GGAs: A History

GGAs: A HistoryGGAs: A History

P BriddonP Briddon

Thomas FermiThomas Fermi

First attempt to write E[n].First attempt to write E[n]. An early DFT.An early DFT. Issue with KE: Used nIssue with KE: Used n5/35/3 Seemed good for absolute energiesSeemed good for absolute energies Not accurate enough for energy Not accurate enough for energy

differences.differences.

Hohenberg and Kohn Hohenberg and Kohn (1964)(1964)

Formal proof that can write E[n].Formal proof that can write E[n]. The real problem: What is the The real problem: What is the

functional?functional? No progress towards the LDANo progress towards the LDA Instead followed on from TF by Instead followed on from TF by

attempting to develop T[n] by attempting to develop T[n] by gradient expansion.gradient expansion.

Kohn-Sham (1965)Kohn-Sham (1965)

Realised that T[n] not accurate Realised that T[n] not accurate enough.enough.

Instead wrote T[n] = TInstead wrote T[n] = Tss[n]+[n]+TT

TTss found from Kohn-Sham states. found from Kohn-Sham states. T incorporated into what is left – T incorporated into what is left –

the exchange correlation energy.the exchange correlation energy.

LDALDA

Used by physicists for 40 years.Used by physicists for 40 years. WriteWrite

xcxc(n) for homogenous electron gas.(n) for homogenous electron gas. exchange-correlation energy exchange-correlation energy per electronper electron

Assumption: grad n is small in some Assumption: grad n is small in some sense.sense.

Accurate for nearly homogeneous Accurate for nearly homogeneous system and for limit of large density.system and for limit of large density.

rrr dnnnE xcxc ][

LimitationsLimitations

Band gap problemBand gap problem Overbinding (cohesive energies 10-Overbinding (cohesive energies 10-

20% error).20% error). High spin states.High spin states. Hydrogen bonds/weak interactionsHydrogen bonds/weak interactions GraphiteGraphite

GEAGEA Early method attempt to go beyond the LDA.Early method attempt to go beyond the LDA. Based on the idea that for slowly varying Based on the idea that for slowly varying

density, we could develop an expansion:density, we could develop an expansion:

221, nnfnnfnnn LDAxc

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 Exchange-Correlation HoleHole

Due to phenomena of exchange there is a Due to phenomena of exchange there is a depletion of density (of the same spin) depletion of density (of the same spin) around each electron.around each electron.

Mathematically described asMathematically described as The exchange correlation energy written asThe exchange correlation energy written as

rr ,xc

rrrr

rrrddxcExc

,

Properties of the holeProperties of the hole

Subject of much research.Subject of much research.

0'',

1'',

0',

',',',

rrr

rrr

rr

rrrrrr

d

d

c

x

x

cxxc

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?Why is this important?

Huge error made to the integral Huge error made to the integral would occur if the hole is not would occur if the hole is not normalised correctly.normalised correctly.

The LDA has this correct – it is the The LDA has this correct – it is the correct expression for a proper correct expression for a proper physical system.physical system.

Gunnarsson and Lundqvist [1976].Gunnarsson and Lundqvist [1976]. In fact, only need the spherical In fact, only need the spherical

average of the hole is needed.average of the hole is needed.

GGA ideaGGA idea A brute force fix.A brute force fix. If If xx((rr,,rr’)>0, set it to zero.’)>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:

rr

r

3/4

3/4

n

ns

dnsFLSDAEGGAE xx

Exchange GGAExchange GGA

Note that sNote that s is large when is large when Gradient is bigGradient is big n is low (exponential tails; surfaces)n is low (exponential tails; surfaces)

ss is small when is small when Gradient is smallGradient is small n is large (including core regions)n is large (including core regions)

Sometimes written as Sometimes written as enhancementenhancement factor.factor.

2 Flavours2 Flavours Chemistry stable: e.g. Becke (B88)Chemistry stable: e.g. Becke (B88)

EmpiricalEmpirical =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.

ss

sF B 1

2

sinh61

A second flavour: PBE96A second flavour: PBE96

The physics stable:The physics stable: Principled, parameter freePrincipled, parameter free Numerous analytic propertiesNumerous analytic properties Slow varying limit should give LDA Slow varying limit should give LDA

response. This requires Fresponse. This requires Fxx → →ss2 2 , , =0.21951=0.21951

Density scaling, n(Density scaling, n(rr)→)→33n(n(rr), E), Exx→→EExx

804.0,/1 2

s

sF

Correlation FunctionalsCorrelation Functionals

Perdew - Zunger 1986Perdew - Zunger 1986 Perdew Wang (1991) Perdew Wang (1991)

Part of parameter free PW91Part of parameter free PW91 Perdew, Burke, Ernzerhof (1996)Perdew, Burke, Ernzerhof (1996)

GGA made simple!GGA made simple! Parameter freeParameter free Simplified constructionSimplified construction Smoother, better behaved.Smoother, better behaved.

Lee Yang ParrLee Yang Parr

Different approach – based on Different approach – based on accurate wave functions for the accurate wave functions for the Helium atom.Helium atom.

No relation to the homogeneous No relation to the homogeneous electron gas at all.electron gas at all.

One empirical parameterOne empirical parameter Often combined with Becke Often combined with Becke

exchange to give BLYP.exchange to give BLYP.

Atomisation energies Atomisation energies (kcal/mol)(kcal/mol)

HFHF LSDLSD PBEPBE EXEXH2H2 8484 113113 105105 109109CH4CH4 328328 462462 420420 419419C2H2C2H2 294294 460460 415415 405405C2H4C2H4 428428 633633 571571 563563N2N2 115115 267267 243243 229229O2O2 3333 175175 144144 121121F2F2 -37-37 7878 5353 3939

Hybrid FunctionalsHybrid Functionals

Why not just add correlation to HF Why not just add correlation to HF calculations? We could write calculations? We could write EEXCXC=E=EXX[exact]+E[exact]+ECC[LSD][LSD]

Try it – error for G2 set is 32 kcal/mol, Try it – error for G2 set is 32 kcal/mol, similar to LDA [HF gives 78; best 5].similar to LDA [HF gives 78; best 5].

Why is this?Why is this?

Hybrid functionals [2]Hybrid functionals [2]

Correct XC hole is localised. Correct XC hole is localised. Exchange and correlation separately Exchange and correlation separately

are delocalised.are delocalised. DFT in LDA and GGA give localised DFT in LDA and GGA give localised

expressions for both parts.expressions for both parts. Sometimes simpler is better!Sometimes simpler is better!

Hybrid functionals [3]Hybrid functionals [3]

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

LSDC

LYPC

BXXC

LSDxc

LYPBxc EccEbEaEEaE 11 03

Today, most common is B3LYPToday, most common is B3LYP

9103 PWC

BX

LSDXXC

LSDxc

Bxc cEbEEEaEE

Gives mean unsigned error of 5 kcal/molGives mean unsigned error of 5 kcal/mol

Hybrid functionals [4]Hybrid functionals [4]

Admixture can be justified theoretically, Admixture can be justified theoretically, the work of PEB (96), BEP (97):the work of PEB (96), BEP (97):

Using PBE96 as the GGA gives the Using PBE96 as the GGA gives the PBE1PBE (or PBE0) functional.PBE1PBE (or PBE0) functional.

Nearly as good as B3LYPNearly as good as B3LYP

GGAX

HFX

GGAXC

hydbridXC EEEE 25.0

Meta GGAsMeta GGAs

Perdew 1999Perdew 1999 Better total energies.Better total energies. Ingredients: , KE densityIngredients: , KE density Very hard to find potential, so cannot do Very hard to find potential, so cannot do

SCF with this.SCF with this. Therefore structural optimisation not Therefore structural optimisation not

possible.possible.

n2

HSE03HSE03

Recent development. Several Recent development. Several motivations:motivations:

B3LYP more accurate than BLYP. B3LYP more accurate than BLYP. Some admixture of exchange needed.Some admixture of exchange needed.

Exact exchange is slow to calculate.Exact exchange is slow to calculate. Linear scaling K-builds don’t scale Linear scaling K-builds don’t scale

linearly in general.linearly in general. Plane wave based (physics) codes can’t Plane wave based (physics) codes can’t

easily find exact exchange.easily find exact exchange.

Screened ExchangeScreened Exchange

Key idea (Heyd, Scuseria 2003):Key idea (Heyd, Scuseria 2003):

First term is short-ranged; second long First term is short-ranged; second long ranged.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?

r

r

r

r

r

erferfc1

HSE03HSE03

SRPBE

xSRHF

xPBEx

LRPBEx

SRPBEx

SRHFx

LRPBEx

SRPBEx

LRHFx

SRHFx

PBEx

HFx

PBEx

EEaE

EEaaE

Ea

EaaEaE

EaaEE

,,

,,,

,

,,,

0

1

1

1

1

Where does this leave Where does this leave us?us?

Need to find short-ranged HF contribution.Need to find short-ranged HF contribution. Linear scalingLinear scaling Parallelism is perfectParallelism 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 Can also do with different splittings with only

minor modification:minor modification:

r

r

r

rxp

r

2222 exp1e1

Where does this leave Where does this leave us?us?

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

rrrr

rrrddxcExc

,

Modify the interaction to short ranged termModify 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 holeThe modified hole

...)](1[)9/4(1

1

erfc,

,9

8

,,

d,

22222

0

ysFsCBy

A

Ayy

AJ

k

yysJJ

dyysyJF

sFss

snE

PBE

F

PBEHSE

PBEPBEx

PBEx

LDAx

PBEx

PBEx

PBEx

rrrrr

rrrr

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

How about the accuracy?How about the accuracy?

Enthalpies of formation (kcal/mol):Enthalpies of formation (kcal/mol):MAE(G2)MAE(G2)

MAE(G3)MAE(G3)B3LYPB3LYP 3.043.04 4.314.31PBEPBE 17.1917.19 22.8822.88PBE0PBE0 5.155.15 7.297.29HSE03HSE03 4.644.64 6.576.57

Conclusion: competitive with hybrids.Conclusion: competitive with hybrids.

How about the accuracy?How about the accuracy?

Vibrational freqs (cm-1); 82 diatomicsVibrational freqs (cm-1); 82 diatomicsMAE(G2)MAE(G2)

B3LYPB3LYP 33.533.5PBEPBE 42.042.0PBE0PBE0 43.643.6HSE03HSE03 43.943.9

Conclusion: competitive with hybrids.Conclusion: competitive with hybrids.

How about the accuracy?How about the accuracy? Band Gaps (eV)Band Gaps (eV)

LDALDA PBEPBE HSEHSE EXPEXPCC 4.234.23 4.174.17 5.495.49 5.485.48SiSi 0.590.59 0.750.75 1.281.28

1.171.17GeGe 0.000.00 0.000.00 0.560.56 0.740.74GaAsGaAs 0.430.43 0.190.19 1.211.21 1.521.52GaNGaN 2.092.09 1.701.70 3.213.21 3.503.50MgOMgO 4.924.92 4.344.34 6.506.50 7.227.22

Has HSE got legs?Has HSE got legs?

Different separations?Different separations? Improved formalism for GGA then Improved formalism for GGA then

possible.possible. Standard applications: ZnO, Ge etc.Standard applications: ZnO, Ge etc. Effect on spectral calculations: EELSEffect on spectral calculations: EELS Possibility of multiplet calculations Possibility of multiplet calculations

for defect centres.for defect centres.