NANO266 - Lecture 5 - Exchange-Correlation Functionals
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.
Published on: Mar 3, 2016
Transcripts - NANO266 - Lecture 5 - Exchange-Correlation Functionals
Shyue Ping Ong
LDA uses local density ρ from homogenous
Next step: Let’s add a gradient of the density!
Generalized gradient approximation (GGA)
]= drρ(r)εxc (∫ ρ↑
Unlike the Highlander,there is more than“one”
• 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
• PBEsol, 2008: Revised PBE for solids
Performance of GGA
GGA tends to correct
• Better bond lengths, lattice
Why stop at the ﬁrst derivative?
Example: TPSS functional
]= drρ(r)εxc (∫ 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.
• 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.
(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 +
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!)
U values used in the Materials
Project, fitted by a UCSD
Chimera from God of War
(memories of times when I was still a carefree graduate student)
Rationale for Hybrids
Semi-local DFT suffer from the dreaded self-
• Spurious interaction of the electron not completely cancelled with
ρi (ri )ρj (rj)
ρi (ri )ρj (rj)
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
• Originally fitted from a set of atomization energies, ionization potentials, proton affinities and
total atomic energies.
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
+ ao (Ex
)+ ax (Ex
where a0 = −0.20, ax = 0.72, ac = 0.81
, ω = 0.2
Do hybrids work?
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?
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/
The Jacob’s Ladder
Which functional to use?
To answer that question,we need to go back to
our trade-off trinity
you only get
Accuracy of functionals – lattice parameters
GGA and meta GGA
largely corrects that
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,
energies too low, i.e.,
Again, GGA does
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.
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
• 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
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
Ref: MIT 3.320 Lectures on Atomistic Modeling of Materials
bcc vs fcc in GGA
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.
Wang, L.; Maxisch, T.; Ceder, G. Oxidation
energies of transition metal oxides within the
GGA+U framework, Phys. Rev. B, 2006, 73,
Atomization energies,ionization energies and
Carried out over G2 test set of molecules (note that PBE1PBE in the
tables below refers to the PBE0 functional)
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.
• GGA better than LSDA
• Hybrids most efficient (good
accuracy comparable to highly
Some well-known problems can be addressed by
judicious ﬁtting to experimental data
Wang, L.; Maxisch, T.; Ceder, G. Oxidation energies
of transition metal oxides within the GGA+U
framework, Phys. Rev. B, 2006, 73, 195107, doi:
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
analysis using the
Materials Project, again
done by a UCSD
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
(1) Kuisma, M.; Ojanen, J.; Enkovaara, J.; Rantala, T. T.
Phys. Rev. B, 2010, 82, doi:10.1103/PhysRevB.
(2) Chan, M.; Ceder, G. Phys. Rev. Lett., 2010, 105,