NANO266 - 05 - Exchange-Correlation Functionals

Transcript

1. Exchange-Correlation Functionals Shyue Ping Ong
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 E xc 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 E xc meta−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.
6. Hybrids NANO266 6 Chimera from God of War (memories of times when I was still a carefree graduate student) HF DFT
7. 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 7 E ee = 1 2 ρi (r i )ρj (r j ) r ij dr i dr j ∫∫ E x HF = − 1 2 ρi (r i )ρj (r j ) r ij dr i dr j ∫∫ Includes interaction of electron with itself! HF Exchange cancels self- interaction by construction
8. 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 8 E xc B3LYP = E x LDA + a o (E x HF − E x LDA )+ a x (E x GGA − E x LDA )+ E c LDA +(E c GGA − E c LDA ) where a 0 = −0.20, a x = 0.72, a c = 0.81 E xc PBE0 = 1 4 E x HF + 3 4 E x PBE + E c PBE E xc HSE = aE x HF,SR (ω)+(1− a)E x PBE,SR (ω)+ E x PBE,LR (ω)+ E c PBE a = 1 4 , ω = 0.2
9. Do hybrids work? NANO266 9 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.
10. Do hybrids work? NANO266 10 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.
11. The Jacob’s Ladder NANO266 11 http://www.sas.upenn.edu/~jianmint/Research/
12. Which functional to use? NANO266 12
13. To answer that question, we need to go back to our trade-off trinity NANO266 13 Choose two (sometimes you only get one) Accuracy Computational Cost System size
14. Accuracy of functionals – lattice parameters LDA overbinds GGA and meta GGA largely corrects that overbinding NANO266 14 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.
15. Cohesive energies LDA cohesive energies too low, i.e., overbinding Again, GGA does much better NANO266 15 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.
16. Bond lengths NANO266 16 Cramer, C. J. Essentials of Computational Chemistry: Theories and Models; 2004.
17. 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 17
18. 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 18 Ref: MIT 3.320 Lectures on Atomistic Modeling of Materials
19. bcc vs fcc in GGA NANO266 19 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.
20. Magnetism NANO266 20 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.
21. 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 21 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.
22. Reaction energies Broad conclusions •  GGA better than LSDA •  Hybrids most efficient (good accuracy comparable to highly correlated methods) NANO266 22
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