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

From any observable:

degrees of freedom:

COST

Note: only electronic degrees of freedom; Born-Oppeheimer approximation

al i t n e n o p x E wall s n o tr c e l #e

Challenge: develop efficient and accurate methods to achieve that goal

Introduction to DFT - Myrta Grüning

Use simpler quantities than wavefunction : all info's about the system: do I really need that?!? reduce # degree of freedom by averaging out information I do not need e.g. density matrices

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 NOTE:

diagonal two-particle density matrix one-particle density matrix electron density

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? Hohenberg-Kohn (1964) 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: Introduction to DFT - Myrta Grüning

Can we use the density for calculations? Ground state energy is a density functional:

one-to-one correspondence + Ritz variational principle:

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? Electronic structure approach whose key quantity is the density based on the Hohenberg-Kohn theorem (ensures that many-particle system in its GS is fully characterized by its GS density)

to calculate GS properties of a system minimizes

Introduction to DFT - Myrta Grüning

Rewrite the total energy as functional of n... We need: We have:

Challenge: Fit many-particle intricacies in such simple object as the density Introduction to DFT - Myrta Grüning

Critical approximation is the kinetic energy Thomas-Fermi (1927)

+ Achievements: qualitative trends for atoms

- Problems: at best qualitative, no chemical bonding i.e.: separate

always lower energy than

Introduction to DFT - Myrta Grüning

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)

Introduction to DFT - Myrta Grüning

Kohn-Sham equations Kohn-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-consistency 2. basis set expansion to get an algebraic problem

Introduction to DFT - Myrta Grüning

Solution through self-consistency

GUESS DENSITY

CALCULATE KS POTENTIAL SOLVE KS EQUATIONS

CHECK CRITERIA

CALCULATE ENERGY/NEW DENSITY

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 Localized basis sets Possible choices:

e.g.: Gaussians, Slater

Delocalized basis sets Plane-waves

Introduction to DFT - Myrta Grüning

Periodic crystals are described in terms of: Direct, real space

Crystal

Primitive Lattice vectors

Unit cell Basis

Translations

Fourier trasform Reciprocal, momentum, k-space translation respresented by

Primitive reciprocal Lattice vectors 1st Brillouin zone:

Reciprocal lattice vectors: with

Wigner-Seitz primitive cell in k-space

with

Introduction to DFT - Myrta Grüning

Eigenvalue/functions of electrons in a crystal Bloch functions Eigenfunctions have symmetry of the system

with Band structure

E(k)

with k varying over the whole Brillouin zone (# k in 1BZ = # unit cells)

Introduction to DFT - Myrta Grüning

k

Planewave basis set Expand: with

Diagonalize:

Matrix elements:

pseudopotential! 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 with

b. number & density of k points used to sample k-space I need to evaluate integrals of the type (e.g. for the charge density)

numerically on a discrete (uniform) grid as:

Introduction to DFT - Myrta Grüning

DFT with PWs in practice: IN: System:

OUT: Physical quantities

unit cell lattice vectors basis

(physical approx.)

xc-approximation (relativistic effects)

Numerical approx: energy cut-off k-points grid pseudopotentials SCF procedure/threshold

total energy and components any GS observable (in principle)

GUESS DENSITY

CALCULATE KS POTENTIAL SOLVE KS EQUATIONS

CHECK CRITERIA

CALCULATE ENERGY/NEW DENSITY

1-particle quantities Kohn-Sham 1-p wavefunctions Kohn-Sham 1-p energies

RUN: Solve KS equations

Energy (eV)

Hamiltonian:

density and related quantities

L

Z

A

Introduction to DFT - Myrta Grüning

L

D

A

Connection KS and physical quantities? 15

"looks like" experimental results

LDA Band Gap (eV)

Energy (eV)

Bandgap problem:

but bandgap systematically too small

10

5

0 0

L

Z

A

L

5

10

15

Experimental Band Gap (eV) D

A

Is it just a xc problem?

Δ

εc

NO: εv

−A −I

KS bandgap

Energy gap

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 approach:

1-p model:

cost:

Hartree

independent particles

N3

Kohn-Sham

independent particles in effective potential

N3

HartreeFock

independent fermions

N4

generalized Kohn-Sham

partially interacting particles in effective potential

N4

Introduction to DFT - Myrta Grüning

References&material&further reading A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram 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 theory Ph.D thesis - Robert van Leeuwen Electronic Structure of Matter – Wave Functions and Density Functionals W. Kohn - Nobel Lecture, January 28, 1999

Introduction to DFT - Myrta Grüning