THRESHOLD PHOTO-IONISATION AND DENSITY FUNCTIONAL THEORY STUDIES OF METAL-CARBIDE CLUSTERS

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1 THRESHOLD PHOTO-IONISATION AND DENSITY FUNCTIONAL THEORY STUDIES OF METAL-CARBIDE CLUSTERS Viktoras Dryza A thesis submitted in total fulfillment of the requirements for the degree of Doctor of Philosophy November, 2008 Department of Chemistry The University of Adelaide

2 This work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being made available in all forms of media, now or hereafter known. Viktoras Dryza November, II

3 Dedicated to my Dad. III

4 Serenity now, serenity now! - Frank Costanza (Seinfeld, Season 9/Episode 3) IV

5 Abstract Neutral gas-phase metal-carbide clusters are generated by laser ablation and are detected in the constructed time-of-flight mass-spectrometer by laser ionisation. Photo-ionisation efficiency (PIE) experiments are performed on the metal-carbide clusters to determine their ionisation potentials (IPs). Complimentary density functional theory (DFT) calculations are performed on the energetically favorable structural isomers of the metalcarbide clusters. Comparison between the calculated IPs of the isomers and the experimental IP allows the carrier of the observed ionisation onset for a metal-carbide cluster to be assigned. The niobium-carbide clusters Nb 3 C y (y = 0 4), Nb 4 C y (y = 0 6) and Nb 5 C y (y = 0 6) are examined by PIE experiments and DFT calculations. The IPs of the niobium-carbide clusters are found to be either left reasonably unchanged from the IPs of the bare metal clusters or moderately reduced. The clusters Nb 3 C 2, Nb 4 C 4, Nb 5 C 2 and Nb 5 C 3 display the largest IP reductions for their corresponding cluster series. The structures assigned to the IPs of the Nb 3 C y (y = 1 3) clusters are based on the carbon atoms attaching to the niobium faces and/or niobium-niobium edges of the triangular Nb 3 cluster. However, for Nb 3 C 4 the ionisation onset is assigned to a low-lying isomer, which contains a molecular C 2 unit, rather than the lowest energy isomer, a niobium atom deficient face-centred cubic (fcc) nanocrystal structure. The structures assigned to the IPs of the Nb 4 C y (y = 1 4) clusters are based on the carbon atoms attaching in turn to the niobium faces of the tetrahedral Nb 4 cluster, developing a fcc nanocrystal structure for Nb 4 C 4. For Nb 4 C 3 two ionisation onsets are observed; one weak onset at low energy and another more intense onset at high energy. It is proposed that the two onsets are due to ionisation from both a metastable 3 A 1 state and the ground 1 A 1 state of the lowest energy isomer. The ionisation onsets of Nb 4 C 5 and Nb 4 C 6 are also proposed to originate from metastable triplet states of the lowest energy isomers, with the transitions from the ground singlet states calculated to be greater than the highest V

6 achievable photon energy in the laboratory. The structures of Nb 4 C 5 and Nb 4 C 6 have one and two carbon atoms in a fcc nanocrystal substituted with molecular C 2 units, respectively. The structures assigned to the IPs of the Nb 5 C y (y = 1 6) clusters are based on the underlying Nb 5 cluster being in either a prolate or oblate trigonal bipyramid geometry; the former has six niobium faces available for carbon addition, while the latter has two niobium butterfly motifs and two niobium faces available for carbon addition. Both the structures of Nb 5 C 5 and Nb 5 C 6 have the underlying Nb 5 cluster in the oblate trigonal bipyramid geometry and contain one and two molecular C 2 units, respectively. The tantalum-carbide clusters Ta 3 C y (y = 0 3), Ta 4 C y (y = 0 4) and Ta 5 C y (y = 0 6) are examined by PIE experiments and DFT calculations. The IPs of the tantalum-carbide clusters in each series show trends that are very similar to the corresponding iso-valent niobium-carbide cluster series, although the IP reductions upon carbon addition are smaller for the former. For the vast majority of tantalum-carbide clusters, the same structural isomer is assigned to the ionisation onset as that assigned for the corresponding niobium-carbide cluster. Bimetallic tantalum-zirconium-carbide clusters are generated using a constructed double ablation cluster source. The Ta 3 ZrC y (y = 0 4) clusters are examined by PIE experiments and DFT calculations. The IP trend for the Ta 3 ZrC y cluster series is reasonably similar to that of the Ta 4 C y cluster series, although the IP reductions upon carbon addition are greater for the former. The structures assigned to the IPs of the Ta 3 ZrC y (y = 1 4) clusters are based on the carbon atoms attaching in turn to the metal faces of the tetrahedral Ta 3 Zr cluster. In summary, the work presented in this thesis demonstrates that the structures of metalcarbide clusters can be inferred by the determination of their IPs through PIE experiments in combination with DFT calculations on candidate structural isomers. VI

7 Acknowledgments First and foremost I would like to thank my supervisor Dr. Greg Metha for giving me this opportunity. Professionally his helpful advice, extensive knowledge and forward thinking attitude to research have been invaluable. Personally he has been a great friend and mentor. I will always remember our stay in Munich; getting lost in a dark snowing forest and throwing snowballs at each other on the way home after a late night out. Also, the many laughing fits at our mutual appreciation of Borat. Thanks must also go to my co-supervisor Prof. Mark Buntine for his smart-arse comments and the occasional helpful one. I am also totally indebted to our post-doc Dr. Jason Gascooke, whose ability to solve problems which seem unsolvable to me is truly amazing. He has also been a great drinking buddy and is the co-inventor of beer-table tennis. Thanks to Matt Addicoat for his help with the computational calculations. The support in the mechanical workshop from Peter Apoefis has been fantastic in converting my caveman-like drawings into reality. I have also had many a fun time in the lab with my partner in crime Olivia Maselli; whether it be dancing around like a fool, discussing our small overlap in musical tastes or laughing at her stories. Also thanks to all the other Honours and PhD students I have shared the lab with. Huge thanks must go to my extremely supportive family (Mum, Dad and Kristina) for the Sunday lunches and for still believing I have some intelligence, despite observing my constant stupidity over the years. Last, but definitely not least, I must thank my beautiful girlfriend Milena. She has always believed in me and has the magical ability to make me happy whenever I m with her (even when my experiments are not working). VII

8 Publications The following publications were produced from the work presented in this thesis: Ionization Potentials of Tantalum-Carbide Clusters: An Experimental and Density Functional Theory Study Dryza, V.; Addicoat, M.A.; Gascooke, J.R.; Buntine, M.A.; Metha, G.F. J. Phys. Chem. A 2005, 109, Threshold Photo-ionization and Density Functional Theory Studies of the Niobium- Carbide Clusters Nb 3 C n (n = 1 4) and Nb 4 C n (n = 1 6) Dryza, V.; Addicoat, M.A.; Gascooke, J.R.; Buntine, M.A.; Metha, G.F. J. Phys. Chem. A 2008, 112, Onset of Carbon-Carbon Bonding in the Nb 5 C y (y = 0 6) Clusters: A Threshold Photoionisation and Density Functional Theory Study Dryza, V.; Gascooke, J.R.; Buntine, M.A.; Metha, G.F. PCCP 2008, (in press). The following publications are currently in preparation from the work presented in this thesis: Onset of Carbon-Carbon Bonding in the Ta 5 C y (y = 0 6) Clusters: A Threshold Photoionization and Density Functional Theory Study Dryza, V.; Gascooke, J.R.; Buntine, M.A.; Metha, G.F. (in preparation). Threshold Photo-ionization and Density Functional Theory Studies of Bimetallic-Carbide Clusters: Ta 3 ZrC y (y = 0 4) Dryza, V.; Gascooke, J.R.; Buntine, M.A.; Metha, G.F. (in preparation). VIII

9 Abbreviations AO C DFT EA ECP ev FC FCC FEL FWHM GTO HOMO IP IR LUMO MO MPD MPI MRCI Nb NBO PES PFI-ZEKE PIE REMPI SPI Ta TOF-MS ZPE Zr Atomic Orbital Carbon Density Functional Theory Electron Affinity Effective Core Potential Electron Volt Frank-Condon Face-centred Cubic Free Electron Laser Full Width at Half Maximum Gaussian Type Orbital Highest Occupied Molecular Orbital Ionisation Potential Infrared Lowest Unoccupied Molecular Orbital Molecular Orbital Multi-photon Dissociation Multi-photon Ionisation Multi-Reference Configuration Interaction Niobium Natural Bond Order Potential Energy Surface Pulsed Field Ionisation Zero Electron Kinetic Energy Photo-ionisation Efficiency Resonance Enhanced Multi-photon Ionisation Single-photon Ionisation Tantalum Time-of-flight Mass-spectrometer Zero-point Energies Zirconium IX

10 Table of Contents Chapter 1: Introduction to Metal-Carbide Clusters Generation and Distribution of Metal-Carbide Clusters The Structures of the Ti 8 C 12 Metcar and Ti 14 C 13 Nanocrystal Potential Applications of Metal-Carbide Clusters Scope of this Thesis Why Determine the Ionisation Potentials of Metal-Carbide Clusters? References 10 Chapter 2: Experimental Approach Laser Ablation Laser Ablation Theory Experimental Design and Operation Photo-ionisation Efficiency (PIE) Photo-ionisation Theory PIE Theory PIE of Transition Metal Containing Clusters Experimental Procedure for PIE Experiments References 31 Chapter 3: Computational Approach Introduction to Density Functional Theory (DFT) Basis Sets Computational Method and Procedure for Metal-Carbide Clusters References 40 Chapter 4: The Time-of-Flight Mass-Spectrometer The Constructed TOF-MS Apparatus References 47 X

11 Chapter 5: Niobium-Carbide Clusters Introduction to Niobium-Carbide and Tantalum-Carbide Clusters Photo-ionisation Efficiency Experiments DFT Calculated Isomers The Nb 3 C y (y = 0 4) Cluster Series The Nb 4 C y (y = 0 6) Cluster Series The Nb 5 C y (y = 0 6) Cluster Series Comparison between Calculated and Experimental IPs The Nb 3 C y (y = 0 4) Cluster Series The Nb 4 C y (y = 0 6) Cluster Series The Nb 5 C y (y = 0 6) Cluster Series Discussion on Niobium-Carbide Clusters Comparison to Previous Spectroscopic Data Onset of Carbon-Carbon Bonding Low-Lying Isomers and Metastable Electronic States IP trends, Molecular Orbitals and HOMO-LUMO gaps Thermodynamic Cycle Summary and Future Directions References 109 Chapter 6: Tantalum-Carbide Clusters Photo-ionisation Efficiency Experiments Comparison between Calculated and Experimental IPs The Ta 3 C y (y = 0 3) Cluster Series The Ta 4 C y (y = 0 4) Cluster Series The Ta 5 C y (y = 0 6) Cluster Series Discussion on Tantalum-Carbide Clusters Comparison to the Niobium-Carbide Clusters Summary References 130 XI

12 Chapter 7: Tantalum-Zirconium-Carbide Clusters Introduction to Bimetallic-Carbide Clusters Double Ablation Cluster Source Design Generation of Tantalum-Zirconium-Carbide Clusters Photo-ionisation Efficiency Experiments DFT Calculated Isomers The Ta 3 ZrC y (y = 0 4) Cluster Series Comparison between Calculated and Experimental IPs Discussion on Tantalum-Zirconium-Carbide Clusters Relative Energetics of the Bimetallic-Carbide Cluster Isomers Electronic Structure of the Bimetallic-Carbide Clusters Comparison of IP Trends for the Ta 3 ZrC y and Ta 4 C y Clusters Summary References 158 Appendix I 159 Appendix II Appendix IV 189 Appendix V 193 Appendix VI 203 Appendix VII 205 XII

13 Tables displaying the geometrical parameters and atomic charges for the neutral and cationic niobium-carbide cluster isomers (B3P86/Nb SDD, C aug-cc-pvtz). Nb 3 I A 2 A C s Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1 = Nb2 = 0.08 Nb3 = Nb 3 + I A + 3 A 1 D 3h Nb-Nb = Å Nb =

14 Nb 3 C II A 2 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-C = Å Nb1 = Nb2 = 0.27 C = 0.77 Nb 3 C + II A + 3 A 2 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1 = Nb2 = Nb1-C = Å C = 0.68 Nb 3 C II B 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-C = Å Nb2-C = Å Nb1 = Nb2 = C = 0.84 Nb 3 C + II B + 1 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1 = Nb2 = Nb1-C = Å Nb2-C = Å C =

15 Nb 3 C 2 III A 2 A C s Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C = Å Nb2-C = Å Nb3-C = Å Nb1 = Nb2 = Nb3 = C = 0.90 Nb 3 C 2 + III A + 1 A 1 D 3h Nb-Nb = Å Nb-C = Å Nb = C =

16 Nb 3 C 3 IV A 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-C1 = Å Nb1-C2 = Å Nb2-C1 = Å Nb2-C2 = Å Nb1-Nb2-Nb1-C1 = Nb1 = Nb2 = C1 = 0.66 C2 = 0.88 Nb 3 C 3 + IV A + 1 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1 = Nb2 = Nb1-C1 = Å Nb1-C2 = Å Nb2-C1 = Å Nb2-C2 = Å C1 = 0.57 C2 = 0.82 Nb1-Nb2-Nb1-C1 = Nb 3 C 3 IV B 2 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb2-C2 = Å Nb2-C3 = Å Nb3-C1 = Å Nb3-C2 = Å Nb3-C3 = Å C1-C2 = Å Nb1 = Nb2 = Nb3 = C1 = 0.55 C2 = 0.52 C3 = 0.70 Nb 3 C 3 + IV B + 3 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb2-C2 = Å Nb2-C3 = Å Nb3-C1 = Å Nb3-C2 = Å Nb3-C3 = Å C1-C2 = Å Nb1 = Nb2 = Nb3 = C1 = 0.41 C2 = 0.50 C3 =

17 Nb 3 C 4 V A 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C3 = Å Nb1-Nb2-Nb1-C1 = Nb2-Nb1-Nb1-C2 = Nb1 = Nb2 = C1 = 0.55 C2 = 0.62 C3 = 0.86 Nb 3 C 4 + V A + 1 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1 = Nb2 = Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C3 = Å C1 = 0.51 C2 = 0.49 C3 = 0.81 C1-C1 = Å Nb1-Nb2-Nb1-C1 = 99.4 Nb2-Nb1-Nb1-C2 = Nb 3 C 4 V B 2 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C2 = Å Nb2-C3 = Å Nb2-C4 = Å Nb3-C1 = Å Nb3-C2 = Å Nb3-C3 = Å Nb3-C4 = Å C1-C2 = Å Nb1 = Nb2 = Nb3 = C1 = 0.60 C2 = 0.50 C3 = 0.83 C4 = 0.66 Nb 3 C 4 + V B + 1 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C2 = Å Nb2-C3 = Å Nb2-C4 = Å Nb3-C1 = Å Nb3-C2 = Å Nb3-C3 = Å Nb3-C4 = Å C1-C2 = Å Nb1 = Nb2 = Nb3 = C1 = 0.49 C2 = 0.39 C3 = 0.73 C4 =

18 Nb 4 VI A 1 A 1 T d Nb-Nb = Å Nb = 0.00 Nb 4 + VI A + 2 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb2-Nb2 = Å Nb1 = Nb2 =

19 Nb 4 C VII A 1 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C = Å Nb2-C = Å Nb1 = Nb2 = Nb3 = 0.27 C = 0.85 Nb 4 C + VII A + 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C = Å Nb2-C = Å Nb1 = Nb2 = Nb3 = C = 0.73 Nb 4 C VII B 1 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb2-Nb2 = Å Nb1-C = Å Nb1 = Nb2 = 0.06 C = 0.70 Nb 4 C + VII B + 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C = Å Nb1 = Nb2 = Nb3 = C =

20 Nb 4 C 2 VIII A 1 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb2-Nb2 = Å Nb1-C = Å Nb2-C = Å Nb1 = Nb2 = C = 0.85 Nb 4 C 2 + VIII A + 2 B 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb2-Nb2 = Å Nb1-C = Å Nb2-C = Å Nb1 = Nb2 = C = 0.77 Nb 4 C 2 VIII B 1 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-C1 = Å Nb1-C2 = Å Nb2-C1 = Å Nb2-C1 = Å Nb1 = Nb2 = C1 = 0.82 C2 = 0.84 Nb 4 C 2 + VIII B + 2 B 2 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1 = Nb2 = Nb1-C1 = Å Nb1-C2 = Å Nb2-C1 = Å Nb2-C1 = Å C1 = 0.83 C2 =

21 Nb 4 C 3 IX A 1 A 1 C 3v Nb1-Nb2 = Å Nb2-Nb2 = Å Nb1-C = Å Nb2-C = Å Nb1 = Nb2 = C = 0.79 Nb 4 C 3 + IX A + 2 A 1 C 3v Nb1-Nb2 = Å Nb2-Nb2 = Å Nb1 = Nb2 = Nb1-C = Å Nb2-C = Å C =

22 Nb 4 C 4 X A 1 A 1 T d Nb-Nb = Å Nb-C = Å Nb = C1 = 0.72 Nb 4 C 4 + X A + 2 B 1 D 2d Nb1-Nb1 = Nb2-Nb2 = Å Nb1-Nb2 = Å Nb1 = Nb2 = Nb1 Nb1-C1 = Nb2-C2 = Å Nb1-C2 = Nb2-C1 = Å C1 = 0.70 C2 = C1 175

23 Nb 4 C 5 XI A 1 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb1-C4 = 2.190Å Nb2-C1 = Å Nb2-C3 = Å Nb3-C1 = Å Nb3-C2 = Å C3-C4 = Å Nb1 = Nb2 = Nb3 = C1 = 0.83 C2 = 0.79 C3 = 0.53 C4 = 0.55 Nb 4 C 5 + XI A + 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb1-C4 = Å Nb2-C1 = Å Nb2-C3 = Å Nb3-C1 = Å Nb3-C2 = Å Nb1 = Nb2 = Nb3 = C1 = 0.76 C2 = 0.72 C3 = 0.46 C4 = 0.48 C3-C4 = Å 176

24 Nb 4 C 6 XII A 1 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb2-Nb2 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C2 = Å C2-C3 = Å Nb1 = Nb2 = C1 = 0.81 C2 = 0.53 C3 = 0.44 Nb 4 C 6 + XII A + 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb1-C4 = Å Nb1-C5 = Å Nb2-C1 = Å Nb2-C2 = Å Nb3-C1 = Å Nb3-C4 = Å Nb1 = Nb2 = Nb3 = C1 = 0.76 C2 = 0.48 C3 = 0.35 C4 = 0.46 C5 = 0.30 C2-C3 = Å C4-C5 = Å Nb 4 C 6 XII B 1 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb1-C4 = Å Nb2-C1 = Å Nb2-C2 = Å Nb3-C1 = Å Nb3-C4 = Å Nb3-C5 = Å C2-C3 = Å C4-C5 = Å Nb1 = Nb2 = Nb3 = C1 = 0.82 C2 = 0.50 C3 = 0.51 C4 = 0.64 C5 = 0.25 Nb 4 C 6 + XII B + 2 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb2-Nb4 = Å Nb3-Nb4 = Å Nb1-C1 = Å Nb1-C3 = Å Nb1-C4 = Å Nb1-C5 = Å Nb2-C2 = Å Nb2-C3 = Å Nb2-C4 = Å Nb2-C5 = Å Nb3-C1 = Å Nb3-C2 = Å Nb3-C3 = Å Nb4-C1 = Å Nb4-C2 = Å Nb4-C5 = Å Nb4-C6 = Å C3-C4 = Å C5-C6 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.77 C2 = 0.75 C3 = 0.46 C4 = 0.45 C5 = 0.60 C6 =

25 Nb 4 C 6 XII C 1 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb2-Nb4 = Å Nb3-Nb4 = Å Nb1-C1 = Å Nb1-C3 = Å Nb1-C5 = Å Nb1-C6 = Å Nb2-C2 = Å Nb2-C3 = Å Nb2-C4 = Å Nb2-C5 = Å Nb2-C6 = Å Nb3-C1 = Å Nb3-C2 = Å Nb3-C3= Å Nb3-C4 = Å Nb4-C1 = Å Nb4-C2 = Å Nb4-C5 = Å C3-C4 = Å C5-C6 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.83 C2 = 0.77 C3 = 0.47 C4 = 0.44 C5 = 0.55 C6 = 0.43 Nb 4 C 6 + XII C + 2 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb2-Nb4 = Å Nb3-Nb4 = Å Nb1-C1 = Å Nb1-C3 = Å Nb1-C5 = Å Nb1-C6 = Å Nb2-C2 = Å Nb2-C3 = Å Nb2-C4 = Å Nb2-C5 = Å Nb2-C6 = Å Nb3-C1 = Å Nb3-C2 = Å Nb3-C3 = Å Nb3-C4 = Å Nb4-C1 = Å Nb4-C2 = Å Nb4-C5 = Å C3-C4 = Å C5-C6 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.76 C2 = 0.70 C3 = 0.45 C4 = 0.33 C5 = 0.48 C6 = 0.42 Nb 4 C 6 XII D 1 A C 2 Nb1-Nb2 = Å Nb1-Nb3 = Nb2-Nb4 = Å Nb1-Nb4 = Nb2-Nb3 = Å Nb3-Nb4 = Å Nb1-C1 = Nb2-C2 = Å Nb1-C3 = Nb2-C5 = Å Nb1-C5 = Nb2-C3 = Å Nb1-C6 = Nb2-C4 = Å Nb3-C1 = Nb4-C2 = Å Nb3-C2 = Nb4-C1 = Å Nb3-C3 = Nb4-C5 = Å Nb3-C4 = Nb4-C6 = Å C3-C4 = C5-C6 = Å Nb1 = Nb2 = Nb1 Nb3 = Nb4 = Nb3 C1 = 0.79 C2 = C1 C3 = 0.50 C4 = 0.47 C5 = C3 C6 = C4 Nb 4 C 6 + XII D + 2 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb2-Nb4 = Å Nb3-Nb4 = Å Nb1-C1 = Å Nb1-C3 = Å Nb1-C5 = Å Nb1-C6 = Å Nb2-C2 = Å Nb2-C3 = Å Nb2-C4 = Å Nb2-C5 = Å Nb3-C1 = Å Nb3-C2 = Å Nb3-C3 = Å Nb3-C4 = Å Nb4-C1 = Å Nb4-C2 = Å Nb4-C5 = Å Nb4-C6 = Å C3-C4 = Å C5-C6 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.72 C2 = 0.73 C3 = 0.45 C4 = 0.40 C5 = 0.45 C6 =

26 Nb 4 C 6 XII E 1 A C s Nb1-Nb1 = Å Nb2-Nb3 = Å Nb1-Nb2 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb1-C4 = Å Nb1-C5 = Å Nb2-C1 = Å Nb2-C3 = Å Nb3-C1 = Å Nb3-C2 = Å C3-C4 = Å C4-C5 = Å Nb1 = Nb2 = Nb3 = C1 = 0.79 C2 = 0.74 C3 = 0.48 C4 = 0.31 C5 = 0.41 Nb 4 C 6 + XII E + 2 A C s Nb1-Nb1 = Å Nb2-Nb3 = Å Nb1-Nb2 = Å Nb1 = Nb2 = Nb3 = Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb1-C4 = Å Nb1-C5 = Å Nb2-C1 = Å Nb2-C3 = Å Nb3-C1 = Å Nb3-C2 = Å C1 = 0.75 C2 = 0.68 C3 = 0.43 C4 = 0.27 C5 = 0.28 C3-C4 = Å C4-C5 = Å 179

27 Nb 5 XIII A 2 B 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb2 = Å Nb2-Nb3 = Å Nb1 = 0.20 Nb2 = Nb3 = Nb 5 + XIII A + 3 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb2 = Å Nb2-Nb3 = Å Nb1 = Nb2 = Nb3 =

28 Nb 5 C XIV A 2 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb2-Nb4 = Å Nb2-Nb5 = Å Nb3-Nb4 = Å Nb3-Nb5 = Å Nb4-Nb5 = Å Nb1-C = Å Nb3-C = Å Nb4-C = Å Nb1 = Nb2 = 0.13 Nb3 = Nb4 = Nb5 = 0.15 C = 0.88 Nb 5 C + XIV A + 1 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb2-Nb4 = Å Nb2-Nb5 = Å Nb3-Nb4 = Å Nb3-Nb5 = Å Nb4-Nb5 = Å Nb1-C = Å Nb3-C = Å Nb4-C = Å Nb1 = Nb2 = Nb3 = Nb4 = Nb5 = C = 0.81 Nb 5 C XIV B 2 A 1 C 2v Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb2 = Å Nb2-Nb3 = Å Nb1-C = Å Nb2-C = Å Nb1 = Nb2 = Nb3 = C = 0.89 Nb 5 C + XIV B + 3 B 2 C 2v Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb2 = Å Nb2-Nb3 = Å Nb1-C = Å Nb2-C = Å Nb1 = Nb2 = Nb3 = C =

29 Nb 5 C 2 XV A 2 B 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb2 = Å Nb1-C = Å Nb2-C = Å Nb3-C = Å Nb1 = Nb2 = Nb3 = C = 0.84 Nb 5 C 2 + XV A + 1 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb2 = Å Nb1-C = Å Nb2-C = Å Nb3-C = Å Nb1 = Nb2 = Nb3 = C = 0.80 Nb 5 C 2 XV B 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb3-Nb4 = Å Nb1-C = Å Nb2-C = Å Nb3-C = Å Nb1 = Nb2 = Nb3 = Nb4 = 0.24 C = 0.82 Nb 5 C 2 + XV B + 3 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb3-Nb4 = Å Nb1 = Nb2 = Nb3 = Nb4 = C = 0.76 Nb1-C = Å Nb2-C = Å Nb3-C = Å 182

30 Nb 5 C 3 XVI A 2 A 1 C 3v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-C = Å Nb2-C = Å Nb1 = Nb2 = Nb3 = 0.26 C = 0.80 Nb 5 C 3 + XVI A + 1 A 1 C 3v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1 = Nb2 = Nb3 = 0.01 Nb1-C = Å Nb2-C = Å C =

31 Nb 5 C 4 XVII A 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb3-Nb4 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C2 = Å Nb3-C1 = Å Nb4-C3 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.83 C2 = 0.77 C3 = 0.87 Nb 5 C 4 + XVII A + 3 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb3-Nb4 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C2 = Å Nb3-C1 = Å Nb4-C3 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.75 C2 = 0.69 C3 = 0.80 Nb 5 C 4 XVII B 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C2 = Å Nb3-C1 = Å Nb3-C3 = Å Nb4-C3 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.80 C2 = 0.80 C3 = 0.85 Nb 5 C 4 + XVII B + 3 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C2 = Å Nb3-C1 = Å Nb3-C3 = Å Nb4-C3 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.74 C2 = 0.73 C3 =

32 Nb 5 C 4 XVII C 2 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb2 = Å Nb1-C1 = Å Nb1-C2 = Å Nb2-C1 = Å Nb2-C2 = Å Nb3-C1 = Å Nb1 = Nb2 = Nb3 = C1 = 0.86 C2 = 0.83 Nb 5 C 4 + XVII C + 1 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb2 = Å Nb1-C1 = Å Nb1-C2 = Å Nb2-C1 = Å Nb2-C2 = Å Nb3-C1 = Å Nb1 = Nb2 = Nb3 = C1 = 0.84 C2 = 0.77 Nb 5 C 4 XVII D 2 A C 2 Nb1-Nb2 = Nb3-Nb4 = Å Nb1-Nb3 = Å Nb1-Nb4 = Nb2-Nb3 = Å Nb1-Nb5 = Nb3-Nb5 = Å Nb2-Nb5 = Nb4-Nb5 = Å Nb1-C1 = Nb3-C2 = Å Nb1-C2 = Nb3-C1 = Å Nb1-C3 = Nb3-C4 = Å Nb2-C1 = Nb4-C2 = Å Nb2-C3 = Nb4-C4 = Å Nb5-C3 = Nb5-C4 = Å Nb1 = Nb2 = Nb3 = Nb1 Nb4 = Nb2 Nb5 = C1 = 0.80 C2 = C1 C3 = 0.84 C4 = C3 Nb 5 C 4 + XVII D + 1 A C 2 Nb1-Nb2 = Nb3-Nb4 = Å Nb1-Nb3 = Å Nb1-Nb4 = Nb2-Nb3 = Å Nb1-Nb5 = Nb3-Nb5 = Å Nb2-Nb5 = Nb4-Nb5 = Å Nb1-C1 = Nb3-C2 = Å Nb1-C2 = Nb3-C1 = Å Nb1-C3 = Nb3-C4 = Å Nb2-C1 = Nb4-C2 = Å Nb2-C3 = Nb4-C4 = Å Nb5-C3 = Nb5-C4 = Å Nb1 = Nb2 = Nb3 = Nb1 Nb4 = Nb2 Nb5 = C1 = 0.73 C2 = C1 C3 = 0.78 C4 = C3 185

33 Nb 5 C 5 XVIII A 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb1-C4 = Å Nb2-C1 = Å Nb2-C3 = Å Nb3-C1 = Å Nb3-C2 = Å Nb4-C2 = Å Nb4-C4 = Å C3-C4 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.83 C2 = 0.86 C3 = 0.58 C4 = 0.55 Nb 5 C 5 XVIII A + 1 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb1-C4 = Å Nb2-C1 = Å Nb2-C3 = Å Nb3-C1 = Å Nb3-C2 = Å Nb4-C2 = Å Nb4-C4 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.77 C2 = 0.81 C3 = 0.53 C4 = 0.52 C3-C4 = Å Nb 5 C 5 XVIII B 2 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb3-Nb4 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C3 = Å Nb3-C1 = Å Nb3-C2 = Å Nb4-C2 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.72 C2 = 0.79 C3 = 0.84 Nb 5 C 5 XVIII B + 1 A C s Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb3-Nb4 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C3 = Å Nb3-C1 = Å Nb3-C2 = Å Nb4-C2 = Å Nb1 = Nb2 = Nb3 = Nb4 = C1 = 0.66 C2 = 0.73 C3 =

34 Nb 5 C 5 XVIII C 2 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb2-Nb4 = Å Nb3-Nb4 = Å Nb3-Nb5 = Å Nb4-Nb5 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C2 = Å Nb2-C4 = Å Nb3-C1 = Å Nb3-C3 = Å Nb3-C4 = Å Nb4-C2 = Å Nb4-C3 = Å Nb4-C4 = Å Nb4-C5 = Å Nb5-C3 = Å Nb5-C5 = Å Nb1 = Nb2 = Nb3 = Nb4 = Nb5 = C1 = 0.82 C2 = 0.81 C3 = 0.86 C4 = 0.76 C5 = 0.61 Nb 5 C 5 + XVIII C + 1 A C 1 Nb1-Nb2 = Å Nb1-Nb3 = Å Nb1-Nb4 = Å Nb2-Nb3 = Å Nb2-Nb4 = Å Nb3-Nb4 = Å Nb3-Nb5 = Å Nb4-Nb5 = Å Nb1-C1 = Å Nb1-C2 = Å Nb1-C3 = Å Nb2-C1 = Å Nb2-C2 = Å Nb2-C4 = Å Nb3-C1 = Å Nb3-C3 = Å Nb3-C4 = Å Nb4-C2 = Å Nb4-C3 = Å Nb4-C4 = Å Nb4-C5 = Å Nb5-C3 = Å Nb5-C5 = Å Nb1 = Nb2 = Nb3 = Nb4 = Nb5 = C1 = 0.76 C2 = 0.75 C3 = 0.86 C4 = 0.70 C5 = 0.54 Nb 5 C 5 XVIII D 2 A C 1 Nb1-Nb2 = Å Nb1-Nb4 = Å Nb1-Nb5 = Å Nb2-Nb3 = Å Nb2-Nb4 = Å Nb2-Nb5 = Å Nb3-Nb4 = Å Nb3-Nb5 = Å Nb4-Nb5 = Å Nb1-C1 = Å Nb1-C4 = Å Nb2-C1 = Å Nb2-C2 = Å Nb2-C3 = Å Nb3-C2 = Å Nb3-C3 = Å Nb3-C5 = Å Nb4-C1 = Å Nb4-C2 = Å Nb4-C4 = Å Nb4-C5 = Å Nb5-C3 = Å Nb5-C4 = Å Nb5-C5 = Å C4-C5 = Å Nb1 = Nb2 = Nb3 = Nb4 = Nb4 = C1 = 0.85 C2 = 0.81 C3 = 0.88 C4 = 0.65 C5 = 0.64 Nb 5 C 5 XVIII D + 1 A C 1 Nb1-Nb2 = Å Nb1-Nb4 = Å Nb1-Nb5 = Å Nb2-Nb3 = Å Nb2-Nb4 = Å Nb2-Nb5 = Å Nb3-Nb4 = Å Nb3-Nb5 = Å Nb4-Nb5 = Å Nb1-C1 = Å Nb1-C4 = Å Nb2-C1 = Å Nb2-C2 = Å Nb2-C3 = Å Nb3-C2 = Å Nb3-C3 = Å Nb3-C5 = Å Nb4-C1 = Å Nb4-C2 = Å Nb4-C4 = Å Nb4-C5 = Å Nb5-C3 = Å Nb5-C4 = Å Nb5-C5 = Å Nb1 = Nb2 = Nb3 = Nb4 = Nb4 = C1 = 0.80 C2 = 0.78 C3 = 0.80 C4 = 0.61 C5 = 0.59 C4-C5 = Å 187

35 Nb 5 C 6 XIX A 2 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb2-Nb2 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb2-C1 = Å Nb2-C2 = Å Nb2-C3 = Å Nb3-C3 = Å C2-C3 = Å Nb1 = Nb2 = Nb3 = C1 = 0.83 C2 = 0.54 C3 = 0.57 Nb 5 C 6 + XIX A + 1 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb2-Nb2 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb2-C1 = Å Nb2-C2 = Å Nb2-C3 = Å Nb3-C3 = Å Nb1 = Nb2 = Nb3 = C1 = 0.75 C2 = 0.50 C3 = 0.54 C2-C3 = Å Nb 5 C 6 + XIX B + 2 B 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb2-C1 = Å Nb2-C2 = Å Nb3-C1 = Å Nb1 = Nb2 = Nb3 = C1 = 0.79 C2 = 0.67 C2-C2 = Å Nb 5 C 6 + XIX B + 1 A 1 C 2v Nb1-Nb1 = Å Nb1-Nb2 = Å Nb1-Nb3 = Å Nb2-Nb3 = Å Nb1-C1 = Å Nb1-C2 = Å Nb2-C1 = Å Nb2-C2 = Å Nb3-C1 = Å Nb1 = Nb2 = Nb3 = C1 = 0.71 C2 = 0.64 C2-C2 = Å 188