Density Functional Theory

Rubén Pérez

Departamento de Física Teórica de la Materia Condensada,

“Métodos Computacionales en Física de la Materia Condensada” Master en Física de la Materia Condensada y Nanotecnología

Curso 2007/8

Departamento de Física Teórica de la Materia Condensada,

Universidad Autónoma de Madrid, Spain

ruben.perez@uam.es

Outline

• Motivation: limitations of the standard approach based on the

wave function.

• The electronic density n(r) as the key variable: Functionals &

Thomas-Fermi theory.

• Density functional theory (DFT): Hohenberg-Kohn (HK) Theorem

• Kohn-Sham equations.

• Total Energy

• Interpretation of the KS eigenvalues

• Exchange-Correlation functional:

• Local density approximation (LDA): Limitations.

• Generalized Gradients Approximations (GGA) and beyond.

•Making DFT practical: Basis sets, Supercells and K-sampling

Motivation

Goals

Evolution of Simulation Methods

(after E. Wimmer, 1998)

Born-Oppenheimer Approximation

Hamiltonian for M nuclei

and N electrons

Mα >> m ⇒ ionic and

(much faster) electronic

motions can be decoupled

• electrons relaxed to GS for a given ionic configuration.

• nuclei move in a potential given by electronic GS energy

Time-independent Schrodinger equation (SE)

Hamiltonian for an N-electron system: Whether it is an

atom, a molecule or a solid depends only on v(ri)

Two possible strategies: direct solution or minimization

Quantum Chemistry approach to solve SEfocus on systematic improvement of the wave function |Ψ>

• Hartree-Fock (HF)

|Ψ> is a Slater determinant

HF includes all the exchange effects

• Configuration Interaction (CI) : includes correlation

The general theory of quantum mechanics is now almost complete. The underlying

physical laws necessary for the mathematical theory of a large part of physics and

the whole of chemistry are thus completely known, and the difficulty is only that the

exact application of these laws leads to equations much too complicated to be

soluble. (Dirac,1929)

• Configuration Interaction (CI) : includes correlation

|Ψ>: linear combination of Slater determinants

Problem: factorial growth of the number of

determinants with the number of electrons.

Condensed Matter Physics Approach: the density n(r) as the key variable

QC

DFT

• DFT provides a viable alternative, less accurate perhaps, but more versatile

• DFT recognizes that systems differ only by their potential v(r) and provides a prescription to deal with T and Vee ⇒ maps the many-body problem with Vee onto a single-body problem without Vee .

• Knowledge of n(r) implies knowledge of Ψ and v(r), and hence of all other observables.

Practical implementation?

Thomas-Fermi theory for atoms (1927-28)

35322 )()3(10

3)( rnrt π=

Kinetic energy density in a uniform electron gas with n = n(r)

Local approximation

GS ⇒ minimization of energy functional with

the constraint:

�rnrd =∫ )(µ = chemical potential (- µ =electronegativity)

• basic description of charge density & electrostatic potential.

• It does not reproduce the atom shell structure !!!

DFT basics: Hohenberg-Kohn Theorem

Practical DFT : Kohn-Sham Equations

“...We do not expect an accurate description of chemical bonding with the Local Density Approximation (LDA)...” (Kohn & Sham, 1965)

• LDA: structural predictive power

(e.g. transition pressure ZB→β-Sn in Si).

• GGA: not too far from chemical accuracy

(1kcal/mole = 0.0434 eV/atom)

(W. Kohn & J. Pople)

Nobel Prize in

Chemistry 1998

The first convincing DFT-LDA calculation

M.T. Yin & M. Cohen,

PRL 45, 1004 (1980)

DFT : Conceptual & Practical Advantages

R.P., M.C. Payne & A.D. Simpson, PRL 75, 4748 (1995)

Hohenberg-Kohn Theorem (1)

Hohenberg-Kohn Theorem (2)

Hohenberg-Kohn Th.: Consequences (1)

(and, thus, the excited states!!)

(1)

(2)

((2) not true in spin-DFT,.. )

Hohenberg-Kohn Th.: Consequences (2)

Fundamental equation in DFT : Minimization of E [n] Fundamental equation in DFT : Minimization of Ev0[n]

with the normalization constraint

Some subtleties...

• How do I know, given an arbitrary function n(r), that it is

a density coming from an antisymmetric N-body wave

function Ψ(r1,...,rN)? N-representability

Solved: any square-integrable nonnegative

function satisfies it

V-representability

• How do I know, given an arbitrary function n(r), that it is

the ground state density of a local potential v(r)?

Not so simple: Constrained-search formalism

(Levy-Lieb), but unicity of the potential is lost.

Thomas-Fermi vs Hartree: A hint...

Thomas-Fermi (1927-28) Hartree (1928)

Hartree describes GS of atoms

much better than TF

(reproduces the shell structure)

Kinetic energy for independent electrons

{ })()(

1111 �rr φφ L

),,(2

1),,(),,(ˆ),,( 1

1

2

111 �

�

i

i���rrrrrrTrr LLLL ψψψψ ∑

=

∇−=

Independent

General expression for the Kinetic energy

{ })()(

!

1)(det,,(

1

111

1

���

�

i�

rr�

rrr

φφ

φψ

L

MOML ==Independent

electrons

)(2

1)(),,(ˆ),,(

1

2

11 rrrrTrri

�

i

i��φφψψ ∑

=

∇−=LL

DFT as an effective single-body theory: Kohn-Sham equations

We know how to relate KE and the density for a non-interacting system!!

{ })(det),,( 1 rrri�

φψ =L

density for a non-interacting system!!

What is the local effective potential?

Kohn-Sham equations

Kohn-Sham equations: Remarks

Total Energy in the Kohn-Sham scheme

+ Eion-ion

+ Eion-ion

HF vs LSD

Physical meaning of the Kohn-Sham

eigenvalues εεεεi KS?

• εi KS are only Lagrange parameters to fullfill the orthogonalithy

constraints of the φi(r) orbitals. Only n(r) has a physical meaning!!.

• BUT, in many situations, εi KS are empirically a good

approximation to the real spectrum (band structure calculations). (Implies taking KS eq as an approximation to real many-body SE ⇒ DFT as a mean-field theory (not a rigorous many-body theory)

Can εi KS be interpreted as excitation energies? (the energy necessary to

remove or add an electron –e.g. what it is measured in photoemission—).

Hartree-Fock: Koopmans Theorem

(assuming that rest of orbitals due not change significantly when the occupation changes)

Physical meaning of the εεεεi KS?

DFT: Koopman’s is not valid; instead Janak’s theorem:

εi KS are not excitation energies; only exception: highest occupied eigenvalue.

εNKS (N) = - I ; I ≡ Ionization energy of the N-body system

εN+1KS (N+1) = - A ; A ≡ electron affinity of the N-body system

• Only valid for the exact Exc functional; test for approximate functionals. (B3LYP works very well)

• works better with extended states; problems with localized states.

I & A can be rigorously calculated as total-energy differences:

I = E0(N-1) – E0(N) A = E0(N) – E0(N+1)

(E0(N) = ground-state energy of the N-body system)

Excitation energies: Dyson’s equation

Comparison of Σ and VXC for the uniform electron gas

Making DFT practical: Approximations

• Building the Exc functional.

• Solution of the Kohn-Sham equations:

Basis set to expand the Kohn-Sham Orbitals.

• Using Bloch´s theorem: Supercells and K-sampling

• Effective implementations for large systems:

Car-Parrinello approach and iterative minimization Car-Parrinello approach and iterative minimization

methods.

Supercells

vacuum

• Artificial periodicity of the unit cell that contains the aperiodic

configuration we want to study (molecules, defects, surfaces,...)

• “vacuum”: avoids the overlap of wavefunctions in neighbouring cells.

• charged or dipolar systems: electrostatic interaction among the images must be corrected (classical multipolar expansion)

Molecule Defect Surface

K-point sampling

H. J. Monkhorst and J. D. Pack PRB 13, 5188 (1976); 16, 1748 (1977)

J. Moreno and J. M. Soler PRB 45, 13891 (1992)

DFT implementations: a quick reminder...

1

(After E. Wimmer, Journal of Computer-Aided Materials Design 1, 215 (1993))

)r(ε)r()r(v)rv(2

1iiiXC

2 φφ =

++∇−

Making DFT practical: Approximations

• Building the Exc functional.

• Local density approximation (LDA)

• Generalized Gradient approximation (GGA)

• Hybrid functionals (including exact exchange)

• meta-GGA functionals (including KE)

• Solution of the Kohn-Sham equations:

Basis set to expand the Kohn-Sham Orbitals

• Effective implementations for large systems:

Car-Parrinello approach and iterative minimization

methods.

Local density approximation (LDA) for EXC

Local density approximation (LDA) for EXC

LDA for EXC including spin (LSD)

J.P. Perdew & A. Zunger, Phys. Rev. B 23, 5048 (1981)

LDA exchange energy

Simple argument: spherical hole of constant depth n/2 around the electron

n/2Rx

Vx atractive due to

the e- charge deficit

LDA correlation energy

DFT &Kohn-Sham equations including spin

LSD: performance

• EX : 5% smaller ; EC : 100% larger (EXC << T, VH, Vne ; but EXC ∼ 100% bonding energy)

• Cohesive (atomization) energies: 15% larger (∼ 1.3 eV overbinding)

• bond lengths: 1% smaller ; bulk moduli (elastic constants) 5 %

• Favors close-packed structures

• Energy barriers: 100% too low (no “chemical accuracy”)

• wrong description of magnetic systems: Fe LDA is fcc paramagnetic (exp: BCC ferromagnetic)

• Poor description of weak bonding (van der Waals, hydrogen bonds).

• Atoms & Clusters

• VXCLSD : exponetial decay with – n(r)**(1/3) instead of -1/r

• negative ions: generally unstable (electron affinities: 20% error)

Beyond LDA: Gradient expansions (GEA)

EXC : some rigorous results...

HK

g(r,r’)≡ pair correlation function

Vee-VH due to charge fluctuations and the self-interaction correction

Relating G[n] to TS[n]. XC-hole (nXC)Coupling constant integration technique relates the non-interacting (λ=0) system with the (λ=1) interacting one; gλ(r,r´) ↔ λ / |r-r’|

nXC describes the effect of e- e- repulsion: the presence of an e-

in r reduces the probability of finding another e- in r´ ⇒electronic charge defect (effective positive charge) ⇒ EXC :

coulomb interaction (attractive) between an e- and its XC-hole

Properties of the XC-hole

nX

nC

nXC = nX + nC

Fermi

hole

Coulomb hole

Why LSD works?

Exact vs LSD results

Jones & Gunnarsson, RMP 61, 689 (1989)

Generalized Gradient Approximations (GGA)

Two different strategies to determine f(n,∇n) ...

• Semiempirical (Becke): fitted to reproduce molecular results (but they fail for delocalized systems) ⇒ Chemistry (BLYP)

• Non-empirical, based on general arguments and capable of describing different types of bonding (Perdew) ⇒ Physics (PBE)

Generalized Gradient Approximation (PBE)

• Forced to retain the correct uniform electron gas limit (good aprox. to Na & Al metals, nXC of a real system) .

• Built from the nXCGEA , removing the spurious long-range parts with a

real-space cutoff, to recover the hole normalization properties.

• spin scaling:

• Satisfy constraints from scaling laws and other independent bounds

Perdew, Burke & Ernzerhof, PRL 77, 3865 (1996)

• Satisfy constraints from scaling laws and other independent bounds

(Older version: PW91; Perdew & Wang, PRB 46, 6671 (1992))

Generalized Gradient Approximation (PBE)

Generalized Gradient Approximation (BLYP)

• EX from Becke (PRA 38, 3098 (1988)): functional form without the

r→∞ divergence of the 2nd order expansion and β, γ fitted to reproduce HF atomic energies.

• EC from Lee, Yang & Parr (PRB 37, 785 (1988)): nC does not

satisfied some basic constraints.

The combination (BLYP) works extremely well

for chemical applications (empirical)

GGA (GGS) performance

• EX : 0.5% ; EC : 5% larger (LDA: EX : 5% ; EC : 100%)

• Cohesive energies: 4% larger (∼ 0.3 eV ) (LDA: 15% l (∼ 1.3 eV))

• bond lengths: 1% larger ; (LDA: 1% shorter)

• improved description of structural properties

• Energy barriers: 30% too low (LDA: 100% too low)

• magnetic systems: Fe GGA is BCC ferromagnetic !!• magnetic systems: Fe GGA is BCC ferromagnetic !!

• improved description of weak bonding (hydrogen bonds).

• Atoms & Clusters

• VXCLSD : still wrong exponential decay

• negative ions: improved electron affinities (10% error)

GGA: major improvement over LDA,

“chemical accuracy” not too far away

The quest for more accurate functionals...

• Hybrid functionals: Mixture of Hartree-Fock exchange with a DFT

exchange functional (Empirical: weight factors are optimized for certain

sets of molecules): B3LYP = B3 (Becke, JCP 98, 5648 (1993)) + LYP

B3LYP: most successful functional for chemical applications

• Orbital functionals: represented directly in terms of single-particle

orbitals instead of the density (e.g. TS[n])⇒ implicit n(r) dependence ⇒indirect approaches to minimize EXC and obtain vXC: EXXindirect approaches to minimize EXC and obtain vXC: EXX

EXX: “Exact Exchange”

M. Stadele et al, PRL 79, 2089 (1997); PRB 59 10031 (1999): Semiconductor Gap !!

• Nonlocal functionals

Self-Interaction correction: SIC (ensure EC[n]=0 , EX[n]=-EH[n] for one-e- system)

ADA, WDA

The quest for more accurate functionals... (2)• Meta-GGAs: depend also on the Kohn-Sham kinetic energy for the

occupied orbitals (Non-Empirical derivation): TPSS

Tao, Perdew, Staroverov & Scuseria, PRL 91, 146401 (2003))

More Tests: Molecules (JCP, 119, 12129 (2003)), Solids (PRB 69, 075102 (2004))

A multi-scale problem: Combination of classical and Quantum-mechanical

methods…

Are electronic structure calculations useful?

… Reuter and his colleagues show just how close science has come to using ab initio

theoretical methods to calculate net catalytic reaction rates on complex solid surfaces. They

have applied an elegant method that they call “ab initio statistical mechanics”, which involveshave applied an elegant method that they call “ab initio statistical mechanics”, which involves

two stages — first, using first-principles quantum mechanics to calculate the activation barriers

and transition-state vibrational frequencies for all the relevant elementary surface reactions;

second, to couple these through statistical mechanical methods involving transition-state theory

and kinetic Monte Carlo simulations of the reaction process.

• Vibrational frequencies … are used to incorporate entropy considerations …

• Kinetic Monte Carlo …, allowing efficient sampling of the tremendous range of different timescalesnecessary to describe all the different elementary steps, and thus simulate the kinetics of the whole system.

News & Views, Nature 432, 282 (2004)

…In this way, Reuter et al. were able to calculate net catalytic reaction rates on

solid surfaces under conditions similar to those used in industrial processes.

• Density-functional theory (DFT) for energetics of all relevant processes: motion

of the gas-phase molecules, dissociation, adsorption, surface diffusion, surface

chemical reactions, and desorption.

• Statistical mechanics problem solved by kinetic Monte Carlo ⇒ Narrow region of

highest catalytic activity: Kinetics builds an adsorbate composition that is not found

anywhere in the thermodynamic surface phase diagram.

K. Reuter et al, PRL 93, 116105 (2004)

Are electronic structure calculations useful?Rate of CO2 formation at T=350 K .

Unexpected agreement: Experiment

(dotted) vs theory (solid) !!!!

Steady-state surface structures

(T=600 K) and map of the turnover

frequencies (TOFs) in cm-2 s-1 .

The quantum method used by Reuter et al. is a version of density functional

theory (DFT) that achieves nearly state-of-the-art accuracy in predicting the

energies of such systems. Nevertheless, the authors admit that this can still be

off by as much as 30 kJ per mol in estimating activation barriers. Given this

drawback, it is surprising that they were able to achieve such impressive

accuracy in their calculated carbon monoxide oxidation rates over RuO2 .They

attribute this agreement with experimental rates to an effect they imply is

generic to catalytic reactions — the combined action of many elementary steps

that simultaneously affect the rate. News & Views, Nature 432, 282 (2004)

References• R.M. Martin. “Electronic Structure: Basic Theory and Practical Methods”.

(Cambridge University Press, Cambridge, 2004).

• W. A. Harrison. “Electronic Structure and The Properties of Solids”. (Dover, New York, 1989).

• A. P. Sutton. “Electronic Structure of Materials”. (Clarendon, Oxford, 1993).

• K. Capelle. “A Bird’s-Eye View of Density Functional Theory”. arXiv:cond-• K. Capelle. “A Bird’s-Eye View of Density Functional Theory”. arXiv:cond-mat/0211443 v5 (2006).

• M. C. Payne et al. “Iterative minimization techniques for ab-initio total-energy calculations”. Review of Modern Physics 64, 1045 (1992).

• R. O. Jones & O. Gunnarson. “The density functional formalism, its application and prospects”. Review of Modern Physics 61, 689 (1989).