Why density functional theory
works and how to improve upon it
Kieron Burke
&
Donghyung Lee, Attila Cangi, Peter
Elliott, John Snyder, Lucian Constantin
UC Irvine Physics and Chemistry
http://dft.uci.edu
Sep 20, 2010
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Outline
a) Overview
b) Some details
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Modest statements
• The most important problem I’ve ever worked on
• Possible payoffs
– Understanding of asymptotic approximations
– Complete transformation of society
• Explains many things about many areas
– Semiclassical expansions
– DFT and approximations like Thomas-Fermi
• Ties together
–
–
–
–
Math
Physics
Chemistry
Engineering
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Insults
• Physicists
– Is it possible that your most precious elegant little theories
(e.g., many-body theory with Feynman diagrams) are a
stupid approach to electronic structure?
• Chemists
– Would you rather continue with LCSF (linear combinations
of successful functionals) or actually derive stuff?
• Applied mathematicians
– Do you want to spend the rest of your life proving things
only 6 people care about, or would you rather do
something useful?
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Electronic structure problem
What atoms, molecules, and solids exist, and what
are their properties?
May 5, 2010
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Properties from Electronic Ground State
• Make Born-Oppenheimer approximation
• Solids:
– Lattice structures and constants, cohesive energies,
phonon spectra, magnetic properties, …
• Molecules:
– Bond lengths, bond angles, rotational and vibrational
spectra, bond energies, thermochemistry, transition states,
reaction rates, (hyper)-polarizabilities, NMR, …
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Big picture
WKB
Gutwiller trace
1d or 2d
TF theory
Lieb et al
Atoms
Exact
conditions
Perdew, Levy
Empiricism
Becke, Truhlar
Modern DFT
Kohn-Sham
EXC[n↑,n↓]
ochemistry
Chemistry
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Condensed
matter physics
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Materials
science
Astrophysics,
protein folding,
soil science,…
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Thomas/Fermi Theory 1926
• Around since 1926, before QM
• Exact energy: E0 = T + Vee + V
– T = kinetic energy
– Vee = electron-electron repulsion
– V = All forces on electrons, such as nuclei and external fields
• Thomas-Fermi Theory (TF):
–
–
–
–
T ≈ TTF = 0.3 (3p)2/3∫dr n5/3(r)
Vee≈ U = Hartree energy = ½ ∫dr ∫dr’ n(r) n(r’)/|r-r’|
V = ∫dr n (r) v(r)
Minimize E0[n] for fixed N
• Properties:
– Exact for neutral atoms as Z gets large (Lieb+Simon, 73)
– Typical error of order 10%
– Teller’s unbinding theorem: Molecules don’t bind.
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Modern Kohn-Sham era
• 40’s and 50’s: John Slater began doing
calculations with orbitals for kinetic energy
and an approximate density functional for
Exc[n] (called Xα)
• 1964: Hohenberg-Kohn theorem proves an
exact E0[n] exists (later generalized by Levy)
• 1965: Kohn-Sham produce formally exact
procedure and suggest LDA for Exc[n]
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Kohn-Sham equations (1965)
 1 2

 2   v s [n]r i r    ii r 
N
nr    i r  =
ground-state density of
interacting system
2
i 1
nr'
v s r   vext r   
dr'  v xc [n]r 
r  r'
E0  TS  V  U  EXC [n]
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Exc
v xc [n]r  
nr 
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He atom in exact Kohn-Sham DFT
Everything
has (at
most) one
KS potential
n(r )
vS (r )
Dashed-line:
EXACT KS potential
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Recipe for exact Exc[n]
• Given a trial density n(r)
– Find the v(r) that yields n(r) for interacting electrons
– Find the vs(r) that yields n(r) for non-interacting
electrons
– Find vH(r) (easy)
– vxc(r)=vs(r)-v(r)-vH(r)
– Can also extract Exc=E-Ts-V-U
• Much harder than solving Schrödinger equation.
• In fact, QMA hard (Schuch and Verstraete. Nature Physics, 5, 732 (2009).)
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Local (spin)density approximation
• Write Exc[n]=∫d3r exc(n(r)), where exc(n) is XC
energy density of uniform gas.
• Workhorse of solid-state physics for next 25
years or so.
• Uniform gas called reference system.
• Most modern functionals begin from this, and
good ones recover this in limit of uniformity.
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Subsequent development
• Must approximate a small unknown piece of the
functional, the exchange-correlation energy Exc[n].
• 70’s-90’s: Much work (Langreth, Perdew, Becke, Parr)
going from gradient expansion (slowly-varying
density) to produce more accurate functionals, called
generalized gradient approximations (GGA’s).
• Early 90’s:
– Approximations became accurate enough to be useful in
chemistry
– 98 Nobel to Kohn and Pople
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Commonly-used functionals
• Local density approximation (LDA)
– Uses only n(r) at a point.
• Generalized gradient approx (GGA)
– Uses both n(r) and |nr)|
– Should be more accurate, corrects overbinding of
LDA
– Examples are PBE and BLYP
• Hybrid:
– Mixes some fraction of HF
– Examples are B3LYP and PBE0
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Too many functionals
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Functional approximations
• Original approximation to EXC[n] : Local density
approximation (LDA)
• Nowadays, a zillion different approaches to
constructing improved approximations
• Culture wars between purists (non-empirical) and
pragmatists.
• This is NOT OK.
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Modern DFT development
It must
have
sharp
steps for
stretched
bonds
May 5, 2010
It keeps
H2 in
singlet
state as
R→∞
Pitt1
It’s tail
must
decay
like -1/r
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Things users despise about DFT
• No simple rule for reliability
• No systematic route to improvement
• If your property turns out to be inaccurate,
must wait several decades for solution
• Complete disconnect from other methods
• Full of arcane insider jargon
• Too many functionals to choose from
• Can only be learned from another DFT guru
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Things developers love about DFT
• No simple rule for reliability
• No systematic route to improvement
• If a property turns out to be inaccurate, can
take several decades for solution
• Wonderful disconnect from other methods
• Lots of lovely arcane insider jargon
• So many functionals to choose from
• Must be learned from another DFT guru
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Difference between Ts and EXC
• Pure DFT in principle gives E directly from n(r)
–
–
–
–
Original TF theory of this type
Need to approximate TS very accurately
Thomas-Fermi theory of this type
Modern orbital-free DFT quest (See Trickey and
Wesolowsi talks)
– Misses quantum oscillations such as atomic shell
structure
• KS theory uses orbitals, not pure DFT
– Made things much more accurate
– Much better density with shell structure in there.
– Only need approximate EXC[n].
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Semiclassics in
Coulomb
potential
Kieron’s trail of tears
Bohr atoms:
Vee=0
Include
exchange
1d particles
in wavy box
V->0 at ∞
TF theory
Lieb et al
Atoms
HF atoms
Langer
uniformization
WKB
Arbitrary 3d
potential
Include
correlation
All electronic
structure
calculations
Real atoms
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Include
turning
points
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The big picture
• We show local approximations are leading
terms in a semiclassical approximation
• This is an asymptotic expansion, not a power
series
• Leading corrections are usually NOT those of
the gradient expansion for slowly-varying
gases
• Ultimate aim: Eliminate empiricism and derive
density functionals as expansion in ħ.
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Basic picture
• Turning points produce quantum oscillations
– Shell structure of atoms
– Friedel oscillations
• There are also evanescent regions
• Each feature produces a contribution to the
energy, larger than that of gradient corrections
• For a slowly-varying density with Fermi level
above potential everywhere, there are no such
corrections, so gradient expansion is the right
asymptotic expansion.
• For everything else, need GGA’s, hybrids, metaGGA’s, hyper GGA’s, non-local vdW,…
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• Many difficulties in
answering this question:
– Semiclassical methods
– Asymptotic expansions
– Boundary layer theory
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What we’ve done so far
Semiclassical
density, Elliott,
PRL 2008
Bohr atoms,
Snyder, in prep
Corrections to
local approx,
Cangi, PRB 2010
Derivation of B88,
Elliott, Can J Chem,
2009
Slowly varying
densities, with
Perdew, PRL 2006
Sep 20, 2010
Ionization in large Z
limit, Constantin,
sub. JCP, 2010
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Exact conditions on
PBEsol, Perdew et
TS, D. Lee et al, PRA,
al, PRL 2008.
2009
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A major ultimate aim: EXC[n]
• Explains why gradient expansion needed to be
generalized (Relevance of the slowly-varying electron gas to atoms, molecules, and solids J.
P. Perdew, L. A. Constantin, E. Sagvolden, and K. Burke, Phys. Rev. Lett. 97, 223002 (2006).)
• Derivation of b parameter in B88 (Non-empirical 'derivation' of
B88 exchange functional P. Elliott and K. Burke, Can. J. Chem. 87, 1485 (2009).).
• PBEsol
Restoring the density-gradient expansion for exchange in solids and surfaces J.P. Perdew, A. Ruzsinszky,
G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100, 136406 (2008))
– explains failure of PBE for lattice constants and
fixes it at cost of good thermochemistry
– Gets Au- clusters right
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Structural and Elastic Properties
Errors in LDA/GGA(PBE)-DFT computed lattice constants and
bulk modulus with respect to experiment
→ Fully converged results
(basis set, k-sampling,
supercell size)
→ Error solely due to
xc-functional
→ GGA does not outperform
LDA
→ characteristic errors of
<3% in lat. const.
< 30% in elastic const.
→ LDA and GGA provide
bounds to exp. data
→ provide “ab initio
error bars”
Blazej Grabowski, Dusseldorf
 Inspection of several xc-functionals is critical to estimate
Seppredictive
20, 2010
power and error bars!
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Essential question
• When do local approximations become
relatively exact for a quantum system?
• What is nature of expansion?
• What are leading corrections?
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Need help
•
•
•
•
•
•
•
Asymptotic analysis
Semiclassical theory, including periodic orbits
Boundary layer theory
Path integrals
Green’s functions for many-body problems
Random matrix theory
E.g., who has done spin-decomposed TF
theory?
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What we might get
• We study both TS and EXC
• For TS:
– Would give orbital-free theory (but not using n)
– Can study atoms to start with
– Can slowly start (1d, box boundaries) and work
outwards
• For EXC:
– Improved, derived functionals
– Integration with other methods
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Outline
a) Overview
b) Some details
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One particle in 1d
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N fermions
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Rough sums
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Inversion
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Higher orders
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Test system
v(x)=-D sinp(mπx)
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Semiclassical density for 1d box
TF
Classical momentum:
Classical phase:
Fermi energy:
Classical transit time:
Elliott, Cangi, Lee, KB, PRL 2008
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Density in bumpy box
• Exact density:
– TTF[n]=153.0
• Thomas-Fermi
density:
– TTF[nTF]=115
• Semiclassical
density:
– TTF[nsemi]=151.4
– DN < 0.2%
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Usual continua
• Scattering states:
– For a finite system, E > 0
• Solid-state: Thermodynamic limit
– For a periodic potential, have continuum bands
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A new continuum
• Consider some simple problem, e.g., harmonic
oscillator.
• Find ground-state for one particle in well.
• Add a second particle in first excited state, but
divide ħ by 2, and resulting density by 2.
• Add another in next state, and divide ħ by 3, and
density by 3
• …
• →∞
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Continuum limit
Leading corrections to local
approximations Attila Cangi,
Donghyung Lee, Peter Elliott,
and Kieron Burke, Phys. Rev.
B 81, 235128 (2010).
Attila Cangi
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Example of utility of formulas
• Worst case (N=1)
• Note accuracy
outside of
turning points
• No evanescent
contributions in
formula
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Getting to real systems
• Include real turning points and evanescent
regions, using Langer uniformization
• Consider spherical systems with Coulombic
potentials (Langer modification)
• Develop methodology to numerically calculate
corrections for arbitrary 3d arrangements
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Classical limit for neutral atoms
• For interacting
systems in 3d,
increasing Z in
an atom,
keeping it
neutral,
approaches
the classical
continuum, ie
same as ħ→0
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Ionization as Z→∞
Lucian
Constantin
Using code of
Eberhard Engel
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Z→∞ limit of ionization potential
• Shows even energy
differences can be found
• Looks like LDA exact for EX
as Z-> ∞.
• Looks like finite EC
corrections
• Looks like extended TF
(treated as a potential
functional) gives some sort
of average.
• Lucian Constantin, John
Snyder, JP Perdew, and KB,
arXiv.
Could we get accurate results with QMC? See Richard
Needs, PRE, 2005.
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Ionization density for large Z
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Ionization density as Z→∞
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Bohr atom
• Atoms with e-e repulsion made infinitesimal
x=Z1/3r,
Z=28
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Exactness for Z→∞ for Bohr atom
Using
hydrogenic
orbitals to
improve
DFT
John C
Snyder
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Orbital-free potential-functional
for C density
4pr2ρ(r)
r
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C+
I=11.26eV
ΔI=0.24eV
I(LSD)=11.67 eV
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Outline
a) Overview
b) Some details
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Third prize
• Will be able to see directly the nature of
semiclassical corrections, and calculate them
for simple systems
• Can build better density functional
approximations which capture these limits
• Remove empiricism in functional construction
– Get parameters from limits
– Knowing which exact conditions to apply
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Second prize
• Able to say what approximation to Green’s
function or wavefunction gives rise to density
functional approximation.
• Able to perform more accurate calculations in
vital part of system, and stitch on to DFT
calculation.
• Know what a DFT approximation means
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First prize
• Extract kinetic energy directly from vS(r)
without solving KS equations
• Extract EXC directly from v(r) without needing
the density
• Replace DFT with potential functional theory
using semiclassical expansions for energies
from potential.
• Speed up all calculations tremendously.
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Conclusions
• All work in progress – Rome was not burnt in a
day
• For EXC:
– Already have bits and pieces
– Beginning assault on EX[n]
• For TS:
– Strongly suggests orbital-free calculations should use
potential not density
– Now have improved formula for getting T directly
from any n[v](r)
– Developing path-integral formulation
• Thanks to students and NSF
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