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

3. The Jackson Living Room Piano Sofa Chair Table Books

4. Piano Sofa Chair Table Books The Jackson Living Room

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

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

7. Random starting structure Local minimum

8. Random starting structure Local minimum

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

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 10 2 – 10 3 times faster Use DFTB (Frauenheim et al.) to probe energy landscape H[  ]  i  i  i

11. Ordering minima: Si 24 0.00 (1) 0.06 (2)0.42 (3) 0.46 (4)0.46 (5) 0.47 (6) 0.50 (7)0.57 (8) 0.58 (9)0.59 (10) 0.31 (3)1.00 (5)2.21 (9)2.31 (10) 1.09 (6) 1.78 (8)0.64 (4) 1.25 (7)0.30 (2)  E in eV (DFTB rank)  E in eV (DFT rank)

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

13. Si 25 100 local minima DFT Relaxed vs DFTB DFT Relaxed vs DFT1

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

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

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

17. C s 3.666 20 21 22 23 24 25 26 27 C 1 3.557 C s 3.551 C 2v 3.583 C 2v 3.565 C s 3.616 C 1 3.600 C 1 3.635 C 1 3.627 C s 3.652 C 1 3.649 C s 3.666 C 2v 3.691 C s 3.687 C 1 3.697 C 1 3.691 Best prolate vs best compact structure: shape evolution of Si N + global minima Stability crossover at n=25: shape transition driven by thermodynamics! Jackson et al., Phys. Rev. Lett. 93, 013401 (2004)

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

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

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

21. Recent DFTB-based work (X. C. Zeng) extending Si structure searches to larger sizes: 1.Bai J, Cui LF, Wang JL, et al., Structural evolution of anionic silicon clusters Si n (20 <= n <= 45) J. Phys. Chem. A 110 (3): 908- 912 JAN 26 2006 2.Yoo S, et al., Structures and relative stability of medium-sized silicon clusters. V. Low-lying endohedral fullerene-like clusters, Si 31 – Si 40 and Si 45. J. Chem. Phys. 124 124 (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) Guvelioglu et al. Phys. Rev. Lett. 94:26103(2005) 10 11 11 12 12 13 Fernandez et al. Phys. Rev. B 70:165403(2004) DFT: no icosahedral ordering; but no agreement on minima

24. 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) Ag N N = 9 - 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 Side view 10 11 12 13 14 15 16 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 can distinguish isomers -Lowest-energy structures generally in best agreement expt Isomer 1 Isomer 2 Isomer 3 IP (eV) Cluster size Cu N : Calculated and measured vertical ionization potentials M. Knickelbein, CPL 192,129(1992)

28. Shape evolution and shell filling: Ag N vs “ultimate jellium” Ag N JNJN = 3*I i /(I 1 + I 2 + I 3 ) Sphere: I 1 = I 2 =I 3 = 1 Prolate: I 1,I 2 > 1 Oblate: I 1,I 2 < 1 J N : 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. WafersPlateletsSpheres N = 5 N = 12 N = 19 N = 16 Shape fluctuations in Cu clusters

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

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

35. Ag N vs Cu N Thirty lowest-energy isomers of Cu 10 vs. corresponding isomers of Ag 10 Excellent correlation: structures found in Cu N search can be used as candidate structures for Ag N M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

36. 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) Ag N N = 9 – 20 (cont’d) 15A(0.00) 15B(0.03) 15C(0.06) 16A(0.00) 16B(0.09) 16C(0.14) M. Yang, K. Jackson, J. Jellinek, J. Chem. Phys. (to appear)

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

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

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

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