DFT approximations: Which one to use ?

P. Blaha and F. Tran Institute of Materials Chemistry

TU Wien

The accuracy of DFT calculations depends on

The particular DFT approximation LDA, various GGAs, meta-GGAs, hybrid-GGAs, vdW-functionals

The particular system and property Solids vs. Molecules specific atoms (heavy or less common element, 3d, 4f systems, …) Binding energy, phase stability geometry electronic structure (gaps, magnetic moments, NMR, IR, …)

The method (code) to solve the KS equations all electron codes pseudopotentials, PAW-versions, …

The person running these calculations convergence of all numerical parameters 2

The total energy in KS-DFT

Ts: kinetic energy of non-interacting electrons EH: electron-electron Coulomb energy Ven: electron-nucleus Coulomb energy Vnn: nucleus-nucleus Coulomb energy Exc = Ex + Ec: exchange-correlation energy

Ex is in principle known (HF-energy) Ec is unknown must be approximated in practice Ex and Ec must be “compatible approximations” (error cancellation)

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The grand challenge:

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Testing solids with Rung 1-4 DFT

Test on 44 strongly bound and17 weakly bound solids 15 GGAs 9 meta-GGAs 7 hybrid-GGAs 5 hybrid-mGGAs DFT-D3 13 vdW-DFT

light-heavy elements (Li - Au) metals-semiconductors-insulators

lattice parameters + cohesive energies + atomization energies of molecules (not presented here)

F. Tran et al., JCP 144, 204120 (2016); F.Tran et al., PRM 3, 063602 (2019)

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MARE of a0 and Ecoh (strongly bound solids)

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GGA m-GGA

hybr-DFT vdW-DFT

TM

MRE of a0 and Ecoh

7MRE a0 (%) MRE a0 (%)

MRE

Ec(%

)

M

RE E

c(%

) GGA

hybr-DFT

meta-GGA

DFT+D3

0.5%

5%

Trends in various groups of solids

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largest errors (reliability):

max. error in a0: GGA < DFT-D3 < hybr-DFT < mGGA (Cs,V)max. error in Ec: DFT-D3 ~ mGGA ~ GGA < hybr-GGA (TM)

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a0 Ec

a0 Eca0 Ec

mGGA

can we understand the trends ?

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accurate GGAs for solids very inaccurate for binding energies of molecules

GGAs have problems comparing open/dense structures

smax ~1-1.5 in solids(2.0 for alkali metals)

smax much larger in finite systems !

success of mGGAs ?

= 0: one and 2-electron regions very close and very far from nuclues = 1: regions of constant density >> 1: regions between closed shell atoms

mGGAs can be more flexible (Cs ??) SCAN is as good as the best GGAs for a0, but gives also

excellent EB for molecules SCAN predicts ground state structures much more reliable

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BE

Efficient first-principles prediction of solid stability: Towards chemical accuracy. Y.Zhang, … J.Perdew etal., npj Comp. Mat. (2018) 4:9

200 binary compounds

CsCl, SiO2

meta-GGAs seem good, but:

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bcc Fe

• SCAN overestimates magnetic moments and magnetic energies of itinerant systems drastically (D.Singh, S.B.Trickey, ..)

• Other m-GGAs (TPSS, TM) do not do this

hybrid-DFT: optimization of

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MRE a0 (%) MARE a0 (%)M

ARE

Ec(%

)

MRE

Ec(%

)

optimization of can further improve hybrid-DFTs

=0.15

and eg.screened hybrid PBEsol (=0.15) would be optimal, but:

=0.15

hybrid-DFT for transition metalsmagnetic moments of Fe (2.8B),

Co (1.9B), and Ni (0.9B) largely overestimated

Pd, Rh, Pt, … become ferro-magnetic using YS-PBE0 (HSE)!

F. Tran et al., PRB 86, 134406 (2012) F. Tran et al., PRMat 2, 023802 (2018) 14

Pt

LDA

LDA

PBE

PBE

NM-HSE

FM-HSE

strongly vs. weakly bound solids:

van der Waals - DFT

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van der Waals - DFT

Nonlocal vdW-density functionals (M.Dion et al., PRL 92,246401 (2004):

many combinations of various GGAs for X and C with different kernels probably the most promising methods in terms of cpu-time/performance Kernels proposed in literature:

DRSLL (Dion et al. 2004): derived from ACFDT, no adjustable parameter (Zab=-0.8491)

LMKLL (Lee et al. 2010): DRSLL kernel with Zab=-1.887 (r)VV10 (Vydrov+Van Voorhis 2010, Sabatini et al. 2013): different analytical

form and 2 parameters C and b

Exc = Ex(m)GGA + Ec

LDA/(m)GGA + EcNL

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Roman-Perez & Soler method (PRL 2009)

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but results converge to all-electron densities

Smoothing of an all-electron density (r)

smoothing depends only on , not on „position“ (core region,..) basis set independent method, forces and potential available

c is the density cutoffn = 1,2, … (the first n derivatives of s are continuous) A = 1 Bohr3n (consistency of units)

which c and n to choose ?

F.Tran et al., PRB 96, 054103 (2017)

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Smoothing of depending on n and c

a plane wave expansion converges faster for n=1 no benefit using n>1

a plane wave expansion converges faster for smaller c but too small values of c may influence the results

(n=1,2,3 (c = 1.5 Bohr−3) c = 1.5, 2.0, 2.5 Bohr−3 (n=1)

Au

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Au

Convergence of c with FFT cutoff Gmax

h-BN (a vdW system):

good convergence already with c ~0.3 Bohr-3 and Gmax>30 Bohr-1 are necessary

tests on many differentsystems show that for usual accuracy we can use globally:

c ~0.3 Bohr-3 and Gmax~30 Bohr-1

which leads to a very fast method. 20

Tests of 13 different vdW functionals

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They differ by their semilocal part and their nl-vdW part,combining hard/soft GGAs with hard/soft vdW functionals.

weakly bound layered solids Graphite, h-BN, TiS2, TiSe2, MoS2, MoSe2, MoTe2, HfTe2, WS2, WSe2

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 MARE (%)

0 2 4 6 8 10 12 14 16 18 MARE (%)

0 20 40 60 80 100 120 140MARE (%)

TM is the only acceptable semilocal functionalseveral modern vdW functionals perform well

and the winner is:

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strongly bound solids weakly bound solids

strongly bound solids: SCAN,SCAN+rVV10L,optB86b,rev-vdW-DF2,PBE+rVV10L,vdW-DF-cx

weakly bound solids: PBEsol+rVV10s,rev-vdW-DF2

probably the most balanced functional: rev-vdW-DF2

band gaps

(testing the xc-potential)

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band gaps

Eg = I - A (Ionization potential - electron affinity)

KS theory: Eg = EgKS + XC XC derivative discontinuity

GKS theory: XC is included due to the non-local potential

GGAs: (often adapted for bandgaps), but neglecting XC LDA, PBE, Sloc (Vx ~0.3); EV93, AK13, HLE16 (Vx*1.5,Vc*0.5), LB94

hybrid-DFT (in various flavors, eg. HSE ~ YS-PBE0) expensive

GW (in various flavors) very expensive

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band gaps

mGGA-potentials (no corresponding EXC exists) Becke-Johnson potential: approximate Vx

OEP in atoms (2006) TB-mBJ: adapted for band gaps (F.Tran and PB, PRL 102, 226401 (2009)

gets recalculated for specific system

GLLB-SC potential (including XC , Kuisma etal. PRB 82, 115106 (2010))

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approximate KLI response term

Exp. vs. theoretical bandgaps

PBE: too small gaps

HSE: good, but too smallgaps for insulators

27D.Koller etal., J. Phys.: Condens. Matter 25 (2013) 435503

variable 0, leads to ~0.6 for LiF and good gaps

Exp. vs. theoretical bandgaps

TB-mBJ: very good gaps forall materials except Cu+

(CuCl, Cu2O,..)

GLLB-SC: very good gaps formany materials, but:

too small gaps (metal) for smallgap semiconductors (InAs,..) orFeO, and

too large gaps for MgS, MgSe,Fe2O3 (exchange splitting)

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Fe2O3

FeO

PBE AK13 GLLB-SC HSE TB-mBJMARE (%) 53 24 24 17 15MAE (eV) 1.99 0.75 0.64 0.82 0.47

Cu2OCuBr

CuCl

NMR chemical shifts: the slope - problem

hybrid-DFT is the standard method in CS calculations of molecules (Gaussian)

for (ionic) solids YS-PBE0 (HSE) gives a much too largecorrection (smaller mixing ??)

original BJ (OEP-X) works muchbetter

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Accuracy of pseudopotential calculations

all-electron vs. pseudopotential (PAW) methods ? We often compare our WIEN2k results with (recent) VASP If VASP was done carefully, usually very good agreement is found for

PBE calculations:

NMR chemical shifts (for light elements): perfect agreement with special PAWs

results for “strange” elements/compounds ?

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VASP min (+1.5%) VASP min (+10%)

WIEN2k WIEN2k

TB-mBJ: WIEN2k vs. VASP

Very good agreement for semiconductors (gap < 5 eV) VASP strongly underestimates mBJ gaps larger than 5 eV

Be carefull when using non-optimized PP (DFT or property)31

PBE mBJ exp.VASP WIEN2k VASP WIEN2k

Si 0.58 0.62 1.20 1.17 1.17

GaAs 0.52 0.57 1.55 1.64 1.52

C 4.14 4.15 4.96 4.93 5.48SiO2 6.01 6.02 8.23 8.98 8.90Ar 8.71 8.71 12.55 13.91 14.15

LiF 9.09 9.09 12.08 12.94 13.60

Ne 11.59 11.58 18.85 22.72 21.7

Summary hybrid-DFT is quite ok, but expensive (improves „correlated“ TMO, but

fails for many metallic TMs (fixed !!) and vdW-systems) m-GGAs are more reliable than GGAs (structure prediction, chemical

accuracy ?), work for finite and infinite systems nl-vdW functionals: many of them give either good a0 or Ecoh for weakly

bound solids, only

rev-vdW-DF2 (Hamada, Phys. Rev. B 89, 121103(R) (2014))seems to be the best functional for strongly and weakly bound solids

band gaps: TB-mBJ still most reliable fast method, GLLB-type potentials may offer an alternative

Thank you for your attention !