Beyond LDA and GGA in practice · Beyond LDA and GGA in terms of many-body perturbation theory:...

Beyond LDA and GGA in practice

Xinguo Ren

Fritz Haber Institute, Berlin

DFT and beyond Hands-on tutorial workshopBerlin, July 12-21, 2011

Xinguo Ren (FHI, Berlin) Beyond LDA and GGA in practice DFT and beyond, 12-21.07.11 1 / 38

Beyond LDA and GGA in terms of many-bodyperturbation theory: theory and numerical practice

Xinguo Ren

Fritz Haber Institute, Berlin

DFT and beyond Hands-on tutorial workshopBerlin, July 12-21, 2011


1 Introduction of the “beyond-LDA/GGA approaches”

2 Implementation based on numeric atom-centered orbitals

3 Recent development of more accurate computational schemes

4 Conclusion


Practical approximations to Exc

Jacob’s ladder in DFT

LDA n(r) N3

GGA ∇n(r) N3

meta-GGA ∇2n(r) N3

hybrids occ. ψn(r) N4

RPA-type unocc. ψn(r) N4-N6

J. P. Perdew and K. Schmidt, in Density

Functional Theory and its Application to

Materials, (AIP, Melville, NY, 2001).

RPA: Random-phase approximation

Quantum chemistry approaches

CCSD(T) N7

6

CCSD N6

6

MP2 N5

6

Hartree-Fock N4

MP2: second-order Moller-Plessetperturbation theory;CCSD: Couple-cluster with singlesand doubles


Shortcomings of explicit density-dependent functionals(LDA, GGAs, meta-GGAs):

Self-interaction error =⇒

– too small band gaps in semi-conductors– underestimated chemical reaction barrier heights– failure to describe localized electrons in both solids and molecules– the exchange-correlation potential decays too fast for finite systems⇒ unbound negative ions, no Rydberg states, etc.

Unable to describe “strong correlations”(e.g., transition metal oxides)

Absence of van der Waals interactions(true also for hybrid functionals)

No access to electronic excited states


On the 4th rung of Jacob’s ladder: hybrid functionals

Ehybxc = (1− α)EGGA

x + βEGGAc + αEEX

x

Generalized Kohn-Sham (gKS) single-particle equation[

−1

2∇2 − vext(r)− vH(r)

]

ψn(r) +

∫

dr′vxc(r, r′)ψn(r

′) = ǫnψn(r)

?

vxc(r, r

′) = (1− α)vGGAx (r)δ(r − r′) + βvGGAc (r)δ(r − r

′) + αvHFx (r, r′)9

KS-DFT

α = 0, β = 1

?

hybrid functionals0 < α < 1 (typically 1/4)0 < β < 1 (typically 1)

XXXXXXXz

Hartree-Fockα = 1, β = 0

A. D. Becke, J. Chem. Phys. 98, 1372 (1993); Seidl et al., 53, 3764 (1996).

Main features of hybrid functionals

+ Self-interaction is reduced, and the localized electronicstates are treated much better.

+ The generalized Kohn-Sham scheme allows forobtaining a much better band gap, particulary forsemiconductors.

- The van der Waals interactions are still absent

- The computational costs are dominated by theevaluation of the exact-exchange part.


On the fifth rung of the Jacob’s ladder

I. Adiabatic-connection fluctuation-dissipation theorem: a formally exactway for constructing EXC

Imagine a continuum of fictitious systems governed by

Hλ = T + vλext + λVee (where 0 ≤ λ ≤ 1),

Hλ|Φλ[n]〉 = Eλ|Φλ[n]〉.

Exchange-correlation (XC) part of the interaction energy

UλXC = 〈Φλ[n]|Vee|Φλ[n]〉 −

1

2

∫

dr

∫

dr′n(r)n(r′)

|r − r′|

Exact EXC: EXC =

∫ 1

0dλUλ

XC = Uλ=1XC + Tc

D. C. Langreth and J. P. Perdew, Phys. Rev. B 15, 2884 (1977).


On the fifth rung of the Jacob’s ladder: RPAFluctuation-Dissipation theorem

UλXC = −

1

2

∫

dr

∫

dr′v(r − r′)

[

−1

π

∫ ∞

0dωImχλ(r, r

′, ω)− n(r)δ(r − r′)

]

Fluctuation Dissipation

χλ(r, r′, t − t

′

) = δn(r, t)/δvλext(r′, t

′

): Response function(the imaginary part describes the dissipation process)

χλ = χ0 + χ0(λv + f λxc)χλ fxc = δvxc/δn

χ0(r, r′, ω) = 2

∑

mn

(fn − fm)ψ∗m(r)ψn(r)ψ

∗n(r

′)ψm(r′)

ω − ǫm + ǫn

0 ≤ fm ≤ 1 : Fermi occupation number

RPA ⇐⇒ f λxc = 0


On the fifth rung of the Jacob’s ladder: GL2/MP2

II. Many-body perturbation theory

H = H0 + H ′

H0 =N∑

i=1

[

−1

2∇2

i + v ext(r) + vMFi

]

H ′ =N∑

i<j

1

|ri − rj |−

N∑

i=1

vMFi

H0|Φn〉 = E(0)n |Φn〉

E0 = E(0)0 + E

(1)0 + E

(2)0 + · · ·

E(0)0 = 〈Φ0|H

0|Φ0〉

E(1)0 = 〈Φ0|H

′|Φ0〉

E(2)0 =

∑

n>0

|〈Φ0|H′|Φn〉|

2

E(0)0 − E

(0)n

vMF = vHF (HF potential) =⇒ Møller-Plesset perturbation theoryvMF = vKS (KS potential) =⇒ Gorling-Levy perturbation theory

A. Gorling and M. Levy, Phys. Rev. B 47, 13105 (1993).

S. Grimme, J. Chem. Phys. 122, 034108 (2006). (Double hybrids)


Diagrammatic (Goldstone) representation of cRPA

Random phase approximation to the correlation energy

ERPAc = + + · · ·

cRPA: “ infinite summation of ring diagrams”

M. Gell-Mann and K. A. Brueckner, Phys. Rev. 106, 364 (1957).

Second-order Møller-Plesset perturbation theory (MP2)

EMP2c = +

MP2: “second-order direct (ring) plus exchange terms”

Advantages:

The long-range vdW interactions are included.

Compatible with exact-exchange, the inclusion of whichcancels the self-interaction term present in the Hartree energy.


Beyond LDA and GGA: hybrid functionals, GL2, RPA, andGW

Basic ingredients required:

Full or a fraction of exact-exchange– Exact-exchange : the first-order term of the perturbation series based onthe Kohn-Sham non-interacting reference Hamiltonian

GL2 (MP2) or RPA-type correlation energy– GL2 (MP2) : the second-order term of the perturbation series based onthe Kohn-Sham (Hartree-Fock) reference Hamiltonian

– RPA: a partial summation of “ring” diagrams to infinite order

GW self-energy (talk by P. Rinke, Wed, Jul. 20)– In analogy to RPA, but for the self-energy.

All can be formulated within the framework of many-body perturbationtheory (either for total energy or the self-energy)





4 Conclusion


Beyond LDA and GGA in practiceNumeric atom-centered basis (NAO):

φi (r) =unl(r)

rYlm(Ω)

Well established for standard functionals (LDA and GGAs) Flexible in shape ⇒ compactness

Natural inclusion of core electrons

Becoming more and more popular recently

Dmol, FPLO, FHI-aims, Siesta, etc.

But how about doing exact-exchange and beyond

(e.g., MP2, RPA et al.)?

Main challenges:four-center Coulomb integrals

(ik |lj) =

∫

drdr′φi (r)φk(r)φl(r

′)φj(r′)

|r − r′|

φk(r)

φi (r)

φj(r′)

φl(r′)

1|r−r′|


Our solution: resolution of identity (RI), or “densityfitting”

GW , RPA (noninteracting response function)

χ(r, r′, iω) =occ∑

m

unocc∑

a

[

ψm(r)ψa(r)ψm(r′)ψa(r

′)2(εm − εa)

ω2 + (εm − εa)2

]

Hartree-Fock (exchange matrix)

Σx

ij =

∫∫

drdr′occ∑

n

φi (r)ψn(r)ψn(r′)φj (r′)

|r − r′|

MP2 (Coulomb repulsion integral)

(ma|nb) =

∫∫

drdr′ψm(r)ψa(r)ψn(r′)φb(r

′)

|r − r′|

ψn(r) =∑

i

φi (r)cin

“Density fitting” φi (r)φj(r) ≈∑

µ

Cµij Pµ(r)

φi (r) : NAO basis; ψn(r) : single-particle KS/HF orbitals


RI with a set of auxiliary basis functions

Observation: Nbasis · (Nbasis + 1)/2 many pair products φi (r)φj(r),where i < j = 1, 2, · · ·Nbasis, are heavily linear dependent

One can construct a linearly independent, auxiliary basis set Pµ(r),(Naux ∼ Nbasis), sufficiently accurate to represent φi (r)φj(r).

φi (r)φj(r) ≈

Naux∑

µ

Cµij Pµ(r)

It follows (RI decomposition)

(ik |lj) ≈∑

µν

Cµik < Pµ|v |Pν > C νlj =

∑

µν

CµikVµνC

νlj

Construct Pµ(r), and determine Cµik


A simple, working procedure to construct Pµ(r)

1. Pµ(r) should also be atom-centered orbitals

P(r) =ξqla(r)

rYlm(Ωa)

(

“normal” basis: φ(r) =unla(r)

rYlm(Ωa)

)

numerically convenient

2. Determine the shape of ξqla(|r − Ra|) For a given species of element s, angular momentum l

(0 ≤ l ≤ lABF-max), form the “On-site” products:un1l1a(s)(|r − Ra|)un2l2a(s)(|r − Ra|), |l1 − l2| ≤ l ≤ l1 + l2

Employing Gram-Schmidt orthonormalization procedure to eliminatelinear dependence, according to a threshold ǫorth

3. Multiply ξqla(|r − Ra|) with Ylm(Ωa) for all angular momentuml < lABF-max and all atoms a.

On-site

Off-site

– “On-site” pairs φi (r − Ra)φj(r − Rb) “exactly”represented by Pµ(r) !

– But how about “off-site” pairsφi (r − Rb)φj(r − Ra)?


Determining Cµik : “RI-SVS” vs “RI-V”

φi (r)φj(r) ≈∑

µ

Cµij Pµ(r);

δρ(r) = φi (r)φk (r) −∑

µ

Cµ

ikPµ(r).

minimizing (δρδρ)

Cµ

ik =∑

ν

(ikν)S−1νµ

(ikν) =

∫

drφi (r)φk (r)Pν(r)

Sνµ =

∫

drPν(r)Pµ(r)

(ik|lj) =∑

µµ′νν′

(ikν)S−1νν′

Vν′µ′S−1µ′µ

(µlj).

“RI-SVS” version

minimizing (δρ|δρ)

Cµ

ik =∑

ν

(ik |ν)V−1νµ

(ik|ν) =

∫

drdr′φi (r)φk (r)v(r − r′)Pν(r

′).

Vνµ =

∫

drdr′Pν(r)v(r − r′)Pµ(r

′).

(ik|lj) =∑

µν

(ik|ν)V−1νµ (µ|lj).

“RI-V” version

“RI-V” gives much better accuracy

A singular value decomposition ǫsvd is introduced in the matrix inversion toprevent the possible illconditioning behavior.

Dunlap, Connolly, and Sabin, JCP 71, 3396 (1979).

Vahtras, J. Almlof, and Feyereisen, CPL 213, 514 (1993).


On the accuracy of the RI approximation (for off-site pairs)

-0.15-0.1

-0.050

0.05

-0.02

-0.01

0

0.01

-0.1

0

0.1

0.2

φ i(r-R

A)φ

j(r-R

B)

ExactRI-V

-0.02

0

0.02

0.04

-1.5 -1 -0.5 0 0.5 1(x,0,0) (angstrom)

-0.15

-0.1

-0.05

0

-1.5 -1 -0.5 0 0.5 1 1.5(x,0,0) (angstrom)

-0.03

-0.02

-0.01

0

0.01

[2s, 2s]

[2px, 2p

x]

[2s, 2px]

N2: d = 1.1Å d = 2.0 Å

NAO tier 2 basis set (4s3p2d1f1g) is used in the auxiliary basisconstruction


Accuracy of the RI approxiamtion: HF total energy for N2

1 2 3 4 5Bond length (Å)

-2970

-2960

-2950

-2940

-2930

HF

tota

l ene

rgy

(eV

) NWchem ("Exact")FHI-aims ("RI-V")

1.00 1.05 1.10 1.15-2966.0

-2965.6

-2965.2

"cc-pVQZ" basis

1 2 3 4 5Bond length (Å)

-0.8

-0.4

0.0

0.4

0.8

Err

or in

HF

tota

l ene

rgy

(meV

)

Gaussian basis set ”cc-pVQZ” is used.

Reference values are obtained with NWchem.

Thresholding parameterslABF-max = lAO-max + 1, ǫorth = 10−2, ǫsvd = 10−4


On the accuracy of RI-HF for light elements

C2H

2C

2H4

C2H

5OH

C6H

6C

H3C

lC

H4

CO

CO

2C

S 2N

a 2C

lFH

Cl

HF

NH

3O

2H

2O P 2P

H3

Si 2H

6S

iH4

-2

0

2

RI-

HF

err

or (

meV

)

-2

0

2

-2

0

2

Total energyBinding energy

εsvd=10

-4, εorth

=10-2

, l ABF-max

=l AO-max

+1

εsvd=10

-5, εorth

=10-3

, l ABF-max

=lAO-max

+1

εsvd=10

-5, εorth

=10-3

, l ABF-max

=2l AO-max

Gaussian cc-pVQZ basis set is used

“RI-free” NWchem results are used as reference


On the accuracy of RI-MP2 for light elements

C2H

2C

2H4

C2H

5OH

C6H

6C

H3C

lC

H4

CO

CO

2C

S 2N

a 2C

lFH

Cl

HF

NH

3O

2H

2O P 2P

H3

Si 2H

6S

iH4-4-2024

RI-

MP

2 er

ror

(meV

)

-4-2024

-4-2024

Total energyBinding energy

εsvd=10

-4, εorth

=10-2

, lABF-max

=lAO-max

+1

εsvd=10

-5, εorth

=10-3

, lABF-max

=lAO-max

+1

εsvd=10

-5, εorth

=10-3

, lABF-max

=2lAO-max

Gaussian cc-pVQZ basis set is used

“RI-free” NWchem results are used as reference


On the accuracy of RI-HF for heavy elements

Cu2

-10

0

10

20

30

∆Eto

t(meV

)

2.2 2.4 2.6 2.8 3.0 3.2 3.43.6 3.8 4.0Bond length (Å)

-0.2-0.10.00.10.2

∆Eb(m

eV)

εorth=10

-2, εsvd

=10-4

εorth=10

-3, εsvd

=10-5

εorth=10

-4, εsvd

=10-5

εorth=10

-5, εsvd

=10-5

Au2

-100

0

100

200

300

∆Eto

t(meV

)2 2.2 2.42.6 2.8 3 3.2 3.43.6 3.8 4

Bond length (Å)

-0.1

0.0

0.1

0.2

∆Eb(m

eV)

εorth=10

-2, εsvd

=10-4

εorth=10

-2, εsvd

=10-5

εorth=10

-3, εsvd

=10-5

εorth=10

-4, εsvd

=10-5

Tighter ǫorth parameter is needed for heavy elements, but meV/atomtotal-energy accuracy can still be achieved with ǫorth = 10−4 and tighter.


Convergence with NAO basis sets

50 100 150 200 250-11.0

-10.5

-10.0

-9.5

-9.0

-8.5

50 100 150 200 250-9.0

-8.5

-8.0

-7.5

-7.0

100 200 300 400Basis size

-0.23

-0.22

-0.21

-0.20

-0.19

Bin

ding

ene

rgy

(eV

)

100 200 300 400-0.22

-0.21

-0.20

-0.19

-0.18

-0.17

N2 N

2

H2O...H

2O H

2O...H

2O

MP2

MP2

RPA@HF

RPA@HF

tier 4

tier 4 + diffuseaug-cc-pV6Z

NAOs for N:tier 1: 3s2p1dtier 2: 4s3p2d1f 1gtier 3: 5s4p3d2f 1gtier 4: 6s5p4d3f 2g(80)

diffuse functions fromaug-cc-pV5Z1s1p1d1f 1g1h (36)

aug-cc-pV6Z:8s7p6d5f 4g3h2i(189)


Convergence with NAO basis sets

0 100 200

-14.8

-14.6

-14.4H

OM

O le

vel (

eV)

0 100 200

-16.6

-16.4

-16.2

-16.0

0 200 400Basis Size

-11.0

-10.8

-10.6

-10.4

-10.2

0 200 400-12.4

-12.0

-11.6

-11.2

G0W

0@PBE

G0W

0@PBE

G0W

0@HF

G0W

0@HF

N2 N

2

H2O...H

2O H

2O...H

2O

tier 4

tier 4 + diffuseaug-cc-pV6Z


Application I: Binding energies for molecules

1 2 3 4 5 6 7 8 9 10 11 12 13 1415 1617 18 19 20 21 22

S22 molecules-150

-100

-50

0

50

100

150

Err

or in

inte

ract

ion

ener

gy (

meV

)

RPA@PBERPA@HFMP2

Hydrogen Dispersion Mixed

S22 molecular set:Jurecka et al., PCCP 8,1985

(2006)

7 hydrogen-bondedmolecules

8 dispersionbonded molecules

7 with mixedcharacter


Application II: Adsorption problem

Delta-XC correction scheme

Eads(XC-better) = Eads(PBE) + limcluster→∞

[E clusterads (XC-better)− E cluster

ads (PBE)]

Hu, Reuter, and Scheffler, PRL 98, 176103 (2007).

PBE0 and RPA calculation forCO@Cu(111)

5 10 15 20 25 Cluster Size N

-0.5

0.0

0.5

1.0

∆Ead

s (eV

) 0.0

0.5

1.0PBE0RPA@PBE0

top site

hollow site

∞

Ren, Rinke, and Scheffler, PRB, 80, 045402 (2009).

Cluster ⇓ modeling


Application III: G 0W

0 quasiparticle energy corrections

G0W

0 correction to KS energy levels

EG0W 0

n = EKSn +ΣG0W 0

nn (EG0W 0

n )− V xcn

Ionization potentials for benzene

5 10 15 20 25 30Experimental Ionization potentials (eV)

10

20

30

The

oret

ical

Ioni

zatio

n po

tent

ials

(eV

)

LDA G

0W

0 @LDA

HFPBE0G

0W

0@HF

G0W

0@PBE0

Ionization potentials for Na clusters

1 2 3 4 5 6 7 8 9 10Cluster size

3

3.5

4

4.5

5

5.5

Ioni

zatio

n po

tent

ials

(eV

)

G0W

0@LDA

Exp. (Herrmann et al., ’78)





4 Conclusion


Exact-exchange plus the correlation energy in the randomphase approximation (EX+cRPA)

⇒ A promising route to go beyond (semi-)local DFT

Self-interaction in (semi-)local DFT is reduced in EX+cRPA;(cRPA is not self-correlation free though)

– Correct asymptotic Kohn-Sham potential for finite systems

cRPA is fully non-local, and van der Waals interactions are includedautomatically;

Applicable to molecules, insulators, and metals with various bondingnatures. (In contrast to second-order perturbation theory like MP2)


RPA caclulations in practice

In practical calculations, RPA is done perturbatively on a LDA/GGAreference (e.g., RPA@PBE)

The “standard” RPA scheme:

ERPA@PBE = EPBE − EPBExc +

(

E exactx + ERPA

c

)

@PBE

=(

Ekin + Eext + EHartree + E exactx + ERPA

c

)

@PBE

= EEX@PBE + E cRPA@PBE

EEX@PBE: non-self-consistent Hartree-Fock energy(exchange-only total energy) evaluated with PBE orbitals.

E cRPA@PBE: RPA correlation energy evaluated with PBE orbitals.

But, the “standard” RPA scheme underestimates the bond strengthsystematically


Rayleigh-Schrodinger perturbation theory

Perturbative correction to the ground state energy:

E0 = E(0)0 + E

(1)0 + E

(2)0 + · · ·

zeroth-order: E(0)0 = 〈Φ0|H

0|Φ0〉

1st order: E(1)0 = 〈Φ0|H ′|Φ0〉

2nd order:

E(2)0 =

∑

n 6=0

|〈Φ0|H ′|Φn〉|2

E(0)0 − E

(0)n

=∑

i,a

|〈Φ0|H ′|Φi,a〉|2

E(0)0 − E

(0)i,a

︸︷︷︸

Single excitations

+∑

ij,ab

|〈Φ0|H ′|Φij,ab〉|2

E(0)0 − E

(0)ij,ab

︸︷︷︸

Double excitations

= ESEc

+ +

Hartree-Fock vs Kohn-Sham reference

H(0) = HHF; |Φ0〉 = |ΦHF〉

E(0)0 + E

(1)0 = EHF

ESEc = 0 (Brillouin theorem)

H(0) = HPBE; |Φ0〉 = |ΦPBE〉

E(0)0 + E

(1)0 = EEX@PBE

ESEc =

∑

ia

|〈a|vHF − vKS-PBE|i〉|2

ǫi − ǫa6= 0

(but not included in the RPA@PBE)

Kohn-Sham many-body perturbation theoryA. Gorling and M. Levy, Phys. Rev. B 47, 13105 (1993);H. Jiang and E. Engel, J. Chem. Phys. 125, 184108 (2006).

ERPA+SEc = ERPA

c + ESEc


Performance of the RPA+SE for Ar2 and Kr2

3.6 4.0 4.4 4.8Bond length (Å)

-15

-10

-5

0

5

10

15B

indi

ng e

nerg

y (m

eV) RPA@HF

RPA@PBE(RPA+SE)@PBETang-Toennies

3.6 4.0 4.4 4.8

Ar2

Kr2


Performance of the RPA+SE for S22

1 2 3 4 5 6 7 8 9 10 11 12 13 1415 1617 18 19 20 21 22

S22 molecules-150

-100

-50

0

50

100

150

Err

or in

inte

ract

ion

ener

gy (

meV

)RPA@PBERPA@HFMP2(RPA+SE)@PBE

Hydrogen Dispersion Mixed

MAE: 18 meV

X. Ren, A. Tkatchenko, P. Rinke, and M. Scheffler, Phys. Rev. Lett. 106, 153003 (2011).





4 Conclusion


Conclusions

The RI technique makes the NAO basis functions a competitiveoption for the implementation of approaches beyond LDA and GGA.

hybrid functionals MP2 RPA and GW

– can treat system size ∼ 100 light atoms · · · .

The many-body perturbation theory offers a useful perspective fordesigning more elaborate functionals in the higher rungs of Jacob’sladder.

On-going workPeriodic extension (Sergey Levchenko, Jurgen Wieferink)

Fully self-consistent GW (Fabio Caruso)


Acknowledgements

Code development Method development

Volker Blum Patrick RinkePatrick Rinke Alexandre TkatchenkoJurgen Wieferink Matthias SchefflerSergey LevchenkoFabio KarusoAndreas SanfillpoMatthias Scheffler

Thank you!


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