DFT functionals in WIEN2k

Transcription

1 d E xc DFT functionals in WIEN2k ] [ 2 1 ] [ xc ext o E d d d V T E 1-electon equations Kohn Sham } 2 1 { 2 V V V i i i xc C ext E F i i 2 vay de/d=0 -Z/ E xc = E x + E c : exchange-coelation enegy V xc = : exchange coelation potential Both, E xc and V xc ae unknown and must be appoximated E xc

2 Appoximations fo E xc Jacob s ladde 1 1. Local density appoximation LDA: xc = f 2. Genealized gadient appoximation GGA: xc = f 3. Meta-GGA: xc = f, = 4. use of occupied obitals e.g., Hatee-Fock hybid-dft PBE0, HSE, YS-PBE0 5. use of unoccupied obitals e.g., RPA 1 J. P. Pedew et al., J. Chem. Phys. 123,

3 GGA functionals A huge numbe of GGA functionals have been poposed: whee F xc is the enhancement facto specialized GGAs WC, PBESOL give much bette lattice paametes than PBE these GGAs ae soft had GGAs RPBE ae much bette fo atomization enegies of molecules PBE is a good compomise fo both quantities HTBS might be bette except alkali atoms Haas,Tan,Blaha,Schwaz, Phys.Rev. B83, RPBE PBE WC HTBS

4 Pefomance of GGAs in solids and molecules WC HTBS PBE RPBE 4

5 Tends of GGAs: Stuctual popeties on aveage GGAs ae bette than LDA, but LDA still best fo 5d-seies Pt, Au. PBE too lage! PBE best fo 3d seies Fe, Ni,... LDA too small! WC, PBEsol, HTBS best compomise fo all elements van de Waals: LDA ovebinds, GGAs undebind sometimes non-bonding! Cohesive popeties: had GGAs much bette than LDA Electonic stuctue: LDA and GGAs vey simila band gaps undeestimated by 50 % TB-mBJ Stongly coelated electons 3d, 4f often LDA and GGA give qualitatively wong esults: metal instead of insulato, too small magnetic moments o even non-magnetic instead of AFM cupates, no too small stuctual distotions, obital ode,.

6 meta-ggas Pedew,Kuth,Zupan,Blaha 1999: E m GGA 2 F[,,, ] xc d use laplacian of, o kinetic enegy density 1 i 2 i 2 analytic fom fo V xc not possible V xc = de xc /d, SCF vey difficult best meta-ggas today descibe stuctual paametes like WC o PBEsol, but have bette atomization enegies: evtpss by Pedew et al. 2009

7 Band gaps by a semi-local potential Becke-Johnson potential J. Chem. Phys. 124, local potential designed to epoduce non-local OEP potentials in atoms modified Becke-Johnson potential F.Tan P.Blaha PRL 102, c depends on the density popeties of a mateial + gaps of GW quality + good fo coelated TM-oxides - NO enegy only V

8 How to un a calculation with the MBJ potential? 1. Pepae the input files fo an usual PBE o LDA calculation 2. un scf cycle + stuctue optimization,. 3. save_lapw case_pbe 3. init_mbj_lapw phase 1, ceates case.inm_vesp, sets R2V in case.in0 4. unsp_lapw -i 1 Run one PBE cycle to ceate case.vesp and case.2v 5. m *.boyd* 6. init_mbj_lapw phase 2 sets indxc=28 MBJ in case.in0 and geneates case.in0_g with indxc=50; select mbj-paametes, see Kolle et al. PRB 85, unsp_lapw -i 80 mbj calculations need moe cycles than PBE

9 semilocal functionals available in WIEN2k Functional Authos Yea indxc case.in0 LDA Diac, Slate, etc GGA: PBE Pedew et al WC Wu, Cohen PBEsol Pedew et al HTBS Haas et al TB-mBJ * Tan, Blaha , 50 meta-gga: evtpss ** Pedew et al * only a potential E xc = LDA ** only E xc V xc = PBE

10 moe non-local functionals beyond DFT Self-Inteaction coection Pedew,Zunge 1981; Svane+ Temmemann vanishes fo Bloch-states, select localized states by hand LDA+U DMFT dynamical mean field theoy appoximate HF fo selected highly-coelated electons 3d,4f,5f empiical paamete U Optimized Exact exchange OEP, simila to HF, but with a local DFT based potential, expensive, numeically poblematic, coelation?? Hatee-Fock neglects coelation, which fo most solids is essential Hybid functionals mixing of DFT+ HF GW method: calculate the quasipaticle self enegy a available fo WIEN2k M. Scheffle et al. vey expensive

11 Hatee-Fock kinetic E + extenal V classic Coulomb enegy of electons exchange enegy This leads to a non-local obital dependent potential. It teats exchange e - -e - inteaction between e - of same spin, Pauli-pinciple, Slate det. exactly, but neglects coelation completely. Coelation can be teated by petubation methods MP2, coupled cluste CCSD, o CI methods, using many Slate deteminants.

12 PBE0: E xc PBE0 [] = E xc PBE [] + E x HF [] E x PBE [] =0.25

13 sceened full-hybid functionals 1/ is decomposed into a shot-ange and long-ange component using an exponential o the eo function HSE06 functional: =0.11 boh -1 =1 exponential with =3/2 is vey simila = YS-PBE0: Tan,Blaha, PRB 83, sceened functionals impove k-mesh convegence damatically

14 2 paametes: mixing and sceening gaps: stong coelation between and fomation enegies: < 0.7 has little influence

15 band gaps of YS-PBE0 with =0.25 stong impovement ove PBE, but gaps of insulatos still stongly undeestimated

16 adapt fo each individual system optimal found by fit to exp. as function of 1/ 0 constant dielectic =0.25 yields much bette gaps

17 speed-up by diagonal-only appoximation

18 band gap compaison standad full hybids fixed wok well fo semiconductos 1/ 0 impoves the esults significantly diagonal-only appoximation woks in most cases speed-up

19 hybid functionals in WIEN2k expensive times moe than LDA k-point and mpi-vesion useful aleady fo medium sized cases fo bigge cases use a educed k-mesh fo the potential must be compatible with full mesh like 4x4x4 and 2x2x2 conside non-scf calculations fo DOS, bands o even the diagonal appoximation PBE0 0 Tan, Physics Lettes A fo setup follow the UG 4.5.8; un_lapw -hf woks well fo semiconductos and TM-oxides mixing paamete like the U in LDA+U should be bigge fo lage gap insulatos should be VERY small fo magn. metals hybids localize TM-d and O-p states stuctual paametes depend mainly on the undelying GGA, but ae always a bit smalle than plain GGA.

20 onsite-hybid-dft fo coelated electons select cetain electons within an atomic sphee mainly 3d o 4f states, since only those valence electons ae sufficiently localized and equie stonge exchange potentials same spiit and speed as LDA+U with paamete instead of U E xc PBE0 [] = E xc PBE [] + E x HF [ sel ] E x PBE [ sel ] cp $WIENROOT/SRC_templates/template.ineece case.ineece unsp -eece Tan et al. PRB 74,

21 Failue of the independent paticle appoximation expect lage excitonic effects when localized electons ae excited into the conduction bands optics, XAS, EELS the emaining hole and the excited electon may inteact stongly with each othe

22 fully elativistic electon-hole inteaction BSE Bethe-Salpete-equation: L12;1 2 solving a 2-paticle e - - h equation of lage dimension N v N c N k ~ single paticle APW WIEN2k,, 2 * * 3 3 * * 3 3,, v d d H W d d H E E H H H H H k c k v ck vk x k c vckv k c k v ck vk di k c vckv kk vv cc k c k v diag x di diag eh eigenvalue diffeence between hole c and electonv state attactive sceened static Coulomb inteaction W; W~ -1 e-h exchange with bae Coulomb potential v

23 Excitons in LiF BSE calculations ae vey expensive code available on equest, needs hundets of coes. + memoy R. Laskowski, P. Blaha, Phys. Rev. B, 81,

24 Ca-L 23 edge in CaF 2 Ca-2p Ca-3d expeiment gound-state DOS coe-hole calc.atio 2:1 p 3/2 p 1/2 BSE with p 1/2 and p 3/2 togethe BSE fo L 2 and L 3 sepaately BSE without diect tem H eh di BSE without exchange tem H eh x