Methods available in WIEN2k for the treatment of exchange and correlation effects. F. Tran
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1 Methods available in WIEN2k for the treatment of exchange and correlation effects F. Tran Institute of Materials Chemistry Vienna University of Technology, A-1060 Vienna, Austria 22th WIEN2k workshop, June 2015, IHCP, Singapore W I E N 2k
2 Outline of the talk Introduction Semilocal functionals: GGA and MGGA mbj potential Keywords The DFT-D3 method for dispersion On-site methods for strongly correlated electrons: DFT+U Hybrid-DFT Hybrid functionals GW
3 Total energy in Kohn-Sham DFT 1 E tot = 1 ψ i (r) 2 d 3 r + 1 ρ(r)ρ(r ) 2 2 r r d3 rd 3 r i }{{}}{{} E T H s + 1 Z A Z B +E xc 2 R A R B A,B A B } {{ } V nn T s : kinetic energy of non-interacting electrons + E H : electron-electron electrostatic Coulomb energy V en : electron-nucleus electrostatic Coulomb energy V nn : nucleus-nucleus electrostatic Coulomb energy E xc = E x +E c : exchange-correlation energy Problem with E xc : the exact form of E c is too complicated to be useable = approximations have to be used in practice 1 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965) v nuc(r)ρ(r)d 3 r } {{ } V en
4 Approximations for E xc (Jacob s ladder 1 ) E xc = ǫ xc (r)d 3 r 1. Local density approximation (LDA): ǫ xc = f (ρ) 2. Generalized gradient approximation (GGA): ǫ xc = f (ρ, ρ) 3. Meta-GGA: ǫ xc = f ( ρ, ρ, 2 ρ,t ), t = 1 2 i ψ i 2 4. The use of occupied orbitals (e.g., Hartree-Fock) 5. The use of unoccupied orbitals (e.g., RPA) 1 J. P. Perdew et al., J. Chem. Phys. 123, (2005)
5 The one-electron Schrödinger equations Minimization of E tot leads to ( 1 ) 2 2 +v en (r)+v H (r)+ ˆv xc (r) ψ i (r) = ǫ i ψ i (r) Two ways of calculating ˆv xc : ˆv xc = δe xc /δρ = v xc (KS method) ˆv xc = (1/ψ i )δe xc /δψ i = v xc,i (generalized KS 1, e.g., HF) Remarks: If E xc is an explicit functional of ρ (LDA and GGA), then both possibilities lead to the same potential v xc. MGGA (with t) and hybrid functionals are usually implemented in the generalized KS method. 1 A. Seidl et al., Phys. Rev. B 53, 3764 (1996)
6 Semilocal functionals: GGA ǫ GGA xc (ρ, ρ) = ǫ LDA xc (ρ)f xc (r s,s) where F xc is the enhancement factor and r s = 1 ( 4 3 πρ) 1/3 (Wigner-Seitz radius) s = ρ 2(3π 2 ) 1/3 ρ 4/3 (inhomogeneity parameter) There are two types of GGA: Semi-empirical: contain parameters fitted to accurate (i.e., experimental) data. Ab initio: All parameters were determined by using mathematical conditions obeyed by the exact functional.
7 Semilocal functionals: GGA F x (s) = ǫ GGA x /ǫ LDA x LDA PBEsol PBE B88 good for atomization energy of molecules good for atomization energy of solids good for lattice constant of solids F x exact for homogeneous electron gas inhomogeneity parameter s
8 Construction of an universal GGA: A failure (up to now) Attempts: RGE2 1, PBEint 2, HTBS 3 8 Results for 44 solids Mean absolute relative error for cohesive energy (%) 6 4 PBEsol WC RGE2 PBE PBEint HTBS 2 PBEsol and WC are the worst for atomization energy of molecules PBEint, RGE2 and PBE not good for atomization energy of molecules HTBS is the best for atomization energy of molecules HTBS very good if alkali and alkali earth are excluded Mean absolute relative error for lattice constant (%) 1 A. Ruzsinszky et al., J. Chem. Theory Comput. 5, 763 (2009) 2 E. Fabiano et al., Phys. Rev. B 82, (2010) 3 P. Haas et al., Phys. Rev. B 83, (2011)
9 Semilocal functionals: meta-gga ǫ MGGA xc (ρ, ρ,t) = ǫ LDA xc (ρ)f xc (r s,s,α) where F xc is the enhancement factor and α = t t W t TF α = 1 where the electron density is uniform α = 0 in one- and two-electron regions α 1 between closed shell atoms = MGGA functionals are more flexible Example: MGGA MS2 1 is as good as the best GGA for atomization energies of molecules as good as the best GGA for lattice constant of solids maybe not too bad for van der Waals systems 1 J. Sun et al., Phys. Rev. Lett. 111, (2013)
10 Semilocal functionals: meta-gga F x (s,α) = ǫ MGGA x /ǫ LDA x LDA PBEsol PBE MGGA MS2 (α=0) MGGA MS2 (α=1) F x inhomogeneity parameter s
11 Semilocal functionals: MGGA MS2 8 Results for 44 solids Mean absolute relative error for cohesive energy (%) 6 4 PBEsol WC MGGA_MS2 PBEint MGGA_MS2 very good for molecules RGE2 PBE HTBS 2 PBEsol and WC are the worst for atomization energy of molecules PBEint, RGE2 and PBE not good for atomization energy of molecules HTBS is the best for atomization energy of molecules HTBS very good if alkali and alkali earth are excluded Mean absolute relative error for lattice constant (%)
12 Semilocal functionals: modified Becke-Johnson potential Modified Becke-Johnson (mbj) potential: 1 vx mbj (r) = cvx BR (r)+(3c 2) 1 5 t(r) π 6 ρ(r) where vx BR is the Becke-Roussel potential, t is the kinetic-energy density and c is given by p c = α+β 1 ρ(r) d 3 r ρ(r) V cell cell 1 F. Tran and P. Blaha, Phys. Rev. Lett. 102, (2009)
13 Band gaps with mbj LDA mbjlda HSE G 0 W 0 GW Theoretical band gap (ev) ScN GaAs SiC FeO AlP CdS GaN ZnO MnO ZnS NiO C BN AlN MgO LiCl Xe Kr LiF Ar 2 Si Ge Experimental band gap (ev)
14 How to run a calculation with the mbj potential? 1. init lapw (choose LDA or PBE) 2. init mbj lapw (create/modify files) 2.1 automatically done: case.in0 modified and case.inm vresp created 2.2 run(sp) lapw -i 1 -NI (creates case.r2v and case.vrespsum) 2.3 save lapw 3. init mbj lapw and choose one of the parametrizations: 0: Original mbj values 1 1: New parametrization 2 2: New parametrization for semiconductors 2 3: Original BJ potential 3 4. Eventually: edit case.inm and choose PRATT with small mixing factor smaller than run(sp) lapw... 1 F. Tran and P. Blaha, Phys. Rev. Lett. 102, (2009) 2 D. Koller et al., Phys. Rev. B 85, (2012) 3 A. D. Becke and E. R. Johnson, J. Chem. Phys. 124, (2006)
15 Input file case.in0: keywords for the xc-functional The functional is specified at the first line of case.in0. Three possibilities: 1. Specify a global keyword for E x, E c, v x, v c : TOT XC NAME 2. Specify a keyword for E x, E c, v x, v c individually: TOT EX NAME1 EC NAME2 VX NAME3 VC NAME4 3. Specify keywords to use functionals from LIBXC 1 : TOT XC TYPE X NAME1 XC TYPE C NAME2 TOT XC TYPE XC NAME where TYPE is the family name: LDA, GGA or MGGA 1 M. A. L. Marques et al., Comput. Phys. Commun. 183, 2272 (2012)
16 Input file case.in0: examples with keywords PBE: TOT XC PBE or TOT EX PBE EC PBE VX PBE VC PBE or TOT XC GGA X PBE XC GGA C PBE mbj (with LDA for the xc-energy): TOT XC MBJ MGGA MS2: TOT XC MGGA MS }{{} κ,c,b All available functionals are listed in inputpars.f and vxclm2.f (SRC lapw0 directory) and in $WIENROOT/SRC lapw0/xc funcs.h for LIBXC (if installed)
17 Dispersion methods for DFT Problem with semilocal functionals: They do not include London dispersion interactions Results are qualitatively wrong for systems where dispersion plays a major role Two common dispersion methods for DFT: Pairwise term 1 : E PW c,disp = A<B n=6,8,10,... f damp n (R AB ) CAB n R n AB Nonlocal term 2 : E NL c,disp = 1 2 ρ(r)φ(r,r )ρ(r )d 3 rd 3 r 1 S. Grimme, J. Comput. Chem. 25, 1463 (2004) 2 M. Dion et al., Phys. Rev. Lett. 92, (2004)
18 The DFT-D3 method 1 in WIEN2k Features of DFT-D3: Very cheap (pairwise) depend on positions of the nuclei (via coordination number) Functional-dependent parameters Energy and forces (minimization of internal parameters) 3-body term C AB n Installation: Not included in WIEN2k Download and compile the DFTD3 package from copy the dftd3 executable in $WIENROOT input file case.indftd3 run(sp) lapw -dftd S. Grimme et al., J. Chem. Phys. 132, (2010)
19 The DFT-D3 method: the input file case.indftd3 Default (and recommended) input file: method bj damping function fn damp func default the one in case.in0 grad yes forces pbc yes periodic boundary conditions abc yes 3-body term cutoff 95 interaction cutoff cnthr 40 coordination number cutoff num no numerical gradient default will work for PBE, PBEsol, BLYP and TPSS. For other functionals, the functional name has to be specified (see dftd3.f of DFTD3 package)
20 The DFT-D3 method: hexagonal BN PBE BLYP PBE+D3 BLYP+D3 Total energy (mry/cell) exp Interlayer distance (Å)
21 Strongly correlated electrons Problem with semilocal functionals: They give qualitatively wrong results for solids which contain localized 3d or 4f electrons Why? The band gap is too small or even absent like in FeO The magnetic moments are too small Wrong ground state The strong on-site correlations are not correctly accounted for by semilocal functionals. (Partial) solution to the problem: Combine semilocal functionals with Hartree-Fock theory: DFT+U Hybrid Even better: LDA+DMFT (DMFT codes using WIEN2k orbitals as input exist)
22 On-site DFT+U and hybrid methods in WIEN2k For solids, the hybrid functionals are computationally very expensive. In WIEN2k the on-site DFT+U 1 and on-site hybrid 2 methods are available. These methods are approximations of the Hartree-Fock/hybrid methods Applied only inside atomic spheres of selected atoms and electrons of a given angular momentum l. On-site methods As cheap as LDA/GGA. 1 V. I. Anisimov et al., Phys. Rev. B 44, 943 (1991) 2 P. Novák et al., Phys. Stat. Sol. (b) 243, 563 (2006)
23 DFT+U and hybrid exchange-correlation functionals The exchange-correlation functional is E DFT+U/hybrid xc = E DFT xc [ρ]+e onsite [n mm ] where n mm is the density matrix of the correlated electrons For DFT+U both exchange and Coulomb are corrected: E onsite = Ex HF +E }{{ Coul } correction Ex DFT E DFT Coul }{{} double counting There are several versions of the double-counting term For the hybrid methods only exchange is corrected: E onsite = αe HF x }{{} corr. where α is a parameter [0,1] αe LDA x }{{} d. count.
24 How to run DFT+U and on-site hybrid calculations? 1. Create the input files: case.inorb and case.indm for DFT+U case.ineece for on-site hybrid functionals (case.indm created automatically): 2. Run the job (can only be run with runsp lapw): LDA+U: runsp lapw -orb... Hybrid: runsp lapw -eece... For a calculation without spin-polarization (ρ = ρ ): runsp c lapw -orb/eece...
25 Input file case.inorb LDA+U applied to the 4f electrons of atoms No. 2 and 4: nmod, natorb, ipr PRATT,1.0 mixmod, amix iatom, nlorb, lorb iatom, nlorb, lorb 1 nsic (LDA+U(SIC) used) U J (Ry) U J (Ry) nsic=0 for the AMF method (less strongly correlated electrons) nsic=1 for the SIC method nsic=2 for the HMF method
26 Input file case.ineece On-site hybrid functional PBE0 applied to the 4f electrons of atoms No. 2 and 4: emin, natorb iatom, nlorb, lorb iatom, nlorb, lorb HYBR HYBR/EECE 0.25 fraction of exact exchange
27 SCF cycle of DFT+U in WIEN2k lapw0 orb -up orb -dn lapw1 -up -orb lapw1 -dn -orb lapw2 -up lapw2 -dn lapwdm -up lapwdm -dn lcore -up lcore -dn v DFT xc,σ +v ee +v en (case.vspup(dn), case.vnsup(dn)) v mm (case.vorbup) v mm (case.vorbdn) ψ nk,ǫ nk (case.vectorup, case.energyup) ψ nk,ǫ nk (case.vectordn, case.energydn) ρ val (case.clmvalup) ρ val (case.clmvaldn) n mm (case.dmatup) n mm (case.dmatdn) ρ core (case.clmcorup) ρ core (case.clmcordn) mixer mixed ρ σ and n σ mm
28 Hybrid functionals On-site hybrid functionals can be applied only to localized electrons Full hybrid functionals are necessary (but expensive) for solids with delocalized electrons (e.g., in sp-semiconductors) Two types of full hybrid functionals available in WIEN2k 1 : unscreened: E xc = E DFT xc screened (short-range), 1 E xc = E DFT xc +α ( E HF x Ex DFT r r e λ r r r r : +α ( E SR HF x ) Ex SR DFT ) screening leads to faster convergence with k-points sampling 1 F. Tran and P. Blaha, Phys. Rev. B 83, (2011)
29 Hybrid functionals: technical details times more expensive than LDA/GGA k-point and MPI parallelization Approximations to speed up the calculations: Reduced k-mesh for the HF potential. Example: For a calculation with a k-mesh, the reduced k-mesh for the HF potential can be: 6 6 6, 4 4 4, 3 3 3, or Non-self-consistent calculation of the band structure Underlying functionals for unscreened and screend hybrid: LDA PBE WC PBEsol B3PW91 B3LYP Use run bandplothf lapw for band structure
30 Hybrid functionals: input file case.inhf Example for YS-PBE0 (similar to HSE06 from Heyd, Scuseria and Ernzerhof 1 ) 0.25 fraction α of HF exchange T screened (T, YS-PBE0) or unscreened (F, PBE0) screening parameter λ 20 number of bands for the 2nd Hamiltonian 6 GMAX 3 lmax for the expansion of orbitals 3 lmax for the product of two orbitals 1d-3 radial integrals below this value neglected Important: The computational time will depend strongly on the number of bands, GMAX, lmax and the number of k-points 1 A. V. Krukau et al., J. Chem. Phys. 125, (2006)
31 How to run hybrid functionals? 1. init lapw 2. Recommended: run(sp) lapw for the semilocal functional 3. save lapw 4. init hf lapw (this will create/modify input files) 4.1 adjust case.inhf according to your needs 4.2 reduced k-mesh for the HF potential? Yes or no 4.3 specify the k-mesh 5. run(sp) lapw -hf (-redklist) (-diaghf)...
32 SCF cycle of hybrid functionals in WIEN2k lapw0 -grr v DFT x (case.r2v), αe DFT x (:AEXSL) lapw0 lapw1 v DFT xc +v ee +v en (case.vsp, case.vns) ψnk DFT,ǫ DFT nk (case.vector, case.energy) lapw2 nk ǫdft nk (:SLSUM) hf lapw2 -hf lcore mixer ψ nk,ǫ nk (case.vectorhf, case.energyhf) ρ val (case.clmval) ρ core (case.clmcor) mixed ρ
33 Calculation of quasiparticle spectra from many-body theory In principle the Kohn-Sham eigenvalues should be viewed as mathematical objects and not compared directly to experiment (ionization potential and electron affinity). The true addition and removal energies ǫ i are calculated from the equation of motion for the Green function: ( 1 ) 2 2 +v en(r)+v H(r) + Σ(r,r,ǫ i )ψ i (r )d 3 r = ǫ i ψ i (r) The self-energy Σ is calculated from Hedin s self-consistent equations 1 : Σ(1,2) = i G(1,4)W(1 +,3)Γ(4,2,3)d(3,4) W(1,2) = v(1,2) + v(4, 2)P(3, 4)W(1, 3)d(3, 4) P(1,2) = i G(2, 3)G(4, 2)Γ(3, 4, 1)d(3, 4) δσ(1,2) Γ(1,2,3) = δ(1,2)δ(1,3) + δg(4,5) G(4,6)G(7,5)Γ(6,7,3)d(4,5,6,7) 1 L. Hedin, Phys. Rev. 139, A769 (1965)
34 The GW and G 0 W 0 approximations GW: vertex function Γ in Σ set to 1: Σ(1,2) = i G(1,4)W(1 +,3)Γ(4,2,3)d(3,4) ig(1,2 + )W(1,2) Σ(r,r,ω) = i G(r,r,ω +ω )W(r,r,ω )e iδω dω 2π G(r,r ψ i (r)ψi (r ),ω) = W(r,r,ω) = v(r,r )ǫ 1 (r,r,ω)d 3 r ω ǫ i=1 i iη i G 0 W 0 (one-shot GW): G and W are calculated using the Kohn-Sham orbitals and eigenvalues. 1 st order perturbation theory gives ǫ GW i = ǫ KS i +Z(ǫ KS i ) ψi KS R(Σ(ǫ KS i )) v xc ψi KS
35 A few remarks on GW GW calculations require very large computational ressources G and W depend on all (occupied and unoccupied) orbitals (up to parameter emax in practice) GW is the state-of-the-art for the calculation of (inverse) photoemission spectra, but not for optics since excitonic effects are still missing in GW (BSE code from R. Laskowski) GW is more accurate for systems with weak correlations
36 FHI-gap: a LAPW GW code 1 Based on the FP-LAPW basis set Mixed basis set to expand the GW-related quantities Interfaced with WIEN2k G 0 W 0, GW Parallelized 1 H. Jiang et al., Comput. Phys. Comput. 184, 348 (2013)
37 Flowchart of FHI-gap
38 How to run the FHI-gap code? 1. Run a WIEN2k SCF calculation (in w2kdir) 2. In w2kdir, execute the script gap init to prepare the input files for GW: gap init -d <gwdir> -nkp <nkp> -s 0/1/2 -orb -emax <emax> 3. Eventually modify gwdir.ingw 4. Execute gap.x or gap-mpi.x in gwdir 5. Analyse the results from: 5.1 gwdir.outgw 5.2 the plot of the DOS/band structure generated by gap analy
39 Parameters to be converged for a GW calculation Usual WIEN2k parameters: Size of the LAPW basis set (RKmax ) Number of k-points for the Brillouin zone integrations GW-specific parameters: Size of the mixed basis set Number of unoccupied states (emax) Number of frequencies ω for the calculation of Σ = GWdω
40 Band gaps LDA mbjlda HSE G 0 W 0 GW Theoretical band gap (ev) ScN GaAs SiC FeO AlP CdS GaN ZnO MnO ZnS NiO C BN AlN MgO LiCl Xe Kr LiF Ar 2 Si Ge Experimental band gap (ev)
41 Thank your for your attention!
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