NANO266 - Lecture 5 - Exchange-Correlation Functionals
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Transcript of NANO266 - Lecture 5 - Exchange-Correlation Functionals
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
NANO266 13 http://www.sas.upenn.edu/~jianmint/Research/
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
NANO266 27 https://www.materialsproject.org/docs/calculations
(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.