Electrons in Materials Density Functional Theory Richard M. Martin

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Electrons in MaterialsDensity Functional
TheoryRichard M. Martin
d orbitals
Electron density in La2CuO4 - difference from sum
of atom densities - J. M. Zuo (UIUC)
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Outline

• Many Body Problem!
• Density Functional Theory Kohn-Sham Equations
  allow in principle exact solution for ground
  state of many-body system using independent
  particle methods Approximate LDA, GGA
  functionals
• Examples of Results from practical calculations
• Pseudopotentials - needed for plane wave
  calculations
• Next Time - Bloch Theorem, Bands in crystals,
  Plane wave calculations, Iterative methods

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Ab Initio Simulations of Matter

• Why is this a hard problem?
• Many-Body Problem - Electrons/ Nuclei
• Must be Accurate --- Computation
• Emphasize here Density Functional Theory
• Numerical Algorithms
• Some recent results

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Eigenstates of electrons

• For optical absortion, etc., one needs the
  spectrum of excited states
• For thermodynamics and chemistry the lowest
  states are most important
• In many problems the temperature is low compared
  to characteristic electronic energies and we need
  only the ground state
• Phase transitions
• Phonons, etc.

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The Ground State

• General idea Can use minmization methods to get
  the lowest energy state
• Why is this difficult ?
• It is a Many-Body Problem
• Yi ( r1, r2, r3, r4, r5, . . . )
• How to minimize in such a large space

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The Ground State

• How to minimize in such a large space
• Methods of Quantum Chemistry- expand in extremely
  large bases - Billions - grows exponentially with
  size of system
• Limited to small molecules
• Quantum Monte Carlo - statistical sampling of
  high-dimensional spaces
• Exact for Bosons (Helium 4)
• Fermion sign problem for Electrons

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Quantum Monte Carlo

• Variational - Guess form for Y ( r1, r2, )
• Minimize total energy with respect to all
  parameters in Y
• Carry out the integrals by Monte Carlo
• Diffusion Monte Carlo - Start with VMC and apply
  operator e-Ht Y to project out an improved ground
  state Y0
• Exact for Bosons (Helium 4)
• Fermion sign problem for Electrons

E0 ? dr1 dr2 dr3 Y H Y
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Density Functional Theory

• 1998 Nobel Prize in Chemistry to Walter Kohn and
  John Pople
• Key point - the ground state energy for the hard
  many-body problem can in principle be found by
  solving non-interacting electron equations in an
  effective potential determined only by the
  density
• Recently accurate approximations for the
  functionals of the density have been found

H Yi (x,y,z) Ei Yi (x,y,z) , H -
V(x,y,z)
D
2
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Density Functional Theory

• Must solve N equations, I 1, N with a
  self-consistent potential V(x,y,z) that depends
  upon the density of the electrons
• Text-Book - Find the eigenstates
• More efficient Modern Algorithms
• Minimize total energy for N states subject to the
  condition that they must be orthonormal
• Conjugate Gradient with constraints
• Recent Order N Linear scaling methods

H Yi (x,y,z) Ei Yi (x,y,z) , H -
V(x,y,z)
D
2
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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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Enthalpy vs pressure

• H E PV - equilibrium structure at a fixed
  pressure P is the one with minimum H
• Transition pressures slightly below experiment
  80 kbar vs 100kbar

Needs and Mujica (1995)
Simple Hexagonal
Cubic Diamond
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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 grahite and diamond
  essentially the same!

0. 3 eV/atom barrier
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A new phase of Nitrogen

• Published in Nature this week. Reported in the
  NY Times - Dense, metastable semiconductor
• Predicted by theory 10 years ago!

Molecular form
Mailhiot, et al 1992
Cubic Gauche Polymeric form with 3 coordination
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The Great Failures

• 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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Conclusions

• The ground state properties are predicted with
  remarkable success by the simple LDA and GGAs.
  Structures, phonons (5), .
• Excitations are NOT well-predicted by the usual
  LDA, GGA forms of DFT The Band Gap
  Problem Orbital dependant functionals
  increase the gaps - agree well with experiment -
  now a research topic