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Quantum Monte-Carlo Studies of B, Al, and C clusters

International Workshop Quantum Monte Carlo in the Apuan Alps III Saturday 21st - Saturday 28th July 2007 The Towler Institute, Vallico Sotto, Tuscany. Quantum Monte-Carlo Studies of B, Al, and C clusters. Ching-Ming Wei Institute of Atomic & Molecular Sciences, Academia Sinica, TAIWAN

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Quantum Monte-Carlo Studies of B, Al, and C clusters

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  1. International Workshop Quantum Monte Carlo in the Apuan Alps III Saturday 21st - Saturday 28th July 2007 The Towler Institute, Vallico Sotto, Tuscany Quantum Monte-Carlo Studies of B, Al, and C clusters Ching-Ming Wei Institute of Atomic & Molecular Sciences, Academia Sinica, TAIWAN In collaboration with: Cheng-Rong Hsing, Hsin-Yi Chen Neil Drummond, Richard Needs

  2. Outline • Motivation • Results • B18 and B20 (July ~ Oct. 2006) • Al13 and Al55 (May ~ June 2007) • C20 (June ~ July 2007) • grapheneribbon (Jan. ~ May 2007) • 3. Summary and Conclusion

  3. Motivation? Quantum Size Effects in Metallic Nanoparticles C. M. Wei1, C. M. Chang2 and C. Cheng3 1 Institute of Atomic and Molecular Sciences, Academia Sinica, Taiwan 2National Dong-Hwa University, Taiwan 3National Cheng-Kung University, Taiwan

  4. Quantum Size Effects in Metallic Nanoparticles: • Possible shell structures of nano particles Decahedral: 10 (111) faces + 5 (100) faces Icosahedral: 20 (111) faces Cubotohedral: 8 (111) faces + 6 (100) faces No. of particles for icosahedral, decahedral & cubotohedral N= 10/3 n3+ 5 n2 +11/3 n+1 N= 13 (n=1) ; 55 (n=2) 147 (n=3) ; 309 (n=4) 561 (n=5) ; 923 (n=6) ………… V & S of 3 structures is basically the same ! Stability & structural transition ?

  5. Cubotohedron Icosahedron

  6. Cohesive energy of metallic nanoparticles Etot = a V + b S = a v0 N + b cV2/3 = a v0 N + b’ (v0 N )2/3 = a v0 N + b’ v0 2/3N2/3 = a’N + b”N2/3 Ecoh(N) = Etot / N = a’ + b” N -1/3

  7. Li et al., Science 2003 Johansson et al. Angew. Chem. Int. Ed. 2004 • The cohesive energy of Au13 deviate from N-1/3 curve is a sign of QSE! • Hollow Au20 & Au32 is stable because lower than N-1/3 curve!

  8. For Lennard-Jones Cluster: Eico < Edeca < Ecubo J. Chem. Soc. Faraday Trans. 87, p215 (1991)

  9. Structure phase transition of Icosahedral  Cubotohedral Mackay transition Acta Cryst. 15, p916 (1962) ico if fcc if s= 0

  10. Barrier heights (~10 meV) of ICO  FCC transition of Pb clusters oscillate with the shell index (or radius of cluster) indicates the possible Quantum Size Effect of the melting points ?

  11. Which Au38 is a more stable structure? Efcc= - 100.40 eV(PBE) EO_h= - 101.86 eV Efcc= - 97.50 eV(GGA) EO_h= - 99.02 eV Efcc= - 131.98 eV (LDA)EO_h= - 130.81 eV QMC needed?

  12. Motivation? Atomic structures of 13-atommetallic clusters by DFT Hsin-Yi,Tiffany, Chen Ching-Ming Wei Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan

  13. 31 complex bcc bcc bcc cubiccomplex 23 73 41 25 bcc hcp fcc hcp bcc hcp hcp hcp fcc hcp bcc hcp tetr hcp hex bcc hcp hcp bcc bcc hcp fcc fcc fcc hcp hcp bcc fcc fcc hcp hcp fcc fcc hcp hcp fcc hcp bcc bcc 27 14 81 44 24 5 11 79 32 56 49 37 80 47 50 30 48 55 29 13 28 22 76 6 78 12 42 20 43 77 72 38 46 75 3 40 57 26 4 45 39 82 74 21 Ga Ta Nb V Mn Li Co Be Na Sr Ru Ca Y Cr Mg La Zr Ba Mo Pb Zn Sc Ti Rb Ni Cu W Cs Ag Sn Ge Hg Re Hf Tc Al Fe C Os B Tl Rh Au Pd In Cd Pt Si Ir dec+hcp bcc+ico bcc+2D bcc tbp(?) 2D-tbp ico ico ico ico 2D-cag ico ico 2D-cag 2D+ico ico 2D 2D-tbp 2D+ico 2D 2D+ico ico 2D-cag hcp(?) dec 2D-tbp dec+hcp 2D-gcl ico dec+hcp 2D-gcl 2D-cag dec(?) ico 2D-bbp ico ico ico 2D-tbp dec(?) 2D 2D-gcl ico Group 1A Group 2A Group 3A bcc 19 K 2D+ico Group 3B Group 4B Group 5B Group 6B Group 7B Group 8B Group 8B Group 8B Group 1B Group 2B Motivation • Motivation • To determine the ground-state structures of 44 metallic (Tab.1) 13-atom clusters • Find out the possible regularity existed and then try to understand the reasons accounting for the regularity. • Two questions we are asking: • (1) If the highest symmetry icosahedral structure would always be the most stable in each element? • (2)Are there any relations betweenclusters and their bulk crystal structures? Tab.1 Selected 13-atom clusters of the Group1A~3A, 3d~5d series and Pb13 in a periodic table (44 elements)

  14. 13 2 1 12 10 3 4 5 8 9 6 7 11 Method & Calculated Materials • Method • Software : Vienna Ab Initio Simulation Program (VASP) • Pseudopotential method :PAW • Compare 3 exchange-correlation functional :LDA, PW91, PBE • K points sampling : gamma point • Supercell Dimensions : 20 Å × 20 Å × 20 Å • Calculated Materials • Chosen elements :Group 1 ~ Group 13, and Pb in the periodic table • 9 available and familiar atomic structures of ground-state from literature searches were calculatedto find out the lowest energy in each element.

  15. dec fcc bcc hcp ico Materials – 9 available 13-atom atomic structures 5 High Symmetry → 3D • icosahedral (ico), Ih • cuboctahedral (fcc), Oh • decahedral (dec), D5v • body-center cubic (bcc) D4h • hexagonal-close packed (hcp), D3v 4 Low Symmetry → 2D • buckled biplanar (bbp), C2v • triangular biplanar (tbp),C3v • garrison-cap layer (gcl) C2v • cage (cag), C1h cage(cag) <ref2> triangular biplanar(tbp) garrison-cap(gcl) buckled biplanar (bbp)<ref1> top view side view side view top view top view side view triangle (3)+ (7) atoms+ triangle (3) (1) atom+ 2 square (12) hexagonal array (7) + central square (4)+(2)side atoms hexagonal array (7)+ triangle (6) <Ref 1> C. M. Chang, M. Y. Chou, Phys. Rev. Lett. 93, 133401 (2004) , <Ref 2> Y. C. Bae, et al, Phys. Rev. B 72, 125427 (2005)

  16. How do we compare 3 exchange correlation functionals? Define the average energy of 9 atomic structures as “reference point” 1 Equ. reference point = Ebbp+Egbp+Ebcc+Edec+Efcc+Ehcp+Eico+Etbp+Ecag)/ 9 2 Define “relative energy”, dE(eV) = Total energy of the atomic structure– reference point Equ. • dE atomic structure= E atomic structure –[(Ebbp+Egbp+Ebcc+Edec+Efcc+Ehcp+Eico+Etbp+Ecag)/ 9] dE bbp = Ebbp– [(Ebbp+Egbp+Ebcc+Edec+Efcc+Ehcp+Eico+Etbp+Ecag)/ 9] remark We use “relative energy” to compare 3 exchange correlation functionals

  17. tbp tbp tbp Consistency in 3 exchange correlation functionals For Ba13 the lowest energyall occur in Icosahedral relative stability Remark Remark For Re13 the lowest energyall occur ingarrison-cap layer (2D-gcl, low symmetry) For In13,∵ the energies of dec and hcp→ too close and competitive ∴Atomic structure of ground-statecould be dec or hcp Remark

  18. bcc bcc bcc 24 42 74 Cr Mo W dec dec(?) dec(?) Group 6B Bulk tbp tbp tbp Cluster (III) dec Consistency and Inconsistencyof LDA, PW91, & PBE occurred in Group 6 BUT consistency gcl LDA bcc 3Mo13 • For PW91 & PBE→ the lowest energy only occur in “dec” Cr13 • For LDA, PW91, & PBE → the lowest energies all occur in “dec” BUT BUT • For LDA → lower energies occur in “dec”&ico” • Inconsistent “relative stabilities” occur in “bcc” & “gcl”

  19. Overall Results of Regularity If all these DFT results are reliable? Group 1 Group 2 Group 13 2D+ico: cluster structure could be“2D Low symmetry” or “ico” bcc hcp Bulk structure hcp 3 4 5 6 Li Be B C dec(?): undetermined structure Cluster structure ico 2D+ico 2D-bbp 2D-tbp: cluster structure is “2D low symmetry--tbp” bcc hcp hcp 11 12 13 14 dec+hcp: cluster structure could be“dec” or “hcp” Na Mg Al Si Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Group 11 Group 12 2D+ico 2D ico bcc fcc hcp hcp bcc bcc bcc hcp fcc fcc hcp cubiccomplex 19 20 21 22 23 24 25 26 27 28 29 30 32 31 complex K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ge Ga 2D+ico ico ico ico bcc+2D dec(?) 2D-tbp(?) ico 2D-tbp ico 2D-gcl 2D dec+hcp bcc fcc hcp hcp bcc bcc hcp hcp fcc fcc fcc hcp tetr 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn 2D+ico ico ico ico bcc dec(?) 2D-tbp 2D-cag 2D-cag 2D-tbp 2D-gcl hcp(?) dec+hcp bcc bcc hex hcp bcc bcc hcp hcp fcc fcc fcc hcp hcp 55 56 57 72 73 74 75 76 77 78 79 80 81 82 Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb ico ico ico bcc+ico dec 2D-tbp 2D-cag 2D-cag 2D 2D-gcl dec+hcp ico (I) ico or 2D+ico except Mg13 (II) bcc (III) dec (IV) 2D (tbp, cag, gcl) (V) dec+hcpcompetitive cluster’s structures are the same as bulks’ only in Group 5

  20. Motivation? DFT is no predict power!!! Ag adsorbed on Graphite ExC Ag+Graphite Ag/Graphite Ead LDA -323.063 eV -323.614 eV 0.551 eV PW91 -295.830 eV -295.876 eV 0.046 eV PBE -295.447 eV -295.481 eV 0.034 eV

  21. trans-stilbene cis-stilbene Ecis= 0.204 eV Etrans= 0.0 eV Ag-Ge(111)-IET

  22. cis-stilbene/Ag-Ge(111) trans-stilbene/Ag-Ge(111) Eads= 1.059 eV (LDA) Eads= 0.887 eV (LDA) LDA agrees expt., but… Eads = 0.40 & 0.20 eV (PW91) and again DFT without any predicting power!!!

  23. QMC results of B18 and B20 • To check if the hollow B18(Oh) will become the most stable cluster? • To check if tube structure will become favor in B20?

  24. Boron18cluster (1) (2) (3) (4) VASP(PBE) 0.717 eV 0.739 eV 0.000 eV CASTEP(PBE) 0.740 eV 0.752 eV 0.000 eV 0.970 eV -1366.606 -1366.594 -1367.346 -1366.376 CASTEP(LDA) 0.601 eV 0.836 eV 0.000 eV 0.192 eV -1362.212 -1361.977 -1362.813 -1362.620 ----------------------------------------------------------------------------------------------------------------------------------------- QMC(dt=0.005) -1361.680(39) (1.55) -1362.316(42) (0.91) -1363.227(35) eV -1361.836(42) (1.39) QMC(dt=0.010) -1361.598(25) (1.51)-1362.148(28) (0.96) -1363.107(27) eV -1361.716(27) (1.39)

  25. Boron20clusters 5 Above 4 structures are described by J. Chem. Phys. 124, 154310 (2006), but 5th structure is found by me recently with a comparable low energy with structures 2, 3, and 4.

  26. 5(C1h) (1) (2) (3) (4) (5) VASP(PBE) 0.00 0.65 0.46 0.59 0.48 CASTEP(PBE) 0.00 0.44 0.57 0.49 -1520.216 -1519.774 -1519.646 -1519.727 CASTEP(LDA) 0.00 0.28 0.39 0.08 -1514.946 -1514.667 -1514.554 -1514.863 ------------------------------------------------------------------------------------------------------------------------------------------------ QMC(dt=0.005) (~100000 steps)-1516.171(36) -1515.417(35) -1515.426(51) -1515.358(45) 0.75 0.74 0.81 QMC(dt=0.010) (~100000 steps) -1515.953(33) -1515.292(31) -1515.267(57) -1515.115(35) 0.66 0.69 0.84 QMC(dt=0.010) (=300000 steps) -1515.961(19) -1515.278(19) -1515.239(23) -1515.125(20) 0.68 0.72 0.84 All calculations were performed using Gaussian 03, Revision C.02 package.24 For neutral clusters, full geometry optimizations were performed using the second-order Møller-Plesset perturbation theory25,26 MP2method as well as DFT methods in generalized gradient approximations GGAswith two hybrid exchange-correlation functionals, namely, B3LYP Ref. 27and PBE1PBE,28 and a recently developed hybrid metafunctional TPSS1KCIS.29 A modest cc-pVDZ basis set30 Dunning’s correlation consistent polarized valence double zeta, contracted 3s2pplus polarization set 1dwas chosen with the MP2 method and a large ccpVTZ basis set30 Dunning’s correlation consistent polarized valence triple zeta, contracted 4s3pplus polarization set 2d1fwith DFTs. Next, the harmonic vibrationalfrequency analyses were carried out to assure that the optimized structures give no imaginary frequencies. To determine the energy ordering, several high-level ab initio molecular-orbital methods were employed to calculate single-point energies of the four neutral isomers with the optimized structures at the MP2/cc-pVDZ level of theory: 1the fourth-order Møller-Plesset perturbation theory31 MP4with cc-pVTZ basis set for neutral isomers; 2a coupled-cluster32 method at the CCSDT1Diag/6-311Gdlevel of theory to examine possible multireference quality for the top-two lowest-energy isomers; and 3the coupledcluster method including single, double, and noniteratively perturbative triple excitations at the CCSDT/6-311Gdlevel of theory.

  27. QMC study of Al13 and Al55 • To answer if DFT can be used to the study of metallic clusters? • Which ExC approximation might be better if LDA, PW91, and • PBE do not give consistent results?

  28. MD simulation at 500 K starting from Al55 ICO structure ========PW91===LDA===PBE========================= 1 ps -3.009 -2.618 -2.772 eV 2 ps -3.357 -2.099 -2.432 eV 3 ps -2.821 -3.116 -3.053 eV 4 ps -2.726 -2.854 -3.044 eV 5 ps -3.131 -3.595 -3.049 eV 6 ps -2.993 -3.678 -2.828 eV 7 ps -3.531 -3.687 -3.354 eV 8 ps -3.402 -3.541 -3.238 eV 9 ps -3.479 -3.670 -3.223 eV 10 ps -3.501 -3.672 -3.211 eV 11 ps -3.257 -3.313 -3.411 eV 12 ps -2.673 -3.297 -3.382 eV 13 ps -3.039 -3.183 -3.383 eV 14 ps -3.408 -3.443 -3.502 eV 15 ps -3.544 -3.454 -3.712 eV 16 ps -3.201 -3.447 -3.396 eV 17 ps -3.128 -3.453 -3.384 eV 18 ps -3.186 -3.450 -3.306 eV 19 ps -2.946 -3.104 -2.970 eV 20 ps -3.481 -3.500 -3.107 eV • Question: if we really find the local minimum? • Relax the structure using the relaxed structure obtained by LDA at 7 ps with PBE potential, then using this relaxed structure but with LDA potential again, it happens the relaxed structure go back to original structure ! • Relax the structure using the relaxed structure obtained by PBE at 15 ps with LDA potential, then using this relaxed structure but with PBE potential again, it happens the relaxed structure go back to original structure ! • Answer : YES

  29. ICO DEC AMOR FCC

  30. QMC study of C20 • To see if DFT with new developed ExC (like PBE) can describe well the energy difference of local minimum structures? • To see if DFT can describe well the energy difference due to Jahn-Teller distortion?

  31. C20 structures bowl C5v 1.24A 1.40~1.43A ring 20h10h 1.28A 1.24A 1.32A cage IhC3 1.44A 1.4~1.5A

  32. PBE does give the correct energy order!

  33. DFT fail to give a correct DE due to Jahn-Teller effect!

  34. QMC results of graphene ribbon FIG. 7. Color online Isovalue surfaces of the spin density for the antiferromagnetic case (a) and ferromagnetic case (b) of ribbon of width N=10. FIG. 1. Color online A monohydrogenated ribbon of width N =5 along y. The system is periodic only along x and the dashed lines delimit the periodic unit cell of length a. For N=5 Ribbon, the energy difference of nanoribbon states obtained by DFT are: ------------------------------------------------------------- Code & ExC DE (NM-AF) DE (FM-AF) ------------------------------------------------------------ CASTEP LDA 32.5 meV 1.5 meV VASP LDA 36.3 meV 2.0 meV VASP PW91 55.6 meV 3.3 meV VASP PBE 79.1 meV 5.7 meV ------------------------------------------------------------------------------------------------------------------------------ Crystal B3LYP 290 meV49 meV ref: Harrison et al. in PRB 75, 2007 ------------------------------------------------------- FIG. 4. Color online Electron density of a nonmagnetic, monohydrogenated, N=10 ribbon contributed by a the states near the Fermi level and b the rest of the occupied states.

  35. For N=5 Ribbon, the energy difference of nanoribbon states obtained by DFT are: ------------------------------------------------------------------------ Code & ExC DE (NM-AF) DE (FM-AF) -------------------------------------------------------------- CASTEP LDA 32.5 meV 1.5 meV VASP LDA 36.3 meV 2.0 meV VASP PW91 55.6 meV 3.3 meV VASP PBE 79.1 meV 5.7 meV ------------------------------------------------------------------------ Crystal B3LYP 290 meV49 meV ref: Harrison et al. in PRB 75, 2007 ------------------------------------------------------------------------ N=5 Ribbon K-point (PBE) DE (NM-AF) DE (FM-AF) ---------------------------------------------------------- 1x1x 6 59.0 meV 3.6 meV 1x1x 9 76.5 meV 5.0 meV 1x1x12 79.1 meV 5.7 meV 1x1x15 70.8 meV 9.9 meV 1x1x30 67.4 meV 10.9 meV 1x1x60 67.5 meV 13.0 meV ---------------------------------------------------------- LDA, PW91, PBE are at least a factor of 4~5 less than B3LYP!!! For N=5 Graphene Ribbon, the QMC energies obtained are : AF (VMC) : Final energy = -1564.659 (38) eV NM (VMC) : Final energy = -1563.803 (41) eV AF (DMC) : Final energy = -1577.092 (31) eV (~9000 CPU hour) NM (DMC) : Final energy = -1576.658 (35) eV (~9000 CPU hour) And the energy difference obtained by QMC are DE (NM-AF) = 856 meV VMC DE (NM-AF) = 434 meV DMC (DT=0.005) (Here E_cut = 600 eV, BLIP = 2.0 and 1x1x6 unit cell) It seems that DMC favors B3LYP!

  36. Summary and Conclusion • In general, DFT should be able to use in the study of the metallic clusters judging from the QMC results of Al13 and Al55 clusters, and PBE is perhaps better! • DFT fails to describe B18 and B20 clusters, and QMC is needed! • DFT fails to describe well C20 clusters, however, PBE can perhaps describe the energy difference of local minimums! • DFT with LDA, PW91, PBE ExC fails to describe the energy difference of AF and NM states of graphene ribbon!

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