NANO266 - Lecture 5 - Exchange-Correlation Functionals
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UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
![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)
NANO266
2
Exc
GGA
[ρ↑
,ρ↓
]= drρ(r)εxc (∫ ρ↑
,ρ↓
, ∇ρ↑
, ∇ρ↓
)](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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NANO266 - Lecture 5 - Exchange-Correlation Functionals
- 2. 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) NANO266 2 Exc GGA [ρ↑ ,ρ↓ ]= drρ(r)εxc (∫ ρ↑ ,ρ↓ , ∇ρ↑ , ∇ρ↓ )
- 3. 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 NANO266 3
- 4. Performance of GGA GGA tends to correct LDA overbinding • Better bond lengths, lattice parameters, atomization energies, etc. NANO266 4
- 5. Why stop at the first derivative? Meta-GGA Example: TPSS functional NANO266 5 Exc meta−GGA [ρ↑ ,ρ↓ ]= drρ(r)εxc (∫ ρ↑ ,ρ↓ , ∇ρ↑ , ∇ρ↓ ,∇2 ρ↑ ,∇2 ρ↓ ) 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.
- 6. 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. NANO266 6 (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 2 ρσ m1,m1 m1 ∑ " # $$ % & ''− ρσ m1,m2 m1m2 ∑ ρσ m2,m1 " # $$ % & '' ) * + + , - . .σ ∑ Penalty term to force on-site occupancy in the direction of of idempotency, i.e. to either fully occupied or fully unoccupied levels
- 7. 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!) NANO266 7 U values used in the Materials Project, fitted by a UCSD NanoEngineering Professor
- 8. Hybrids NANO266 8 Chimera from God of War (memories of times when I was still a carefree graduate student) HF DFT
- 9. Rationale for Hybrids Semi-local DFT suffer from the dreaded self- interaction error • Spurious interaction of the electron not completely cancelled with approximate Exc NANO266 9 Eee = 1 2 ρi (ri )ρj (rj) rij dri drj∫∫ Ex HF = − 1 2 ρi (ri )ρj (rj) rij dri drj∫∫ Includes interaction of electron with itself! HF Exchange cancels self- interaction by construction
- 10. 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 NANO266 10 Exc B3LYP = Ex LDA + ao (Ex HF − Ex LDA )+ ax (Ex GGA − Ex LDA )+ Ec LDA +(Ec GGA − Ec LDA ) where a0 = −0.20, ax = 0.72, ac = 0.81 Exc PBE0 = 1 4 Ex HF + 3 4 Ex PBE + Ec PBE Exc HSE = aEx HF,SR (ω)+(1− a)Ex PBE,SR (ω)+ Ex PBE,LR (ω)+ Ec PBE a = 1 4 , ω = 0.2
- 11. Do hybrids work? NANO266 11 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.
- 12. Do hybrids work? NANO266 12 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.
- 14. Which functional to use? NANO266 14
- 15. To answer that question,we need to go back to our trade-off trinity NANO266 15 Choose two (sometimes you only get one) Accuracy Computational Cost System size
- 16. Accuracy of functionals – lattice parameters LDA overbinds GGA and meta GGA largely corrects that overbinding NANO266 16 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.
- 17. Cohesive energies LDA cohesive energies too low, i.e., overbinding Again, GGA does much better NANO266 17 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.
- 18. Bond lengths NANO266 18 Cramer, C. J. Essentials of Computational Chemistry: Theories and Models; 2004.
- 19. 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. NANO266 19
- 20. 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 NANO266 20 Ref: MIT 3.320 Lectures on Atomistic Modeling of Materials
- 21. bcc vs fcc in GGA NANO266 21 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.
- 22. Magnetism NANO266 22 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.
- 23. 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) NANO266 23 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.
- 24. Reaction energies Broad conclusions • GGA better than LSDA • Hybrids most efficient (good accuracy comparable to highly correlated methods) NANO266 24
- 25. Some well-known problems can be addressed by judicious fitting to experimental data NANO266 25 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.
- 26. 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 NANO266 26 https://www.materialsproject.org/docs/calculations
- 27. Band gaps In 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.