Density Functional Theory The Basis of Most Modern Calculations

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Density Functional Theory The Basis of Most
Modern Calculations
Richard M. Martin University of Illinois at
Urbana-Champaign
Lecture II Behind the functionals limits and
challenges
Lecture at Workshop on Bridging Time and Length
Scales in Materials Science and BiophysicsIPAM,
UCLA September, 2005
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Density Functional Theory The Basis of Most
Modern Calculations
Hohenberg-Kohn Kohn-Sham 1965 Defined a new
approach to the many-body interacting electron
problem

• Yesterday
• Brief statement of the Hohenberg-Kohn theorems
  and the Kohn-sham Ansatz
• Overview of the solution of the Kohn-Sham
  equations and the importance of pseudopotentials
  in modern methods

• Today
• Deeper insights into the Hohenberg-Kohn theorems
  and the Kohn-sham Ansatz
• The nature of the exchange-correlation functional
• Understanding the limits of present functionals
  and the challenges for the future
• Explicit many-body methods and improved DFT
  approaches

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Interacting
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The basis of most modern calculationsDensity
Functional Theory (DFT)

• Hohenberg-Kohn (1964)

• All properties of the many-body system are
  determined by the ground state density n0(r)
• Each property is a functional of the ground state
  density n0(r) which is written as f n0
• A functional f n0 maps a function to a result
  n0(r) ? f

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The Hohenberg-Kohn Theorems
n0(r) ? Vext(r) (except for constant)
? All properties
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The Hohenberg-Kohn Theorems
Minimizing En for a given Vext(r) ? n0(r) and E
In principle, one can find all other properties
and they are functionals of n0(r).
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The Hohenberg-Kohn Theorems - Proof
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The Hohenberg-Kohn Theorems - Continued

• What is accomplished by the Hohenberg-Kohn
  theorems?

• Existence proofs that properties of the
  many-electron system are functionals of the
  density

• A Nobel prize for this???

• The genius are the following steps to realize
  that this provides a new way to approach the
  many-body problem

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The Kohn-Sham Ansatz - from Lecture I

• Kohn-Sham (1965) Replace original many-body
  problem with an independent electron problem
  that can be solved!
• The ground state density is required to be the
  same as the exact density

• The new paradigm find useful, approximate
  functionals

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Meaning the functionals?
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Functional Excn in Kohn-Sham Eqs.

• How to find a approximate functional Excn
• Requires information on the many-body system of
  interacting electrons

• Local Density Approximation - LDA
• Assume the functional is the same as a model
  problem the homogeneous electron gas
• Exc has been calculated as a function of
  densityusing quantum Monte Carlo methods
  (Ceperley Alder)
• Gradient approximations - GGA
• Various theoretical improvements for electron
  density that is varies in space

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What is Excn ?

• Exchange and correlation ? around each electron,
  other electrons tend to be excluded x-c hole
• Excis the interaction of the electron with the
  hole involves only a spherical average

Very non-spherical! Spherical average very
closeto the hole in a homogeneouselectron gas!
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Exchange-correlation (x-c) hole in silicon

• Calculated by Monte Carlo methods

Exchange
Correlation
Hole is reasonably well localized near the
electron Supports a local approximation
Fig. 7.3 - Hood, et. al. 349
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Exchange-correlation (x-c) hole in silicon

• Calculated by Monte Carlo methods

Exchange-correlation hole spherical average
Bond Center
Interstitial position
Comparison to scale
x-c hole close to that in the homogeneous gas in
the most relevant regions of space Supports
local density approximation !
Fig. 7.4 - Hood, et. al. 349
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The Kohn-Sham Equations

• Assuming a form for Excn
• Minimizing energy (with constraints) ? Kohn-Sham
  Eqs.

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Comparisons LAPW PAW - - Pseudopotentials
(VASP code)
(Repeat from Lecture I)

• a lattice constant B bulk modulus m
  magnetization
• aHolzwarth , et al. bKresse Joubert cCho
  Scheffler dStizrude, et al.

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What about eigenvalues?

• The only quantities that are supposed to be
  correct in the Kohn-Sham approach are the
  density, energy, forces, .
• These are integrated quantities
• Density n(r ) Si Yi(r )2
• Energy Etot Si ei Fn
• Force FI - dEtot / dRI where RI
  position of nucleus I
• What about the individual Yi(r ) and ei ?
• In a non-interacting system, ei are the energies
  to add and subtract Kohn-Sham-ons
  non-interacting electrons
• In the real interacting many-electron system,
  energies to add and subtract electrons are
  well-defined only at the Fermi energy

• The Kohn-Sham Yi(r ) and ei are approximate
  functions - a starting point for meaningful
  many-body calculations

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Electron Bands

• Understood since the 1920s - independent
  electron theories predict that electrons form
  bands of allowed eigenvalues, withforbidden gaps
  
• Established by experimentally for states near
  the Fermi energy

Silicon
Extra added electronsgo in bottom of conduction
band
Empty Bands
Gap
Missing electrons(holes) go in top of valence
band
Filled Bands
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Comparison of Theory and ExperimentAngle
Resolved Photoemission (Inverse
Photoemission)Reveals Electronic Removal
(Addition) Spectra
Germanium
Many-body Th. (lines) Experiment (points)
LDA DFT Calcs. (dashed lines)
Silver
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Explicit Many-body methods

• Present approximate DFT calculations can be the
  starting point for explicit many-body
  calculations
• GW - Greens function for excitations
• Use DFT wavefunctions as basis for many-body
  perturbation expansion
• QMC quantum Monte Carlo for improved treatment
  of correlations
• Use DFT wavefunctions as trial functions
• DMFT dynamical mean field theory
• Use DFT wavefunctions and estimates of parameters
• Combine Kohn-Sham DFT and explicit many-body
  techniques
• The many-body results can be viewed as
  functionals of the density or Kohn-Sham
  potential!
• Extend Kohn-Sham ideas to require other
  properties be described
• Recent extensions to superconductivity E.K.U.
  Gross, et al.

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Explicit Many-Body Methods

• Excitations
• Electron removal (addition)
• Experiment - Photoemission
• Theory QuasiparticlesGW ApproximationGreens
  functions, . . .
• Electron excitation
• Experiment Optical Properties
• Theory ExcitonsBethe-Salpeter equation (BSE)

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Explicit Many-Body Methods

• Excitations
• Electron removal (addition)
• Experiment - Photoemission
• Theory QuasiparticlesGW Approximation
• Greens functions, . . .
• Electron excitation
• Experiment Optical Properties
• Theory ExcitonsBethe-Salpeter equation (BSE)

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Comparison of Theory and ExperimentAngle
Resolved Photoemission (Inverse
Photoemission)Reveals Electronic Removal
(Addition) Spectra
Germanium
Many-body Th. (lines) Experiment (points)
LDA DFT Calcs. (dashed lines)
Silver
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Explicit Many-Body Methods

• Excitations
• Electron removal (addition)
• Experiment - Photoemission
• Theory QuasiparticlesGW Approximation
• Greens functions, . . .
• Electron excitation
• Experiment Optical Properties
• Theory ExcitonsBethe-Salpeter equation (BSE)

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Optical Spectrum of Silicon
Many-body BSE calculation correctsthe gap and
the strengths of the peaks- excitonic effect
Gap too small inthe LDA
Photon energy
From Lucia Reining
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Strongly Correlated Systems

• All approximate functionals fail at some point!
• Simple density functionals, e.g., LDA, GGAs,
  etc. fail in many cases with strong interactions
• Atoms with localized electronic states
• Strong interactions
• Transition metals -- Rare earths
• Open Shells
• Magnetism
• Metal - insulator transitions, Hi-Tc materials
• Catalytic centers
• Transition metal centers in Biological molecules
  . . .

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Conclusions I

• Density functional theory is by far the most
  widely applied ab intio method used in for
  real materials in physics, chemistry, materials
  science
• Approximate forms have proved to be very
  successful
• BUT there are failures
• No one knows a feasible approximation valid for
  all problems especially for cases with strong
  electron-electron correlations

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Conclusions II

• Exciting arenas for theoretical predictions
• Working together with Experiments
• Realistic simulations under real conditions
• Molecules and clusters in solvents, . . .
• Catalysis in real situations
• Nanoscience and Nanotechnology
• Biological problems
• Beware -- understand what you are doing!
• Limitations of present DFT functionals
• Use codes properly and carefully
• Critical issues to be able to describe relevant
  Time and Length Scales