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High resolution electronic spectra of some new transition metal-bearing molecules Rixon, Scott John 2004

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HIGH RESOLUTION ELECTRONIC SPECTRA OF SOME NEW TRANSITION METAL-BEARING MOLECULES by SCOTT JOHN RIXON B. Sc., Memorial University of Newfoundland, 1993 M . Sc., University of Victoria, 1996 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (DEPARTMENT OF PHYSICS A N D ASTRONOMY) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A July 2004 © SCOTT JOHN RIXON, 2004 11 Abstract Spectra of zirconium monocarbide (ZrC), zirconium methylidyne (ZrCH) and lanthanum imide (LaNH) are reported for the first time. Al l species have been produced under jet-cooled conditions by reaction of gaseous metal atoms with methane or ammonia and examined via laser-induced fluorescence. ZrC has several electronic states below 2500 c m - 1 : X 3 2 + and three l2Z+ states, which appear to represent all possible arrangements of two electrons in the nearly degenerate 11a and 12a orbitals (from Zr 55a + C 2pa). For X 3 £ + (r 0 = 1.807 A), the small spin-spin constant (Xq = 0.514 cm - 1 ) and large Fermi contact parameter of 9 1 Z r 1 2 C ( I Z r = 5/2) indicate the configuration (1 la) 1(12a) 1. The tightly bound al~L+ state (T 0 = 187.83 cm" 1, r 0 = 1.739 A) has the configuration (11a)2, while anomalous vibrational intervals and 1 2 C / 1 3 C isotope shifts of the other two states reveal their isosymmetry. This orbital scheme also accounts for the complex structure above 16000 c m - 1 ; identical, strongly interacting manifolds result from promoting either a electron to the same orbital. Z rCH has a 2n - X 2 Z + electronic system in the visible region analogous to that seen for isoelectronic ZrN, but they otherwise share little similarity. For X 2 E + , high-resolution rotational analyses give r 0 bond lengths, while dispersed fluorescence has characterised its vibrational structure. Conversely, many details remain unclear for the excited state. Upper state P = A+f+S values and Zr isotope shifts have been obtained for many subbands of ZrCH and ZrCD; 1 2 C / 1 3 C shifts have also been determined for some ZrCH bands. Unexpectedly small values of the spin-orbit constant and Zr -C stretching frequency, and the large Renner-Teller splitting in the 010-000 band are believed to be due to strong vibronic coupling with 2 E and 2 A states. The electronic structure of LaNH is similar to isoelectronic LaO, with the added complication of vibronic coupling between the A 2 n and B 2 E + states. This is apparent from anomalies in the B 2 X + - X 2 E + hyperfine intensity profiles (JL a = 7/2), the Hund's case (a) coupled B 2 E + (0u 2 0) level structure and the large B 2 2 + state bending frequency. Ill Table of Contents Abstract 1 1 Table of Contents iii List of Tables ix List of Figures xiv List of Commonly Used Abbreviations, Acronyms and Symbols xviii Acknowledgements xxiii Chapter 1 Introduction 1 Chapter 2 Theoretical Background 3 2.1 Introduction 3 2.2 The Molecular Hamiltonian and the Born-Oppenheimer Approximation 3 2.2(a) The Born-Oppenheimer Approximation 3 2.2(b) Vibronic Coupling 6 (i) Selection rules for the Renner-Teller effect 6 2.3 Nuclear Part of the Molecular Hamiltonian 9 2.3(a) The Separation of Vibration and Rotation 9 2.3(b) Molecular Vibration 11 (i) Diatomic molecules 11 (ii) Polyatomic molecules 12 2.4 Angular Momentum and Spherical Tensor Operators 13 2.4(a) Introduction 13 2.4(b) Properties of Angular Momentum Operators 13 2.4(c) Addition of Angular Momenta: The Wigner 3-, 6- and 9-j Symbols 15 (i) The Wigner 3-j symbol 15 Table of Contents iv (ii) The Wigner 6-j symbol 15 (iii) The Wigner 9-j symbol 16 2.4(d) Coordinate Rotations: The Wigner Rotation Matrix 17 2.4(e) Spherical Tensor Operators 18 (i) Definition of a spherical tensor 18 (ii) Compound spherical tensor operators 19 (iii) The Wigner-Eckart Theorem 19 2.5 The Effective Molecular Hamiltonian 23 2.5(a) The Rotational Hamiltonian .23 2.5(b) The Fine Structure Hamiltonian . .25 (i) The spin-orbit Hamiltonian 25 (ii) The spin-rotation Hamiltonian 27 (iii) The spin-spin Hamiltonian 28 2.5(c) The Hyperfine Hamiltonian 29 (i) Magnetic hyperfine structure 30 (ii) Electric quadrupole hyperfine structure 33 2.6 Hund's Coupling Cases and Hamiltonian Matrix Elements 34 2.6(a) Hund's Coupling Cases 34 (i) Hund's case (a) 35 (ii) Hund's case (b) 36 (iii) Hund's case (c) 37 (iv) Modified Hund's cases: nuclear spin effects 37 2.6(b) Matrix Elements of the Hamiltonian 40 (i) Hamiltonian matrix elements evaluated in a Hund's case (ap) basis 40 (ii) Hamiltonian matrix elements of a 2 2 state evaluated in a Table of Contents v Hund's case (bp§) basis 47 2.7 Symmetry, Parity and A-type Doubling 49 2.7(a) Symmetry Properties of Linear Molecules and e//*Parity Labels 49 2.7(b) A-type Doubling in Degenerate Electronic States 51 2.7(c) Effect of A-type Doubling on Hyperfine Structure 53 2.8 Selection Rules and Intensities 53 2.9 Fitting of Data and the Hellmann-Feynman Theorem 58 2.9(a) Method of Combination Differences 58 2.9(b) Non-linear Iterative Least Squares Fitting via the Hellmann-Feynman Theorem 58 Chapter 3 The Electronic Spectrum of Zirconium Monocarbide, ZrC, in the 16000 - 19000 cm- 1 Region: Analysis of Singlet and Triplet Structure 61 3.1 Introduction 61 3.2 Experimental Apparatus and Techniques 62 3.2(a) Overview 62 3.2(b) Preparation of ZrC in a Free Jet Expansion 62 (i) The vacuum chamber 62 (ii) The ablation laser 64 (iii) The gas handling system 64 (iv) The "Smalley" expansion source 65 3.2(c) The Two Probe Laser Systems 67 (i) The pulsed dye laser 67 (ii) The cw ring dye laser 68 3.2(d) Signal Detection and Data Acquisition 70 3.2(e) Description of the Experiments 71 (i) Wavelength-selected fluorescence (WSF) 71 Table of Contents vi (ii) Dispersed fluorescence (DF) 74 3.2(f) Calibration of WSF Spectra 75 3.3 The Visible Spectrum of Zirconium Monocarbide, ZrC 77 3.3(a) Motivation 77 3.3(b) Review of Relevant Literature 79 3.3(c) Description of the Spectrum 82 3.3(d) The Low-lying Electronic States of ZrC: X 3 E + , a 1 !* b 1 ^ and c 1 ^ 92 (i) Vibrational analysis of the low-lying electronic states of ZrC 92 (ii) Rotational analysis of the X 3 E + and a1!;4" states of ZrC 101 (iii) Internal hyperfine perturbations in the X 3 2 + , v = 0 state of 9 1 Z r C 107 (iv) Discussion: A molecular orbital model of ZrC 122 3.3(e) Electronic States of ZrC in the 16000-19000 cm" 1 Region 130 (i) T h e P 6 . 2 ] 3 I L - X 3 £ + ( 0 , 0 ) band 130 (ii) Higher-lying levels of ZrC 140 3.4 Conclusions 149 Chapter 4 Laser Spectroscopy of Zirconium Methylidyne: The ZrCH, ZrCD and Z r 1 3 C H Isotopomers 151 4.1 Background 151 4.2 Experiment 154 4.3 The Visible Spectrum of Zirconium Methylidyne 156 4.3(a) Description of the Spectrum 156 4.3(b) The X 2 E+ State 159 (i) Vibrational analysis 159 (ii) Rotational analysis 172 4.3(c) Levels of the A 2 n State in the 15100-15800 cm" 1 Region 179 Table of Contents vii (i) Assignment of the A 2 n 1 / 2 and A 2 n 3 / 2 , v = 0 levels 179 (ii) Remaining levels up to 15800 c m - 1 : Vibrational structure of the A 2 n state 186 4.3(d) Levels above 15800 cm" 1 194 (i) The 15800-16250 cn r 1 region 194 (ii) Evidence for the B 2 Z + state: the 16547 cm" 1 band 196 4.4 Conclusions 200 Chapter 5 Laser Spectroscopy of Lanthanum Imide (LaNH and LaND) 202 5.1 Background 202 5.1 (a) Introduction 202 5.1(b) Electronic Structures of ScO, ScNH, YO, Y N H , LaO, LaS and LaNH 202 5.2 Experiment 204 5.3 Description of the WSF Spectrum 205 5.4 The X 2 E + State of LaNH and LaND 210 5.4(a) Vibrational Analysis 210 5.4(b) Rotational, Fine and Hyperfine Structure of v = 0 Level 213 5.4(c) The o 2 = 1 Level of the Ground State of L a N H 223 5.5 The Excited Electronic States, A 2 n and B 2 E+ 226 5.5(a) The A2Ur State 226 5.5(b) The B 2 Z + State 229 (i) Pure bending vibrational structure of the B 2 S + state 231 (ii) Anomalous intensity patterns in B 2 2 + - X 2 2 + system bands 234 5.6 Conclusions 241 Bibliography 242 Appendix I Rotational assignments and line measurements of triplet bands of ZrC 257 Appendix II Rotational assignments and line measurements of singlet bands of ZrC 317 Table of Contents viii Appendix III Hyperfine assignments and line measurements of triplet bands of 9 1 Z r C . . . . 322 Appendix IV Rotational assignments and line measurements of bands of Z rCH 325 Appendix V Rotational assignments and line measurements of bands of ZrCD 338 Appendix VI Hyperfine and rotational assignments and line measurements of B 2 S+ - X 2 E + system bands of LaNH and LaND 351 ix List of Tables 3.1 Comparison of ground state symmetries of some transition metal-containing monocarbides and their isoelectronic mononitrides 82 3.2 Low-lying vibrational levels of ZrC identified by dispersed fluorescence 95 3.3 Rotational and fine structure Hamiltonian matrix elements for a 3 E + state 102 3.4 ZrC, X 3 E + : Rotational, fine structure and vibrational constants 103 3.5 ZrC, a1^4": Rotational constants of v = 0 level 106 3.6 9 0 Z r C , X 3 2 + , u = 1: Rotational and fine structure constants 106 3.7 Nonvanishing hyperfine Hamiltonian matrix elements for a 3 Z + state 114 3.8 9 1 Z r C , X 3 Z + , v = 0: Rotational, fine and hyperfine constants 117 3.9 Manifolds of electronic states arising from various two-electron configurations . . . 125 3.10 ZrC, [16.2]3ni, u = 0: Effective rotational and fine structure constants 137 3.11 Constants from the attempted deperturbation of the nearly degenerate Q. = Oe and 1 levels of 9 0 Z r C mear 17833 cm" 1 147 4.1 Measurements of assigned excited vibrational levels in the X 2 Z + ground states of ZrCH, ZrCD and Z r 1 3 C H 170 4.2 Molecular constants and structure of the X 2 S + , v = 0 levels of various isotopomers of zirconium methylidyne 175 4.3 Comparison of spin-rotation constants for X 2 2 + states of various isovalent species 177 4.4 Some fine structure constants of several molecules isovalent to Z rCH 182 4.5 Calculated and observed isotope shifts of co3 for various states of ZrCH(D) 192 4.6 Rotational linestrengths for Q' - Q" = 1/2 - 1/2 transitions 199 5.1 Characteristic molecular constants of some isovalent Group 3 molecules 203 5.2 Measurements of assigned excited vibrational levels in the X 2 S + ground states of LaNH and LaND 212 5.3 Hamiltonian matrix elements for a 2£ +(bps> I = 7/2) electronic state 215 List of Tables x 5.4 Molecular constants and structure of the X 2 £ + , v = 0 levels of L a N H and LaND . 2 2 1 5.5 Bond lengths of the Group 3 monoxides, monosulphides and imides 223 5.6 Molecular constants of the X 2 E + , v2 = 1 level of L a N H 226 5.7 Molecular constants of the B 2 S + states of LaNH and LaND 231 5.8 Molecular constants of the o 2 = 1 and 2 levels of the B 2 S + state of L a N H 234 A I . l Rotational assignments and line measurements of the 16028/61 c m - 1 band of ZrC 258 AI.2 Rotational assignments and line measurements of the 16178 c m - 1 band of ZrC . . . 259 AI.3 Rotational assignments and line measurements of the 16306 c m - 1 band of ZrC . . . 261 AI.4 Rotational assignments and line measurements of the 16488/502 c m - 1 band of ZrC : 263 AI.5 Rotational assignments and line measurements of the hot 16626 c m - 1 band of ZrC 265 AI.6 Rotational assignments and line measurements of the 16643 c m - 1 band of ZrC . . . 266 AI.7 Rotational assignments and line measurements of the 16655/80 c m - 1 band of ZrC 267 AI.8 Rotational assignments and line measurements of the hot 16655/58 c m - 1 band of ZrC 269 AI.9 Rotational assignments and line measurements of the 16681 c m - 1 band of ZrC . . . 269 ALIO Rotational assignments and line measurements of the 16694 c m - 1 band of ZrC . . . 271 A L U Rotational assignments and line measurements of the 16702 c m - 1 band of ZrC . . . 273 AI.12 Rotational assignments and line measurements of the 16910 c m - 1 band of ZrC . . . 275 AI.13 Rotational assignments and line measurements of the 16911 c m - 1 band of ZrC . . . 277 AI.14 Rotational assignments and line measurements of the 16935/41 c m - 1 band of ZrC 278 A I . l 5 Rotational assignments and line measurements of the 17089 c m - 1 band of ZrC . . . 280 A l . 16 Rotational assignments and line measurements of the 17112 c m - 1 band of ZrC . . . 281 A I . l 7 Rotational assignments and line measurements of the 17342 c m - 1 band of ZrC . . . 283 A l . 18 Rotational assignments and line measurements of the hot 17458 c m - 1 band of List of Tables xi ZrC 285 A l . 19 Rotational assignments and line measurements of the 17494/506 c m - 1 band of ZrC 286 AI.20 Rotational assignments and line measurements of the 17535/38 c m - 1 band of ZrC 288 AI.21 Rotational assignments and line measurements of the 17673 c m - 1 band of ZrC . . . 290 AI.22 Rotational assignments and line measurements of the 17689 c m - 1 band of ZrC . . . 292 AI.23 Rotational assignments and line measurements of the 17690/700 c m - 1 band of ZrC 294 AI.24 Rotational assignments and line measurements of the 17815 c m - 1 band of ZrC . . . 296 AI.25 Rotational assignments and line measurements of the 17832 c m - 1 band of ZrC . . . 298 AI.26 Rotational assignments and line measurements of the 17833 c m - 1 band of ZrC . . . 300 AI.27 Rotational assignments and line measurements of the 17908 c m - 1 band of ZrC . . . 300 AI.28 Rotational assignments and line measurements of the 17912 c m - 1 band of ZrC . . . 302 AI.29 Rotational assignments and line measurements of the 18093 c m - 1 band of ZrC . . . 304 AI.30 Rotational assignments and line measurements of the hot 18101 c m - 1 band of ZrC , 304 AI.31 Rotational assignments and line measurements of the 18107 c m - 1 band of ZrC . . . 305 AI.32 Rotational assignments and line measurements of the 18159/82 c m - 1 band of ZrC 307 AI.33 Rotational assignments and line measurements of the 18169 c m - 1 band of ZrC . . . 308 AI.34 Rotational assignments and line measurements of the 18338 c m - 1 band of ZrC . . . 309 AI.35 Rotational assignments and line measurements of the 18467/79 c m - 1 band of ZrC 311 AI.36 Rotational assignments and line measurements of the 18569 c m - 1 band of ZrC . . . 313 AI.37 Rotational assignments and line measurements of the 18616 c m - 1 band of ZrC . . . 314 AI.38 Rotational assignments and line measurements of the 18981 c m - 1 band ofZrC . . . 315 AII . l Rotational assignments and line measurements of the 17485 c m - 1 band of ZrC . . . 318 AII.2 Rotational assignments and line measurements of the 17501 c m - 1 band of ZrC . . . 318 A l l . 3 Rotational assignments and line measurements of the 17502 and 17584 c m - 1 bands of List of Tables xii ZrC 319 AII.4 Rotational assignments and line measurements of the 17720 c m - 1 band of ZrC . . . 320 AII.5 Rotational assignments and line measurements of the 17724 c m - 1 band of ZrC . . . 320 AIL 6 Rotational assignments and line measurements of the 17975 c m - 1 band of ZrC . . . 321 AIII. 1 Hyperfine and rotational assignments and line measurements of the 16488/502 c m - 1 band of 9 ' Z r C 323 AIII.2 Hyperfine and rotational assignments and line measurements of the 17089 c m - 1 band of 9 ' Z r C 323 AIII. 3 Hyperfine and rotational assignments and line measurements of the 17342 c m - 1 band of 9 ' Z r C 324 A I V . l Rotational assignments and line measurements of the 15179 c m - 1 band of Z r C H . .326 AIV.2 Rotational assignments and line measurements of the 15428 c m - 1 band of ZrCH . .327 AIV.3 Rotational assignments and line measurements of the 15621 c m - 1 band of ZrCH . .328 AIV.4 Rotational assignments and line measurements of the 15680 c m - ' band of ZrCH . .329 AIV.5 Rotational assignments and line measurements of the 15705 c m - ' band of Z r C H . .330 AIV.6 Rotational assignments and line measurements of the 15988 c m - ' band of Z r C H . .331 AIV.7 Rotational assignments and line measurements of the 16112 c m - ' band of Z r C H . .333 AIV.8 Rotational assignments and line measurements of the 16529 c m - ' band of Z r C H . .334 AIV.9 Rotational assignments and line measurements of the 16548 c m - 1 band of Z r C H . .335 AIV.10 Rotational assignments and line measurements of the 16661 c m - ' band of Z r C H . .336 A V . l Rotational assignments and line measurements of the 15189 c m - ' band of ZrCD . .339 AV.2 Rotational assignments and line measurements of the 15441 c m - 1 band of ZrCD . .340 AV.3 Rotational assignments and line measurements of the 15576 c m - ' band of ZrCD . .341 AV.4 Rotational assignments and line measurements of the 15623 c m - ' band of ZrCD . .342 AV.5 Rotational assignments and line measurements of the 15851 c m - ' band of ZrCD . .343 AV.6 Rotational assignments and line measurements of the 15941 c m - ' band of ZrCD . .344 AV.7 Rotational assignments and line measurements of the 16007 c m - ' band of ZrCD . .345 List of Tables xiii AV.8 Rotational assignments and line measurements of the 16022 c m - 1 band of ZrCD . . 346 AV.9 Rotational assignments and line measurements of the 16057 c m - 1 band of ZrCD . .347 A V . 10 Rotational assignments and line measurements of the 16437 c m - 1 band of ZrCD . . 349 AVI . 1 Hyperfine and rotational assignments and line measurements of the B2S+(000, 2 Z + ) - X2Z+(000,2S+) band of LaNH near 15198 cm" 1 352 AVI.2 Hyperfine and rotational assignments and line measurements of the B 2 £+(001, 2E+) - X 2 S + (000, 2 2 + ) band of LaNH near 15889 c i r r 1 353 AVI . 3 Hyperfine and rotational assignments and line measurements of the B 2Z+(02°0, 2 S + ) - X 2 Z + ( 0 0 0 , 2Z+) band of LaNH near 16412 cm" 1 358 AVI . 4 Rotational assignments and line measurements of the B^+fOliO, 2n3/2) - X 2S+(000, 2E +) band of LaNH near 15803 cm- 1 359 AVI. 5 Rotational assignments and line measurements of the B2??(p\lG, 2n1/2) - X 2 S + (000, 2 £ + ) band of L a N H near 15821 cm" 1 360 AVI . 6 Rotational assignments and line measurements of the B 2 Z+(02 2 0 , 2 A 5 / 2 ) -X 2 I+(000, 2 S + ) band of LaNH near 15961 crrr 1 360 AVI . 7 Rotational assignments and line measurements of the B^+fO^O, 2 A 3 / 2 ) -X 2S+(000, 2 E + ) band of LaNH near 15989 cm" 1 361 AVI . 8 Hyperfine assignments and line measurements of the B^E+YOliO, 2n3/2) - X 2 I + ( 0 0 0 , 2 E + ) band of LaNH near 15803 cm" 1 362 AVI . 9 Hyperfine assignments and line measurements of the B 2S +(0110, 2n 1 / 2) - X 2 I + ( 0 0 0 , 2 2 + ) band of LaNH near 15821 c n r 1 363 AVI . 10 Hyperfine assignments and line measurements of the B2E+(000, 2 Z + ) - X 2S +(000, 2Z+) band of LaND near 15160 c n r 1 . 363 xiv List of Figures 2.1 Mechanism of vibronic coupling between a II electronic state and a higher-lying I electronic state 8 3.1 Schematic diagram of experimental apparatus 63 3.2 Top face and side views of the "Smalley" source 66 3.3 Proposed structures of the metallo-carbohedrene M g C 1 2 78 3.4 Stick spectrum of 9 0 Z r C WSF bands 84 3.5 A portion of the [16.2]3n2 - X 3 I + (0,0) band of ZrC 86 3.6 High-resolution spectrum of the [17.67]Q=1 - alI+ band of ZrC 87 3.7 Effect of 1 2 C -> 1 3 C substitution on the 17908 c m - 1 WSF band of Z r 1 2 C and on the DF of its upper state 89 3.8 DF from the 17908 cm" 1 level of Z r 1 2 C and its Z r 1 3 C counterpart 91 3.9 Dispersed fluorescence (DF) spectrum of the [16.2]3n2, v = 0 level of ZrC 93 3.10 DF spectra from the upper states of the 18512 c m - 1 band of Z r 1 2 C and the corresponding 18467 c m - 1 band of Z r 1 3 C 94 3.11 Vibrational structure of low-lying electronic states of Z r 1 2 C and Z r 1 3 C 96 3.12 Rotational structure of fluorescence from an Q. = 1 state to various electronic states . . 98 3.13 Rotationally resolved DF features from the 18512 c m - 1 band of ZrC 99 3.14 Electron spin structure of the X 3 E+, o = 0 level of 9 0 Z r C 104 3.15 Hyperfine structure of the S,(0) line of the 17342 c m ' 1 band of 9 1 Z r C 108 3.16 A portion of the 17342 c m - 1 band of ZrC 109 3.17 Hyperfine structure in 9 1 Z r C : the Si(5) line of the 17342 c m - 1 band and the P 3(5) line of the 16502 cm ' 1 band 110 3.18 Internal hyperfine perturbations in the Si(2) and Si (4) lines of the 17342 c m - 1 band of 9 1 Z r C I l l 3.19 Electron spin and hyperfine structure of the X 3 S + , v = 0 level of 9 1 Z r C 118 List of Figures xv 3.20 Enlarged portions of Fig. 3.19 119 3.21 Q-form head of the 16488 crrr 1 band of 9 1 Z r C 121 3.22 The P 2(4) line of the 16502 cm" 1 band of 9 1 Z r C 123 3.23 Calculated hyperfine energy level patterns for the F 2 and F 3 electron spin components of the X 3 £+, v = 0 level of 9 1 Z r C from N = 11 - 34 124 3.24 M O diagrams and low-lying electronic states of ZrC and Y N 128 3.25 Vibrational levels of 9 0 Z r C identified in the 16000-19000 cm" 1 region 131 3.26 WSF spectrum of the [16.2]3TL - X 3 S + (0,0) group of ZrC bands recorded at low resolution 132 3.27 Ri branch head of the [16.2]3n1 - X 3 S + (0,0) band of ZrC at high-resolution 134 3.28 DF spectrum of the [16.2]3n,, u = 0 level of ZrC 136 3.29 Reduced rotational energies of the Q = Oe and Q = 2 spin-orbit components of the [16.2]3nr,u = 01evelof 9 0 ZrC 139 3.30 Calculated v = 1 - 5 levels and A E ( 9 0 Z r C - 9 4 Z r C ) isotope shifts of the [16.2] 3IT(a) state of 9 0 Z r C 141 3.31 Reduced rotational energies of the 16681 c m - 1 group of levels of 9 0 Z r C , 9 2 Z r C and 9 4 Z r C 143 3.32 Anomalous D-type splitting of the Q = 1 level of 9 0 Z r C at 16681 cm- 1 144 3.33 Reduced rotational energies of selected 9 0 Z r C levels 146 3.34 Attempted deperturbation of the nearly degenerate Q = Oe and 1 levels of 9 0 Z r C near 17833 cm" 1 148 4.1 Stick spectrum of 9 0 Z r C H and 9 0 Z r C D WSF bands 158 4.2 DF spectra of the A 2 T I 1 / 2 , u = 0 levels of ZrCH at 15179 c n r 1 and ZrCD at 15189 cm- 1 161 4.3 DF spectra of the 15988 c m - 1 level of ZrCH and the corresponding 15980 crrr 1 level of Z r 1 3 C H 163 4.4 DF spectrum of the 16461 cm" 1 level of ZrCH 164 List of Figures xvi 4.5 DF spectra of the 15705 cm- 1 level of ZrCH 166 4.6 DF spectrum from the 16680 cm" 1 level of Z rCH 167 4.7 DF spectra from various levels of ZrCH and ZrCD 169 4.8 Ground state vibrational levels of ZrCH, ZrCD and Z r 1 3 C H 171 4.9 Low-N lines in the 1/2P branch of the 15179 cm" 1 band of Z r C H 173 4.10 Stick spectrum of cold WSF bands of 9 0 Z r C H and 9 0 Z r C D in the 15100-15800 c m - 1 region 180 4.11 A-type doubling in the 15179 cm" 1 level of 9 0 Z r C H 181 4.12 Reduced rotational energies of the 15179 and 15428 cm- 1 levels of 9 0 Z r C H 181 4.13 DF spectra of the 15428 cm- 1 level of Z rCH and the 15441 cm" 1 level of ZrCD . . . 184 4.14 Vibronic energy levels of a linear triatomic molecule in a 2Ilr electronic state for u 2 = 0, 1, 2 and 3 188 4.15 DF spectra from the 15680 cm- 1 level of ZrCH and the 15576 cm" 1 level of ZrCD 190 4.16 DF spectra from the 15621 cm- 1 level of ZrCH and the 15623 era"1 level of ZrCD 193 4.17 ZrCH(D) bands in the 15800-16250 c m ' 1 region 195 4.18 A portion of the high-resolution spectrum of the 16548 c m - 1 band of Z rCH 197 5.1 Portions of the B 2£+(001) - X 2 £ + (000) band of L a N H 207 5.2 Hyperfine structure of the 1/2P4(13) line of the B 2 S + (001)- X 2 I + (000) band of LaNH 209 5.3 DF spectra of the B 2 S + , n = 0 levels of LaNH at 15198 cm" 1 and LaND at 15157 cm" 1 211 5.4 Ground state ( X 2 Z + ) vibrational levels of LaNH and LaND 214 5.5 Hyperfine energy level pattern for N = 18 of the ground state of L a N H 216 5.6 Hyperfine structure of the rj = 0 level of the X 2 L + state of LaNH 220 5.7 The B 2Z+(02 20, 2 A 3 / 2 ) - X2Z+(0110, 2n(b)) hot band of L a N H 224 5.8 Excitation spectra of LaNH and LaND observed below 15000 c m - 1 228 5.9 Excitation spectra of LaNH and LaND observed above 15000 c m - 1 230 List of Figures xvn 5.10 The Av2 = +1 sequence of the B 2 E+ - X 2 E + system of LaNH 232 5.11 Results of hyperfine intensity profile calculations for the 1/2P4(13) line in Fig. 5.2 236 5.12 Hyperfine intensities calculated for 1/2P(1) F - G" features in P = 1/2 - 1/2 transitions of the B - X system of LaNH 237 5.13 Effect of relative phase on calculated rotational profiles for a P = 1/2 - 1/2 transition 239 5.14 High-resolution spectrum of the B2Z+(01 *0, 2n1/2) - X 2 £ + (00°0 , 2 2 + ) band of L a N H 240 XV111 List of Commonly Used Abbreviations, Acronyms and Symbols A spin-orbit (or spin-vibration) parameter <xe vibration-rotation constant of a diatomic molecule vibration-rotation constant of a polyatomic molecule with respect to vibration Vj a nuclear spin-electron orbit hyperfine parameter B rotational constant B magnetic field b nuclear spin-electron coupling hyperfine parameter b F Fermi contact hyperfine parameter B O A Born-Oppenheimer Approximation c hyperfine dipolar coupling parameter c speed of light d hyperfine parity doubling parameter D centrifugal distortion parameter DF dispersed fluorescence <D Wigner rotation matrix element h Kronecker delta function 5(x) Dirac delta function e magnitude of electron charge * M o s t b o l d f a c e q u a n t i t i e s t h a t r e p r e s e n t a n a n g u l a r m o m e n t u m h a v e b o t h s p a c e - f i x e d a n d m o l e c u l e - f i x e d c o m p o n e n t s w r i t t e n i n r e g u l a r t y p e f a c e a n d d e n o t e d b y X , Y a n d Z a n d x , y a n d z s u b s c r i p t s r e s p e c t i v e l y , a n d a c o r r e s p o n d i n g q u a n t u m n u m b e r w r i t t e n i d e n t i c a l l y i n r e g u l a r t y p e f a c e ; t h e s e a r e o m i t t e d f r o m t h i s l i s t f o r b r e v i t y . L i k e w i s e o m i t t e d a r e s y m b o l s d e n o t i n g c e n t r i f u g a l d i s t o r t i o n c o r r e c t i o n s t o v a r i o u s m o l e c u l a r c o n s t a n t s ; t h e s e a r e u s u a l l y d e n o t e d b y t h e s a m e s y m b o l a s f o r t h e c o r r e c t e d c o n s t a n t w i t h a n a d d e d D s u b s c r i p t . F o r s y m b o l s w i t h m o r e t h a n o n e m e a n i n g , t h e m e a n i n g s a r e g i v e n i n d e c r e a s i n g f r e q u e n c y o f u s e i n t h i s t h e s i s . List of Commonly Used Abbreviations, Acronyms and Symbols xix E energy E * space-fixed inversion operator E electric field e e-type parity label eQq 0 hyperfine electric quadrupole coupling parameter 8ijk Levi-Civita symbol F total angular momentum including nuclear spin F Wilson's force field matrix Fi ith electron spin component of a rotational energy level Fi(J) rotational term energy of Fj / /-type parity label Coriolis coupling constant G(n) vibrational term energy A G u+l/2 energy interval between vibrational levels o and VJ+1 G pseudospin angular momentum (sum of I and S) G Wilson's inverse reduced mass matrix Se electron g-factor, 2.0023... §22 anharmonicity constant corresponding to vibrational angular momentum S N nuclear spin g-factor Y spin-rotation operator r generic irreducible representation H Hamiltonian (total energy) operator h Frosch and Foley hyperfine parameter h Planck's constant hfs hyperfine structure List of Commonly Used Abbreviations, Acronyms and Symbols xx I moment of inertia I nuclear spin IHP internal hyperfine perturbation J total angular momentum excluding nuclear spin j generic angular momentum j ± ladder operator j x ± i) y of j k rank of a tensor k Boltzmann constant K rank of a compound tensor t (or I) vibrational angular momentum 1 single electron orbital angular momentum L total electron orbital angular momentum LTF laser-induced fluorescence X effective spin-spin coupling parameter XgQ second order spin-orbit coupling parameter A,§R true (first order) spin-spin coupling parameter A projection of L onto linear molecular axis nig electron mass M L magnetic quantum number corresponding to angular momentum quantum number L M n nuclear mass p reduced mass of diatomic molecule p dipole moment operator P B Bohr magneton P N nuclear magneton N rotational angular momentum including electron orbital angular momentum List of Commonly Used Abbreviations, Acronyms and Symbols xxi Vj ith vibrational mode (polyatomic) o A-type doubling parameter for 2S+ln electronic state, S = 1, 3/2, 2, ... p A-type doubling parameter for 2S+ln electronic state, S = 1/2, 1, 3/2, ... q A-type doubling parameter for 2S+ln electronic state, S = 0, 1/2, 1, ... q e set of electron coordinates Q n set of nuclear coordinates, OR nth vibrational normal coordinate r e (A-B) equilibrium bond length between atoms A and B R end-over-end molecular rotational angular momentum R2PI resonant two-photon ionisation p 2 ratio of reduced masses of diatomic isotopomers S electron spin angular momentum s single electron spin angular momentum ov(xz) operator for reflexion in xz plane £ projection of S onto linear molecular axis T k spherical tensor of rank k T e electronic term energy T e total electron kinetic energy operator T e v electronic plus vibrational term energy T n total nuclear kinetic energy operator T temperature v vibrational quantum number (diatomic molecule) Uj quantum number of ith vibrational mode (polyatomic molecule) v velocity cpn nuclear part of molecular eigenfiinction \|/e electronic part of molecular eigenfiinction List of Commonly Used Abbreviations, Acronyms and Symbols *F total molecular wavefunction WSF wavelength-selected fluorescence Xy vibrational anharmonicity constant (polyatomic molecule) yjjk vibrational anharmonicity constant (polyatomic molecule) co e vibrational frequency (diatomic molecule) <Dj frequency of ith vibrational mode (polyatomic molecule) co e x e vibrational anharmonicity constant (diatomic molecule) co e y e vibrational anharmonicity constant (diatomic molecule) Q, projection of J onto linear molecular axis XXlll Acknowledgements In a letter to Robert Hooke, Isaac Newton once commented that he was able to see farther by standing on the shoulders of giants. It remains to be seen how the results of this dissertation will be viewed in four hundred years; nevertheless, I, too, must thank a number of giants who have allowed me to see farther. First among them is my research supervisor, Prof. Anthony Merer. I consider it a privilege to have pursued molecular spectroscopic research in the laboratory of one of its foremost practitioners, and have learned a great deal from him. For this, and for continued financial support for the duration of my studies, I thank him. This thesis marks the end of an era; as the final graduate student to earn a Ph. D. under his guidance, I would like to take this opportunity to wish him many happy years of retirement. I am also grateful to Prof. Irving Ozier. I thank him for undertaking many administrative responsibilities, which allowed me, as a physics student, the opportunity to pursue my research interests in the Department of Chemistry. I would also like to acknowledge him for our countless lunchtime conversations on many wide and varied topics, ranging from the Lions' Grey Cup hopes to matters more scientific. Last, but not least, I would also like to acknowledge his generous donation of a sample of 1 3 CH4, which generated much useful data in the studies of ZrC and ZrCH reported herein. I would also like to thank Prof. Michael Gerry and all the members of the microwave team, past and present, whose time at U B C overlapped with mine. There are too many to name individually, but over the years, they have been a great source of advice, constructive criticism and muscle power in helping move those plasma tubes between floors, and have my gratitude. Several post-doctoral fellows and research associates have been among my contemporaries in the Merer lab and also deserve my thanks (in chronological order): Dr. David A. Gillett, Dr. James R. D. Peers, Dr. K. ("Peggy") Athanassenas, Dr. Shuenn-Jiun ("S-T') Tang, Dr. Gretchen Rothschopf and Dr. Pradyot K. Chowdhury. I would also like to acknowledge Chris T. Kingston, who, for the duration of my time here, was my only fellow Acknowledgements xxiv Ph.D. student in the lab and a source of many discussions. Summer undergraduate students were always a welcome addition to the Merer lab; I would like to thank Ms. Francesca Setiadi, Ms. Vivian Yip and Ms. Connie Aw, all of whom offered valuable assistance in sundry tasks. I wish them every success in their future endeavours. Several support staff members at U B C also deserve my thanks. Mr. Chris P. Chan left the Merer lab much too soon, only months after I had arrived; nevertheless, a great deal of the electronics in the lab, used routinely in my experiments, were of his construction. I would also like to thank, from the electronics shop, Martin Carlisle, Dave Bains and Jason Gozjolko, all of whom coped with various electronic and computer maladies. Al l of the members of the mechanical shop were helpful in nursing various malfunctioning apparatus back to health and occasionally lathing away those nuisance oxide coatings from my zirconium and lanthanum rods. Various office staff in both the Chemistry and Physics & Astronomy Departments, in particular, Tony Walters and his predecessor Janet Johnson, were also supportive in smoothing out many of the wrinkles associated with my departmental duality. The Merer lab was home to several visiting scientists in recent years, from all of whom I have benefited greatly. Dr. Leah C. O'Brien of Southern Illinois University at Edwardsville afforded me my first opportunity at international travel by inviting me to Kitt Peak National Observatory in Arizona to assist in some of her experiments. Dr. Dennis J. Clouthier of the University of Kentucky was a valuable mentor in the lab and was instrumental in setting up the pulsed dye amplifier system. Dr. Allan G. Adam of the University of New Brunswick (Fredericton) also deserves acknowledgement; the installation of a transmission grating to remove A S E from the pulsed dye laser system was done at his suggestion. Without it, the projects reported in this thesis would have been nearly impossible to pursue. Finally, Dr. Thomas D. Varberg of Macalester College (St. Paul, MN) was of considerable help in the early stages of the LaNH project; his vision at near infrared wavelengths is far superior to my own! Other research colleagues from outside U B C have also been helpful. I would like to Acknowledgements xxv acknowledge Dr. Timothy C. Steimle of the University of Arizona for his interest in the LaNH project, which led his research group to undertake Stark spectroscopic experiments on this molecule. I also thank him for pointing out the utility of pyrromethene 556 laser dye; the performance improvement over other dyes in the same region was quite immense and doubtless improved the data quality accordingly. I would also like to thank Dr. Michael D. Morse of the University of Utah for his cordial e-mail correspondences and for forwarding portions of the Ph. D. thesis of his former student, Ryan S. DaBell. Finally, I would like to thank the many friends I have made in Vancouver over the years who helped make the time outside the lab just as enjoyable as the time inside. With memories of Christmas skits, Blenheim parties, Lions games at B C Place, concerts, and soccer and curling matches (before the leg injuries exacted their toll!), my U B C years will always be remembered with fondness. You can't spell loveliness without lines. 1 Chapter 1 Introduction While the ability of a molecule to emit and absorb discrete frequencies of electromagnetic radiation in a specific pattern, known as a spectrum, was recognised over a century ago, the classical Newtonian mechanics of that time could not account for such spectra. Only with the advent of quantum mechanics in the 1920s did satisfactory accounts begin to emerge; the quantum mechanical theory interprets the discrete nature of molecular spectra as radiative transitions between discrete energy levels. Substantial insights into the structural and bonding properties of the carrier molecules of these spectra were quickly achieved, thus pioneering the field of molecular spectroscopy that still flourishes almost a century later. Molecules, composed of atoms with electrons and nuclei, have several internal motions through which they store discrete amounts of energy; some examples can be outlined. The orbiting motion of the electrons around the nuclei gives rise to an electronic state; infinitely many bound electronic states of various energies can be realised depending on the details of this electron motion. The nuclei may also vibrate relative to one another; for each electronic state, a series of discrete vibrational energy levels results. One may further consider rotation of the entire molecular frame; consequently, each vibrational energy level subdivides into a set of rotational energy levels. Chapter 2 develops these ideas in more complete detail. A molecular spectrum can be defined, for purposes of this thesis, as a collection of radiative transitions between its various energy levels and classified by the type of transitions comprising it. Each type often appears in a specific region of the electromagnetic spectrum. Electronic spectra arise from transitions between electronic states and often, but not always, occur in the visible region; they are also called optical spectra. An infrared spectrum can arise from transitions between vibrational levels of the same electronic state. Transitions can also Chapter 1 Introduction 2 occur between rotational levels of the same vibrational level of the same electronic state; these are often called microwave spectra because they were first studied in detail in the microwave region. In this thesis, only electronic spectra have been studied; three molecules have been examined for the first time. All of them have the general form MR, where M and R respectively represent a transition metal (TM) atom and a monatomic or diatomic non-metal ligand. Such molecules enjoy a wide range of application in several fields of research. For example, TMs with several open shell d electrons are prominent constituents of catalytic cycles, in which intermediate M R species often form. Spectroscopic analyses of these can identify their properties and help advance understanding of these catalytic processes. MR-type molecules are also known to exist in the atmospheres of cooler (-2000-3000 K) stars. While this knowledge has advanced the understanding of stellar evolution, the identification of such stellar atmospheric constituents has only been possible by comparing stellar spectra with carefully recorded and analysed laboratory spectra. This thesis is organised as follows. Chapter 2 discusses the theoretical background required in analysing the data presented in the three following chapters. Chapter 3 describes the experimental apparatus and the procedures used to acquire spectral data. It also presents results from a study of the electronic spectrum of zirconium monocarbide (ZrC); singlet and triplet electronic states were identified in this work. This molecule was serendipitously discovered during studies of zirconium methylidyne (ZrCH). The latter species is the latest of several transition metal methylidynes first studied by the Merer lab; it is discussed in Chapter 4. Finally, Chapter 5 presents the analysis of the visible and near-infrared systems of another triatomic molecule, lanthanum imide (LaNH). Each polyatomic species shows, in its spectrum, evidence for strong interactions between its observed excited states. Chapter 2 Theoretical Background 3 2.1 Introduction This chapter reviews the theory used to analyse the data presented in Chapters 3-5. Section 2.2 presents the Schrbdinger wave equation of the molecule and the Born-Oppenheimer Approximation that separates its electronic and nuclear motion; vibronic coupling is also examined. Separation of nuclear motion into vibration and rotation is discussed in Section 2.3. Angular momentum and spherical tensor operators, in terms of which the Hamiltonian operator for the total energy of the molecule can be written, are described in Section 2.4; Section 2.5 identifies various terms of the effective Hamiltonian. Hund's coupling cases and their use in evaluating Hamiltonian matrix elements are discussed in Section 2.6. A-type doubling and its effect on hyperfine structure are reviewed in Section 2.7, and Section 2.8 evaluates the matrix elements of the dipole moment operator; these lead to transition intensity formulae and selection rules. Finally, use of the Hellmann-Feynman Theorem in the fitting of the data in Chapters 3-5 is discussed in Section 2.9. 2.2 The Molecular Hamiltonian and the Born-Oppenheimer Approximation 2.2(a) The Born-Oppenheimer Approximation Experimentally, one finds that a molecule cannot have an arbitrary amount of energy; rather, it possesses a set of quantised energy levels. The energies of these levels are determined by solving the Schrbdinger wave equation # - | ¥ > = E | ¥ > , [2.1] where Jfis the Hamiltonian (total energy) operator, E is one of the aforementioned energy levels or eigenvalues, and is an eigenfiinction corresponding to E, written in Dirac notation. Chapter 2 Theoretical Background 4 To develop Eq. [2.1] further, the following notation is used. In general, quantities pertaining to electrons are written in lower case with an e subscript, while those for nuclei appear in upper case with an n subscript. Thus, m e , q e and p e will refer to the mass, Cartesian co-ordinates and conjugate linear momentum of an electron, while the corresponding nuclear properties will be M n , Q n and P n . The symbol V ( q e , Q n ) will denote the total electrostatic energy of the molecule, i.e., the sum of all pairwise Coulombic interactions among the entire collection of electrons and nuclei, including all electron-electron repulsions, all nuclear-nuclear repulsions and all electron-nuclear attractions. With these definitions: 2 7 ^ I P e + X ^ + V ( q e , Q n ) 2 m e r n 2 M n | ¥ > = E | ¥ > . [2.2] Eq. [2.2] neglects contributions due to electron spin and nuclear spin; these are discussed in Section 2.5. The first summation on the left, taken over all electrons, represents the total electron kinetic energy operator T e of the molecule. Likewise, the second summation represents the total nuclear kinetic energy operator T n . Since Eq. [2.2] cannot be solved analytically, approximations must be introduced. Because electrons are much lighter than nuclei, they move much more rapidly within the molecule. Born and Oppenheimer (1) argued that the electrons thus adapt themselves instantaneously to a given nuclear configuration, and that Eq. [2.2] can be solved for a fixed configuration Q n . This is the essence of the Born-Oppenheimer Approximation (henceforth abbreviated as BOA), which allows the electron and nuclear motions to be treated independently. The total wavefunction |SP> of the molecule can then be written as a summation of functions, each a product of an electronic part, IVeOle'Qn)) > a n c * a nuclear part, l < P n ( Q n ) > : l ^ ) = Z | v i / 1 e ( q e , Q n ) > | c p 1 n ( Q n ) > - [2.3] i This summation is taken over all eigenfunctions IVefae'Qn)) o r * m e electronic Hamiltonian, Chapter 2 Theoretical Background % = ^ - Z P e + V ( q e , Q n ) , 2m f [2.4] *e e whose eigenvalues are E e ( Q n ) . |y e (q e ,Qn)> c a n b e called a fixed-nuclei electronic eigenfunction because the total molecular Hamiltonian Hreduces to J{Q if the nuclei are fixed at some configuration Q n . In the remainder of this section, the coordinates q e and Q n are suppressed for brevity. Substituting Eqs. [2.3] and [2.4] into the Schrodinger equation, 6 n 2 M n ZlVe>l<Pn> =EX|v(/ e>|cPn>-[2.5] Since P„ acts on both |\|/e) and |cp^>, the product rule is required: Pn IVe>l<Pn> = (Pnlvi»lq>n> + 2 (P n |v|/e>)(Pn |(pn>) + l^X^n l<Pn»• Using this expression and the fact that #g|v|/e) = Eg | \ | / e ) , Eq. [2.5] becomes: EEi| V i>|q>i,> + Z ^ r [ ( P n | v i ' e » + 2(PnlM/e»Pn+|v|/e>Pn]l<Pn> i i,n ^ M n = E£ivL>i<pii>. i Premultiplying by (\|/g | and using its orthonormality properties, this becomes [2.6] [2.7] Ee1cp n >+I i,n <V|/e|PnlVe>+iVe|PnlVe>Pn 2 M r i 2 , „ k A + I ^ ^ = E | < p n > . [2-8] n 2 M n If the second term on the left is neglected, ^ n l c p „ > = Z % ^ + E e k | cp n >=E| (pk > ; n 2 M n [2.9] where Hn can be regarded as the nuclear Hamiltonian operator, and a wave equation for the nuclei moving in a potential Eg results. Thus, if the fixed-nuclei electronic eigenfunctions |vj/g > are known, the total eigenfunction \^) can be expanded as in Eq. [2.3] with |cpk> given by Eq. [2.9], provided the first summation on the left of Eq. [2.8] is negligible. Of the two terms in this summation, the first is essentially the matrix element of the nuclear kinetic energy operator T n taken between a pair of distinct electronic eigenstates. This depends on nuclear mass and leads to a small isotope-dependent correction to the potential curve Eg (mainly about 5 c m - 1 or less) that is largely negligible. The second is usually more Chapter 2 Theoretical Background 6 significant, particularly in the case of polyatomic molecules, and is discussed below. 2.2(b) Vibronic Coupling The second summand in the neglected term of Eq. [2.8] contains the matrix element |Pn|v|/g>, which introduces an interaction between the electronic eigenstates |v|/g) and |\|/e> via the nuclear momentum operator P n = -in d/dQn. This term, the neglect of which essentially amounts to the BOA, represents a coupling between the vibrational and electronic motions of these electronic states and is therefore called vibronic coupling. While this approximation is valid for well-separated electronic states, non-negligible effects can arise under certain conditions for close-lying electronic states. In such cases, the total molecular wavefunction IT) can no longer be written as in Eq. [2.3]. In linear molecules, the only type studied in this thesis, vibronic coupling is referred to as the Renner-Teller effect (2). Conditions for this effect are now described. 2.2(b)(i) Selection rules for the Renner-Teller effect This derivation of the selection rules for the Renner-Teller effect is restricted to linear triatomic molecules. Vibronic coupling of and ) requires that the matrix element M v c = |Pn|\)/e) be non-vanishing. Discrete group theory gives the required condition: r ( M v c ) = r ( | V J » ® r (Q n ) ® r a y i » => r T S . [2.10] T(a) is the irreducible representation (or symmetry species) of a in the point group of the molecule, and T T § refers to the totally symmetric representation of this point group. The symmetry of M v c is given by the direct product of the symmetries of the two electronic states and that of the operator coupling them; T(Q n ) = T(P n) since Q n and P n form a coordinate and linear momentum conjugate pair. An equivalent statement of Eq. [2.10] is that the direct product of r(|\j/e )) and r(|\|/e>) must contain T(Q n ) (where Q n is taken as a vibrational normal coordinate): r ( | v ^ » ® r ( |v | / i» => r(Q n ) . [2.ii] Chapter 2 Theoretical Background 7 Linear triatomic molecules of the form A B C belong to the C ^ point group. These have three vibrational normal modes: two totally symmetric stretches, denoted by V i (o + ) and v 3 (o + ) , and one degenerate bending vibration, denoted by v2(7t). The Renner-Teller selection rule for the a + vibrations is that the two electronic states must have the same symmetry, but only the case of the bending (n) vibration is relevant to this thesis. Its selection rule can be stated arithmetically since the irreducible representation of the electronic state identifies its electron orbital angular momentum projection quantum number A and v2(7t) carries one unit of vibrational angular momentum: | A k - A i | = 1. [2.12] In other words, electronic states whose A values differ by one unit can vibronically couple through the bending vibration. This vibronic coupling can drastically affect the bending vibrational energy level manifolds of both states. As an example, Fig. 2.1 shows potentials for two electronic states of IT and E + symmetry, with the latter being more energetic. These are plotted as functions of the bending normal coordinate Q 2 (coordinates Qi and Q 3 are fixed). Renner-Teller coupling between them steepens the S + potential, thus increasing its bending frequency from the unperturbed value (the A' designation refers to the electronic symmetry in the Cs point group of the bent molecule). The situation is slightly different for the II state, whose orbital degeneracy is now lifted, creating separate A' and A" states in the bent geometry. The A" component is essentially unperturbed by the vibronic coupling and is shown displaced slightly above for clarity. Only the A' component is shifted downward by the higher-lying 2 + (A') state. The weak coupling depicted in the figure reduces the bending vibrational frequency of the n(A') potential. Sufficiently strong coupling can force this potential into a double minimum shape; the resultant non-linear equilibrium geometry effects a partial reordering of the bending vibrational energy levels, as discussed by Jungen et al. (3-6). Chapter 2 Theoretical Background Figure 2.1 Mechanism of vibronic coupling between a II electronic state and a higher-lying S electronic state. Dashed curves denote the unperturbed bending potentials; the coupling connects the A' component (C s geometry) of the orbitally degenerate IT state and the E (A) state and increases the bending frequency of the latter. The A" component of the IT state is not affected by the perturbation and is shown displaced slightly above for clarity. Chapter 2 Theoretical Background 9 Two good examples of molecules with vibronically coupled electronic states are Y O H (7) and LaNH; Chapter 5 of this thesis discusses the latter of these. 2.3 Nuclear Fart of the Molecular Hamiltonian 2.3(a) The Separation of Vibration and Rotation In Section 2.2(a), the electronic and nuclear motions of the molecule were separated using the BOA. Hn specifies approximately the nuclear motion: where the potential E e ( Q n ) depends on the nuclear configuration and is an eigenvalue of 9fe. The next step is to separate the vibrational and rotational motions of the nuclei; this has been done by Wilson et al. (8) and Margenau and Murphy (9) as follows. Since no potential energy is associated with the rotational motion, the separation of vibration and rotation amounts to a separation of the kinetic energy term of Eq. [2.13]. This is more easily done if the momentum representation of this equation is replaced with a coordinate representation, although care must be taken in the choice of coordinate system. In particular, each coordinate should be defined using either space-fixed axes, with respect to which the molecule rotates, or molecule-fixed axes, which rotate in space with the molecule. The quantities required to separate the vibration and rotation are the following: R, the position vector of the centre of mass of the molecule in space-fixed axes, r{, the position vector of the z'th atom in molecule-fixed axes, a;, the equilibrium position of the z'th atom in molecule-fixed axes and dj, the displacement of the z'th atom from equilibrium in molecule-fixed axes, such that dj = rj - aj. The velocity Vj of the z'th atom (of mass mj), measured in the space-fixed frame, is simply the velocity of the centre of mass of the molecule, R , plus the velocity of the atom relative to this * n = Z n n + E e ( Q n ) , [2.13] Chapter 2 Theoretical Background 10 centre of mass, r; (the overdots indicate temporal differentiation): Vi=R + ri . [2.14] The velocity rj, measured in the molecule-fixed frame, is simply dj, but the rotation of this frame, at an angular velocity co (referred to the space-fixed axes), means that r j ^ d i + w x r ; [2.15] in the space-fixed frame. By definition, the molecular kinetic energy is T = | Z m i v i v i , [2.16] i Using Eqs. [2.14] and [2.15]: T = y £ m i ( R + d i +ojxrj)-(R + di + coxrj). [2.17] i The summation can be expanded and then simplified with the Eckart conditions (10) £ 1 ^ 1 4 = 0 [2.18a] i and £ m i ( a i x d i ) = 0, [2.18b] i which stipulate that the origin of the molecule-fixed axes be the molecular centre of mass, and that the molecule has no net angular momentum in its equilibrium configuration, measured in the molecule-fixed axis system. When this is done, the result is 2T = R - R X m j +Xmi(»xn)-(G) x ri) + Z m i d i • dj + 2co• Z m i ( d i x d i ) t 2 1 9 J i i i i The four terms on the right are readily interpreted physically. The first one represents translational energy of the molecular frame, which can be ignored for molecules in free space. The second term corresponds to rotational energy, while the third can be regarded as the vibrational kinetic energy. The final term represents a Coriolis coupling of vibration and rotation that involves the vibrational angular momentum; as a result, the rotational and vibrational motions cannot strictly be completely separated. However, this Coriolis interaction is often negligible, so that rotational and vibrational energies determined from independent Chapter 2 Theoretical Background 11 quantum mechanical analyses can be added to yield the total nuclear kinetic energy to an adequate level of approximation. Townes and Schawlow (11) have discussed the general problem of determining the rotational energy levels of a molecule, which can be readily solved with the use of angular momentum and spherical tensor operators. These are considered in Section 2.4, after which the rotational Hamiltonian is discussed in detail in Section 2.5(a). The rest of this section deals with vibrational structure. 2.3(b) Molecular Vibration The solution of the Schrbdinger equation for the vibrational energy levels and eigenfunctions of a molecule has been discussed by Pauling and Wilson (12) and Dunham (13); the results are summarised by Herzberg (14, 15) and will be recapitulated here for diatomic and polyatomic molecules. 2.3(b)(i) Diatomic molecules The diatomic molecule has only one vibrational quantum number u = 0, 1, 2, ... ; its energy levels G(u) correspond to those of an anharmonic oscillator: G(u) = (Be(u +1) - co ex e(u +1) 2 + coeye(u +1) 3 + • • •. [2.20] In this equation a > e = - . £ [2.21] c \ p is the classical harmonic vibrational frequency measured in c m - 1 units, where c is the speed of light, k is the force constant of the bond and u = JHlE2_ [2.22] mj + m 2 is the reduced mass of the molecule; n^ and m 2 are its nuclear masses. Vibrational constants co'e, (o'qXq and ®'eyle for another isotopomer of the molecule are related to those of the reference isotopomer by (13): Chapter 2 Theoretical Background 12 G>e=P<°e> [2.23a] C0ex'e =p 2 co e x e [2.23b] and co ey e = p 3 © e y e , [2.23c] where p 2 = p /p ' [2.24] is the reduced mass ratio of the two isotopomers. The anharmonicity constants, © e x e and coeye, are generally much smaller than a>e. If these are neglected, then the wavefunctions associated with the eigenvalues G(o) of Eq. [2.20], are = N B exp( - l o x ^ H ^ c V a x ) , [2.25] where the normal coordinate x = ^/p ( r - r e ) is the mass-weighted change in bond length from its equilibrium value, a = 27tca>e/h and N v = /•^2vv\ is a normalisation factor. a H 1 ) (Vax) is the nth degree Hermite polynomial, which has u nodes 2.3(b)(ii) Polyatomic molecules Non-linear (linear) polyatomic molecules with N atoms have 3N - 6 (3N - 5) normal modes of vibration, each with its own vibrational quantum number Oj. The vibrational structure can be approximated as a superposition of slightly anharmonic oscillators, so the vibrational energy G(u iv j 2 - ) can be written as an expression similar to Eq. [2.20] summed over all the vibrational modes of the molecule, with additional anharmonic terms to account for interactions between various vibrational modes: G(n,n 2- • •) = Y>i(Ui +1) + 2>ij(Ui + £)0>j + ^ ) i j * i + Z yijkO>i+i)0>j+i)0)k+i) + - - t 2 - 2 6 ] k>j>i An alternative definition of G ^ O j •••) sets the zero-point energy at zero: G ( o 1 o 2 - ) = Z Q ) i 1 ) i + Z x i j , } i v j + Z yijkUiOjO k+--- t 2 - 2 7 ] i j>i k>j>i In the harmonic approximation, the eigenfunctions ^PYx^x^ •••) corresponding to G ( o i 0 2 - ) Chapter 2 Theoretical Background 13 are written as a product of the single oscillator eigenfunctions from Eq. [2.25]: ^ U l x a ) 2 - ) = n ^ D i ( x i ) ' p - 2 8 ] i although in this case, the normal coordinates Xj no longer relate simply to the corresponding geometrical structural parameters (i.e., bond lengths and bond angles). Moreover, the vibrational frequencies of polyatomic molecules have more complicated isotopic scaling relations than those of diatomic species; they involve the use of Wilson's matrices for the inverse reduced mass G and force field F and require knowledge of not only all vibrational frequencies of the molecule but also all structural parameters. Assumptions about the force constants of the various bonds are sometimes necessary. The treatment essentially determines the force constants of one isotopomer and assumes they are isotopically invariant to estimate vibrational frequencies of other isotopomers. Further discussion appears in Wilson et al. (8). 2.4 Angular Momentum and Spherical Tensor Operators 2.4(a) Introduction This section describes various aspects of angular momentum and spherical tensor operators, the Wigner 3-j, 6-j and 9-j symbols, the Wigner rotation matrix, the Wigner-Eckart Theorem and their applications. Much of this discussion can be found in Edmonds (16) and Zare (17). 2.4(b) Properties of Angular Momentum Operators An angular momentum operator is defined as a quantum mechanical operator J whose Cartesian components, Jx, Jy a n < ^ Jz> °bey J i J j - J j J ^ / h X s i j k J k , [2.29] k where 8p, the Levi-Civita symbol, equals +1 (-1) if ijk is a cyclic (anticyclic) permutation of X Y Z , and vanishes if any index is repeated. The quantity on the left side of Eq. [2.29] is called the commutator of Jj and J; and is usually written as [Jj, J;]; Jj and J; are said to commute if their Chapter 2 Theoretical Background 14 commutator vanishes. While no component of J commutes with any other component, it can be shown that each component Jj commutes with J 2 , i.e., [J 2 , Jj] = 0, i = X , Y o r Z . [2.30] With this result, simultaneous eigenfunctions of J 2 and one of Jx, J Y o r c a n D e f ° u n d- Jz is usually chosen because of its simple differential form in spherical polar coordinates, and the eigenfunctions are written as |JM>, where J and M are the quantum numbers respectively associated with J 2 and J^. Using only Eq. [2.30] and the commutation relations of Eq. [2.29], the eigenvalues of | JM> with respect to both J 2 and can be determined. The derivation can be found in several textbooks (see, for example, Zare (17) and Sakurai (18)); the results are J 2 |JM> =h2J(J+l)|JM> [2.31a] and J Z |JM> =hM|JM>. [2.31b] J is a non-negative integer or half-integer that specifies the total angular momentum; M , the projection quantum number, can assume any one of the 2J + 1 values ranging from - J to J in steps of one. If needed, T x | J M ) and Jy |JM) are easily obtained in terms of the ladder operators: J ± = J X ± / J Y , [2.32] which transform |JM) into a multiple of | J M ±1): L J J M ) = hVj(J + l ) - M ( M + l ) | J M ± l > . [2.33] The following matrix elements are easily obtained from Eqs. [2.31] and [2.33] < T M | J 2 1JM > = J (J +1) 5 J J . 5 M M - , [2.34a] < J ' M ' | J Z | J M > = M 6 J J . 5 M M . [2.34b] and < J' M ' | J ± | J M > = V ± ( J , M ) S J J . 5 M , M ± 1 , [2.34c] where V ±(a,b) =Va(a + l ) - b ( b ± l ) ; [2.34d] in Eq. [2.34d], a is generally an angular momentum quantum number with a projection Chapter 2 Theoretical Background 15 quantum number b. The Kronecker delta symbol, Sy, equals unity if i = j and vanishes otherwise; for convenience, units for which h = 1 have been introduced. The phase convention of Zare (17), for which the matrix elements of Jx are real and positive, has been adopted in Eq. [2.34]. 2.4(c) Addition of Angular Momenta: The Wigner 3-, 6- And 9-j Symbols 2.4(c)(i) The Wigner 3-j symbol The coupling of two angular momenta jj and j 2 to form a sum J can be represented by two different but equivalent types of wavefunctions. The first, called the coupled representation, have the form | J i J 2 J M ) (sometimes abbreviated as |JM)); these are simultaneous eigenfunctions of the four mutually commuting operators jj , j 2 , J = G l + J2) and J z =Jiz+J2Z- The alternative uncoupled representation | j i m i ; j 2 m 2 ) = | j i t t i i ) | j 2 m 2 > are simultaneous eigenfunctions of jj , j i z , j 2 and j 2 Z . These two equivalent representations are related by a unitary transformation |JM>= S l J i m i ; J 2 m 2 > < J i m i ; J 2 m 2 l J M > » P - 3 5 ] m]m 2 where J is one of + j 2 , j i + j 2 - 1, ... , | j i - j 2 | , and M = mi + m 2 . The Clebsch-Gordan coefficients ( j i m i ; j 2 m 2 | J M ) , easily calculated from these restrictions and the use of the ladder operators J± = j ] ± + j 2 ± , appear in the literature, but are more usefully recast in terms of the Wigner 3-j symbol (17): < j 1 m 1 ; j 2 m 2 | JM> = ( - l ) J i - j 2 + M V 2 j T T f J l J 2 [ ) . [2.36] ^m, m 2 -M) This is advantageous because the Wigner 3-j symbols have higher symmetry; the most common ones are widely available in closed algebraic form. Eq. [2.35] can be rewritten: f : : t A |JM>= X ( - 1 ) J 1 _ J 2 + M V 2 J + 1 JI J2 J mj m 2 - M j 1 m 1 ; j 2 m 2 > . [2.37] 2.4(c)(ii) The Wigner 6-j symbol The wavefunctions for the sum J = + j 2 + J3 of three angular momenta can be formed Chapter 2 Theoretical Background 16 by adding the momenta in two stages using 3-j symbols in each step as discussed above, but this cannot be done uniquely. Any two of them can be combined into an intermediate angular momentum (j 1 2, J23 or j i 2 , with obvious definitions) to which the third can be added. The three coupling schemes are equivalent and related by unitary transformations of the type IGl h) J12 J3 J M > = S< Jl 0 2 J3) J23 M I Ox h) Jl2 J3 JM>I Jl 0 2 J3) J23 M > • [ 2- 3 8l J23 The recoupling coefficient <ji (J2 J3)J23 M lOi J2)Jl2 J3 J M ) c a n D e written as a product of four 3-j symbols, but is more commonly expressed in terms of the Wigner 6-j symbol: fj l J2 J12]_ ( - l ) J l + J 2 + J 3 + J <JlC2 J3)J23 M I G l J2)J12 J3 M > - [2.39] U3 J J23J V(2J12 +1X2J23 +1) Like the 3-j symbol, the 6-j symbol has many symmetry properties and has been evaluated algebraically for the most commonly encountered cases. 2.4(c)(iii) The Wigner 9-j symbol The logical extension of the preceding discussion is the recoupling of four or more angular momenta; the number of possible coupling schemes increases quickly with the number of angular momenta added. If there are four angular momenta, the Wigner 9-j symbol is used to relate one coupling scheme to another. The transformation in atoms from Russell-Saunders (LS) coupling to jj-coupling is a typical example. The two schemes are 11 + 12 = L ;s ! + s2 = S ; L + S = J [2.40a] and li + s 1 =j 1 ; li + s2 = j 2 ; J i + J 2 = J> t 2 - 4 0 b ] and the transformation between them is l ( l l S i ) J i ( l 2 S 2 ) j 2 J > - Z l ( l i l 2 ) L ( S l s 2 ) S J > LS :V(2L + l)(2S + l)(2j 1+l)(2j 2+l) h 12' L Jsi s 2 S ; [2.41] J l J2 J . the quantity in braces is a Wigner 9-j symbol, usually evaluated as a sum of products of three 6-j symbols or of six 3-j symbols. Chapter 2 Theoretical Background 17 2.4(d) Coordinate Rotations: The Wigner Rotation Matrix To describe the coupling of an angular momentum defined in a rotating reference frame (such as that of a freely rotating molecule) to another defined in a space-fixed frame, one must be transformed into the reference frame of the other. Great care is required to do this because of the anomalous commutation relations discussed by Van Vleck (19): if well-defined space-fixed components of an angular momentum operator obey Eq. [2.29], then its components referred to the rotating molecular frame do not obey Eq. [2.29]. Rather, they obey JiJj - Jj Jj = - i h £ e i j k J k . [2.42] k This "anomalous sign of i" means that care must be taken in evaluating the matrix elements of such operators. This can be done by defining the operators in a space-fixed frame and transforming them back to the molecule-fixed frame by a coordinate rotation. If J k is an angular momentum Cartesian component, the rotation operator R k(a) = exp(-/a J k) [2.43] describes a rotation about the k axis by an angle a. According to results derived by Euler in the 18th century, there exist three angles a, P and y such that three such rotations can bring the space-fixed axes into coincidence with the molecule-fixed axes as follows: R(aPy) = exp(-/a J z ) exp(-z' P J Y ) exp(-/'y J z ) . [2.44] Al l rotations are taken about the original space-fixed axes, and a, P and y, the Euler angles, are referred to these. The rotation operator R(aPy) transforms the angular momentum eigenfiinction |JM) defined in the space-fixed system into a mixture of |JM'> eigenfunctions defined in the molecule-fixed system: R(aPy)|JM> = £ © ^ M ( a p Y ) | JM'> • t 2 - 4 5 ! M ' The coefficients ( D ^ ' M ^ P Y ) are the matrix elements of R(aPy): ^ M ( a P Y ) = < J M ' | R | J M > ; [2.46] Chapter 2 Theoretical Background 18 these have a number of useful properties. The complex conjugate of an element is © £ K ( a p Y ) = ( - l ) M - K © J_M _ K ( a p y ) [2.47] It can be shown that the angular momentum wavefunction for a symmetric top molecule is |JKM>= E ± l [2.48] V 8tc2 where, in this case, M and K are the angular momentum projection quantum numbers in the molecule- and space-fixed frames respectively; a> refers to the set of Euler angles. Another useful property, of which use is made later, relates the integral of a product of three (D functions to a product of Wigner 3-j symbols: V l ^ M ^ K ' ^ ^ ^ ^ K ^ d C D = ( - I ) ™ 8tc2 k 0 f J' k V - M ' P V - K ' P K > . [2.49] 2.4(e) Spherical Tensor Operators 2.4(e)(i) Definition of a spherical tensor An irreducible spherical tensor of rank k is defined as a set of 2k+l quantities, denoted by Tq where q = - k , -k+1 , ... ,k , which transform under a coordinate rotation according to R ^ R - V ) = X © q \ ( ( D ) T q k . . [2.50] q' This equation states that coordinate rotation by the set of Euler angles o transforms the k k spherical tensor component T q into a linear combination of the 2k + 1 components T q ' , with Wigner rotation matrix elements as expansion coefficients. Operators of rank zero, such as the Hamiltonian operator, are referred to as scalar operators because they are invariant under rotation. Racah's original definition (20) of the spherical tensor operator is based on the commutation relations of its components with those of the angular momentum operator J: [J Z >T q k ] = qT q k [2.51a] and [ J ± , T k ] = Vk(k + l ) - q ( q ± l ) T q < ± 1 . [2.51b] From this definition, it follows that J itself is a first rank spherical tensor operator, whose Chapter 2 Theoretical Background 19 spherical components can be expressed in terms of the Cartesian components: To(J) = J z [2.52a] and Tl 1 (J)= + - ^ J ± - + - ^ ( J X ± / J Y ) . [2.52b] 2.4(e)(ii) Compound spherical tensor operators Since all first rank spherical tensors have the form of Eq. [2.52], the rules governing the addition of angular momenta apply to the construction of compound spherical tensor operators. For example, a kth rank compound tensor operator acting on a coupled system can be formed from two single spherical tensor operators: T^ x ( l ) of rank k i , which acts on the first part of the coupled system, and T^ 2(2) of rank k 2 , which acts on the second part. The spherical components of this compound tensor, denoted by [T k l(l)<8>T k2 (2)]q, can be written in terms of T ^ Q ) and T k2(2) by analogy with Eq. [2.37]: [ T k l ( l ) ® T k 2 ( 2 ) ] k = X ( - l ) k l _ k 2 + q V 2 k + l Qi92 % k 2 k ^ T k 1 ( 1 ) T k 2 ( 2 ) [ 2 5 3 ] ^qi q2 ~v If two tensors of equal rank k couple to form a zero rank tensor (i.e., a scalar), this reduces to [ T k ( l ) ® T k (2) ]° = -^Lx(-l)q T k ( l ) T ^ ( 2 ) , [2.54] V 2k +1 q where the summation on the right represents the conventional scalar (dot) product. 2.4(e)(iii) The Wigner-Eckart Theorem The expression of operators as spherical tensors is advantageous because the Wigner-Eckart Theorem, proved in many angular momentum textbooks, can be used to evaluate their matrix elements in the basis of angular momentum eigenfunctions | r | JM). The theorem states that < n T M ' | T k h J M > =(-1)J'-M( J ' k J \ r i ' J ' | |T k | | r , J> , [2-55] where r| represents the remaining quantum numbers of the system of interest. The Wigner-Eckart Theorem factorises the matrix element of Tq into two terms: the first depends on the Chapter 2 Theoretical Background 20 projection quantum numbers M and therefore on the geometry of the system, while the other, the reduced or "double bar" matrix element, is independent of such details and represents its physical properties. The Wigner-Eckart Theorem is typically applied by first determining independently a simple <r|TM'|Tq |r|JM> element and then solving Eq. [2.55] for the reduced element, after which the same equation is used to evaluate elements for other M values. This section concludes with some important reduced matrix elements often used in molecular spectroscopy. Reduced matrix element of the angular momentum operator The first element of interest is the double-bar element <J'||T1(J)|| J > . The matrix element for T-(J) = J z follows from Eq. [2.34b]: <J'M|T<5(J)|JM> = <J'M|J Z |JM> = M 5 j r . [2.56] The left side can be rewritten using the Wigner-Eckart Theorem: ( " ^ ( L \ A J r V , | |T 1 ( J ) | | J>=M5 J J , [2.57] ^ - M 0 MJ and the required 3-j symbol can be substituted: < J | | T 1 ( J ) | | J > =5JJVJ(J+1)(2J+1) . [2-58] Reduced matrix element of the (D functions Another reduced matrix element of interest is that of (D^ (co), which projects internal (molecule-fixed, corresponding to the q subscript) angular momenta expressed in space-fixed coordinates (corresponding to p) back into the molecule-fixed reference frame. The Wigner-Eckart theorem can be applied to ©pq^*(o) matrix elements evaluated in the basis of symmetric top eigenfunctions: <J 'K 1 M , |© p ( q ) *(co) |JKM>=(-l )J ' -M( T k J W ' | | © « 1 | J K > , [2.59] where the dot subscript on the right means that the value of p need not be specified since the matrix element has been reduced with respect to its space-fixed components. The left side of Chapter 2 Theoretical Background 21 this equation can be recast by substituting Eq. [2.48] for the symmetric top eigenfunctions: ( r K ' M l ^ ^ a O I J K M ) - V ( 2 J + 1 ) f J ' + 1 ) J < i , ( » ) < > ) © < & ) d « > P.«>l 8 7 T The integral is given by Eq. [2.49]. Eliminating < J T C M ' I ^ ^ Q ^ J K M ) between the resulting equation and Eq. [2.59] leads to an expression for the reduced matrix element: < J ' K ' 1 1 0 . ^ * 1 1 J K > = (-1)J'-K' V(2J + 1)(2J'+1) r T I k J [2.61] - K ' q K Reduced matrix elements of compound tensor operators in a coupled basis From the Wigner-Eckart Theorem, the matrix elements of a compound spherical tensor XQ (1 ,2) = [ T K i ( 1 ) ® T K 2 ( 2 ) ] Q evaluated in the coupled basis | j i j 2 J M > are as follows: <Jl ' J2 , J 'M l |X^( l ,2) | j 1 j 2 JM> (-l)J' ' - M ' J' K J <Jl'J2'J , | |XK(l,2)||j 1j 2J>. [2.62] -M* Q M , Evaluation of the reduced matrix element requires conversion from a coupled to an uncoupled basis, several applications of the Wigner-Eckart Theorem and various properties of the Wigner symbols. Details of the derivation are available in several sources (16, 17, 21); the result is: <Jl , J2'J ' | |X K ( l ,2) | | j 1 j 2 J> = <Ji1|Tki (1)|| J iXJ2 ' l |T k 2 (2)|| j 2> Jl Jl k l )2 J2 k 2 T J K x > /(2J+l)(2J ,+l)(2K+l) [2.63] Eq. [2.63] applies to any compound spherical tensor operator X of any rank K, but is rarely required in spectroscopy since such operators appearing in the Hamiltonian are scalar (i.e., K = 0), in which case the 9-j symbol in Eq. [2.63] collapses to a multiple of a 6-j symbol. A frequently encountered special case of Eq. [2.63] is that of a scalar product of two tensor operators of the same rank, T k ( l ) -T k (2) = X ( - i ) q T q (1)T^(2), identified inEq. [2.54]. q Its matrix element evaluated in the coupled basis is Chapter 2 Theoretical Background 22 < T i , j 1 ' j 2 7 ' M 1 T k ( l ) - T k ( 2 ) | T 1 j J 2 J M > = ( - l ) J i + J 2 , + J 5 J J . 5 M M , j ^ J. 2 J l l l k J l J2j x EOYJl 'H T k ( l ) II T,"JiXr,"J2'll T k (2 ) || T, j 2 > , [2.64] where r| represents the remaining quantum numbers of the system. Two other special cases of Eq. [2.63] can be identified. The first of these entails a single operator acting on only one part of a coupled basis, in which either T k l (1) or T k 2 (2) can be regarded as the zeroth rank identity operator. If the latter, then j 2 ' = j 2 k 2 = 0, K = k] and Eq. [2.63] reduces to <Jl ,J2J , | |T k l(l)IIJlJ2J> = (_l)Ji'+J2+J+ki^ ( 2J + i)( 2J'+l) j J ; ' J ' ^ W l l T ^ O H J i ) . [2.65a] U J l klJ Alternatively taking T k l (1) as the identity operator, j , ' =ji k x = 0 and K = k 2 so that <JlJ2 , J , | |T k 2(2) | | j 1 j 2 J> = ( - l ) J i + J 2 + J ' + k 2 ^ ( 2 J + l)(2J'+l) \]] J , J l j<j2 , | |T k 2 (2) | | j 2 >. [2.65b] [ J J2 k 2 j Of particular note here is that the phases in Eqs. [2.65a] and [2.65b] are not the same, i.e., the order of the coupling of jj and j 2 governs the sign of the reduced matrix elements. The other special case of Eq. [2.63] is that for which both parts of the compound tensor X Q (1 ,2 ) act on the same system. In this instance, the reduced element of Eq. [2.63] becomes <r i , J1 |X K | hJ> = (- l ) K + >J 'V2KTTxj k . 1 k? 1^ ri"j"l J J J J x <r|']'l|T k l | |Ti"j"><Ti"j , , | |T k2 |h j> . [2.66] Finally, a mechanism is required to transform spherical tensor operators between space-fixed and molecule-fixed coordinate systems. To express the molecule-fixed components T q in terms of space-fixed components Tp , the following transformation is used: Tqk = I © p k q ( < ° ) T p k - t 2 - 6 T l P The inverse transformation is found by multiplying Eq. [2.67] by <D^ (©), summing over q Chapter 2 Theoretical Background 23 and using orthogonality properties of the Wigner rotation matrix elements. The result is: Tp =Z<Z)pT(ffi)Tq-q Eq. [2.68] is required when evaluating matrix elements for an internal operator written in a space-fixed axis system; it projects the operator into the molecule-fixed axis system. The reduced matrix elements for (D^*((o) are also required in such evaluation; they are given by an extension of Eq. [2.61]. Eqs. [2.58], [2.61], [2.64], [2.65], [2.66] and [2.68] are some of the most useful for evaluating Hamiltonian matrix elements. The individual terms of the Hamiltonian are presented in the next section; their matrix elements are evaluated in Section 2.6(b). 2.5 The Effective Molecular Hamiltonian 2.5(a) The Rotational Hamiltonian The Hamiltonian operator for a rigid rotating molecule can be written as R 2 R 2 T J 2 n g l d 2I X 2I y 2I Z where R x , R y and R z are the Cartesian components of the rotational angular momentum R, taken along the principal axes of the molecule, and I x , I y and I z are the corresponding principal moments of inertia, given by I i = Z m j r j i ' i = x>y>z> [2-70] j where nij and are respectively the mass and distance of the jth atom from the i axis. This thesis is concerned only with linear molecules, for which R is perpendicular to the molecular, i.e., z axis. Rotations about the z axis cannot be distinguished in this case, so R z = 0 [2.71] and I x = Iy = I. [2.72] For diatomic and linear triatomic molecules, the moment of inertia I can be expressed in terms of the bond length(s) of the molecule. For diatomics: Chapter 2 Theoretical Background 24 I = UJ-2, [2.73] where r is the bond length and p is the reduced mass from Eq. [2.22]. For linear triatomics: j = m 1m 2r 1 2 2 + mim 3 ( r 1 2 + r 2 3 ) 2 + m 2 m 3 r 2 3 [ 2 ? 4 ] mi + m 2 + m 3 where the subscript 2 denotes the central atom of the molecule and ry denotes the bond length between atoms i and j . In each case Eq. [2.69] simplifies to R ? + R ? R 2 Eq. [2.75] does not accurately specify the rotational Hamiltonian for a real molecule because with rotation of the molecule, centrifugal forces increase I; a more complete treatment yields (14, 15): Hroi = B R 2 - D R 4 . [2.76] In this expression D, the centrifugal distortion constant, is typically about 10 6 times less than B. An accurate estimate of D is made from the Kratzer relations (14, 15): 4 B 3 D = ^ - , [2.77a] or D = 4 B 3 (fl r 2 \ Z2_L + 223_ 2 2 [2.77b] These expressions apply to diatomic and linear triatomic molecules respectively. In Eq. [2.77b], the £ 2 i parameters describe the Coriolis coupling of the bending vibration with the 2 2 4 two stretching vibrations, and are normalised such that C 2 i + C 2 3 = 1- The R dependence of the distortion term comes from the general prescription for evaluating the centrifugal distortion correction of any term in the Hamiltonian: Xd = \ C [ # R 2 + R 2 # ] = \ C[H ,R2]+, [2.78] where C is the experimentally determinable constant and the anticommutator form [x, y ] + retains the Hermicity of 7^, as required of any Hamiltonian operator. R in Eq. [2.76] is one of many angular momenta that occur in a molecule; others Chapter 2 Theoretical Background 25 discussed in later sections include the electron orbital angular momentum L and the electron spin angular momentum S. The sum of these is J, the total angular momentum neglecting nuclear spin: J - R + L + S, [2.79] which leads to the form of R required in Section 2.6(b) for the evaluation of the matrix elements of Kro{. R = J - L - S . [2.80] 2.5(b) The Fine Structure Hamiltonian Electron spin contributions to the Hamiltonian have thus far been ignored. Spin arises in quantum mechanical systems as a relativistic effect; it is an intrinsic angular momentum of electrons (and many nuclei) with no classical analogue. Like other angular momenta, it produces a magnetic moment, [2.81] n directly proportional to itself. In this expression, g e is the relativistic gyromagnetic ratio or "g-factor", approximately equal to 2.0023, p B = eh/2me is the Bohr magneton, the unit of electron magnetic moment, and s is the spin angular momentum of magnitude ^/s(s +1) h where s = 1/2. A minus sign appears in Eq. [2.81] because the electron is negatively charged; ji s and s point in opposite directions. Four important effects arise if the total spin angular momentum S is nonzero. A nonzero nuclear spin angular momentum, if present, can couple with the electron spin angular momentum; this represents one of a class of several nuclear spin effects collectively referred to as hyperfine structure. The hyperfine Hamiltonian is discussed in Section 2.5(c). The other three effects, referred to as fine structure, are now examined. 2.5(b)(i) The spin-orbit Hamiltonian The positively charged nuclei of the molecule produce an electric field through which Chapter 2 Theoretical Background 26 the electrons travel. This orbital motion generates a magnetic field; using Maxwell's equations of electromagnetic theory to describe the interaction of the electron spin magnetic moment with this magnetic field (22), the following Hamiltonian can be obtained: f "\ „ .. 1 ^ dV g e p B v 1 dV g e ^ B y 1 9 Z-i . 'en' Se ^ i 2 2hc m e e,nren V^ren j hc^ e,nren V°^ren J (renXVn)-Se- [2.82] Some of the quantities have already been defined; others require clarification: r e n denotes the displacement from a nucleus to an electron, V denotes the potential of the nuclei, I e n is the electron orbital angular momentum around a nucleus, and s e is the electron spin. The expression is summed over all electrons and nuclei in the molecule. The second term in this expression is #~sr, the spin-rotation interaction; it is discussed in Section 2.5(b)(ii). The first term is Jfso, the spin-orbit interaction, which describes the interaction of the electron orbital and spin angular momenta. A form of the spin-orbit Hamiltonian can be derived by considering the form of the potential V , which must take electron-electron repulsions into account. Van Vleck (19) has shown that the screening effect of these repulsions can be approximated adequately by a Coulombic potential that treats each nucleus n as an effective point charge Zn( eg)e: V = Z " ( e £ ° e . [2.83] 47ts 0r e n With this potential and the definition of pg, the first term of Eq. [2.82] becomes: 43 v-> 2 2 1 Anz^h c e,n ren By defining the microscopic spin-orbit parameter a e according to s 4 T Z — P . » 5 1 47180 ft c n ren the spin-orbit Hamiltonian becomes e which is often approximated macroscopically as Chapter 2 Theoretical Background 27 #so = A L - S , [2.87] where A is the macroscopic spin-orbit coupling constant and L and S are respectively the total electron orbital and spin angular momenta: e and S = £ s e . t 2 - 8 8 b ] e Eq. [2.87] is normally sufficient to describe isolated, unperturbed electronic states, while Eq. [2.86] is required to model interactions between electronic states of different S or A. 2.5(b)(ii) The spin-rotation Hamiltonian The microscopic form of the spin-rotation Hamiltonian Hsx was given in the previous section as the second term in Eq. [2.82]: f ^ geHB sr 2 he e,n ren dV V^ren j ( r e n x v n ) - S e . [2.89] As with the spin-orbit Hamiltonian, the effective potential of Eq. [2.83] can be substituted. In addition, the nuclear velocity v n is given by v n = w x r n = (Ico) x r n / I = Rx r n / I , [2.90] where o> is the rotational angular velocity of the (linear) molecule, and R and I are the rotational angular momentum and moment of inertia as given in Section 2.5(a). With these modifications, ^ - - J ^ Z ^ I ' - x ( * « „ ) ] - , . . [2.9!] 47is0^c Ie,n r e n This can be written as a sum of two terms by using the triple cross product identity A x ( B x C ) = (C-A)B-(B-A)C. [2.92] Rewritten in this fashion, the second resulting term of Eq. [2.91] operates between different electronic states and is generally much smaller than the first; it is usually omitted. Therefore « - s r = - ^ Z ^ f I ( r „ r e n ) R S e . [2.93] 47t8Q c^ Ie,n r e n Chapter 2 Theoretical Background 28 This can be simplified in a manner analogous to the microscopic spin-orbit Hamiltonian of Eq. [2.86]; by defining b e = - ^ B i _ I ^ f - ( r n . r e n ) , [2.94] 47CS()ftc I n r e n Eq. [2.93] can be written as * s r=Xb e R-s e , [2.95] e whose approximate macroscopic form is #sr = y R - S , [2.96] where S is as defined in Eq. [2.88b]. This represents in effect an interaction of the rotational and total electron spin angular momenta with coupling constant y. This direct spin-rotation parameter is often dominated by a larger indirect spin-rotation coupling that arises from off-diagonal matrix elements of the spin-orbit Hamiltonian. Using second order perturbation theory, Henderson has shown (23) that this contribution has the same form as the direct spin-rotation coupling of Eq. [2.89]. Consequently, the experimentally determined y parameter is actually an inseparable sum of the direct spin-rotation constant of Eq. [2.96], renamed y S R , and the second-order contribution y s o : y = YSR + YSfJ [2.97] where y s o is usually the dominant contribution in all but the lightest molecules. 2.5(b)(iii) The spin-spin Hamiltonian The final contribution to the fine structure Hamiltonian arises from the interaction of the spin magnetic moment of each electron with the magnetic field produced by the remaining electron spins. This spin-spin interaction is . . r Si Sj 3 (S J -i-jiXsj -i-ji) ^ = _ 8 ^ Z | ^ ( o ) | 2 s i _ S j + g e2 p i 3n i>j n i>j r3. v5. Ji Ji [2.98] Some identification of parameters is necessary. The Sj denote the various electron spins, rjj denotes the distance between the ith and jth electrons and \|/j(0) refers to the wavefunction of Chapter 2 Theoretical Background 29 the jth electron evaluated at the location of the ith electron. In the first term of Eq. [2.98], |v|/j(0)|2 is the probability that electrons j and i have the same coordinates; this term is thus constant and can be absorbed into the Born-Oppenheimer potential. The second term has the same form as the classical Hamiltonian for the interaction between dipoles; it is called the dipolar electron spin-spin interaction. It is usually written in the following macroscopic form as a direct spin-spin coupling: S z denotes the molecule-fixed projection of the total spin S and X is the electron spin-spin interaction parameter. Just as off-diagonal matrix elements of the spin-orbit Hamiltonian introduce an indirect second-order correction to the direct spin-rotation interaction of Eq. [2.96], they also introduce a second-order correction to the direct spin-spin Hamiltonian. As discussed by Levy (24) and Lefebvre-Brion and Field (25), the form of this correction is identical to the direct spin-spin Hamiltonian, so that the experimentally determinable spin-spin constant X has a form analogous toEq. [2.97]: where X s s represents the direct contribution of Eq. [2.98] and X s o represents the indirect contribution that arises from second-order spin-orbit interactions. Once again, the latter, when present, is usually the dominant contribution to the experimental value of X. 2.5(c) The Hyperfine Hamiltonian Like electrons, the protons and neutrons that comprise nuclei each possess an intrinsic one-half unit of spin angular momentum. The interaction of these spins within a nucleus results in a total nuclear spin I, whose quantum number I (not to be confused with the moment of inertia I introduced in Section 2.5(a)) is restricted to nonnegative multiples of one-half and is, for the purposes of this thesis, constant for a given nucleus. This nuclear spin, like the electron ^ s s _ 3^(3S 2 - S 2 ) ; [2.99] [2.100] Chapter 2 Theoretical Background 30 spin, possesses a magnetic moment that is free to interact with those associated with other angular momenta in the molecule. Accordingly, the molecular energy levels are split into magnetic hyperfine structure components. Nuclei with I > 1 have anisotropic charge distributions; the resultant electric quadrupole moments contribute additional complexity in the form of electric quadrupole hyperfine structure. Both phenomena, discussed below in further detail, are called hyperfine effects because the energy corrections they introduce are typically very small; in electronic spectra, their effects are discernible only at very high resolution. 2.5(c)(i) Magnetic hyperfine structure In the 1950s, Frosch and Foley (26) recognised the correspondence of the magnetic moments of the nuclear and electron spins, and thereby formulated the first general theory for the magnetic hyperfine structure of linear molecules. Their formalism was thus essentially identical to that presented above for the electron spin Hamiltonian, with the nuclear spin magnetic moment replacing that of the electron. As was the case for electron spin, several interactions of the nuclear spin magnetic moment can be identified, namely: (1) the spin magnetic moments of the electrons, (2) the orbital magnetic moments of the electrons, (3) the rotational magnetic moment of the molecule and (4) the spin magnetic moments of other nuclei. The last of these four interactions is far too weak to detect in electronic spectra and will not be considered here; only the remaining three are examined. The appropriate Hamiltonian in each case can be derived from the formalism of Section 2.5(b) by simply replacing the various electron parameters with those for the relevant nuclei. An additional hyperfine effect similar to A-type doubling of rotational structure is discussed in Section 2.7(c). Nuclear spin-electron spin interaction The Hamiltonian for the nuclear spin-electron spin interaction has the same form as Eq. Chapter 2 Theoretical Background 31 [2.98], except that one of the two electron spin magnetic moments is replaced by a nuclear magnetic moment g n ^ N 1 Hi h [2.101] where g n is the g-factor of the nucleus, p^ = eh/2m p is the nuclear magneton and m p is the proton mass; the minus sign of Eq. [2.81] no longer appears since the nucleus is positively charged. With this change, the nuclear spin-electron spin Hamiltonian is % = M M £ g J v , e ( 0 ) f , n . s < 3h e , n g e W N h2 e , n ^ n s e 3 ( I n • r e n ) (s e • r e n ) [2.102] ' e n ' e n Once again, two terms arise as in Eq. [2.98], but in this case the first term is not a constant; it represents the Fermi contact interaction between the electron and the nucleus when the electron is in an orbital with non-vanishing amplitude at the nucleus. This occurs only when an unpaired electron occupies a molecular orbital with atomic s character. The second term is analogous to that of Eq. [2.98] and represents the dipolar interaction between the magnetic moments of the nuclear and electron spins. If, as is the case for all molecules studied in this thesis, only one nucleus has a large magnetic moment, no summation over n is needed. In addition, the Fermi contact parameter, bT 8 7 1 g e P - B U N g n • r m | 2 J F , e - — I V e C°) I 3 h2 can be introduced, so that Eq. [2.102] becomes fllS=X>F>eI.se - X g e g n f W N h2 I s e 3 ( I r e ) (s e - r e ) [2.103] [2.104] Nuclear spin-electron orbital angular momentum interaction Once again, the treatment follows that of the electron spin-electron orbital angular momentum interaction, with nuclear spin parameters replacing those of the electron spin. The Hamiltonian is given by Chapter 2 Theoretical Background 32 ^ = Z ^ M - | - | ^ I - ' e = Z a e I - l e ; [2-105] e 2hc m e dV where only one nuclear spin is assumed and the middle part of the equation defines a e. Nuclear spin-rotation interaction The nuclear spin-rotation interaction can be derived in similar fashion to the electron spin-rotation interaction, giving n w l dV ^ I n J = Z ^ n I n - J = C l I - J . [2.106] V r V U 1 n In the term following the first equals sign, the quantity I is the moment of inertia, not to be confused with the nuclear spin I. The last term applies if only one nuclear spin is present; J is used instead of R because the effect of the nuclear spin interaction with the electron spin and orbit angular momenta have already been considered in the preceding discussion. This very weak interaction is generally not significant for electronic spectra; it was not observed in the present work and will not be discussed further. The total magnetic hyperfine Hamiltonian Like the nuclear spin-rotation interaction, the other terms in the magnetic hyperfine Hamiltonian are usually expressed in terms of macroscopic hyperfine parameters: # m a g h f = a I - L + bI-S + cI z S z . [2.107] In Eq. [2.107], only the diagonal part of the dipolar nuclear spin-electron spin Hamiltonian is taken. The hyperfine parameters are the nuclear spin-electron orbit constant a= [2.108] the Fermi contact parameter b F = 8 7 L g e g n W N | v ) / ( 0 ) | 2 [ 2 1 0 9 ] 3 ti and the dipolar parameter „ _ 3 g e g n p B H N /3cos 9-1 x m i n i 2 h2 \ r 3 } - [ 2 1 1 0 ] Chapter 2 Theoretical Background 33 InEq. [2.107], b = b F - ^ c . [2.111] Of these constants, a, b and c are the experimentally determinable parameters. The constant b F is more fundamental than b because it gives the value of the electronic wavefiinction at the location of the spinning nucleus, but it can only be determined indirectly from Eq: [2.111]. 2.5(c)(ii) Electric quadrupole hyperfine structure While magnetic hyperfine effects are generally important only for open shell electronic states, another type of hyperfine effect arises even for closed shell electronic states, if an I > 1 nucleus is present. The anisotropic charge distribution in such a nucleus creates an electric quadrupole moment that interacts with the electric field gradient of the electron charge distribution. This effect, called the electric quadrupole hyperfine interaction, is the first non-vanishing term in the multipole expansion of the nuclear-electron Coulombic interaction. Many early workers (27-33) have examined the theory of electric quadrupole hyperfine structure for various types of molecules; Bardeen and Townes (34) offered the first general formalism. The electric quadrupole hyperfine Hamiltonian HQ can be expressed macroscopically as the scalar product of two second rank tensors: # Q = - T 2 ( V E ) - T 2 ( Q ) , [2.112] namely, the electric field gradient and the nuclear quadrupole tensor. By convention, the nuclear quadrupole moment operator eQ is given by e Q = Z q n r n ( 3 c o s 2 e n - l ) . [2.113] n The summation is over all nucleons, each with a charge q n and spherical polar coordinates r n and 6 n as measured from the nuclear centre. The corresponding nuclear quadrupole moment Q is defined by e Q = < L M I = I | Q | L M I = I > , [2.114] where I and M T are the quantum numbers for the nuclear spin and its axial projection. Q cannot Chapter 2 Theoretical Background 34 be measured independently from molecular spectra; the experimentally determinable parameters that specify the strength of the quadrupole interaction in linear molecules are the products eQqo and eQq2- As explained below in Section 2.6(b)(1), only the former is needed in this thesis. By convention, the value of qo is specified by the axial component of the electric field gradient (or equivalently, the negative Laplacian of the potential) evaluated at the quadrupolar nucleus: q 0 = <J, M j = J | ( - V E ) z 0 | J , M j = J> = <J, M j = J | (d2V/dz2)0 \ J , M j = J) . [2.115] 2.6 Hund's Coupling Cases and Hamiltonian Matrix Elements The formulation of the Hamiltonian Jfis the first step in determining the energy level structure for a molecular electronic state of interest. The next step is the evaluation of the matrix elements of 9f, followed by the diagonalisation of the resulting matrix to yield the energy eigenvalues of the state. Always implicit in this step is the representation, or choice of basis functions used to evaluate the matrix elements; since the eigenvalues are invariant to the representation, any valid, convenient basis set can be used. In part (a) of this section, the most commonly used basis sets are discussed. Part (b) evaluates the matrix elements using two of these representations. 2.6(a) Hund's Coupling Cases In 1926, Hund (35) considered four coupling situations that could arise for different strengths of the couplings between the rotation and electron angular momenta, which subsequently became known as Hund's coupling cases (a), (b), (c) and (d). Shortly thereafter, Mulliken (36) proposed a fifth coupling case that he labelled Hund's case (e). None of these considered the various types of nuclear spin angular momentum interactions because hyperfine effects had rarely been observed at that time, but the first three cases, the only ones used to describe the present data, were later so extended by Frosch and Foley (26) and Townes and Chapter 2 Theoretical Background 35 Schawlow (11). These three schemes and their extensions for nuclear spin are now discussed. The Hund's coupling cases are choices of basis functions that generate a convenient representation of the Hamiltonian matrix, usually the one that is either the most nearly diagonal or the one for which the matrix elements are algebraically simplest. The starting point is the spin-orbit and rotational Hamiltonian, #- = A L - S + J3R2, [2:116] where R = J - L - S. Expanding B R 2 gives 3f = B ( J 2 + L 2 + S2) + (A+2B) L S - 2B J L - 2B J S, [2.117] The choice of basis is made according to which of the final three terms in Eq. [2.117] is dominant over the other two in the state of interest. 2.6(a)(i) Hund's case (a) Hund's case (a) is an uncoupled representation, for which the electron orbital and spin angular momenta L and S are separately quantised along the molecular z-axis, with projections A and 2 respectively. Its basis functions are written | n A ; S E ; J Q > , [2.118] where n refers to the radial part of the electronic wavefiinction. This ket denotes independent electron orbital and spin factors, |(L) A) and | SS>, where L is written in parentheses since it is strictly not a good quantum number; since there is no molecular rotation about the z-axis, the projection Q. of the total angular momentum J (exclusive of nuclear spin) onto this axis comes from electrons only: £2 = A + Z. [2.119] The possible values of A and S are A = 0 , ± l , ± 2 , ± 3 , ... , [2.120a] and 2 = +S, +S-1, +S-2, ... -S ; [2.120b] the A values correspond respectively to the irreducible representations (symmetries) E, n, A, O, Chapter 2 Theoretical Background 36 ..., of the electronic wavefunction in the C ^ y point group of the molecule. The various quantum numbers are usually assembled into the term symbols 2 S + 1 A Q ; the quantity 2S+1 is called the spin multiplicity and Q. is given by Eq. [2.119]. A superscript of either + or -denotes the Kronig symmetry for Z states (not to be confused with the quantum number £ of Eqs. [2.119] and [2.120b]), i.e., whether the electronic part of the molecular wavefunction remains unchanged or reverses sign upon reflexion in the molecular plane. Hund's case (a) is the most diagonal representation if L and S are moderately coupled through the spin-orbit operator (A+2B) L-S, with much smaller couplings from the last two operators in Eq. [2.117]. This coupling mixes case (a) basis functions for which A A = - A S , so that A and S are strictly no longer good quantum numbers, but can usually be approximated as such; their sum £1 always remains good if the L-S coupling is sufficiently strong. Very large L-S coupling thoroughly mixes various A (and various Z ) , so that Hund's case (c), described below, applies. Even if the electronic state of interest does not conform closely to Hund's case (a) coupling, it is often the best choice of basis for data fitting purposes because its matrix elements are algebraically simplest. Since the projection of J is characterised by £2, the quantum number J is restricted: J = |Q| , + | Q | + 2, ... . [2.121] 2.6(a)(ii) Hund's case (b) Hund's case (b) applies if the -2B J-S term in Eq. [2.117] dominates the two preceding it. Since this interaction increases with rotation while (A+2B) L-S is independent thereof, in principle this will always occur at sufficiently high J values; this process is called "spin-uncoupling". In case (b), the vector sum of R and L is called N: N = R + L. [2.122] Since R is perpendicular to the molecular axis, A can be regarded as the projection quantum number of both L and N. By analogy with Eq. [2.121], N is bounded below by | A | : Chapter 2 Theoretical Background 37 N = | A | , | A | + 1, | A | + 2, ... . [2.123] The total angular momentum J (exclusive of nuclear spin) is then J = N + S. [2.124] Thus if orbital angular momentum effects are negligible, the rotational energy is B R 2 ~ B N 2 = B(J - S) 2 = B t J 2 + S 2) - 2B J S, [2.125] The appropriate basis functions, when the effects of the - 2B J-S term dominate, represent the vector coupling of N and S as per Eq. [2.124], and follow from Eq. [2.37]: O S N ^  v n - z - A |r|A;SZ;jn>. [2.126] |n ;NASJ> = Q V 2 N + l | The symbol 2 S + 1 A , identical to the case (a) notation but with the superfluous Q. subscript removed, denotes an electronic state for which Hund's case (b) applies. 2.6(a)(iii) Hund's case (c) As explained earlier, Hund's case (c) applies if the spin-orbit coupling of L and S through the operator (A+2B)L-S is so strong that the projections A and £ lose their meanings. The vector coupling in this case is J a = L + S, [2.127] and the basis functions are simply \r\; J a£2 ; JI2), where Q is the projection of both J and J a . This situation often occurs in the electronic states of molecules containing heavy atoms, even in states with nominal Z symmetry. In Hund's case (c) coupling a 2 Z ± state becomes a n i i = 1/2 state without a Kronig symmetry, and a state becomes three case (c) states, one each with Q. - 0 + , 1 + and 1~ (although the last two are normally regarded as a single CI = 1 state). The rotational energy level patterns are essentially the same for case (a) and case (c). The allowed J values follow Eq. [2.121] and (for Q > 0) are doubly degenerate in the absence of A - or D-type doubling effects. 2.6(a)(iv) Modified Hund's cases: nuclear spin effects Frosch and Foley (26) and Townes and Schawlow (11) were the first to extend Hund's Chapter 2 Theoretical Background 38 coupling cases to account for nuclear spin effects. In their work, Hund's coupling cases (a), (b) and (c) are sub-classified depending on whether the nuclear spin angular momentum I is coupled more strongly to the molecular axis or to another angular momentum within the molecule. The appropriate coupling case is modified by a subscript; in the former case, a is used, while the latter uses a P subscript. In practice, the former situation is never encountered since the nuclear spin magnetic moment is so small compared to those of the other angular momenta that their interactions are too weak to couple I to the molecular axis; only cases (ap), (bp) and (cp) are generally observed. Hund's case (ap) is a simple extension of case (a), in which the nuclear spin couples to the total angular momentum excluding nuclear spin, J, to form the total angular momentum F. This coupling scheme can be written as J = R + L + S ; F = J + I. [2.128] The total angular momentum quantum number F has the 2 • min(J,I) + 1 allowed values F = J + I, J + I - l , ... , | J - I | [2.129] (where min(a,b) is the lesser of a and b), and the appropriate basis functions are written as | n A ; S 2 ; . [2.130] Hund's case (bp) is further subclassified by an extra subscript of N , J or S according to which of the corresponding angular momenta is most strongly coupled to the nuclear spin I. The coupling to I of N is generally much weaker than that of J or S; no examples of case (bpN) coupling appear in the literature and it will not be discussed here. Of the remaining pair, case (bpj) is the more commonly encountered; it is a simple extension of case (b) analogous to case (ap). Once again, the total angular momentum excluding nuclear spin, J, couples with the nuclear spin I to form the total angular momentum F; the coupling scheme for this case is: N = R + L ; J = N + S ; F = J + I. [2.131] Hund's case (bpj) basis functions are written in the form Chapter 2 Theoretical Background 39 | T | ; N A S J I F > . [2.132] The other coupling case, called Hund's case (bps), occurs if the interaction between the nuclear and electron spins {i.e., the Fermi contact interaction) is the largest electron spin interaction term in the Hamiltonian. Under these circumstances, the coupling of I and S forms a resultant angular momentum G that is sometimes called the pseudospin. Its quantum number G takes the values G = I + S , I + S - 1 , ... , 11 — S |, [2.133] in accord with the usual rules of vector coupling. G in turn couples with N to form the total angular momentum F ; the coupling scheme can be written as N = R + L ; G = I + S ; F = G + N , [2.134] This represents a less common coupling scheme for which J is not a good quantum number. Hund's case (bps) basis functions are written in the form |nA;(IS)GNF>; [2.135] in analogy with Eq. [2.129], the values of F range from | N - G | to N + G. Of the electronic states reported in the literature that follow Hund's case (bps) coupling, the vast majority have 2 Z symmetry, for which spin-spin and spin-orbit couplings do not occur. These couplings can be present for states of other symmetry and, if so, are usually much larger than the Fermi contact interaction. The only other electron spin interaction that appears in 2 S states is the spin-rotation interaction, which increases with rotation. In principle, every electronic state that begins in Hund's case (bps) coupling at low N eventually reaches a sufficiently high N value at which the spin-rotation interaction overtakes the (constant) Fermi contact interaction and a Hund's case (bpj) description becomes more appropriate. This is similar to the spin-uncoupling phenomenon, in which case (a) coupling converts to case (b) at high rotation. Finally, Hund's case (cp) is also possible. This extension of Hund's case (c) is Chapter 2 Theoretical Background 40 analogous to the extension of Hund's case (a) to Hund's case (ap). Once again, the total angular momentum F is formed from a coupling of J and I; the basis functions are written as |r,; J a Q;Jf i IF> . 2.6(b) Matrix Elements of the Hamiltonian Various contributions to the molecular Hamiltonian were identified in Section 2.5. This section presents the evaluation of their matrix elements using Hund's case (ap) and (bp§) basis sets; the latter are for a 2 2 state, the only type for which these are needed in this thesis. 2.6(b)(i) Hamiltonian matrix elements evaluated in a Hund's case (ap) basis The total molecular Hamiltonian operator can be written as X = e^v + r^ot + ^so + s^r + ^ss + ^mhfs + ^Q + ^ld- t 2 1 3 6 ] The terms on the right of Eq. [2.136] are the operators for, respectively, the electronic-vibrational energy, the rotation, the spin-orbit interaction, the spin-rotation interaction, the spin-spin interaction, the magnetic hyperfine structure, the electric quadrupole hyperfine structure and the A-doubling interaction. The final term has not yet been discussed; this is done in Section 2.7. The electronic-vibrational Hamiltonian needs little discussion since in any rotational basis it merely contributes the vibrational energy T v for its electronic state of energy T e ; thus < A ; S S ; J O I F | ^ e v | A ; S E ; JQIF) = T e + T v = T e v , [2.137] where r\ is suppressed since only matrix elements within an electronic state are considered in this discussion. The remaining matrix elements are now evaluated. Some are independent of I and F; for brevity, these are suppressed from the basis functions for such elements. The rotational Hamiltonian: the B 1 } term The rotational Hamiltonian of a linear molecule is given by Eq. [2.76]; with R written in terms of J, L and S, Hm = B v (J - L - S) 2 - Dy (J - L - S)4 [2.138] Chapter 2 Theoretical Background 41 where the x> subscripts of the rotational and distortion constants specify the vibrational level of interest. The first term can be expanded: By (J - L - S)2 = B v (J 2 + L 2 + S 2 - 2 J-S + 2 L-S - 2 J-L). [2.139] The x and y components of the L operator connect electronic states with different A values; their effects are neglected here and incorporated into the A-type doubling operator. Thus, if the centrifugal distortion term is ignored for the moment, the rotational Hamiltonian becomes #-rot = Bv [(J2 + L 2 - J 2 - S 2 - (J+S_ + J_S+)], [2.140] where -Bv (J+S_ + JLS+) is the aforementioned spin-uncoupling operator. This form of the rotational Hamiltonian is useful because the Hund's case (a) basis functions \r\ A ; SZ ; Jfi} are eigenfunctions of J 2 , J z , S2, S z and L z , and the matrix elements can be evaluated using Eq. [2.34]: < A ; S Z ; J Q | # r o t | A ; S Z ; j n > = BV[J(J+1) + S (S +1) - Q 2 - Z 2 ] , [2.141a] and < A ; S Z ; JQ|#" r o t | A ; S Z ± 1 ; JO±l>= -BvV±(J,n)V±(S,i:), [2.141b] where V±(a,b) is given by Eq. [2.34d]. The off-diagonal matrix elements are derived with the implicit understanding that the ladder operators J ± behave like J+ because of the anomalous commutation rules for angular momentum operators in a molecule-fixed axis system. The rotational constant Bv strictly does not correspond to the inertial moment at the absolute minimum of the potential surface (i.e., at equilibrium) of the electronic state of interest, but represents instead an effective value for the vibrational level u. Rotational constants from at least two vibrational levels must be experimentally determined to extract the equilibrium moment of inertia. B^ is usually expressed as a power series in (UJ +-|): B v = B e -Xa i (v J i +l) + ..-. [2.142] i The summation is over all the vibrational modes; B e is the equilibrium rotational constant and the O j parameters are the vibration-rotation interaction constants. For a diatomic molecule, B e Chapter 2 Theoretical Background 42 corresponds to the equilibrium bond length r e; when written in c m - 1 units, B - h e 2 2 ' 87c cpr e [2.143] where p is the reduced mass of the molecule. For jet-cooled spectra of the type reported in this thesis, the v = 0 level of the ground state is usually the only one significantly populated, and the rotational constants of higher vibrational levels cannot be determined; in this case, r e is approximated by r 0 , corresponding to B 0 . Diatomic molecules have only one vibration-rotation interaction constant, called a e . The Pekeris relation (37) gives an approximate value of a e: where coe and coexe are the vibrational constants of Eq. [2.20]. For polyatomic species, the extraction of geometrical structural parameters from rotational constants is not always trivial. For a linear triatomic molecule, for example, the equilibrium constant B e depends on both bond lengths. These are usually determined by isotopic substitution; rotational constants are independently measured for the A B C and A B C isotopomers. In the BOA, the equilibrium bond lengths r e (A-B) and r e (B-C) are independent of isotopomer, so the B e (ABC) and B e ( A B C ) values can be used to give these. If only the zero-point vibrational level constants, B 0 , can be determined, then an r 0 structure emerges. As with diatomic molecules, this only approximates the equilibrium (re) structure, even more so since r 0 (B-C) differs for A B C and ABC ' . The Laurie correction (38) offers an improvement, the difference in r 0 (B-C) for the diatomic species BC and B C , if known independently, can approximate the corresponding difference in A B C and A B C and improve the r 0 values extracted from the B 0 constants. Centrifugal distortion: the D v term The matrix elements of the distortion term need not be given here; they can be found simply by squaring the coefficient matrix of the Bv term and reversing its sign. The centrifugal [2.144] Chapter 2 Theoretical Background 43 distortion constant D v , like the rotational constant B^, can be expressed as a rapidly converging power series in (u, + ): Do = D e + £ B i O > i + i ) + - - [2.145] i In the Kratzer relations of Eq. [2.77], the B and D constants are strictly the equilibrium constants of Eqs. [2.142] and [2.145], although the 04 and pj are sufficiently small compared to B e and D e that their zero-point counterparts are often used in Eq. [2.77] to good approximation. Eq. [2.78] gives the centrifugal distortion correction to any rotational operator in terms of the operator itself, so that the matrix elements of the centrifugal distortion corrections can be determined by straightforward matrix algebra. For brevity, therefore, such matrix elements will not be derived here, although they are quoted as needed in subsequent chapters. Also, for convenience, a subscript v identifying the vibrational level of interest has been suppressed from all determinable parameters in the matrix elements below. The spin-orbit Hamiltonian The spin-orbit Hamiltonian of Eq. [2.87] can be expanded into Cartesian components: Xso = A [ L Z S Z + \ (L + S_ + LJS+)], [2.146] with diagonal elements <A; S E ; J£2 | Jfs0 | A ; SE; J£2> = A A E , [2.147] Matrix elements with A A = - A E = ±1 exist for this operator, but were not used in this thesis. The spin-rotation Hamiltonian The spin-rotation Hamiltonian of Eq. [2.96] must be expressed in terms of the J , L and S angular momenta for evaluation of its matrix elements in a Hund's case (a) basis: Hsr = y ( J - L - S) • S = y[S2Z - S 2 +^(J+S_ + J_S +)], [2.148] where terms involving the x and y components of L have been neglected in the expression after the second equals sign. Thus <A;SE ;JQ|#- s r |A ;SE ;J f i> = y [E 2 -S (S+ l ) ] [2.149a] Chapter 2 Theoretical Background 44 and < A ; S £ ; J Q | # s r | A ; S E ± 1 ; JO+l ) = ^yV ±(J,£2)V ±(S,S). [2.149b] The spin-spin Hamiltonian The spin-spin Hamiltonian of Eq. [2.99] has only diagonal elements when evaluated in a Hund's case (a) basis, and is only required for states of at least triplet multiplicity: < A ; S E ; J Q | ^ S S | A ; S E ; JQ> = § X [3Z 2 - S(S+1YJ. [2.150] The magnetic hyperfine Hamiltonian The magnetic hyperfine Hamiltonian, Eq. [2.107], can be written in spherical tensor form: *mag.hf = a H O D • T l (L) + b Tl(I)• Tl(S) + cTj(I)T 0(S). [2.151] This formalism provides the simplest means of evaluating the matrix elements of ^niag.hf- The derivations are somewhat lengthy; only those for the second term are presented here. The operator TL(T) • TL(S) is a scalar product of two commuting first rank tensor operators; therefore, its matrix elements follow from Eq. [2.64]: < A ; SE ; JOTE | T^I) • T^S) | A ; SE' ; J'QTF > • ( - t ^ J ^ A ^ ^ I I T ^ A ^ m X ! , , ^ , , ! ! ) . [ 2 , 5 2 , Reduced matrix elements for T^I) and T*(S) appear in this equation. The former is given by Eq. [2.58]. The latter requires projection of T^S) from space- to molecule-fixed axes; from Eq. [2.68] (the p subscript is suppressed since only the reduced element of T^S) is needed): Tl(S)=X© ( q 1 ) *(»)T q 1 (S). [2.153] q Therefore < A ; S E ; JO || T :(S) || A ; SE*; J'O') = £ < S E | T<J(S) | SZ'XJn || © . ^ " ( ( D ) II J'Q'> . [2.154] q The Wigner-Eckart Theorem and Eq. [2.67] are needed to evaluate the two matrix elements. The final result reads: Chapter 2 Theoretical Background ( A ; SE ; J O J F I ^ O Q - T ^ I A ; SE' ; J'QTF) 45 (_l)J'+J+I+F-n ^ s (S + 1)(2S +1) 1(1 + 1)(2I + 1)(2J + 1)(2J' +1) f S 1 S'N F J i (-i) s-s E q E' [2.155] - Q q OJ The evaluation of the matrix elements for the other two terms in Eq. [2.151] follows similar methods. The symmetry properties of the 3- and 6-j symbols determine the selection rules on J, Q. and E for the three terms; only four non-vanishing elements exist for -T^iag.hf <A ; SE ; JQTF | XmagM | A ; SE ; JDJF) _ hQR(J) and 2J(J + 1) ' <A ; S E ; JOJF | ^ m a g M \ A ; S E ; J - l , O I F ) = hVj 2 -Q 2 P(J)Q(J) 2 j V 4 J 2 - l < T i A ; S E ; J Q I F | ^ m a g h f | nA ; SE + 1; JQ + 1,IF> _ bR(J)V ±(J,Q)V ±(S,Z) 4J(J + 1) < A ; S E ; J Q I F | ^ m a g h f | A ; SE ± 1 ; J -1 ,Q±1,IF> _ _ bV(j+n)(j+n-i)p(j)Q(j)v±(s,£) 4 j V 4 J 2 - l [2.156a] [2.156b] [2.156c] [2.156d] In Eq. [2.156], h, sometimes called the Frosch and Foley parameter (not to be confused with Planck's constant), is a linear combination of the hyperfine parameters in Eq. [2.151]: h = a A + (b + c)E, [2.157] and the functions P, Q and R are given by [2.158a] [2.158b] [2.158c] P(J) = V(J-I + F)(F + J + I + 1): Q(J) = V(J + I - F X F - J + I + 1), and R(J) = F(F+1)-1(1+1)-J(J+1). The electric quadrupole hyperfine Hamiltonian The evaluation of the matrix elements of the quadrupole Hamiltonian in Eq. [2.112] Chapter 2 Theoretical Background 4 6 follows from Eq. [ 2 . 6 4 ] by taking = J, j 2 = I and J = F: < A ; S E ; J O T F | ^ Q | A';S£;J'fl 'IF> = ( _ 1 ) J + I + F J F ^' j|<A;jn||-T2(yE)||A';J'n'>ai|T2(Q)||I). [ 2 . 1 5 9 ] The field gradient tensor must be projected into molecule-fixed axes using Eq. [ 2 . 6 8 ] : T2(VE) = X < D ( q 2 ) V ) T q 2 ( V E ) ; [ 2 . 1 6 0 ] q the dot subscript is used since the value of p need not be specified for the reduced element in Eq. [ 2 . 1 5 9 ] , which is now < A ; j n | | - T 2 ( V E ) | | A ^ m > = £ < ^ q K-i/^VaJ+^r+i) q J' 2 J - f l ' q Q <A' | | -T a 2 (VE) | |A>. [ 2 . 1 6 1 ] Eq. [ 2 . 6 7 ] has been used in the second step for the evaluation of the reduced Wigner rotation matrix elements. In this thesis, only states with A = 0 , i.e., E states, had observable quadrupolar hyperfine structure, for which the only non-vanishing field gradient matrix element has q = 0 . Thus from Eq. [ 2 . 1 1 5 ] , and the relation TQ (VE)=^(VE) Z : <A;Jfl| | - T 2 ( V E ) | | A ; J'Q> = lq0(-l)J'-QV(2J + l)(2J'+l) J' 2 J fl 0 fl [ 2 . 1 6 2 ] The evaluation of (I ||T ( Q ) | | I) requires the Wigner-Eckart Theorem, which can be applied to the M T = I component of the unreduced element, whose value is taken from Eq. [ 2 . 1 1 4 ] : e Q = < L M T = I | 2 T 0 2 ( Q ) | I , M T = I> = 2 I 2 I - I 0 I <I||T2(Q)||I>. [ 2 . 1 6 3 ] Thus, <I||TZ(Q)||I> = i e Q I 2 I [ 2 . 1 6 4 ] and the conversion to spherical tensor components again introduces a factor of y . Eqs. [ 2 . 1 6 0 ] - [ 2 . 1 6 4 ] , taken together, give the quadrupole matrix elements as ( A ; S E ; J Q I F | # Q | A ; S E ; J 'O IF> = \ e Q q 0 ( - 1 ) J + J ' + I + F _ Q V ( 2 J + 1 ) (2J '+1) fF y i]f r 2 J Y I 2 r- 1 2 I J • n o n - i o i [ 2 . 1 6 5 ] Chapter 2 Theoretical Background 47 This matrix element has selection rules | A J | = 0, 1 and 2; written explicitly: < A ; S Z ; J O J F | # Q | A ; S S ; J O J F > = e Q q 0 [ 3 n 2 - J ( J + l)]{3R(J)[R(J) + l ] - 4 J ( J + l)I(I + l)} 81(21 -1) J (J +1)(2 J -1)(2 J + 3) <A;SS; jn iF |^Q |A;SE;J - l ,OJF> [2.166b] [2.166c] = 3QeQq 0 [R(J)+J + l ]Vj 2 -Q 2 P(J)Q(J) 8 J (J -1) (J +1) I (21 - 1)V4J2 -1 and <A;S£ ; JQIF |#Q |A;SS;J-2 ,OIF> _ 3 eQqp yj [ J 2 - Q 2 ] [ ( J - 1 ) 2 -"a 2] P(J) Q (J)P(J -1) Q (J -1) 161 (21 -1) J (J -1) (2 J -1) V(2 J - 3)(2 J +1) where P(J), Q(J) and R(J) are given by Eq. [2.158]. 2.6(b)(ii) Hamiltonian matrix elements of a 2 £ state evaluated in a Hund's case (bps) basis Problems associated with eigenvalue sorting occasionally necessitated evaluation of the Hamiltonian matrix elements in an alternative basis set. For example, the close conformity of the 2 Z + ground state of LaNH to Hund's case (bp§) required use of this basis set; the resulting matrix elements are now presented, for which the following coupling has been taken: G = I + S ; F = N + G. [2.167] For a 2 S state, several terms can be omitted from the Hamiltonian. The spin-spin interaction term only applies for states of at least triplet multiplicity; the coupling terms for the electron spin-orbit and nuclear spin-electron orbit interactions vanish since A = 0. For the same reason, N can be used in place of R. The following Hamiltonian remains: X = B N 2 - D N 4 + y H ( N ) • T!(S) + b F T^I) • T^S) + i c { 3 T „ ( I ) T u ( S ) - T 1 ( I ) • Ti(S)} - T 2 ( V E ) - T 2 ( Q ) . [2.168] The rotational Hamiltonian The rotation and distortion terms are diagonal in all quantum numbers: < N ( I S ) G F | B N 2 - D N 4 |N(IS)GF> = BN(N+1)-D[N(N+1)] 2 . [2.169] Chapter 2 Theoretical Background 48 Determination of the remaining matrix elements requires the spherical tensor formalism. The spin-rotation Hamiltonian The spin-rotation Hamiltonian y T ^ N ) • T^S) is the scalar product of two commuting first rank tensors in the coupled basis N + G = F. Eq. [2.64] applies: {F G' N ' l : \ 1 N GJ x<N'||T 1(N)||N><(IS)G ,F||T 1(S)||(IS)GF> [2.170] The first reduced matrix element, for T^N) , is given by Eq. [2.58]. The second one involves an operator acting on the second half of a coupled basis, so from Eq. [2.65b]: <(IS)G'F||T 1(S)||(IS)GF> K S || T^S) || S> S G' II = (_i)I+S+G'+l ^ S(s + !)(2S + 1)(2G + 1)(2G'+1) | Q g ^ [2.171] where Eq. [2.58] has been used once again. Combining Eqs. [2.170] and [2.171]: < N 1 (IS) G F | y T 1 (N) • T 1 (S) | N(IS) G F > x ^ N ( N + 1)(2N +1) S (S + 1)(2S + 1)(2G + 1)(2G'+1) . [2.172] The magnetic and electric quadrupole hyperfine Hamiltonian The evaluation of the hyperfine matrix elements is similar to that of the spin-rotation Hamiltonian; only the results are shown here. The Fermi contact term, like the rotational term, is diagonal in all quantum numbers: < N 1 (IS) GF | b F Tl(I) • TL(S)\N(IS) GF> = b F (-1 )I+S+G^(T + i)(2I +1)S(S + 1)(2S +1) | ^ ^ g | . [2.173] The dipolar term is much more complicated, with off-diagonal elements in both N and G: Chapter 2 Theoretical Background < N ' ( I S ) G ' F | \c{3T<J(I)TQ1 ( S ) -T l(l) • T ^ S ) } | N ( I S ) G F > Vjo c ( _ 1 ) N + N ' + G ' + F ^ / l ( l + i)(2I + 1)S(S + 1)(2S +1) ' N ' 2 N x ^ /(2N + 1)(2N'+1)(2G + 1)(2G'+1) [F G' N ' 2 N G v o 0 o y I I 1 S S 1 G' G 2 Finally, the electric quadrupole term is < ISP (IS)G' F | T 2 ( V E ) • T 2 (Q) | N(IS) GF> = l e Q q 0 ( - l ) ' N ' 2 N 0 0 0 N+N'+I+S+G+G'+F I 2 I - I 0 I V ( 2 N + 1)(2N'+1)(2G + 1)(2G'+1) F G* N ' H l G Si 2 N G l l G * I 2 |" The exact algebraic forms of these matrix elements will be tabulated in Chapter 5. 49 [2.174] [2.175] 2.7 Symmetry, Parity and A-type Doubling 2.7(a) Symmetry Properties of Linear Molecules and e/f Parity Labels The molecules investigated in this thesis are either linear triatomics of the form A B C , or simply diatomic. Both types belong to the C ^ point group since they have C ^ rotational symmetry about the molecular axis, as well as an infinite number of o v reflexion symmetry planes containing the molecule. The effect of this operator on the total molecular wavefunction or portions thereof leads to various symmetries by which the wavefunction can be classified. If the molecular plane is taken to be xz, then the effect of the reflexion operator, ov(xz) on the molecule- and space-fixed coordinates of the various electrons and nuclei can be written as follows (16): o v(xz) (X j, yh Z j ) = (x i ; - y i ; z4), [2.176a] and a v (xz)(Xi, Y i , Z i ) = ( - X i , - Y i , - Z i ) . [2.176b] The molecule-fixed ov(xz) operator and the space-fixed inversion operator, E*, have the same effect on space-fixed coordinates: Chapter 2 Theoretical Background 50 E * ( X i 3 Y i ; Zj) = (-Xj, - Y i ; -Zi) . [2.176c] Lefebvre-Brion and Field (25) and Larsson (39) have examined the effects of these operators on the | T | A ) , |SS) and | J Q > functions that comprise the Hund's case (a) basis functions: E * | J Q > = ( - 1 ) J - ^ | J - Q > , [2.177a] E * | S £ > = (-1)S-2|S-Z>, [2.177b] ov(xz)|SZ> = ( - l ^ i s - S ) [2.177c] and c v(xz)|nA> = ( -1) A + S | n - A > , [2.177d] where s = 1 for Ir states and vanishes otherwise. While none of the functions in Eq. [2.177] are eigenfunctions of the operators acting on them, it is clear that the eigenvalues of both operators must be ±1 since two applications of each one must yield the original function. Two linearly independent eigenfunctions can be constructed for each operator by taking a linear combination of the relevant basis function in Eq. [2.177] and another one identical except with the signs of its projection quantum numbers reversed; this is often called a Wang transformation. The normalised eigenfunctions for E * are | V | / ^ T I | A | ; S | I | ; J | Q | > = ^ { | q A ; S Z ; J T 2 > ± | n - A ; S - 2 ; J - Q > } , [2.178a] so that E * | V | / ± ; T 1 | A | ; S | S | ; J | 0 | > = ± ( - l )J -S+s | v ± ; T i |A | ;S |Z | ;J |n |> , [2.178b] with s defined as in Eq. [2.177]. Similarly, a Wang transformation of the electron orbital portion |qA) yields the following eigenfunctions of ov(xz): | y ; ; r , | A | > = ^ { J n A ) ± | n - A > } . [2.179a] From this definition of |\|/* ,r\ | A | ) : o v (xz) |yt ;r) | A | > = ±(- l )A + s | V ± ;r, | A | > [2.179b] The ly* ; T J | A | ; S | Z | ; J | Q | ) eigenfiinction of E* given by Eq. [2.178a] has a definite rotational parity eigenvalue of ± ( - l ) J - S + s that alternates with J for a given set of |A|, |E| and \Q\ values. A more convenient notation, originally proposed by Kopp and Hougen (40) and Brown Chapter 2 Theoretical Background 51 et al. (41), denotes a J level of rotational parity ( -1 ) J _ ° or - ( - 1 ) J _ ° with an e or / parity label respectively, where a = or 0 for states of even or odd spin multiplicity. When |Q| or |A| are nonzero, the rotational levels are doubly degenerate in the absence of interactions with other electronic states; the degenerate pairs consist of an e parity and an / parity level. For £ (A = 0) electronic states, the rotational levels of a given electron spin component all have the same elf parity, which depends in a complicated way on the spin and Kronig symmetry of the state. To be exact, e and / are merely labels rather than true parities, and the AJ = ±1 matrix elements of the hyperfine operators connect e levels with/levels. The invariance of the Hamiltonian under all symmetry operations requires conservation of elf parity for all interactions between two electronic states for which J is a good quantum number. In particular, interactions between the nondegenerate J levels of a £ state and the degenerate J levels of a n state can perturb only those n levels with the same elf parity as the corresponding £ levels. This degeneracy lifting, known as A-type doubling, occurs in degenerate states (FL, A, etc.); it is discussed in Section 2.7(b). Both | r | A ; S £ ; J Q ) and ly* ; r | |A | ;S |£ | ; J |Q | ) are considered Hund's case (a) basis functions; they are distinguished respectively as signed basis functions (since A, £ and D. all carry signs) and e/f parity or simply parity basis functions, for which only the absolute values of A, £ and Q. are significant. 2.7(b) A-type Doubling in Degenerate Electronic States Within the limit of the BOA, electronic states with A ^ 0 are orbitally degenerate, although strictly speaking, the interaction of molecular rotation and electronic motion removes this degeneracy. This effect, which introduces a splitting between energy levels differing only in elf parity, is known as A-type doubling or simply A-doubling; it originates from interactions of the |+A> and | - A ) components of the state with a comparatively distant £ (nondegenerate) state or states via the following Hamiltonian: Chapter 2 Theoretical Background 52 # i d = -2B J-L+Zailj-Sj, [2.180] i Even though this interaction occurs between different electronic states, it can be expressed equivalently as a direct coupling of the two A components of the state of interest (42), and can therefore be considered a |2A|th order effect. This effective Hamiltonian and its matrix elements have been examined for n and A states (43, 44); only the former, by far the more commonly encountered type of A-doubling, is considered here. Brown and Merer (43) have proposed the following effective Hamiltonian for the A-doubling of a n state that conforms to Hund's case (a) coupling: #id = \(P + P + q) (S+ + S 2 . ) - \(v + 2q)(J +S + + J_S_) + | q ( J 2 + J 2 ) , [2.181] where the angular momentum operators have their usual meaning (e.g., S ± = S x ± z'Sy) and are defined in molecule-fixed axes. The A-doubling parameters o, p and q are experimentally determinable only in the combinations (o + p + q), (p + 2q) and q that appear as coefficients in Eq. [2.181]; they can be derived algebraically via second order perturbation theory applied to the matrix elements of the Jf^ in Eq. [2.180]. This approach leads to complicated expressions identical to those listed by Brown and Merer (43); they will not be presented here. The formalism of Section 2.4(b), and Eq. [2.34c] in particular, can be used to evaluate the matrix elements of the effective for a n state in a signed case (a) basis set | A = ± 1 ; SS; J£2), although allowance must be made for the anomalous sign of i since molecule-fixed axes are used in Eq. [2.181]. The matrix elements are all diagonal in J and S with A A = ±2: < T l ; S Z ± 2 ; J Q | # - l d | ± 1 ; S I ; J Q > = \(o + p + q)V ± (S, 2)V±(S, 2±1), [2.182a] < T 1 ; S I ± 1 ; J Q + 1 | # - U | ± 1 ; S Z ; J Q > = - ± ( p + 2q)V ±(S, S )V T ( J ,Q) [2.182b] and <+l ;SE; JQ+2\J{LD | ± 1 ; S 2 ; J Q > = | q V+(J,Q) V+(J,Q+1), [2.182c] where V±(a,b) is given by Eq. [2.34d]. Chapter 2 Theoretical Background 53 2.7(c) Effect of A-type Doubling on Hyperfine Structure A hyperfine interaction similar to the (o + p + q) term in the A-doubling Hamiltonian is important for n electronic states. Its Hamiltonian is generally written as (26, 45, 46): #idhfs= id[exp(2z(p)LS_ + exp(-2/(p)I+S+], [2.183] where (p is the azimuthal angle about the molecular axis, and the spin ladder operators have their usual meanings. The nonvanishing matrix elements of this operator are off-diagonal in A when evaluated in a signed case (a) basis | A ; SS; J£2IF): <A±2; SE + l ; JQ±1,IF | 7/" l d h f s | A ; SE ; J£HF> _ dR(J)V ± (J ,Q)V T (S,E) 4J(J + 1) and <A±2; SE + l ; J-1,Q± 1,D717f l d h f s | A ; S E ; JQU7> [2.184a] _ _ dV(J + Q)(J + 0 - l )P(J )Q(J)V ± (S ,E) [ 2 1 g 4 b ] 4jV4J 2 - l where P(J), Q(J) and R(J) are given by Eq.- [2.158]. The determinable hyperfine doubling parameter d has a microscopic form similar to those of the other magnetic hyperfine parameters, a, b and c: d _ 3 g e g n p B p N / s i n 2 9 \ [2 185] 2 h2 \ r 3 / ' The selection rules for the A-doubling of the hyperfine structure are not exactly the same as for those of the A-doubling of the rotational structure, so that the two effects are not completely analogous; for example, only the latter appears for lU states. In fact, hyperfine A-doubling is important only for the Q = 1/2 components of even-multiplicity n states, since these are the only states for which the Wang-transformed matrix elements of 7/Jd nf s appear on the main diagonal of the Hamiltonian matrix when written in the elf parity case (a) basis. 2.8 Selection Rules and Intensities For the purposes of this thesis, a molecular spectrum can be loosely defined as a Chapter 2 Theoretical Background 54 collection of radiative transitions, or spectral lines (either absorptions or emissions of photons), between the eigenstates of the molecule; these spectral lines occur at discrete frequencies where E' and E" are the energies of respectively the upper and lower eigenstates connected by the transition, and all quantities are in c m - 1 units. The discussion so far has addressed only the determination of the energy eigenstates producing the spectrum. This section considers the theory that governs (a) whether a radiative transition can occur between two energy eigenstates (i.e., the selection rules), and (b) if so, its relative intensity. Two factors determine the relative intensity of a spectral line. One of these is the population of each level between which the transition occurs. The high-resolution data collected in the present work are- essentially absorption spectra of highly cooled molecules for which the upper level population is zero; only the lower state population need be considered here. The molecules were prepared at thermal equilibrium, so the Boltzmann population distribution Nj applies: where gj and Ej are respectively the degeneracy and energy of the jth eigenstate, k is the Boltzmann constant and T is the temperature of the molecules associated with a specific degree of freedom. The apparatus described in Section 3.2 was designed to cool all degrees of freedom in the prepared molecules; usually, only the lowest 20 or so rotational levels of the ground state zero-point vibrational level were appreciably populated. Data from excited low-lying vibrational levels and electronic states were also occasionally obtained; this is discussed in the later chapters. The other factor governing the relative intensity of a spectral line is the matrix element of the interaction operator that mediates the transition. For this thesis, the only such operator O of concern is the interaction of the electric field E of the radiation with the electric dipole v = F - E", [2.186] Nj oc g j exp[ -Ej / kT ], [2.187] Chapter 2 Theoretical Background 55 moment of the molecule: O = - M . E = -Tl( | i)-Tl(E) = - S H ^ T ^ - T i p C E ) . [2.188] P The matrix elements of this interaction must be evaluated between the two states of interest. If both states follow Hund's case (ap) coupling, they can be written in the following form: <TV;miFMF' l 0 | r|; JOIFM F> = - £ ( - i ) P T l p ( E ) <T|'; J 'Q'IF'Mp'l T^p) | r,; JDJFMF>, [2.189] P where |r|) = | u ; A S £ ) represents the vibrational and electronic parts of the total wavefunction. Neither of the states need necessarily follow Hund's case (ap) coupling, since the eigenvectors of the Hamiltonian for each state can always be expanded as a sum of such basis functions. The Wigner-Eckart Theorem can be applied to the dipole moment matrix element to remove its M F dependence: < r | ' ; m i F M y i Tpfji) | rj; JQIFM F> = ( _ 1 ) F ' - M F { F 1 F 1<TI';PO'IPIITV) II TI; jmF>. [2.190] ^ - M F p M F J The 3-j symbol yields the selection rule AF = 0, ±1, but can otherwise be ignored; its M F dependence is irrelevant since no external fields were used in this thesis. Only the reduced element of the dipole moment, hereafter called (Tl(\i)), is of concern. TA(u) must be projected back into the molecular frame with the Wigner rotation matrix: Tl(u) = X C ^ T l f u ) = S S © . ^ h " ; J"fi"IF'><ri";rn"IF»| T ^ u ) , [2.191] q q " where the double primed terms comprise the identity operator. The (D.^ operator is diagonal in r|, since it only operates on the | JO> part of the wavefunction; conversely, T q(p) operates only on | ri) and is diagonal in the remaining parameters. Thus <Tl(u)> = X<miF|| © ( Q 1 }* || JOJFXTYU TjOi) |h>, [2.192] q where ( T I ' I I Tq(n) ||rj) is called the transition moment, hereafter referred to as Rq. Chapter 2 Theoretical Background 56 Since (D.^* acts on only the first part of the J + I = F coupled basis, its matrix element can be simplified using Eq. [2.65a], and the resulting reduced element is given by Eq. [2.67]: <j,n,iF,ll©.(q1)*l|jaiF> = ^ly+I+F+l >/(2F + 1)(2F'+1) j ^ ' j j j a ' Q I I ^ I I J Q ) = (-l)2J'+I+F-0'+l ^ (2F + 1)(2F'+1)(2J + 1)(2J'+1) Jr F \\f r I r X [ F J q CI/ The spectral line strength S(J'fiTF'; JQTF) is proportional to K T 1 ^ ) ) ! 2 : S(J'QTF'; JQ.W) oc (2J+1)(2J'+1)(2F+1)(2F*+1) 1 [2.193] | J ' F ' II f T I F J I Rr [2.194] - O' q O y This expression must be weighted by the Boltzmann distribution from Eq. [2.187] to describe the intensity of an observed spectrum. The selection rules for J, Q. and F can be read immediately from Eq. [2.194] using symmetry properties of 3- and 6-j symbols: AJ = J ' - J = 0, ±1, [2.195a] AQ = n'-Q = 0 ,± l , [2.195b] and AF = F ' - F = 0 ,±1 ; [2.195c] the last of these was already obtained from Eq. [2.190]. Selection rules for the remaining good quantum numbers require consideration of the transition moment Rq. The wavefunction |r|> can be factored by the B O A into electronic and vibrational terms, of which the dipole moment operator acts on only the former: Rq= <ir||TjG0||Ti> = <e'|Tj(u)|e> <u'|u>, [2.196] where (e'| Tq(u)|e) is called the electronic transition moment and (v'\v) is the well-known Franck-Condon overlap integral (47). The only restriction on Ao = v'-v is that the resulting (U'|D) be nonvanishing. Herzberg Chapter 2 Theoretical Background 57 (14) describes sufficient conditions for this situation in diatomic molecules in terms of the change in bond length, Ar e , between the states of the transition. If an electronic band system has very small Ar e , then Au = 0 bands are by far the most intense. A band system with moderate Ar e distributes its transition moment among bands of various Au with an intensity maximum at some Au * 0. Finally, a system with very large A r e distributes its transition moment among bands over an even larger range of An, with no sharp intensity maximum. For polyatomic molecules, similar arguments relate the various geometrical parameter changes to the Arjj at which the most intense bands appear (48). Selection rules for <e'|Tq(u.) |e) are derived with group theory arguments similar to those applied in Section 2.2(b)(i) to the Renner-Teller effect. The aforementioned dipole moment operator, defined in the molecule-fixed system, acts upon only the orbital part of the electronic wavefunction and has £ + or Tl symmetry in CooV. Selection rules for S, £ and A follow immediately: AS = S ' - S = 0, [2.197a] AX = r - 2 = 0, [2.197b] and AA = A' - A = 0, ±1. [2.197c] A A = 0 transitions are said to be parallel, those with A A = ±1, perpendicular. Selection rules for the rotational parity can also be considered. The space-fixed dipole moment operator transforms as Ir (or TL) in linear molecules, so only levels of opposite rotational parity combine; this is expressed symbolically as + <—>_ ) + <-A-+ ,_<- / - •_ . [2.198] In terms of elf parity, these rules become (41) e <—• e and f<—> f for AJ = ±1 [2.199a] and e<—>/forAJ = 0. [2.199b] Chapter 2 Theoretical Background 58 2.9 Fitting of Data and the Hellmann-Feynman Theorem 2.9(a) Method of Combination Differences In fitting the spectral data to determine molecular constants, either one of two avenues can be pursued. One involves the direct fitting of observed spectral line measurements. For the present work, this method is not favoured because in most cases the upper states of the observed bands are too strongly perturbed to model by a simple Hamiltonian matrix, even though one ordinarily exists for the lower (usually ground) state. An alternative method, the use of combination differences, overcomes this difficulty. Two spectral lines with different lower levels of energies E a " and E b " from the same electronic state, but the same upper level with energy E c ' , have measurements v c a = E c ' - E a " [2.200a] and v c b = E c * - E b " . [2.200b] The combination difference A v a b = v c b - v c a = E a " - E b " [2.200c] between these line measurements depends only on the lower state (in principle, combination differences can also be formed for the upper state). A fit of only lower state combination differences overcomes the perturbative difficulties described above. Redundancies are also easily removed; all bands with the same lower state have the same set of combination differences, so duplicate measurements can be averaged before fitting. This lower (ground) state combination difference technique was the first step in the rotational analysis of most of the molecules in the present work. 2.9(b) Non-linear Iterative Least Squares Fitting via the Hellmann-Feynman Theorem If matrix diagonalisation is needed to determine the energy levels of a molecule, the least squares optimisation of the fitting parameters becomes non-linear. In this case, the calculation of the derivatives dE^/dX^ is best performed using the Hellmann-Feynman Chapter 2 Theoretical Background 59 Theorem (49, 50): [2.201] U T ^ U In this expression, J€ is the Hamiltonian matrix expressed as a function of the molecular constants X k , Ej are its eigenvalues, U is the unitary matrix of column eigenvectors that diagonalises JT, and U is the transpose of U . This theorem is useful because the Hamiltonian matrix is usually a linear function of the molecular constants even though its eigenvalues need not be: #-=XXp#p, [2.202] P where the #"p are the matrices of the Hamiltonian operators corresponding to the molecular constants X p . This linearity allows the following iterative scheme for the determination of the X p molecular constants (25). An initial estimate Xp0-* of the X p parameters is used to diagonalise the Hamiltonian and determine both U and the eigenvalues E , . From Eq. [2.201], SEj / 5 X p is now completely specified since the Hamiltonian derivatives on the right are simply #"p, from Eq. [2.202]. The residuals AE, are simply the difference between E ° b s , the energies observed from the data, and the calculated E j . The X ^ estimates can be improved by adding the approximate correction A X p given by AEi = Z D i p A X p , [2.203a] P where D i p = a E j / d X p [2.203b] are the known eigenvalue derivatives. Eq. [2.203a] can be rewritten in matrix/vector notation AE = D A X [2.204a] and solved for A X : A X = (DTBO-iDTAE. [2.204b] Al l quantities on the right side of this equation are known since they are based on the X*-0-* Chapter 2 Theoretical Background 60 estimates of the molecular constants. The correction A X thus found is added to X*-0-* for new estimates of X , and the process of determining A X is repeated until the root-mean-square (rms) of AE converges to a minimum value and successive estimates of X are essentially identical. In practice, this convergence is very rapid; usually no more than two or three iterations are required if the parameters to which the data are most sensitive are reasonably estimated. The standard deviation (AE) T (AE) [ 2 2 Q 5 ] \ N - m measures the fit quality, where N - m is number of degrees of freedom of the fit (number of data points minus number of fitted parameters). An acceptable fit has o approximately equal to the measurement uncertainty of the input data. The uncertainties in the fitted parameters are the square roots of the diagonal matrix elements of the variance-covariance matrix 0 : 0 = a2(DTT))-1. [2.206] The correlation matrix 0 Cij= , ' J , [2.207] whose elements range from -1 to +1, measures the extent to which the fitted parameters depend on one another. |Cy| values approaching unity indicate strong correlation between Xj and X J ; only their sum or some other specific combination can be measured accurately from the data. A Fortran programme performed the above non-linear least squares iterative procedure, taking as input initial estimates of the various parameters and the available spectral data with their relevant quantum number assignments. To determine the eigenvalue residuals AE correctly at each iteration of the routine, great care was required to ensure quantum numbers were properly assigned to the nearly degenerate energy eigenvalues determined by each diagonalisation; as noted in Section 2.6(b), this occasionally influenced the choice of basis set (i.e., Hund's coupling case) in which the Hamiltonian matrix elements were evaluated. 61 Chapter 3 The Electronic Spectrum of Zirconium Monocarbide, ZrC, in the 16000 -19000 cm-1 Region: Analysis of Singlet and Triplet Structure1 3.1 Introduction Diatomics, the simplest molecules, have interested spectroscopists since the dawn of quantum mechanics in the 1920s. While the last 75 or so years have seen the identification and systematic study of a large number of such molecules, initial data for many species, particularly those containing transition metals (TMs), have only appeared in the last twenty years or so. Among the reasons for this is the sheer complexity of their spectra. Their abundant unpaired d electrons cause not only their familiar catalytic behaviour, but also a high density of electronic states with large electron spin and orbital angular momenta, even for diatomics. Transitions between such states are often complicated, with many branches in each vibrational band. Some TMs have several isotopes, often with nonzero nuclear spins, further complicating their spectra with isotopic and hyperfine structure. Historically, the highly refractory nature of TMs and their compounds required high temperature sources (such as a King furnace) to prepare them in gaseous form. Under such conditions, the lowest rotational levels, which carry information about the symmetries of the states, are scarcely populated, yielding spectra dominated by mostly redundant lines of high rotational quantum number. Analysis was therefore slow and laborious. Advances in the last twenty years have simplified matters. The development of laser ablation and supersonic jet expansion methods have allowed preparation of sample molecules usually just in their electronic ground states, and at low rotational temperatures. State lA portion of this chapter has been refereed and accepted for publication: Rixon, S. J., Chowdhury, P. K . and Merer, A . J. (2004). "Nuclear Hyperfine Structure in the X 3 S + State of 9 1 Z r C " , Journal of Molecular Spectroscopy (in press). Chapter 3 Visible Spectrum of ZrC 62 assignments in these much simpler, colder spectra are relatively straightforward. The concurrent availability of high-resolution tuneable lasers has made recording of electronic spectra with resolved isotope and hyperfine structure virtually routine. Finally, modern computers have automated the data acquisition process, and permit calculation of complicated energy level patterns by numerical diagonalisation of the relevant Hamiltonian matrices. The rest of this chapter presents the analysis of the electronic spectrum of zirconium monocarbide, ZrC. Section 3.2 describes the apparatus and procedures used to acquire the spectra. This section also applies to similar studies of ZrCH and LaNH presented respectively in Chapters 4 and 5; only major differences in technique are discussed therein. The first part of Section 3.3 provides motivation for the study of ZrC and reviews some relevant literature; three further subsections give an overview of the ZrC spectrum and details of the low-lying and higher electronic states. Conclusions are given in Section 3.4. 3.2 Experimental Apparatus and Techniques 3.2(a) Overview The main component of the apparatus is a diffusion pumped vacuum chamber, in which a free jet expansion containing ZrC is prepared by reaction of laser-ablated zirconium atoms with a small amount of methane diluted in helium. A tuneable laser excites the ZrC molecules; the resulting laser induced fluorescence (LIF) is digitally recorded on a PC. Fig. 3.1 shows a schematic diagram of the apparatus. The remainder of the section gives details. 3.2(b) Preparation of ZrC in a Free Jet Expansion 3.2(b)(i) The vacuum chamber The free jet expansion occurs in a stainless steel vacuum chamber about 40 cm (length) x 40 cm (width) x 30 cm (height) in size. A rotary pump (Edwards, model E2M40) and a Roots blower (Edwards, model EH500A) provide backing for a water-cooled diffusion pump Chapter 3 Visible Spectrum of ZrC 63 delay generator 3.2(d) Tuneable bandpass (33 A or less) Vacuum Chamber 3.2(b)(1) 355 nm N d : Y A G 3.2(b)(ii) 50 psi gas line| 3.2(b)(iii) low-resolution probe laser < Av = 0.1 c m - 1 355/532 nm Pulsed laser 3.2(Q(i) N d : Y A G 3.2(c)(1) high-resolution A probe laser 308 nm XeCl 4.2 A relative calibration P D A 4.2 A 750 M H z etalon 3.2(f) A Av = 500 kHz 514.5 nm cw dye laser 3.2(c)(ii) A r + laser 3.2(c)(ii) Figure 3.1 Schematic diagram of experimental apparatus. The number given with each component identifies the section of text where it is described. Chapter 3 Visible Spectrum of ZrC 64 (Balzers, model DIF 320) mounted underneath the chamber. A pneumatic valve (Parker Hannifin Corp., Skinner Electric Valve Div., model TJ4S) opens and closes the diffusion pump to the chamber; the chamber pressure is about 2X10 - 6 torr at stagnation and about one to two orders of magnitude higher during free jet expansion. Several windows allow laser beams and fluorescence signals to enter and exit the chamber. A quartz window on one side of the chamber admits the U V ablation laser beam. The probe lasers, described in Section 3.2(c), enter through another window mounted at Brewster's angle (to minimise reflexion losses) near the ablation window; an identical exit window on the other side of the chamber assists in their alignment. The LIF signal is collected through a window on the top of the chamber. An additional large window (20 cm diameter) facing the free jet expansion allows viewing inside the chamber while it is under vacuum. 3.2(b)(ii) The ablation laser ZrC molecules were prepared by the reaction of gaseous zirconium atoms and methane in an environment of helium. The metal atoms are ablated from the surface of a Zr rod (Goodfellow, 5 mm diameter x 100 mm length, 99.8% purity) by pulses of 355 nm radiation from a N d : Y A G laser (Lumonics, model HY400) operating at 15 Hz. Second harmonic generation via a KD*P (potassium dideuterium phosphate) crystal partially converts its 1064 nm fundamental to 532 nm. A C D * A (caesium dideuterium arsenate) crystal combines these two beams via sum-frequency generation to produce a frequency-tripled 355 nm beam, which is tightly focussed by a 50 cm focal length quartz lens onto the rod surface. The ablation process requires a minimum threshold of laser power, but since higher powers contribute considerable noise to the ZrC LIF signal, the ablation power was kept just above threshold. 3.2(b)(iii) The gas handling system The helium/methane gas mixture flowed into the chamber through a 0.5 mm pulsed nozzle (General Valve Corporation, series 9). A custom-built power supply allows user control Chapter 3 Visible Spectrum of ZrC 65 over the duty cycle (about 0.2%). Soot, presumably from backflow of Z r / C H 4 reaction products, accumulates inside the nozzle after a few days of operation and degrades the ZrC signal; this necessitates periodic cleaning. Gas mixtures were prepared in a stainless steel cylindrical "bomb", 30 cm long x 10 cm diameter, connected to the nozzle via P V C tubing. Typically, the bomb was filled to 250 psi; a regulator placed beyond it maintained the nozzle at a constant pressure (usually 50 psi). The bomb was connected in parallel with the same rotary pump and Roots blower that backs the diffusion pump, allowing ready evacuation as needed; the pneumatic valve was closed at these times to prevent backflow into the reaction chamber through the parallel link. Closing a manually operated valve located between the diffusion pump and the backing pumps also helped to counter this problem. The ZrC signal was essentially constant for methane concentrations in helium in the range 1-8% (by pressure); however, to quench production of ZrCH (see Chapter 4), ZrC experiments normally used a 1% methane concentration. Some spectra were also taken with C D 4 and 1 3 CH4, in order to confirm the carriers and vibrational assignments of certain bands. 3.2(b)(iv) The "Smalley" expansion source In a "Smalley source" a gas expands adiabatically from a pressure of several atmospheres to the vacuum of the apparatus. The rotational and vibrational kinetic energy of the molecules converts almost entirely to translational energy, and they emerge as a narrow cone-shaped gas flow, with very low rotational temperature. Collisions between molecules no longer occur, effectively stabilising reactive species such as ZrC. In the present experiments a probe laser beam crosses the expansion about 5 cm downstream from the nozzle, and the resulting fluorescence is collected, again at right angles, from the centre of the cone. Fig. 3.2 shows the Smalley source used in the experiments. The pulsed gas nozzle is attached to a baseplate; gas from the nozzle enters at point C (Fig. 3.2(b)). The gas then flows Chapter 3 Visible Spectrum of ZrC 66 (a) (b) D 1 7 mm A (J c 7 cm 2 cm Figure 3.2 Top face (a) and side (b) views of the "Smalley" source located inside the vacuum chamber. The Nd: Y A G laser enters through aperture A and ablates the rotating Zr rod placed in aperture B . The resulting gas phase metal atoms react with a methane/helium gas mixture admitted through aperture C; ZrC molecules enter the chamber via aperture D. Dimensions shown are approximate. Chapter 3 Visible Spectrum of ZrC 67 through a small channel and over the Zr rod, which is mounted vertically in hole B , at the point where the ablation laser beam strikes it, having come in through the 1 mm diameter hole A. The ablation beam, the gas flow and the rod axis are mutually perpendicular. The metal atoms and methane react as the mixture flows toward point D, from which it expands into the vacuum chamber. Soot accumulation inside the source, as in the nozzle, necessitates periodic cleaning. Two factors maintain the ZrC signal steadily with time. One is a fresh rod surface. To this end, a Motor Mike (Oriel Corp., model 18040), secured to one end of the rod by a small rubber tube, effects a screwing motion of the rod. Adjustable micro-switches set the range of translation, and a custom-built power supply controls the angular speed and direction of the Motor Mike. Over time, the laser cuts a thread into the rod, which can be removed with a lathe. The other is precession-free rotation of the rod about its axis; otherwise, the focus of the ablation laser spot and the signal vary with rod rotation. To achieve this, the diameters of the rod and of the rod aperture of the Smalley source were matched as closely as possible; the rod was further secured by several turns of Teflon tape wrapped around its non-ablated portions. 3.2(c) The Two Probe Laser Systems 3.2(c)(i) The pulsed dye laser For the low- to medium-resolution wavelength-selected fluorescence (WSF) and dispersed fluorescence (DF) spectra described in Section 3.2(g), a pulsed dye laser (Lumonics, model HD500) probed the ZrC species. Two identical such lasers are available; one is pumped by the 532 nm harmonic of a N d : Y A G laser (Lumonics, FTY400) virtually identical to the ablation laser, the other by the 355 nm harmonic. The 532 and 355 nm harmonics are both vertically polarised and exit the N d : Y A G laser head from different apertures, so that switching between the two prealigned dye lasers reduces to a simple tuning of the pump laser crystals. Each pulsed dye laser has a reflexion grating (1800 or 2400 groove/mm) and tuning mirror assembly, two dye cells (an oscillator and an amplifier), a water-cooled circulator to Chapter 3 Visible Spectrum of ZrC 68 distribute dye in series to the two cells, and a wavelength scanning control touchpad. The pump laser excites the dye in the oscillator cell and induces isotropic, broadband spontaneous emission, some of which falls on the grating at grazing incidence and is dispersed toward the tuning mirror. Fluorescence at a wavelength determined by the orientation of this mirror (as specified by the touchpad) retroreflects to the oscillator cell and initiates stimulated emission, i.e., laser radiation. Rotation of the mirror scans this wavelength. The dye laser radiation, like that of the pump laser, is vertically polarised. The amplifier cell, about 40 cm beyond the oscillator, strengthens the laser signal with further N d : Y A G pumping, although the spontaneous emission from the oscillator cell is also amplified at this stage. Typically, this amplified spontaneous emission (ASE) amounts to at least 10-15% of the total output, as determined from power measurements with the reflexion grating blocked and unblocked. A transmission grating (1016 groove/mm) placed downstream mostly removes the A S E from its first order laser radiation output. A 386 computer is available to interface with either pulsed dye laser; commercial software from Stanford Research Systems (SRS) allows computer control of laser scanning and digital recording of spectra. Sample linewidths are limited to 0.1 c m - 1 by the oscillator output; power broadening from the amplifier stage further increases this linewidth. 3.2(c)(ii) The cw ring dye laser The strong bands of ZrC were studied at high resolution with a continuous wave (cw) tuneable ring dye laser (Coherent, model 899-21) pumped by an argon ion laser (Coherent, Innova Sabre). In the visible region, where the ZrC bands lie, the ring laser gain medium is an organic dye, but for work in the near IR a Ti: sapphire crystal is also available. In "dye mode", either the 488.0 nm or 514.5 nm line of the A r + laser excites fluorescence from the organic dye in a ring-shaped optical cavity. Tuning elements in the cavity select one of the cavity modes and provide frequency tuning. The concept of all these tuning elements is the same: tilting it Chapter 3 Visible Spectrum of ZrC 69 about an axis perpendicular to the optical path tunes the cavity length slightly; this in turn tunes the frequency of the selected cavity mode. The cavity modes of the ring laser are less than 300 M H z apart, but it is possible to select just one of them by the use of a three element system. A "thick etalon", consisting of two triangular prisms whose separation can be varied piezo-electrically, and whose free spectral range is about 10 GHz, transmits a "comb" of frequencies separated by 10 GHz; its finesse is high enough that only one cavity mode of the laser resonator is transmitted at each interval of 10 GHz. A "thin etalon" with a free spectral range of 220 GHz, in series with it, then restricts the transmission to single cavity modes separated by 220 GHz; the point is that a single etalon with such a free spectral range would need an extremely high finesse to distinguish individual cavity modes, which is difficult to attain with wide wavelength coverage. Finally a Lyot filter (a birefringent crystal quartz plate that transmits light at a wavelength governed by the tilt of the crystal axis relative to the axis of polarisation) with a transmission bandwidth of about 300 GHz, selects just one out of the "comb" of cavity modes separated by 220 GHz that was transmitted by the two etalons. An additional quartz plate mounted inside the cavity at Brewster's angle can be tilted to tune the cavity length very slightly and the cavity mode frequencies accordingly. Electronic adjustment of the etalon tilts maintains transmission of the changing frequency of the chosen cavity mode. The ring laser has a linewidth of about 500 kHz, but residual Doppler broadening from the cone-shaped free jet expansion limits sample linewidths to about 150 M H z (0.0050 cm - 1 ) . Since the photon flux of the ring laser is much less than that of the pulsed dye laser, only the strongest bands of ZrC were recorded with it. A pulsed dye amplifier, which increases the photon flux at the expense of a modest increase in laser linewidth, allows recording of weaker bands, but this system was not used for ZrC. It is described in Chapter 4, which reports on the study of ZrCH. Chapter 3 Visible Spectrum of ZrC 70 3.2(d) Signal Detection and Data Acquisition A spatial filter consisting of an iris placed between a pair of confocal convex lenses (each of focal length / = 5 cm) collects laser-induced fluorescence (LIF) from the target molecules at right angles to both the probe laser beam and the free jet expansion. Optimising the height of this filter above the centre of the free jet expansion and its iris diameter reduces the spectral linewidths from 250 to 150 MHz. Various attempts to enhance the LIF signal were made, either by reflecting the probe laser beam back through the chamber or by placing a concave mirror underneath the jet expansion to collect light travelling in the opposite direction; these gave only modest improvements at the expense of considerable trouble and were therefore abandoned. The LIF signal from the spatial filter passes through a window at the top of the chamber, is steered horizontally by a 45° mirror and focussed onto the entrance slit of a 0.75 m monochromator (Spex, model 1702). With its 1200 groove/mm grating, the monochromator has a first order dispersion of 1.1 nm/mm; with fully open slits (3 mm), this corresponds to a band pass of 40 c m - 1 at 900 nm or 120 c m - 1 at 500 nm, the shortest wavelength used in the ZrC experiments. An uv-blocking optical filter (Corning 3-73) removes stray ablation laser radiation (355 nm) that would otherwise be transmitted in second order at 710 nm. The LIF signal is collected by a Hamamatsu, model R943 photomultiplier tube (PMT), housed in a Pelletier cooling unit (Products For Research, model TE104RF), and held at between 900 and 1600 V by a Keithley, model 237 power supply depending on the strength of the detected LIF signal. The current from the PMT is amplified by a custom-built unit (gain = 880) and sent to a boxcar integrator (SRS, model SR250). Typically, the signals from thirty ablation shots are averaged using a gate width of approximately 1 ps for low-resolution experiments; this is increased to 5 ps for high-resolution experiments. A digital oscilloscope (Tektronix, TDS 340A) connected in parallel with the boxcar Chapter 3 Visible Spectrum of ZrC 71 integrator displays the intensity and timing of the LIF signal relative to the integration gate. This is important for optimising the boxcar settings and for maximising the LIF signal strength. The integrated voltage is sent to a display module (SRS, model SR280) and then to a PC via a computer interface (SRS, model SR245). A digital delay generator (SRS, model DG535) controls the event timing. It delivers trigger pulses at 15 Hz to the gas nozzle, to the ablation laser and, when in use, to the pulsed probe laser. Typical delay settings for the various pulses are as follows: the reference pulse triggers the nozzle, the ablation laser is triggered 200 ps later and the pulsed probe laser is triggered 35 ps before the ablation laser. This implies that the probe laser pulse arrives in the chamber before the free jet expansion, but the timings specified by the delay generator are not absolute since they do not account for non-electronic factors such as the mechanical delay in the opening of the gas nozzle and the time taken for the gas to travel through the apparatus. 3.2(e) Description of the Experiments The scanning capabilities of the probe lasers and the monochromator allow two types of experiments described below: wavelength-selected fluorescence and dispersed fluorescence. 3.2(e)(i) Wavelength-selected fluorescence (WSF) In this thesis, the term wavelength-selected fluorescence (WSF) describes the detection of laser-induced fluorescence (LIF) at only one wavelength (chosen with the monochromator) that is offset from that of the probe laser by an amount corresponding to a known (usually vibrational) interval of molecular energy. Thus, as the probe laser scans, the monochromator also scans to maintain this offset. This detection technique has two benefits: it suppresses scattered probe laser light, and since the energy interval is unique to the carrier molecules (notwithstanding coincidental near-degeneracies), the monochromator simultaneously acts as a molecule-selective filter. This latter point is important because impurity molecules are often present in the free jet expansion and can produce their own L U signals; the WSF spectrum Chapter 3 Visible Spectrum of ZrC 72 becomes contaminated if these are not separated from the target signal. Moreover, this technique allows concurrent recording of spectra from whatever isotopomers of the target molecule are abundant in the free jet expansion, provided the isotopic variation of a characteristic energy interval is smaller than the bandpass of the monochromator (which is usually the case for the type of molecules studied in this thesis). This is useful because isotopic data can provide checks on the validity of the analysis. Clearly, WSF experiments on a new molecule of unknown vibrational frequencies are not always trivial to undertake; they must be done using an estimated frequency or with the monochromator tracking the laser frequency itself. This produces a noisy spectrum containing resonance fluorescence from all the different molecules present. If the spectra of the other likely species present are known, strong bands of a new molecule can often be recognised without much difficulty; its characteristic frequencies can then be determined by dispersed fluorescence, as described below. However, it is entirely possible to generate simultaneously new spectra from two previously uncharacterised molecules. The two molecules discussed in this chapter and the next one, ZrC and ZrCH, present a good example of this more challenging scenario (see, for example, Fig. 4.6). The monochromator wavelength can be fixed at the target LIF frequency during a high-resolution cw ring laser scans since the 1 c m - 1 maximum range covered is much less than the 40-120 c m - 1 bandpass of the monochromator (vide supra); it need only be updated at 5 to 10 c m - 1 intervals of the cw laser frequency. In pulsed laser work, however, survey WSF scans typically cover 8 nm, far exceeding the monochromator bandpass, so that the monochromator wavelength must be scanned to preserve molecule selectivity. Although the probe laser can scan linearly in wavenumber or wavelength, the monochromator scans linearly in wavelength only. Only linear wavelength scanning of both pieces of apparatus is practical. Clearly, equal scan rates for the laser wavelength, X L , and the monochromator Chapter 3 Visible Spectrum of ZrC 73 wavelength, XM, cannot maintain a constant frequency difference between them over an entire scan; exact coincidence can occur only once in this case. As an example, the detection of ZrC in the 580-588 nm range, monitoring LIF at the X 3 E + , o = 0-1 interval of 880 cm" 1 below the laser, may be considered. Initially the monochromator is set 880 c m - 1 to the red of 580 nm, i.e., at 611.20 nm (ignoring the vacuum correction for discussion purposes). When the laser has reached 588 nm, the monochromator, scanning at the same wavelength rate, will have reached 619.20 nm, offset 856.9 c m - 1 in frequency from 588 nm. Since the v 3 vibration of ZrCH lies at 863 c m - 1 , the monochromator now detects ZrCH rather than ZrC. This effect can be reduced by scanning the monochromator so that the desired 880 c m - 1 offset occurs at the centre of the scan, but another impurity species endangers the molecule-specificity in this case. ZrO is especially troublesome in ZrC work because its extremely strong triplet manifold spectrum lies in the same wavelength region; "hot" bands with large v" readily appear under jet-cooled conditions, even without an added oxygen precursor. Although the first vibrational interval, AGy2, of its low-lying a 3 A state is 932 c m - 1 (1), the intervals decrease with vibration such that the v = 3-4 interval is only 912 c m - 1 . ZrO impurity bands are often seen in spectra recorded in this fashion. A refinement is possible, however, if the laser and monochromator are scanned at slightly different rates. While the monochromator scans only at certain fixed rates, s^, the laser scan rate, sL, is arbitrary and may be chosen to produce equal frequency offsets at the beginning and end of a scan. For the above example, the maximum variation in monochromator offset is less than 1 c m - 1 . Straightforward algebra gives the required S L . If the laser scan range is X - L to XL+AA, (in nm) and the monochromator offset by Av (in cm - 1 ) , then S L and Sjyj (both in nm/s), are related by [3.1] s L = 1 0 _ 7SM AX 10' XL + AX -1 - Av ' l O 7 Av Chapter 3 Visible Spectrum of ZrC 74 For the example chosen, if s^ = 0.02 nm/s, SL = 0.01799 nm/s to four significant figures. The small deviation in Av over the course of the scan, never more than 1 c m - 1 from the desired 880 c m - 1 value, allows for a much narrower slitwidth (say, 1 mm) and the removal of almost all impurity features. WSF scans so recorded take about 10% longer than normal to complete but the ability to maintain Av within a very small tolerance greatly outweighs this disadvantage. 3.2(e)(ii) Dispersed fluorescence (DF) In a dispersed fluorescence (DF) experiment, the pulsed probe laser is tuned to a known absorption wavelength A,L of the molecule, and the monochromator wavelength XM is scanned. The resulting spectrum displays LIF intensity as a function of A,M . At the outset of a DF scan, the lower state of the absorption band is generally unknown, so the start value of Xjyj is chosen to allow for LIF features that may lie up to 1000 c m - 1 to the blue of X.L. Such features may appear for a "hot" absorption band, i.e., one whose lower state is not the ground vibrational level, as emission to lower-lying levels. Scans typically ranged over 200 nm at 0.2 nm/s with averaging of LIF signals from thirty ablation pulses. DF spectra represent state-selected emission spectra from the upper levels of the various absorption bands. Since the relative intensity patterns follow the Franck-Condon principle, they potentially indicate the vibrational assignments of the upper states, although these are not always easy to extract. The energy displacements of the peaks from the laser wavelength give the positions of the low-lying vibrational and electronic levels of the molecule. Obviously the narrower the monochromator slit, the narrower the DF features, so that DF spectra should be taken with the narrowest slits compatible with intensity considerations. Typically a 1 mm slit is used, though for very strong bands this can be reduced. The DF spectra cannot be easily calibrated with external wavelength standards; instead this is done approximately by linear interpolation between the start and end wavelengths of the Chapter 3 Visible Spectrum of ZrC 75 scan, using the position of the very intense Rayleigh scattering peak at X L t 0 allow for any possible offset. Accurate measurement of this peak requires care that it not saturate the PMT; the vertically polarised pulsed dye laser output is easily attenuated with a polarisation filter. While XM is within 4 nm of X L , the filter is placed in the laser beam with its transmission axis tilted sufficiently from the vertical to produce a sharp unsaturated Rayleigh peak. Occasionally in the present work, no such attenuation was required, particularly in the red region of the visible, in accord with the well-known v 4 proportionality of Rayleigh scattering intensity. The absolute error in this calibration is quite small for low-lying vibrational levels, where it can be checked from the positions of hot bands measured to cw laser accuracy, but rises with vibrational energy because it is not easy to synchronise the scanning of the monochromator to the data acquisition exactly. At 3000 c m - 1 the uncertainty is probably ±10 cm - 1 . Nevertheless the reproducibility of the measurements is generally better than this, with the limiting factor being the widths of the LIF features. 3.2(f) Calibration of W S F Spectra For both low- and high-resolution WSF spectra, calibration against secondary wavelength standards is provided by atomic spectra recorded optogalvanically. A low reflectivity beamsplitter directs a small portion of the probe laser beam to a uranium hollow cathode lamp filled with either argon or neon (Cathodeon, model 3UAXU). For low-resolution experiments, the U/Ar lamp is used and generally only the Ar lines are of sufficient strength to record. A custom-built amplifier (gain = 1000) collects the optogalvanic signal from the hollow cathode lamp and delivers it to a boxcar integrator identical to the one described previously. Air wavelengths of argon with an uncertainty of about ±0.005 nm are taken from Norlen (2). For high-resolution experiments, either the U/Ar or U/Ne lamp is used, but only the uranium lines, which are much narrower than the inert gas lines, are used in the calibration Chapter 3 Visible Spectrum of ZrC 76 process. A chopper and lock-in amplifier (Princeton Applied Research, model 128, or SRS, model SR510) are required to observe the uranium lines; the reference signal is detected by a photodiode (Pomona Electronics, model 2397) from a cw diode laser. Wavenumbers of the uranium lines, quoted to nine significant figures, are available from the Fourier transform measurements of Palmer et al. (3). Further calibration is recorded by directing a small portion of the ring laser output through an I 2 cell. A PMT, powered by a voltage supply (Fluke, model 4313) of typically about 650 V , detects fluorescence. The resulting current flows to a custom-built amplifier and is digitally recorded. Since the relative accuracy of the I 2 wavenumbers listed by Gerstenkorn and Luc (4) is only ±0.002 cm - 1 , this spectrum is used only to help establish proper overlap of consecutive cw laser scans. The uranium lines used for calibrating high-resolution WSF spectra are typically spaced about 5 c m - 1 apart, while the molecular lines are generally far more closely spaced. To interpolate between the uranium lines, a small portion of the cw ring laser output is sent through a confocal Fabry-Perot etalon (Burleigh, model CF100) that is locked to a frequency stabilized He-Ne laser (Spectra Physics model 117A). An evacuated, temperature-controlled housing isolates the etalon from variations in laboratory pressure and temperature, allowing for very trouble-free locking of the etalon to the laser. The free spectral range of the etalon is always in the range 0.024973 ± 0.000001 c m - 1 . Transmission maxima from the ring laser are detected by a photodiode (Pomona Electronics, model 2397) and recorded simultaneously with the other calibration spectra. This etalon system was developed by the Merer group in the late 1980s; a paper by Adam et al. (5) offers further details on its operation. In all, four spectra are simultaneously recorded during high-resolution WSF experiments: (a) the sample spectrum, (b) the I 2 spectrum which, together with readings from a Burleigh WA20-VIS wavemeter, ensures proper concatenation of adjacent scans, (c) the Chapter 3 Visible Spectrum of ZrC 11 optogalvanic spectrum of the U/Ar or U/Ne hollow cathode lamp and (d) the spectrum of interpolation markers from the etalon. Using the latter two, calibration of sample spectra proceeds in two steps. In the first step, each etalon marker is assigned a relative order number n and the measurement of each uranium line in units of n is determined. The wavenumber v of each uranium line is then fitted by least squares to a polynomial in n; a simple linear fit with the coefficient of n given by the free spectral range of the etalon usually suffices. In the second step, each spectral line is measured in units of n and its wavenumber is determined from the polynomial. In this manner, strong unblended spectral lines over some 250 c m - 1 can be measured to nine significant figures with a relative precision of about 10 MHz. Care must be taken with the line shapes of the uranium calibration lines; incorrect settings on the lock-in amplifier can lead to unsymmetrical profiles and absolute calibration shifts. 3.3 The Visible Spectrum of Zirconium Monocarbide, ZrC 3.3(a) Motivation As discussed in Chapter 1, TM-containing molecules often appear as atmospheric constituents of cooler stars containing recycled supernova material. Well-known examples include ZrO (6, 7), TiO (7, 8), V O (7,8), ZrS (7, 9) and FeH (9). The strength of ZrO and ZrS features in the spectra of cool stars makes ZrC a plausible atmospheric constituent of carbon-rich stars and thus worthy of spectroscopic study. Further motivation stems from the 1992 discovery, via mass spectrometry, of the exceptionally stable M 8 C 1 2 family of metallo-carbohedrenes, or "metcars", for M = Ti, Zr, V and H f (10, 11). Fig. 3.3(a) shows the cage structure originally proposed to account for their stability (10). Each equivalent M atom lies at one of eight cube vertices; the twelve C atoms lie in pairs atop each of the six cube faces. Later ab initio (12-15) and experimental (16, 17) work (a) (b) Figure 3.3 Proposed structures of the metallo-carbohedrene M g C 1 2 (C atoms, open circles; M atoms, filled circles). The cube-based (broken lines) structure (a) was first proposed to account for its stability; structure (b) is currently more accepted. In (b) the inner metal atoms M 1 form a tetrahedron (heavy solid lines) that is capped (broken lines) by the four outer metal atoms M°; each M° is bridged to each of the others (light solid lines) by a C 2 dimer. Chapter 3 Visible Spectrum of ZrC 79 suggested the alternative structure of Fig. 3.3(b), for which only four metal atoms of type M 1 or M° are equivalent. The M ' form a regular tetrahedron (heavy lines), with an M° capping each of its faces to form four additional tetrahedra with r ^ i - M 1 ) ^ r(Mi-M°); the M° are bridged by C 2 dimers. An understanding of the properties of these metcars requires some knowledge of the TM-carbon bond, which gas phase spectroscopic studies of the smaller diatomic metal carbide species can offer. The literature review below shows the scarcity of such data before the discovery of metcars; for ZrC in particular, the void remained until the present work. This lack of prior data, especially for its ground state, makes ZrC interesting in its own right. Spectroscopic analyses of a wide range of T M monocarbides (18-54), reviewed below, suggest that TMs bond to carbon quite differently than to either nitrogen or oxygen. However, the behaviour of zirconium in this regard was unknown, since ZrC had never been studied. 3.3(b) Review of Relevant Literature Although there are no previous studies on ZrC, either experimental or theoretical, other T M monocarbides have received considerable attention (18-55). Three experimental papers on the isoelectronic Y N molecule have also appeared (56-58). This review describes just the more detailed experimental work, with only occasional reference to ab initio calculations. References to much of the theoretical research are given in the experimental papers. Scullman and co-workers were the first to study spectra of T M monocarbides; they examined electronic data on PtC (18-21), RhC (22-24), IrC (25, 26) and RuC (27, 28) in the mid-1960s and early 1970s. The samples were prepared at high rotational temperatures in a King furnace, which impeded definitive symmetry assignments in some of the spectra; later work with jet-cooled sources resolved some of these difficulties (vide infra). No further spectroscopic studies of T M monocarbides were undertaken until after the discovery of metcars; in 1994, Simard et al. (29) reported the first spectrum of Y C . A ground state of 4n, was assigned, from the electron configuration (10a)2(llo)1(12a)1(57c)3, in accord Chapter 3 Visible Spectrum of ZrC 80 with earlier ab initio calculations of Shim et al. (55). This result contrasted with the (10o)2(57t)4(lla)1 2 E + ground state symmetry that would be expected for a T M monoxide with the same number of electrons (59), showing clearly the profound effect of the different ionisation potentials (IPs) of carbon and oxygen on the nature of the bonding. Such considerations have stimulated a great deal of subsequent research on other T M monocarbides. Many papers, both extending the analyses of the Scullman group and describing new molecules, have appeared in rapid succession since the Y C study (30-54). The Morse group at the University of Utah has been at the forefront of these advances, applying laser ablation and jet-cooling techniques to prepare samples of FeC (30), MoC (31, 32), RuC (32, 33), PdC (32, 34), WC (35) and NiC (36). These six species were investigated primarily via resonant two-photon ionisation (R2PI); as the name implies, the molecule of interest is ionised by two sequential single photon absorptions. A tuneable dye laser supplies the first photon to promote the molecule to an excited state; the second photon, from a fixed wavelength laser, promptly ionises it from the intermediate state. As the first laser scans, a mass spectrometer detects ions produced by the second laser; the resulting signal mimics the absorption spectrum of the neutral species corresponding to the first photon. Dispersed fluorescence spectra similar to those described in Section 3.2(e)(ii) above were also recorded for MoC, PdC and RuC (32), providing rich insights into their low-lying vibrational and electronic structures. While Scullman could not determine any symmetries for the electronic states of RuC (27, 28), R2PI studies of supersonically cooled samples of this species by the Morse group showed unambiguously that its ground state is A £ + (33). Further unpublished R2PI data by the Morse group on V C , CrC and TiC are mentioned in their work on FeC (30); the latter molecule was first studied by Balfour et al. (37). Several recent studies of FeC in the microwave, infrared and visible regions by various workers (38-43) have made it the most widely Chapter 3 Visible Spectrum of ZrC 81 investigated of all T M monocarbides. Other groups have also studied T M carbides. The Ziurys group at the University of Arizona has recorded pure rotational spectra in the mm-wave region for NiC and CoC (44) as well as FeC (38, 43); for these studies the molecules were prepared in a high-temperature Broida-type oven by reacting the relevant metal vapour with methane. The Steimle group at Arizona State University has recorded both Stark and field-free high-resolution electronic spectra to provide, for the first time, electric dipole moment measurements in various electronic states of FeC (42), IrC (45), RuC (46, 47) and PtC (48, 49), as well as hyperfine analyses of these species, where appropriate (45, 46, 49, 50). Additional data on other T M monocarbides have also appeared since 1994. Two LIF studies of CoC have been published, the first by Barnes et al. (51) in the 12800 - 14700 c m - 1 region and the second by Adam and Peers (52) in the 14000 - 19100 c m - 1 region. A report by Simard et al. (53) on the electronic spectrum of NbC represents the only gas phase work on this species, while Balfour et al. (54) have reinvestigated RhC using jet-cooled LIF spectroscopy; they revised the symmetries of some of the states observed by the Scullman group (22-24) and identified some further states. The above discussion identifies all the gas phase spectroscopic studies of T M monocarbides published to date; with the current study of ZrC, AgC is now the only non-radioactive 4d T M monocarbide for which no gas phase spectrum has been reported. As expected from the results on Y C (vide supra), it has emerged that the ground states of the T M monocarbides are not necessarily the same as those of the isoelectronic T M mononitrides; Table 3.1 lists several examples. The differences follow from the difference in IPs of nitrogen (14.5 V) and carbon (11.3 V). Al l the metals in the molecules of Table 3.1 have IPs in the range 6.5 to 8.0 V, so that the valence atomic orbitals of carbon are more nearly degenerate with those of the metal. This rearranges the energy order of the valence molecular orbitals Chapter 3 Visible Spectrum of ZrC 82 (MOs) compared to the corresponding mononitrides and alters the most energetically favourable electron configuration and ground state symmetry of the molecule. Table 3.1 Comparison of ground state (GS) symmetries of some transition metal-containing monocarbides and their isoelectronic mononitrides Monocarbide GS Reference(s) Isoelectronic G S Reference NbC 2 A r 53 ZrN 2X+ 60 MoC 3 Z - 31 NbN 3Ar 61 CoC 2Z+ 51,52 FeN 2 A ; - 62 ZrC 3 £+ present work Y N lZ+ 56 The molecule described in this chapter, ZrC, is the latest such example: whereas prior work on isoelectronic Y N (56-58) establishes a 1 S + ground state, this state is found to lie 188 c m - 1 above the X 3 E + state in ZrC. The latter is shown to arise from the configuration (10a)2(57t)4(lla)1(12a)1; adapting the proposed M O scheme of ZrC to Y N , its ground state comes from the closed shell configuration (10a)2(57i)4(l l a ) 2 . 3.3(c) Description of the Spectrum Survey wavelength-selected fluorescence (WSF) spectra of ZrC have been recorded over the 16000-19000 c m - 1 region using normal isotopic methane (99% 1 2 C H 4 ) . These were recorded by monitoring fluorescence at four intervals of energy below the probe laser. Two of them, the v = 0-1 and 0-2 vibrational intervals of the X 3 S + ground state, are about 885 and 1755 c m - 1 . The al2Z+ state, which lies just 188 c m - 1 above X 3 S + , has very nearly the same vibrational intervals; this state was assigned from dispersed fluorescence and high-resolution WSF data discussed later in this Chapter. Since both states are populated in the free jet expansion, monitoring one of these two vibrational intervals generates the sum of the absorption spectra of the two spin manifolds, and it is not initially obvious at low resolution Chapter 3 Visible Spectrum of ZrC 82 (MOs) compared to the corresponding mononitrides and alters the most energetically favourable electron configuration and ground state symmetry of the molecule. Table 3.1 Comparison of ground state (GS) symmetries of some transition-metal containing monocarbides and their isoelectronic mononitrides Monocarbide GS Reference(s) Isoelectronic Reference v ' monomtnde NbC 2 \ 53 ZrN 2 E + 60 MoC 3 Z - 31 NbN 3 A r 61 CoC 2 Z + 51, 52 FeN 2 A ; - 62 ZrC 3 E + present work Y N 12+ 56 -The molecule described in this chapter, ZrC, is the latest such example: whereas prior work on isoelectronic Y N (56-58) establishes a 1 Z + ground state, this state is found to lie 188 c m - 1 above the X 3 E + state in ZrC. The latter is shown to arise from the configuration (10o)2(57i)4(lla)1(12a)1; adapting the proposed M O scheme of ZrC to Y N , its ground state comes from the closed shell configuration (10o)2(57c)4(l l a ) 2 . 3.3(c) Description of the Spectrum Survey wavelength-selected fluorescence (WSF) spectra of ZrC have been recorded over the 16000-19000 c m - 1 region using normal isotopic methane (99% 1 2 C H 4 ) . These were recorded by monitoring fluorescence at four intervals of energy below the probe laser. Two of them, the v = 0-1 and 0-2 vibrational intervals of the X 3 E + ground state, are about 885 and 1755 c m - 1 . The a.lIf state, which lies just 188 c m - 1 above X 3 E + , has very nearly the same vibrational intervals; this state was assigned from dispersed fluorescence and high-resolution WSF data discussed later in this Chapter. Since both states are populated in the free jet expansion, monitoring one of these two vibrational intervals generates the sum of the absorption spectra of the two spin manifolds, and it is not initially obvious at low resolution Chapter 3 Visible Spectrum of ZrC 83 from which lower state a given band originates. To identify spin-contaminated excited states that connect to both X 3 E + and a 1 ^ , WSF scans monitoring two other intervals were also recorded. Levels that connect to the v = 0 levels of both states were identified by maintaining the monochromator 188 c m - 1 below the laser, while those that connect to X 3 Z + , VJ = 1 and alI+, v = 0 were monitored by maintaining the monochromator 692 c m - 1 below the laser. For these scans, the detection scheme unambiguously defines the lower state symmetry of any observed band. The resolution in the spectra generated from these four scans is about 15 cm - 1 , so neither rotational structure, nor structure from the naturally occurring isotopes of Zr ( 9 0 Zr, 51.45%; 9 1 Z r , 11.22%; 9 2 Z r , 17.15%; 9 4 Z r , 17.38% and 9 6 Z r , 2.80%), could be discerned; about 40 well-separated band heads appear. Dispersed fluorescence (DF) spectra have been collected for nearly all of them. A limited supply of 1 3 C H 4 was available to record a few key features of Z r 1 3 C at low resolution. Since the ZrC molecules were jet-cooled, only upper state vibrational progressions should appear in the WSF spectra, but these are not obvious at low resolution. Ffigh-resolution data, recorded for almost every band in the survey spectrum, show that many features in the survey data that are apparently single actually contain structure from two or more close-lying vibrational levels. In all, 56 bands have been identified, although few unambiguous D' assignments were possible. Fig. 3.4 gives a stick spectrum of the positions and relative intensities of these bands for the most abundant isotopomer, 9 0 Z r C ; corresponding bands of 9 2 Z r C and 9 4 Z r C are observed for all but the weakest of these. The bands have rotational envelopes that span typically about 50 c m - 1 and often have considerable overlap. The right inset of Fig. 3.4, near 16650 c m - 1 , shows nine bands that lie within 90 c m - 1 of one another. The strongly perturbed rotational structure in this region is discussed in Section 3.3(e)(ii). 19000 18500 18000 17500 laser wavenumber / cm" 17000 1 .90 16500 16000 Figure 3.4 Stick spectrum of ZrC WSF bands. The plot indicates the location of the band head and its intensity as measured from its 3 + • 1 + 3 + maximal fluorescence wavelength. Bands arising from X I , o = 0, a S , u = 0 and X E , u = 1 (marked by asterisks) appear in the spectrum. The band density is high in some areas, with eight or nine bands occasionally appearing within 100 cm - 1 , as shown in the insets (both have the same intensity scale). oo 4^ Chapter 3 Visible Spectrum of ZrC 85 A few isolated features of 9 6 Z r C were identified in the strongest bands, but no detailed assignments were attempted. While 9 1 Z r C might seem sufficiently abundant for its lines to be assigned easily, the spinning 9 1 Z r nucleus (I = 5/2) distributes the rotational linestrength among six or more hyperfine components, each about as strong as a 9 6 Z r C line. Nevertheless, some 9 1 Z r hyperfine structure could be assigned in a few of the strongest bands, allowing a hyperfine analysis of the ground state. These results are presented in Section 3.3(d)(iii). The high resolution data show that the vast majority of the absorption bands come from the same 3 Z lower level, which is assumed to be the ground state v = 0 level because of the jet cooling; such bands are considered "cold". One of these appears near 16306 c m - 1 , a portion of which is shown in Fig. 3.5. The various branches in this band can be classified by their value of J '-N", which ranges from +2 ("S-form") to -2 ("O-form"); there are nine of them: one S-, two R-, three Q-, two P- and one O-form. Subscripts of 1, 2 or 3 indicate whether the branch involves the F ^ J = N+l), F 2 (J = N) or F 3 ( J = N - l ) electron spin component of the 3 Z state. The first line of each branch has J' = 2; the structure is that of a 3 n 2 - 3 S transition. Rotational lines from the 9 0 Z r C , 9 2 Z r C and 9 4 Z r C isotopomers have been assigned in this band; only the 9 0 Z r C features have been labelled for clarity. Five weak "hot" bands from the X 3 Z , v = 1 level have also been observed (for 9 0 Z r C only). The vibrational assignment is clear because each of these has the same upper state combination differences as another band from X 3 E , v = 0 that appears about 880 c m - 1 above it; the rotational analyses of these two levels, discussed in Section 3.3(d)(ii), are consistent with their vibrational assignments. Further excitation bands in the high-resolution spectrum originate from a single vibrational level of the aforementioned aAS state. Fig. 3.6 shows a typical example of these bands. The rotational structure of this band is much simpler than the previous one in Fig. 3.5. Only three branches, labelled R, Q and P, appear; respectively, these have AJ = +1, 0 and -1 , Q 2 (N) Ql(N) 2 Q 3 (N) 2 R 2 (N) Ri(N) 13 3 + spin splitting of N=5 in X Z 14 15 9 16305.5 16305.0 16304.5 laser wavenumber / cm -1 16304.0 16303.5 3 3 + 90 Figure 3.5 A portion of the [16.2] Yl2 - X I -(°>0) band of ZrC. Lines from five of the nine branches are labelled for ZrC; unlabelled lines are from 9 2 Z r C and 9 4 Z r C . The Q-form branches connect to all three spin components of the ground state, as shown for N = 5. 00 R(J) 4 5 Q(J) 8 P(J) 2 4 10 12i 4^  ikJ ™lilnrr | TO* s a. o TO *P TO TO 17488 17486 17484 17482 -1 17480 17478 laser wavenumber / cm 1 + - 9 0 92 Figure 3.6 High-resolution spectrum of the [17.67]Q=1 - a £ band of ZrC. For clarity, only lines of ZrC are labelled. The ZrC and 9 4 Z r C R heads lie near 17484.47 and 17481.07 c m - 1 . Unlabelled lines above the 9 2 Z r C band head originate from the much weaker [17.69]Q=1-a 1Z +band. 00 Chapter 3 Visible Spectrum of ZrC 88 and first lines with J" = 0, 1 and 2, indicating an Q = 1 - 0 (or LIJ - iE) transition. Once again, only lines from the most abundant 9 0 Z r C isotopomer are labelled for clarity. Most of the bands of the type shown in Fig. 3.6 have upper levels that also connect to X 3 £ , v = 0; the resulting bands appear about 188 c m - 1 to the blue. Two important conclusions follow from the rotational analyses of bands that connect to these two lower levels. One is that the X 3 Z and al2Z states have the same Kronig symmetry. This is apparent by observing, for example, that the R 2 (N) - P 2 (N) upper state combination differences of the "triplet" bands match the R(J) - P(J) differences of the "singlet" bands. From the proposed molecular orbital scheme discussed later in the Chapter, this symmetry is positive. The second conclusion is that the a^ 4 " level observed in absorption has u = 0, which follows from its very small observed A E ( 9 0 Z r C - 9 4 Z r C ) = 0.068 cm" 1 isotope shift (referred to the X 3 E+, v = 0 level). Since the vibrational frequencies of the X 3 Z + and &l2Z+ states are both about 900 cm - 1 , this shift would be at least 2 c m - 1 in magnitude for an excited vibrational level. The vibrational assignment is further substantiated by the observed carbon isotope shift of the X 3 S + , v = 0 - alIf, u = 0 interval observed in dispersed fluorescence. An example is shown in Fig. 3.7. Trace (a) in this figure shows a low-resolution WSF spectrum of Z r 1 2 C , recorded by monitoring fluorescence 188 c m - 1 below the probe laser. Ffigh-resolution data of the 17908 c m - 1 band (marked by. a heavy vertical line) show that it is excited from X 3 S + , v = 0. The DF spectrum observed from its upper state appears in trace (b); three features appear below the laser peak. The corresponding data for Z r 1 3 C are shown in traces (c) and (d). The WSF band shifts 26 c m - 1 to the red, to 17882 cm - 1 . In the DF spectrum, the 188 c m - 1 interval is essentially unchanged, which indicates that the two peaks have the same u; i.e., v = 0 for both peaks as shown in the figure. If a vibrational frequency of 880 c m - 1 is assumed, the isotope shift for i> = 1 predicted from the scaling factor p = [ p ( 9 0 Z r 1 2 C ) / p ( 9 0 Z r 1 3 C ) ] 1 / 2 = 0.9654 (to T i i i i | i i i i | i i i i 1 1 1 r 1 1 1 1 1 1 1 r 17900 17850 17800 0 -500 -1000 laser wavenumber / c m - 1 displacement from laser / c m - 1 12 13 —1 12 Figure 3.7 Effect of C substitution on the 17908 cm WSF band of Zr C and on the DF of its upper state. Laser pumping of 1 the band marked by a vertical line in WSF trace (a) gives the DF trace (b). Traces (c) and (d) are the corresponding Zr C data. For both WSF scans, LIF to the a1!4", u = 0 level lying 188 c m - 1 above X 3 I + , o = 0 in both species was monitored. The isotope shifts of the various levels in traces (b) and (d) confirm the vibrational assignments in trace (b). Chapter 3 Visible Spectrum of ZrC 90 four significant figures, using the most abundant of the five naturally occurring isotopes of zirconium) is about -30 c m - 1 . This is in excellent agreement with the observed -29 c m - 1 shifts of the other two peaks. The different relative intensities of these three peaks, for Z r 1 2 C and Z r 1 3 C , reflect the isotopic dependence of the spin mixing in the excited triplet states of the absorption bands; the singlet peaks are relatively weak for Z r 1 2 C , while all three peaks are about equally intense for Z r 1 3 C . At high resolution, the rotational structure of the Z r 1 2 C absorption band shows strong absorption to a nominally 3 n 2 upper state level accompanied by weaker absorption to a lU level less than 5 c m - 1 higher, induced by weak mixing with the 3 n 2 level. This mixing also induces weak singlet DF features as the 3 n 2 level is pumped. No high-resolution data were recorded for the corresponding absorption band of Z r 1 3 C , but its 3 TI 2 and lIl levels must be essentially degenerate and thoroughly mixed, since the singlet and triplet features in the DF spectrum are equally intense. Vibrational features from both spin manifolds are observed in other DF spectra of both Z r 1 2 C and Z r 1 3 C , indicating a strongly mixed set of excited states, consistent with the confused appearance of the WSF spectrum and a very complicated electronic structure. It is clear from the DF data in Fig. 3.7 that the 17908 (17882) cm- 1 level of Z r 1 2 C (Zr 1 3 C) can be populated by excitation from a 1 2 + , u = 0. The DF spectra generated in this fashion are shown in Fig. 3.8. Aside from the laser scatter peak, the appearance of a DF spectrum is independent of the upper state population mechanism, so these spectra are the same as those in Fig. 3.7. The electronic interval is preserved and the shifts and relative intensities of the v = 1 peaks are essentially the same. The sharp features at the red ends of the WSF spectra in parts (a) and (c) of Fig 3.7 are worth mentioning. These are the a 3 P 0 —» z 3 S f and a 3 P 2 —» z 3 S ° atomic zirconium lines, lying at 17788.1 and 17777.3 c m - 1 respectively (63); they appear in the WSF spectra because of a Chapter 3 Visible Spectrum of ZrC 91 u = 0 u = 0 1 1 1 X 3 E + Z r 1 2 C 0 -500 -1000 displacement from laser / c m - 1 Figure 3.8 DF from (a) the 17908 cm 1 level of Z r 1 2 C and (b) its Z r 1 3 C counterpart. These levels are the same as those in Fig. 3.11(b) and (d) respectively; the only difference is that here, 1 + 3 + they are populated by absorption from a Z , u = 0 instead of X S , u = 0. As expected, the two spectra for each isotopomer are essentially identical aside from laser scatter effects. Chapter 3 Visible Spectrum of ZrC 92 coincidental degeneracy between the energy intervals of Zr and ZrC. The metastable a 3 P state has an unusual spin-orbit structure: the J = 0, 1 and 2 energies are respectively 4186.1, 4376.3 and 4196.9 c m - 1 , so that the "central" J = 1 component appears about 185 c m - 1 above the other two nearly degenerate components. For this reason, WSF scans that monitor fluorescence 188 c m - 1 to the red of the laser, such as those in parts (a) and (c) or Fig. 3.8, also detect Zr lines from either a 3 P 0 or a 3 P 2 if selection rules also allow fluorescence to the a3J*i level. Indeed, many Zr lines were observed in such WSF survey scans spanning 15000 - 19000 c m - 1 . This also necessitated, for each DF spectrum containing a peak displaced about 185 c m - 1 in either direction from its corresponding laser absorption, careful checking against known Zr lines to verify the carrier of this peak. Two other accidental degeneracies generate Zr lines in WSF scans that monitor the 692 c m - 1 interval between a^"1", v = 0 and X 3 E + , v = 1. The a5Fi level of Zr lies at 4870.5 cm - 1 , or about 680 c m - 1 above the metastable a 3 P 0 2 levels discussed above. Also, the a 3 F ground state of Zr has J = 2 and 4 levels that lie about 670 c m - 1 apart; J = 3 excited states can connect to both of them, generating impurity features in the DF spectra. This again required careful scrutiny to avoid confusion by atomic features. The rest of Section 3.3 is divided into two parts that discuss the low- and higher-lying electronic states of ZrC observed in the present work. 3.3(d) The Low-lying Electronic States of ZrC: X 3 E + , a JL:+, b 1 ^ and c1^ 3.3(d)(i) Vibrational analysis of the low-lying electronic states of ZrC DF spectra were recorded for nearly every feature observed in the survey WSF spectrum. Two of these were given in Fig. 3.7 and 3.8. Further examples are shown in Figs. 3.9 and 3.10. The DF spectrum in Fig. 3.9, from the upper state of the "triplet" excitation band shown in Fig. 3.5, is very simple; it contains only a single series of peaks corresponding to the u = 1, 2 and 3 levels of the ground state. The DF spectra shown in Fig. 3.10 are among the X 3 E + , u = 0 Attenuated pump laser scatter G(u)= 880 1753 2615 cm -500 -1000 -1500 -2000 -2500 Displacement from laser / cm -1 Figure 3.9 Dispersed fluorescence (DF) spectrum of the [16.2]3n2, u = 0 level of ZrC. The level is populated by pumping from u of the ground state; fluorescence to u = 1, 2 and 3 is clearly observed. J Chapter 3 Visible Spectrum of ZrC 94 ( a ) Z r 1 2 C u = 1 u = 0 1 2 3 i i 1 1 X 3 S + a 1 ^ - i — i — | — i — i — i — i — | — i — i — i — i — | — i — i - - | 1 1 1 1 1 1 1 1 r i 1 r-( b ) Z r 1 3 C — i — i -•1500 -i 1 r--2000 -1 -500 -1000 -2500 -3000 displacement from laser / cm Figure 3.10 DF spectra from the upper states of the (a) 18512 c m - 1 band of Z r 1 2 C and the (b) —1 13 corresponding 18467 cm band of Zr C. Tie lines between traces join corresponding features. No LIF to X £ , u = 0 occurs; two peaks in each trace (marked with asterisks) that do not belong to X 3 Z + or a*E + are assigned to the b X E + and c 1 S + states. See text for details. Chapter 3 Visible Spectrum of ZrC 95 most complicated. These are from the upper levels of a Z r 1 2 C band at 18512 c m - 1 and its Z r 1 3 C counterpart at 18467 c m - 1 ; these display even more features, which are assigned to two further low-lying electronic states, bl2Z+ and c1!"1", as discussed in more detail below. Table 3.2 Low-lying vibrational levels of ZrC identified by dispersed fluorescence3 Vibrational Energy / cm assignmentb Zrl2C Z r l 3 C X 3 E+, u = 0 0 0 a 1 ^ , v = 0 188(16) 187(13) X 3 E+, v = 1 880(11) 851(6) a A Z + , u = 1 1078(14) 1049(11) X 3 Z+, v = 2 1753(11) 1691(14) b!E+ 1846(15) 1802(11) a 1 2 + , v = 2 1944(14) 1907 c 1 E + 2463(12) 2409(3) X 3 Z+, u = 3 2615(6) 2528(18) a 1 ^ , v = 3 2792(14) 2775(10) b ! Z + 2888 X 3 E+, v = 4 3470(3) a 1 ^ , u = 4 3599 b 1 ^ 3735(22) 3684 X 3 Z + , rj = 5 4316 Quoted uncertainties are three times the standard deviation of the given weighted mean of all measurements; energies quoted without uncertainties were measured only once from data. bAssignments of some of the higher-lying singlet levels are tentative due to strong mixing of the a J S + , b : E + and c 1 ^ states and are only intended to represent the dominant character of the level. For all DF scans, careful measurements were made of both the displacement of each peak from the laser scatter and its signal to noise ratio (S/N). For each low-lying vibrational level, an average of all the independent displacement measurements, weighted by their S/N, was determined. Table 3.2 lists these mean values and their uncertainties, taken as three standard deviations of the mean; an energy level diagram is shown in Fig. 3.11. The energies quoted for the a 1! 4", u = 0 and X 3 E + , v = 1 levels agree well with those obtained for 9 0 Z r C from the rotational analyses described in Section 3.3(d)(ii). The ground state vibrational intervals A G u + i / 2 = G(u+1) - G(o) decrease approximately Chapter 3 Visible Spectrum of ZrC 96 E/10 3 cm 1 4 H 2 H 0 J S55 (-87) 3_ 862 (-63) 2_ 4_ 807 (-17) 3 850 (-37) 2_ 864 (-29) 1 1 (-51) 3735 2888 (-54) 2463 (-44) 1846 b 1 S + and cL2Z+ u = 0 890 ^ ^ - T 0 = 188cm- 1 X 3 E + a 1!* 12 13 Figure 3.11 Vibrational structure of low-lying electronic states of Zr CandZr C. The anomalous C -> C shifts (in parentheses in cm where known) suggest strongly mixed a 1 S + , b*E + and c 1 S + states. Energies are given instead of u for the levels at the far right. Chapter 3 Visible Spectrum of ZrC 97 linearly, as predicted by Eq. [2.20]; fitting them to this equation gives coe = 888.84(83) and coexe = 4.22(41) c m - 1 . The 1 2 C —> 1 3 C shifts, about -29 c m - 1 per vibrational quantum, also scale linearly with u. The vibrational structure of the a^2Z+ state, however, has somewhat anomalous behaviour. The carbon isotope shifts for u = 0 and 1, already observed from Figs. 3.7 and 3.8, are reasonable, but deviate greatly from linearity thereafter. The vibrational quanta decrease with v, but not as smoothly as in the ground state, indicating the presence of strong perturbations. The source of these perturbations may be the four relatively poorly characterised levels in the rightmost column of Fig. 3.11, which must all therefore have the same symmetry as the a1^4" state. The lowest of these, at 1846 c m - 1 , clearly represents a level of a third electronic state, which can be labelled b 1 !^ . The next one, at 2463 c m - 1 , seems too low to be its next vibrational level and is probably from a fourth electronic state, c^"1". The 2888 c m - 1 level is a more satisfactory candidate for the next level of bl2Z+. No u's are given since the states are too strongly mixed to have good vibrational quantum numbers. The supersonic jet-cooling precluded absorption spectra from these higher levels. However, the 1846 c m - 1 peak in the DF spectrum of the 18512 c m - 1 band (which appears at 1846 - 188 = 1658 c m - 1 in Fig. 3.10(a)) is strong enough to be recorded at higher resolution by using narrower monochromator slits and with the laser tuned to single unblended lines in the tail of the 18512 c m - 1 band. The hypothetical examples of Fig. 3.12 show how this technique reveals the electronic symmetry assignment of the peak simply from the number of rotational features that appear if a positive or negative parity level is populated. For example, only for Z states do the number of features produced change with upper state parity. Fig. 3.13 shows the results of exciting the Q(9) and P(10) lines of the 18512 c m - 1 band, which populate, respectively, the J' = 9 + and 9~ levels. Trace (a) in the figure is the emission pattern to X 3 Z + , v = 2 given by the Q(9) line. Q = 1, J = 9 N 10 + 9 -N 1 0 ± : 9+: 8 ± : N 10+-9 - " 8+-N 11 -10 + 9 -8 + 7 - E F2(J=N>; F3(J=N-1), F,(J=N+1) 1 0 - • 9 + 8 -N 11 +z 1 0 - = 9+ = 8 - z 7+ E F2(J=N), F3(J=N-1), F1(J=N+1) =ff= I U> s TO TO TO 1 s -(a) (b) Figure 3.12 Rotational structure of fluorescence from an Q = 1 state to various electronic states. The J = 9 upper level is pumped by one of the two absorptions at the far left. The stick spectra in row (a) are fluorescence from J = 9~ to the electronic state given directly above in the figure. Those in row (b) represent fluorescence from J = 9 + . Fine structure within each N is unresolved for the S states. The uniqueness of these five patterns indicates the lower state symmetry of the fluorescence. Chapter 3 Visible Spectrum of ZrC 99 i 1 1 1 • i 1 i 30 20 10 0 -10 -20 -30 relative wavenumber / c m - 1 Figure 3.13 Rotationally resolved DF features from the 18512 cm 1 band of ZrC. Trace (a) 3 + is the X S , u = 2 feature generated by pumping the Q(9) line of the laser band. The three peaks expected from Fig. 3.12 appear. Traces (b) and (c) are of the strong feature in Fig. 3.10 marked by an asterisk; these are obtained by pumping Q(9) and P(10) respectively of the laser band. The appearance of one and two peaks in these traces indicates 1 E + lower state symmetry. Chapter 3 Visible Spectrum of ZrC 100 This shows that the experiment can resolve rotational features, although with difficulty because of the weakness of the band and the comparatively low J value. The number of features that appear (three, labelled as S-, Q- and O-form lines) is significant. By comparison with row (b) in Fig. 3.12, this shows that the X 3 £ + and a1!"*" states have the same Kronig symmetry (two features would have appeared if the states had opposite symmetries). The absolute symmetries are not established though, because reversing all of the rotational parities in the figure reverses the Kronig symmetries of all the S states while conserving the energy level patterns. The absolute (positive) symmetry derives from the interpretation of the ground state rotational and hyperfine analyses discussed in Section 3.3(d)(iv). Trace (b), also from Q(9), shows that the emission from J' = 9 + to the 1846 c m - 1 level consists of a single rotational feature. Trace (c) also shows emission to the 1846 c m - 1 level, but from J' = 9~ (by pumping P(10)). The appearance of one and two peaks in these traces, respectively, though very weak, gives further confirmation of the bl2Z+ symmetry. Unfortunately, experiments of this type were not possible for the three higher levels, for intensity reasons, but the irregular 1 3 C isotope shifts are consistent with strong homogeneous perturbations between the a, b and c states, suggesting the same symmetry for all three. The 1 3 C shifts for the vibrational levels of the X 3 S + state are about -30 c m - 1 per quantum, as expected for a vibrational frequency of 880 cm - 1 . The al2Z+ state has an almost identical AGj/2, so its 1 3 C shifts should be similar. The 1 3 C shifts for v = 2 and b : Z + , v = 0 sum to -81 c m - 1 , somewhat higher in magnitude than the -60 c m - 1 value expected for a total of two quanta in states whose average vibrational frequency is about 880 c m - 1 . This suggests that the observed levels are strongly mixed and that at least one of them has a vibrational frequency much higher than 880 c m - 1 . Consistent with this, the centrifugal distortion constant from the rotational analysis of the a!S +, v = 0 level, discussed in Section 3.3(d)(ii) below, shows that the unperturbed Chapter 3 Visible Spectrum of ZrC 101 vibrational frequency of the al~Z+ state is about 930 c m - 1 , much larger than the observed A G 1 ( / 2 value. It can be argued that the lower of the two levels near 1900 c m - 1 must have more al~L+, u = 2 character because of its larger carbon shift. This may be true; the assignment in Fig. 3.11 has been made only so that A G 3 / 2 for the al~Z+ state is close to the 850 c m - 1 value extrapolated from the o>e and A G i / 2 values. 3.3(d)(ii) Rotational analysis of the X 3 L + and a 1 !^ states of ZrC As discussed above, 56 vibronic transitions of ZrC have been identified from the high-resolution WSF spectra, most of which are "cold", i.e., with X 3 £ + , v = 0 lower states. Seven electronically "hot" bands, originating from al2Z+, v = 0, and five vibrationally "hot" bands, originating from X 3 2 + , v = 1, have been observed; the upper states of most of these hot bands also appear in cold bands. Over 8500 rotational lines have been assigned for 9 0 Z r C , 9 2 Z r C and 9 4 Z r C , the three most abundant isotopomers, though only unblended and well-measured lines were used for the data analysis (the assignments and measurements of these lines are listed in Appendices I and II). A systematic approach, making maximal use of this vast quantity of data and taking full advantage of all upper state linkages among the observed bands, has given accurate least squares rotational constants for the X 3 Z + , v = 0 and 1 and al2Z+, v = 0 levels. A description of this procedure, which makes extensive use of spreadsheets for the tedious arithmetic, now follows. Data fitting procedure Data from the three principal isotopomers, 9 0 Z r C , 9 2 Z r C and 9 4 Z r C , have been fitted independently. Initially, all possible combination differences for the X 3 Z + , v — 0 level were extracted from the data as explained in Section 2.9(b). Some combination differences have been measured many times, since there are many available cold bands. Averages of these values were fitted by least squares using the standard Hamiltonian for a 3 S + state; Table 3.3 presents its matrix elements evaluated using Hund's case (a) unsigned (i.e., elf parity) basis Chapter 3 Visible Spectrum of ZrC 102 functions. Combination differences with only one available measurement were deweighted or discarded according to the quality of their fit. The rotational constants obtained from these fits were then used to calculate the X 3 2 + , o = 0 rotational energies. Table 3.3 Rotational and fine structure Hamiltonian matrix elements for a 3 E + state, evaluated using Hund's case (a) elf parity basis functions, where A = 0 and x = J(J+1) |J, Q=l+) |J, O = 0> |J,Q=l-> <J, 1 + | B x - y + 27J3 -D(x2 + 4x) -2 V x [B -y /2 - 2D(x+l)] 0 < J, 01 B(x+2) - 2y - 4 X / 3 - D ( x 2 + 8x + 4) 0 < J, 1-1 symmetric Bx - y + 21/3 - D x 2 Least squares fitting for the al~L+, v = 0 level followed, using data from the electronically "hot" bands for which corresponding "cold" bands had been analysed. The upper state rotational energies of the cold bands were found by adding the X 3 Z + , v = 0 rotational energies to the line measurements, and averaging where appropriate. The line measurements from the electronically hot bands were subtracted from the upper state energies, again averaging as appropriate, to determine the rotational energies of the a 1 ^ , v = 0 level. Rotational constants for this level were determined from a least squares fit of the rotational energies to the standard llf rotational energy expression: FV(J) = Tv + VCJ+l) - IV^J+l) 2 . [3.2] Five weak absorption bands from the X 3 S + , v = 1 level of 9 0 Z r C were also assigned in the high-resolution data; corresponding cold bands appeared for all of these. Rotational energies of the 9 0 Z r C , X 3 Z + , v = 1 level were thus evaluated, with appropriate averaging, in similar fashion to those of a1!"1", o = 0. These were also fitted to the matrix in Table 3.3, except that the vibrational energy Tj was added to the diagonal elements. Results of the fits The high-resolution data set for the most abundant isotopomer, 9 0 Z r C , contains over Chapter 3 Visible Spectrum of ZrC 103 4000 combination differences from the X 3 E + , v = 0 level; after averaging, about 200 unique values remain. The rotational constants obtained from them by least squares are given in Table 3.4; the uncertainties quoted are three standard deviations in units of the last quoted significant figure. Centrifugal distortion parameters yrj and XQ, initially included in the fit, were not kept because they were poorly determined and reduced the rms error only marginally. The vibrational constants determined from the A G v + i / 2 intervals for v = 0 - 4 are also given. Table 3.4 ZrC, X 3 Z + : Rotational, fine structure and vibrational constants3 9 °ZrC 9 2 Z r C (calc.)b 9 2 Z r C 9 4 Z r C (calc.)b 9 4 Z r C p 2 = p/p ; 0.9974374 0.9949822 B 0 0.4878743(29) 0.4866241 0.4866219(41) 0.4854253 0.4854289(39) 107D0 5.860(55) 5.830 5.66(11) 5.784 5.76(10) Yo --0.0228714(42) -0.0228128 -0.0228109(65) -0.0227563 -0.0227539(62) *0 0.513958(79) 0.513958 0.51390(10) 0.513958 0.513911(99) r 0 / A 1.8065904(54) 1.8065904 1.8065935(76) 1.8065904 1.8065847(73) rms 0.0002428 0.0002792 0.0002686 4041 / 199 2025 / 178 1946/ 178 N d 1 N max 28 22 21 Vibrational constants: oo e = 888.84(83), o o ^ = 4.22(41) a A l l constants in c m - 1 units except as indicated, with 3 a errors in parentheses bValues isotopically scaled from 9 0 Z r C constants cNumber of total and average combination differences used in least squares fit dHighest value of quantum number N appearing in data set Fig. 3.14 shows how the spin structure of the X 3 S + , v = 0 level of 9 0 Z r C varies with the rotational quantum number N . The quantity plotted is the calculated energy from the fit, less the purely rotational energy B 0N(N+1) - D 0[N(N+1)] 2. The separation of the Fi(J = N + 1) and F 3 ( J = N - 1) spin components approaches 2yN at high N ; F 3 lies above F j because y < 0. The F 2 (J = N) component lies above the mean of the other two by an amount asymptotic to X. Of particular note is the crossing of the F i and F 3 curves at N = 2, where the energy levels lie only 0.0071 c m - 1 apart. Another crossing occurs between the F 2 and F 3 components Chapter 3 Visible Spectrum of ZrC 104 Figure 3.14 Electron spin structure of the X S , u = 0 level of ZrC. The quantities plotted are the eigenvalues of the matrix in Table 3.3 calculated using the least squares molecular 2 2 constants of Table 3.4, less the purely rotational energy E r o t = B 0N(N+1) - D 0 N (N+l) . The F 3 curve crosses the F 1 and F 2 curves at N = 2 and 23 respectively. Chapter 3 Visible Spectrum of ZrC 105 near N = 23. These are not avoided crossings since the J values in each case differ for the two curves and the Hamiltonian matrix is diagonal in J. However, avoided crossings do occur here for the 9 1 Z r C species since 9 1 Z r has a non-zero nuclear spin, I = 5/2; matrix elements of the hyperfine Hamiltonian off-diagonal in J exist. These internal hyperfine perturbations in the ground state of 9 1 Z r C are discussed in Section 3.3(d)(iii). Constants for the less abundant 9 2 Z r C and 9 4 Z r C species are included in Table 3.4. For comparison, isotopically scaled values, calculated from the 9 0 Z r C constants with B* = p 2 B, D ' = p 4 D, yl = p 2y and X1 = X, are also given. In virtually all cases the observed and calculated values agree to within the quoted uncertainties; the only exception is the D 0 constant of 9 2 Z r C . The isotopic scaling applies strictly only to equilibrium constants, but this should not be a major factor since D 0 must be very similar to D e . It is more likely due to the nature of the data. The coefficient multiplying D 0 in the energy level expression is N 2(N+1) 2 , so the uncertainty in D 0 depends strongly on N m a x , the highest value of N available in the data. From Table 3.4, N m a x ( 9 2 Z r C ) is somewhat less than N m a x ( 9 0 Z r C ) ; presumably the observed value of D 0 ( 9 2 Z r C ) would agree better with the isotopically scaled value if higher-N data were available. Also, the calculated values have uncertainties about equal to those of the observed values of 9 0 Z r C , so the ranges of calculated and observed values overlap significantly. Table 3.5 lists the constants of the v = 0 levels of 9 0 Z r C , 9 2 Z r C and 9 4 Z r C determined by least squares. Again, those of 9 2 Z r C and 9 4 Z r C agree with the values isotopically scaled from 9 0 Z r C . It is instructive to estimate coe for the al2Z+ state from these constants using the Kratzer relation. Only for 9 0 Z r C are both B 0 and D 0 determined to sufficient accuracy for this purpose; from these, the vibrational frequency is about 930 cm - 1 . The vibrational frequency can also be estimated from the observed A E ( 9 0 Z r - 9 4 Zr ) isotope shift of the, a^If, v = 0 level and the known co e (X 3 2 + ) value, and a similar result is obtained. Since the A G 1 / 2 = 890 c m - 1 value is much smaller, the alIf state must be highly anharmonic, Chapter 3 Visible Spectrum of ZrC 106 even in the absence of the perturbations discussed above in Section 3.3(d)(i). The estimated value of coexe for this state is 20 c m - 1 . Table 3.5 ZrC, a.lZ+: Rotational constantsa of v = 0 level 90ZrC 9 2 Z r C (calc.)b 9 2 Z r C 9 4 Z r C (calc.)b 9 4 Z r C p 2 = p/p' 0.9974374 0.9949822 187.83053(42) 187.79451(49) 187.76263(60) B 0.526267(10) 0.524918 0.524924(17) 0.523626 0.523641(23) 10 7D 6.78(45) 6.75 7.50(113) 6.71 7.52(170) r/A 1.739445(17) 1.739445 1.739434(28) 1.739445 1.739419(38) rms 0.0003070 0.0002733 0.0002631 #d 145 / 17 87/13 74/12 N e 1 > max 17 12 11 a A l l constants in cm 1 units except as indicated, with 3 a errors in parentheses bValues isotopically scaled from 9 0 Z r C constants cMeasured with respect to fictitious F 2(N=0) level of X 3 E + , u = 0 dNumber of total and averaged term values used in least squares fit eHighest value of quantum number N appearing in data set Table 3.6 9 0 Z r C , X 3 £ + , v = 1: Rotational and fine structure constants* J C 880.40170(29) B 0.4851992(67) 10 7D 5.97(29) y -0.023682(25) X 0.51475(25) r/A 1.811564(13) rms 0.0003217 #c 184/40 N c 1 ^ max 16 a A l l constants in c m - 1 units except as indicated, with 3 a errors in parentheses Equilibrium constants: B e = 0.4892119(55), a e = 0.0026751(73), r^A = 1.804119(10) These have the same meaning as in Table 3.5. Table 3.6 lists constants for the X 3 E + , o = 1 level of 9 0 Z r C . The equilibrium values B e , a e and r e can be determined from the constants for X 3 £ + , v = 0 and 1. With the coe and B e Chapter 3 Visible Spectrum of ZrC 107 values from Tables 3.4 and 3.6 respectively, the Kratzer estimate for the distortion parameter is 5.928(11) x 10 - 7 c m - 1 , consistent with the D 0 and D i values observed for 9 0 Z r C . 3.3(d)(iii) Internal hyperfine perturbations in the X 3 L + , v = 0 state of 9 l Z r C Description of the hyperfine patterns Hyperfine features of the minor isotopomer 9 1 Z r C (11.22% natural abundance) were assigned in a few of the strongest bands. To improve their signal-to-noise, some of them were rescanned at a slower speed and with greater signal averaging. Fig. 3.15 shows one such scan, recorded for the Si(N=0) rotational line of the Q' = 2 - X 3 S + band near 17342 cm" 1. The lower level of an Si(0) line has J = 1 and therefore consists of 2J + 1 = 3 hyperfine components with total angular momenta F = 3/2, 5/2 and 7/2. The upper level, with J = 2, has 5 hyperfine components with F = 1/2, 3/2, 5/2, 7/2 and 9/2. Al l nine hyperfine transitions expected from the AF = 0, ±1 selection rules are observed and assigned in the figure, although one of these is blended with a 9 2 Z r C line. The hyperfine structure in each of the upper and lower states follows a Lande-type pattern, for which the spacing of adjacent levels, F and F+l, goes as F+l (64). The high density of rotational and isotope structure in the centres of the bands mostly blends the 9 1 Z r C hyperfine patterns beyond recognition. To some extent the well resolved spectrum of Fig. 3.15 is a lucky accident, because Si(0) lies fortuitously in the gap between the R i and R 2 heads of 9 2 Z r C . This is illustrated by Fig. 3.16, which shows the surrounding region under normal scanning conditions (i.e., faster and with less averaging); had Si(0) appeared only slightly to either side, much information would have been lost. It is usually only in the S branches, where the line density is sparsest, that the hyperfine patterns can be clearly identified. Further examples of 9 1 Z r C hyperfine structure appear in Figs. 3.17 and 3.18. The Si(5) line of the 17342 c m - 1 band is shown in the top trace of Fig. 3.17. In this case, the six r-type (AF = +1) "main" components, with F" = 7/2 through 17/2, give a clear Lande-type pattern; Figure 3.15 Hyperfine structure of the S^O) line of the 17342 cm 1 (Q' = 2 - X 3 E + , u = 0) band of 9 1 Z r C . Al l nine components save q(7/2) are resolved. o -1 r~ - i 1 1 r-17343.6 -1 r~ I U> a: TO TO TO 17344.2 17344.0 Figure 3.16 A portion of the 17342 cm 1 band of ZrC. Nine asterisks mark the fortuitously located hyperfine components of the 17343.8 laser wavenumber / cm" 17343.4 .91 SjfO) line of ZrC shown in Figure 3.15. o Chapter 3 Visible Spectrum of ZrC 110 c--3/2 5/2 Figure 3.17 Hyperfine structure in 9 1 Z r C : the S1(5) line of the 17342 cm 1 band (fi' = 2) (top -1 trace) and the P 3(5) line of the 16502 cm band (Q! = 0) (bottom trace). The F assignments 3 + increase in opposite directions because at N = 5, the X £ hypermultiplets are inverted for the F, electron spin component and regular for F 3 . Both traces have the same dispersion. Chapter 3 Visible Spectrum of ZrC 111 r(F) 15/2 13/2 11/2 9/2 7/2 5/2 Figure 3.18 Internal hyperfine perturbations in the S,(2) (top trace) and S,(4) (bottom trace) —1 91 lines of the 17342 cm band (fl' = 2) of ZrC. F 3 " features are induced by the perturbations as labelled on each trace; the heavy tie lines join the features with coupled F , " and F 3 " lower levels. Both traces have the same dispersion. Chapter 3 Visible Spectrum of ZrC 112 since the line has J" = 6 and AJ = +1, the p- and q-type "satellites", with AF = -1 and 0 respectively, are too weak to appear. In a Hund's case (b) coupled 3 E + state, the hyperfine width of the F^ electron spin component barely changes with N (64); for this reason the hyperfine widths of the Si(0) and Si(5) lines of Figs. 3.15 and 3.17, about 0.2 c m - 1 , are nearly the same. The lower trace of Fig. 3.17 shows the six main (p-type) hyperfine components of the P 3(5) line of the 3TIo e - X 3 S + band near 16502 cm - 1 ; the sense of the Lande-type pattern is reversed compared to the lines, consistent with the opposite energy order of the hyperfine components in the F j and F 3 components of a 3 S + state. The top trace of Fig. 3.18 shows the hyperfine structure of the Si(2) line of the 17342 c m - 1 band. This is very different from the previous patterns; there is no obvious Lande pattern, so the F assignments are obfuscated. The reason is that an internal hyperfine perturbation (IHP) in the N " = 2 level confuses the hyperfine pattern, causing shifts and splittings in the structure. Such perturbations are well documented for high multiplicity 2 electronic states, having previously been analysed in various states of 5 l V O ( C 4 E - and X 4 2 _ , I = 7/2) (65, 66), 5 5 M n O ( X 6 S + , I = 5/2) (67, 68), 5 5 M n S ( X 6 2 + ) (69), 5 5 M n H (X 7E+) (70, 71), 5 5 M n F (a 5S +) (72) and 5 1 V S (C 4 E") (73). On the other hand, states with other symmetries, even 3 E states, rarely exhibit IHPs because they seldom satisfy the conditions described in the next paragraph required to produce them. Ubachs et al. (74) have observed an IHP in the A 2 A state of C H ( I H = 1/2); it is the only known A ^ 0 state for which this phenomenon occurs and arises from its small spin-orbit coupling. Ahmed et al. (75), in their analysis of the a 3 2 + - X 1 S + system of 6 3 C u F (I =3/2), cited irregular rotational structure in the F 2 electron spin component near J = 50 as evidence for an IHP in a 3 S + . Unfortunately the resolution of the data was too low to verify this hypothesis directly from hfs. Brazier et al. (76) later analysed the hfs of this system, but at a much lower rotational temperature; they could only verify the IHP indirectly by extrapolating the fit of their Chapter 3 Visible Spectrum of ZrC 113 colder data. The corresponding state of CuBr also almost certainly exhibits an DTP, but the currently available data (77) are again incomplete and inconclusive. To the author's knowledge, the HTP in the ground state of 9 1 Z r C reported here is the first to be directly observed and analysed for a 3 Z state. An HTP is an interaction that occurs among electron spin components of an electronic state with the same value of N but different values of J. If these happen to lie at similar energies because of the specific values of the rotational and spin parameters, the nearly-degenerate hyperfine levels then interact strongly via the matrix elements of the hyperfine Hamiltonian that are diagonal in N and F, but off-diagonal in J. The electron spin structure of the X 3 2 + state of 9 0 Z r C is illustrated in Fig. 3.14; a similar pattern occurs for 9 l Z r C , so this figure can be used to understand its level structure as well. 9 0 Z r C has no IHPs because I( 9 0Zr) = 0, but these occur for 9 1 Z r C because I( 9 1Zr) = 5/2. The perturbations are observed in the data only if the hyperfine structures of the Fj and F 3 electron spin components overlap, i.e., only for N = 2, 3 and 4. The anomalous patterns in Fig. 3.18 reflect what happens for N = 2 and 4. The Si(0) line of Fig. 3.15 is unperturbed because the N " = 0 level has only one spin component, F i ( J = 1). The Si(5) line in Fig. 3.17 is also unperturbed because at N = 5 the and F 3 spin components are too distant for their hyperfine manifolds to overlap. Unfortunately the perturbed part of the Si(3) line of this band is blended with stronger lines of other isotopomers, but Si(4) lies in the clear, as shown in the bottom trace of Fig. 3.18; its F" = 5/2 component is doubled. The full explanation is given in the next Sections. Suffice to say here that at N = 4, the high energy end of the F i hyperfine manifold slightly overlaps the low energy end of the F 3 hyperfine manifold, so that only the F" = 5/2 hyperfine components of the two manifolds lie close enough in energy to interact significantly. Matrix elements of the hyperfine Hamiltonian The rotational and fine structure Hamiltonian matrix elements required in the analysis Chapter 3 Visible Spectrum of ZrC 114 appear in Table 3.3. The hyperfine Hamiltonian matrix elements are given in Table 3.7, evaluated in a Hund's case (ap) parity basis. The various terms can be identified by the hyperfine parameters they contain: the Fermi contact parameter, b, the dipolar electron spin-nuclear spin constant, c, and the electric quadrupole parameter eQqQ. The matrix elements are diagonal in F; the basis functions are | JO*) , where | Or) means 2 _ 1 / 2 [ | fl = +1) ± | fl - -1) ], or | fl = 0) and the final ± symbol refers to the parity; the matrix elements have been evaluated using those from Section 2.6(b)(i), for which signed basis functions were used. Table 3.7 Nonvanishing hyperfine Hamiltonian matrix elements for a 3 2 + state, evaluated using Hund's case (ap) parity basis functions. The symbols are defined in the text. <J1+I <J0 <J 1 i <J-1 1+| <J-i o <J-11-<J-2 1+ <J-2 0 <J-2 1-J O = l+> a J e J h J , l J f l = 0) C J bj gj ^ ,0 J Q = l-> a J e J l T , i The functions aj, etc., in the matrix elements of Table 3.7 are as follows: a J (b + c)R(J) 2x + eQq 0 (3 - x){3R(J)[R(J) +1] - 4x1(1 +1)} 8xI(I- l)(2J-l)(2J + 3) bj = - eQq 0 3R(J)[R(J) + 1]-4x1(1 + 1) 81(1 -1)(2J - 1)(2J + 3) C j = jbR(J)/V^, e J •1P(J)Q(J) 2J>/4J2 -1 (b + c) + eQq 0 (R(J) + J + l) 41(21 -1)(J 2 -1) [3.3] [3.4] [3.5] [3.6] Chapter 3 Visible Spectrum of ZrC 115 fj 2bP(J)Q(J)Vx" [3.7] W4J 2 -1 gj = -bP(J)Q(J)Vj(J^l) [3.8] 2jV4J 2 -1 and h J Q 3 e Q q 0 A / ( J - l ) 2 - Q^-y/j2 - Q 2 P(J)Q(J)P(J-1)Q(J-1) [3.9] 161(21 -1)J(2 J - 1)^/(21+ 1)(2J-3) where x = J(J+1) and the functions P(J), Q(J) and R(J) are given by Eq. [2.164]. The size of the matrix depends on the total angular momentum F for F < 7/2; these cases are discussed below. For F > 7/2, the matrix is of order (2I+1)(2S+1) = 18, although judicious choice of the basis functions | J, £2*) labelling its rows and columns can block-diagonalise it into two matrices half as large. One can be obtained using the functions |F + 5/2,l +>, | F + 5/2,0>, |F + 3/2,r>- | F + l / 2 , l + >, | F + l/2,0>, | F - 1 / 2 , T > , IF —3/2, l+>, IF —3/2, 0) and | F - 5 / 2 , 1 " ) . The other matrix can be formed from the remaining nine basis functions: |F + 5/2,l">, | F + 3 /2 , l + >, |F + 3/2,0>, |F + 1/2,1">, | F - l / 2 , l + > , | F - l / 2 , 0 ) , | F - 3 / 2 , T > , | F - 5 / 2 , l + ) and | F - 5 / 2 , 0) . It is clear from Tables 3.3 and 3.7 that none of the first nine basis functions connect with any of the latter nine, so two 9 x 9 submatrices result; these have well-defined rotational parities of ( - 1 ) F + 1 / 2 and - ( - 1 ) F + 1 / 2 respectively. For a S + state in case (b) coupling, the two matrices correlate to even or odd values of N , respectively. For example, if F = 7/2, the values of J in the basis functions range from 1 to 6, but the first matrix gives eigenvalues corresponding to N = 0, 2, 4 and 6 ("low-N"), with positive rotational parity, while the eigenvalues of the second matrix correspond to N = 1, 3, 5 and 7 ("high-N"), with negative rotational parity. For F = 1/2, 3/2 and 5/2 the orders of the matrices are 3, 6 and 8 respectively. The matrices for F = 1/2 and 3/2 are obtained from the general 9 x 9 matrices by retaining only the upper left 3 x 3 and 6 x 6 blocks respectively. If F = 5/2, the two matrices have the same form as for F > 7/2, except that the non-existent basis functions | F - 5 / 2,1 +) and | F - 5 / 2,1~) Chapter 3 Visible Spectrum of ZrC 116 are deleted to give a pair of 8 x 8 matrices. Because the spin-spin parameter X is quite small, a Hund's case (bpj) basis would give an almost diagonal matrix for the rotation and the electron spin effects. This is advantageous when IHPs cause problems with eigenvalue sorting for purposes of iterative least squares fitting, but comes at the expense of considerably greater algebraic complexity of the matrix elements. Case (bpj) offers no particular advantage here since the IHPs are localised for the available data; a case (ap) basis was adequate. The low-N IHPs found in ZrC have A N = 0, AF = 0 and AJ = ±2. Initially the AJ = ±2 perturbations were thought to require a term in the Hamiltonian with matrix elements of this type, namely the electric quadrupole interaction, but the mechanism is actually more subtle. Although the Fj(J = N+l) and F 3 ( J = N - l ) electron spin components for a given N value differ in J by two units, they both interact with the F 2 level (J = N) through AJ = ±1 matrix elements of the Fermi contact interaction. Since the Fermi contact interaction is by far the largest of the hyperfine interactions in the X 3 E + state of ZrC, and the F 2 level lies only 0.5 c m - 1 above the Fj and F 3 levels, the IHP arises in second order. The perturbing levels (from Fj and F 3 electron spin components) have no direct interaction; rather, they interact indirectly through a nearby third level (from F 2 ) . Nevertheless, the effects are non-negligible, as can be seen in Fig. 3.18. This mechanism is equivalent to that proposed by Launila and Simard (72) to describe similar perturbations in the a 5 S + state of MnF. Data fitting procedure and results Compared to the more abundant 9 0 Z r C , 9 2 Z r C and 9 4 Z r C species, only a small number of rotational lines were recorded and assigned for 9 1 Z r C . The assignments and measurements of these are given in Appendix III. No ground state rotational combination differences could be identified from them; the only available combination differences were between hyperfine components of the same N level. For least squares fitting, it was thus necessary to fix the non-Chapter 3 Visible Spectrum of ZrC 117 hyperfine parameters of 9 1 Z r C to values isotopically scaled and averaged, with appropriate weighting, from those of 9 0 Z r C , 9 2 Z r C and 9 4 Z r C given in Table 3.4. The available data gave 37 hyperfine combination differences from rotational levels with N < 9; the hyperfine constants obtained from them by least squares are listed in Table 3.8. Only the Fermi contact parameter, b, and the dipolar parameter, c, were determinable. As described above, the quadrupole parameter eQqo was initially believed responsible for the observed AJ = ±2 perturbations, but this hypothesis was rejected since it is too small to determine from the data. The significance of the b and c parameters is discussed in Section 3.3(d)(iv) in connection with the electron configuration of the ground state. Table 3.8 9 1 Z r C , X 3 Z+, u = 0: Rotational, fine and hyperfine constants3 B 0.4872417[20]b 10 7D 5.809[44]b y -0.0228406[31]b 0.513928[53]b b -0.03133(15) c -0.00123(37) rms 0.0004312 #c 37 N d 9 a A l l constants in c m - 1 units except as indicated, with 3 a errors in parentheses b Fixed value calculated as weighted mean of isotopically scaled constants of 9 0 Z r C , 9 2 Z r C and 9 4 Z r C ; uncertainty given in brackets is based on propagation of uncertainties in these observed values cNumber of combination differences used in least squares fit dHighest value of quantum number N appearing in data set The hyperfine structure of the X 3 E + , v = 0 level of 9 1 Z r C is plotted against N in Fig. 3.19; as in Fig. 3.14, subtraction of the rotational energy magnifies the details. Enlargements of two interesting regions are given in Fig. 3.20. The level patterns corresponding to the fflPs of Fig. 3.18 are shown in the top half of Fig. 3.20. Because of the different range of F values in the F L and F 3 electron spin components (from F = N+7/2 to F = N-3/2 and from F = N+3/2 to F Chapter 3 Visible Spectrum of ZrC 118 0.4 0.2 -\ 0.0 i -0.2 -\ -0.4 -0.6 F 2 ( J = N) N - l ) F 1 ( J = N+1) F=N+5/2 F=N+3/2 F=N+l/2 F=N-l /2 F=N-3/2 F=N-5/2 F=N-7/2 F=N-3/2 F=N-l /2 F=N+l/2 F=N+3/2 F=N+5/2 F=N+7/2 0 10 N 3 + 91 Figure 3.19 Electron spin and hyperfine structure of the X E , u = 0 level of ZrC. The quantities plotted are the eigenvalues of the matrix in Table 3.7 calculated using the least squares fitted constants in Table 3.8, less the purely rotational energy E r o t defined as in Fig. 3.14. In the N = 2 - 4 range, internal hyperfine perturbations produce avoided crossings in the four hyperfine components of F 1 and F 3 for which AF = 0. Larger plots of the boxed regions appear in Fig. 3.20. Chapter 3 Visible Spectrum of ZrC 119 0.0 -0.1 a w2 -0.2 i -0.3 H N = 1 0.39 7 0.38 a o w2 0.37 i m 0.36 0.35 N = 3 i 7 8 F=N+3/2 F=N+l/2 F=N-l /2 F=N-3/2 F=N-5/2 F=N-7/2 F=N-3/2 F=N-l /2 F=N+l/2 F=N+3/2 F=N+5/2 F=N+7/2 F=N-l /2 F=N+l/2 F=N-3/2 F=N-5/2 F=N+3/2 — F=N+5/2 Figure 3.20 Enlarged portions of Fig. 3.19. Top panel: Internal hyperfine perturbations in the 3 + 91 X S , u = 0 level of ZrC. The J values of the Fj and F 3 electron spin components differ by two units, so only four of the six hyperfine components are mutually perturbed in each of F L and F 3 . Bottom panel: unusual structure of the F 2 electron spin component. The six hyperfine components sort into pairs with the same |F - N | value. See text for details. Chapter 3 Visible Spectrum of ZrC 120 = N-7/2 respectively), only four of the six hyperfine series interact. The avoided crossing in the F = N+3/2 series (if regarded as continuous rather than quantised) occurs almost exactly at the N = 2 level, which explains the doubled F" = 7/2 component of Si(2) in Fig. 3.18. Similarly the avoided crossing in the F = N-3/2 series occurs almost exactly at N = 4, for which N - 3/2 is the lowest value of F for the F] spin component; as a result Si(4) shown in Fig. 3.18 has its F" = 5/2 component doubled. The avoided crossings in the F = N+l/2 and F = N - l / 2 hyperfine series occur near N = 3, but not close enough to disrupt its hyperfine pattern significantly, or to induce obvious extra lines. As described above, the Si(3) line of the 17342 c m - 1 band is too severely blended to give any useful information. The DHPs are clearest in the S-form branches, because for these, only the Fj electron spin component carries zero-order oscillator strength. This means that for these branches, IHPs due to levels from the F 3 electron spin component induce obvious splittings and shifts. In the Q branches, by contrast, the effect of the IHPs is less obvious because all three electron spin components carry their own oscillator strength, even in the absence of the IHP. Fig. 3.21 illustrates this point for the Q branches of the 16488 c m - 1 band (the lines of the more abundant isotopomers, saturated in the figure, are fairly sparse, so that the 9 1 Z r C lines are, for once, mostly quite well resolved). The Qi(2) and Q 3(2) lines, for example, are affected by the same perturbation that induces extra features in the Si(2) line of Fig. 3.18, but it is not obvious at first glance that anything is irregular about them. The hyperfine structure of the F 2 electron spin component is plotted against N in the bottom half of Fig. 3.20. Somewhat surprisingly, the hyperfine energies are not proportional to F(F+1), as might be expected for magnetic hyperfine interactions. Instead the trend with increasing N is towards a pattern of three doubly-degenerate pairs, with F = N±5/2, F = N±3/2 and F = N±l/2. This is a direct reflexion of the fact that the F 2 levels are mixed through the Fermi contact interaction with the Fj and F 3 levels, and can be regarded as a partial transition —1 91 3 + Figure 3.21 Q-form head of the 16488 cm band of ZrC (Q.' = Of- X £ ). Al l three ground state electron spin components carry their own hyperfine linestrength, but pairs of Q x and Q 3 lines with the same N and F are shifted apart by the AJ = 2 internal hyperfine perturbation; the two lines marked by asterisks are an example of this, for which N = 2 and F = 7/2. Filled (open) circles 90 92 94 mark unblended (blended) lines. Saturated lines are from the more abundant ZrC, ZrC and ZrC species. Chapter 3 Visible Spectrum of ZrC 122 to Hund's case (bps) coupling. In this limit the dominant spin effect is the Fermi contact interaction, bl-S, which couples the electron and nuclear spins, S and I, to form a resultant G. In the present instance S = 1 and I = 5/2, so the possible values of G are 7/2, 5/2 and 3/2; since the spin energies are proportional to G(G+1), there will be three groups of levels. The double degeneracy results (78) from the specific forms of the matrix elements of the electron spin-spin interaction in case (bps) coupling. Fig. 3.22 shows the P 2(4) line of the 16502 c m - 1 band. At this low N value the grouping of the hyperfine components into three pairs is not complete: the two highest-F components are clearly resolved, while comparison with the calculated structure in Fig. 3.20 shows that the other four components are blended into the strong feature on the low frequency side of the line. The IHPs shown in Figs. 3.19 and 3.20 follow the selection rule AJ = ± 2, and may be classed as "second order" since the coupled levels do not interact directly but are both coupled to a nearby third level through AJ = ± 1 matrix elements of the Fermi contact operator. A "first order" DTP, for which the hyperfine levels belonging to two electron spin components interact directly through the AJ = ±1 matrix elements, occurs near N = 23 in the X 3 S + state of 9 1 Z r C . It was not observed in the present work due to the low abundance of 9 1 Z r C and the jet-cooling, but the calculated hyperfine level pattern is noteworthy; it is shown in Fig. 3.23. In this case the F 2 and F 3 electron spin components cross through each other, with internal perturbations affecting five of the six hyperfine series for each component. The F = N+5/2 series of the F 2 component and the F = N-7/2 series of the F 3 component pass through the perturbation region unscathed. The reason is that the ranges of F values in the two interacting electron spin components differ by one unit; therefore the highest- and lowest-F series only occur once in the group of levels, and are unaffected since there is nothing to perturb them. 3.3(d)(iv) Discussion: A molecular orbital model for ZrC Tables 3.2, 3.4, 3.5, 3.6 and 3.8 summarise all the data currently available for the four Figure 3.22 The P 2(4) line of the 16502 cm 1 band of 9 1 Z r C (OJ = Oe - X 3 S + ) . A head is formed by the F levels of the ground state F 2 electron spin component; the total hyperfine width (0.035 cm - 1 ) is much less than in Fj (0.168 cm - 1 ) or F 3 (0.125 cm 1 ) . 0.6 S o I 0.4 0.2 0.0 i F N + 5/2 N + 3/2 N + 1/2 N - 1/2 N - 3 / 2 N - 5 / 2 N - 7 / 2 F 2 (J = N) F 3 (J = N - l ) 10 15 20 25 30 35 N Figure 3.23 Calculated hyperfine energy level patterns for the F 2 and F 3 electron spin components of the X 3 L + , u = 0 level of 9 1 Z r C for N = 11 - 34. The plot is similar to Fig. 3.14, with E r o t identically defined. Levels with a common F - N value are joined by a curve; those with intermediate F - N values appear twice and have avoided crossings (internal hyperfine perturbations) near N = 20; the F - N = -7/2 and +5/2 curves appear only once each and are therefore unperturbed. to Chapter 3 Visible Spectrum of ZrC 125 low-lying electronic states of ZrC. Taken together, they allow a satisfactory molecular orbital (MO) scheme to be devised for this molecule. The starting point is the identification of the ground state as 3 £ + rather than 3 Z~, which follows from its very small electron spin-spin interaction parameter, X. Table 3.9 Manifolds of electronic states arising from various two-electron configurations Configuration Electronic states ca' • 3E+ nii 3E+, 12- 3S- l A , 3 \ 55' 3s+, 12-, 3 z - i r , 3 r r Tl2 3 s - l A 52 3E- i r If electron configurations with only two valence electrons are considered, 3 Z states arise if the two electrons have the same orbital angular momentum projection quantum number X, e.g., cc', mi, 88', n2 or 8 2 (Table 3.9 lists the set of electronic states arising from each of these). Except for the cc' configuration, the 3E states are always accompanied by states of opposite Kronig symmetry (for example, 3 2 r and 1 S + ) , between which strong spin-orbit interactions always occur (79). The result is that the £1 = 0 and 1 components of a non-rotating 3 S state coming from one of these configurations may be separated by about 30-100 c m - 1 in a molecule containing a 3d or 4d transition metal atom. An example is the n2 X 3 £~ state of NiO (59), where interaction with the isoconfigurational 1E+ state pushes the Q = 0 component 52 c m - 1 below Q = 1. An even more extreme example, admittedly from a 5d system, is the 8 2 X 3 S _ state of ReN, where the separation is no less than 2630 c m - 1 (80-82). Similar arguments apply to 3E states from configurations with four or more valence electrons (79), so that no generality is lost here. The £2 = 1 - 0 separation is, strictly speaking, a second-order spin-orbit interaction, but Chapter 3 Visible Spectrum of ZrC 126 is indistinguishable from the electron spin-spin interaction if the state is described by an effective Hamiltonian, since it has the same quantum number dependence. As a result, the electron spin-spin interaction parameter, X, includes the second-order spin-orbit interaction, and in fact is dominated by it in molecules containing heavier atoms. In a 3 2 state, the Q. = 1 component is defined as lying above the Q = 0 component by an amount 2X, so that values of X of the order of 15-50 c m - 1 can be expected for 3d and 4d systems. However, as previously indicated, this does not apply to the configuration aa', for which no spin-orbit interaction occurs; in this case the value of X more nearly represents the true spin-spin interaction, with at most higher order spin-orbit effects, and should not exceed 1 c m - 1 . The very small observed value, X = 0.514 c m - 1 , is therefore clear proof that the 3 E ground state of ZrC comes from the configuration aa', and must be a 3 Z + state. In fact, for the ground states of two-electron systems, 3 E + states are rare compared to 3 S _ states because stringent conditions are needed for them: there must be two o-type valence orbitals present, and they must lie close enough in energy that the exchange interaction of the aa' configuration places the 3 E + state below the closed shell l2Z+ (a 2) electronic state. Of all diatomic molecules studied so far, only ZrC appears to meet these conditions, although the isovalent TiC molecule, currently under study in the Morse laboratory, represents another possible example. At this time it is not clear whether its ground state is 3 2 + or lIf (30), although the complexity of the moderately resolved jet-cooled absorption data suggest its ground state is also 3 2 + (83, 84). The comparatively large Fermi contact parameter, bp = b + c/3 = -0.03174(19) cm - 1 , further confirms the Kronig symmetry because it indicates the presence of an unpaired electron in a molecular orbital (MO) derived from the Zr 55a atomic orbital, which, from Table 3.9, can only be the case for the aa' configuration. Its value can be estimated from Ho 47ihc ' l > V 2 S y xgg#B^{^5 2 s(0)} . [3.10] Chapter 3 Visible Spectrum of ZrC 127 The ab initio value ( ^ ( O ) ) = 5.283 a ^ 3 (85) and the IUPAC recommended value for the magnetic moment of the 9 1 Z r nucleus (86) give b F = 0.00318625 x — x (1/2) x V 3 J •1.30362 x 5.283 = -0.03677 cm" 1. [3.11] 5/2 J 1 1 This value, about 16% larger than the experimental value, indicates that the ab initio value of (Y 5 2 s (0)) is too large, as has generally been found for such Hartree-Fock calculations. The bp value for the 9 1 Z r atom is -0.0446 c m - 1 (87); the value for ZrC is about 83% of this, indicating that one of the a MOs is predominantly Zr(5s) in character. At this stage an M O diagram can be drawn for the ground state of ZrC, based on that for the diatomic 3d oxides (59); this is shown in Fig. 3.24(a). The a and a' orbitals (1 l a and 12c in the figure) are constructed from the Zr 5so and C 2po atomic orbitals; each holds one electron, giving the configuration (\\d)\\2ti)^. With the electron configuration of the ground state established, those of the other low-lying states may be considered. It is tempting to regard the a}l+ state as being isoconfigurational to the ground state, but the considerably different r 0 bond lengths of the two states do not support this conjecture. More reasonably, the much shorter bond length of al2Z+ suggests that it originates from the 11a 2 closed shell configuration. The proximity of the differently configured X 3 2 + (aa ' ) and a 1 S + (a 2 ) electronic states, less than 200 c m - 1 apart, indicates that the a and a' MOs must be very similar in energy. This implies that the two 1 E + states from the aa' and a' 2 configurations should also lie very low in the ZrC manifold, so that only slightly above the ground state there should be three close-lying 1 E + states, among which mutual homogeneous perturbations can occur. This is precisely the case (see Section 3.3(d)(i)), and supports the assignment of the b 1 ^ and c1!"*" states in the third column of Fig. 3.11. The only arrangement of the MOs consistent with all of the experimental facts is that shown in Fig. 3.24(a). Because there are no low-lying Tl or A states, the two a MOs, 1 l a and (a) Zr (4d25s2) 4d 5s 4do 4d% 4db 5sa 11a 13a 671 25 12CT 4-2p% 2pa 4 H H 5 ^ 2sa 10a—It ^ C (2s22p2) 2p 2s (c) (4c?15s2) 4d 5s 4da 4dn 4dh 5sa 13a 671 25 12a 10a 2/771 2pa l l a \ — 4 > -M M 15n 2scs N (2s22p3) 2E. 2s | U> TO TO TO (b) 12a 11a 12a lie* TQ/ cm 2463 1846 188 b 1 S + a 1 S + ( r 0 = 1.74 A) X 3 S + ( r 0 = 1.81 A) (d) 12a z 11a 12a TQ / cm -1 3883 2496 B 1 I + (not yet observed) L L 3^+ A T ( r e = 1.82 A) a J I ( r 0 = 1.87 A) X : I + (r = 1.80 A) Figure 3.24 M O diagrams and low-lying electronic states of ZrC (parts a and b) and Y N (parts c and d). The difference in ligand IPs leads to different ground states for the isoelectronic ZrC and Y N species. Spectroscopic data for Y N are taken from Refs. 55 - 57, all ZrC data are from the present work. to 00 Chapter 3 Visible Spectrum of ZrC 129 12o, must lie between the nonbonding 25 and the bonding 5JI MOs. In part (b) of the figure the three electron configurations responsible for the four low-lying 2 states are illustrated; the 3 E + ground state comes from the configuration (10a)2(57i)4(llo)1(12a)1 and the &*2Z+ state from (10a)2(57t)4(llc)2. The bl2Z+ and c 1 ^ states originate from the (lQa)2(5ji)*(Ua)l(12c)1 and (10a)2(5jt)4(12a)2 configurations, but which is which cannot be determined without bond length measurements. The perturbations apparent in Fig. 3.11 indicate extensive mixing of the three electronic states. As discussed in Section 3.3(b), the ground state of the isoelectronic species Y N is known to be lZ+ (56-58), so that its M O scheme is clearly different. Its ground state must come from the configuration (10c) 2(5u) 4(l l a ) 2 , while the low-lying 3 E + state at 2496 c m - 1 (58) must be the analogue of the ground state of ZrC. A second singlet state at 3883 cm - 1 , A ! Z + (56), must correspond to either blZ+ or clIf in ZrC, but obviously it cannot yet be decided which. Fig. 3.24(c) shows how the M O diagram must be modified for Y N , and the resulting set of low-lying states, of which three have been observed (56-58), are shown in Fig. 3.24(d). The basic difference is that the nitrogen has a higher IP than carbon, so that the N atomic orbitals lie lower in the diagram than those of C; the 11a and 12a MOs are more separated in Y N than in ZrC. The ground state equilibrium bond length of Y N (1.80405 A) (56) is nearly identical to that of ZrC (1.80412 A), although a more insightful comparison would be with the analogous a^lf state of ZrC, which has a much shorter bond length of r 0 = 1.7394 A. The better energy match of the metal 5sc and ligand 2pa atomic orbitals in ZrC stabilises the 11a orbital with greater bonding character. The same reasoning should apply to the bond lengths of the ground state of ZrC and the analogous a 3 S + state of Y N . The latter, r e ~ 1.87(8) A, has been measured from medium resolution dispersed fluorescence data (58); its uncertainty is too large to draw definitive conclusions, but its support of the above argument does appear promising. Higher Chapter 3 Visible Spectrum of ZrC 130 resolution data of this state would be helpful. Based on the proximity of the 11a and 12a orbitals in ZrC, the complexity of the WSF spectrum in the 16000-19000 c m - 1 region can be understood. Any promotion of the 11a electron produces a manifold of electronic states that will be accompanied by an identical, nearly degenerate, manifold arising from an equivalent promotion of the 12a electron. Strong interactions occur between the two manifolds, producing the unusually disorganised spectrum described in Section 3.3(c). 3.3(e) Electronic States of Z r C in the 16000-19000 cm" 1 Region Rotational analyses have been carried out at high resolution for 56 bands in the 16000 -19000 c m - 1 region. The first lines of the branches and the upper state H-doubling patterns give unambiguous Q! assignments of 0+e, 0~f 1 or 2. Most of the bands arise from the v = 0 level of the X 3 £ + state, though a few come from X 3 E + , v = 1 and alIf, v = 0. Fig. 3.25 plots the observed upper state vibrational energies; they are grouped into columns by their Q. values. The partitioning of the £ 2 = 1 levels into 3TTi and lTL states is somewhat arbitrary as they are all to some extent mixtures of the two; the symmetry label indicates the dominant character of the level. The isotope shifts, A E ( 9 0 Z r C - 9 4 ZrC) , given in parentheses, are the shifts in the band origins of the transitions from the X 3 E + , v - 0 level; for those few upper levels that do not combine with this level, the shifts are referred to the alE+, u = 0 level. The upper state levels are readily divided into two categories: (a) the four lowest levels between 16000 and 16400 era"1, comprising the upper state of the [16.2]3TL, - X 3 I + (0,0) band, which are relatively well-understood, and (b) the remaining levels between 16400 and 19000 c m - 1 , whose properties are obscured by severe perturbations. 3.3(e)(i) The [16 .2 ] 3 H r - X 3 E + (0,0) band Fig. 3.26 shows the four absorption bands in the 16000-16400 c m - 1 region as they Chapter 3 Visible Spectrum of ZrC E / c m - 1 19000 18500 18000 17500 17000 16500 16000 (5.7) (5.8) (9.1) (3.6) (3_pl (2.3) (-0.1) u = 0 (10-5) (5.4) (2.9) -(9.7)= (2.7) (103) (4.5) i l l ) (6.6) i !p\4) (1.2) (-0.2) u = 0 (13.6) n = 2(3n2) Q = l(3Ul and 1!!) (6.8) (3.9) (15.9) (11.9) (6.9) (1.5) 0 2 ) u = 0 131 (7.5) (4.5) (18-3) (6.1) (103) (9.3) (3.2) u = 0 (-0.1) Q-Oe (3n0e) Q = o/(3n0/) Figure 3.25 Vibrational levels of 9 0 Z r C identified in the 16000 - 19000 cm Region. Where available, isotope shifts A E ( 9 0 Z r C - 9 4 ZrC)/cm~ 1 are given in parentheses and show few obvious trends. The levels appear with high density and mostly follow Hund's case (c) coupling, 3 1 with predominantly Tl and n parentage; nearly all of them are globally perturbed. Q • = 2 Q ' = 1 Si' = Of i 1 1 1 1 1 1 1 1 < 1 1 ' 1 1 r 16300 16200 16100 16000 laser wavenumber / c m - 1 3 3 + Figure 3.26 WSF spectrum of the [16.2] TLr - X I (0,0) group of ZrC bands recorded at low resolution. Portions of the Q' = 2 and 1 bands, recorded at high resolution, appear in Figs. 3.5 and 3.27 respectively. Of all the observed bands, these are the only four for which unambiguous u' assignments can be made. Two ZrCH features near 16200 c m - 1 and a Zr atomic line near 16090 c m - 1 also appear; these are marked with asterisks. Chapter 3 Visible Spectrum of ZrC 133 appear at low resolution. They arise from the X 3 £ + , u = 0 level and are the lowest energy bands found in the search region (15000-19000 cm - 1 ) for their D! values. Some minor perturbations are present, but these are only apparent at high resolution. A detail of the high-resolution spectrum of the Q! = 2 band was shown in Fig. 3.5. Fig. 3.27 shows the rotational structure of the R-form head of the Q.' = 1 band as seen at high resolution. F T " and F 2 " spin components might be expected to appear in this R-form structure on the basis of the AJ and parity selection rules. However, it is well known that of the nine branches in a 3Ui - 3 £ + band that satisfy these rules, the three involving the F 2 " spin component are vanishingly weak if the upper state has sufficiently large spin-orbit coupling (88), as is the case here. Accordingly, only F i " components appear in the R-form head of this band, although other Q' = 1 cold bands in the spectrum exhibit F 2 " branches; the relative strength of these branches is regarded as a measure of the extent of upper state 3Ui ~ lU mixing. The 9 0 Z r / 9 2 Z r / 9 4 Z r isotope structure of this R-form branch is also shown in the figure. The Av( 9 0 Zr - 9 4 Z r ) isotope shift of this band, and of the others, where known, is very small and negative (approximately -0.20 cm - 1 ) . The spacing of the four bands in Fig. 3.26, their similar isotope structures and their Q' ordering with increasing energy indicate that they are the Hund's case (a) spin-orbit components of the [16.2]3n>. - X 3 Z + (0,0) band, where the upper state is labelled using the convention of Linton et al. (89). The upper state A-doubling parameter (o + p + q) (90) of the Q' = 0 level, corresponding to half the T y - T e splitting at zero rotation, is very large, at about -16 c m - 1 . Moreover, while the intensity of the 3 I % - X 3 £ + band is similar to those of 3 n x - X 3 E + and 3 n 2 - X 3 S + , the 3 IIo e - X 3 E + band is much weaker. Both observations indicate that the 3IIo level interacts with a £ electronic state, although no such states were identified above the four low-lying £ + states previously discussed. 16182.2 16182.0 16181.8 16181.6 16181.4 laser wavenumber / c m - 1 Figure 3.27 R } branch head of the [16.2] - X E (0,0) band of ZrC at high resolution. Diagonal tie lines join corresponding features of 9 0 Z r C , 9 2 Z r C and 9 4 Z r C , as shown for the N = 3 lines. Some lines are blended because the isotope shift is coincidentally equal either to the branch spacing or to the separation of lines on the outgoing and returning arms of the branch. £ Chapter 3 Visible Spectrum of ZrC 135 The Q.' = 2 band, shown in Fig. 3.5, has an R-form head at N " = 6, indicating a moderate increase in bond length on excitation. The same conclusion follows from the steadily decreasing intensity pattern in the DF spectrum of the upper level of this band (see Fig. 3.9). Moreover, the lack of a local minimum in this pattern indicates a nodeless upper state vibrational wavefunction, i.e., v' = 0, as already concluded from the Av( 9 0 Zr - 9 4 Z r ) isotope shift. The DF spectra for the two Q! = 0 bands behave likewise, while that for the Q' = 1 band, shown in Fig. 3.28, shows additional features corresponding to the low-lying states discussed in Section 3.3(d). These weak features indicate that the Q' = 1 level, though nominally 3 I I 1 , must also have a small amount of singlet character. Its most likely source is second-order spin-orbit mixing with an isoconfigurational, higher energy *n state, analogous to that described in Section 3.3(d)(iv) for a 3 E + / 1 E _ or 3 S _ / 1 E + pair of states. This mutual interaction pushes the 3TI1 and lH levels apart, producing an asymmetry in the spin-orbit structure of the 3n(a) state. The Q 1 = 1 - 2 splitting is about 127 cm - 1 , and if the location of the Q' = 0 level is taken from the e parity component, then the Q! = 0 - 1 splitting is only about 118 c m - 1 . Arguments fashioned after Lefebvre-Brion and Field (79) using the positions of the observed bands locate the *n perturbing state approximately 3500 c m - 1 above the 3 I I 1 level, near 19500 c m - 1 , which is unfortunately beyond the region of investigation and cannot be observed, even upon excitation from a 1 Z + , v = 0. This calculation represents the low energy limit of the lH perturber; setting the Q.' = 0 location any lower decreases the spin-orbit asymmetry which in turn increases the *n - 3TIi separation. While this second order spin-orbit interaction decreases the energy of the [16.2]3IIi, u = 0 level, its rotational energy level pattern is unaffected. In fact, this level is one of the few observed above 16000 c m - 1 for which a least squares fit of its rotational energies is feasible; these can be fitted to the standard Hund's case (c) expression for an Q = 1 level: X 3 Z + , u = 0 attenuated laser scatter I x s to TO -1000 displacement from laser / cm -2000 -3000 Figure 3.28 DF spectrum of the [16.2] n l 3 u = 0 level of ZrC. Weak fluorescence features to the low-lying singlet states, marked 3 1 by asterisks, indicate a small amount of singlet character in the excited state, probably due to isoconfigurational f l j ~ f l mixing. Os Chapter 3 Visible Spectrum of ZrC 137 F(J) = T + BJ(J+1) - D[J(J+1)]2 ± 1 {qJ(J+l) + qD[J(J+l)]2}, [3.12] where the positive (negative) sign is taken for / (e) parity rotational levels. The results of these fits are shown in Table 3.10 for the three most abundant isotopomers. Table 3.10 ZrC, [16.2]3TIi, v = 0: Effective rotational and fine structure constants3 90ZrC 9 2 Z r C (calc.)b 92ZrC 9 4 Z r C (calc.)b 9 4 Z r C 2 = p/p* 0.9974374 0.9949822 10-4TC 1.61784261(3) 1.61785261(6) 1.61786243(5) B 0.4260501(69) 0.424958 0.424958(20) 0.423912 0.423902(12) 10 6D 1.004(31) 0.999 0.99(13) 0.994 0.989(60) 103q --1.139(10) -1.136 -1.140(16) -1.133 -1.121(16) 107qD 4.70(56) 4.68 4.68d 4.65 4.5(10) r/A 1.933229(16) 1.933229 1.933229(45) 1.9332294 1.933252(27) rms 0.0003106 0.0004726 0.0004035 #e 34 21 27 j f J max 20 13 17 Kratzer estimate of vibrational frequency, 9 0 Z r C : coe = 555 a A l l constants in c m - 1 units except as indicated, with 3 a errors in parentheses bValues isotopically scaled from 9 0 Z r C constants cMeasured with respect to fictitious F 2(N=0) level of X 3 S + , v = 0 d Fixed in the least squares fit eNumber of term values used in least squares fit fffighest value of quantum number J appearing in data set While the constants of Eq. [3.12] must strictly be regarded as effective fitting parameters, their isotopic scaling within experimental error lends validity to their interpretation as true mechanical constants. For example, the B Q value can be regarded as deriving from the r 0 bond length. In addition, from the 9 0 Z r C constants and the Kratzer relation, the vibrational frequency estimated for this electronic state is 555 c m - 1 , about 38% less than the 889 c m - 1 value for the ground state. Positions and Av( 9 0 Zr - 9 4 Zr ) isotope shifts of higher vibrational levels can be estimated for this electronic state and compared against the levels plotted in Fig. 3.25. This point is discussed in Section 3.3(e)(ii). Chapter 3 Visible Spectrum of ZrC 138 Inasmuch as the [16.2]3EL. - X 3 Z + (0,0) band can be described by the promotion of a single electron, the regular spin-orbit structure of the upper state rules out the promotion of a 5% electron to one of the a orbitals. Clearly, either the 11a or 12a electron is promoted, most likely to the antibonding 6% orbital. The stronger bonding character of the 11a orbital and the moderate increase in bond length favour a 12a —» 6% promotion. The resulting (10a)2(57c)4(lla)1(67t)1 configuration yields both the [16.2]3fL state and the lU state responsible for its asymmetric spin-orbit splitting some 3500 c m - 1 above it. Additional perturbations appear in the rotational energy level structure of the Q! = Oe and Q' = 2 bands. Figs. 3.29(a) and 3.29(b) show the upper state reduced rotational energy plots for the 9 0 Z r C isotopomer. The Q! = Oe level exhibits an avoided crossing near J = 6, although this perturbation does not appear in the Q' = 0/level. For the £1' = 2 level, the nearly parabolic curvature of the plot in Fig. 3.29(b), with a minimum near J = 6, represents a severe global perturbation. This cannot be explained, for example, by the normal 3 n 2 spin-uncoupling effect that arises from the off-diagonal Hund's case (a) matrix elements of the B(J - L - S) 2 term of the Hamiltonian. These matrix elements merely change the effective rotational constant of the 3 n 2 level from the mechanical B value to an effective value B e f f given by the Mulliken relation: B e f f = B ( l + 2 B S / A A ) , [3.13] so that a plot of the energies versus J(J+1) should remain linear. This level is further disrupted near J = 11 by another perturbation which must derive from a degenerate (Q > 0) electronic state since it affects both e and/ parity components in essentially identical fashion. Since these are the lowest energy bands observed in the spectrum for their respective Q.' values, the source of their perturbations must be high-u levels of lower-lying electronic states. The presence of these perturbations is not surprising in view of the great number of anomalies already identified in Sections 3.3(c) and 3.3(d). Moreover, the perturbed nature of even the Chapter 3 Visible Spectrum of ZrC 139 2.0 1.6 3 1.2 0.8 0.4 20 40 60 J(J+1) 80 100 120 2.4 ~ 2.0 'e o ? 1 6 3 l o H I H 1.2 0.8 H Figure 3.29 Reduced rotational energies of the (a) Q = 0e and (b) O = 2 spin-orbit components of the [16.2]3nr, u = 0 level of 9 0 Z r C . The Cl = 0e level has an avoided crossing near J = 7; the Q = 2 level is globally perturbed in both e and/ parities. In the plotted quantity, T Q = 16061 (a) and 16305 c m - 1 (b); k = 0.427 c m - 1 in both plots. Chapter 3 Visible Spectrum of ZrC 140 [16.2]3n, u = 0 level, the lowest-lying upper state found in the electronic spectrum, suggests that perturbations will be the rule rather than the exception for the remainder of the higher-lying levels discussed below. 3.3(e)(ii) Higher-lying levels of ZrC in the visible region The upper states of all bands recorded at high-resolution have been classified according to their £2' values and are plotted as an energy level diagram in Fig. 3.25. The numbers of upper state levels identified with £2' = 0+e, 0"f 1 and 2 are in the approximate ratio of 1:1:2:1. The simplest interpretation of these levels is that they represent the Hund's case (a) spin-orbit components of a collection of 3n levels, while the extra £2 = 1 levels are isoconfigurational lH states. All of the observed £ 2 = 1 levels are then mixtures of the 3ni and *n states. Constants from the least squares fit of the [16.2]3ni, v = 0 rotational energies of 9 0 Z r C predict coe = 555 c m - 1 for this state, much less than the 889 c m - 1 value for the ground state. On this basis, estimated energies and Av( 9 0 Zr - 9 4 Z r ) isotope shifts for higher levels of the [16.2]3n state up to v = 5 have been plotted in Fig. 3.30 (dashed lines) with the observed levels (solid lines) from Fig. 3.25. The predicted isotope shifts are independent of £2 and for clarity are given only for the 3 n 2 levels. The spin-orbit structure of the calculated levels was assumed to be identical to that observed for rj = 0, and (HQX^COQ = 0.0047 was chosen to match the ground state. These calculations show only the most modest agreement to the observed levels. For example, the calculated £2 = 2 levels are typically about 50 c m - 1 from the closest matching observed levels, and the observed isotope shifts are generally larger. These anomalies no doubt result from the presence of at least one other 3n level in this region. In fact, the six £2 = 1 levels in the 17650 - 17950 c m - 1 region lie too close together for any pair of them to belong to the same electronic state, so at least this many electronic states must be producing the complicated energy level pattern in Figs. 3.25 and 3.30. Chapter 3 Visible Spectrum of ZrC E / c m - 1 19000 18500 18000 17500 17000 H 16500 A 16000 H [6.3] (5.7) (5.8) [5.1] (9.1) (3.6) [3.8] (3.6) (2.3) [2.5] [1.2] (-0.1) u = 0 (10.5) (5.4) (2.9) -(9.7), (2.7) (10.3) (1.2) (5A) (6.6) (16.4) (13-6) (-0.2) u = 0 (6.8) i3_9 l (15.9) (11.9) (6.9) (15) (2.2) o = 0 141 (7.5) (4.5) (18-3) (61) (10-3) (9.3) (3-2) u = 0 (-0.1) a = 2 (3n2) a -1 ( 3n : and lu) a = oe (3n0e) a = o/(3n0/) on QA Figure 3.30 Calculated u = 1 - 5 levels (dashed lines) and AE( ZrC - ZrC) isotope shifts (in 3 90 brackets for the Q = 2 components) of the [16.2] 11(a) state of ZrC, extrapolated from the observed u = 0 levels shown (see text for details of calculations). Strong perturbations in these levels result in poor agreement with observed levels (solid lines, taken from Fig. 3.25). Chapter 3 Visible Spectrum of ZrC 142 Not surprisingly, virtually every vibrational level in the 16000-19000 c m - 1 region has global rotational perturbations; the J(J+Independence of their rotational energies is often very non-linear. The cold bands in the 16640 - 16710 c m - 1 region, shown in the right inset of Fig. 3.4, are a good example of this; their reduced upper state rotational energies are plotted in Fig. 3.31 for the three most abundant isotopomers of ZrC. Of the four £ 2 = 1 states in this region, the three highest are obviously interacting strongly. In 9 0 Z r C , the 16681 c m - 1 level has particularly bizarre £2-type doubling. The normal ATy e = qJ(J+l) behaviour is replaced by a strongly non-linear splitting, as shown in Fig. 3.32; it is already rather large at J = 1 and 2, then decreases until it changes sign, immediately reverses sign once again and rapidly increases. It appears that, in addition to the interactions with the other two £2=1 states, the / parity levels of this state are also interacting with the £2 = 0/ state that lies just 0.3 c m - 1 beneath it. The perturbed nature of the three £ 2 = 1 states is also clear from the isotope dependence in the intensity distribution patterns in their cold absorption bands. In 9 0 Z r C , the three band origins are at 16681, 16694 and 16702 c m - 1 , and the first of these bands is by far the strongest of the three, with the other two each about five times weaker; this can be expressed as a 30:6:6 intensity ratio. This ratio changes both for 9 2 Z r C , for which these band origins lie at 16676, 16687 and 16693 cm- 1 , and for 9 4 Z r C , where they appear at 16670, 16681 and 16686 cm- 1 . Extending the previous intensity ratio scheme to these species leads to an 8:8:3 ratio for 9 2 Z r C and a 1:5:10 ratio for 9 4 Z r C . The sum of the intensities is fairly consistent with the natural abundance of the isotopomers, but their distribution changes in the three species. Evidently, the £2 = 1 levels are composed of three strongly interacting zero-order basis states with very different transition moments; their composition is isotope-sensitive, and the intensities of the above bands are transformed accordingly. With increasing Zr mass, intensity is transferred from the lowest to the highest of the three levels. Other than the [16.2]3n(a), v = 0 levels already discussed, those shown in Fig. 3.31 are 16710 16700 16690 'g 16680 o + 16670 H OS 1 16660 16650 16640 16630 16702 (0 = le) 16694 (0 = 1/0 16681 (0 = 1/) 16681 (0= le) 16680 (O = Of) 16655 (O = Oe) 16643 (O = If) 16643 (0= le) 90 ZrC 0 = 1/ 0 = le 0=1/ O = 0e 0=1/ 0= le 92 ZrC 0= le 0 = 1/ ' 0 = 1 / 0= le 0 = 1 / 0 = le O = 0e 0 = 1 / 0 = le 94 ZrC 0 100 200 300 J(J+1) 400 500 0 100 200 J(J+1) 300 0 100 200 JYJ+1) 300 Figure 3.31 Reduced rotational energies of the 16681 cm 1 group of levels of 9 0 Z r C , 9 2 Z r C and 9 4 Z r C . Corresponding O levels have the same energy order in all isotopomers. Strong interactions occur among the three highest-lying O = 1 levels. Chapter 3 Visible Spectrum of ZrC 144 T , - T / c m - 1 200 J(J+1) 90 —1 Figure 3.32 Anomalous Q-type splitting of the Q = 1 level of ZrC at 16681 cm . Two turning points are produced by interactions with the other nearby levels plotted in Fig. 3.31. Chapter 3 Visible Spectrum of ZrC 145 among the lowest-lying levels of the observed upper states, and underscore the complexity of the ZrC energy level pattern. In fact, the large A E ( 9 0 Z r C - 9 4 Z r C ) isotope shifts of some of the levels in Fig. 3.31, at least 10 c m - 1 in magnitude, suggest that these are high-u levels from lower-lying states than those presently observed. It seems likely that low-o' bands from these states should appear below those shown in Fig. 3.25; however, WSF survey scans down to 15000 c m - 1 revealed none of these. Perturbations appear in nearly all of the upper states shown in Fig. 3.25. One example of a perturbed Q, level, for each of Q. = Oe, Of, 1 and 2, is shown in Fig. 3.33 in the form of a reduced rotational energy plot. While numerous perturbations appear, it is interesting that very few doubled rotational levels could be assigned. A rare instance of this is the Q = 2 level at 17342 c m - 1 , shown in Fig. 3.33(d), whose J = 7/ and Se levels are both split into two components; in each case, the splitting is about 0.5 c m - 1 . More generally, in most of the bands the line intensities become very weak at the onset of a perturbation, beyond which the rotational structure cannot be followed. The perturbations are so pervasive that there is little point in attempting least squares fits of the rotational levels of any of the excited states. For most of these states, there is no simple model to which the data can be fitted. Even where such a model exists, it is essentially phenomenological, does not always describe the full range of observed rotational levels and offers little physical insight, especially since the results, with their limited success, would only apply to a scattered few out of the many levels studied. This section concludes with an example of an attempted deperturbation of the Q. = Oe and 1 levels near 17833 c m - 1 . To fit the data, a 3 x 3 Hamiltonian matrix was devised, with standard Hund's case (c) level expressions for the diagonal elements: F F L (J , / / e ) = BQJ(J+1) - D Q J 2 ( J + 1 ) 2 ± i 5 A I [ Q I J ( J + l ) + q D 1 J 2 ( J + l ) 2 ] , [3.14a] while only one off-diagonal element was used to model the heterogeneous interaction between Chapter 3 Visible Spectrum of ZrC 10.0 146 1.0 0.8 0.6 0.4 4 0.2 (b)Q = 0/ 40 80 120 160 J(J+1) —i 1 1 1 1 1 0 50 100 150 200 250 300 J(J+1) 1.1 1.0 T3 l 0 > 0.9 4 0.8 0.8 0.4 0.0 n r 0 50 100 150 200 250 300 J(J+1) T 1 1 r 0 200 400 600 800 1000 J(J+1) Figure 3.33 Reduced rotational energies of selected ZrC levels. Each is strongly perturbed. The quantity E + kJ(J+l) has been subtracted from the energies to expand the vertical scale, where E and k are (in cm"1): (a) 17700 and 0.430, (b) 16488 and 0.434, (c) 17689 and 0.412 and (d) 17342 and 0.431. Open and filled circles respectively mark e and/parity levels. Chapter 3 Visible Spectrum of ZrC 147 the Q. = Oe level and the e parity component of the Q. — 1 level: Ho e/i e(J) = HoiVJ(J + l ) . [3.14b] With this matrix, only 30 of the 48 observed rotational energies from the two states could be fitted with an rms error commensurate with their uncertainties. The constants determined from the fit are given in Table 3.11. Fig. 3.34 is a plot of the observed energy levels; the calculated patterns are plotted as solid curves. It is clear that Eq. [3.14] does not accurately describe the data at high J. In the Q. = le component, for example, there is an avoided crossing for which the matrix cannot give account. Even if Eq. [3.14] were valid in this case, the results of the fit show a model deficiency in the physically unreasonable negative D 0 and D i constants. Table 3.11 Constants3 from the attempted deperturbation of the nearly degenerate Q = Oe and 1 levels of 9 0 Z r C near 17833 cm" 1 To 17833.0247(17) B 0 0.38911(20) 10 6 D 0 -24.0(14) T i 17832.9780(11) B i 0.42625(12) 10 6D, -2.13(83) qi -0.000181(21) 1 0 5 q D l 1.47(16) Hoi 0.59043(20) rms 0.0004722 #b 30 J b 11 J max a A l l constants in c m - 1 units except as indicated, with 3 a errors in parentheses bThese have the same meaning as in Table 3.5. Figs. 3.33 and 3.34 are typical of the complicated rotational level patterns encountered in the upper states of the observed ZrC bands; it is unfortunate that so few of them can be analysed as successfully as the low-lying levels. Fligh-level ab initio calculations may be helpful in interpreting the observed level structure, but the density of the observed electronic Chapter 3 Visible Spectrum of ZrC 148 -1 1 1 1 1 1 1 1 : 1—1 0 50 100 150 200 250 300 350 J(J+1) 90 Figure 3.34 Attempted deperturbation of the nearly degenerate fl = Oe and 1 levels of ZrC near 17833 c m - 1 . The curves are the calculated eigenvalues of the matrix in Eq. [3.14], using constants in Table 3.11; the constants were generated from a fit of the data plotted as filled circles. Open circles are observed data that cannot be fitted to an accuracy commensurate with their measurement uncertainty; the deperturbation is successful only at low J. Chapter 3 Visible Spectrum of ZrC states offer little hope that such calculations will be reliable. 149 3.4 Conclusions This chapter has described the experimental details of the jet-cooled spectroscopic studies of three molecules presented in this thesis. In the first of these studies, the electronic spectrum of zirconium monocarbide, ZrC, was recorded at high resolution in the 16000-19000 c m - 1 region. A total of 56 bands, most with perturbed upper states, have been examined in the first spectroscopic investigation of this molecule, and a large number of electronic states have been identified. ZrC has a rare X 3 Z + ground state, unlike the isoelectronic species Y N whose ground state is Xi'Z+; its analogue in 9 0 Z r C lies at T 0 = 187.83 c m - 1 and is readily populated even under jet-cooled conditions. Rotational analyses of the X 3 Z + ( r 0 = 1.8065 A, a>e = 889 cm" 1 for 9 0 Z r 1 2 C ) and a 1 ^ (r 0 = 1.7394 A, A G 1 / 2 = 890 cm- 1) states reveal a much shorter bond length for the latter. Two additional 1 E + states lying below 2500 c m - 1 have been identified in dispersed fluorescence spectra; anomalous vibrational intervals and 1 2 C / 1 3 C isotope shifts observed in dispersed fluorescence in the three low-lying states show that they all have the same symmetry. This evidence, plus the consistency of the small effective spin-spin constant = 0.5140 c m - 1 of the ground state with a aa' electron configuration, indicates that the four lowest electronic states, X 3 E + , a 1!^, blI.+ and c 1 ^ represent all possible arrangements of two electrons in the nearly degenerate 11a and 12a orbitals (both formed from Zr 55a + C 2pa). In particular, the tightly bound alIf state originates from the closed shell 11a 2 configuration. Hyperfine structure in the ground state of 9 1 Z r C (11.22% natural abundance, I = 5/2) has been studied; internal hyperfine perturbations (IHPs) have been analysed in a 3 Z + electronic state for the first time. The data show that a second order IHP occurs where the Fj and F 3 spin components cross near N = 2, and suggest that a first order IHP should appear near N = 23, Chapter 3 Visible Spectrum of ZrC 150 where the F 3 and F 2 spin components cross. Both fflPs arise via coupling of the fine structure components by the large Fermi contact interaction (bp = -0.0317 cm - 1 ) and further confirm the Zr(5sa) character of the ground state electron configuration. The confused level structure above 16000 c m - 1 , consisting of at least four close-lying, strongly perturbed II states (two singlets and two triplets, including the [16.2]3TL, state), is also consistent with this molecular orbital scheme and can be explained by the promotion of either a electron to a n orbital. Large A E ( 9 0 Z r - 9 4 Zr ) isotope shifts and rotational perturbations near the bottom of this level structure suggest further electronic structure below 16000 cm - 1 . Investigations of this region are warranted to improve understanding of the current data. High level ab initio calculations may also provide further insight. Chapter 4 151 Laser Spectroscopy of Zirconium Methylidyne (ZrCH, ZrCD and Zr 1 3 CH) 4.1 Background The significance of free radicals, in particular those containing transition metals (TMs), has motivated a large body of research on their microwave, infrared and electronic spectra in the last 75 or so years. Diatomic species, the simplest TM-bearing molecules and easiest to prepare with traditional high temperature sources, have received the overwhelming majority of this attention (1, 2). However, the dwindling number of such systems that remain unstudied and the wider applicability of laser ablation and jet-cooling techniques have begun shifting the focus to polyatomic (especially triatomic) systems. Analyses of their spectra are doubtless far more challenging than for similar diatomic molecules; added complexity arises from the much higher density of vibrational and rotational states and, for linear species, the Renner-Teller effect. For TM-containing triatomic molecules, the amount of rotationally analysed electronic data is still sufficiently small that it can be reviewed completely. The first rotational analysis of a TM-containing triatomic molecule appeared in 1983, when Trkula and Harris (3) reported 1 A " - X^A! electronic spectra from the bent species CuOH(D). The molecules were prepared by reaction with peroxide ( H 2 0 2 or D 2 0 2 ) of copper atoms from a hollow cathode sputtering source; r 0 structures were derived for each electronic state. Virtually all subsequent research on such species has employed jet-cooled techniques, beginning with work on the visible spectra of the Y N H , Y N D and Y 1 5 N H isotopomers of linear yttrium imide, first reported in 1990 by Simard et al. (4). Two later papers from the same lab described the rotational, fine and hyperfine structures of the (0,0) bands of its B 2 E + - X 2 Z + (5), A 2 n - X 2 Z + and A" 2n - X 2 S + systems (6). Numerous rotational perturbations were identified in these bands. Chapter 4 Visible Spectra of ZrCH, ZrCD and Zr13CH 152 In a series of publications, Steimle and co-workers (7-9) provided a detailed account of the near infrared A 2 A j - X 2 A j system of Y C 2 , including the ground state vibrational structure, as well as the geometry, fine and hyperfine structures and dipole moments of both states; they also reported corresponding results for the A 2 n - X 2 S + system of ScNH (10). While the latter species is linear, the strongly bent C 2 v geometry of Y C 2 is more properly regarded as T-shaped, with the metal ionically bonded to the side of the relatively stable C 2 moiety. High-resolution electronic data have also appeared recently for the coinage metal dichloride family. The Brown group at Oxford has produced the first rotational analyses (11, 12) of the well-known 460 nm and 360 nm systems of N i C l 2 (13-19), providing the first conclusive evidence of its X 3 Z g ~~ ground state symmetry. A similar body of low- to medium-resolution studies on C u C l 2 dating from the early 1960s (13, 14, 20-22) long presaged extensive subsequent rotational analyses and identification of its inverted X 2 n g ground state (23-30). Very recently, the Brown group has also reported a preliminary rotational analysis (31) of the 310 nm band system of C o C l 2 (14) indicating that this molecule has an X 4 A g ground state. The isomeric possibilities from bonding the C N ligand to TMs, i.e., M C N and M N C , the linear cyanide and isocyanide, and the T-shaped structure, raise the question of the most energetically favourable conformer; recent electronic data provides answers for two cases. Lie and Dagdigian (32) observe an isocyanide geometry for iron based on the F e N 1 2 C —> F e N 1 3 C rotational isotope effect; its X 6 A ground state is analogous to those of several iron monohalides. The most favourable conformer for nickel was recently established here in the Merer lab as the cyanide via 1 4 N —* 1 5 N isotopic substitution (33). These are the first high-resolution optical studies of a TM-containing isocyanide and cyanide respectively. Further work in the Merer lab has generated data for many triatomic TM-containing species. The electronic spectrum of lanthanum imide, LaNH(D), the third and final Group 3 Chapter 4 Visible Spectra of ZrCH, ZrCD and Zr13CH 153 imide studied under high resolution, appears mostly in the red and near infrared and is discussed in Chapter 5 of this thesis; the results of this analysis were used by Steimle et al. to measure dipole moments of its X 2 £ + and B 2 £ + states (34). A search for the still unobserved yttrium methylidyne (YCH) species led to the serendipitous discovery of Y O H (35), making it the first spectroscopic study of a linear T M hydroxide. Vibronic coupling effects are strongly pronounced for both Y O H and LaNH. The T M methylidyne species M C H have been of interest to the Merer group for about a decade now, and while attempts to observe methylidynes of yttrium, molybdenum and rhenium have thus far been unfruitful, studies with other TMs have met with resounding success. As of this writing, LIF spectra have been observed for the methylidynes and deuterated counterparts of vanadium (36), tungsten (37), titanium (38), chromium (39), niobium (40), tantalum (40) and zirconium. These species are linear in all observed electronic states, and in most cases have ground state symmetries identical to those of the corresponding isoelectronic T M mononitrides; the last of these forms the subject of this Chapter. Relevant to the present results is work on isoelectronic ZrN. The first analysis of this molecule by Bates and Dunn (41) identified two electronic systems, the A 2 n - X 2 2 + yellow system (origin near 17400 cm - 1 ) and the B 2 S + - X 2 S + violet system (origin near 24700 cm - 1 ) . This work and extensive subsequent studies of both systems (42-50) reveal numerous excited state rotational perturbations, as is the case for the isoelectronic species Y N H (5-6). Similarities in the electronic structures of ZrCH, ZrN and Y N H are expected; in fact, these three species are part of a larger family of isovalent TM-bearing species, most of whose B 2 E + , A 2 n and X 2 E + states have been studied in detail. Some of these have already been reviewed above; others are identified in the later discussion. This Chapter presents studies of the ZrCH, ZrCD and Z r 1 3 C H isotopomers of zirconium methylidyne. The next Section discusses the experimental apparatus and techniques used in the Chapter 4 Visible Spectra of ZrCH, ZrCD and Zr13CH 154 current work; these are very similar to those described in detail in Chapter 3, so only major differences are explained. Section 4.3 describes the appearance of the recorded spectra, the vibrational and rotational structure of the ground state, and the excited state vibronic and rotational structure. A discussion and conclusions are given in Section 4.4. 4.2 Experiment All the apparatus used in the ZrC study of Chapter 3 was also used here for the methylidyne experiments; the only additional component, a pulsed dye amplifier (PDA), is described later in this section. Since essentially the same laser ablation and LIF techniques of that study were also used, only major differences need be described. In fact, in the earliest (low-resolution) WSF experiments, ZrC and ZrCH often appeared in the same scan, so the complete lack of prior spectroscopic data on either of these species posed two challenges. The first was establishing the spectral carrier for each of the various new WSF bands. The second was to minimise such ambiguities by identifying what experimental conditions, if any, favoured the yield of one molecule over the other in the plasma chemistry. Extensive trial and error with various experimental parameters revealed two discriminating factors. One was the methane concentration of the CH 4 /He gas mixture. While the ZrC yield was steady and high over a wide range of C H 4 concentration (1-8%), maximum ZrCH production required the full 8% concentration. Occasional contamination of ZrCH signals by ZrC occurred as a result. Fortunately, the visible spectra of these molecules do not completely overlap; while the ZrC spectrum appears in the 16000-19000 c m - 1 region, ZrCH bands were found in the 15100-17600 c m - 1 region, above which no searches were conducted for this species. Otherwise, dispersed fluorescence (DF) data and deuteration (replacement of C H 4 by C D 4 ) were sufficient at low resolution to distinguish between ZrC and ZrCH bands. None of the high-resolution methylidyne data were blended by ZrC features. Chapter 4 Visible Spectra of ZrCH, ZrCD and Zr13CH 155 The other factor was the length of the reaction channel in the Smalley source. The best ZrC signal appeared with a 7 mm long channel, as shown in Fig. 3.2, while shorter channels gave progressively better methylidyne signals. In fact, the best Z rCH results came with essentially no reaction channel; it was machined from the top of the source to expose about 50% of the rod diameter to easy view from point D in Fig. 3.2. It should be noted, however, that some ZrC signal remained even under these conditions, while the 7 mm channel almost completely quenched the ZrCH signal. Z rCH was first identified from WSF scans monitoring resonance fluorescence. DF scans of new ZrCH features located its five lowest ground state vibrational levels, i.e., (001), (002), (010), (020) and (011). The (t)it) 2 u3) numbering here identifies quanta of the V i ( a + , C -H stretch), v2(7t, bend) and v 3 (o + , Zr -C stretch) vibrations. Low-resolution WSF scans monitoring fluorescence to each of these levels were systematically recorded. This was repeated for ZrCD using deuterated methane in the gas mixture. The five sets of WSF scans so recorded for each isotopomer cover the 14700-17600 c m - 1 region. Those recorded for ZrCH while monitoring LIF to (001) required the slightly slower laser scan rate given by Eq. [3.1]; this helped remove spurious signals caused by the coincidental near degeneracy of this interval with the AG1/2 quanta of both the X 3 E + and a 1 ^ states of diatomic ZrC (862 versus about 885 cm - 1 ) , and with vibrational intervals of the metastable a 3 A state of ZrO. As discussed in Chapter 3, this allowed a narrower monochromator slitwidth (1 mm versus 3 mm for the other WSF scans) to preserve molecular selectivity. DF spectra were recorded for all observed WSF features. Corresponding Z r 1 3 C H data were also recorded for some key bands, albeit with somewhat poorer quality as the limited availability of 1 3 C H 4 required some sacrifice of S/N optimisation. The strong WSF bands of ZrCH and ZrCD at the red end of the spectrum were also recorded at high resolution. They are generally weaker than those of ZrC; the cw ring laser Chapter 4 Visible Spectra of ZrCH, ZrCD andZr13CH 156 alone was insufficient to record some of them. A P D A (Lambda Physik FL2003) was therefore added to the experiment; its laser output gave a reasonable compromise between the low-flux, narrow-linewidth cw laser output and the high-flux, broad-linewidth output of the pulsed laser described in Section 3.2(c)(i). This apparatus is now described. The working principle of the P D A is almost identical to that of the pulsed dye laser, with only minor differences. Whereas the pulsed dye laser emission is self-stimulated with a reflexion grating tuned to a wavelength within the spontaneous broadband emission of the dye, the P D A amplifies external laser light, in this case the cw ring laser. Three dye cuvettes (two pre-amplifiers and one main amplifier) lie along the propagation axis of the cw beam; these are each pumped by a 308 nm pulsed (15 Hz) XeCl excimer laser (Lambda Physik, Compex 102) fed into each cuvette at right angles to the cw beam. A cylindrical lens focusses the excimer beam into a plane to match the dye volume through which the two lasers propagate. The cw beam stimulates pulsed laser emission from the first cuvette, which the other two cuvettes amplify. The resulting high-flux, low-linewidth pulsed output (Av ~ 200 MHz, slightly greater than the residual Doppler linewidth of spectral lines recorded with the cw laser alone) has the same wavelength as the cw input. Depending on the type and quality of dye used in the PDA, the intensity ratio of laser emission to A S E can be up to 100. As with the low-resolution pulsed dye laser, the digital delay generator (described in Section 3.2(d)) controls the XeCl pulse timing relative to other events in the pulsing sequence. The broadened output requires that the cw laser be scanned at about half the normal rate for maximum spectral resolution. 4.3 The Visible Spectrum of Zirconium Methylidyne 4.3(a) Description of the Spectrum More than 50 vibrational bands have been observed by low-resolution WSF and DF for Chapter 4 Visible Spectra of ZrCH, ZrCD andZr13CH 157 each of Z rCH and ZrCD in the 15100-17600 c m - 1 region; stick spectra for both species, given in Fig. 4.1, show the locations and intensities of the heads of the cold bands, i.e., those that arise from the zero-point vibrational level. The intensities approximate the signal-to-noise (S/N) ratio of the strongest feature in the DF spectrum from the upper state of each band. Numerous other bands arising from excited vibrational levels, i.e., hot bands, also appear; these are assigned via identification of corresponding cold bands in Fig. 4.1 and on the basis of their DF spectra as described in Section 4.3(b)(i). At this resolution, the zirconium isotope structure ( 9 0 Zr, 51.45%; 9 1 Z r , 11.22%; 9 2 Z r , 17.15%; 9 4 Z r , 17.38% and 9 6 Z r , 2.80%) is not resolved and the bands are all very similar in appearance, with rather weak P-type structure accompanied by two R-type heads. One R-head is generally much stronger than and lies slightly to the red of the other, although their relative intensities vary somewhat among the bands. Eleven of the strongest bands of each of ZrCH and ZrCD, from Fig. 4.1, lying mostly at the red end of the spectrum, were recorded at high (rotational) resolution; rotational assignments and line measurements are given for these in Appendices IV and V. For each of these bands, the first lines of each of its branches for the 9 0 ZrCH(D), 9 2 ZrCH(D) and 9 4 ZrCH(D) isotopomers can be assigned, from which the X 2 E+ ground state electronic symmetry can be unambiguously identified, as in ZrN and Y N H . In addition, the projection quantum number was determined for the upper state of each of these bands. Here, I is the bending vibrational angular momentum quantum number; for a given bending vibrational quantum number u 2 : The minimum value -|(1 - ( - 1 ) ° 2 ) is 0 or 1 as u 2 is even or odd, and the energy dependence of the € components is given by g 2 2 £ 2 , where the anharmonicity constant g 2 2 is generally much less than the bending frequency a>2. The cold bands have P" = 1/2, so the selection rule p = A + e + s , [4.1] |£| = n 2 , u 2 - 2 , u 2 - 4 , I ( i - ( - l ) 0 * ) . [4.2] s TO TO TO N 8 (a) (b) 17000 16500 16000 15500 laser wavenumber / cm Figure 4.1 Stick spectrum of (a) 9 0 Z r C H and (b) 9 0 Z r C D WSF bands. The plot indicates the location of the band head and its intensity 2 + as measured from its wavelength of maximal fluorescence. Only cold bands (i.e., those arising from X I , u = 0) are shown. oi 00 Chapter 4 Visible Spectra of ZrCH, ZrCD and Zr13CH 159 AP = 0, ±1 [4.3] limits the allowed P' values to 1/2 or 3/2. Faint hints of 9 6 ZrCH(D) and of the anticipated hyperfine structure of 9 1 ZrCH(D) (I = 5/2) occasionally appear in the rotational structure, but no analyses of these data were attempted. In fact, only for the most abundant 9 0 ZrCH(D) species do the branches extend high enough in N for satisfactory least squares fitting; nevertheless, isotope shifts of rotational lines of 9 2ZrCF£(D) and 9 4 ZrCH(D) proved valuable in assigning the upper state spin-orbit and vibrational structure in the observed bands. Nearly every band of ZrCH and ZrCD recorded at high resolution suffers to some extent from local rotational perturbations. In most cases the interacting states are dark and have very different rotational constants, as usually only one or two upper J levels are doubled by a perturbation; occasionally multiple local perturbations occur at various J values in a particular vibrational level, rendering least squares fitting of the data impossible. Some upper state progressions in the V2(bend) and V3(Zr-C stretch) vibrations were assigned from the WSF data, while the DF spectra provided corresponding ground state data. Only a slight amount of activity in the V j ( C - H stretch) vibration appeared in a few of the DF spectra, indicating that the observed electronic transitions are dominated by promotions of metal-based electrons. The rest of this Section discusses the vibrational and rotational structure of the ground electronic state, excited levels in the 15100-15800 c m - 1 range and higher lying excited levels. 4.3(b) The X 2 E + State 4.3(b)(i) Vibrational analysis Extensive DF data of ZrCH and ZrCD have mapped many ground state vibrational levels for these species; no evidence of other low-lying electronic states was found. Some typical DF spectra appear in Figs. 4.2-4.7; these are now discussed individually. Chapter 4 Visible Spectra of ZrCH, ZrCD and Zr13CH 160 Fig. 4.2 shows the patterns from the 15179 c m - 1 level of ZrCH in trace (a) and from its 15189 c m - 1 ZrCD counterpart in trace (b); these are the lowest-lying upper state levels found for their respective isotopomers. Based on the blue shift upon deuteration, the P' values obtained from high-resolution data and analogy with the A - X system origin of ZrN at 17400 c m - 1 , the cold absorption bands to these levels are assigned as A 2 ni/ 2 - X 2 E + (0,0); these are discussed in more detail in Section 4.3(c). The principal features of trace (a) form a progression in the V3 vibration, whose frequency is about 862 c m - 1 . This assignment is based on deuteration shifts (vide infra), analogy with similar results on previously studied methylidynes (36-40) and comparison with the known ground state value of co e = 889 c m - 1 of ZrC from Chapter 3. Its intensity distribution also supports the vibrational assignment of the laser band; the absence of a local minimum can be attributed to a nodeless upper state vibrational wavefunction, which is the case for the zero-point vibrational level only. Since the only emission from the A 2ni/ 2(00°0) level allowed by the vibrational selection rules is to totally symmetric (even u 2 with I = 0) levels, the weak appearance in this trace of the v 2 bending fundamental at 596 c m - 1 suggests some degree of vibronic coupling in the A 2 n state, i.e., the Renner-Teller effect. The At = 1 selection rule of this feature and the anticipated presence of a nearby B 2 S + state favour this interpretation. More evidence of vibronic coupling in this state appears later in the Chapter. The corresponding ZrCD spectrum in trace (b) differs slightly. The features identified for ZrCH are all present with a similar intensity distribution, but an additional progression in v 3 built on the overtone of v 2 also appears. A Fermi resonance that mixes levels within each (0u 2*U3) polyad, for which u 2 + 2U3 and I are constant, induces these features. A similar Fermi resonance enhancement upon deuteration was also observed for the ground states of T iCH and TiCD (38). In both cases, the mixing is stronger in M C D than in M C H because the v 3 and v 2 vibrational frequencies are more nearly in a 2:1 ratio, although even in the deuterated species, 13, Chapter 4 Visible Spectra of ZrCH, ZrCD and Zr CH 161 (001) laser scatter (a) (010) (002) (003) 1 — 1 — 1 — < ~ (001) laser scatter (010) ( b ) (002) (020) (021) ( 0 0 3 ) ( o 2 2 ) 1 r --2000 -500 -1000 -1500 -2500 displacement from laser / cm -1 Figure 4.2 DF spectra of the A 2 n 1 / 2 , u = 0 levels of (a) ZrCH at 15179 cm 1 and (b) ZrCD at 15189 c m - 1 . A Fermi resonance in the ground state of ZrCD enhances the (02u3) progression. Chapter 4 Visible Spectra of ZrCH, ZrCD and Zr]3CH 162 the relative intensities of the principal (00° 1) and induced (02°0) features, shown in Fig. 4.2(b) for ZrCD in particular, show sufficiently weak mixing that u 2 and 03 may still be regarded as approximately good quantum numbers. As expected, the principal progression that carries the oscillator strength and the induced progression have essentially the same intensity profile. The assignments of the v 2 and V3 vibrations are supported by their isotope shifts; v 2 should have the larger shift as it is essentially a hydrogen wagging motion. The richest DF spectrum of ZrCH is from the 15988 c m - 1 level, as shown in Fig. 4.3(a). Fluorescence to no less than 10 ground state vibrational levels occurs; features marked by asterisks are from the frequently encountered ZrO impurity. Once again, both totally symmetric and non-totally symmetric fluorescence features betray the presence of excited state vibronic coupling. The (020) peak is somewhat broader than the others due to unresolved splitting of the € = 0 and 2 levels. In some of the other DF spectra, this feature is sharp and unblended with measurements that cluster around two averages of 1164 and 1189 cm - 1 , suggesting an I splitting of about 25 cm - 1 . An example of a DF spectrum for which £ structure is resolved is presented shortly. The limited available supply of 1 3 C H 4 was sufficient to locate the corresponding band head of Z r 1 3 C H near 15980 c m - 1 (among others) and record its DF spectrum; the latter is the only DF spectrum recorded for this isotopomer and appears in Fig. 4.3(b). Some broad laser artefacts and weak unassigned features appear in this spectrum, but most of the vibrational levels from Fig. 4.3(a) can be identified. Since the bending vibration involves little carbon motion, its frequency is not very sensitive to isotopic substitution on this atom, while 0D3 drops by 4%. In Fig. 4.4, the DF spectrum of the 16461 c m - 1 level of Z rCH shows resolved I components of the features with u 2 = 2; however, without high-resolution absorption spectra from these levels, unambiguous €" assignments, which establish the sign of g 2 2 , cannot be 13, Chapter 4 Visible Spectra of ZrCH, ZrCD andZrJCH 163 (022) * (013) -500 -1000 -1500 -2000 -1 -2500 -3000 displacement from laser / cm Figure 4.3 DF spectra of the (a) 15988 cm 1 level of Z rCH and (b) the corresponding 15980 —1 13 cm level of Zr CH. Asterisks in trace (a) mark ZrO features. Nine of the ten levels in ZrCH 13 can be identified in Zr CH. T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 0 -500 -1000 -1500 -2000 -2500 -3000 displacement from laser / cm - 1 Figure 4.4 DF spectrum of the 16461 c m - 1 level of ZrCH. The /-type splittings of the (02° ' 2 l ) and (02°' 22) peaks are resolved in this spectrum. Chapter 4 Visible Spectra of ZrCH, ZrCD and Zr13CH 165 made. The (02° 1) and (0221) features are discernible; one appears as a weaker shoulder slightly to the red of the other with about a 26 c m - 1 separation. The nature of the DF scans allows better resolution of the (02^2) features. They typically cover a monochromator wavelength range of AXM = 200 nm and the AX = 1.1 nm bandpass is constant in wavelength, so the wavenumber resolution Av = A A A , M 2 improves with X M . Thus the 4 g 2 2 = 28 c m - 1 splitting is best resolved in the highest members of the (02*1)3) progression. DF spectra of ZrCD offer a clue in establishing the sign of g 2 2 . Since the Fermi resonance mechanism is diagonal in £ and the (00° 1) level appears below (02°0) in this isotopomer, the separation of the £ components of the bending overtone level increases upon deuteration if g 2 2 is negative, or decreases if g 2 2 is positive. No £ splittings were observed in any DF spectra of ZrCD, which suggests that the Fermi resonance suppresses them and that g 2 2 = +7.0 cm- 1 . Several hot absorption bands from the (01*0) and (00°1) levels appeared in the WSF spectra; these were generally rather weak, consistent with the cold molecular beam, and could usually be identified from DF spectra of their upper states on the basis of a (00°0) feature appearing respectively 596 or 862 c m - 1 to the blue of a laser band. An example is shown in Fig. 4.5. Trace (a) shows DF from the upper state of the 15109 c m - 1 Z rCH band. Its lowest wavelength feature appears 596 c m - 1 to the blue of the laser band, which represents an upper state appearing at 15705 c m - 1 . Such a level does indeed exist; its DF spectrum is shown as trace (b). The two traces are essentially identical, except that trace (a) has a better S/N ratio. Evidently, the Franck-Condon factor for absorption to this level is so much larger for (010) than for (000) that it overcomes the low population of the excited vibrational level. Weak features in some DF spectra suggest v^ activity. The clearest evidence is from the 16680 c m - 1 level of ZrCH; its DF spectrum is shown in Fig. 4.6. Unfortunately, this region of the absorption spectrum also has many closely-spaced ZrC bands (see the right ins