Density Functional Theory Richard M' Martin University of Illinois

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Density Functional TheoryRichard M. Martin
University of Illinois
Cud orbitals
Electron density in La2CuO4 - difference from sum
of atom densities - J. M. Zuo (UIUC)
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Outline

• DFT is an approach to Interacting Many-Body
  Problems
• Hohenberg-Kohn Theorems Levy-Lieb Construction
• Kohn-Sham Ansatz allows in principle exact
  solution for ground state of many-body system
  using independent particle methods
• Classes of functionals LDA, GGA, OEP, .
• Examples of Results
• Locality Principles and linear scaling
• Electric polarization in crystals - deep issues
  that bring out stimulating questions about DFT,
  and the differences between the Hohenberg-Kohn
  and Kohn-Sham approaches

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Questions for you

• Why were orbitals mentioned on the introductory
  slide and not simply density
• Can you tell whether La2CuO4 is an insulator or a
  metal just by looking at the density?If so, what
  aspects of the density?
• Is Kohn-Sham theory the same as Density
  Functional Theory?
• If not, what is the difference? What did
  Kohn-Sham add? What did they subtract?
• Do locality principles in independent particle
  methods carry over to the real many-body world?
• Is the electric polarization of a ferroelectric
  an intrinsic ground state property? Is it
  determined by the density?

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Assumesnon-degenerateground state
H-K Functional
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What have we gained so far?

• Apparently Nothing!

• The only result is that the density determines
  the potential

• We are still left with the original many-body
  problem

• But the proofs suggest(ed) the next step

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Kohn-Sham Ansatz

• If you dont like the answer, change the
  question

• Replace the original interacting-particle problem
  with a different problem more easily solved

• Kohn-Sham auxiliary systemnon-interacting
  electrons assumed to have the same density as
  the interacting system

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Auxiliary System
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• Replace interacting problem with auxiliary
  non-interacting problem

• Each term in figure is uniquely relatedto each
  other term!

• The ansatz has been shown to be fulfilled in
  several simple cases but not in general

• We will proceed assuming the ansatz is justiified

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Negative energyelectron positive hole
Kinetic energypositive
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Exchange-correlation hole in homogeneous
electron gas

• Exchange dominates at high density (small rs)
• Correlation dominates at low density (large rs)

Gori-Giorgi, Sacchetti and Bachelet,PRB 61, 7353
(2000).
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Exchange hole in Ne atom
Gunnarsson, et al, PRB 20, 3136 (79).

• Spherical average close to LDA!

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Exchange hole in Si Crystal

• VariationalMonte Carlo

Hood, et al, PRB 57, 8972(98).
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Examples of Results

• Hydrogen molecules - using the LSDA (from O.
  Gunnarsson)

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Examples of Results

• Phase transformations of Si, Ge
• from Yin and Cohen (1982)
  Needs and Mujica (1995)

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Graphite vs Diamond

• A very severe test
• Fahy, Louie, Cohen calculated energy along a path
  connecting the phases
• Most important - energy of graphite and diamond
  essentially the same!

0. 3 eV/atom barrier
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Less compressible than Diamond

• Bulk Modulus B (Gpa) Exp Th
  (LDA)C 444 467Os 462 444

Cynn, et al, PRL March 14 (2002)
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Phonons - LDA and GGA
Baroni, et al, RMP 73, 515 (2000).

• Calculated by response function method

LDA
GGA
Exp
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The Band Gap Problem

• Often said that the eigenvalues have no meaning
  just Lagrange multipliers
• Energy to add or subtract an electron in the
  non-interacting system - not an excitation energy
  of the interacting system
• Naïve use of the eigenvalues as exciation
  energies is the famous band gap problem
• To understand the effcets we first examine the
  potential

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Exchange potential in atoms

• 2-electron systems
• LDA Vxc is too shallow

Almbladh and Pedroza, PR A 29, 2322 (84).
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The Band Gap Problem

• Excitations are NOT well-predicted by the
  standard LDA, GGA forms of DFT

The Band Gap Problem
Orbital dependent DFT is more complicated but
gives improvements - treat exchange better,
e.g, Exact Exchange
Ge is a metal in LDA!
M. Staedele et al, PRL 79, 2089 (1997)
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Status of Band Gap Problem

• It should be possible to calculate all excitation
  energies from the Kohn-Sham approach
• But not clear how close Kohn-Sham eigenvalues
  should be to true excitation energies
• Not clear how much of the band gap problem is
  due to approximate functionals
• Size of derivative discontinuity?

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Locality and Linear Scaling

• DFT provides a fundamental basis for
  nearsightedness (W. Kohn) -- if properties in a
  region are determined only by densities in a
  neighborhood -- so that an Order N method must
  be possible
• Used, e.g., by W. Yang in his divide and conquer
  method
• Orbital picture in Kohn-Sham method provides the
  concrete methods

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Linear Scaling Order-N Methods

• Computational complexity N number of atoms
  (Current methods scale as N2 or N3)
• Divide and Conquer
• Greens Function
• Fermi Operator Expansion
• Density matrix purification
• Generalized Wannier Functions
• Spectral Telescoping(Review by S. Goedecker in
  Rev Mod Phys)

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Example of Our workPrediction of Shapes of Giant
Fullerenes
S. Itoh, P. Ordejon, D. A. Drabold and R. M.
Martin, Phys Rev B 53, 2132 (1996).See also C.
Xu and G. Scuceria, Chem. Phys. Lett. 262, 219
(1996).
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Simulations of DNA with the SIESTA code

• Machado, Ordejon, Artacho, Sanchez-Portal, Soler
• Self-Consistent Local Orbital O(N) Code
• Relaxation - 15-60 min/step ( 1 day with
  diagonalization)

Iso-density surfaces
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Conclusions - I

• DFT is a general approach to interacting
  many-body problems Kohn-Sham approach makes it
  feasible
• Ground state properties are predicted with
  remarkable success by LDA and GGAs. Structures,
  phonons (5), .
• Excitations are NOT well-predicted by the LDA,
  GGA approximations The Band Gap
  Problem Orbital dependant functionals
  increase the gaps - agree better with
  experiment Derivative discontinuity
  natural in orbital functionals

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Conclusions - II

• Locality inherent for properties of a region that
  depend only on the density in a
  neighborhood Forces, stress, .. Order N
  linear scaling method should be possible
• Density matrix shows the locality in the quantum
  system Several feasible methods for insulators
• Carries over to interacting many-body system
• Some propreties are not local in real
  space Fermi surface of a metal, etc. But states
  near Fermi energy have universal behavior that
  should make linear scaling possible
• When is the functional an extremely non-local
  functional of the density? A polarized
  insulator, where the Kohn-Sham theory must be
  fundamentally revised