Why density functional theory works and how to improve upon it

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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
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Outline

1. Overview
2. Some details

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Modest statements

• The most important problem Ive 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?

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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
Modern DFT Kohn-Sham EXCn?,n?
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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 E0n for fixed N
• Properties
• Exact for neutral atoms as Z gets large
  (LiebSimon, 73)
• Typical error of order 10
• Tellers unbinding theorem Molecules dont bind.

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Modern Kohn-Sham era

• 40s and 50s John Slater began doing
  calculations with orbitals for kinetic energy and
  an approximate density functional for Excn
  (called Xa)
• 1964 Hohenberg-Kohn theorem proves an exact
  E0n exists (later generalized by Levy)
• 1965 Kohn-Sham produce formally exact procedure
  and suggest LDA for Excn

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Kohn-Sham equations (1965)
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He atom in exact Kohn-Sham DFT
Everything has (at most) one KS potential
Dashed-line EXACT KS potential
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Recipe for exact Excn

• 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 ExcE-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 Excn?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
  Excn.
• 70s-90s Much work (Langreth, Perdew, Becke,
  Parr) going from gradient expansion
  (slowly-varying density) to produce more accurate
  functionals, called generalized gradient
  approximations (GGAs).
• Early 90s
• 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 EXCn 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
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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 EXCn.

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Kierons trail of tears
Include turning points
Real atoms
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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 h.

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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 GGAs, hybrids,
  meta-GGAs, hyper GGAs, non-local vdW,

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Pandora

• Many difficulties in answering this question
• Semiclassical methods
• Asymptotic expansions
• Boundary layer theory

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What weve done so far
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A major ultimate aim EXCn

• 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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Improvements of PBEsol
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 lt3 in lat.
const. lt 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
• predictive 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
• Greens 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

1. Overview
2. 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(mpx)
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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
• TTFn153.0
• Thomas-Fermi density
• TTFnTF115
• Semiclassical density
• TTFnsemi151.4
• DN lt 0.2

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Usual continua

• Scattering states
• For a finite system, E gt 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 h by 2, and resulting density by 2.
• Add another in next state, and divide h by 3, and
  density by 3
• ?8

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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 (N1)
• 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 h?0

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Ionization as Z?8
Lucian Constantin
Using code of Eberhard Engel
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Z?8 limit of ionization potential

• Shows even energy differences can be found
• Looks like LDA exact for EX as Z-gt 8.
• 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?8
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Bohr atom

• Atoms with e-e repulsion made infinitesimal

xZ1/3r, Z28
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Exactness for Z?8 for Bohr atom
Using hydrogenic orbitals to improve DFTJohn C
Snyder
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Orbital-free potential-functional for C density
4pr2?(r)
r
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C
I11.26eV ?I0.24eV
I(LSD)11.67 eV
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Outline

1. Overview
2. 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 Greens
  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 EXn
• For TS
• Strongly suggests orbital-free calculations
  should use potential not density
• Now have improved formula for getting T directly
  from any nv(r)
• Developing path-integral formulation
• Thanks to students and NSF