Beyond LDA and GGA - Hartree-Fock, hybrid functionals, MP2 ... · Beyond LDA and GGA - Hartree-Fock, hybrid functionals, MP2, and RPA in FHI-aims: theory and a numerical orbital practice

Beyond LDA and GGA - Hartree-Fock, hybridfunctionals, MP2, and RPA in FHI-aims: theory and a

numerical orbital practice

Xinguo Ren

Hands-on tutorial on ab-initio molecular simulationsBerlin, June 25, 2009

Xinguo Ren (FHI, Berlin) Beyond LDA and GGA - Hartree-Fock, hybrid functionals, MP2, and RPA in FHI-aims: theoFHI-aims workshop, 25.06.09 1 / 38

Problems with LDA and GGA

Total energy in Kohn-Sham DFT

Etot[n(r)] = −1

2

∑

i

< ψi |∇2|ψi > +

∫

vext(r)n(r)dr

+1

2

∫

dr

∫

dr′n(r)n(r′)

|r − r′| + Exc[n(r)]

Exc[n(r)] has to be approximated !

LDA and GGA: very useful, but have problems in certain applications.

Deficiencies of LDA and GGA:◮ Self-interaction error (delocalization error [1]) =⇒

– Underestimated band-gaps, wrong dissociation bebavior of molecularions, etc.– Failure to describe localized electrons in both solids and molecules.

◮ Absence of van der Waals interaction (A. Tkatchenko’s talk)

[1] A. J. Cohen, P Mori-Sanchez, W. Yang, Science, 321, 792 (2008).


LDA and GGA errors for atomization energy

Atomization energy

∆E =

−[

Emol −∑

i

Eatom(i)

]

Desired accuracy: 1 kal/mol

Data from

Perdew, Burke, Erzenhof,

Phys. Rev. Lett. 77, 3565 (1996).

Unit: kcal/mol (= 43.4 meV)

Molecules LDA PBE-GGA EXPH2 113 105 109LiH 60 52 58CH4 462 420 419NH3 337 302 297OH 124 110 107H2O 267 234 232HF 162 142 141Li2 23 19 24LiF 153 136 139Be2 13 10 3C2H2 460 415 405C2H4 633 571 563HCN 361 326 312CO 299 269 259N2 267 243 229NO 199 172 153O2 175 144 121F2 78 53 39P2 142 120 117Cl2 81 63 58Mean abs. error 31.4 7.9


Wave-function based methods: systematically increasing

accuracyac

cura

cy

6

com

puta

tion

tim

e

(43.4 meV)CCSD(T) – Gold standard,

higher order correlations

MP2 – Correlation is treated up to2nd order of Coulomb interaction

Hartree-Fock – Exchange treated exactly,but no correlation

6

6

Quantum Chemistryapproach

MP2: 2nd order Møller-Plesset (many-body) Perturbation Theory

CCSD(T): Coupled-Cluster Theory with Single, Double and perturbative Triple excitations


Perdew’s dream: Jacob’s ladder in DFT

1 n(r), LDA

2 ∇n(r), GGA (e.g., PBE)

3 τ(r), meta-GGA (e.g., TPSS)

4 occupied ψn(r), hybrid functional (e.g.,PBE0)

5 unoccupied ψn(r), e.g., ACFD-RPA

accu

racy

?

6

com

puta

tion

tim

e(43.4 meV)

τ(r) : KS kinetic energy density

ACFD: Adiabatic Connection Fluctuation-Dissipation theorem

RPA: Random Phase Approximation


Underlying principle of constructing Jacob’s ladder

”Adiabatic Connection”: a formally exact way for constructing EXC

Imagine a continuum of fictitious systems governed by

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

N∑

i<j

1

|r − r′|

Hohenberg-Kohn theorem holds for every λ =⇒ vλext(r) can be chosen such that

nλ(r) = nλ=1(r) = n(r)

Hλ|Φλ[n] >= Eλ|Φλ[n] > .

Exact EXC: E exactXC =

∫ 1

0 dλUλXC

Exchange-correlation (XC) potential energy

UλXC =< Φλ[n]|Vee|Φλ[n] > −1

2

∫

dr

∫

dr′ n(r)n(r′)

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

O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976).


Two limits of the connection

λ = 0, vλ=0ext (r) = vKS(r) (KS system)

Uλ=0XC = < Φ0|Vee|Φ0 > −1

2

∫

dr

∫

dr′n(r)n(r′)

|r − r′|

= −1

2

∑

mn

∫

dr

∫

dr′ψ

∗m(r)ψn(r)ψn(r

′)∗ψm(r′)

|r − r′| := E exactx

Exchange-only limit, LDA/GGA are least accurate. =⇒ Instead exactexchange is needed

λ = 1, vλ=1ext = vext (Full interacting system)

LDA/GGA work well.Correlation is crucial, and exact exchange will not work!

EhybXC = αE exact

x + (1 − α)EGGAx + EGGA

c

A. D. Becke, J. Chem. Phys. 98, 1372 (1993).


Exact exchange in KS-DFT vs Hartree-Fock (HF) exchange

E exactx = −1

2

N∑

mn

∫

dr

∫

dr′ψ

∗m(r)ψn(r)ψn(r

′)∗ψm(r′)

|r − r′|

How ”exact” is exact exchange?

The orbitals for evaluating E exactx are different!

◮ HF orbitals → HF exchange◮ KS orbitals → exact exchange in DFT

Numerical technique is the same!


Contents

1 Hartree-Fock theory and its implementation in FHI-aims

2 Hybrid functionals, MP2

3 ACFD-RPA


Hartree-Fock Theory

N-electron interacting hamiltonian

H = −N

∑

i

[

1

2∇2

i + vext(ri )

]

+

N∑

i<j

1

|ri − rj |

HΦexact0 (r1, r2, · · · , rN) = E exact

0 Φexact0 (r1, r2, · · · , rN)

Hartree-Fock approximation (single Slater determinant)

ΦHF(r1, r2, · · · , rN) =1√N!

∣

∣

∣

∣

∣

∣

∣

∣

ψ1(r1) ψ2(r1) · · · ψN(r1)ψ1(r2) ψ2(r2) · · · ψN(r2)· · · · · · · · · · · ·

ψ1(rN) ψ2(rN) · · · ψN(rN)

∣

∣

∣

∣

∣

∣

∣

∣

ψm(r) = |m >: spin-orbital

Ec = E exact0 − EHF = E exact

0 − < ΦHF|H |ΦHF >


Hartree-Fock (HF) vs Kohn-Sham (KS) DFT

HF equation: δEHF[{ψn(r)}]/δψn(r) = 0

(

−1

2∇2 + vext(r) + vH(r)

)

ψn(r) +

∫

dr′vx(r, r

′)ψn(r′) = ǫHF

n ψn(r)

Nonlocal HF exchange potential: vx(r, r′) = −

N∑

m=1

ψm(r)ψ∗m(r′)

|r − r′|

KS equation: δEKS[n(r)]/δn(r) = 0(

−1

2∇2 + vext(r) + vH(r)

)

ψn(r) + vxc(r)ψn(r) = ǫKSn ψn(r)

Local KS XC potential: vxc(r) =δExc[n(r)]

δn(r)

Exc[n(r)]: DFT exchange-correlation (XC) energy functional


Solving HF equation in practice

Introduce a set of basis function φi(r)

ψn(r) =∑

i

cinφi (r).

HF Eq. becomes a matrix (Roothaan) Eq.

∑

j

Fijcjn = ǫHFn

∑

j

Sijcjn. (Fij : Fock matrix)

Fij = < φi |HHF|φj >=< φi | −1

2∇2 + vext(r) + vH(r)|φj >

− < φi |vx(r, r′)|φj >

Sij = < φi |φj >

Exchange matrix requires a special treatment


Evaluating HF exchange matrix

vx,ij = − < φi |vx(r, r′)|φj >

= −N

∑

m=1

∫

dr

∫

dr′φ

∗i (r)ψm(r)ψ∗

m(r′)φj (r′)

|r − r′| = −∑

kl

Dkl(ik|lj)

Dkl = −N

∑

m=1

ckmclm. (density matrix)

Two electron Coulomb repulsion integral

(ik|lj) =

∫

dr

∫

dr′φ

∗i (r)φk(r)φ∗l (r

′)φj(r′)

|r − r′|φk(r)

φi (r)

φj(r′)

φl(r′)

1|r−r′|


Two electron Coulomb integral in practice

Plane waves φ(r) = e ik·r

(ik|lj) =4π

|ki + kl − kk − kj |2

Gaussian-type orbital φ(r) = x lymzne−αr2

- Gaussian product theorem

e−α1|r−RA|2

e−α2|r−RB |2

= exp[

−α1α2|RA − RB |2/γ]

exp[

−γ|r − P|2]

γ = α1 + α2, P = (α1RA + α2RB) /γ

- Analytical evaluation

Numerical atomic orbital φ(r) = Rnl(r)Ylm(ϑ,ϕ)

(ik|lj) : N4basis many 6-dimensional integrals: =⇒

very expensive to calculate and to store !

But, do we have to do this ?


Auxiliary basis to two electron Coulomb integral

(Resolution of Identity (RI))

Observation:

– N2basis many pair products {φi (r)φj (r)}, where i , j = 1, 2, · · ·Nbasis,

are heavily linear dependent

There must exist an auxiliary basis set {Pµ(r)}◮ Linear independent, (hence Naux << N2

basis)◮ Sufficiently accurate to represent {φi (r)φj (r)}

φi (r)φk (r) ≈∑

µ

CµikPµ(r)

◮ It follows

(ik |lj) ≈∑

µν

Cµik < Pµ|v |Pν > Cν

lj =∑

µν

CµikVµνCν

lj

Cνlj =

∑

µ

Oµlj Sµν

−1, where Oµlj =

∫

drφi (r)φj (r)Pµ(r), and

Sµν =∫

drPµ(r)Pν(r)


Construction of auxiliary basis

A very simple procedure, but it works!

Should also be atom-centered orbitals

P(r) = ξql (r)Ylm(ϑ,ϕ) (“normal” basis: φ(r) = Rnl (r)Ylm(ϑ,ϕ))

numerically easy to manipulate

Determine the shape of ξql (r)

Rn

1l1(r)R

n2l2(r)

For every atom, l

|l1-l

2| <= l <= |l

1+l2|

Gram-Schmidt Orthonormalization

ξql

– {Pµ(r)} are orthonormal on eachatom, but nonorthogonal betweendifferent atoms.

On-site

Off-site

– “On-site” pairsφi (r − RA)φj (r − RA) “exactly”represented by Pµ(r) !

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


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

A B(H2: d = 0.7 A)

φ1s(x − XA)φ1s(x − XB)

-2.0 -1.0 0.0 1.0 2.0x coordinate (angstrom)

0

0.02

0.04

0.06

0.08

ExactRI approx.

φ2s(x − XA)φ2px (x − XB)

-2.0 -1.0 0.0 1.0 2.0x coordinate (angstrom)

0

0.1

0.2

0.3

0.4

ExactRI approx.


On the accuracy of RI approximation

HF binding energy curve for H2

0.5 1 1.5 2 2.5 3Bond length (angstrom)

-4

-2

0

2

4

Bin

ding

ene

rgy

(eV

)

FHI-aims (RI)Gaussian (exact)

cc-pVQZ basis


On the convergence of NAO basis set



-4

-2

0

2

4

Bin

ding

ene

rgy

(eV

)

FHI-aims/Tier1(5)G03/cc-pVQZ(30)


On the convergence of NAO basis set



-4

-2

0

2

4

Bin

ding

ene

rgy

(eV

)

FHI-aims/Tier1(5)FHI-aims/Tier2(15)G03/cc-pVQZ(30)


Contents



3 ACFD-RPA


Hybrid functional

General principle:

Mixing a fraction of “exact change” with GGA semilocal exchange

B3LYP (semi-empirical)

EB3LYPXC = ELSDA

XC + a0(EexactX − ELSDA

X ) + aX∆EB88X + aC∆ELYP

C

a0 = 0.20, aX = 0.72, aC = 0.81

PBE0 (non-empirical)

EPBE0XC = 0.25 ∗ E exact

X + 0.75 ∗ EPBEX + EPBE

C

A. D. Becke, J. Chem. Phys. 98, 5648 (1993).

J. P. Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. 105, 9982 (1996).


Binding energy for N2

0.8 1 1.2 1.4 1.6Bond length (Angstrom)

-10

-5

0

Bin

ding

ene

rgy

(eV

) HFPBEExpPBE0 (aims/Tier2)

Vibrational frequency(cm−1)

HF: 2729.7 (371.1)PBE: 2347.9 (-10.7)PBE0: 2478.3 (119.7)EXP: 2358.6

PBE0 improves over PBE for the atomization energy, but not thevibrational frequency


Argon dimer: importance of the vdW interaction

3 3.5 4 4.5 5 5.5 6Bond length (angstrom)

-20

-10

0

10

20

30

Bin

ding

ene

rgy

(meV

) HFPBEPBE0Accurate

”Accurate” result:K. T. Tang and J. P. Toennies, J. Chem. Phys.

118, 4976 (2003).

4.5 5 5.5 6Bond length (angstrom)

-6

-4

-2

0

2

Bin

ding

ene

rgy

(meV

)

C6/R

6

Nonlocal correlation is needed for

describing the aymptotic behavior.


Second Order Møller-Plesset perturbation theory (MP2)

Rayleigh-Schrodinger perturbation theory

H = H(0) + V; H(0)|n >= E(0)n |n >

E(1)0 =< 0|V|0 >, E

(2)0 =

∑

n 6=0

| < 0|V|n > |2

E(0)0 − E

(0)n

, · · ·

MP2

H(0) = HHF; EHF = E(0)0 + E

(1)0

EMP2c =

1

4

occ∑

ij

vir∑

ab

[(ia|jb)∗ − (ib|ja)∗] (ia|jb)

ǫi + ǫj − ǫa − ǫb

Nonlocal correlation =⇒ van der Waals (vdW) interaction

EMP2c −→CMP2

6

R6AB

when RAB → ∞B

R AB

A


MP2 calculations for a benchmark database S22

S22: Benchmarkdatabase containing 22weakly bondedmolecular complex [1]

The complete basis setlimit is approachedwith Tier4 basis set(remaining error about5% by average).

FHI-aims calculations performed by Alexandre Tkatchenko[1] Jurecka, Sponer, Cerny, and Hobza, Phys. Chem. Chem. Phys. 8 , 1985 (2006)


MP2 for Argon dimer


-20

-10

0

10

20

30

Bin

ding

ene

rgy

(meV

) HFPBEPBE0MP2Accurate

”Accurate” result:K. T. Tang and J. P. Toennies, J. Chem. Phys. 118,

4976 (2003).


-6

-4

-2

0

2

Bin

ding

ene

rgy

(meV

)

C6/R

6

Nonlocal correlation in MP2is crucial for a quantitativedescription.


Successes and limitations of MP2

Pros

+ “Cheapest” ab-initio post-HFmethod for incorporatingnon-local correlation effect;

+ Good for properties of organicmolecule

+ Very good for hydrogen bonding

+ Good for charge transfer barriers(which LDA/GGA stronglyunderestimate)

Cons

- Bad for small-gap systems,molecules containing transitionmetal/rare earth ions, andcompletely fails (diverges) for3D metals

- Reason: HF reference, andperturbation based on bareCoulomb interaction


Contents



3 ACFD-RPA


Adiabatic connection fluctuation dissipation theorem

Exact XC energy via “adiabatic connection”

E exactXC =

∫ 1

0dλUλ

XC

UλXC is related to the density-density fluctuation of the system.

Fluctuation-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′

) = ∂n(r, t)/∂vλext(r

′, t′

): Response function(the imaginary part of which describe the dissipation process)

=⇒ Opens a new route to construct Exc

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

O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976).


RPA formulated within DFT framework

Dyson equation for χλ (in TDDFT)

χλ = χ0 + χ0(λv + f λxc)χλ

f λxc(r, r

′, t − t′

) =δv lambda

xc (r, t)

δn(r′, t ′)

RPA: fxc = 0

ERPAxc = E exact

x + ERPAc

ERPAc =

1

2π

∫ ∞

0dωTr [ln(1 − χ0(iω)v) + χ0(iω)v ]

Tr =∫

dr∫

dr′

RPA is done non-self-consistently as a post-correction to LDA/GGA(in this talk, RPA based on PBE orbitals)


Some attractive features of RPA

The application of RPA to realistic systems is just at the begining. Itsfurther development and systematic assessment is underway.

ERPAtot = Ts + Eext + EH + E exact

x + ERPAc

“Exact exchange” incorporated, self-interaction error isdramatically reduced.

Nonlocal vdW interactions are included automatically(and seamlessly).

Screening effect is taken into account, thus works formetals/small gap systems, in contrast to MP2.


RPA in FHI-aimsRPA correlation energy

ERPAc =

1

2π

∫ ∞

0dωTr [ln(1 − χ0(iω)v) + χ0(iω)v ]

Noninteracting response function in real space

χ0(r, r′, iω) = 2

∑

mn

(fn − fm)ψ∗m(r)ψn(r)ψ

∗n(r′)ψm(r′)

iω − ǫm + ǫn

0 ≤ fm ≤ 1 : Fermi occupation number

Matrix representation with auxiliary basis

χ0,µν =< Pµ|χ0|Pν >= 2∑

mn

(fn − fm)OµmnO

νnm

iω − ǫm + ǫn

where Oµmn =

∫

drψ∗m(r)ψn(r)Pµ(r).

vµν =

∫

dr

∫

dr′Pµ(r)Pν(r′)

|r − r′|Xinguo Ren (FHI, Berlin) Beyond LDA and GGA - Hartree-Fock, hybrid functionals, MP2, and RPA in FHI-aims: theoFHI-aims workshop, 25.06.09 33 / 38

Atomization energy for small moleculesTable: Unit in eV

Molecule PBE PBE0 RPA∗ EXP

H2 4.54 4.53 4.73 4.73

N2 10.58 9.75 9.67 9.88

O2 6.24 5.36 4.86 5.03

F2 2.30 1.48 1.30 1.65

CO 11.70 11.09 10.58 11.23

HF 6.16 5.93 5.74 6.11

H2O 10.17 9.84 9.67 10.06

C2H2 18.00 17.54 16.52 17.56

mean abs. error 0.48 0.17 0.45

* Gaussian basis extrapolated to basis set limit

The accuracy achieved by hybrid functionals for atomization energiesis hard to beat. A simple RPA does not improve over PBE0.


RPA for Argon dimer


-20

-10

0

10

20

30

Bin

ding

ene

rgy

(meV

) HFPBEPBE0MP2RPAAccurate


-6

-4

-2

0

2

Bin

ding

ene

rgy

(meV

)

Both RPA and MP2 give the correctasymptotic behavior.


Binding energy of two graphene layers

2.5 3 3.5 4 4.5 5 d (Å)

-80

-60

-40

-20

0

20E

b (A

B)

(m

eV/a

tom

)

LDA PBE PBE0MP2RPA

exp

Calculations done by A. Sanfilippo with FHI-aims


Summary

Quantum chemistry method HF, MP2

and in the DFT world

1 LDA

2 GGA

3 meta-GGA

4 hybrid functionals

5 ACFD-RPA

accu

racy

?

6

com

puta

tion

tim

e

(43.4 meV) Post-LDA/GGA approachesimprove things in one way oranother, but they have theirown limitations.

Present higher rungs (hybridfunctionals and RPA) arepromising, but the improvementis not (yet) systematic. Morework are needed to constructthese rungs, and FHI-aimsprovides such a platform.


Many thanks to

Volker Blum and Patrick Rinke

Andrea Sanfilippo and Karsten Reuter

Alexandre Tkatchenko

Martin Fuchs and Hong Jiang

Matthias Scheffler


Documents

Beyond LDA and GGA - Hartree-Fock, hybrid functionals, MP2 ... · Beyond LDA and GGA - Hartree-Fock, hybrid functionals, MP2, and RPA in FHI-aims: theory and a numerical orbital practice