GGAs: A History P Briddon

Thomas Fermi First attempt to write E[n]. First attempt to write E[n]. An early DFT. An early DFT. Issue with KE: Used n 5/3 Issue with KE: Used n 5/3 Seemed good for absolute energies Seemed good for absolute energies Not accurate enough for energy differences. Not accurate enough for energy differences.

Hohenberg and Kohn (1964) Formal proof that can write E[n]. Formal proof that can write E[n]. The real problem: What is the functional? The real problem: What is the functional? No progress towards the LDA No progress towards the LDA Instead followed on from TF by attempting to develop T[n] by gradient expansion. Instead followed on from TF by attempting to develop T[n] by gradient expansion.

Kohn-Sham (1965) Realised that T[n] not accurate enough. Realised that T[n] not accurate enough. Instead wrote T[n] = T s [n]+  T Instead wrote T[n] = T s [n]+  T T s found from Kohn-Sham states. T s found from Kohn-Sham states.  T incorporated into what is left – the exchange correlation energy.  T incorporated into what is left – the exchange correlation energy.

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

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

GEA Early method attempt to go beyond the LDA. Early method attempt to go beyond the LDA. Based on the idea that for slowly varying density, we could develop an expansion: Based on the idea that for slowly varying density, we could develop an expansion: In fact the first order term is zero. In fact the first order term is zero. Made things much worse. Made things much worse. Why? Why?

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

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

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

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

Exchange GGA Note that s  is large when Note that s  is large when Gradient is big Gradient is big n is low (exponential tails; surfaces) n is low (exponential tails; surfaces) s  is small when s  is small when Gradient is small Gradient is small n is large (including core regions) n is large (including core regions) Sometimes written as enhancement factor. Sometimes written as enhancement factor.

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

A second flavour: PBE96 The physics stable: The physics stable: Principled, parameter free Principled, parameter free Numerous analytic properties Numerous analytic properties Slow varying limit should give LDA response. This requires F x →  s 2,  = Slow varying limit should give LDA response. This requires F x →  s 2,  = Density scaling, n(r)→ 3 n( r), E x →  E x Density scaling, n(r)→ 3 n( r), E x →  E x

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

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

Atomisation energies (kcal/mol) HFLSDPBEEX H CH C2H C2H N O F

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

Hybrid functionals [2] Correct XC hole is localised. Correct XC hole is localised. Exchange and correlation separately are delocalised. Exchange and correlation separately are delocalised. DFT in LDA and GGA give localised expressions for both parts. DFT in LDA and GGA give localised expressions for both parts. Sometimes simpler is better! Sometimes simpler is better!

Hybrid functionals [3] Chemists approach: take empirical admixtures. e.g. Becke 1993: Chemists approach: take empirical admixtures. e.g. Becke 1993: Today, most common is B3LYP Today, most common is B3LYP Gives mean unsigned error of 5 kcal/mol Gives mean unsigned error of 5 kcal/mol

Hybrid functionals [4] Admixture can be justified theoretically, the work of PEB (96), BEP (97): Admixture can be justified theoretically, the work of PEB (96), BEP (97): Using PBE96 as the GGA gives the PBE1PBE (or PBE0) functional. Using PBE96 as the GGA gives the PBE1PBE (or PBE0) functional. Nearly as good as B3LYP Nearly as good as B3LYP

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

HSE03 Recent development. Several motivations: B3LYP more accurate than BLYP. Some admixture of exchange needed. B3LYP more accurate than BLYP. Some admixture of exchange needed. Exact exchange is slow to calculate. Exact exchange is slow to calculate. Linear scaling K-builds don’t scale linearly in general. Linear scaling K-builds don’t scale linearly in general. Plane wave based (physics) codes can’t easily find exact exchange. Plane wave based (physics) codes can’t easily find exact exchange.

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

HSE03

Where does this leave us? Need to find short-ranged HF contribution. Need to find short-ranged HF contribution. Linear scaling Linear scaling Parallelism is perfect Parallelism is perfect Will not be time consuming for large systems. Will not be time consuming for large systems. Can also do with different splittings with only minor modification: Can also do with different splittings with only minor modification:

Where does this leave us? Need short ranged part of PBE exchange energy. Approach this from the standard expression: Need short ranged part of PBE exchange energy. Approach this from the standard expression: Modify the interaction to short ranged term Modify the interaction to short ranged term Need explicit expression for the hole. Need explicit expression for the hole. Provided by work of EP (1998). Provided by work of EP (1998).

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

How about the accuracy? Enthalpies of formation (kcal/mol): Enthalpies of formation (kcal/mol): MAE(G2)MAE(G3) B3LYP PBE PBE HSE Conclusion: competitive with hybrids.

How about the accuracy? Vibrational freqs (cm-1); 82 diatomics Vibrational freqs (cm-1); 82 diatomicsMAE(G2) B3LYP33.5 PBE42.0 PBE043.6 HSE Conclusion: competitive with hybrids.

How about the accuracy? Band Gaps (eV) Band Gaps (eV) LDAPBEHSEEXP C Si Ge GaAs GaN MgO

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