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.

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)
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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
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4. Performance of GGA
GGA tends to correct
LDA overbinding
• Better bond lengths, lattice
parameters, atomization
energies, etc.
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5. Why stop at the first derivative?
Meta-GGA
Example: TPSS functional
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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.
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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
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!)
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U values used in the Materials
Project, fitted by a UCSD
NanoEngineering Professor

8. Hybrids
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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
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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
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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?
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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.

12. 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.

13. The Jacob’s Ladder
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http://www.sas.upenn.edu/~jianmint/Research/

14. Which functional to use?
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15. 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

16. 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.

17. 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.

18. Bond lengths
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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.
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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
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Ref: MIT 3.320 Lectures on Atomistic Modeling of Materials

21. 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.

22. 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.

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)
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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.

24. Reaction energies
Broad conclusions
• GGA better than LSDA
• Hybrids most efficient (good
accuracy comparable to highly
correlated methods)
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25. 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.

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
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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
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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.