Basics of APW-like methods. Lars Nordström Uppsala University

Transcription

1 Basics of APW-like methods Lars Nordström Uppsala University

2 APW-like methods All electron (no pseudo-potential) Full-potential Robust Accurate the benchmark to test against But, not too time consuming...

3 Outline: Chronological order... Slater s APW method Andersen s Linearisation Singh s local orbitals APW+lo Some nitty-gritties... Full-potential All electrons Linearisation energies

4 c b a Muffin-tin approximation

5 Augmented Plane Waves { e ik G r r I χ G (r)= L a G L f l(r α,e)y L (ˆr) r MT α Slater 1937: f l solution to spherical potential MT MT MT Interstitial MT MT MT

6 Rayleigh formula expansion of PW around a center (atom at rα) lm e ik G r =e i(k+g) r =4πi l e i(k+g) r α j l ( k + G r r α )Y lm(k + G)Y lm (r r α ) Matching at muffin-tin radius: a G lm =4πi l f l (R)j l ( k + G R)Y lm(k + G)

7 DFT... One-electron equation H = { 2 + v( r)} Kinetic and potential energies Hamiltonian and Overlap matrices H GG = χ(g) H χ(g ) O GG = χ(g) χ(g )

8 Augmented Plane Waves { e ik G r r I χ G (r)= L a G L f l(r α,e)y L (ˆr) r MT α Slater 1937: f l solution to spherical potential { r 2 2 r r + v 0 (r)+ l(l + 1) r 2 } E f l (r, E) = 0 MT MT Interstitial MT MT MT

9 original APW: Energy dependent basis det [ H (E) EO(E) ] detm (E)=0 bcc Nb 1e!18 The APW secular determinant 0e E (Ry)

10 Augmented Plane Waves { e ik G r r I χ G (r)= L a G L f l(r α,e)y L (ˆr) r MT α Slater 1937: f l solution to spherical potential Andersen 70 s: linearization f l (r α, E) = φ l (r α, E ν ) + b G l φ l (r α, E ν ) Full potential FP-LAPW/FLAPW

11 LAPW To match 2 coefficients (a & b) Both value and derivative continuous at MT Most important: No energy dependence det (H εo) = 0 generalised eigen-value problem

12 Augmented Plane Waves { e ik G r r I χ G (r)= L a G L f l(r α,e)y L (ˆr) r MT α Slater 1937: f l solution to spherical potential Andersen 70 s: linearization f l (r α, E) = φ l (r α, E ν ) + b G l φ l (r α, E ν ) Full potential FP-LAPW/FLAPW Singh 91: adds local orbitals (lo) Sjöstedt 00: APW+lo

13 Energy window LAPW+lo Essentially Taylor expansion: valid within an energy range E Problem for low lying states (semi-core) 3d states in Zn or 4p states in Y Add a flexibility local orbitals (Singh) new appropriate linearisation energy E2 χ i (r) = { 0 r I L [ai L φ l(r α,e ν )+b i L φ l (r α,e ν )+c i L ψ l(r α,e 2 )]Y L (ˆr ) r MT α

14 Alternative linearization: LAPW APW+lo f l (r α, E) = φ l (r α, E ν ) + b G l φ l (r α, E ν ) f l (r α, E) = φ l (r α, E ν ) + b l φl (r α, E ν )

15 APW+lo APW { e ik G r r I χ G (r) = L ag L φ l(r α, E ν )Y L (ˆr) r MT α + lo χ i (r) = { 0 r I L [ai L φ l(r α, E ν ) + b i L φ l (r α, E ν )]Y L (ˆr) r MT α Sjöstedt et al., SSC 114, 15 (2000)

16 Kinks of APW 0.10 Orbital solutions, k=(0.450, 0.223, 0.150), G= MT I 0.00!0.05 E p =0.5 Ry apw s " p " d " f! r!mesh r MT

17 Kinks of APW 0.10 Orbital solutions, k=(0.450, 0.223, 0.150), G= MT I 0.00!0.05 E p =0.5 Ry lapw s " p " d " f! r!mesh r MT

18 Convergence eigenvalues (Ry) Cu, convergence Random k!point E(0.403, 0.173, 0.173)a.u.!1 tot versus plane wave cut!off apw+lo lapw Converged LAPW APW+lo ref. (LAPW)! r MT G max

19 4APW w. semicore Mo BCC a=5.95 a.u APW+LO NAPW APW+LO energies NAPW energies Fermi Energy 1 Energy (htr) p -1 Γ Δ H G N Σ Γ Λ P D N P F H

20 Nitty Gritties { ρ( r) = G ρ G e i G r r I L ρ L(r)Y L (ˆr) r MT For FFT s need a structure function Θ( r) = { 1 r I 0 r MT so FFT Θρ

21 Full-potential Basis set exact for MTA non-mta corrections treated variationally V full = V MTA + V non MTA H non MTA = χ(g) V non MTA χ(g )

22 All electrons

23 Important parameters Plane wave part cutoff R MT k + G < RKGMAX describes spherical harmonics RKGMAX l+6 Partial waves linearisation energies experience or automagic...

24 4APW w. semicore Mo BCC a=5.95 a.u APW+LO NAPW APW+LO energies NAPW energies Fermi Energy 1 Energy (htr) p -1 Γ Δ H G N Σ Γ Λ P D N P F H

25 General method...

26 (L)APW codes Wien-2k (small fee) Fleur (free) Elk elk.sourceforge.net (free) Main advantages: Stable, general, and accurate.

27 ELK code (electrons in k-space) A new modern APW-like implementation Main developer Kay Dewhurst Open source community (GPL etc.) Developer-friendly User-friendly Still in beta-phase use with caution... but please try: elk.sourceforge.net

28 elk.in for fcc Al tasks 0 10 : DOS avec scale 3.75 atoms 1 : nspecies 'Al.in' : spfname 1 : natoms : atposl, bfcmt ngridk 4 4 4

29 Al.in 'Al' 'aluminium' : spname : spzn : spmass E : sprmin, rmt, sprmax, nrmt 6 : spnst T : spn, spl, spk, spocc, spcore T F F F F 1 :apw + lo F 0 3 : no loc orb 0 2 : l F : lorbe0, lorbdm, lorbve F 1 2 : lorbl, lorbord F : lorbe0, lorbdm, lorbve F 1 2 : lorbl, lorbord F : lorbe0, lorbdm, lorbve T

30 Collaborators Elisabeth Sjöstedt, Uppsala Fredrik Bultmark, Uppsala Francesco Cricchio, Uppsala Kay Dewhurst, Edinburgh David Singh, Oak Ridge Ole Krogh Andersen, Stuttgart