GGAs: A History

1
GGAs A History

• P Briddon

2
Thomas Fermi

• First attempt to write En.
• An early DFT.
• Issue with KE Used n5/3
• Seemed good for absolute energies
• Not accurate enough for energy differences.

3
Hohenberg and Kohn (1964)

• Formal proof that can write En.
• The real problem What is the functional?
• No progress towards the LDA
• Instead followed on from TF by attempting to
  develop Tn by gradient expansion.

4
Kohn-Sham (1965)

• Realised that Tn not accurate enough.
• Instead wrote Tn TsnDT
• Ts found from Kohn-Sham states.
• DT incorporated into what is left the exchange
  correlation energy.

5
LDA

• Used by physicists for 40 years.
• Write
• exc(n) for homogenous electron gas.
• exchange-correlation energy per electron
• Assumption grad n is small in some sense.
• Accurate for nearly homogeneous system and for
  limit of large density.

6
Limitations

• Band gap problem
• Overbinding (cohesive energies 10-20 error).
• High spin states.
• Hydrogen bonds/weak interactions
• Graphite

7
GEA

• Early method attempt to go beyond the LDA.
• Based on the idea that for slowly varying
  density, we could develop an expansion

• In fact the first order term is zero.
• Made things much worse.
• Why?

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Exchange-Correlation Hole

• Due to phenomena of exchange there is a depletion
  of density (of the same spin) around each
  electron.
• Mathematically described as
• The exchange correlation energy written as

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Properties of the hole

• Subject of much research.

• The LDA must obey these.
• The GEA does not need to.

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Why is this important?

• Huge error made to the integral would occur if
  the hole is not normalised correctly.
• The LDA has this correct it is the correct
  expression for a proper physical system.
• Gunnarsson and Lundqvist 1976.
• In fact, only need the spherical average of the
  hole is needed.

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GGA idea

• A brute force fix.
• If rx(r,r)gt0, set it to zero.
• If sum rule violated, truncate the hole.
• Resulting expressions look like

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Exchange GGA

• Note that ss is large when
• Gradient is big
• n is low (exponential tails surfaces)
• ss is small when
• Gradient is small
• n is large (including core regions)
• Sometimes written as enhancement factor.

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2 Flavours

• Chemistry stable e.g. Becke (B88)
• Empirical
• b0.0042, fitted to exchange energies of He ...
  Rn.
• Gives correct asymptotic form in exponential
  tails.

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A second flavour PBE96

• The physics stable
• Principled, parameter free
• Numerous analytic properties
• Slow varying limit should give LDA response. This
  requires Fx ?ms2 , m0.21951
• Density scaling, n(r)?l3n(lr), Ex?l Ex

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Correlation Functionals

• Perdew - Zunger 1986
• Perdew Wang (1991)
• Part of parameter free PW91
• Perdew, Burke, Ernzerhof (1996)
• GGA made simple!
• Parameter free
• Simplified construction
• Smoother, better behaved.

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Lee Yang Parr

• Different approach based on accurate wave
  functions for the Helium atom.
• No relation to the homogeneous electron gas at
  all.
• One empirical parameter
• Often combined with Becke exchange to give BLYP.

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Atomisation energies (kcal/mol)

• HF LSD PBE EX
• H2 84 113 105 109
• CH4 328 462 420 419
• C2H2 294 460 415 405
• C2H4 428 633 571 563
• N2 115 267 243 229
• O2 33 175 144 121
• F2 -37 78 53 39

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Hybrid Functionals

• Why not just add correlation to HF calculations?
  We could write EXCEXexactECLSD
• Try it error for G2 set is 32 kcal/mol, similar
  to LDA HF gives 78 best 5.
• Why is this?

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Hybrid functionals 2

• Correct XC hole is localised.
• Exchange and correlation separately are
  delocalised.
• DFT in LDA and GGA give localised expressions for
  both parts.
• Sometimes simpler is better!

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Hybrid functionals 3

• Chemists approach take empirical admixtures.
  e.g. Becke 1993

• Today, most common is B3LYP

• Gives mean unsigned error of 5 kcal/mol

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Hybrid functionals 4

• Admixture can be justified theoretically, the
  work of PEB (96), BEP (97)
• Using PBE96 as the GGA gives the PBE1PBE (or
  PBE0) functional.
• Nearly as good as B3LYP

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Meta GGAs

• Perdew 1999
• Better total energies.
• Ingredients , KE density
• Very hard to find potential, so cannot do SCF
  with this.
• Therefore structural optimisation not possible.

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HSE03

• Recent development. Several motivations
• B3LYP more accurate than BLYP. Some admixture of
  exchange needed.
• Exact exchange is slow to calculate.
• Linear scaling K-builds dont scale linearly in
  general.
• Plane wave based (physics) codes cant easily
  find exact exchange.

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Screened Exchange

• Key idea (Heyd, Scuseria 2003)
• First term is short-ranged second long ranged.
• w0 gives full 1/r potential.
• How to incorporate into a functional?

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HSE03
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Where does this leave us?

• Need to find short-ranged HF contribution.
• Linear scaling
• Parallelism is perfect
• Will not be time consuming for large systems.
• Can also do with different splittings with only
  minor modification

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Where does this leave us?

• Need short ranged part of PBE exchange energy.
  Approach this from the standard expression

• Modify the interaction to short ranged term
• Need explicit expression for the hole.
• Provided by work of EP (1998).

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The modified hole
Essentially, fits into code as at present, but e
needs to be evaluated via an integral.
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How about the accuracy?

• Enthalpies of formation (kcal/mol)
• MAE(G2) MAE(G3)
• B3LYP 3.04 4.31
• PBE 17.19 22.88
• PBE0 5.15 7.29
• HSE03 4.64 6.57
• Conclusion competitive with hybrids.

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How about the accuracy?

• Vibrational freqs (cm-1) 82 diatomics
• MAE(G2)
• B3LYP 33.5
• PBE 42.0
• PBE0 43.6
• HSE03 43.9
• Conclusion competitive with hybrids.

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How about the accuracy?

• Band Gaps (eV)
• LDA PBE HSE EXP
• C 4.23 4.17 5.49 5.48
• Si 0.59 0.75 1.28 1.17
• Ge 0.00 0.00 0.56 0.74
• GaAs 0.43 0.19 1.21 1.52
• GaN 2.09 1.70 3.21 3.50
• MgO 4.92 4.34 6.50 7.22

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Has HSE got legs?

• Different separations?
• Improved formalism for GGA then possible.
• Standard applications ZnO, Ge etc.
• Effect on spectral calculations EELS
• Possibility of multiplet calculations for defect
  centres.