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Introduction to DensityFunctional Theory
# electronsCO
ST Exponential
wall
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Goal: determine material properties directly from fundamental equations
Challenge: develop efficient andaccurate methods to achieve that goal
# electronsCO
ST Exponential
wall
any observable:
From
degrees of freedom: Note: only electronic degrees of freedom; Born-Oppeheimer approximation
Challenge of electronic structure problem
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Use simpler quantities than wavefunction
reduce # degree of freedom by averaging out information I do not need
e.g. density matrices
: all info's about the system: do I really need that?!?
The simplest of them all: density
electron density: just 3 degrees of freedom!
?Which information is contained in the density
?Can we use the density to calculate materials properties
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Which information does the density contain?
Formal proof: H-K theorem (reductio ad absurdum)
one-to-one correspondence:
ground state unique, universal functional of the density:
any ground state observable is a density functional:
Hohenberg-Kohn (1964)
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Can we use the density for calculations?
one-to-one correspondence + Ritz variational principle:
Ground state energy is a density functional:
With this minimum principle we can develop a computationalmethod to calculate GS properties of a system
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What is density-functional theory?
based on the Hohenberg-Kohn theorem
Electronic structure approach whose key quantity is the density
to calculate GS properties of a system
(ensures that many-particle system in its GS is fully characterized by its GS density)
minimizes
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Note: HK theorem has clear limits
It considers: * nondegenerate* nonmagnetic (only scalar potential!), spinless (only spin unpolarized!)* ground-states
z
y
x
S
L
SL = +
SN
Extensions: spin-density, current-density, B-density, density-polarization von Barth-Hedin 1972, Vignale-Rasolt 1988, Grayce-Harris 1994, Gonze-Ghosez-Godby 1995....list is not complete
spin-orbit coupling external magnetic field
++ + + + +
systems with no ground-state(anti)ferromagneticity
Not described*:
* for B=0, m = m[n] but unknown!
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Generalization to spin-polarized systems
2x2 electron density:
spin-dependent 1-e potential: 2x2 Hermitian matrices
e.g.: scalar potential + magnetic field/magnetic moment coupling (zero if no B ext)scalar potential + spin-orbit coupling (relativistic effects)
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Can HK theorem be generalized to 2x2 case?
Similar proof to H-K theorem (reductio ad absurdum)
NO one-to-one correspondence:
ground state unique, universal functional of the 2x2 density:
any ground state observable is a spin-density functional:
von Barth-Hedin (1972)
X
+variational principle
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How do we put the HK theorem in practice?
Challenge:
Fit many-particle intricacies in such simple object as the density
We need:
We have:
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Get large part of T via non-interacting system
Kohn-Sham (1965)
Physical system (N-body problem)
Kohn-Sham system (N X 1-body problems)
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Kohn-Sham equations
Kohn-Sham (1965)
Defining the exchange-correlation energy functional
+ applying Hohenberg-Kohn II ( minimize E) for both systems:
{
Exchange-correlation energy functional defined in the same way
but dealing with a 2x2 external potential, thus xc potential
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von Barth-Hedin (1972)
Spin-polarized Kohn-Sham equations
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How do we approximate the xc functional
Introduction to DFT - Myrta Grüning
How to solve the KS equations in practice
{Nonlinear, integro-differential equations
1. solution through self-consistency2. basis set expansion to get an algebraic problem
{ }
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Solution through self-consistency
GUESSDENSITY
CALCULATEKS POTENTIAL
SOLVE KSEQUATIONS
CALCULATEENERGY/NEW DENSITY
CHECKCRITERIA
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Basis set expansion to get algebraic problem
Expansion in a convenient basis set
Hamiltonian (overlap) matrix elements
Solve (generalized) eigenproblem
Possible choices:Localized basis sets
e.g.: Gaussians, Slater
Delocalized basis sets Plane-waves
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Periodic crystals are described in terms of:
Crystal
Unit cell
Primitive Lattice vectors
Basis
(for simplicity 2D example, trivially extended to 3D)
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Electrons in a crystal potential
Free-electron periodic potential
a
Ene
rgy
k
Ene
rgy
k
1/k
Plane waves
Bloch-states
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Periodic crystals are described in terms of:
Reciprocal lattice vectors:
1st Brillouin zone:
Primitive reciprocalLattice vectors
translation respresented by with
with
Wigner-Seitz primitive cell in k-space
Direct, real spaceFourier trasform
Reciprocal,momentum, k-space
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Planewave basis set
pseudopotential:
Expand:
Diagonalize:
Matrix elements:
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You need to "converge" wrt parameters
a. energy cutoff used to define the size of the planewave basis set
I need to evaluate integrals of the type(e.g. for the charge density)
numerically on a discrete (uniform) grid as:
b. number & density of k points used to sample k-space
with
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Choice of the Hamiltonian
Schrödinger Hamiltonian is not 'exact': relativistic effects (QED/Dirac)
Visible relativistic effects (splitting of order of tenths of eV) from third-row semiconductors: spin-orbit coupling
Example optical absorption in GaSb
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?
Hamiltonian with spin-orbit coupling
What changes to include spin-orbit coupling:
where Pauli matrices:
2x2 external potential : 4-density (noncollinear spin formalism) we considered earlier
Main differences with spinless approach:
(spinor)
spinless: non-collinear spin:
full-relativistic pseudopotentials
4-vxc (or 2x2 ) xc approximations e.g. spin-LDA
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DFT with PWs in practice:
System:
Hamiltonian: (physical approx.)
Numerical approx:
Physical quantities
1-particle quantities
GUESSDENSITY
CALCULATEKS POTENTIAL
SOLVE KSEQUATIONS
CALCULATEENERGY/NEW DENSITY
CHECKCRITERIA
Solve KS equations
IN: OUT:
RUN:
xc-approximationso-interaction(relativistic effects)
unit celllattice vectorsbasis
energy cut-offk-points gridpseudopotentialsSCF procedure/threshold
density and related quantities
total energy and components
any GS observable (in principle)
Kohn-Sham 1-p wavefunctions
Kohn-Sham 1-p energies
Energ
y (
eV
)
L LZ A D A
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Connection KS and physical quantities?
Can be related to the fundamental band gap?
Ek
e-
Ei
Ef
hv
e-Ei
Evac
Efhv
Ek
EF
e-
e-
with (charged excitation)
Connection KS and physical quantities?
Can be related to neutral excitations of the systems?
hv
Ei
Ef
EF
hv
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In conclusions:
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1-p model:independent
particles Hartree
independentfermions in
effective potential
independentfermions
Hartree-Fock
partially interactingparticles in
effective potential
generalizedKohn-Sham
cost:approach:
N3
N3
N4
N4
Kohn-Sham
The Kohn-Sham energies/wavefunctions do not have a precise physical meaning
However KS is a privileged systems of independent particles:
* Reproduce (in principle) GS density * with relatively low computational effort
That is why it is often used as starting point for many-body
perturbation theory calculations
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References&material&further reading
A Chemist’s Guide to Density Functional Theory. Second EditionWolfram Koch, Max C. Holthausen (2001) Wiley-VCH Verlag GmbH
A Primer in Density Functional Theory
Eds. C. Fiolhais F. Nogueira M. Marques (2003) Springer-Verlag Berlin Heidelberg Density Functional Theory - An Advanced Course
Eberhard Engel · Reiner M. Dreizler (2001) Springer-Verlag Berlin Heidelberg
Kohn-Sham potentials in density functional theoryPh.D thesis - Robert van Leeuwen
Electronic Structure of Matter – Wave Functions and Density FunctionalsW. Kohn - Nobel Lecture, January 28, 1999
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1. Many-body perturbation theory calculations using the yambo code Journal of Physics: Condensed Matter 31, 325902 (2019)2. Yambo: an ab initio tool for excited state calculations Comp. Phys. Comm. 144, 180 (2009)