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. Introduction to Density
FunctionalTheory
Shyue Ping Ong

2. Two broad approaches to solving the Schrödinger
equation
Variational
Approach
Expand wave function as a linear
combination of basis functions
Results in matrix eigenvalue problem
Clear path to more accurate answers
(increase # of basis functions,
number of clusters / configurations)
Favored by quantum chemists
Density
Functional Theory
In principle exact
In practice, many approximate
schemes
Computational cost comparatively low
Favored by solid-state community
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3. NANO266
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The birth place of DFT
> 20,000 citations each! (Web of Science, March 2015)

4. NANO266
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http://www.nature.com/news/the-top-100-papers-1.16224

5. The Hohenberg-KohnTheorems
The Hohenberg-Kohn existence theorem
• For any system of interacting particles in an external potential
Vext(r), the density is uniquely determined (in other words, the
external potential is a unique functional of the density).
The Hohenberg-Kohn variational theorem
• A universal functional for the energy E[n] can be defined in terms
of the density. The exact ground state is the global minimum
value of this functional.
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Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas, Phys. Rev., 1964, 136, B864, doi:10.1103/
PhysRev.136.B864.

6. Proof of H-KTheorem 1
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Assume there are two different external potentials, Va and Vb,
(with corresponding Hamiltonians Ha and Hb )
consistent with the same ground state density, ρ0.
Let the ground state wave function and energy for each Hamiltonian
be ψ0 and E0. From the variational theorem:
E0,a < ψ0,b Ha ψ0,b
E0,a < ψ0,b Ha − Hb ψ0,b + ψ0,b Hb ψ0,b
E0,a < ψ0,b Va −Vb ψ0,b + E0,b
E0,a < (Va −Vb )ρ0 (r)dr∫ + E0,b
Similarly,
E0,b < (Vb −Va )ρ0 (r)dr∫ + E0,a
Summing the two, we have
E0,a + E0,b < E0,b + E0,a

7. Consequence of H-K theorems
Schrodinger equation in 3N electronic
coordinates reduced to solving for electron
density in 3 spatial coordinates!
In theory, H-K theorems are exact.
Unfortunately
• No recipe for what the functional is
• In other words, beautiful theory, but practically useless (until one
year later…)
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8. A switch of notation
Electronic Hamiltonian (atomic units)
From H-K theorem, energy (and everything) is a functional
of the density. Therefore,
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H = −
1
2
∇i
2
i
∑
2
−
Zk
rikk
∑
i
∑ +
1
rijj
∑
i
∑
H = T +Vne +Vee
E[ρ(r)]= T[ρ(r)]+Vne[ρ(r)]+Vee[ρ(r)]

9. The Kohn-ShamAnsatz
Fictitious system of electrons that do not interact
and live in an external potential (Kohn-Sham
potential) such that ground-state charge density
is identical to charge density of interacting system
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E[ρ(r)]= Tni[ρ(r)]+Vne[ρ(r)]+Vee[ρ(r)]+ ΔT[ρ(r)]+ ΔVee[ρ(r)]
Non-interacting KE Corrections to non-
interacting KE and Vee

10. The Kohn-Sham Equations
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E[ρ(r)]= −
1
2
ψi
*
(r)∇2
ψi (r)dr∫ − ψi
*
(r)
Zk
rik
ψi (r)dr∫
k
∑
%
&
'
(
)
*
i
∑
+
1
2
ρ(r')
ri − r'
ψi (r)
2
dr∫∫ 'dr
i
∑
+Exc[ρ(r)]
hi
KS
= −
1
2
∇2
−
Zk
rik
+
k
∑
1
2
ρ(r')
ri − r'
dr∫∫ '+Vxc[ρ(r)]
KS one-electron operator
hi
KS
ψi (r) =εiψi (r)
ρ(r) = ψi (r)
i
∑
2

11. Solution of KS equations
Follows broadly the general concepts of HF SCF
approach, i.e., construct guess KS orbitals within
a basis set, solve secular equation to obtain new
orbitals (and density matrix) and iterate until
convergence
Key differences between HF and DFT
• HF is approximate, but can be solved exactly
• DFT is formally exact, but solutions require approximations (Vxc)
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12. Flowchart for KS
solution
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13. Limits of KSTheory
Eigenvalues are not the energies to add / subtract electrons, except
the highest eigenvalue in a finite system is the negative of the
ionization energy.
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Silicon bandstructure from
www.materialsproject.org
Exp bandgap: 1.1eV
But KS orbitals and energies can
be used as inputs for other many-
body approaches such as quantum
Monte Carlo.

14. Exchange-Correlation
Thus far, we have constructed an elegant system, but we
have convenient swept all unknowns into the mysterious
Vxc. Unfortunately, the H-K theorems provide no guidance
on the form of this Vxc. With approximate Vxc, DFT can be
non-variational.
What’s the simplest possible assumption we can make?
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E[ρ(r)]= Tni[ρ(r)]+Vne[ρ(r)]+Vee[ρ(r)]+Vxc[ρ(r)]

15. Local DensityApproximation (LDA)
Independent particle kinetic energies and long-range Hartree
contributions have been separated out => Remaining xc term can be
reasonably approximated (to some degree) as a local or nearly local
functional of density
LDA: XC energy is given by the XC energy of a homogenous electron
gas with the same density at each coordinate
With spin (LSDA):
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Exc
LDA
[ρ]= ρ(r) εx
hom
(ρ)+εc
hom
(ρ)!" #$∫ dr
Exc
LSDA
[ρ↑
,ρ↓
]= ρ(r) εx
hom
(ρ↑
,ρ↓
)+εc
hom
(ρ↑
,ρ↓
)#$ %&∫ dr

16. LDA,contd
For a homogenous electron gas (HEG), the exchange energy can be
analytically derived as:
Correlation energy for HEG has been accurately calculated using
quantum Monte Carlo methods
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Ex
σ
[ρ]= −
3
4
6
π
ρσ"
#
$
%
&
'
1/3

17. Does LDA work?
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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.
Over-binding
evident
Radial density of the Ne atom, both exact and
from an LDA calculation

18. Error in LDA xc energy density of Si
Exchange energies are too low and correlation energies that are too
high => Cancellation of errors!
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Hood, R.; Chou, M.; Williamson, a.; Rajagopal, G.; Needs, R. Exchange and correlation in silicon,
Phys. Rev. B, 1998, 57, 8972–8982, doi:10.1103/PhysRevB.57.8972.

19. Phases of Si from LDA – an early success story
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Yin, M. T.; Cohen, M. L. Theory of
static structural properties, crystal
stability, and phase
transformations: Application to Si
and Ge, Phys. Rev. B, 1982, 26,
5668–5687, doi:10.1103/
PhysRevB.26.5668.