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