Koblar Jackson Physics Department Central Michigan University

1. Wafers, platelets, rods and spheres: Using DFTB to determine the structural minima of atomic clusters Koblar Jackson Physics Department Central Michigan University

2. The structure problem Bulk fragment Cluster Bulk Si: diamond structure • Bulk fragments not favorable for clusters due to surface dangling bonds • What happens when chemical intuition doesn’t work???

3. Piano Books Chair Sofa The Jackson Living Room Table

4. Table Books Chair Piano Sofa The Jackson Living Room

5. Table Books Chair Piano Sofa The Jackson Living Room “NP hard problem”: the number of local minima grows exponentially with cluster size

6. Probing the energy surface Gradient optimization – following forces Energy vs structure (R) Energy Global minimum Local minimum R

7. Random starting structure Local minimum

8. Random starting structure Local minimum

9. Efficiency vs Box size: Lennard-Jones clusters • Optimal volume compression ~ (1/5)3 • Ground states found up to N = 105 • As efficient as genetic algorithm up to N = 40

10. Role of DFTB in Search Process • Need quantum description: good vs bad bonds; electron kinetic energy • DFT (PBE-GGA) accurate, but computationally demanding • DFTB mimics DFT, but is 102 – 103 times faster • Use DFTB (Frauenheim et al.) to probe energy landscape H[]Yi = ei Yi

11. DE in eV (DFTB rank) DE in eV (DFT rank) Ordering minima: Si24 0.00 (1) 0.06 (2) 0.42 (3) 0.46 (4) 0.46 (5) 0.00 (1) 0.31 (3) 1.00 (5) 2.21 (9) 2.31 (10) 0.47 (6) 0.50 (7) 0.57 (8) 0.58 (9) 0.59 (10) 1.09 (6) 1.78 (8) 0.64 (4) 1.25 (7) 0.30 (2)

12. DFT vs DFTB energy surfaces E DFTB DFT1 DFT Q • DFT1 energy ordering improves DFTB A

13. DFT Relaxed vs DFT1 Si25 100 local minima DFT Relaxed vs DFTB

14. Compressed Geometries & DFTB relaxation - Done in parallel ~1 x 10^6 local minima ~2 x 10^3 stored Reorder using DFT1 • Approximate DFT ordering • lowest ~300 structures -Exact DFT ordering ~30 lowest structures Full DFT optimization Big Bang Search Methodology: Parallel method for finding global minima Jackson et al., Comp. Mat. Sci. 35, 232 (2006)

15. Sample # of clusters # of clusters Drift Tube Laser Drift Time (ms) Drift Time (ms) SiN Shape Transition: ExperimentHudgins et al, Journal of Chemical Physics 111, 7864 (1999) Abrupt change in cluster shape across 24-28

16. Si21+ +0.39 eV +0.26 eV 0.00 eV +0.37 eV +0.08 eV +0.45 eV Rich structural variety: unbiased search

17. 24 20 C1 3.649 Cs 3.551 Cs 3.652 C1 3.557 25 21 C2v 3.565 Cs 3.666 C2v 3.583 Cs 3.666 26 22 C2v 3.691 C1 3.600 Cs 3.616 Cs 3.687 23 27 C1 3.697 C1 3.635 C1 3.627 C1 3.691 Best prolate vs best compact structure: shape evolution of SiN+ global minima Stability crossover at n=25: shape transition driven by thermodynamics! Jackson et al., Phys. Rev. Lett. 93, 013401 (2004)

18. Predicted vs observed ion mobilities Stretched Expt minor Expt major Th local min Th ground state Prolate Compact Lowest-energy isomers reproduce data across transition region Hudgins et al., J. Chem. Phys. 111, 7865 (1999) Jackson et al., Phys. Rev. Lett. 93, 013401 (2004)

19. Dissociation Energy ( E(M+) + E(N-M) ) – E(N+) Fragments Theory ED Local Expt Global M+ N+ Minimum-energy structures reproduce dissociation E data N-M Expt: Jarrold and Honea, J. Phys. Chem. 95, 9181 (1991)

20. Structural Families n=22 Compact: 1 subunit Prolate: 2 subunits Stretched: 3 subunits

21. Recent DFTB-based work (X. C. Zeng) extending Si structure searches to larger sizes: • Bai J, Cui LF, Wang JL, et al., Structural evolution of anionic silicon clusters Sin (20 <= n <= 45) J. Phys. Chem. A 110 (3): 908-912 JAN 26 2006 • Yoo S, et al., Structures and relative stability of medium-sized silicon clusters. V. Low-lying endohedral fullerene-like clusters, Si31 – Si40 and Si45. J. Chem. Phys. 124124 (16): 164311 APR 28 2006

22. Cu, Ag clusters Empirical/semi-empirical predictions: icosahedral growth pattern Tight-Binding Molecular Dynamics Search Kabir et al. Phy. Rev. A 69:43203(2004)

23. Cu Clusters (N = 10 – 15): DFT Predictions (limited sampling) 10 11 11 12 12 13 Fernandez et al. Phys. Rev. B 70:165403(2004) Guvelioglu et al. Phys. Rev. Lett. 94:26103(2005) DFT: no icosahedral ordering; but no agreement on minima

24. AgN N = 9 - 14 9A(0.00) 9B(0.02) 9C(0.02) 10A(0.00) 10B(0.09) 10C(0.20) 11A(0.00) 11B(0.07) 11C(0.08) 12A(0.00) 12B(0.08) 12C(0.13) 13A(0.00) 13B(0.01) 13C(0.07) 14A(0.00) 14B(0.13) 14C(0.14) M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

25. Ground-state structures of Cu clusters N = 10 – 16: “platelets” Top view 10 11 12 13 14 15 16 Side view M. Yang, K. Jackson, C. Koehler, Th. Frauenheim, and J. Jellinek, J. Chem. Phys. 124, 024308 (2006)

26. Ground-state structures of Cu clusters N = 17 – 20 : “spheres” 17 18 19 20 Icosahedral core M. Yang, K. Jackson, C. Koehler, Th. Frauenheim, and J. Jellinek, J. Chem. Phys. 124, 024308 (2006)

27. IP (eV) expt Isomer 1 Isomer 2 Isomer 3 Cluster size CuN: Calculated and measured vertical ionization potentials M. Knickelbein, CPL 192,129(1992) • IP can distinguish isomers • Lowest-energy structures generally in best agreement

28. AgN JN Shape evolution and shell filling: AgN vs “ultimate jellium” <Ii> = 3*Ii/(I1+ I2+ I3) Sphere: I1= I2=I3 = 1 Prolate: I1,I2 > 1 Oblate: I1,I2 < 1 JN : M. Koskinen et al., Z. Phys. D:At., Mol. Clusters 35, 285 (1995). M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

29. Summary • DFTB plays essential role in structural search algorithm: scan energy surface for likely structures • Search methodology yields structures consistent with known expt data • Clusters display an array of shapes at small sizes: wafers, platelets, rods, and spheres

30. Thanks to: • J. Barra, J. Boike, J. Juen, I. Rata, A. Balakrishnan (students) • M. Yang, M. Horoi (CMU) • Frauenheim, Seifert, Koehler, Hajnal (DFTB friends) • A. Shvartsburg (PNNL) • J. Jellinek (ANL)

31. Support • This work is supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, U. S. Department of Energy, under contract DE-FG02-03ER15489

32. Wafers Platelets Spheres N = 16 N = 19 N = 12 N = 5 Shape fluctuations in Cu clusters

33. Cohesive energy of layered and compact Cu clusters Cohesive energy (eV) layered compact Cluster size M. Yang, K. Jackson, C. Koehler, Th. Frauenheim, and J. Jellinek, J. Chem. Phys. 124, 024308 (2006)

34. VDE (eV) layered compact I measured Cluster size Calculated and measured vertical detachment energies of Cu anions • Cha et al. J. Chem. Phys. 99:6308(1993)

35. AgN vs CuN Thirty lowest-energy isomers of Cu10 vs. corresponding isomers of Ag10 Excellent correlation: structures found in CuN search can be used as candidate structures for AgN M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

36. AgN N = 9 – 20 (cont’d) 15A(0.00) 15B(0.03) 15C(0.06) 16A(0.00) 16B(0.09) 16C(0.14) 17A(0.00) 17B(0.24) 17C(0.26) 18A(0.00) 18B(0.06) 18C(0.13) 19A(0.00) 19B(0.11) 19C(0.18) 20A(0.00) 20B(0.05) 20C(0.24) M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

37. + PES TH (neutral) TH (anion) AgN HOMO-LUMO Gaps M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

38. ECoh (eV) Ecoh = [NE(1) – E(N)]/N DE(2) (eV) DE(2) = 2E(N) – E(N+1) – E(N) N AgN: Impact of shape on properties M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

39. Planar to layered Layered to compact Shape vs dipole polarizability M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

40. 12A 13 11 16A 14 17 CuN- PES: expt vs theory