Methods available in WIEN2k for the treatment of exchange and correlation effects
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This document summarizes methods available in the WIEN2k software for treating exchange and correlation effects beyond semilocal density functional theory. It discusses the semilocal generalized gradient approximation and meta-GGA functionals, the modified Becke-Johnson potential for improving band gaps, dispersion correction methods, and on-site corrections like DFT+U and hybrid functionals for strongly correlated materials. Input parameters and keywords for selecting these methods in the WIEN2k code are also outlined.
![DFT+U and hybrid exchange-correlation functionals
The exchange-correlation functional is
xc
EDFT+ U/hybrid DFT onsite= Exc [ρ] + E [nmm′ ]
where nmm′ is the density matrix of the correlated electrons
◮ For DFT+U both exchange and Coulomb are corrected:
Eonsite = EHF DFT DFT
x + ECoul −Ex −ECoul
s
corr
¸
ec
¸
tion
x s
double
¸
c
¸
ounting
x
There are several versions of the double-counting term
◮ For the hybrid methods only exchange is corrected:
Eonsite = αEHF − αELDA
x
d
s
.c
¸
o
¸
unt
x
.
x
s
c
¸
or
¸
r.
x
where αis a parameter ∈ [0,1]](https://image.slidesharecdn.com/ft-xc-effects-170116212154/85/Methods-available-in-WIEN2k-for-the-treatment-of-exchange-and-correlation-effects-23-320.jpg)
Methods available in WIEN2k for the treatment of exchange and correlation effects
- 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 23rd WIEN2k workshop, 4-7 June2016, McMaster University, Hamilton, Canada I W E N 2k
- 2. Outline of the talk ◮ Introduction ◮ Semilocal functionals: ◮ GGA and MGGA ◮ mBJ potential (for band gap) ◮ Input file case.in0 ◮ The DFT-D3 method for dispersion ◮ On-site methods for strongly correlated electrons: ◮ DFT+U ◮ Hybrid functionals ◮ Hybrid functionals ◮ GW
- 3. Total energy in Kohn-Sham DFT 1 Et ot = 1 2 . i ¸ i s x¸ T ¸ s 2 3 |∇ψ (r)| d r + 2 1 ¸ ¸ ρ(r)ρ(r′ ) 3 3 ′ s |r − r′| d rd r + ¸ en 3 v (r)ρ(r)d r x s x E ¸ e ¸ e E ¸ e ¸ n A,B Aƒ= B 2 |RA − RB| s x E ¸ n ¸ n + 1 . ZAZB +Exc ◮ Ts : kinetic energy of the non-interacting electrons ◮ Eee : electron-electron electrostatic Coulomb energy ◮ Een : electron-nucleus electrostatic Coulomb energy ◮ Enn : nucleus-nucleus electrostatic Coulomb energy ◮ Exc = Ex + Ec : exchange-correlation energy Approximations for Exc have to be used in practice =⇒ The reliability of the results depends mainly on Exc! 1 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)
- 4. Approximations for Exc (Jacob’s ladder 1) Exc = ¸ ǫxc (r)d 3r The accuracy, but also the computational cost, increase when climbing up the ladder 1 J. P. Perdew et al., J. Chem. Phys. 123, 062201 (2005)
- 5. The Kohn-Sham Schr¨odinger equations Minimization of Etot leads to . 1 . − ∇2 + vee(r) + ven(r) + vˆxc(r) ψi(r) = ǫiψi(r) 2 Two types of vˆxc: ◮ Multiplicative: vˆxc = δExc/δρ = vxc (KS method) ◮ LDA ◮ GGA i ,◮ Non-multiplicative: vˆxc = (1/ψi )δExc/δψ∗ = vxc i (generalized KS) ◮ Hartree-Fock ◮ LDA+U ◮ Hybrid (mixing of GGA and Hartree-Fock) ◮ MGGA ◮ Self-interaction corrected (Perdew-Zunger)
- 6. Semilocal functionals: trends with GGA ǫGGA xc LDA(ρ, ∇ρ)= ǫx (ρ)Fxc(rs,s) where Fxc is the enhancement factorand rs = 1 . 4 3πρ .1/3 (Wigner-Seitz radius) s= |∇ρ| 2(3π2)1/3 ρ4/3 (inhomogeneity parameter) There are two types of GGA: ◮ Semi-empirical: contain parametersfitted to accurate (i.e., experimental) data. ◮ Ab initio: All parametersweredetermined byusing mathematical conditions obeyed by the exact functional.
- 7. Semilocal functionals: GGA Fx(s) = ǫGGA/ǫLDA x x 0 0.5 3 0.9 1 1.4 1.3 1.2 1.1 1.5 1.6 1.7 1.8 1 1.5 2 2.5 inhomogeneity parameter s Fx LDA PBEsol PBE B88 good for atomization energy of molecules good for atomization energy of solids good for lattice constant of solids good for nothing
- 8. Construction of an universal GGA: A failure Test of functionals on 44 solids1 0 2 0 4 6 8 PBEsol WC RGE2 PBE PBEint PBEalpha SG4 0.5 1 1.5 Mean absolute percentage error for latticeconstant Meanabsolutepercentageerrorforcohesiveenergy 2 The accurate GGA for solids (cohesive energy/lattice constant). They are ALL very inaccurate for the atomization of molecules 1 F. Tran et al., J. Chem. Phys. 144, 204120 (2016)
- 9. Semilocal functionals: meta-GGA ǫMGGA xc LDA(ρ, ∇ρ, t) = ǫxc (ρ)Fxc(rs,s,α) where Fxc is the enhancement factorand ◮ α= t−tW t T F ◮ α= 1 where the electron density is uniform ◮ α= 0 in one- and two-electron regions ◮ α≫1between closed shell atoms =⇒ MGGA functionals are more flexible Example: SCAN1 is ◮ as good as the best GGA for atomization energies of molecules ◮ as good as the best GGA for lattice constant of solids 1 J. Sun et al., Phys. Rev. Lett. 115, 036402 (2015)
- 10. Semilocal functionals: meta-GGA 0 0.5 2.5 3 1.2 1.1 1 0.9 1.3 1.4 1.5 1 1.5 2 inhomogeneity parameter s Fx Fx(s,α) = ǫMGGA/ǫLDA x x 1.6 LDA PBE PBEsol SCAN (=0) SCAN (=1) SCAN (=5)
- 11. Semilocal functionals: MGGA MS2 andSCAN Test of functionals on 44 solids1 0 0 2 4 6 8 PBEsol WC PBEint PBEalpha RGE2 PBE SG4 MGGA_MS2 SCAN Meanabsolutepercentageerrorforcohesiveenergy The accurate GGA for solids (cohesive energy/lattice constant). They are ALL very inaccurate for the atomization of molecules MGGA_MS2 and SCAN are very accurate for the atomization of molecules 0.5 1 1.5 2 Mean absolute percentage error for latticeconstant 1 F. Tran et al., J. Chem. Phys. 144, 204120 (2016)
- 12. Semilocal potential for band gap: modified Becke-Johnson ◮ Standard LDA and GGA functionals underestimate the band gap ◮ Hybrid and GWaremuchmoreaccurate, but also muchmore expensive ◮ A cheapalternative is to usethe modified Becke-Johnson(mBJ) potential: 1 vmBJ x BR(r) = cvx (r) + (3c − 2) 1 . . 5 t(r) π 6 ρ(r) xwherev BR is the Becke-Roussel potential, t is the kinetic-energy density and c is given by c = α+ β V cell ¸ cell 1 |∇ρ(r)| ρ(r) 3 d r p mBJ is a MGGA potential 1 F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009)
- 13. Bandgapswith mBJ 0246810121416 0 2 4 6 8 10 12 14 16 Experimentalbandgap(eV) Theoretical band gap (eV) Kr Xe Si Ar LiF LiCl Ge ScN GaAs SiC FeO AlP CdS C AlN BN MgO ZnOGaN MnO NiO ZnS LDA mBJ HSE GW00 GW
- 14. How to run a calculation with the mBJ potential? 1. init lapw (choose LDA or PBE) 2. init mbj lapw (create/modify files) 1. automatically done: case.in0modified and case.inm vresp created 2. run(sp) lapw -i 1 -NI (creates case.r2v and case.vrespsum) 3. save lapw 3. init mbj lapw and chooseoneof the parametrizations: 0: Original mBJ values1 1: New parametrization2 2: New parametrization for semiconductors2 3: Original BJ potential3 4. run(sp) lapw ... 1 F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009) 2 D. Koller et al., Phys. Rev. B 85, 155109 (2012) 3 A. D. Becke and E. R. Johnson, J. Chem. Phys. 124, 221101 (2006)
- 15. Input file case.in0: keywords for the xc-functional The functional is specifiedat the 1st line of case.in0.Three different ways: 1. Specifya global keyword for Ex, Ec, vx, vc: ◮ TOT XC NAME 2. Specify a keyword for Ex, Ec, vx, vc individually: ◮ TOT EX NAME1 EC NAME2 VX NAME3 VCNAME4 3. Specify keywords to use functionals from LIBXC1: ◮ 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) http://www.tddft.org/programs/octopus/wiki/index.php/Libxc
- 16. Input file case.in0: examples with keywords ◮ PBE: TOT XC PBE or TOT EX PBE EC PBE VX PBE VCPBE 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 0.504 0.14601 4.0 s x κ ¸ ,c ¸ ,b All available functionals are listed in tables of the UG. 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 ◮ Resultsarequalitatively wrong for systemswhere dispersion plays a majorrole Two common dispersion methodsfor DFT: ◮ Pairwise term1: EPW c,disp = − . . A<B n=6,8,10,... dampfn (RAB ) CAB n Rn AB ◮ Nonlocal term2: ENL c,disp 2 1 ¸ ¸ = ρ(r)φ(r, r′)ρ(r′)d3rd3r′ 1 S. Grimme, J. Comput. Chem. 25, 1463 (2004) 2 M. Dion et al., Phys. Rev. Lett. 92, 246401 (2004)
- 18. The DFT-D3 method1 in WIEN2k ◮ Features of DFT-D3: ◮ Very cheap (pairwise) ◮ CAB n depend on positions of the nuclei (via coordination number) ◮ Functional-dependent parameters ◮ Energy and forces (minimization of internal parameters) ◮ 3-body term ◮ Installation: ◮ Not included in WIEN2k ◮ Download and compile the DFTD3 packagefrom http://www.thch.uni-bonn.de/tc/index.php copy the dftd3 executable in $WIENROOT ◮ input file case.indftd3 (if not present a default one is copied automatically) ◮ run(sp) lapw -dftd3 . . . ◮ case.scfdftd3 is included in case.scf 1 S. Grimme et al., J. Chem. Phys. 132, 154104 (2010)
- 19. The DFT-D3 method: the input file case.indftd3 Default (and recommended) input file: damping function f damp n the one in case.in0∗ forces method func grad pbc abc cutoff cnthr num bj def ault yes yes yes 95 40 no periodic boundary conditions 3-body term interaction cutoff coordination number cutoff numerical gradient ∗default will work for PBE, PBEsol, BLYP and TPSS. For other functionals, the functional namehasto bespecified(seedftd3.f of DFTD3 package)
- 20. The DFT-D3 method: hexagonal BN1 3 3.2 3.4 3.6 4.4 4.6 4.8 5 0 1.4 1.2 1 0.8 0.6 0.4 0.2 1.6 1.8 2 exp 3.8 4 4.2 Interlayer distance (A˚) Totalenergy(mRy/cell) PBE BLYP PBE+D3 BLYP+D3 1 F. Tran et al., J. Chem. Phys. 144, 204120 (2016)
- 21. Strongly correlated electrons Problem with semilocal functionals: ◮ They give qualitatively wrong results for solids which contain localized 3d or 4f electrons ◮ The band gap is too small or even absent like in FeO ◮ The magnetic moments are too small ◮ Wrong ground state Why? ◮ The strong on-site correlations arenot correctly accountedfor by semilocal functionals. (Partial) solution to the problem: ◮ Combine semilocal functionals with Hartree-Fock theory: ◮ DFT+U ◮ Hybrid Even better: ◮ LDA+DMFT (DMFT codesusing WIEN2k orbitals asinput exist)
- 22. On-site DFT+U and hybrid methods in WIEN2k ◮ Forsolids,the hybrid functionals arecomputationally very expensive. ◮ In WIEN2k the on-site DFT+U 1 and on-site hybrid2,3 methods areavailable. Thesemethodsareapproximations of the Hartree-Fock/hybrid methods ◮ Applied only inside atomic spheresof selectedatomsand 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. Nov´ak et al., Phys. Stat. Sol. (b) 243, 563 (2006) 3 F. Tran et al., Phys. Rev. B 74, 155108 (2006)
- 23. DFT+U and hybrid exchange-correlation functionals The exchange-correlation functional is xc EDFT+ U/hybrid DFT onsite= Exc [ρ] + E [nmm′ ] where nmm′ is the density matrix of the correlated electrons ◮ For DFT+U both exchange and Coulomb are corrected: Eonsite = EHF DFT DFT x + ECoul −Ex −ECoul s corr ¸ ec ¸ tion x s double ¸ c ¸ ounting x There are several versions of the double-counting term ◮ For the hybrid methods only exchange is corrected: Eonsite = αEHF − αELDA x d s .c ¸ o ¸ unt x . x s c ¸ or ¸ r. x where αis a parameter ∈ [0,1]
- 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.ineecefor 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 . .. Foracalculation without spin-polarization (ρ↑ = ρ↓): runsp c lapw -orb/eece . ..
- 25. Input file case.inorb LDA+U applied to the 4f electrons of atomsNo. 2 and 4: 1 2 0 PRATT, 1. 0 2 1 3 4 1 3 1 0.61 0.07 0.61 0.07 nmod, natorb, i p r mixmod, amix iatom, nlorb, lorb iatom, nlorb, lorb nsic (LDA+U(SIC) used) U J (Ry) U J (Ry) nsic=0 for the AMF method (lessstrongly correlated electrons) nsic=1 for the SIC method nsic=2 for the HMF method
- 26. Input file case.ineece On-site hybrid functional PBE0applied to the 4f electrons of atomsNo. 2 and 4: -12.0 2 2 1 3 4 1 3 HYBR 0.25 emin, natorb iatom, nlorb, lorb iatom, nlorb, lorb HYBR/EECE fraction of exact exchange
- 27. SCF cycle of DFT+U in WIEN2k lapw0 xc,σ ee en→ v D F T + v + v (case.vspup(dn), case.vnsup(dn)) orb -up mm′→ v↑ (case.vorbup) orb -dn mm′→ v↓ (case.vorbdn) lapw1 -up -orb nk nk → ψ↑ , ǫ↑ (case.vectorup, case.energyup) lapw1 -dn -orb nk nk → ψ↓ , ǫ↓ (case.vectordn, case.energydn) lapw2 -up va l → ρ↑ (case.clmvalup) lapw2 -dn va l → ρ↓ (case.clmvaldn) lapwdm -up mm′ lapwdm -dn mm′ → n↑ (case.dmatup) → n↓ (case.dmatdn) lcore -up core lcore -dn core → ρ↑ (case.clmcorup) → ρ↓ (case.clmcordn) mixer σ→ mixed ρ and nσ 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 WIEN2k1: ◮ unscreened: Exc = EDFT xc HF DFT+ α . Ex −Ex . − λ |r− r′| |r−r′| |r−r′| ◮ screened (short-range), 1 → e : Exc = EDFT xc SR−HF SR−DFT+ α . Ex −Ex . screening leads to faster convergence with k-points sampling 1 F. Tran and P. Blaha, Phys. Rev. B 83, 235118 (2011)
- 29. Hybrid functionals: technical details ◮ 10-1000 times more expensive than LDA/GGA ◮ k-point and MPI parallelization ◮ Approximations to speed up the calculations: ◮ Reduced k-meshfor the HF potential. Example: For a calculation with a 12× 12× 12 k-mesh, thereduced k-meshfor the HF potential can be: 6× 6× 6, 4 × 4× 4, 3 × 3× 3, 2× 2 × 2 or 1× 1× 1 ◮ 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 Ernzerhof1 ) 0.25 T 0. 165 20 6 3 3 1d-3 fraction α of HF exchange screened (T, YS-PBE0) or unscreened ( F , PBE0) screening parameter λ number of bands for the 2nd Hamiltonian GMAX lmax for the expansion of o r b i t a l s lmax for the product of two o r b i t a l s r a d i a l i n t e g r a l s below t h i s value neglected Important: The computational time will dependstrongly on the number of bands, GMAX, lmax and the number of k-points 1 A. V. Krukau et al., J. Chem. Phys. 125, 224106 (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) 1. adjust case.inhf according to your needs 2. reduced k-meshfor the HF potential? Yes or no 3. specify the k-mesh 5. run(sp) lapw -hf (-redklist) (-diaghf) ...
- 32. SCF cycle of hybrid functionals in WIEN2k lapw0 -grr → v DFT (case.r2v), αEDFT (:AEXSL) x x lapw0 xc → vDFT + vee + ven (case.vsp, case.vns) lapw1 nk nk → ψDFT, ǫDFT (case.vector,case.energy) lapw2 nk → . n k ǫDFT (:SLSUM) hf lapw2 -hf → ψnk, ǫnk (case.vectorhf, case.energyhf) → ρval (case.clmval) lcore mixer → ρcore (case.clmcor) → mixed ρ
- 33. Calculation of quasiparticle spectra from many-body theory ◮ In principle the Kohn-Shameigenvalues should beviewed as mathematical objects and not compareddirectly to experiment (ionization potential and electron affinity). ◮ The true addition and removal energiesǫiarecalculated from the equation of motion for the Green function: 2 ¸ , ¸ 1 − ∇2 + ven(r) + vH(r) + Σ(r, r′ , ǫi)ψi(r′ )d3 r′ = ǫiψi (r) ◮ The self-energy Σ is calculated from Hedin’s self-consistent equations1: 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) δG(4,5) Γ(1, 2, 3) = δ(1, 2)δ(1, 3) + ¸ δΣ(1, 2) 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 G0W0 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) 2π ∞ − ∞ Σ(r, r′, ω)= i ¸ G(r, r′, ω+ ω′)W(r, r′, ω′)e−iδω′ dω′ ′ G(r, r ,ω) = ∞ . i i ∗ ′ψ(r)ψ (r ) i=1 ω− ǫi − iηi W (r, r′, ω) = ¸ v(r, r′′)ǫ−1(r′′, r′, ω)d3r′′ ◮ G0W0 (one-shot GW ): Gand W arecalculated using the Kohn-Shamorbitals and eigenvalues. 1st order perturbation theory gives i ǫGW K S K S K S K S= ǫi + Z (ǫi )(ψi |ℜ(Σ(ǫi K S)) − vxc|ψi )
- 35. A few remarks on GW ◮ GW calculations require very large computational ressources ◮ Gand W dependon all (occupied and unoccupied) orbitals (up to parameter emax in practice) ◮ GWis the state-of-the-art for the calculation of (inverse) photoemissionspectra, but not for optics sinceexcitonic 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 code1 ◮ Based on the FP-LAPW basis set ◮ Mixed basis set to expand the GW-related quantities ◮ Interfaced with WIEN2k ◮ G0W0, GW0 @LDA/GGA(+U ) ◮ Parallelized ◮ http://www.chem.pku.edu.cn/jianghgroup/codes/fhi-gap.html 1 H. Jiang et al., Comput. Phys. Comput. 184, 348 (2013)
- 38. How to run the FHI-gap code? 1. Run a WIEN2k SCF calculation (in w2kdir) 2. In w2kdir, executethe script gapinit to preparethe 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: 1. gwdir.outgw 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 unoccupiedstates (emax) ◮ Number of frequencies ωfor the calculation of Σ = ¸ GWd ω
- 40. Band gaps 0246810121416 0 2 4 6 8 10 12 14 16 Experimentalbandgap(eV) Theoretical band gap (eV) Kr Xe Si Ar LiF LiCl Ge ScN GaAs SiC FeO AlP CdS C AlN BN MgO ZnO GaN MnO NiO ZnS LDA mBJ HSE GW00 GW
- 41. Some recommendations Before using a method or a functional: ◮ Read a few papers concerning the method in order to know ◮ why it has been used ◮ for which properties or types of solids it is supposedto be reliable ◮ if it is adapted to your problem ◮ Do you have enough computational ressources? ◮ hybrid functionals and GW require (substantially) more computational ressources (and patience)