lda vs gga vs mgga

7/29/2019 lda vs gga vs mgga

1/25

Comparison of

exchange-correlation functionals:

from LDA to GGA and beyond

Martin Fuchs

Fritz-Haber-Institut der MPG, Berlin, Germany

Density-Functional Theory Calculations for Modeling Materials and Bio-Molecular Properties and Functions - A Hands-On

Computer Course, 30 October - 5 November 2005, IPAM, UCLA, Los Angeles, USA


2/25

Density-functional theory

DFT is an exact theory of the ground state of an interacting many-particle system:

E0 = minnNEv[n] E0 = minN|Hv| by Hohenberg-Kohn theorem: v(r) v[n(r)]

Total energy density functional (electrons: w(r, r) = 1/|r r|)

Ev[n] = minn|

T +

W| +Z

n(r

)v(r

)d = F[n] +Z

n(r

)v(r

) d

Non-interacting case: w(r, r) 0

F[n] minn|T| = [n]|T|[n] = Ts[n] [n] is a Slater determinant

Interacting case rewritten:

Ev[n] = Ts[n] + Exc[n] + 12

Zn(r)w(r, r)n(r)dd +

Zn(r)v(r) d

XC energy functional: kinetic correlation energy & exchange and Coulomb correlation energy

Exc[n] =: [n]|T|[n] Ts[n] + [n]|W|[n] 12R

n(r)w(r, r)n(r)dd


3/25

Kohn-Sham scheme

Minimization of Ev[n] with n(r) N determines groundstate:

Ev[n]n(r)

= F[n]n(r)

+ v(r) = Ts[n]n(r)

+ vKS([n]; r) =

. . . effective non-interacting system Kohn-Sham independent-particle equations:

h

2

2 + vKS(r)i

i(r) = ii(r) with vKS

(r) =Z

w(r, r

)n(r

)d

+Exc[n]

n(r) + v(r) ,

density n(r) =P

i fi|i(r)|2,

occupancies i < : fi = 1, i = : 0 fi 1, i > : fi = 0 (aufbau principle).

non-interacting kinetic energy treated exactly Ts[n] =RP

i fi|i(r)|2d

classicalelectrostatics (e-n & e-e) treated exactly

onlyneed to approximate Exc[n] and vxc([n]; r) =Exc[n]n(r) determines accuracy in practice!

applied is spin density functional theory: n, treated as separate variables

beware: vKS(r) is a local operator. Direct approximation of whole F[n] possible too: DFT with

beware: generalized Kohn-Sham schemes1 and nonlocal effective potentials, e.g. Hartree-Fock eqs.

1 Seidl, G orling, Vogl, Majewski, Levy, Phys Rev B 53, 3764 (1996).


4/25

Local-density approximation

ELDAxc [n] =

Zn(r)e

homxc (n(r))d

0.1 1.0 10.0 100.0

rs

(bohr)

30.0

20.0

10.0

0.0

exc

(Ry)

local dependence on density or Wigner-Seitz radiusrs = (

43 n)

1/3

XC energy per electron is that of the homogeneous electron gas,exc[n] = e

homxc (n)|n=n(r)

exchange part ehomx (n) known analytically correlation part ehomc (n) known

analytically for rs 0 and rs numericallyexact for 2 < rs < 100 from QMC data

1

. . . as parametrization interpolating over all rsPW91,Perdew-Zunger,VWN2

workhorse in DFT applications to solids

real systems are far from jellium-like homogeneity

. . . why does LDA work at all?

. . . where does it fail?

. . . how to improve beyond it?

1 Ceperley, Alder, Phys Rev Lett 45, 566 (1980).2 Perdew, Wang, Phys Rev B 45, 13244 (1992); Perdew, Zunger (1980); Vosko, Wilk, Nussair (1980).


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Performance of the LDA

structural, elastic, and vibrational properties often good enough crystal bulk lattice constants accurate to within 3%, usually underestimated

bulk moduli somewhat too large, > 10% error not uncommon for d-metals phonons somewhat too stiff

binding energies are too negative (overbinding), up to several eV# cohesive energies of solids, but formation enthalpies often o.k.

# molecular atomization energies, mean error (148 molecules G2 set) 3 eV

activation energies in chemical reactions unreliable# too small/absent, e.g. for H2 on various surfaces (Al, Cu, Si, ...)

relative stability of crystal bulk phases can be uncertain

# SiO2 high pressure phase more stable than zero pressure phase# underestimated transition pressure e.g. for diamond -tin phase transitions in Si & Ge# magnetic phases

electronic structure can be usefully interpreted (density of states, band structures),except for band gaps (a more fundamental issue than LDA!)


6/25

View on XC through the XC hole

Definition of the XC energy by a coupling constant integration of the e-e interaction1:

E([n]; ) = [n]|

T +V +

W|[n] = minn ...

by Hohenberg-Kohn theorem: n(r) v([n]; r) for any 0 1

= 0 non-interacting case/ Kohn-Sham potential, v=0([n]; r) = vKS([n]; r)

= 1 external potential, v=1([n]; r) = v(r)

groundstate energy: E([n]; 1) = E([n]; 0) +

Z10

d E([n]; )

dd

Exc[n] = R1

0[n]|W|[n] d

12 Rn(r)w(r, r

)n(r)dd

X, XX =1

2

Zw(r, r

)n(r)n(r

)[n]|

Pp,q

p(r)

q(r

)q(r)p(r)|[n]

n(r)n(r)dd

XC in terms of the pair correlation function g([n];

r

,r

)

XC in terms of the adiabatic connection integrand

(scaled e-e potential energy)

may distinguish exchange and correlation, e.g. Exc = Ex + Ec

g(r, r)r

r

1 cf. DFT books by Dreizler and Gross & Yang and Parr


7/25

. . . coupling constant averaged XC hole

-integration implies coupling constant averaged pair correlation function

g([n]; r, r) =

Z10

g([n]; r, r)d

Identify the XC energy as

Exc[n] =1

2

Zd n(r)

Zd

n(r

){g([n]; r, r

) 1} w(r, r

)

interpretation: the electron density n(r) interacts with the

electron density of the XC holenxc([n]; r, r

) = n(r){g([n]; r, r) 1}

Pauli exclusion principle & Coulomb repulsion

local density approximation corresponds to

nxc([n]; r, r) = n(r){g

hom(n(r); |r r|) 1}

always centered at reference electron & spherical 0 1 2 3 4 5 6

0.5

0.0

XCh

ole

(n)

exchange

+ correlation

homogeneous electron gas

kF|r r|

(NB: correlation part varies with kF)


8/25

From the XC hole to the XC energy

Focus on Exc[n] = Rn(r)exc([n]; r)d (... component of the total energy) w =1

|rr|= 1|u|

for exc only the angle averaged XC hole matters (and LDA hole is always spherical)

nxc(r, r + u) =Xlm

nxclm(r, u)Ylm(u)

Exc[n] =1

2

Zn(r)

Znxc(r, r + u)

udu d

1

2

Zn(r)

Z0

nxc00(r, u)

uu2dud

for Exc only the system & angle averaged XC hole matters

nxc(u) =1

N

Zn(r)n

xc00(r, u)d

XC energy in terms of averaged XC hole1

Exc[n] =N

2

Z0

nxc(u)

uu2

du = N average XC energy per electron

LDA & GGA approximate average holes rather closely work mostly o.k.

1 Perdew et al., J Chem Phys 108, 1552 (1998).


9/25


10/25

Generalized Gradient Approximation for Exc

Gradient expansion of XC energy and (later) hole: nxc([n]; r, u) nxc(r, u = 0) +unxc|u=0 + ...,and generalized to imposing constraints to meet

e.g. nx(r, u) < 0,R

nx(r, u)dr = 1, by real space cutoffs numerical GGA scaling relations and bounds on Exc by analytic approximation to numerical GGA parameter-free GGA by Perdew-Wang PW91

simplified in PBE GGA

contains LDA & retains all its good features

Earlier: analytic model or ansatz + empirical parameter(s)

Langreth-Mehl (C), Becke 86(X) + Lee-Yang-Parr(C) BLYP, ...

Alternative: analytic ansatz + (many) fitted parameters (e.g. fit to thermochemical data)

see accuracy limit of GGA functionals, can be better & worse than PBE

Generic GGA XC functional

EGGAxc [n] = Z n eLDAx (n) FGGAxc (n, s) n=n(r)d, s =

|n|

2kFnn=n(r)

Enhancement factor Fxc(n, s) over LDA exchange: function of density and scaled gradientunderstandinghow GGAs work

Calculations with GGAs are not more involved than with LDA,except that vxc[n; r] = vxc(n,in,ijn)|n(r)


11/25

Differences in present GGAs

PBE

0 1 2 31.0

1.2

1.4

1.6

1.8

rs=100

50

10

5

PBE XC

rs

=0

1

enhancementfactor

scaled gradient

Zhang et al. revised revPBE

0 1 2 31.0

1.2

1.4

1.6

1.8

rs=100

50

10

5

1 Zhang

enhancementfactor

scaled gradient

BLYP

0 1 2 31.0

1.2

1.4

1.6

1.8

rs=100

50

10

5

1 BLYP

enh

ancementfactor

scaled gradient

. . .

WC-PBE Wu & Cohen 05

xPBE Xu & Goddard 04

HCTH Handy et al. 01

RPBE Hammer et al. 99

revPBE Zhang et al. 98PBE GGA Perdew et al. 96

PW91 Perdew & Wang 91

PW91 Perdew & Wang 91

BP Becke & Perdew 88

BLYP Becke & Lee et al. 88. . .

revPBE & BLYP more nonlocal than PBE GGA

molecules: more accurate atomization energies

solids: bondlengths too large, cohesive energies too small

LYP correlation incorrect for jellium

more local GGAs will make lattice constants smaller(e.g. Tinte et al. PRB 58, 11959 (1998)),

but make binding energies more negative

h GGA h l i


12/25

... how GGAs change total energiesAnalysis in terms of selfconsistent LDA density shows how GGAs work:

EGGAtot [n

GGA] = E

LDAtot [n

LDA] + E

GGAxc [n

LDA] E

LDAxc [n

LDA] +O(n

GGA nLDA)2

Spectral decomposition in terms of s:

EGGAxc E

LDAXC =

Z Zs=0

heGGAxc (r) e

LDAxc (r)i (s s(r))dds

0 1 2 3 4

scaled gradient s

0.05

0.00

0.05

energyperatom(

a.u.)

correlation

exchange

PBE rev PBE

X+C

Al fcc

0 1 2 3 4

scaled gradient s

0.5

0.2

0.0

energyperatom

(eV)

PBE

revPBE

differential

integrated

fcc Al

-0.3 (0.5) eV

-0.5 (0.7) eV

only 0 s 4 contribute, similar analysis can be made for n(r)

for more see e.g. Zupan et al., PRB 58, 11266 (1998)

C h i i i GGA


13/25

Cohesive properties in GGA

Bulk lattice constants GGA increase due tomore repulsive core-valence XC.

Na

NaC

l

Al C Si Ge

SiC

AlAs

GaA

s

Cu

5

0

5

erro

r(%)

PW91GGA

inconsistent GGA

LDA

expt.

Cohesive energies GGA reduction mostlyvalence effect.

Na

NaC

l

Al C Si Ge

SiC

AlAs

GaA

s

Cu

0.5

0.0

0.5

1.0

1.5

erro

r(eV)

PW91GGA

inconsistent GGALDA

expt.

for comparison of LDA, GGA, and Meta-GGA see Staroverov, Scuseria, Perdew, PRB 69, 075102 (2004)

b i H C (111)


14/25

... energy barriers: H2 on Cu(111)

barrier to dissociative adsorption

- PW91-GGA 0.7 eV- LDA


15/25

Phase transition -tin - diamond in Si

diamond-structure

-tin structure

a a

c

diamond: a0 = 10.26 bohr, semiconductor

-tin: c/a = 0.552, metallic

Spectral decomposition

in terms of n

0 1 2 3 4

Density parameter rs

[bohr]

0.0

0.5

1.0

1.5

2.0

Rela

tivevolume/r

s

-tin

diamond

Phase transition tin diamond in Si


16/25

Phase transition -tin - diamond in Si

Gibbs construction: Et + ptVt = E

diat + ptV

diat , p

expt = 10.3 . . . 12.5 GPa

0.6 0.7 0.8 0.9 1.0 1.1

relative volume V/Vexp

0.0

0.2

0.4

0.6

energy(eV/atom)

tin diamond

LDA

GGA

0.2 0.3 0.4 0.5 0.6

energy change (eV/atom)

5

10

15

20

25

transitionpressure(GPa)

LDA

PBE

revPBE

PW91

BP

BLYP

expt

GGA increases transition pressure,inhomogeneity effect use of LDA-pseudopotentials insufficient

Moll et al, PRB 52, 2550 (1995); DalCorso et al, PRB 53, 1180 (1996); McMahon et al, PRB 47, 8337 (1993).

Performance of PBE GGA vs LDA


17/25

Performance of PBE GGA vs. LDA

atomic & molecular total energies are

improved

GGA corrects the LDA overbinding:average error for G2-1 set of molecules:

+6 eV HF

1.5 eV LSDA0.5 eV PBE-GGA

0.2 eV PBE0 hybrid0.05 eV goal,

better cohesive energies of solids

improved activation energy barriers in

chemical reactions (but still too low) improved description of relative stability of

bulk phases

more realistic for magnetic solids

useful for

electrostatic

hydrogen bonds

# GGA softens the bonds

increasing lattice constants

decreasing bulk moduli no consistent improvement

LDA yields good relative bond energies forhighly coordinated atoms,

e.g. surface energies, diffusion barriers on surfaces

GGA favors lower coordination (larger gradient!),not always enough where LDA has a problem,

e.g. CO adsorption sites on transition metal surfaces

significance of GGA ?

# GGA workfunctions for several metals turn out somewhatsmaller than in LDA

one-particle energies/bands close to LDA

# Van der Waals (dispersion) forces not included!

Comparing LDA and (different) GGAs gives an idea about possible errors!

Beyond GGA: orbital dependent XC functionals


18/25

Beyond GGA: orbital dependent XC functionals

Kohn-Sham non-interacting system make i[n] density-functionals: n vKS[n; r] i

LDA and GGA are explicit density-functionals Exc[n]

Implicit density-functionals formulated in terms of i[n]? more flexible for further improvements

self-interaction free: Exc[n]|N=1 = 0,vxc(|r| ) =

1r ,

nonspherical nxc(r, r) ... ?

Start with exact exchange :

Exchange-energy Ex[{i}] =1

2

ZPi,j i(r)j(r

)i(r)j(r)

|r r|dd

v same as in Hartree-Fock, but 2

2 vKS(r)|i(r) = ii(r) KS and HF orbitals different!

Groundstate ...

in KS-DFT: optimized effective potential OEP method, local KS potential

in Hartree-Fock: variation with respect to orbitals yield HF eqs., nonlocal effective potential

EHF0 EKSEXX0

Hybrid functional: mixing exact exchange with LDA/GGA


19/25

Hybrid functional: mixing exact exchange with LDA/GGA

XC = EXX + LDA correlation is less accurate (underbinds) than XC = LDA for molecules

challenge: correlation functional that is compatible with exact exchange?

hybrid functionals tointerpolateadiabatic connection, Uxc[n]

Exc[n] =

Z10

Uxc[n]() d Uxc[n]() = [n]|W|[n] UHartree[n]

molecular dissociation: Uxc = molecules - atoms ... looks like

couplingstrength

Uxc()10

Ex

Ec

Tc

=0 = KSKS system

=1physical system

LDA,

GGA

good

LDA,GGA

too negative

. . . hybrid functionals


20/25

. . . hybrid functionalsHow they are defined:

use exact exchange for = 0and alocal functional for = 1

Hybrid functional = interpolation

Ehybxc = E

GGAxc + a nEx EGGAx o

mixing parameter a = 0.16 . . . 0.3from fitting thermochemical data

a = 1/4 by 4th order perturbation theory PBE0 = PBE1PBE

B3LYP 3-parameter combination ofBecke X-GGA, LYP C-GGA, and LDA

molecular dissociation energies on averagewithin 3 kcal/mol 0.1 eV(but 6 times larger errors can happen)

How hydbrids and GGAs work:

Adiabatic connection N2 2N

0 0.5 1

coupling strength

-6

-4

-2

0

2

U

xc

()

(eV)

exchange only

PBE GGA

PBE0 hybrid

"exact"

XC contributions to binding:

(eV) Eb Ex Ec

PBE -10.58 -1.82 -2.49PBE0 -9.62 -1.01 -2.49

exact -9.75 1.39 -5.07

... X and C errors tend to cancel!

Becke, J Chem Phys 98, 5648 (1993); Perdew et al, J Chem Phys 105, 9982 (1996).comprehensive comparison: Staroverov, Scuseria, Tao, Perdew, J Chem Phys 119, 12129 (2003)

XC revisited: role of error cancellation between X and C


21/25

XC revisited: role of error cancellation between X and C

see e.g. Baerends and Gritsenko, JCP 123, 062202 (2005)

... or: when GGAs or hybrids are not enough

... orbital dependent functionals


22/25

... orbital dependent functionals

Meta-GGAs

XC energy functional

EMGGAxc [n] =

Zn(r)e

MGGAxc (n,n, ts)

n(r)

d

kinetic energy density of non-interacting electrons ts(r) =12

Pocci fi|i(r)|

2

OEP or HF style treatment ofEMGGAxc [i]

i(r)= ...

1

2

exc

tsi

TPSS: Tao, Perdew, Stavroverov, Scuseria, Phys Rev Lett 91, 146401 (2003): + vx finite

PKZB: Perdew, Kurth, Zupan, Blaha, Phys Rev Lett 82, 2544 (1999): XCnon-empirial, LDA limitVan Voorhis, Scuseria, J Chem Phys 109, 400 (1998): XChighly fitted, no LDA limit

Colle, Salvetti, Theoret Chim Acta 53, 55 (1979): C, no LDA limit BLYP GGA

Becke, J Chem Phys 109, 2092 (1998): XC + exact exchangefitted

TPSS accomplishes a consistent improvement over (PBE) GGA

PKZB improved molecular binding energy, but worsened bond lengths in molecules & solids

hybrid functionals on average still more accurate for molecular binding energies

TPSS provides sound, nonempirical basis for new hybridsTPSSh

next step: MGGA correlation compatible with exact exchange?

ACFDT XC: including unoccupied Kohn-Sham states


23/25

C XC c ud g u occup ed Ko S a states

Adiabatic connection: KS system = 0 physical system = 1

Exc[n] = R1

0 |Wee|d U[n]

Fluctuation-dissipation theorem:

Wxc() =: |Wee| =1

2

Ze2

|r r|

Z0

(iu, r, r) du n(r)(r r

)

drdr

...using dynamical density response

From noninteracting Kohn-Sham to interacting response by TD-DFT

0(iu, r, r) = 2P

i,j

i

(r)

j

(r)

j

(r)

i

(r)

iu(ji) i[n] . . . KS eigenvaluesi([n], r) ...KS orbitals

(iu) = 0(iu) + 0(iu) Khxc (iu) (iu) Dyson equation6dim.

. . . using Coulomb and XC kernel from TD-DFT

In principle ACFDT formula gives exact XC functional

In practice starting point for fully nonlocal approximations

RPA: Khxc = |r r|1 and zero XC kernel ... yields exact exchange and London dispersion

forces combine with XC kernels, hybrids with usual XC functionals, split Coulomb interaction . . .

Status of RPA type functionals


24/25

yp

Molecular dissociation energies

RPA for all e-e separations

H 2 N 2 O 2 F 2 Si 2 HF CO CO

2

C 2H 2

H 2O

-2

-1

-0.5

0

0.5

1

2

errorin

dissociation

energy

(eV)

RPA+

RPA

PBE0 hyb

PBE GGA

... no consistent improvement over GGA

... not too bad, without X and C error cancellation

... better TD-DFT XC kernels needed!

Furche, PRB 64, 195120 (2002),

Fuchs and Gonze, PRB 65, 235109 (2002)

Stacked benzene dimer

(Van der Waals complex)short-range: GGA + long-range: RPA

3 3.5 4 4.5

Separation (A)

4

3

2

1

0

1

2

3

4

5

In

teractionenergy(kcal/mol)

CCSD(T)

MP2

vdWDF

GGA(revPBE)GGA(PW91)

... includes dispersion forces!

Dion et al., PRL 92, 246401 (2004)

Summary


25/25

y

LDA & GGA are de facto controlled approximations to the average XC hole

GGA remedies LDA shortcomings w.r.t. total energy differences but may also overcorrect(e.g. lattice parameters)

still can & should check GGA induced corrections for plausibility by

. . . simple arguments like homogeneity & coordination

...results fromquantum chemicalmethods (Quantum Monte Carlo, CI, . . . )

. . . depends on actual GGA functional

hybrid functionals mix in exact exchange (B3LYP, PBE0, ... functionals)

orbital dependent, implicit density functionals:exact Kohn-Sham exchange, Meta-GGA & OEP method,functionals from the adiabatic-connection fluctuation-disspation formula

Always tell what XC functional you used,e.g. PBE-GGA (not just GGA)... helps others to understand your results

... helps to see where XC functionals do well or have a problem

lda vs gga vs mgga

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