GGAs: A History

1. GGAs: A History P Briddon

2. Thomas Fermi • First attempt to write E[n]. • 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 E[n]. • The real problem: What is the functional? • No progress towards the LDA • Instead followed on from TF by attempting to develop T[n] by gradient expansion.

4. Kohn-Sham (1965) • Realised that T[n] not accurate enough. • Instead wrote T[n] = Ts[n]+DT • 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?

8. 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

9. Properties of the hole • Subject of much research. • The LDA must obey these. • The GEA does not need to.

10. 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.

11. GGA idea • A brute force fix. • If rx(r,r’)>0, set it to zero. • If sum rule violated, truncate the hole. • Resulting expressions look like:

12. 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.

13. 2 Flavours • Chemistry stable: e.g. Becke (B88) • Empirical • b=0.0042, fitted to exchange energies of He ... Rn. • Gives correct asymptotic form in exponential tails.

14. A second flavour: PBE96 • The physics stable: • Principled, parameter free • Numerous analytic properties • Slow varying limit should give LDA response. This requires Fx →ms2 , m=0.21951 • Density scaling, n(r)→l3n(lr), Ex→l Ex

15. 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.

16. 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.

17. 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

18. Hybrid Functionals • Why not just add correlation to HF calculations? We could write EXC=EX[exact]+EC[LSD] • Try it – error for G2 set is 32 kcal/mol, similar to LDA [HF gives 78; best 5]. • Why is this?

19. 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!

20. 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

21. 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

22. 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.

23. 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 don’t scale linearly in general. • Plane wave based (physics) codes can’t easily find exact exchange.

24. Screened Exchange • Key idea (Heyd, Scuseria 2003): • First term is short-ranged; second long ranged. • w=0 gives full 1/r potential. • How to incorporate into a functional?

25. HSE03

26. 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:

27. 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).

28. The modified hole Essentially, fits into code as at present, but e needs to be evaluated via an integral.

29. 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.

30. 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.

31. 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

32. 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.