Spin-Polarized GLLB-SC potential and efficient real time LCAO ...

Bandgaps and speedSpin-Polarized GLLBSC potentialand LCAO-TDDFT for largesystemsPossibilities with GPAW codeMikael KuismaTampere University of Technology

Idea of model potentials! Usuallyv xc (r) =E xcn(r)! Instead of defining complex E xc and even morecomplex functional derivative of it just approximatethe potential direcltyv xc (r) ⇡E xcn(r)

Model potentials! All xc-potentials can be divided to two parts (detailsomitted)v xc (r) =v scr (r)+v resp (r)! v scr (r)is the Coulomb potential of exchange correlationholei.e. Slater potential.v resp (r)! is the response of the xc-hole potential todensity variation. Contains the discontinuity.! The parts can be approximated separately. There existsseveral potentials with different approximations to theseparts.

Response potential! In OEP: Correct.V resp, (r) =Zdr 0 Zdr 00 X 0 00n 0(r 0 )n 00(r 00 ) g 0 (r0 ,r 00 )n (r)|r 0 r 00 |! In GLLB Bapproximated asw iv resp (r) = X i! In Becke-Johnssonv resp (r) =K| i (r)| 2w i , w GLLB pi = K g ✏f ✏ in(r)r⌧ = X ⌧|r i | 2n(r) , i

Common model potentials✏ i , ⌧! What if one would use both?! Is screening potential of GLLB sufficient? Whatabout V s (r)+Vresp GLLB (r) or V BR (r)+VrespGLLB (r) .Potential v scr (r) v resp (r) incgredientsOEP-EXX V s (r) exact ✏ i , ✏ a , i , aKLI V s (r)Pi wKLI i | i |/n iLHF V s (r)Pij wLHF ij ⇤ i j /n iGLLB 2✏ B88 (r)Pi wGLLB i | i |/n ✏ i , iGLLB-SC 2✏ PBEsol (r) vrespGLLBBecke-Roussel v BR (r) - r 2 , ⌧Becke-Johnson v BR (r) K p ⌧/n r 2 , ⌧Tran-Blaha v BR (r) (Ka bC) p ⌧/n r 2 , ⌧

Derivative discontinuity! On integer occupation numbers, the xc-potentialjumps.! Local and semilocal xc-functionals do not have thisproperty.! OEP-EXX, KLI, GLLB and GLLB-SC have thispropertyxc =lim!0v xc (r)| N+lim!0v xc (r)| N,! This contributes to quasiparticle band gapE QP = I A = E KS + xc

Results for GLLB-SCCastelli et. al., Energy Environ. Sci., 2012, 5, 5814

ComparisonPROS• Avoid OEP equations, when E xc is orbitaldependent ! huge speed up.• Still good properties of orbital dependentfunctionals• Finite xc ! good band gaps• 1/r asymptotic behaviour ! good{✏ i }• Almost “Accuracy of GW with speed of GGA”! can be used to screen promisingmaterials for further study (see et. al.)CONS• Lose total energy! No geometry orenergetics• Depending on v xc maylose size consistency.• Relatively rare (at themoment) ! PBE is wellstudied in 10000s ofpublications, less in knownabout model potential

Spin polarized GLLB-SCwith Transition Metal Oxides! LDA/GGA are known to fail with stronly correlatedtransition metal oxides! More accurate treatment of exchange (e.g. EXX)seem to improve description! We use spin-polarized extension to GLLB-SCpotential for study.! Also some ferromagnetics and single half-metal

Spin Polarized band gapsfor transition metal oxides! In predicting the semiconducting state GLLB-SCperforms as badly as LDA: CoO and FeO metallic.! MnO and NiO improved.! Self-interaction error? Could improving thescreening part of GLLB help.PBEsol GLLB GLLBSC exp.MnO 0.65 4.02 3.52 3.9NiO 0.66 2.95 2.89 4.0CoO 0.00 0.00 0.00 2.5FeO 0.10 0.09 0.00 2.4

Magnetic moments! Improves local magnetic moments forantiferromagnetic metal oxides! Co2FeSi also is improved. DOS of minority spin issplit to two parts as with LDA+U. (however, stillmetallic, not semi-metallic).! Overestimates the magnetic moment offerromagnetic metalsCo2FeSi Fe Co Ni MnO FeO CoO NiOLDA 5.02 2.17 1.59 0.61PBEsol 5.29 2.02 1.62 0.65 4.29 3.43 2.41 1.33GLLB 6.06 N/A N/A 0.78 4.71 3.81 2.76 1.65GLLBSC 6.17 3.08 2.01 0.83 4.61 3.82 2.77 1.67exp 6 2.22 1.7 0.7 4.6-4.8 4.2 3.4-4.0 1.6-1.9

Conclusions for spinpolarized GLLB-SC! Doesn’t work as well as with spin-pairedsemiconductors.! The role of self interaction error should be checked(and more knowledge of connections betweenBecke-Johnsson and GLLB is required also)! Code Slater potential to GPAW/Code GLLB toOctopus/Code Becke-Roussel/Use with SIC etc.! Nevertheless, band gaps are improved on somesystems with still only the effort of GGA.

Optical Properties of MetalClusters! Optical properties important (used as markers)! Have strong absorption due to plasmon resonance! Problems: Large number of valence electrons (~10000),ab initio programs are at their limits.! Must use real time propagation.! Must use small basis sets.! This is implemented to gpaw branch lcaotddft. Resultsare promising, Au931 with 20000 orbitals was easy.

Time propagation! Propagation is done with a basis set, hamiltonianand overlap are “small” matrices.! Stable! Timesteps can be larger than with grid.! Uses full matrix algebra parallerized withscaLAPACK.! Crank-Nicholson: U(dt) = O 1 2 iHdtO + 1 2 iHdt(t + dt) =U(dt, H(t)) (t) ! H(t + dt)H(t + dt/2) ⇡ 1 2 H(t)+1 H(t + dt)2(t + dt) =U(dt, H(t + dt/2)) (t)

Basis set issues! Good TD-DFT response requires good dynamic polarizabilityfrom basis sets.! Basis sets are optimized for good static polarizability !partially optimizes dynamic polarizability also.! Thus, organic molecule excitations are in general betweendelocalized orbitals of partially occupied s and p-states(“intraband”). Represented well with valence lcao (say dzp).! Noble atoms have a low lying atomic (“interband”) excitations.Ie. Au: 6s " 6p, Ag: 5s " 5p. ! Adding atomic excited statesto basis set is easy and improves spectrum a lot (more thandzp ! tzdp).

Organic molecule! 250 atom graphene nano flake, LCAO-RT-TDDFT(dzp) vs. GRID-RT-TDDFT! 50x speed up, essential properties identical,quantitative differences.! 250 atoms with dzp is plausible with just singleprocessor!! In Au55 cluster, the speedup even larger

140120Absorption spectrum of graphene flakereal space gridLCAO basis dzp100806040200−200 5 10 15 20 25 30eV

Metallic excitations: Needfor excited state basis20Photoabsorption spectrum of Au55 clusterdzp+6pgridtzdpdzp1510500 1 2 3 4 5 6 7 8eV

References! PBE relaxed 55-atom AuAgclusters with PBETDDFT absorptionspectrum.! Comparison withoctopus grid code.! Fast! One day witha single core.

Icosahedral Au923 +! dzp-basis + 6p atomic state! 10152 valence electrons, 19383 basis functions! GRID-RT-TDDFT practically impossible(?)! 512 cores (32x16, 128 BLACS grid) Scaling nottested, might be inefficient configuration.! 20fs in 36 wall-hours. Only 18000 cpu-hours.! O(N^3) scaling from full matrix linear solverdominates.

200+ALDA absorption spectrum for Au 923with dzp+6p180160140120100806040200300 400 500 600 700 800 900 1000nm

Conclusions! Spin Polarized GLLB-SC needs more investigation.Different approaches should be analyzed properly.! LCAO-TDDFT enables optical calculations whichwere not previously available with GPAW. Upperlimit is still unknown. Basis set benchmarkingimportant.! Furthermore: GLLB-SC TDDFT could be interesting.Derivative discontinuity appears as counteractingxc-field, which in is provided by the GLLB responsepotential.

Further questions to: mikael.kuisma@tut.fiThanks!

Spin-Polarized GLLB-SC potential and efficient real time LCAO ...

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