Density Functional Theory

Curso de Doctorado “Métodos Computacionales en Física de la Materia Condesada”, Diciembre 2003

Rubén Pérez

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 thewave 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 ApproximationHamiltonian 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

Quantum Chemistry approach to solve SE

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)

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

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(103

)( rnrt π=

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

Local approximation

GS ⇒ minimization of energy functional with the constraint:

Nrnrd =∫ )(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!!)((2) not true in spin-DFT,.. )

(1)

(2)

Hohenberg-Kohn Th.: Consequences (2)

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

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)?

Solved: any square-integrable nonnegative function satisfies it

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)

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

We know how to write KE 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

HF vs LSD

Physical meaning of the Kohn-Sham eigenvalues εi

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 εi 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.

• 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

LDA for EXC including spin (LSD)

DFT &Kohn-Sham equations including spin

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

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)

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

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

GEA , 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

(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 ther→∞ 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 !!

• 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: EXX

EXX: “Exact Exchange”

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

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

ADA, WDA

The quest for more accurate functionals... (2)

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

• Meta-GGAs: depend also on the Kohn-Sham kinetic energy for the occupied orbitals (Non-Empirical derivation): TPSS

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 methods.

Supercells

• 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

vacuum

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...

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

)r(e)r()r(v)rv(21

iiiXC2 φφ =

++∇−