Exchange-Correlation Functionals

Shyue Ping Ong

What’s next?

LDA uses local density ρ from homogenous electron gas

Next step: Let’s add a gradient of the density!

Generalized gradient approximation (GGA)

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ExcGGA[ρ↑,ρ↓]= drρ(r)εxc (∫ ρ↑,ρ↓, ∇ρ↑ , ∇ρ↓ )

Unlike the Highlander, there is more than “one” GGA

•  BLYP, 1988: Exchange by Axel Becke based on energy density of atoms, one parameter + Correlation by Lee-Yang-Parr

•  PW91, 1991: Perdew-Wang 91Parametrization of real-space cut-off procedure

•  PBE, 1996: Perdew-Burke-Ernzerhof (re-parametrization and simplification of PW91)

•  RPBE, 1999: revised PBE, improves surface energetics

•  PBEsol, 2008: Revised PBE for solids

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Performance of GGA

GGA tends to correct LDA overbinding

•  Better bond lengths, lattice parameters, atomization energies, etc.

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Why stop at the first derivative?

Meta-GGA

Example: TPSS functional

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Excmeta−GGA[ρ↑,ρ↓]= drρ(r)εxc (∫ ρ↑,ρ↓, ∇ρ↑ , ∇ρ↓ ,∇2ρ↑,∇2ρ↓)

Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Climbing the density functional ladder: nonempirical meta-generalized gradient approximation designed for molecules and solids., Phys. Rev. Lett., 2003, 91, 146401, doi:10.1103/PhysRevLett.91.146401.

Orbital-dependent methods

DFT+U1,2,3

•  Treat strong on-site Coulomb interaction of localized electrons, e.g., d and f electrons (incorrectly described by LDA or GGA) with an additional Hubbard-like term.

•  Strength of on-site interactions usually described by U (on site Coulomb) and J (on site exchange), which can be extracted from ab-initio calculations,4 but usually are obtained semi-empirically, e.g., fitting to experimental formation energies or band gaps.

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(1)  Anisimov, V. I.; Zaanen, J.; Andersen, O. K. Phys. Rev. B, 1991, 44, 943–954. (2)  Anisimov, V. I.; Solovyev, I. V; Korotin, M. A.; Czyzyk, M. T.; Sawatzky, G. A. Phys. Rev. B, 1993, 48, 16929–16934. (3)  Dudarev, S. L.; Botton, G. A.; Savrasov, S. Y.; Humphreys, C. J.; Sutton, A. P., Phys. Rev. B, 1998, 57, 1505–1509. (4)  Cococcioni, M.; de Gironcoli, S., Phys. Rev. B, 2005, 71, 035105, doi:10.1103/PhysRevB.71.035105.

EDFT+U = EDFT +Ueff

2ρσm1,m1

m1

∑"

#$$

%

&''− ρσ

m1,m2m1m2

∑ ρσm2,m1

"

#$$

%

&''

)

*++

,

-..σ

∑

Penalty term to force on-site occupancy in the direction of of idempotency, i.e. to either fully

occupied or fully unoccupied levels

Where do I get U values

1.  Fit it yourself, either using linear response approach or to some experimental data that you have for your problem at hand

2.  Use well-tested values in the literature, e.g., for high-throughput calculations (though you should use caution!)

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U values used in the Materials Project, fitted by a UCSD NanoEngineering Professor

Hybrids

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Chimera from God of War (memories of times when I was still a carefree graduate student)

HF

DFT

Rationale for Hybrids

Semi-local DFT suffer from the dreaded self-interaction error

•  Spurious interaction of the electron not completely cancelled with approximate Exc

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Eee =12

ρi (ri )ρ j (rj)rij

dri drj∫∫

ExHF = −

12

ρi (ri )ρ j (rj)rij

dri drj∫∫

Includes interaction of electron with itself!

HF Exchange cancels self-interaction by construction

Typical Hybrid Functionals

B3LYP (Becke 3-parameter, Lee-Yang-Parr)

•  Arguably the most popular functional in quantum chemistry (the 8th most cited paper in all

fields) •  Originally fitted from a set of atomization energies, ionization potentials, proton affinities and

total atomic energies.

PBE0:

HSE (Heyd-Scuseria-Ernzerhof) (2006):

•  Effectively PBE0, but with an adjustable parameter controlling the range of the exchange

interaction. Hence, known as a screened hybrid functional •  Works remarkably well for extended systems like solids

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ExcB3LYP = Ex

LDA + ao(ExHF −Ex

LDA )+ ax (ExGGA −Ex

LDA )+EcLDA + (Ec

GGA −EcLDA )

where a0 = −0.20, ax = 0.72, ac = 0.81

ExcPBE0 =

14Ex

HF +34Ex

PBE +EcPBE

ExcHSE = aEx

HF,SR (ω)+ (1− a)ExPBE,SR (ω)+Ex

PBE,LR (ω)+EcPBE

a = 14

, ω = 0.2

Do hybrids work?

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Heyd, J.; Peralta, J. E.; Scuseria, G. E.; Martin, R. L. Energy band gaps and lattice parameters evaluated with the Heyd-Scuseria-Ernzerhof screened hybrid functional., J. Chem. Phys., 2005, 123, 174101, doi:10.1063/1.2085170.

Do hybrids work?

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Chevrier, V. L.; Ong, S. P.; Armiento, R.; Chan, M. K. Y.; Ceder, G. Hybrid density functional calculations of redox potentials and formation energies of transition metal compounds, Phys. Rev. B, 2010, 82, 075122, doi:10.1103/PhysRevB.82.075122.

The Jacob’s Ladder

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Which functional to use?

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To answer that question, we need to go back to our trade-off trinity

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Choose two (sometimes you only get

one)

Accuracy

Computational Cost

System size

Accuracy of functionals – lattice parameters

LDA overbinds GGA and meta GGA largely corrects that overbinding

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Haas, P.; Tran, F.; Blaha, P. Calculation of the lattice constant of solids with semilocal functionals, Phys. Rev. B - Condens. Matter Mater. Phys., 2009, 79, 1–10, doi:10.1103/PhysRevB.79.085104.

Cohesive energies

LDA cohesive energies too low, i.e., overbinding

Again, GGA does much better

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Philipsen, P. H. T.; Baerends, E. J. Cohesive energy of 3d transition metals: Density functional theory atomic and bulk calculations, Phys. Rev. B, 1996, 54, 5326–5333, doi:10.1103/PhysRevB.54.5326.

Bond lengths

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Cramer, C. J. Essentials of Computational Chemistry: Theories and Models; 2004.

Conclusion – LDA vs GGA

LDA almost always underpredicts bond lengths, lattice parameters and overbinds GGA error is smaller, but less systematic. Error in GGA < 1% in many cases Conclusion

•  Very little reason to choose LDA over GGA since computational cost are similar Note: In all cases, we assume that LDA and GGA refers to spin-polarized versions.

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Predicting structure

Atomic energy: -1894.074 Ry Fcc V : -1894.7325 Ry

Bcc V : -1894.7125 Ry

Cohesive energy = 0.638 Ry (0.03% of total E)

Fcc/bcc difference = 0.02 Ry (0.001% of total E)

Mixing energies ~ 10-6 fraction of total E

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Ref: MIT 3.320 Lectures on Atomistic Modeling of Materials

bcc vs fcc in GGA

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Green: Correct Ebcc-fcc

Red: Incorrect Ebcc-fcc

Note: Based on structures at STP

Wang, Y.; Curtarolo, S.; Jiang, C.; Arroyave, R.; Wang, T.; Ceder, G.; Chen, L. Q.; Liu, Z. K. Ab initio lattice stability in comparison with CALPHAD lattice stability, Calphad Comput. Coupling Phase Diagrams Thermochem., 2004, 28, 79–90, doi:10.1016/j.calphad.2004.05.002.

Magnetism

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Wang, L.; Maxisch, T.; Ceder, G. Oxidation energies of transition metal oxides within the GGA+U framework, Phys. Rev. B, 2006, 73, 195107, doi:10.1103/PhysRevB.73.195107.

Atomization energies, ionization energies and electron affinities

Carried out over G2 test set of molecules (note that PBE1PBE in the tables below refers to the PBE0 functional)

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Ernzerhof, M.; Scuseria, G. E. Assessment of the Perdew-Burke-Ernzerhof exchange-correlation functional, J. Chem. Phys., 1999, 110, 5029–5036, doi:10.1063/1.478401.

Reaction energies

Broad conclusions •  GGA better than LSDA •  Hybrids most efficient (good

accuracy comparable to highly correlated methods)

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Some well-known problems can be addressed by judicious fitting to experimental data

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Wang, L.; Maxisch, T.; Ceder, G. Oxidation energies of transition metal oxides within the GGA+U framework, Phys. Rev. B, 2006, 73, 195107, doi:10.1103/PhysRevB.73.195107.

Stevanović, V.; Lany, S.; Zhang, X.; Zunger, A. Correcting density functional theory for accurate predictions of compound enthalpies of formation: Fitted elemental-phase reference energies, Phys. Rev. B, 2012, 85, 115104, doi:10.1103/PhysRevB.85.115104.

If you know what you are doing, results can be pretty good

High-throughput analysis using the Materials Project, again done by a UCSD NanoEngineering professor

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https://www.materialsproject.org/docs/calculations

Band gapsIn a nutshell, really bad in semi-local DFT. But we knew this going into KS DFT…

Hybrids fare much better

New functionals and methods have been developed to address this problem

•  GLLB functional1

•  ΔSCF for solids2

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(1)  Kuisma, M.; Ojanen, J.; Enkovaara, J.; Rantala, T. T. Phys. Rev. B, 2010, 82, doi:10.1103/PhysRevB.82.115106.

(2)  Chan, M.; Ceder, G. Phys. Rev. Lett., 2010, 105, 196403, doi:10.1103/PhysRevLett.105.196403.