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 International Workshop Quantum Monte Carlo in the Apuan Alps III Saturday 21st - Saturday 28th July 2007 The Towler Institute, Vallico Sotto, Tuscany

Outline 1. Motivation 2. Results B 18 and B 20 (July ~ Oct. 2006) Al 13 and Al 55 (May ~ June 2007) C 20 (June ~ July 2007) graphene ribbon (Jan. ~ May 2007) 3. Summary and Conclusion

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

 Possible shell structures of nano particles Icosahedral: 20 (111) faces Decahedral: 10 (111) faces + 5 (100) faces Cubotohedral: 8 (111) faces + 6 (100) faces Quantum Size Effects in Metallic Nanoparticles: No. of particles for icosahedral, decahedral & cubotohedral N= 10/3 n n 2 +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 ?

Icosahedron Cubotohedron

E tot = a V + b S = a v 0 N + b cV 2/3 = a v 0 N + b’ (v 0 N ) 2/3 = a v 0 N + b’ v 0 2/3 N 2/3 = a’ N + b” N 2/3 E coh (N) = E tot / N = a’ + b” N -1/3 Cohesive energy of metallic nanoparticles

 The cohesive energy of Au 13 deviate from N -1/3 curve is a sign of QSE!  Hollow Au 20 & Au 32 is stable because lower than N -1/3 curve! Johansson et al. Angew. Chem. Int. Ed Li et al., Science 2003

For Lennard-Jones Cluster: E ico < E deca < E cubo J. Chem. Soc. Faraday Trans. 87, p215 (1991)

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

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 ?

Which Au 38 is a more stable structure? E fcc = eV(PBE) E O_h = eV E fcc = eV(GGA) E O_h = eV E fcc = eV (LDA) E O_h = eV QMC needed?

Atomic structures of 13-atom metallic clusters by DFT Hsin-Yi,Tiffany, Chen Ching-Ming Wei Institute of Atomic & Molecular Sciences, Academia Sinica, Taiwan 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. Tab.1 Selected 13-atom clusters of the Group1A~3A, 3d~5d series and Pb 13 in a periodic table (44 elements) Group 1A Group 2A Group 3A K 19 bcc 2D+ico Ba 56 bcc ico Sc 21 hcp ico Ti 22 hcp ico V 23 bcc bcc+2D Cr 24 bcc dec(?) Mn 25 cubic complex tbp(?) Ru 44 hcp 2D-cag Co 27 hcp 2D-tbp Ni 28 fcc ico Cu 29 fcc 2D-gcl Zn 30 hcp 2D Ga 31 complex dec+hcp Pb 82 hcp ico Rb 37 bcc 2D+ico Na 11 bcc 2D+ico Mg 12 hcp 2D Li 3 bcc ico Be 4 hcp 2D+ico Sr 38 fcc ico Ca 20 fcc ico Y 39 hcp ico La 57 hex ico Zr 40 hcp ico Hf 72 hcp ico Nb 41 bcc Ta 73 bcc bcc+ico Mo 42 bcc dec(?) W 74 bcc dec Tc 43 hcp 2D-tbp Re 75 hcp 2D-tbp Fe 26 bcc ico Os 76 hcp 2D-cag Ir 77 fcc 2D-cag Rh 45 fcc 2D-cag Pd 46 fcc 2D-tbp Pt 78 fcc 2D Ag 47 fcc 2D-gcl Cd 48 hcp hcp(?) In 49 tetr dec+hcp Au 79 fcc 2D-gcl Tl 81 hcp dec+hcp B 5 hcp 2D-bbp Al 13 hcp ico C 6 Si 14 Ge 32 Sn 50 Cs 55 bcc Hg 80 Group 3B Group 4B Group 5B Group 6B Group 7B Group 8B Group 1B Group 2B 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 between clusters and their bulk crystal structures?

Method & Calculated Materials 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 calculated to find out the lowest energy in each element. 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

Materials – 9 available 13-atom atomic structures fccico dec 5 High Symmetry → 3D  icosahedral (ico), I h  cuboctahedral (fcc), O h  decahedral (dec), D 5v  body-center cubic (bcc) D 4h  hexagonal-close packed (hcp), D 3v buckled biplanarbuckled biplanar (bbp) garrison-cap(gcl) 4 Low Symmetry → 2D  buckled biplanar (bbp), C 2v  triangular biplanar (tbp), C 3v  garrison-cap layer (gcl) C 2v  cage (cag), C 1h hcp bcc triangular biplanartriangular biplanar(tbp ) hexagonal array (7) + central square (4)+(2)side atoms triangle (3) + (7) atoms + triangle (3) C. M. Chang, M. Y. Chou, Phys. Rev. Lett. 93, (2004), Y. C. Bae, et al, Phys. Rev. B 72, (2005) hexagonal array (7) + triangle (6) cagecage(cag) (1) atom + 2 square (12) top view side view top view side view top view side view

How do we compare 3 exchange correlation functionals? Define the average energy of 9 atomic structures as “reference point”  dE atomic structure = E atomic structure – [(E bbp +E gbp +E bcc +E dec +E fcc +E hcp +E ico +E tbp +E cag )/ 9] reference point = E bbp +E gbp +E bcc +E dec +E fcc +E hcp +E ico +E tbp +E cag )/ 9 Define “relative energy”, dE(eV) = Total energy of the atomic structure– reference point Equ. remark We use “relative energy” to compare 3 exchange correlation functionals 1 2 dE bbp = E bbp – [(E bbp +E gbp +E bcc +E dec +E fcc +E hcp +E ico +E tbp +E cag )/ 9 ]

tb p Consistency in 3 exchange correlation functionals For Ba 13 the lowest energy all occur in Icosahedral Remark For Re 13 the lowest energy all occur in garrison-cap layer (2D-gcl, low symmetry) For In 13, ∵ the energies of dec and hcp → too close and competitive ∴ Atomic structure of ground-state could be dec or hcp Remark tb p Remark relative stability

tbp Consistency and Inconsistency of LDA, PW91, & PBE occurred in Group 6 Cr 13  For LDA, PW91, & PBE → the lowest energies all occur in “dec” Bulk Cr 24 bcc dec(?) Mo 42 bcc dec(?) W 74 bcc dec Group 6B (III) dec Cluster consistency LDA 3 Mo 13  For PW91 & PBE → the lowest energy only occur in “dec” bcc gcl tbp BUT  For LDA → lower energies occur in “dec” & ico”  Inconsistent “relative stabilities” occur in “bcc” & “gcl” BUT

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

Ag adsorbed on Graphite ExC Ag+Graphite Ag/Graphite E ad LDA eV eV eV PW eV eV eV PBE eV eV eV Motivation? DFT is no predict power!!!

trans-stilbenecis-stilbene E trans = 0.0 eV E cis = eV Ag-Ge(111)-IET

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

QMC results of B 18 and B 20 To check if tube structure will become favor in B 20 ? To check if the hollow B 18 (O h ) will become the most stable cluster?

Boron 18 cluster (1) (2) (3) (4) VASP(PBE) eV eV eV CASTEP(PBE) eV eV eV eV CASTEP(LDA) eV eV eV eV QMC(dt=0.005) (39) (1.55) (42) (0.91) (35) eV (42) (1.39) QMC(dt=0.010) (25) (1.51) (28) (0.96) (27) eV (27) (1.39)

Boron 20 clusters 5 Above 4 structures are described by J. Chem. Phys. 124, (2006), but 5 th structure is found by me recently with a comparable low energy with structures 2, 3, and 4.

(1) (2) (3) (4) (5) VASP(PBE) CASTEP(PBE) CASTEP(LDA) QMC(dt=0.005) (~ steps) (36) (35) (51) (45) QMC(dt=0.010) (~ steps) (33) (31) (57) (35) QMC(dt=0.010) (= steps) (19) (19) (23) (20) (C 1h ) 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 MP2 method as well as DFT methods in generalized gradient approximations GGAs with two hybrid exchange-correlation functionals, namely, B3LYP Ref. 27 and 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 3s2p plus polarization set 1d was chosen with the MP2 method and a large ccpVTZ basis set30 Dunning’s correlation consistent polarized valence triple zeta, contracted 4s3p plus polarization set 2d1f with 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: 1 the fourth-order Møller-Plesset perturbation theory31 MP4 with cc-pVTZ basis set for neutral isomers; 2 a coupled-cluster32 method at the CCSDT1Diag/6-311Gd level of theory to examine possible multireference quality for the top-two lowest-energy isomers; and 3 the coupledcluster method including single, double, and noniteratively perturbative triple excitations at the CCSDT/6-311Gd level of theory.

QMC study of Al 13 and Al 55 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?

MD simulation at 500 K starting from Al 55 ICO structure ========PW91===LDA===PBE========================= 1 ps eV 2 ps eV 3 ps eV 4 ps eV 5 ps eV 6 ps eV 7 ps eV 8 ps eV 9 ps eV 10 ps eV 11 ps eV 12 ps eV 13 ps eV 14 ps eV 15 ps eV 16 ps eV 17 ps eV 18 ps eV 19 ps eV 20 ps eV Question: if we really find the local minimum? 1.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 ! 2.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

DECICO FCC AMOR

QMC study of C 20 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?

cage I h  C A 1.4~1.5A C 20 structures ring 20 h  10 h 1.28A 1.24A 1.32A bowl C 5v 1.24A 1.40~1.43A

PBE does give the correct energy order!

DFT fail to give a correct  E due to Jahn-Teller effect!

QMC results of graphene ribbon For N=5 Ribbon, the energy difference of nanoribbon states obtained by DFT are: Code & ExC  E (NM-AF)  E (FM-AF) CASTEP LDA 32.5 meV 1.5 meV VASP LDA 36.3 meV 2.0 meV VASP PW meV 3.3 meV VASP PBE 79.1 meV 5.7 meV Crystal B3LYP 290 meV 49 meV ref: Harrison et al. in PRB 75, 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. 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. 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.

For N=5 Graphene Ribbon, the QMC energies obtained are : AF (VMC) : Final energy = (38) eV NM (VMC) : Final energy = (41) eV AF (DMC) : Final energy = (31) eV (~9000 CPU hour) NM (DMC) : Final energy = (35) eV (~9000 CPU hour) And the energy difference obtained by QMC are  E (NM-AF) = 856 meV VMC  E (NM-AF) = 434 meV DMC (  T=0.005) (Here E_cut = 600 eV, BLIP = 2.0 and 1x1x6 unit cell)  It seems that DMC favors B3LYP! N=5 Ribbon K-point (PBE)  E (NM-AF)  E (FM-AF) x1x meV 3.6 meV 1x1x meV 5.0 meV 1x1x meV 5.7 meV 1x1x meV 9.9 meV 1x1x meV 10.9 meV 1x1x meV 13.0 meV For N=5 Ribbon, the energy difference of nanoribbon states obtained by DFT are: Code & ExC  E (NM-AF)  E (FM-AF) CASTEP LDA 32.5 meV 1.5 meV VASP LDA 36.3 meV 2.0 meV VASP PW meV 3.3 meV VASP PBE 79.1 meV 5.7 meV Crystal B3LYP 290 meV 49 meV ref: Harrison et al. in PRB 75, LDA, PW91, PBE are at least a factor of 4~5 less than B3LYP!!!

Summary and Conclusion In general, DFT should be able to use in the study of the metallic clusters judging from the QMC results of Al 13 and Al 55 clusters, and PBE is perhaps better! DFT fails to describe B 18 and B 20 clusters, and QMC is needed! DFT fails to describe well C 20 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!