Introduction to density functional theory - YAMBO · PDF fileIntroduction to DFT - Myrta...
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Introduction to DFT - Myrta Grüning Introduction to density functional theory What is density functional theory How does it work in practice How does DFT fit in this workshop
Transcript of Introduction to density functional theory - YAMBO · PDF fileIntroduction to DFT - Myrta...
Introduction to DFT - Myrta Grüning
Introduction to density functional theory
What is density functional theory
How does it work in practice
How does DFT fit in this workshop
Introduction to DFT - Myrta Grüning
Challenge of electronic structure problem
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
Introduction to DFT - Myrta Grüning
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?!?
Introduction to DFT - Myrta Grüning
Examples of density matrices
diagonal two-particle density matrix
electron density
one-particle density matrix
Introduction to DFT - Myrta Grüning
Examples of density matrices
diagonal two-particle density matrix
one-particle density matrix
electron density
NOTE:
Introduction to DFT - Myrta Grüning
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
Introduction to DFT - Myrta Grüning
Which information does the density contain?
cusps: nuclear positions & charges
Space integration: total number of electrons
Number of electrons, nuclear position/charges uniquely define the Hamiltonian(*)
Once the Hamiltonian is known, we can in principle SOLVE IT!
Introduction to DFT - Myrta Grüning
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)
Introduction to DFT - Myrta Grüning
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 computational method to calculate GS properties of a system
Introduction to DFT - Myrta Grüning
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
Introduction to DFT - Myrta Grüning
Rewrite the total energy as functional of n...
Challenge:Fit many-particle intricacies in such simple object as the density
We need:
We have:
Introduction to DFT - Myrta Grüning
Critical approximation is the kinetic energy Thomas-Fermi (1927)
i.e.: separate always lower energy than
Problems: at best qualitative, no chemical bonding -Achievements: qualitative trends for atoms +
Introduction to DFT - Myrta Grüning
Get large part of T via non-interacting systemKohn-Sham (1965)
Physical system (N-body problem)
Kohn-Sham system (N X 1-body problems)
Introduction to DFT - Myrta Grüning
Kohn-Sham equationsKohn-Sham (1965)
Defining the exchange-correlation energy functional
+ applying Hohenberg-Kohn II ( minimize E) for both systems:
{
Introduction to DFT - Myrta Grüning
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
{ }
Introduction to DFT - Myrta Grüning
Solution through self-consistency
GUESSDENSITY
CALCULATEKS POTENTIAL
SOLVE KSEQUATIONS
CALCULATEENERGY/NEW DENSITY
CHECKCRITERIA
Introduction to DFT - Myrta Grüning
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 setsPlane-waves
Introduction to DFT - Myrta Grüning
Periodic crystals are described in terms of:
Crystal Unit cellPrimitive Lattice vectors
Basis
Reciprocal lattice vectors:
Direct, real space
1st Brillouin zone:
Fourier trasformReciprocal, momentum, k-space
Translations
Primitive reciprocalLattice vectors
translation respresented by
with
with
Wigner-Seitz primitive cell in k-space
Introduction to DFT - Myrta Grüning
Eigenvalue/functions of electrons in a crystal
Eigenfunctions have symmetry of the systemBloch functions
Band structure
with
with k varying over the whole Brillouin zone(# k in 1BZ = # unit cells)
k
E(k)
Introduction to DFT - Myrta Grüning
Planewave basis set
pseudopotential!
Expand:with
Diagonalize:
Matrix elements:
Introduction to DFT - Myrta Grüning
You need to "converge" wrt these 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
You need to "converge" wrt these parameters
with
Introduction to DFT - Myrta Grüning
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-approximation(relativistic effects)
unit celllattice vectorsbasis
energy cut-offk-points gridpseudopotentialsSCF procedure/threshold
density and related quantitiestotal energy and componentsany GS observable (in principle)
Kohn-Sham 1-p wavefunctionsKohn-Sham 1-p energies
Energ
y (
eV
)
L LZ A D A
Introduction to DFT - Myrta Grüning
Connection KS and physical quantities?
Energ
y (
eV
)
L LZ A D A
10
0
5
10
15
0 5 15
LD
A B
an
d G
ap
(eV
)
Experimental Band Gap (eV)
Bandgap problem:
"looks like"experimental
results
but bandgapsystematically
too small
Is it just a xc problem?
NO:
v
c
ε
ε−A
−IKS bandgap Energy gap
Δ
Introduction to DFT - Myrta Grüning
Starting point for excitations in MB system
n-particle propagator =
n-particle propagator in the independent particle system
+Term that brings in
correlation
Introduction to DFT - Myrta Grüning
Choices for 1-particle wavefunction/energy
1-p model:independent
particlesHartree
independentfermions
independentparticles in
effective potential
Hartree-Fock
Kohn-Sham
partially interactingparticles in
effective potential
generalizedKohn-Sham
cost:approach:
N3
N3
N4
N4
Introduction to DFT - Myrta Grüning
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 TheoryEds. C. Fiolhais F. Nogueira M. Marques (2003) Springer-Verlag Berlin Heidelberg
Density Functional Theory - An Advanced CourseEberhard 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
