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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
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 !!!
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
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!!
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)
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.
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
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)
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
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 (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))
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).
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