Density Functional Theory: a first look

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Density Functional Theory a first look

• Patrick Briddon

Theory of Condensed Matter Department of Physics,
University of Newcastle, UK.
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Contents

• Density Functional Theory
• Hohenberg Kohn Theorems
• Thomas Fermi Theory
• Kohn-Sham Equations
• Self Consistency
• Approximations for Exc.

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Density Functional Theory
Work with n(r) instead of ??
Standard approach of QM
DFT work in terms of density
N.B. few IFs and BUTs here
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3 Important Questions

• Three important questions
• Can we really write En?
• If so, how can we find n(r)?
• What is the functional En ?

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1st Hohenberg Kohn Theorem

• The external potential V(r) is determined to
  within a constant by the ground state charge
  density of a system.

i.e. one-to-one relationship
This is an astonishing statement! Why?
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1st Hohenberg Kohn Theorem
Proof Suppose we have two systems Hamiltonians
H1, H2 External potentials V1 , V2 GS
wavefunctions Y1, Y2 But the same GS density n(r)
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We clearly have
But
So
Swap 1 and 2
Contradiction!
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Our starting point was wrong! We cannot have two
different systems with the same GS
density. Importance of this we can write
En. Now move on the second question how can
we find the density? Hohenberg Kohns second
theorem.
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2nd Hohenberg Kohn Theorem
The true ground state charge density is that
which minimises the total energy. An
equivalent to the usual variational principle of
quantum mechanics.
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Proof
We have (variational principle)
Define
Then
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Some problems
V-representability
(only minimise over densities which can arise
from GS wavefunctions of real systems
Levy showed that the densities need only satisfy
a weaker condition that they can be obtained from
an antisymmetric wavefunction (N-representable).
n must be ve, continuous, normalised
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Some extensions
Spin dependent potentials e.g. magnetic
fields En ? En, n? - n? or En?, n? Main
advantage better description of open shell
systems.
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3 Important Questions

• Three important questions
• Can we really write En?
• If so, how can we find n(r)?
• What is the functional En ?
• Now for the last question.
• What is the formula!

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What is the functional?
Difficult still not answered exactly!
Problem is that other two terms are very large -
any attempt at approximation must be good.
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Thomas-Fermi Method
Classical expression for electron- electron term
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Thomas-Fermi Method
Statistical idea for KE based on uniform electron
gas result
KE per electron
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Thomas-Fermi Method
What about a non uniform system? A. Assume that
things vary slowly
DV
r
Total is thus
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Thomas-Fermi Method
Final energy is thus
How useful is this?
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Thomas-Fermi Method

• What is the conclusion?
• Energies quite good (error lt 1).
• Difference of energies not good enough to
  describe bonding.
• How can we improve this?

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Thomas-Fermi Method

• Add exchange/correlation (missing do far).
• Try to take account of non-uniform system.
• Write Tn as Tn, grad n
• All failures!

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Kohn-Sham methodPhys Rev 140, 1133A (1966)
Realised that approximation must be made to terms
that are small KE is big!
Improving Tn did not work. Need a completely
different approach. Second half of HK paper
therefore discarded.
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Kohn-Sham methodPhys Rev 140, 1133A (1966)
Introduce a system which 1. Is
non-interacting 2. Has same n(r) as the real
system.
non-interacting N-representability - an
assumption!
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Kohn-Sham contd.
where Tsn is the KE of the non-interacting
system and the final term, ?T, is small.
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Kohn-Sham contd.

• Excn includes both ?T and contributions to
  el-el energy beyond the Hartree term.
• The key hope is that this is
• small
• less sensitive to external potential
• These mean differences are accurate.

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We now have two questions (a) how to find Tsn
? (b) what is Excn ?
For a non-interacting system it is exactly true
that the many electron wavefunction is a single
Slater determinant.
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For this
and
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The ??(r) must be found from a self consistent
solution of
These are called the Kohn-Sham equations. Solve
iteratively
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Guess
Construct
Solve
Find new density
Look at
Form a better input and continue.
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Self Consistent Cycle

• This process is called the self-consistent cycle.
• Starting guess is a superposition of atomic
  charge densities (or a restart dump).
• AIMPRO produces output showing how the energy
  converged and how the input and output densities
  come together.

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AIMPRO SCF

• etot,echerr 1 -1.1289007706 0.0547166884
  0.884981 2.95 106.1 120.7
• etot,echerr 2 -1.1319461182 0.0303263020
  0.488911 3.05 106.1 120.7
• etot,echerr 3 -1.1361047998 0.0000338275
  0.000689 3.00 106.1 120.7
• etot,echerr 4 -1.1360826939 0.0001723143
  0.002700 2.99 106.1 120.7
• etot,echerr 5 -1.1361076063 0.0000002649
  0.000004 3.00 106.1 120.7

• The numbers are
• Total energy (reduces to converged value)
• 2 measures of
• Time taken per iteration
• Current memory being used (MB)
• Max memory used so far (MB)

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Kohn-Sham Levels
We got the Kohn Sham eqn
Q what exactly are the el and jl? A the
eigenvalues and eigenfunctions of a ficticious
non-interacting system which has the same density
as the real system.
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KS Levels Contd.
They are not the energies of quasiparticles. Typi
cal semiconductor results Lattice constant to
1 Bulk modulus to 1 Phonon frequencies to
5 LDA gap for Si is 0.6eV 0.1 eV for Ge!
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KS Levels Contd.
Bandstructures are qualitatively correct.
(Scissors operator). Physical nature of the KS
eigenfunctions sensible. P in Si - get state
just below conduction band Dangling bonds -
localised states in mid gap.
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AIMPRO and KS levels

• spin, kpoint 1 1
• 1 -10.0938 1.2658 3.7422 3.7422 5.9056
  8.7912
• 2.0000 0.0000 0.0000 0.0000 0.0000
  0.0000

The KS levels in eV. Used in bandstructure
plots. Occupancies also given (this is a spin
averaged calculation)
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3 Important Questions

• Three important questions
• Can we really write En?
• If so, how can we find n(r)?
• What is the functional En ?
• All remaining questions are in Excn. Now
  finally we look at what this is.

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Exchange correlation energy

• Our DFT total energy is

What about the last term?
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The Local Density Approximation (LDA)
Write
where exc(n) is the exchange-correlation energy
per electron for a uniform electron gas.
This seems a bit rough and very similar to Thomas
Fermi, but this term is now very much smaller.
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Exc for Homogeneous electron gas

• By simple analytical treatment for the exchange
  energy.
• Using many body perturbation theory (for various
  limits of correlation energy)
• By looking at quantum Monte-Carlo calculations
  and parametrising them
• Intelligent interpolation between these

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Exc for Homogeneous electron gas contd.
Simple analysis for exchange part gives

• Correlation is harder, see
• Perdew Zunger (1981)
• Vosko, Wilk, Nusair (1980)
• Perdew, Wang (1992)

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Simple Tests Molecules
Example water H20
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Simple tests solids

• Standard bulk calculations
• lattice constant (Si 1)
• bulk modulus (Si 2)
• phonon spectra (2 )
• formation energies (LDA 20 )
• excitation energies (50 )

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Phonon Spectrum
Material GaAs. All frequencies in cm-1
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How to Improve?
Next step is to move beyond the LDA
The Gradient expansion Approximation (GEA).
Early history of these was bad. Calculations made
worse, not better.
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Generalised Gradient Approximations (GGA)
Idea is to ensure that

• Sum rules are obeyed correctly
• scaling behaviour of exchange correlation energy
  correct
• Various limiting forms
• Bounds (Lieb-Oxford)

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Popular GGAs

• B88 (empirical, chemistry)
• BLYP (chemistry)
• PW91 (physics, poor form)
• PBE96 (physics, easier to use)

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Atomisation energies (kcal/mol)(1 eV 23
kcal/mol)
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Generalised Gradient Approximations (GGA)

• In general, GGA weakens bonds slightly.
• It improves results for
• binding energies of molecules
• description of surfaces
• H-bonding

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THE CONCLUSION

• An absolutely huge success
• 1988 two groups in UK doing DFT
• Cambridge (TCM)
• Exeter
• Today every department?
• Chemistry/engineering too!
• Applications in huge variety of areas.

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Work to do!

• Kittel Ch 6 Free Electron Fermi Gas
• Hohenberg and Kohn, PR (1964)
• Kohn-Sham, PR (1965)
• Perdew Zunger, PRB (1981)
• Perdew, Wang, PRB (1992)
• Perdew, Burke, Enzerhoff PRL (1996)