Models and strategies for metallic systems

Transcription

1 Models and strategies for metallic systems Klaus Doll Molpro Quantum Chemistry Software Institute of Theoretical Chemistry Pfaffenwaldring 55 D Stuttgart, Germany MW-MSSC 2017, Minneapolis, July 2017

2 Motivation: explanation of experimental results (experiments performed e.g. at a synchrotron) importance of topics such as catalysis, corrosion technical: test of local basis set for metallic systems Outline: computational parameters structure optimisation with gradients bulk metals surfaces adsorption

3 1. Computational Parameters Method: usually density functional theory; functionals: PWGGA, PBE: most popular LDA: overbinds RHF, B3LYP: underbind see e.g. M. Sgroi, C. Pisani, M. Busso, Thin solid films 400, 64(2001) in general: Fock exchange gives vanishing density of states at Fermi energy see Ashcroft & Mermin, Solid State Physics, chapter 17 basis set: diffuse functions are necessary: enlarge basis, reoptimize exponents k-points: typically ( ) for bulk; (16 16) for surface studies finite temperature scheme used: keyword SMEAR, typical value 0.01 E h or lower; very low values for magnetic materials required; e.g. Ni: Curie temperature is 631 K ˆ= E h E(T = 0) = 1 2 (E(T ) + F (T )) (E0 in output) makes numerical integration more stable

4 k-point convergence and smearing: bulk Lithium shrinking factor number of k-points energy not really converged: problems with energy differences, e.g. surface energies converging converging + good approximation for E(0) K. Doll, N. M. Harrison, V. R. Saunders, J. Phys.: Condens. Matter 11, (1999)

5 PROBLEM, especially for metals SCF Convergence general rule: start from a simple problem if you have never done a calculation with CRYSTAL: start with an insulator, e.g. NaCl, reproduce data from literature metals: 1. step: bulk 2. step: thin surface 3. step: thick surface 4. step: adsorbate + surface convergence tools: - always use FMIXING, often very large mixing required (> 90%) - usually no level shifting - speedup possible with keywords ANDERSON, BROYDEN (avoid level shift together with these keywords!) verify solution!!! check: Mulliken population, binding energy, surface energy..., compare schemes: try FMIXING only (safest, but slow), try FMIXING+ANDERSON...

6 2. Structure optimisation with gradients Gradients with respect to nuclear positions: K. Doll, V. R. Saunders and N. M. Harrison, Int. J. Quant. Chem. 82, 1 (2001) K. Doll, Comp. Phys. Comm. 137, 74 (2001) Gradient with respect to the cell parameter: 3 D periodic systems: K. Doll, R. Dovesi, R. Orlando, Theo. Chem. Acc. 112, 394 (2004) 1 D, 2D periodic systems: K. Doll, R. Dovesi, R. Orlando, Theor. Chem. Acc. 115, 354 (2006)

7 Input for a full optimisation minimum input: OPTGEOM ENDOPT default: full geometry optimisation (atom positions, unit cell) for 3 D periodic systems, the stress tensor is printed in output file e.g. geometry) and SCFOUT.LOG (following geometries) input.out (first σ lm = 1 V E ɛ lm = 3 i=1 E a il a im in the case of hydrostatic pressure, with a pressure p: σ = p p p (1) pressure p available as an analytical derivative, and also the enthalpy H = E + pv K. Doll, Mol. Phys. 108, (2010) optimisation with external hydrostatic pressure: keyword EXTPRESS alternatively CVOLOPT to constrain the volume

8 Gradient for metals Question: is there an extra term for metals, due to the shape of the Fermi surface, compared to the case of insulators? Answer: at zero temperature, there is no extra term at finite temperature (keyword SMEAR), the gradient is consistent with the free energy M. Weinert and J. W. Davenport, Phys. Rev. B 45, (1992) R. M. Wentzcovitch, J. L. Martins, and P. B. Allen, Phys. Rev. B 45, (1992)

9 Test: compare numerical and analytical gradients Example 1: Cu bulk, LDA (at a = 3.4 Å) smearing temperature (E h ) E a (numerical, E h a 0 ) F a (numerical, E h a 0 ) F a (analytical, E h a 0 ) Example 2: Cl/Cu(111) smearing temperature: E h K. Doll, Chem. Phys. Lett. 535, 187 (2012)

10 2. Bulk metals Basis set reoptimization is recommended: start from an existing basis set e.g. Cu: M. D. Towler, R. Dovesi and V.R. Saunders Phys. Rev. B 52, (1995) reoptimize diffuse exponents (e.g. exponents with value smaller than 1) keep inner exponents fixed reoptimization means: determination of exponent with lowest energy of reference system (here: copper bulk)

11 Reoptimization: How does it work? e.g. copper outermost exponents were: sp : and d : good for Cu 2+, e.g. KCuF 3 1) add one sp shell, start with e.g. sp 0.6, 0.2 and d : (guess) 2) reoptimize tight sp, keep others fixed 3) reoptimize diffuse sp, keep others fixed 4) reoptimize d, keep others fixed is energy converged? - if not return to step 2 typically: 2-3 iterations of this type necessary Result: sp 0.596, d (PWGGA level) metals: outermost diffuse sp exponent usually in the range

12 Copper bulk: input file Copper Metal CRYSTAL space group 3.63 lattice constant copper END basis set

13 reoptimized exponent reoptimized/new exponent reoptimized exponent 99 0 END... end of basis set input

14 DFT PW-functional EXCHANGE PWGGA CORRELAT PWGGA END SHRINK k-point sampling MAXCYCLE enable more SCF cycles 60 FMIXING convergence tool 70 NOSHIFT no level shifting (0.6 hartree level shift default in CRYSTAL14) SMEAR finite temperature scheme 0.01 PPAN Mulliken population END use keyword FIXINDEX when comparing calculations at various lattice constants (smooth potential curve)

15 Band structure calculation (properties-part) BAND Copper band structure G-X X-W W-L L-G G-K-X END visualise with: xmgrace -nxy BAND.DAT 5: 5 lines in Brillouin zone 8: divisor, e.g. X is ( ) but is expressed as (4 0 4 ) ˆ= ( ) 500: number of points computed along path 10 15: first, last band to be plotted: Cu, 1s, 2s, 2p, 3s, 3p: 9 bands, flat, core-like Cu 4s, 3d: 6 bands, the interesting ones 1 0: output options note the different paths G-X and G-K-X ; X = ( ) and X =(4 8 lattice vector: ( ) = (0 1 0) + ( ) ) differ by a reciprocal

16 Brillouin zones: see e.g. : C. J. Bradley and A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Press, Oxford, 1972) W. Setyawan, S. Curtarolo, Comp. Mat. Sci. 49, 299 (2010) incorrect Brillouin zone is a frequent source of error! Cohesive properties: a [Å] E coh [E h ] B[GPa] HF LDA PWGGA exp K. Doll, N. M. Harrison, Chem. Phys. Lett. 317, 282 (2000)

17 3. Surfaces model: finite number of layers not repeated in the third dimension no vacuum layers difference in geometry input: SLABCUT cut slab (111) surface layers use e.g. keyword ANDERSON to achieve convergence nice exercise: try as many layers as possible

18 Compute surface energy: Two ways of calculating surface energies: [1] E surface = 1 2 (E slab(n) ne bulk ) [2] E surface = 1 ( 2 Eslab (n) [E slab (n) E slab (n m)] m) n Cu(111): computed 2.01 J/m 2 (PW); exp.: 1.83 J/m 2

19 4. Adsorption on metallic surfaces Cl/Cu(111): Coverage: one third of a monolayer exp.: structure is 3 3 R30 (LEED): P. J. Goddard and R. M. Lambert, Surf. Science 67, 180 (1977) fcc hollow as adsorption site (SEXAFS): M. D. Crapper, C. E. Riley, P. J. J. Sweeney, C. F. McConville, and D. P. Woodruff, Europhys. Lett. 2, 857 (1986)

20 Input Chlorine on copper CRYSTAL space group 3.63 lattice constant copper SLAB 3 layers, (111) surface SUPERCEL supercell ATOMINSE add chlorine adsorbate, hcp site ATOMDISP relaxation of top Cu layer relaxation of top Cu layer relaxation of top Cu layer END

21 Cl basis set: on the web, modified: enlarged (d-exponent) slightly different diffuse exponents

22 END Cu basis set, on the web...

23 DFT EXCHANGE PWGGA CORRELAT PWGGA END SHRINK SCFDIR MAXCYCLE 100 FMIXING 90 ANDERSON NOSHIFT SMEAR 0.01 PPAN END PW91 functional convergence tools

24 Results of the optimization: Site d Cl Cu E adsorption Cl coordination number fcc 2.40 Å ev 3 hcp 2.41 Å ev 3 bridge 2.33 Å ev 2 top 2.17 Å ev 1 exp.: fcc 2.39 ± 0.02 Å 2.6 ev (SEXAFS) (therm. desorption) Rules: higher coordination number higher binding energy lower coordination number shorter bond length (only one bond which is strong) K. Doll, N. M. Harrison, Chem. Phys. Lett. 317, 282 (2000).

25 Cl/Ag(111) Site d Cl Ag E adsorption fcc 2.62 Å ev hcp 2.62 Å ev bridge 2.54 Å ev top 2.38 Å ev exp. a,b : fcc 2.48 Å; 2.70 Å 2.4 ev CRYSTAL: K. Doll, N. M. Harrison, Phys. Rev. B 63, (2001) a A. G. Shard and V. R. Dhanak, J. Phys. Chem. B 104, 2743 (2000) b G. M. Lamble, R. S. Brooks, S. Ferrer, D. A. King and D. Norman, Phys. Rev. B 34, 2975 (1986) CRYSTAL results confirmed by VASP-calculation: fcc 2.64 Å ev hcp 2.64 Å ev bridge 2.56 Å ev top 2.39 Å ev VASP: L. Jia, Y. Wang, and K. Fan, J. Phys. Chem. B 107, 3813 (2003)

26 Compute effective chlorine radius: subtract radius of metal : r Me = a/ 8 substrate a r Cl Cu Ag Ni 3.63 Å 1.12 Å 4.10 Å 1.17 Å 3.53 Å 1.09 Å compare with data from Kittel s book: radius of Cl: 0.99 Å radius of Cl : 1.81 Å consistent with Mulliken charge: Cl carries always only small negative charge ( e )

27 Charge of chlorine consider: Cl on Ag(111) site charge 3s level, relative to E F fcc hcp bridge top note: Mulliken charge increases 3s level gets destabilized (nuclear charge less well screened because more electronic charge on chlorine)

28 Potassium as adsorbate Cu(111)(2 2)-K: top site occupied! S. Å. Lindgren, L. Walldén, J. Rundgren, P. Westrin and J. Neve, Phys. Rev. B 28, 6707 (1983). Simulations prove: Cu atom under potassium adsorbate is pushed into substrate atoms 1,2,3 upwards, relative to clean surface

29 Substate rumpling crucial: Site d K Cu nn E adsorption per Cl atom with rumpling (without rumpling) fcc 3.11 Å ev (1.249) hcp 3.11 Å ev (1.243) bridge 3.04 Å ev (1.243) top 2.83 Å ev (1.227) exp. a : top (SEXAFS) 3.05 ± 0.02 Å a D. L. Adler, I. R. Collins, X. Liang, S. J. Murray, G. S. Leatherman, K.-D. Tsuei, E. E. Chaban, S. Chandravarkar, R. McGrath, R. D. Diehl and P. H. Citrin, Phys. Rev. B 48, (1993) energy gain by substrate rumpling: 0.02 ev for fcc, hcp, bridge, but 0.06 ev for top site! K. Doll, Eur. Phys. J. B 22, 389 (2001).

30 two coverages considered: K/Ag(111): (2 2) ( 3 3)R30 coverage K-K distance recuced stronger repulsion Mulliken charge on K depolarisation bond length [Å] larger radius and bond length exp. a [Å] 3.27 ± ± 0.02 binding energy [ ] E h K atom K 3s, 3p levels (relative to E F ) 3s core levels destabilized 3p a G. S. Leatherman, R. D. Diehl, P. Kaukasoina and M. Lindroos, Phys. Rev. B 53, (1996) K. Doll, Phys. Rev. B 66, (2002)

31 CO/Pt(111) Photograph: Wikimedia Commons Standard functionals give wrong site: fcc (PW91) versus top (experiment) The CO/Pt(111) puzzle P.J. Feibelman, B. Hammer, J. K. Nørskov, F. Wagner, M. Scheffler, R. Watwe, R. Dumesic, J. Phys. Chem. 105, 4018 (2001)

32 Possible solution when adsorbed: 1) C gives charge to Pt 2) Pt back donates charge: more in the case of the hollow site, less in the case of the top site Standard functionals favor charge transfer too strong (gaps too small), prefer hollow site Suggestions: B3LYP, Cluster+extrapolation A. Gil, A. Clotet, J. M. Ricart, G. Kresse, M. García-Hernández, N. Rösch, and Ph. Sautet, Surf. Sci. 530, 71 (2003) LDA+U: G. Kresse, A. Gil, and Ph. Sautet, Phys. Rev. B 68, (2003)

33 Periodic B3LYP calculation Standard-functional (PW91) B3LYP 3 fold hollow site - - bridge site ev ev top site ev ev CO charge: B3LYP: describes band gaps and excitation energies better CO/Pt(111): K. Doll, Surf. Sci. 573, 464 (2004) CO/Cu(111): M. Neef and K. Doll, Surf. Sci. 600, 1085 (2006)

34 Work function Work function: electrostatic potential at infinity, minus Fermi energy But: Cu(111), input as early in the talk: Φ = E h exp.: E h strongly basis set dependent, e.g. when varying the outermost diffuse exponent! other properties are much less basis set dependent!

35 solution: use basis functions in the vacuum (keyword GHOSTS) 1-2 ghost layers are sufficient: without ghosts: E h 1 ghost layer on each side: E h 2 ghost layers on each side: E h experiment: E h K. Doll, Surf. Sci. 600, L321 (2006) see also: P. J. Feibelman, Phys. Rev. B 51, (1995)

36 5. Summary geometry and energetics agree with experiment and plane-wave results typical properties: total energy, geometry, Mulliken population (surprisingly reliable!), core levels, charge+spin density maps, overlayer density of states basis sets must be carefully chosen, possibly readjusted outermost sp is always necessary, sp 0.20 is not diffuse enough work function: use ghosts in the vacuum region above the surface B3LYP versus standard functional shows: HOMO and LUMO position important for CO adsorption Gaussian type orbitals suitable for all types of structures, also metals, all-electron calculations are possible review: K. Doll, Ab initio calculations with a Gaussian basis set for metallic surfaces and the adsorption thereon, in Quantum Chemical Calculations of Surfaces and Interfaces of Materials, edited by Vladimir Basiuk and Piero Ugliengo, American Scientific Publishers, 2009, p

37 Basis sets In the following, some of the basis sets used are given. Note that they are calibrated for systems where the atom is neutral. For systems where the atom is positively charged, it may be necessary to remove the outermost diffuse exponents, or shift them to higher values, possibly reoptimise.

38 Lithium K. Doll, N. M. Harrison, V. R. Saunders, J. Phys.: Condens. Matter 11, (1999) note: calibrated for Li metal - for a system where Li is positively charged, it may be necessary to remove the outermost diffuse exponent, or shift it to a higher value, possibly reoptimise.

39 Copper K. Doll, N. M. Harrison, Chem. Phys. Lett. 317, 282 (2000) note: calibrated for Cu metal - for a system where Cu is positively charged, it may be necessary to remove the outermost diffuse sp exponent, or shift it to a higher value, possibly reoptimise.

40 Nickel K. Doll, Surf. Sci. 544, (2003) note: calibrated for Ni metal - for a system where Ni is positively charged, it may be necessary to remove the outermost diffuse sp exponent, or shift it to a higher value, possibly reoptimise.

41 Silver INPUT K. Doll and N. M. Harrison, Phys. Rev. B 63, (2001) based on the pseudopotential and basis set from: D. Andrae, U. Häussermann, M. Dolg, H. Stoll, and H. Preuss, Theor. Chim. Acta 77, 123 (1990) note: calibrated for Ag metal - for a system where Ag is positively charged, it may be necessary to remove the outermost diffuse exponents, or shift them to higher values, possibly reoptimise

42 Platinum INPUT K. Doll, Surface Science 573, (2004). based on the pseudopotential and basis set from: D. Andrae, U. Häussermann, M. Dolg, H. Stoll, and H. Preuss, Theor. Chim. Acta 77, 123 (1990) note: calibrated for Pt metal - for a system where Pt is positively charged, it may be necessary to remove the outermost diffuse exponents, or shift them to higher values, possibly reoptimise