracy is easily treated

COST 2004 – p. 2/12

SLIDE 5

Introduction

Electronic correlation can be treated by DFT Wave function methods

“dynamical” correlation effects

are well treated

a long expansion of determinants is

needed to treat “dynamical” correla- tion

near-degeneracy is not explicitly

treated

near-degeneracy is easily treated

= ⇒ Combination of the two approaches to take advantage of both of them.

COST 2004 – p. 2/12

SLIDE 6

Adiabatic connection

EKS ˆ T + ˆ VKS

|Φ = EKS|Φ

E

COST 2004 – p. 3/12

SLIDE 7

Adiabatic connection

EKS ˆ T + ˆ VKS

|Φ = EKS|Φ

E ˆ T + ˆ Vee + ˆ Vne

|Ψ = E|Ψ

COST 2004 – p. 3/12

SLIDE 8

Adiabatic connection

Continuously switching on the electron-electron interaction ˆ V µ

ee according to

a parameter µ at constant density n EKS ˆ T + ˆ VKS

|Φ = EKS|Φ

E ˆ T + ˆ Vee + ˆ Vne

|Ψ = E|Ψ

µ = 0 µ → ∞

COST 2004 – p. 3/12

SLIDE 9

Adiabatic connection

Continuously switching on the electron-electron interaction ˆ V µ

ee according to

a parameter µ at constant density n EKS ˆ T + ˆ VKS

|Φ = EKS|Φ

E ˆ T + ˆ Vee + ˆ Vne

|Ψ = E|Ψ

µ = 0 µ → ∞ Eµ µ ˆ T + ˆ V µ

ee + ˆ

V µ |Ψµ = Eµ|Ψµ

COST 2004 – p. 3/12

SLIDE 10

Adiabatic connection

Continuously switching on the electron-electron interaction ˆ V µ

ee according to

a parameter µ at constant density n EKS ˆ T + ˆ VKS

|Φ = EKS|Φ

E ˆ T + ˆ Vee + ˆ Vne

|Ψ = E|Ψ

µ = 0 µ → ∞ Eµ µ ˆ T + ˆ V µ

ee + ˆ

V µ |Ψµ = Eµ|Ψµ ˆ V µ =

i vµ(ri)

COST 2004 – p. 3/12

SLIDE 11

Ground-state electronic energy

Mono-determinantal DFT

E = min

Φ

Φ| ˆ

T + ˆ Vne|Φ + U[nΦ] + Exc[nΦ]

Multi-determinantal DFT

E = min

Ψ

Ψ| ˆ

T + ˆ V µ

ee + ˆ

Vne|Ψ + ¯ U µ[nΨ] + ¯ Eµ

xc[nΨ]

COST 2004 – p. 4/12

SLIDE 12

Ground-state electronic energy

Mono-determinantal DFT

E = min

Φ

Φ| ˆ

T + ˆ Vne|Φ + U[nΦ] + Exc[nΦ]

Multi-determinantal DFT

E = min

Ψ

Ψ| ˆ

T + ˆ V µ

ee + ˆ

Vne|Ψ + ¯ U µ[nΨ] + ¯ Eµ

xc[nΨ]

E = Ψµ| ˆ

T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

COST 2004 – p. 4/12

SLIDE 13

Ground-state electronic energy

Mono-determinantal DFT

E = min

Φ

Φ| ˆ

T + ˆ Vne|Φ + U[nΦ] + Exc[nΦ]

Multi-determinantal DFT

E = min

Ψ

Ψ| ˆ

T + ˆ V µ

ee + ˆ

Vne|Ψ + ¯ U µ[nΨ] + ¯ Eµ

xc[nΨ]

E = Ψµ| ˆ

T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

Ψµ is the multi-determinantal ground-state wave function of ˆ

T + ˆ V µ

ee + ˆ

V µ

COST 2004 – p. 4/12

SLIDE 14

Ground-state electronic energy

Mono-determinantal DFT

E = min

Φ

Φ| ˆ

T + ˆ Vne|Φ + U[nΦ] + Exc[nΦ]

Multi-determinantal DFT

E = min

Ψ

Ψ| ˆ

T + ˆ V µ

ee + ˆ

Vne|Ψ + ¯ U µ[nΨ] + ¯ Eµ

xc[nΨ]

E = Ψµ| ˆ

T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

Ψµ is the multi-determinantal ground-state wave function of ˆ

T + ˆ V µ

ee + ˆ

V µ

¯

U µ[n] = U[n] − U µ[n] is the complement Hartree energy

¯

Eµ

xc[n] = Exc[n] − Eµ xc[n] is the complement exchange-correlation energy

COST 2004 – p. 4/12

SLIDE 15

Long-range/short-range separation

ˆ V µ

ee =

i<j

vµ

ee(rij) : long-range part of the Coulomb interaction

COST 2004 – p. 5/12

SLIDE 16

Long-range/short-range separation

ˆ V µ

ee =

i<j

vµ

ee(rij) : long-range part of the Coulomb interaction

E = Ψµ| ˆ T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

¯

Eµ

xc[n]: short-range interaction ⇒ well approximated by (semi-)local

approximations

COST 2004 – p. 5/12

SLIDE 17

Long-range/short-range separation

ˆ V µ

ee =

i<j

vµ

ee(rij) : long-range part of the Coulomb interaction

E = Ψµ| ˆ T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

¯

Eµ

xc[n]: short-range interaction ⇒ well approximated by (semi-)local

approximations

Ψµ: reduced interaction ⇒ well approximated by a few determinants

corresponding to the near-degenerate states

COST 2004 – p. 5/12

SLIDE 18

Possible long-range interactions

erf interaction:

vµ

ee(r) = erf(µr)

r 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

COST 2004 – p. 6/12

SLIDE 19

Possible long-range interactions

erf interaction:

vµ

ee(r) = erf(µr)

r

erfgau interaction:

v˜

µ ee(r) = erf(˜

µr) r − 2˜ µ √π e− 1

3 ˜

µ2r2

For comparison, ˜ µ ≈ 3µ 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

COST 2004 – p. 6/12

SLIDE 20

The exchange-correlation functional ¯

Eµ

xc[n]

The short-range functional ¯

Eµ

xc[n] can be divided as

¯ Eµ

xc[n] = ¯

Eµ

x[n] + ¯

Eµ

c [n]

COST 2004 – p. 7/12

SLIDE 21

The exchange-correlation functional ¯

Eµ

xc[n]

The short-range functional ¯

Eµ

xc[n] can be divided as

¯ Eµ

xc[n] = ¯

Eµ

x[n] + ¯

Eµ

c [n]

Local density approximation (LDA) for ¯

Eµ

xc[n]:

¯ Eµ

xc[n] =

n(r)¯

ε unif,µ

xc

(n(r))dr ¯ ε unif,µ

xc

(n): short-range exchange-correlation energy per particle of a uniform electron gas with modified interaction

COST 2004 – p. 7/12

SLIDE 22

The exchange-correlation functional ¯

Eµ

xc[n]

Performance of the LDA for the Be atom: Short-range exchange energy ¯ Eµ

x[n]

2 4 6 8 Μ 2.5 2 1.5 1 0.5 E

x

Μ

exact LDA erf

COST 2004 – p. 8/12

SLIDE 23

The exchange-correlation functional ¯

Eµ

xc[n]

Performance of the LDA for the Be atom: Short-range exchange energy ¯ Eµ

x[n]

2 4 6 8 Μ 2.5 2 1.5 1 0.5 E

x

Μ

exact LDA exact LDA erf erfgau

COST 2004 – p. 8/12

SLIDE 24

The exchange-correlation functional ¯

Eµ

xc[n]

Performance of the LDA for the Be atom: Short-range exchange energy ¯ Eµ

x[n]

Short-range correlation energy ¯ Eµ

c [n]

2 4 6 8 Μ 2.5 2 1.5 1 0.5 E

x

Μ

exact LDA exact LDA erf erfgau

2 4 6 8 Μ 0.2 0.15 0.1 0.05 E

c

Μ

exact LDA erf

COST 2004 – p. 8/12

SLIDE 25

The exchange-correlation functional ¯

Eµ

xc[n]

Performance of the LDA for the Be atom: Short-range exchange energy ¯ Eµ

x[n]

Short-range correlation energy ¯ Eµ

c [n]

2 4 6 8 Μ 2.5 2 1.5 1 0.5 E

x

Μ

exact LDA exact LDA erf erfgau

2 4 6 8 Μ 0.2 0.15 0.1 0.05 E

c

Μ

exact LDA exact LDA erf erfgau

COST 2004 – p. 8/12

SLIDE 26

The exchange-correlation functional ¯

Eµ

xc[n]

Performance of the LDA for the Be atom: Short-range exchange energy ¯ Eµ

x[n]

Short-range correlation energy ¯ Eµ

c [n]

2 4 6 8 Μ 2.5 2 1.5 1 0.5 E

x

Μ

exact LDA exact LDA erf erfgau

2 4 6 8 Μ 0.2 0.15 0.1 0.05 E

c

Μ

exact LDA exact LDA erf erfgau

Asymptotic expansions for µ → ∞: ¯ Eµ

x[n] ≈ − A

µ2

n(r)2dr + · · ·

¯ Eµ

c [n] ≈ B

µ2

n2,c(r, r)dr + · · ·

COST 2004 – p. 8/12

SLIDE 27

Ground-state electronic energy of the Be atom

E = Ψµ| ˆ T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

COST 2004 – p. 9/12

SLIDE 28

Ground-state electronic energy of the Be atom

E = Ψµ| ˆ T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

CI in limited configurational spaces

COST 2004 – p. 9/12

SLIDE 29

Ground-state electronic energy of the Be atom

E = Ψµ| ˆ T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

CI in limited configurational spaces LDA

COST 2004 – p. 9/12

SLIDE 30

Ground-state electronic energy of the Be atom

E = Ψµ| ˆ T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

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

exact 1s2s erf

COST 2004 – p. 9/12

SLIDE 31

Ground-state electronic energy of the Be atom

E = Ψµ| ˆ T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

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

exact 1s2s 1s2s2p erf

COST 2004 – p. 9/12

SLIDE 32

Ground-state electronic energy of the Be atom

E = Ψµ| ˆ T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

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

exact 1s2s 1s2s2p erf erfgau

COST 2004 – p. 9/12

SLIDE 33

Ground-state electronic energy of the Be atom

E = Ψµ| ˆ T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

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

exact 1s2s 1s2s2p erf erfgau

COST 2004 – p. 9/12

SLIDE 34

Ground-state electronic energy of the Be atom

E = Ψµ| ˆ T + ˆ V µ

ee + ˆ

Vne|Ψµ + ¯ U µ[n] + ¯ Eµ

xc[n]

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

exact PBE 1s2s 1s2s2p erf erfgau

COST 2004 – p. 9/12

SLIDE 35

Short-range PBE functional

Extension of PBE exchange-correlation functional to the modified interaction: same ansatz with µ-dependent theoretical constants.

COST 2004 – p. 10/12

SLIDE 36

Short-range PBE functional

Extension of PBE exchange-correlation functional to the modified interaction: same ansatz with µ-dependent theoretical constants. Short-range PBE exchange energy ¯ Eµ

x[n] for the Be atom:

2 4 6 8 Μ 2.5 2 1.5 1 0.5 E

x

Μ

exact LDA PBE erfgau

COST 2004 – p. 10/12

SLIDE 37

Short-range PBE functional

Short-range PBE correlation energy ¯ Eµ

c [n] for the Be atom:

2 4 6 8 Μ 0.2 0.15 0.1 0.05 E

c

Μ

exact LDA PBE erfgau

COST 2004 – p. 11/12

SLIDE 38

Conclusions

Separation of electron-electron interaction:

a rigorous way to combine wave function methods and DFT,
an understanding of functional approximations in term of the range of the

interaction.

COST 2004 – p. 12/12

SLIDE 39

Conclusions

Separation of electron-electron interaction:

a rigorous way to combine wave function methods and DFT,
an understanding of functional approximations in term of the range of the

interaction. In particular, this work shows that

a better long-range/short-range interaction separation (erfgau) improves the

accuracy of the method,

short-range correlation energy can be accurately approximated by

gradient-corrected functionals,

short-range exchange energy needs more accurate approximations.

COST 2004 – p. 12/12