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Hyperfine and internal rotation effects in the microwave spectra of some gaseous molecules Hensel, Kristine D. 1993

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HYPERFINE AND INTERNAL ROTATION EFFECTS IN THE MICROWAVE SPECTRA OF SOME GASEOUS MOLECULES  By Kristine D. Hensel B. Sc., Queen’s University, 1989  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES CHEMISTRY  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  December 1993  ©  Kristine D. Hensel, 1993  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Chemistry The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date:  Abstract  The microwave spectra of the following gaseous molecules have been observed and an alyzed, using Stark-modulated microwave spectroscopy and cavity microwave Fourier transform (MWFT) spectroscopy. Bromine ‘ 0-isocyanate, BrNC’ 8 0 8  The microwave spectrum of BrNC O has been 18  measured in the frequency region 23-52 GHz, using a Stark-modulated microwave spec trometer. Because the spectrum is that of a prolate near-symmetric rotor with strong a-type and weak b-type transitions, perturbations in the quadrupole hyperfine patterns of Br were used to improve the precision of A . The geometry of the molecule has been 0 determined; in particular, the NCO chain has been found to have a bend of  -S.’  8° away  from Br. C1 SiH Dichiorosilane, 2  The b-type rotational spectrum of C1 35 has been re SiF1 28 2  measured in the frequency region 10-16 GHz (J=1—10) using a cavity MWFT spectrome ter. The MWFT technique has permitted resolution of the complex hyperfine patterns observed for this molecule, which in turn has allowed the precise determination of the Cl nuclear quadrupole coupling constants. In particular, perturbations in the transition have been analyzed to obtain a value for  Xab  .  918-827  The quadrupole coupling  tensor has been diagonalized to yield principal values, and the results are discussed in terms of the bonding in 2 C1 SiH .  11  Tetrolyl fluoride, CH -CC-COF 3  The microwave spectrum of the unstable molecule  tetrolyl fluoride has been observed for the first time. The a-type rotational spectrum ob served with a Stark-modulated microwave spectrometer is very dense, owing to internal rotation of the methyl group. The spectrum has also been measured in the frequency range 9-17 GHz using a pulsed jet cavity MWFT spectrometer. Cooling in the jet has removed all internal rotation states other than m =0 and m =1, permitting assign ment of the microwave spectrum. The threefold barrier to internal rotation has been confirmed to be very low (V =2.20(12) cm 3 .) 1 Metal Halides: AgCI, Aid, CuC1, mCi, InBr, InF, YC1  An apparatus has  been constructed to produce metal compounds using laser ablation and to investigate their rotational spectra with a microwave Fourier transform (MWFT) cavity spectrome ter. Metal halides have been produced by ablation of metal rods in the presence of a halogen-containing gas, using a Q-switched Nd:YAG laser (532 nm). The first seven such compounds that have been studied are silver chloride, aluminum (I) chloride, copper (I) chloride, indium (I) chloride, indium (I) bromide, indium (I) fluoride, and yttrium (I) chloride; the pure rotational spectrum of YC1 is reported here for the first time. Nuclear spin-rotation coupling constants have been determined for the first time for A1C1, CuC1, InCl, InBr, and YC1, as has eQq(Cl) of YC1. Where possible, nuclear spin-rotation cou pling constants have been used to examine the electronic structures of the molecules, and  eQq(Cl) of YC1 has been interpreted in terms of the ionicity of the Y-C1 bond. Values of the rotational and nuclear quadrupole coupling constants have also been improved for the metal halides.  111  Table of Contents  Abstract  ii  List of Tables  vii  List of Figures  ix  Acknowledgement  Xi  1  Introduction  1  2  Theory  4  2.1  Energy levels of the rigid rotor  4  2.2  Selection rules for rotational transitions of a rigid rotor  2.3  Centrifugal distortion  10  2.4  Structure determination  14  2.5  Nuclear quadrupole hyperfine interactions  20  2.6  Nuclear spin-rotation interactions  31  2.7  Internal rotation  34  Experimental Methods  39  3  3.1  3.2  Stark-modulated microwave spectroscopy  .  7  .  .  .  39  3.1.1  Theory  39  3.1.2  Instrumentation  41  Microwave Fourier transform spectroscopy  iv  43  4  S  6  3.2.1  Theory  3.2.2  Instrumentation  .  43 49  0 8 The Microwave Spectrum of BrNC’  57  4.1  Introduction  57  4.2  Experimental methods  58  4.3  Observed spectrum and analysis  59  4.3.1  Harmonic force field and structure  67  4.3.2  Discussion and conclusions  71  C1 75 2 The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, SiH 5.1  Introduction  75  5.2  Experimental methods  76  5.3  Observed spectrum and analysis  77  5.4  Discussion  86  The Microwave Spectrum of Tetrolyl Fluoride  90  6.1  Introduction  90  6.2  Experimental methods  91  6.3  Results and discussion  91  6.3.1  Prediction of transition frequencies  92  6.3.2  Selection rules  94  6.3.3  Assignments  96  7 Microwave Spectra of Metal Halides Produced Using Laser Ablation 103 7.1  Introduction  103  7.2  Experimental methods  104  7.3  Observed spectra  108 v  7.4  7.3.1  AgC1  108  7.3.2  A1C1  110  7.3.3  CuC1  110  7.3.4  mCi  113  7.3.5  InBr  115  7.3.6  InF  118  7.3.7  YC1  122  Analysis and discussion  125  7.4.1  147  Conclusions  Appendices  149  A Spherical Tensor Relations  149  Bibliography  151  vi  List of Tables  2.1  2 and symmetry species of rotational Character table for the point group D and direction cosine fullctions  10  2.2  Selection rules for rigid asymmetric top rotational transitions  11  4.1  O (in MHz) 18 BrNC Measured transitions of 79  63  4.2  0 (in MHz) 8 BrNC’ Measured transitions of 81  65  4.3  Spectroscopic constants of BrNCO  68  4.4  Structural parameters of bromine isocyanate  70  4.5  Centrifugal distortion constants of BrNCO  71  4.6  Spectroscopic constants of the ground state average structure of BrNCO  72  4.7  Principal values of the bromine quadrupole coupling tensor  73  5.1  5 Measured transitions of 2 C1 3 SiH 28  84  5.2  Chlorine nuclear quadrupole coupling constants of 2 C1 SiH  87  5.3  Cl principal quadrupole coupling constants and Si-Cl Comparison of 35 bond lengths  89  6.1  Measured transitiolls of tetrolyl fluoride  99  6.2  Spectroscopic constants of tetrolyl fluoride  100  6.3  Measured transitions of ‘ C-tetrolyl fluoride 3  101  7.1  Measured transitions of AgC1  109  7.2  Measured hyperfine components of the J=1-0 transition of A1C1  112  7.3  Measured transitions of CuCl  114 vii  116  7.4  Measured transitions of mCi  7.5  Measured transitions of lnBr  7.6  15 ‘ F 9 Measured hyperfine components of the J=l-0 transition of 1n’  7.7  Measured transitions of YC1  124  7.8  Spectroscopic Constants of AgC1  127  7.9  Spectroscopic Constants of A1C1  128  7.10 Spectroscopic Constants of CuC1  129  7.11 Spectroscopic constants of mCi  130  7.12 Spectroscopic constants of InBr  130  F 9 1n’ 7.13 Spectroscopic constants of 115  131  7.14 Spectroscopic constants of YC1  131  7.15 Ratio of Cl quadrupole coupling constants  133  7.16 Ratios of nuclear spin-rotation coupling constants  136  .  .  119  viii  .  122  List of Figures  2.1  Correlation between free (V =O) and hindered (V 3 3  —*  co) threefold internal  rotor states  37  3.1  Schematic cross-section of a Stark cell  42  3.2  Schematic diagram of the Stark-modulated microwave spectrometer.  3.3  Schematic circuit diagram for the cavity MWFT spectrometer  51  3.4  Schematic pulse sequence diagram for the cavity MWFT spectrometer  55  4.1  Rotational energy levels of 79 0 8 BrNC’  61  4.2  The molecular structure of BrNCO, given in the principal inertial axis  .  44  system of 79 O 16 BrNC  74  5.1  Rotational energy level diagram of 2 C1 3 SiH 28 5  79  5.2  Schematic diagram of the  81  5.3  Portion of the  6.1  The m =0, 44-33 transition of tetrolyl fluoride  7.1  Schematic diagram of the arrangement of the cavity MWFT spectrometer  —  918 —  827  transition of 2 C1 3 SiH 28 5  35 SiH 28 2 827 rotational transition of C1  82 97  and the Nd:YAG laser  105  7.2  Schematic diagrams of the nozzle caps  107  7.3  A portion of the J=1-0 rotational spectrum of 27 C1 37 A1  111  7.4  The J=2-1 rotational transition of 115 Br and 81 79 1n 1n 5 “ Br  117  ix  7.5  The F =3.5-4.5 hyperfine component of the J=1-O rotational transition of 1 121  F 9 1n’ 115 7.6  The overlapped F=2.5-1.5,3.5-2.5 hyperfine components of the J=2-1 ro tational transition of C1 35 Y 89  126  7.7  Schematic diagram of the energies of the valence molecular orbitals of A1C1 138  7.8  Schematic diagram of the energies of the valence molecular orbitals of CuC1 141  7.9  Schematic diagram of the energies of the valence molecular orbitals of an indium (I) halide, InX  143  7.10 Schematic diagram of the energies of the valence molecular orbitals of YC1. 146  x  Acknowledgement  This thesis would not have been possible without the guidance and encouragement of my supervisor, Mike Gerry, who was occasionally more certain about the eventual production of this thesis than I was. I would like to thank him for being a great supervisor and a nice person. I’ve been fortunate to work in a lab constantly filled with experienced (and Germanspeaking) post-docs. I would like to thank all of them for their help: Wolfgang Jäger, for fixing everything that wouldn’t go and for trying hard on my Deutsch 200 Hausaufgaben; Christian Styger, for helping to set up the first laser ablation experiments; Nils Heineking, for his advice on tetrolyl fluoride; and Holger Muller, for living up to his nickname. I would also like to thank Yunjie Xu for setting a good example, and Beth Gatehouse and Kaley Walker for any times they asked me questions that I could answer, making me feel like I’d learned something over the last four years. I would also like to thank those people outside the lab who contributed to this work: Anthony Merer, for taking the time to help me sort out where nuclear spin-rotation cou pling constants come from; Mark Barnes and Photos Hajigeorgiou, for trying to explain those electronic spectroscopy papers to a l+state spectroscopist; Chris Chan, for deal ing with unhappy computers and overheated electronics; Ilona Merke, for trans-Atlantic fits of the dichlorosilane data in the midst of finishing her own thesis; Mike Pungente, for making sure that the synthesis of tetrolyl fluoride produced more than a black tar; Bill Henderson of the Mechanical Shop, for making things correctly even when my instruc tions were wrong; and Gordon Burton, for advice on thesis production from the front lines. I would like to thank NSERC for financial support. xi  I thank my family, for their mostly-unquestioning support and because this is the only part of the thesis they will read. I thank Anne Dunlop, for living with me for four years, taking me dancing, and thrilling me with Fun Facts, linguistic and otherwise. I thank my e-mail friends for providing a tenuous link to the outside world. I thank the graduate students of the Astronomy Department, who adopted me as one of their own, although Ted is still in trouble for wimping out of German. Finally, I thank Brad, for making not working on this thesis so much fun.  xii  Chapter 1  Introduction  In molecular spectroscopy, discrete molecular energy levels are investigated via transitions between levels, which occur when a molecule emits or absorbs electromagnetic radiation. Microwave spectroscopy is that branch of molecular spectroscopy which uses radiation with frequencies between  GHz and —lOO GHz. For molecules in the gas phase, most  of the transitions which fall within this frequency range are between rotational energy levels, and so microwave spectroscopy is also referred to as rotational spectroscopy. Since the positions of these levels depend on the masses of the atoms and their geometrical arrangements in a molecule, microwave spectroscopy is an important source of structural information for gaseous molecules. Most microwave spectroscopic studies can probe the ground electronic and vibrational states, and this structural information is very useful for theoretical calculations of potential energy surfaces near the potential minimum. The Hamiltonian operator which governs the positions of the energy levels is param eterized, and the ‘molecular constants’ are those values of the parameters which lead to the best representation of the experimentally-determined positions of the energy levels. These constants are obtained by least-squares fitting of the observed transition frequen cies to an appropriate Hamiltonian. Often a ‘bootstrap’ method is used, which involves fitting to observed transitions and predicting other transitions; the predicted transitions can then be measured and included in the fit to improve the precision of the molecular constants. Rotational energy levels may also be split or shifted by interactions between the  1  Chapter 1. Introduction  2  rotational angular momentum J and another angular momentum. Hyperfine effects are commonly observed in rotational spectra, the result of interactions between the rotation of the electron ‘cloud’ and one or more nuclear spins I. ‘Coupling’ of J and I may occur via different mechanisms, to produce nuclear quadrupole hyperfine effects and/or much smaller nuclear spin-rotation hyperfine effects. Another possible interaction is that between the overall rotation of the molecule and some rotation which is internal to the molecule. In the studies described in this thesis, both hyperfine and internal rotation effects in rotational spectra have been investigated. In Chapter 4, a microwave spectroscopic investigation of the unstable molecule bromine 0-isocyanate, BrNC’ 8 ‘ 0, is reported. This study follows an earlier one from this labora 8 tory [1, 2] in which the microwave spectrum of BrNC’ 0 was investigated. The structure 6 of BrNCO could not be determined unambiguously in Ref. [2], and the present study, using isotopically-labelled bromine isocyanate, has permitted a structure determination. This determination was greatly simplified by the observation of rotational transitions which were perturbed by second order nuclear quadrupole hyperfine interactions [3]. The study of BrNC’ 0 was a fairly conventional microwave spectroscopic investi 8 gation, using the well-established technique of Stark modulation [4, 5] to improve the signal-to-noise ratio of weak microwave absorption signals. The other investigations de scribed here have been motivated, at least in part, by the development of other microwave techniques, especially that of pulsed molecular beam cavity microwave Fourier transform (MWFT) spectroscopy. This technique, first described in the early 1980s [6], allows for much greater resolution of spectral lines than is possible in Stark spectroscopy. As is detailed in Chapter 5, the microwave spectrum of dichiorosilane, 2 C1 has been re SiH , investigated in order to resolve the copious nuclear quadrupole hyperfine structure which could not be fully resolved in an earlier Stark-modulated spectroscopic study [7]. The complete Cl nuclear quadrupole coupling tensor has been determined, which has in turn  3  Chapter 1. Introduction  C SiH [ 8]. 1 provided information regarding the nature of the Si-Cl bond in 2 In addition to improved resolution, the supersonic expansion of the molecular beam employed in cavity MWFT spectroscopy allows the study of molecules at low rotational and vibrational temperatures.  As such, cavity MWFT spectroscopy is a very useful  technique when a microwave spectrum is potentially very crowded because of rotationally and vibrationally excited molecules which are undergoing rotational transitions. This problem is commonly encountered when molecules with unhindered internal rotation or 3 torsion are studied. In Chapter 6, the microwave spectrum of tetrolyl fluoride, CR CC-COF, is reported. As is described in that chapter, the spectrum of tetrolyl fluoride is very dense when recorded at room temperature using a Stark-modulated microwave spectrometer, and analysis would have been virtually impossible without some method of selectively removing some of the torsionally-excited rotational transitions. Using the molecular beam, only the lowest torsional levels were populated, and the few remaining lines in the microwave spectrum could be assigned with relative ease, permitting an approximate determination of the barrier to internal rotation of the methyl group in tetrolyl fluoride. Finally, laser ablation has been used to create metal-containing molecules in the gas phase, as described in Chapter 7. Seven diatomic metal halides were formed using this technique, from solid metal target rods and halogen-containing gas samples. The molecules were studied using cavity MWFT spectroscopy, and so even unstable metal halides could be stabilized in the cold molecular beam and observed. With this technique, the pure rotational spectrum of yttrium (I) chloride has been observed for the first time. The cavity MWFT study has resulted in improved nuclear quadrupole coupling constants, and many nuclear spin-rotation coupling constants have been determined for the first time. Where possible, these latter constants have been related to perturbations of the electronic ground state caused by excited electronic states [9].  Chapter 2  Theory  2.1  Energy levels of the rigid rotor  The Hamiltonian for a rigid rotating molecule may be expressed as (2.1)  Hrot  where J, J,, and J refer to components of the rotational angular momentum J along the x, y, and z principal inertial axes, respectively. The rotational constants B 9 (g=x,y,z)  are related to the principal moments of inertia 1 of the molecule, and are  given in frequency units 1 as 9 B  (2.2) .  The principal axes are usually labelled a,b,c such that Ia  I,  I. In this case, the  rotational constants are rewritten as Ba = A, Bb = B, B = C. All of the symmetric rotors discussed in this work are prolate (cigar-shaped), where the principal inertial axis a has both the largest rotational constant and the highest symmetry; in the discussion which follows, x,y,z will be transformed to a,b,c as z  —*  a,x  —*  b, y  —*  c. In molecules  which are oblate (disc-shaped) symmetric rotors, the c axis has the highest symmetry, and the usual transformation is z  —*  c,x  —*  a, y  —*  b [10].  Matrix elements of the rotational Hamiltonian are calculated using symmetric rotor basis functions JKM >, where K is the quantum number associated with the projection of J along the molecule-fixed z axis and M is associated with the projection of J along Unless otherwise stated, all energies in this thesis will be given in frequency units. 1 4  Chapter 2.  Theory  5  the space-fixed Z axis. Both K and M may take integral values from —J to +J. In the symmetric rotor basis, the non-zero matrix elements of the squared angular momentum operators are given as: <JKM  IJ 2  JKM>  2 K  (2.3)  IJKM> = <JKMIJ 2 <JKMIJ JKM> 2  =  [( +1)  -  K2]  (2.4)  IJKM> = —<JK+2MIJ 2 <J,K+2,MIJ JKM> 2 [J(J +1)  —  K(K+1)] [J(J +1)  —  . 4 (K±1)(K+2)] (2.5)  The matrix elements of Hrot are thus given by <JKM Hrot JKM> = AK 2+ <J,K±2,MHrotJKM> =  BC  B+C  [J(J + 1)  —  K2]  (2.6)  [J(J+1)K(K±1)]  [J(J+1)—(K±1)(K+2)]. (2.7) In the absence of an external electromagnetic field, the energy levels of a rigid rotor are independent of M. For rigid linear molecules, there is no rotational angular momentum about the za symmetry axis, and hence K = 0. Since B=C, the off-diagonal elements of Hrot disap pear, and the energy levels may be given as a function of J: tIJ>BJ(J+l). 0 Ej<JIHr  (2.8)  For a rigid prolate symmetric top, an A dependence is introduced to the rotational energy since K may now be non-zero. Again, Hrot is diagonal since B=C, and the energy levels  Chapter 2.  Theory  6  depend on both J and K: EJK  (2.9)  = <JK Hrot JK> = BJ(J + 1) + (A  —  (2.10)  . 2 B)K  For an asymmetric rigid rotor, ABC and matrix elements of Hrot which are off-diagonal by 2 in K are non-zero. Hrot may be simplified by applying a Wang trans formation [11], which results in a new set of basis functions, related to the symmetric top basis functions as  I  J0>  [  JK> =  IJK>  (2.11)  JO>  =  JK > + J, —K>]  (2.12) (2.13)  HJK>—IJ,—K>].  In this new basis, Hrot consists of four smaller block diagonals, which may more easily be diagonalized. As in the original Hamiltonian, elements which are even and odd in K do not interact, and so the four submatrices which must be diagonalized may be denoted E+,E,0+, and 0-, where E and 0 stand for even and odd K, respectively; E+ and 0+ involve symmetric combinations of symmetric top basis functions (see Eq. 2.12) and E  and 0- involve antisymmetric combinations (Eq. 2.13). Since the Hrat matrix is not diagonal in K for an asymmetric rigid rotor, K is  no longer a ‘good’ quantum number.  For a near-symmetric prolate or oblate rotor,  eigenfunctions of Hrot will strongly resemble the symmetric top basis functions. This correspondence is used to label the 2J+1 asymmetric top wavefunctions which exist for each value of J. A common notation is prolate case (z  —*  JKaKc  where Ka is the value of K in the limiting  a) and K is the value of K in the limiting oblate case (z  —*  c).  The degree to which a molecule resembles a symmetric rotor may be represented in various ways. The degree of asymmetry may be represented by the Wang asymmetry  Chapter 2.  Theory  7  parameter, b, where [11] C-B bp=2ABC;  (2.14)  b=0 corresponds to a prolate symmetric rotor, and b= —1 corresponds to an oblate symmetric rotor. For a prolate near-symmetric rotor (b0), the rotational Hamiltonian may be rewritten in terms of b as Hrot  = B + CJ 2  + [A  —  B H- Cj  Hwang (br),  (2.15)  where Hwg (l)) = a 2 + b(J 2  2). —  b  (2.16)  The reduced energy W(b) corresponding to Hwg (br) may be expressed in terms of an infinite series in b, which depends only on b and the quantum numbers for a given asymmetric rotor energy level. Analytical expressions for and values of W(b) have been tabulated [10, 12]. For more asymmetric molecules, rotational energies must be calculated by diagonalizing the Hrot matrix.  2.2  Selection rules for rotational transitions of a rigid rotor  The molecular rotational transitions reported in this thesis are caused by a coupling of the electric component of an external electromagnetic radiation field with the electric dipole moment  t  of the molecule. When the external radiation oscillates at a frequency  mn, corresponding to the frequency of a molecular rotational transition from state m to 1 state n, the probability that the molecule in state m will undergo this transition is given by Prn-n = p(i’mn)Bm_*n,  (2.17)  8  Chapter 2. Theory  where P(Vmrj is the density of the radiation at frequency mn  Bm÷n is the Einstein  coefficient for absorption for that transition, and is given by (2.18) The matrix elements <n  [1FI  m> (F=X, Y Z) thus depend on the projections of p  onto the space-fixed F axes, and are given in the basis of rotational eigenfunctions of the molecule. However, since the electric dipole moment is determined by the positions of electrons and protons in the molecule, it is a molecule-fixed quantity, and its space-fixed components will change with rotation. It is thus necessary to transform matrix elements defined with respect to the space-fixed axis system to elements defined with respect to the molecule-fixed axis system, using direction cosine matrix elements. If the components of p projected onto the principal inertial axes of the molecule are considered to be constant with rotation, then <fl  where  ‘Fg  FgI>  lIFIm >  g =x,y,z,  (2.19)  is the direction cosine between the space-fixed F axis and the molecule-fixed  g axis. Using symmetric top basis functions, the direction cosine matrix elements may be factored as follows: <J’K’M’  JKM>  =  <  J FgI J>< J’K’  9 F 4  JK >< J’M’  JM>. (2.20)  Analytical expressions for the values of the matrix elements on the right-hand side of Eq. 2.20 have been tabulated [13]. Most microwave spectroscopic experiments use plane-polarized radiation, and thus the molecule’s electric dipole moment will interact with radiation oscillating in only one space-fixed direction. If this direction is chosen to be the Z axis, then the properties of  Chapter 2. Theory  9  the direction cosine matrix elements place the following constraints on allowed rotational transitions for a rigid symmetric top [13]: zSJ = 0, +1 =  (2.21)  0  (2.22)  0.  (2.23)  Transitions with LJ= --1, 0 and +1 are known as F-,  Q-  and R-branch transitions,  respectively. For a rigid linear molecule, K is always zero, and hence transitions with are trivial; the selection rules for a linear molecule are then zJ=±1, zM=0. Note that  t  p for both linear molecules and symmetric tops, where z is the symmetry  axis. Selection rules for rotational transitions in asymmetric molecules are somewhat more complicated. J and M are still good quantum numbers, and the selection rules zSJ=0,±1 and zM=0 still hold. However, the selection rule zK=0 has no meaning, since K is no longer a good quantum number. As was stated in Section 2.1, the asymmetric rotor Hamiltonian may be factored into four submatrices upon application of the Wang transformation. The asymmetric rotor wavefunctions form an irreducible representation in the group D , whose operations 2 correspond to an identity operation (A) and rotations of 1800 about each of the three principal inertial axes (C, C, and Cf). Upon application of a Wang transformation to the Hrot matrix, each of the four submatrices is of a different symmetry species according to this group. The character table for D 2 is given in Table 2.1, along with the symmetries of the submatrices for both even and odd values of J. The symmetries of the rotational states may also be determined according to whether Ka and K are even (e) or odd (o), as shown in Table 2.1. In order that a given rotational transition J’ K K  .‘  J Ka K be allowed, the  Chapter 2. Theory  A Ba Bb B  10  E 1 1 1 1  ca2  rib  riC  ‘—2  ‘—‘2  Jeven  Jodd  1 1 —1 —1  1 1 1 —1  1 1 —1 1  E E 0 0  E— 0 0  Ka e  KC  C  0  Za  °  °  Zb  e  Zc  e  Fg —  Table 2.1: Character table for the point group D 2 and symmetry species of rotational and direction cosine functions matrix element <J’KK  I  JKaKc > must be both non-zero and symmetric with  respect to all operations of the group D . Since 2 <J’IçK  ILzI  i’a< J’KK IZaI J’a’c > +  JKaKc>  +Itc< J’KK  IzcI  [Lb<  J’IcK IZbI  JKaKc>  (2.24)  j1’a’c>,  this requires both that 1 u be non-zero and that <J’KK  I  I  jKaKc>  be totally  symmetric for g=a, b, or c (note that all three components of it may be non-zero in an asyn-u-netric top.) For example, if a molecule has a component of its permanent electric dipole moment along its a principal axis, then the symmetry of must be considered. Since JKaKc> and  I  Za  Za  matrix elements  is of symmetry Ba, if the product of the symmetries of  J’KK> is also Ba, then the direction cosine matrix element will be  non-zero and a transition will be allowed. Because the intensity of this transition will depend on the value of [ a, it is known as an ‘a-type’ transition. As can be seen from t Table 2.1, allowed a-type transitions will be of the form KaKc = ee  —*  eo, oe  -+ 00.  Selection rules for rotational transitions in rigid asymmetric rotors are given in Ta ble 2.2.  2.3  Centrifugal distortion  Real molecules are not rigid, and therefore the rigid rotor model for molecular rotation is only an approximation. Interatomic bonds will stretch as the molecule rotates and angles  11  Chapter 2. Theory  Ka Kc tLaO 1LbO j&LO  ee—eo,oe-+oo  ee—*oo,eo-+oe ee+.-*oe,oo—eo  Table 2.2: Selection rules for rigid asymmetric top rotational transitions between the bonds will change, thereby distorting the molecule. This centrifugal distor tion increases with increasing rotational angular momentum, and must be accounted for by including terms of higher order in J in the rotational Hamiltonian. Up to fourth order in J, the semirigid rotor Hamiltonian may be written as [14] Hsemirigid  (2.25)  =  Hrot + Ha  =  2 + BJ BJ 2 + BJ  , 5 JJJJ 5 r  2+  where Br, B, and B are as given in Eq. 2.2, and c,/3,-y,6 = are 81 possible  rs;  x,y,  or  z.  (2.26)  In principle, there  however, symmetry considerations reduce the number of meaningful rs  to nine [14]. Commutation rules for angular momenta may then be applied to eliminate terms of the form  Tag,  folding their effects into the other coefficients. Hsemirigid may  then be written as Hsernirigici = BJ 2 +  2  2  + BJ + >TcrJcr  j 2 2  (2.27)  where B = B+ B, = B + = B+  —  —  —  —  —  —  (2.28) (2.29) (2.30)  and T  =  (2.31)  Chapter 2.  12  Theory  + 2r)  Ta =  (2.32)  B, B,, and B are effective rotational constants, whose differences from Bc,,, B, and B,, contribute only a very small amount to uncertainties in derived structural parameters [10] (see Section 2.4.) In order to simplify the notation in what follows, the primes will be dropped from the effective rotational constants. As was pointed out by Watson [15, 16, 17], not all the coefficients in Hsemirigici may be determined from observed energy levels. In order to decide which coefficients or com binations of coefficients are determinable, he proposed subjecting Hsem.jrjgjd to a unitary transformation, which by definition leaves the eigenvalues of Hsejñgja unchanged. Those combinations of coefficients which are changed only to a very small extent by the trans formation are eigenvalues of Hsemirigid and can hence be determined from the observed energy levels. A ‘reduced’ Hamiltonian is obtained by choosing a unitary transformation which leaves only determinable combinations of coefficients. These combinations were found to be [15]  13,,  =  B  =  B  = B,,  —  —  —  2T  (2.33)  2T  (2.34)  2T  (2.35) (2.36)  (2.37) (2.38)  T,, 1 T  =  (2.39)  2 T  =  (2.40)  Since there are only eight determinable combinations of coefficients to fourth order in J, only eight of the nine possible linear combinations of the rotational and centrifugal  Chapter 2.  13  Theory  distortion constants in Eq. 2.27 are independent. By choosing the unitary transformation correctly, one of these linear combinations can be set to zero. Watson’s first choice of unitary transformation gave the following reduced Hamiltonian [15]: Hrot  =  Hd  =  (2.41) —LJJ  ‘  6K[Jz  —  2  /-JKJ  (J 2  jz 2  j 2)  2 —  —  LKJz  + (J  26J  ‘  2 —  j  )J 2  (JT 2  2  2],  —  j,  2)  (2.42)  This is known as the Watson A (asymmetric) reduction, which is the most commonly used reduction. However, the A reduction breaks down as the molecule becomes near symmetric and B  —  B approaches zero. By choosing a different unitary transformation  which eliminates a different linear combination of Bas and Ts, One obtains the Watson S reduction, which may be used for nearly symmetric molecules [18]: Hrot H,d.  (S)  =  2 2 + BJ + BJ BJ  (S)  =  —DJ”  —  J DJKJ 2  2 —  2  (2.43)  4 DKJ  (J +J— 4 , J_ J+ )+d ( 2 + ) J2 1 +d  (2.44)  where  J_  =  Jx  =  Jx + jJ.  —  iJ  (2.45) (2.46)  Centrifugal distortion constants are more than empirical parameters used to improve the fitting of spectroscopic data. If the potential V governing intramolecular motions is assumed to be harmonic, V can be expressed as [10] V=  (2.47)  Chapter 2. Theory  fj  where  14  is a harmonic force constant and R, and R, are internal displacement coordi  nates; depending on the nature of R, and J?,  fj  is the force constant for a bond stretch,  an angle bend, or some combination of interactions between stretching and bending modes. For a non-linear molecule, there are 3N —6 internal displacement coordinates, as well as 3N Cartesian displacement coordinates. R is related to the Cartesian coordinates 3 by the B matrix, and X  X. 1 R=B  (2.48)  If the molecule is rotating but not vibrating, the only effects of V will be in the restoring potential forces which balance the centrifugal forces caused by rotation, and so centrifugal distortion constants are directly linked to the harmonic force constants. If is an element of the inverse moment of inertia tensor for a non-rigid molecule, and (i)  •  is the derivative of it with respect to R, then (2.49)  =  (f’)  is an element of the matrix which is inverse to the matrix of force constants  f  [10]. Centrifugal distortion constants may thus be used along with information obtained from vibrational spectra to determine harmonic force fields [19].  2.4  Structure determination  As was shown in Eq. 2.2, the rotational constants obtained in microwave spectroscopy are inversely proportional to the principal moments of inertia of a molecule. These principal moments are obtained by diagonalizing the inertia tensor, whose components are related to the centre-of-mass system coordinates (xj,yj,zj) of the atoms as follows: I  =  mj(y+z)  (2.50)  Chapter 2.  Theory  15  I,  = m(x + z)  I,  =  >  m(x + y)  (2.51) (2.52)  I = I=—rnxy,  (2.53)  myzj  (2.54)  =  —  I = I = —rnxz.  (2.55)  It is therefore possible to use the principal moments of inertia to determine atomic coordinates. However, although rotational constants and hence moments of inertia are determined to high precision by microwave spectroscopy, molecular structures determined from these constants are of much lower precision. This is a result of the corrections that need to be made for vibration-rotation interactions in the molecule. Since rotational transitions are measured for a particular vibrational state, the mo ments of inertia obtained represent effective values of r , where r is an atomic position, 2 in that state; at the vibrational potential minimum, rotational constants depend on r, where re is an equilibrium atomic position at the minimum. It is the equilibrium structure which is generally of most interest, since it may be used to improve vibrational potential functions which are obtained theoretically. However, the structure of the molecule in all vibrational states will differ from the re structure for two reasons: < r >re, since vibra tional potentials are anharmonic, and < r > L< r 2 2 > when the molecule is vibrating. Even in the ground vibrational state, some ‘zero-point’ vibrational effects are present. In principle, it is possible to obtain rotational constants which are free of vibrational effects. For a diatomic molecule, the rotational constant as a function of the vibrational level is given by a power series in (v+1/2), where v is the vibrational quantum number. For a polyatomic molecule, the rotational constants are given by +d / 2) +... A = Ae —>Za(vs 8  (2.56)  Chapter 2.  16  Theory  with corresponding expressions for B and G; summation is over all vibrational modes s, where d 5 is the degeneracy of the mode. If enough vibrational levels are studied, a values can be determined and equilibrium rotational constants obtained. In practice, however, the number of vibrational modes that must be examined is so large that this is rarely done for polyatomics, although it is commonly done for diatomics [20]. Instead, the effects of vibration are accounted for in various ways when a molecular 0 structure structure is reported. The simplest structure to obtain is the ‘effective’ or r (or r structure for a general vibrational state v.) This may be determined directly from the rotational constants if the number of structural parameters to be determined is less than or equal to the number of rotational constants obtained; for example, the bond length r in a diatomic molecule may be determined from the rotational constant using the relation B= where  t  8K ,ir2’  (2.57)  is the reduced mass of the molecule, given by =  2 m 1 2 1 +m m  (2.58)  0 structure for an asymmetric molecule In general it is not possible to determine an r from the three rotational constants available from microwave spectroscopy, unless as sumptions are made concerning some structural parameters. However, if rotational con stants can be obtained for different isotopomers of the molecule, it may be possible to determine a ‘substitution’ or r 8 structure using a series of equations developed by Kraitchman [21]. Bond distances and angles must first be assumed to be unchanged when isotopes are substituted at different atomic positions. Components of the inertia tensor for the substituted molecule (S) may then be calculated in terms of the principal inertial axes of the ‘parent’ molecule (F, usually the most abundant isotopomer). If substitution occurs at the atom with centre-of-mass coordinates x, y, and z in F, then  17  Chapter 2. Theory  the components are given by [21]  where  ) 2 I = I + p(y + z  (2.59)  ) 2 I = I + p(x + z  (2.60)  2+y ) 2 = I + p(x  (2.61)  I = —pxy  (2.62)  I =  —pxz  (2.63)  =  —pyz,  (2.64)  is a component of the inertia teilsor of S,  ‘g  is a principal moment of inertia of  F, and p is the reduced mass for isotopic substitution, given in terms of the total masses of the two molecules by = Mp(Ms— Mp)  (2.65)  The inertia tensor is then diagonalized to obtain principal moments of inertia for S in terms of the coordinates of the substituted atom and the principal moments of P. Changes in the principal moments of inertia with atomic substitution may then be used to determine the centre-of-mass coordinates of the substituted atom in F. For example, the x and y coordinates of a substituted atom in a planar asymmetric top (z=0) are given by  I I  =  [()  (2.66) (i+  =  [(Ii  ;  ‘ (1+  .  (2.67)  Five moment equations exist for the parent planar asymmetric top: I =  (2.68)  Chapter 2.  18  Theory  (2.69)  I  =  I,  =  —>Zmxy  =  0  (2.71)  =  0,  (2.72)  >mjyj  =  (2.70)  0  and hence x and y coordinates may be determined for two of the N atoms if the positions of the remaining atoms are known. A minimum of (N  —  2) isotopic substitutions at  different atomic sites is thus required to determine the ‘full’ substitution structure. For a non-planar asymmetric top, nine moment equations exist (Eqs. 2.50-2.55 and three centre-of-mass conditions), and so (N  —  3) substitutions are required. The number of  required substitutions is also reduced if the positions of some atoms are constrained by molecular symmetry. Substitution structures are based on the assumption that the molecule is rigid, an assumption that can be seen to break down when two different isotopes are substituted at the same position, giving two slightly different substitution structures. Using the planarity condition to constrain the solutions is also problematic [22]. The inertial defect /, defined for a planar molecule in the xy plane as (2.73)  is zero only at the vibrational potential minimum; in the ground vibrational state (v=0),  z is usually small and positive. This means that there are three combinations to sub stitute into Eqs. 2.68 and 2.69: I and I; I and I, corresponding to I and (I + so); and I and I, corresponding to I and (I + z ). It is not possible to determine a priori 0 which combination will give the most accurate substitution structure. Costain [22] has shown that most of the uncertainty in substitution structures arises from forcing the structures to reproduce the measured moments of inertia (given for the  19  Chapter 2. Theory  planar asymmetric top in Eqs. 2.68-2.69) which are valid only for the parent isotopomer. A much more precise structure is obtained if isotopic substitution is carried out at each atom in the molecule and the centre-of-mass coordinates for all atoms are determined independently. Substitution structures determined in this way have also been found to be nearly independent of the choice of isotopic species used [22]. However, this is a timeconsuming task, and is not possible if, for example, an atom has only one stable isotope. In addition, the effects of isotopic substitution at an atom near a principal axis are small, resulting in poorly-determined coordinates for that atom. Zero-point vibrational effects tend to cancel when a structure is determined using the  3 structure is usually closer to the equilibrium structure substitution method, and an r 0 structure: for a diatomic molecule [22], than an r Te 8 r  +  To.  (2.74)  However, the r 3 structure itself has no clear physical meaning. On the other hand, the ‘ground state average’ or r (< r >) structure has a well-defined physical meaning: it is the structure corresponding to the average positions of the nuclei in the ground vibrational state. If the molecule had a harmonic vibrational potential, the r structure  0 structure would be the same as the equilibrium structure, but would still differ from the r z4< r 2 2 >. The o constants because of the harmonic vibration effects which make < r >  in Eq. 2.56 may be divided into harmonic and anharmonic components as [20] o=ah+aanh,  (2.75)  and so for a general rotational constant A we can write =  0+ A  ch, d 8  Ae_d81  (2.76) (2.77)  Chapter 2.  20  Theory  An r structure may thus be obtained from an effective structure if harmonic force con stants are known (see Section 2.3), from which values of  2.5  ah  are calculated [19].  Nuclear quadrupole hyperfine interactions  The electrostatic Hamiltonian for a system of interacting nuclear and electronic charge densities is given classically as [23] Heiectrostatic  =  j  pn(rn)  [J  dVe]  (2.78)  dv.  where p(r) is the nuclear charge density of the nuclear volume element dv at position r, and pe(re) is the electron charge density of volume element dye at position re; for convenience, the centre of the nucleus under consideration is taken as the origin of the coordinate system, and interactions of this nucleus with other nuclei in the molecule and with the extramolecular surroundings are ignored. Eq. 2.78 may be expanded in terms of Legendre polynomials Pk (cos 0): Heiectrostatic  =  j pn(rn) [j pe(re)  (  i  Pk (cos 0)) dVe] dv,  (2.79)  which may in turn be expanded in terms of the spherical harmonics Y)(0, q): co Heiectrostatic =  k q(_)  2k  1  f  r Y(0,  ) dv j  Y(0e, çe) dV. (2.80)  Since Y° =  (4K)_h/2,  the k=0 term is simply equivalent to the large Coulombic in  teraction between point charges, separated by distance re. The k=1 term accounts for interactions between electric dipoles in the nucleus and the surrounding electric field of the molecule; this term disappears for nuclei in their ground states, which do not possess a permanent electric dipole moment.  Chapter 2.  21  Theory  In microwave spectra, only those effects due to the k=2 term of the electrostatic Hamiltonian are observed, as a result of interactions between the electric field gradi ent surrounding the nucleus and the nuclear quadrupole moment. The Hamiltonian for nuclear quadrupole coupling is given by HQ = q=—2  f  p, r Y(O,  “  ) dv,  f  ere  )(Oe, 2 Y  e)  (2.81)  dVe.  HQ may also be written as the scalar product of two second-rank spherical tensors, HQ  =  ).V( 2 Q( )  =  (2.82)  where  = Q ) 2  (2.83)  and  2= v  (2.84)  Y(Oe, ) dv  represent components of the nuclear quadrupole moment tensor and the electric field gradient tensor, respectively. Neglecting centrifugal distortion effects, the microwave spectrum of a molecule with nuclear quadrupole coupling may be analyzed using a Hamiltonian which is the sum of the rotational Hamiltonian and the quadrupolar Hamiltonian, H  (2.85)  Hrot +HQ.  In order to transform the classical expression for HQ given in Eq. 2.81 into a quantum mechanical expression, it is necessary to derive expressions for  Q  (2)  and V  (2)  which  depend on quantum mechanical expectation values rather than classical charge densities. Relations which will be particularly useful in calculating spherical tensor matrix elements, including the Wigner-Eckart Theorem, are given without proof in Appendix A [24].  Chapter 2.  22  Theory  Consider first the nuclear quadrupole tensor, which will depend on the nuclear spin quantum number I.  By the Wigner-Eckart theorem (Eq. A.3, Appendix A), it suf (2),  Q  fices to calculate only one component of  from which the rest may easily be de  O, q) is simply Y ( rived. If the q=O component is considered, the spherical harmonic 2 (3 (5/161r)h 2 1  2O cos  —  1), where O, is usually defined with respect to the symmetry axis  of the (classical) nuclear charge distribution. The nuclear quadrupole moment operator  Q  is conventionally defined as  Q  =  jpn(rn)r(3cos2On_1)dvn,  (2.86)  with the corresponding expectation value  Q  =  =  (2.87)  <I,Mj=IQII,Mi=I> II  I 2 Q  Note that the nuclear quadrupole moment  Q  (2.88)  II>.  is defined for the state in which the (con  stant) nuclear spin I is in its state of maximum alignment along a space-fixed Z axis. The shape of the nuclear charge distribution is determined by the value of pends on the average value of (3 cos 2 O,  —  1);  Q,  which de  Q is positive for prolate charge distribution,  zero for spherical charge distribution, and negative for a nucleus in which the charge is distributed in an oblate fashion about the symmetry axis. Similarly, the appropriate operator may be chosen to correspond to the electric field gradient about the nucleus, which will depend on both the electronic structure and the rotational state of the molecule. The electric field gradient coupling constant  qj’j  is  defined as qj’j =  where  I  r, J, M  =  <r’, J’, M’  =  J Vzzi r, J, M  =  J>,  (2.89)  J> is the rigid rotor basis function with maximum projection of J  along a space-fixed Z axis. The electric field gradient operator Vzz is defined as the  Chapter 2.  23  Theory  second derivative of the potential at the nucleus, surrounded by a generalized electron charge density distribution pe(re):  =  ()  =  jPe) (3cos O— 1)de. 2  (2.90)  Thus from Eq. 2.84 the electric field gradient coupling constant may be defined in terms of the q=0 component of the electric field gradient tensor: qj’j  = 2< r’J’J  IvI rJJ>.  (2.91)  For coupling of one quadrupolar nucleus to rotation, matrix elements of HQ are calculated using basis functions of the form TJIFMF>, where I is the nuclear spin, F is the resultant angular momentum produced by coupling I and J as I+J =F, and MF is the projection of F on a space-fixed Z axis; r refers to all other relevant quantum numbers. Using Eq. A.5 given in Appendix A, matrix elements of HQ are given in this basis by <T’J’IF’M  IHQ I  TJIFMF> I F  J  (I  J  Q  <  =  2J  (2)  >< r’J’  y  (2)  rJ>.  (2.92) (2)  Q  The reduced matrix elements <  I  I> and <r’J’  I  (2)  I  rJ> may be calcu  II>  (2.93)  V  lated using the Wigner-Eckart theorem as  <  I Q  I  (2) =  I  —1  I  2 I’\  I  0 I)  < II  >I 2 IQ  Chapter 2.  24  Theory  —1  /  if  —I 2  =  2 i\  I  o  _  i  I  —1  I  <r’J’  v  (2)  rJ>  (_)J’J  =  (2.94)  eQ  I J’  2  f  rJJ> (2.95)  <r’J’J  0 J) —1  (_Y  =  if ‘—  J’  I  2 J’\  I  (2.96)  qj’j.  \—J 0 J) 2  Matrix elements of HQ are thus given by <T’J’IFMF HQ  I  TJIFMF> —1  /  ill =  6FFlMFM(—)  21\  —1  /  fj’  2J\  I—IOI)I I\\—JOJ)I  /  IJ’IF (I  J  eQqj’j.  2 (2.97)  Since the 3-j symbols in Eqs. 2.94 and 2.96 are zero for I < 1 and J < 1, respectively, no nuclear quadrupole coupling is observed for nuclei with spins I  =  0 or 1/2 or when  J=0 or 1/2. However, it must be recognized that  qj’j  on the rotational state of the molecule, since  is not a constant, but is instead dependent qj’j  is defined in terms of a space-fixed Z  axis: rotation will caused the electric field due to the electrons in the molecule to rotate in the space-fixed frame. Hence,  qj’j  must be transformed to the molecular frame, using  one of two approaches. When using symmetric top basis functions  I  JKM>, it is often  defined in the space-fixed frame, to components  convenient to transform components  which are defined in the molecule-fixed frame and which are independent of the rotational state, using the relation [23] =  Dr  (2.98)  25  Theory  Chapter 2.  where D is a (2k+1)-dimensional matrix representation of the rotation operator, and depends on the relative orientation of the space-fixed and molecule-fixed frames. The electric field gradient coupling ‘constant’ of molecule-fixed components of V’  qj’j  = 2  IJ’2 J  J’\  0 —J  )  (2)  ,  qj’j  is then given in terms of expectation values  [23]:  -K-v  ,  /(2J’ + l)(2J + 1  q’  IJ’  I  2  —K’ q’ K  I v’  /  (2.99) where q’=K’ nents of  v’  —  (2)  K. In the molecular frame, with molecule-fixed axes x, y, z, the compo are given in a Cartesian representation as: (2.100)  = V’ =  (2.101)  + i14)  V’ =  —  (2.102)  V + 2iV),  where Vfg represents a component of the electric field gradient, evaluated at the nucleus: 9 Vf  —  —  (2.103)  i’9fögJ 1  The molecule-fixed axes are usually taken to be the principal inertial axes. Terms of the 9 form eQVf  (f, g  = x, y, z) will appear in HQ, and are often abbreviated to the nuclear  quadrupole coupling constants Often molecular symmetry permits elimination of certain Cartesian components of  v,  (2)  It is also helpful to recall that Laplace’s equation dictates that the field gradient  vanish at the nucleus, i.e. (2.104)  V + V, + Vi,. = 0.  As a result of this relationship, only two of the three diagonal coupling constants are independent; spectroscopic data are commonly fit to Xzz and X  —  x.  For linear  Chapter 2.  Theory  26  molecules, or for symmetric tops with the coupling nucleus on the molecular symmetry axis, the field gradient will be cylindrically symmetric about the molecular symmetry axis, and so xxx  —  x,=O. The molecular constant q is defined as q  =  V  (2.105)  and eQq is the pertinent spectroscopic constant. If the electronic charge distribution surrounding a quadrupolar nucleus (i.e. a nucleus with I1, QE0) is not symmetric about the x,y,z principal inertial axes, as in an asym metric molecule or a symmetric molecule where the nucleus does not lie on the symmetry axis, v,  (2)  will have non-zero off-diagonal elements. The ‘principal’ values of v’  (2)  are then obtained by diagonalizing the tensor. Alternatively, if the shape of the charge distribution is known a priori, the principal values may be estimated in the x’, y’, z’ principal axis system of the tensor. For example, if a quadrupolar nucleus is bonded to only one atom, the nucleus will be affected most by the charge distribution in that bond, and the z’ axis may be taken to lie approximately along the bond axis. If the charge distribution is then cylindrically symmetric about that bond (i.e. the bond is a c bond), then xx’x’  xy’y’.  To calculate quadrupole coupling energies using first order perturbation theory (i.e. considering only J’=J, K’=K), straightforward relationships exist between  qj’j  qjj  and q for linear molecules and symmetric tops: [25]  qjj =  2j+ 3 [J(J 1) —  =  2J+3  —  i]  for symmetric tops for linear molecules:K  =  (2.106) 0.  (2.107)  For a symmetric top with one quadrupolar nucleus on the symmetry axis, the first order quadrupole coupling energy may be calculated for a state KJIFMF> as E)  =  <KJIFMFIHQIKJIFMF>  27  Chapter 2. Theory  =  1  eQq  [c(c + 1) J(2J  —  —  J(J + 1)1(1 + 1)] 1)1(21 1) —  31(2 J 2J +3 J(J + 1)  —  1  ,  (2.108)  where C = F(F+ 1) —J(J+ 1) —1(1+1).  (2.109)  In an asymmetric rotor basis, it is instructive to use direction cosines to transform Vzz, and hence  qj’j,  to the molecule-fixed frame. Using this approach, + Vyzy + Vzzyz  VZ =  (2.110)  V 2 + ’ 4 zxzy xy + 2VyzzyFzz +  where Vjg is defined as in Eq. 2.103. Unlike the direction cosines, the VfgS are molecu lar constants and hence are unchanged by rotation, and calculation of  qj’j  amounts to  evaluating direction cosine matrix elements for the appropriate rotational states. This calculation is greatly simplified if the symmetry properties of the direction cosines and the asymmetric rotor basis functions JKaKcM > are considered, especially if first order perturbation theory is sufficient to account for nuclear quadrupole coupling in a given molecule. If this is the case, then only diagonal matrix elements of HQ are calculated, and the relevant electric field gradient coupling constants are given by qjj  = Vjg< JKaKcM ZfZgI JKaKcM>  f,g =x,y,z.  (2.111)  fg  Only those matrix elements which are totally symmetric will be non-zero; since the matrix elements are diagonal, this amounts to requiring that the products of direction cosine operators, ZfJZg, be totally symmetric. Since only species A in the point group D 2 (see Table 2.1), qjj  =  Vccx < JKaKcM  I  +Vzz < JKaKcM  qjj  and  are of symmetry  reduces to  JKaKcM> +V <JI(KM I zy I JKaKcM>  I  JKaKcM>.  (2.112)  Chapter 2.  28  Theory  The expectation values of the squared direction cosines in a given rotational state may be related to the expectation values of squared components of the rotational angular momentum operator J [10]: 2>  <g  >  +2J+3  (J+ 1)(2J+3)  g = x,y,z.  (2.113)  Using Eq. 2.113, the reduced energy Wb corresponding to Hwg (Eq. 2.16), Laplace’s equation (Eq. 2.104), and qjj  =  VZZ  J 2J +3  <j  2  >= J(J + 1), one obtains  (3<J > 2 — J(J + 1)  ‘  +  bV  (2.114)  J(J + 1)  For a given state I TJIFMF>, the first order quadrupole coupling energy E) is given by  < TJIFMF  IHQ  TJIFMF >, and may be calculated using Eqs. 2.97 and 2.114.  The terms in Vz which are not included in calculating the first order nuclear (2)  quadrupole coupling energy contribute to the second order energy correction EQ  .  By  non-degenerate perturbation theory, E(JKaKcIFMF) <J’KK  + VyzZyZz + VzxzI JKaKc> E (J’IçKI’F’M) E (JKaKc1FMF) VxyZxZy  2  —  +terms in  (2.115)  g,  where E (JKaKcIFMF) is the rotational energy for the state  JKaKc  >  (without  nuclear quadrupole coupling) and E(JKaKcIFMF) is the second order correction. Summation is carried out over all states J’IçKI’F’M > 1  I  J1aKcI1’MF>. Again,  molecular symmetry may allow elimination of V, V, or V. The most significant contributions to E(JKaKcIFMF) will be due to perturbing levels  J’K.KI’F’M>  which are nearly degenerate with JKaKcIFMF >, and which are of appropriate symme try so that the matrix elements containing the products of direction cosines are non-zero.  Chapter 2.  Theory  29  For a molecule whose plane of symmetry is the ab plane, this latter restriction places the following constraints on interacting states  I  JKaKcIFMF> and  J’K,KI’F’M >:  J’—J = O,+l,+2 IcK (see Table 2.1.)  ÷-  KK  ee  =  4-*  oe, eo  —* 00  Another restriction arises upon consideration of matrix elements of  the quadrupolar Hamiltonian HQ , whose matrix elements <r’J’I’F’M  Q  (2)  V  (2)  TJIFMF> are non-zero only for LF=LMF=O. Perturbations to the first order quadrupole coupling energy may therefore be predicted upon examination of the energy levels for a given asymmetric top molecule in order to find near-degeneracies between levels of appropriate symmetry. These perturbations may be used to determine off-diagonal elements Xfg  (f  g) of the quadrupole coupling tensor.  These off-diagonal elements will be larger if the principal axes of the coupling tensor are well-separated from the principal inertial axes. A large quadrupole moment  Q  will also  amplify the effects of perturbations. Hrot must be augmented by a centrifugal distortion Hamiltonian in the final analysis, as small shifts in the rotational energy levels can cause large changes in the magnitude of the second order perturbation, and hence in  [261.  For a molecule containing two quadrupolar nuclei, both nuclear spins are coupled to rotation, as well as to each other. The coupling scheme used to determine matrix elements of HQ depends on the nature of the two nuclei. For two nuclei having similar or equal nuclear quadrupole coupling constants, it is convenient to use the coupling scheme Ii +12 I+J  = I =F,  where I and 12 are the nuclear spins of the two nuclei, in order to create basis functions of the form  IJFMF>. The effect of the two quadrupolar nuclei is additive, and 2 1 I  Chapter 2.  Theory  30  HQ may be written as HQ  =  HQ (1) + HQ (2).  (2.116)  From application of Eqs. A.5 and A.6 of Appendix A, the matrix elements of HQ (1) are given by  Q  <T’I I 2 I 1 ’J’F’M =  (2)  (1). V  FF,6MFM(_)21+J+  •< r’J’  V  (2)  (1)W  (2)  (1)1  IJFMF> 2 I 1 TI l)(21’+ 1  11+12j+  rJ >< I’  W Q  (2)  (1)W I  1)(21’ +  +1+2+  =  { {  ‘  ‘  F’  >,  ‘  ‘  }{ }{  1 i  P i 2  eQ(1)qjij(1), 0 Ii  } }  (2.117)  (2.118)  I  where the reduced matrix elements have been evaluated as in the case of only one coupling nucleus. An equivalent formula applies to the matrix elements of HQ (2), with 1 and  ‘2  interchanged. For a molecule containing two nuclei with very different quadrupole coupling con stants, it is more convenient to use the coupling scheme +J 1  =  1 F  1 +12 F  =  F.  In order to evaluate HQ (1) for the nucleus with much stronger coupling, basis functions of the form r’ J I F M 1 > may be used, leading to a simplified expression for matrix elements which determine the predominant part of the quadrupole interaction: <r’J’I F 1 M  Q  (2)  (1). V  (2)  (1)  MF > F 1 TJI  Theory  Chapter 2.  31  (—)  —  1F1 F 6 MF M. 6  —  +F’1 1 J’+1  I  J’ Ff  Ii  (J  2  I  —1,  I  2 J’  (Ii  2 i\  0 J)  ‘i  0 1)  I  The smaller effect of coupling  eQ(1)qjij(1).  I  I  (2.119)  and F 1 must be evaluated using basis functions of the  ‘2  form F 2 F 1 r’J’I ’Mj I >: 2 F 1 <r’J’I F ’M I =  Q  8 F 6 M FM() F’  (2)  (2). V  Fi+1 + 2 F’  =  V  (2)  6FF,6MFM(_)2F1  <  =  F 1 r’J’I  r’J’  V  (2)  (1)I  12  F’  1  1 F  2  F 1 (1)11 rJI  ><  TJ  ><  1Q  1  (  Q  ‘2  2+F+J+I1( F 2 +  8FF,MFM(_)2F1+I2+F+J+h1V(2F  1  (2)  (2)  ‘2  (1)11  (1)11  12)  0  ‘2  (2.120)  >  F  12  F’  ‘2  1 F  2 J(F 1  J’  F I J  2 (2.121)  12 >  1 +1 + 1)(2F  2  ‘2  1 + i){ 1)(2F  —1  J’  12  2.6  (2) TJI FMF> 2 I F 1  F  ( <  (2)  {  eQ(2)qjij(2).  ‘2  1 F  F’  J’  2 J(F 1  F I J  2 (2.122)  Nuclear spin-rotation interactions  The Hamiltonian for the interaction of a magnetic dipole p with a magnetic field H is given by Hmag =  1L  H.  (2.123)  In a rotating molecule, such an interaction exists between the magnetic dipole moments of the nuclei and the magnetic field induced by the rotating protons and electrons. The  Chapter 2. Theory  32  nuclear magnetic moment depends on the spin I of the nucleus, as well as on the dimen sionless gyromagnetic ratio gi for that nucleus: (2.124)  1tI= gI1NI,  where  PN  is the nuclear magneton. The precession of I is slow relative to the rotation of  the molecule [10], so that u 1 interacts with an average magnetic field Heff in the direction of J. Heff may be expressed as [10] Heff where  <  H  >  <  H>  (2.125)  ,  /J(J+ 1)  depends on the molecule-fixed components of the magnetic field hgg  (g=x,y,z) as  <H  > >  J(J +  +%  <j 2 >  +h <J  2  >].  (2.126)  The nuclear spin-rotation Hamiltonian can then be written as Hnuc.spin_rot = Mrjl  J  ,  (2.127)  where the nuclear spin-rotation coupling constants MTJ depend on the rotational state  rJ> under consideration, and are defined as MTJ  (2.128)  In addition to the rotation of positive and negative charges, contributions to the magnetic field about the nucleus arise from mixing of the ground electronic state and excited states with electronic angular momentum, a mixing which is brought about by rotation of the molecule. This effect may be represented as a second order perturbation of the electronic states, and is discussed further in Section 7.4. For a diatomic molecule in the ground vibrational state and a ‘E electronic state, there is no electronic angular momentum about the interatomic axis z, and so h=0; in  Chapter 2.  Theory  33  2  addition, symmetry about z makes h=h and <J,,  >=0. The nuclear spin-rotation  coupling constant for a nucleus with spin I is then given by M  =  J(J  2 )kXX(< 1  > +  <j2>)  (2.129) (2.130)  =  M.  (2.131)  M± is a molecular constant which is independent of J. The coupling of I and J brought about by nuclear spin-rotation interactions may be described by the vector equation I+J =F. Generally this coupling is weak, with splittings of  IJF> levels often too small to  be resolved by microwave spectroscopy, and I and J may be considered to be good quantum numbers. However, the coupling of I and J also occurs via much stronger nuclear quadrupole interactions (see Sec. 2.5), and transitions between  I  IJF > levels  have the same selection rules for both mechanisms. Nuclear spin-rotation effects may then be measured as small shifts of the expected nuclear quadrupole hyperfine structure patterns. An even smaller hyperfine effect which is sometimes observed in microwave spectra is nuclear spin-spin coupling [27], which arises from a magnetic dipole-magnetic dipole interaction between two nuclei in a molecule. However, since this effect was not observed in any of the studies described in this thesis, it will not be discussed here.  Chapter 2. Theory  2.7  34  Internal rotation  In addition to the overall rotation of a molecule, parts of the molecule may rotate rel ative to each other, a motion that is often described as a torsion. If a symmetric sub group undergoes such a torsion relative to the rest of the molecule, interconverting struc turally equivalent forms, this internal rotation may couple with the overall rotation of the molecule to affect noticeably the observed microwave spectrum. The effects of internal rotation on rotational energy levels depend on the potential energy function V(a) which governs the rotation of the subgroup through an angle a relative to the frame of the molecule. V(a) must be periodic in a, and the potential function is usually expanded as V(a)  =  ajcos(kNa),  (2.132)  k  where N is the number of equivalent configurations. As the internal rotor of interest in this thesis is the threefold-symmetric methyl group, the remainder of this section will be developed to treat the case where N=3. The reference energy level is usually shifted so that V(a) may be written as V(a)  =  (1  —  cos3a) + (1  —  cos6a) +...,  (2.133)  where V 3 is the threefold barrier to internal rotation and V 6 is the sixfold barrier. Usually 6 << V V , and so the leading term in Eq. 2.133 is sufficient. 3 The Schrödinger equation for internal rotation may then be written as —  U(a) 2 d  + [(i  —  cos3a)  —  E} U(a)  =  0,  (2.134)  where U(a) are the eigenfunctions and E is the total energy of internal rotation.  ‘r  is  the reduced moment of inertia for the relative motion of the internal rotor (‘top’) and  Chapter 2.  35  Theory  the rest of the molecule (‘frame’), and is given by =  ‘top’frame  (2.135)  ,  ‘top +  ‘frame  where I is the moment of inertia of the internal rotor about its symmetry axis, and ‘frame  is the moment of inertia of the rest of the molecule about that same axis.  If internal rotation is unhindered (V =O), Eq. 2.134 has solutions 3 in  ‘ —  E where m=O, ±1, ±2  =  e ima (2.137)  m2,  The free rotor states are doubly degenerate (except for m=O),  corresponding to the two directions of internal rotation. If the barrier to internal rotation is infinitely high (V 3  —*  oo), a will be infinitely  small, and cos 3a may be expanded in a Taylor series. V(a) is usually taken to be equal to the leading term in the expansion: V(a)  = a2.  (2.138)  Eq. 2.134 then has solutions which resemble those for the harmonic oscillator (where /2), and the energy of internal rotation is given by 2 V(x)=kx E where v=O,1,2  = h/3(v  + )‘  (2.139)  For a threefold barrier, each torsional state v is triply degenerate,  since the torsional motion is restricted to one of three equivalent potential wells. If V 3 is finite, then the possibility of tunnelling exists and there are three types of solutions to Eq. 2.134, one of A symmetry and two of E symmetry. The A states are non degenerate, and correspond to internal motions localized in one of the potential wells. The E states are degenerate, and correspond to internal motions with some tunnelling  Chapter 2.  Theory  36  character. The solutions for low and high barrier heights may be related, as is shown in Figure 2.1. Each v state splits into an A state and an E state as V 3 falls, and each m state must correspond to either one E state or two A states. As can be seen in Figure 2.1, the +m degeneracy is lifted for free rotor states with  m =3k (k=1,2,3,...) as a torsional  barrier is introduced. Transitions between A and E states are forbidden [28]. The model usually taken to treat the coupling of internal and overall rotation is that of a rigid symmetric top attached to a rigid frame; the top rotates about the bond connecting it to the frame. In most cases, the remaining vibrational degrees of freedom may be ignored [28]. The Hamiltonian used to treat a molecule with both internal and overall rotation is usually separated into three parts: H = Hrot + Htorsion + Hrot_torsion  ,  where Hrot resembles the usual rigid rotor Hamiltonian (see Section 2.1),  (2.140) de  scribes the rotation of the top as described above, and Hrot_toijon is an interaction term, usually treated as a perturbation. If the product of eigenfunctions of Hrot and Htorsion are used as basis functions, then only Hrot_torsion will contribute off-diagonal elements. These elements can be minimized by proper choice of coordinate system.  For a  slightly asymmetric molecule with a light frame, the axis system is often chosen so that the symmetry axis of the top is one of the coordinate axes: this is known as the internal axis method (lAM) [29]. Rotation-torsion coupling is then eliminated or minimized by coordinate transformations. However, since the principal axis system is not used, terms involving products of inertia I (Eqs. 2.53-2.55) will remain, even in Hrot Another approach is the principal axis method (PAM [30, 31, 32]), in which the principal axes of the molecule are chosen as the axis system. Off-diagonal rotation-torsion terms are then larger than in the JAM, but the principal moments of inertia, which are unchanged by internal rotation, may be used. If j is the total angular momentum of the  Chapter 2. Theory  37  2fl  V  3  2  ±5  ±4 ±3 ±2 ±1 0  O 3 v  Figure 2.1: Corre1atjo between free (J’O) and hindered (J rotor States  3 V  threefold internal  Chapter 2.  Theory  38  top about its symmetry axis, then H is given by [32] H  J) 3 +J  =  ij  —2FpJj + Fj  2  + V(ct)  i,j  =  x,y,z,  (2.141)  where A’, B’, and C’ are defined in terms of the rotational constants A, B, and C as A’  =  A+Fp  (2.142)  B’  =  B+Fp  (2.143)  C’  =  C+Fp  (2.144)  and F  h 2 0 2rIt  =  (2.145) j2 —  —  , 4 , t 1 op x  —  ytop  —  z  g =  with  top  2 146  y  g=x,y,z,  (2.147)  9  the direction cosine between the g principal axis and the symmetry axis of the  top. H may then be divided as Hrot  =  2 B’J + C’J 2 + Fpp(J 1 J + J J) A’J + 2  (2.148)  ij  Htorsion Hrot_torsion  = Fj =  2  —2F  + V(a) pJ j.  (2.149) (2.150)  This division is particularly convenient for computational purposes, as symmetric rotor basis functions may be used to calculate Hrot matrix elements, while the free rotor basis functions of Eq. 2.136 may be used to calculate matrix elements of Htorsjon.  Chapter 3  Experimental Methods  Stark-modulated microwave spectroscopy  3.1 3.1.1  Theory  When microwave radiation is passed through a gas sample and molecules in the sam ple undergo rotational transitions, only a small fraction of the radiation is absorbed. Microwave spectroscopy is an inherently low-sensitivity branch of spectroscopy for two reasons: radiation densities p(v) vary as  and microwave frequencies are relatively low;  and rotational energy levels are closely spaced, resulting in a ‘smeared’ Boltzmann dis tribution of molecules over many rotational levels, leaving few molecules in any one level to undergo a given transition. A method commonly used to increase the signal-to-noise ratio in microwave spectroscopy is Stark modulation, whereby only those signals which are affected by an external electric field  are recorded.  The Stark effect is the result of interactions between  and the permanent molecular  electric dipole moment p Hstk  =  —it  £.  (3.1)  The application of an external electric field breaks partially or fully the 2J+1-fold de generacy of M, the projection of J onto a space-fixed axis Z; the result is that rotational energy levels are split by the Stark effect. Since  has a fixed direction in space and  jt  is  defined in terms of the rotating molecular frame, p is conventionally given in terms of its space-fixed components, calculated using direction cosines as in Eq. 2.19. If e is assumed 39  40  Chapter 3. Experimental Methods  to be of constant magnitude and pointing in the space-fixed Z direction, then HStark =  —e  (3.2)  itLgzg.  g=x,y,z  In general, Stark effect energies are calculated to sufficient accuracy using non-degenerate perturbation theory, where H5tark is considered to perturb the eigenstates  I  JrM > of  Hrot  The first order Stark effect energy is given by Erk =  JrM zg 1 JrM>.  [tg<  —6  gz,y,z  (3.3)  For asymmetric molecules, <JrM 4 9 JrM >=0 for all states, since this matrix el z 2 (see ement cannot be totally symmetric (i.e. of symmetry species A) in the group D Table 2.1); a first order Stark effect is not observed. For linear molecules and symmetric tops, [L f and z==O, so 2 Ek  jLE<  =  IzzI  JKM  JKM>  (3.4)  ueKM 1  —  35  J(J+1)’  —  where the direction cosine matrix element has been evaluated from Ref. [13]. Linear molecules will have no first order Stark effect, since K=0. The second order Stark effect energy may be calculated as (2) Stark  <JTM  —  11 J’r’M> I Stark  2  (.)  —  -‘-‘Jr  —  V’ Ld  V’  JrIJT  g  2  6 LdIg  —  1 J’r’M> 12 21 < JrM Zg  where EJT is an eigenvalue of Hrot. Since EJ  p  p  -‘-‘Jr  —  ‘  (.)  —  is generally large relative to ,ue,  Ek<E)k for a molecule in which both effects are present. In that case, relatively low electric fields are required to observe Stark splittings, and the magnitude of the splitting  Chapter 3. Experimental Methods  varies linearly with  .  41  For asymmetric tops, where only Erk is non-zero, higher fields  are required and splittings vary with  . 2 E  However, when a near-degeneracy exists between  the asymmetric top rotational energy levels, non-degenerate perturbation theory breaks down, and Ek cannot be calculated as in Eq. 3.7; rather, the Stark splittings must be evaluated by diagonalization of the (Hrot +Hstk) matrix, or by transforming that matrix in such a way that the effects of near-degeneracies may be ignored in an approximate perturbation treatment [33].  3.1.2  Instrumentation  A schematic cross-section of a Stark cell is shown in Fig. 3.1. The electric field is intro duced by applying a voltage to the electrode, which sits in the middle of the cell and which is insulated from the grounded cell walls by Teflon strips. The electric field direc tion Z is as indicated in the figure, although some distortion of this field occurs at the edges of the electrode; the Stark field is parallel to the electric field of the microwaves. For the experiments described in this work, a 6-foot Hewlett-Packard X-band Stark cell was used. The cell is sealed at both ends with mica windows, and so acts as both a sample cell and a waveguide. Gas samples are introduced into the cell from an attached vacuum line, to static gas pressures of 5-25 mTorr. The Stark-modulated microwave spectrometer is shown schematically in Fig. 3.2, and follows the general design first given by Hughes and Wilson [4, 5]. Microwave radiation is provided by a microwave synthesizer, which can be computer controlled [34]. For the experiments described in this work, a Watkins-Johnson 1291A synthesizer was used to create fundamental frequencies in the range 8-18 GHz, accurate to ‘1O Hz, which were then doubled or tripled as required. The microwave radiation transmitted through the Stark cell is detected by a point-contact diode detector and the resulting signals are  Chapter 3. Experimental Methods  42  2ll  tz F  trode  Teflon  Figure 3.1: Schematic cross-section of a Stark cell.  43  Chapter 3. Experimental Methods  pre-amplified before being sent to the lock-in amplifier (LIA). The square-wave generator produces a square wave of varying voltage (0-2000 V), which is then applied to the Stark electrode in the presence of the gas sample. The frequency of the square wave is 100 kHz; this frequency is also sent to the LIA. The square wave is zero-based, so that the gas sample alternately experiences e=O and  E=é5t&.  When e=0, the sample will absorb microwave radiation at its characteristic rotational transition frequencies  ‘rot;  when  E=EStk,  the sample absorbs at the frequencies  /Stk 1  of all the M components of the J’r’-Jr transition. The LIA detects only those signals which are modulated at 100 kHz, and so spectral noise which is not modulated is filtered out, improving the signal-to-noise ratio. The LIA is phase-sensitive, so the absorptions occurring at e=0 and  Stark 6  are displayed on opposite sides of the baseline.  The limiting contribution to the linewidths is modulation broadening, a result of finite observation periods as the molecules alternate rapidly between =0 and  =Stark  states.  Wall broadening and pressure broadening also contribute, and linewidths observed in this work using the Stark-modulated spectrometer were typically —250 kHz FWHM (full width half maximum).  3.2 3.2.1  Microwave Fourier transform spectroscopy Theory  In MWFT spectroscopy, a short pulse of microwave radiation interacts with the dipole moments of a large number of molecules, creating a macroscopic polarization of the sample. This quantum mechanical interaction may be treated macroscopically using density matrix formalism [35]. For a molecule with only two rotational levels of energies Ea and Eb (Eb > Ea), the rotational wavefunction may be expanded in terms of the  Chapter 3. Experimental Methods  to vacuum line  Figure 3.2: Schematic diagram of the Stark-modulated microwave spectrometer.  44  Chapter 3. Experimental Methods  rotational eigenstates ç and  cb  45  as =  +  Caa  (3.8)  Cbbb.  The density matrix for this two-state system is defined as p  Since  q!r.  =  Ib><’b  =  I  +  (3.9) ><  Cbb  Ca/fa  +  Cbb  I.  (3.10)  and q are orthonormal, CaC*  a  CaC*  CbC  CbCb  b  (3.11)  and ptj  =  crc.  (3.12)  For an ensemble of N molecules with the same two rotational eigenstates, Pij =  (3.13)  The density matrix simplifies the calculation of the average value of a quantum mechan ical operator A and observable < A > over the ensemble, as <A >= Tr(A p),  (3.14)  i.e. the average value of < A > is the trace of the matrix obtained by multiplying A and p. This will hold no matter the basis chosen to represent A and p. The time dependence of a system governed by a Hamiltonian H is given by z ---=[H,pj.  (3.15)  If the two-level system described above is subjected to a microwave pulse of frequency w, then the Hamiltonian may be given as H =Hrot +Hpert,  (3.16)  46  Chapter 3. Experimental Methods  where Hrot is given in the  {& b}  basis as EaO  Hrot =  (3.17)  .  6 0E Hpert describes the interaction of the molecular dipole moment ,u with the electric field produced by the microwave pulse. In the {g, elements L  [Lab  qi,}  basis, only the off-diagonal matrix  of p are non-zero, as transitions between rotational eigenstates depend on  Hpert is thus given by 2[tabE cos(.t +  0  Hpert =  [tab6 cos(wt 2  +  4)  )  ,  (3.18)  0  where the microwave field is of amplitude 2, angular frequency  ,  and phase g.  With this Hamiltonian, the time dependence of the elements of the density matrix is given by Paa 0  Paa = Pab  =  Pba  =  iX(PbaPab)cO5(Wt+)  ZWoPab  +  ‘OPba iX(Pba  pbb  Paa) cos(wt + q)  iX(pbb  + X(Paa —  Pab)  pbb)  cos(wt +  (3.19) (3.20)  cos(wt + q ) 5  (3.21)  4),  (3.22)  0 is the angular frequency of the rotational transition, (EbEa)/h. where X=2[I abE/h and w  The matrix elements pj are usually transformed to elements pj, using Paa  =  Paa  (3.23)  Pbb  =  Pbb  (3.24)  Pab  =  pabe  (3.25)  Pba  =  pae_t  (3.26)  47  Chapter 3. Experimental Methods  The elements  4 are defined in a frame of reference rotating at angular frequency w and  are therefore without an explicit time dependence. The real quantities .s, u, v, and w are then defined as 8  W  =  Paa+Pbb=’  (3.27)  =  pab  + p’  (3.28)  =  (Pa  =  Paa  —  —  Pab)  (3.29) (3.30)  Pbb  Neglecting terms in 2w (known as the rotating wave approximation [36]) and choosing the phase of the microwave pulse such that çz=O,  th where Zw=w 0  —  =  0  (3.31)  =  —viw  (3.32)  =  u/sw  =  xv  —  xw  (3.33) (3.34)  w.  Before an excitation pulse is applied (t0), Hpert =0 and p is diagonal in the  q}  basis. The difference in population between the two rotational levels is given by N(p  —  pbb)=Nw(0)=Nwo, and u(O)=v(0)=0. A microwave pulse of length t, is then applied. The microwave frequency is chosen to be near-resonant to the rotational transition, and so Zw is small. The amplitude of the microwaves is then chosen such that x  abE = 2  >>  (3.35)  In this case, Eqs. 3.32-3.34 may be approximated as =  0  (3.36)  Chapter 3. Experimental Methods  48  —xw  =  (3.37)  th = xv.  (3.38)  As v(0)=0, the coupled differential equations of Eqs. 3.37 and 3.38 may be solved to give v(t) = —w 0 sin(xt)  (3.39)  w(t)  (3.40)  wocos(xt);  during the excitation pulse, v(t) and w(t) oscillate between —w 0 and w 0 at an angular frequency x. The microwave field is then turned ofF at time t, whereupon x becomes zero. Ne glecting relaxation process, Eqs. 3.32-3.34 become IL  =  —v/sw  (3.41)  =  uZw  (3.42) (3.43)  th = 0, and so during the ‘observation’ period after u(t)  =  —v sin(LSt)  (3.44)  cos(L.ct)  (3.45)  v(t) = w(t)  (3.46)  ,. 1 wt  The macroscopic polarization of the sample is given by P(t) = N <  t>  (3.47)  =  NTr(pp)  (3.48)  =  NTr  =  0  ab  Paa  Pab  ILab  0  Pba.  Pbb  N,LLab(pab  + Pba).  (349) (3.50)  Chapter 3. Experimental Methods  49  Using the definitions of Pab, Pba, u, and v, P(t)  = N,Uab[U(t)  cos(.t)  —  v(t) sin(t)].  (3.51)  During the observation period, P(t)  cos(wt) + cos(Lwt) sin(wt)]  =  N,LtabVtp[Sfl(/Wt)  =  N/JabVtp sin[(& + w)t]  =  NIIabVtp  sin(wot).  (3.52) (3.53) (3.54)  The macroscopic polarization will thus oscillate at the frequency of the rotational transi tion. The electric field produced by this oscillating polarization will also oscillate at w , 0 in quadrature to P(t). Since the amplitude of the field is proportional to v,, optimum signals are obtained if the microwave field is turned off when v is at its maximum or minimum. This is known as the ‘ir/2’ condition: optimized microwave pulses satisfy Xtp2[LabEtp/h =  nir/2, where n is an integer. Assuming e to be constant, a molecule  with a smaller transition dipole moment will require a longer microwave pulse to produce an optimized signal. However, the signal intensity is proportional to I-tab, rather than to ,u as in conventional absorption spectroscopy (Eq. 2.18), and so time-domain mi crowave spectroscopy is well-suited to the study of molecules with small transition dipole moments. 3.2.2  Instrumentation  Microwave cavity The pulsed molecular beam, pulsed microwave, cavity Fourier transform spectrometer used in this work is of the type first developed by Balle and Flygare [6]. The Fabry Perot microwave cavity is formed by two spherical aluminum mirrors, of diameter 28 cm  Chapter 3. Experimental Methods  50  and radius of curvature 38.4 cm. The mirrors sit within a vacuum chamber, separated by -.30 cm. The cavity must be adjusted so that the exciting microwave radiation can form a standing wave pattern between the antennae centred in the two mirrors; this is accomplished by moving one of the microwave mirrors with a micrometer screw until the cavity is ‘tuned’ into resonance. When operated at 10 GHz, the cavity has a bandwidth of -1 MHz. During a search over a given frequency region, the exciting microwave frequency is scanned in discrete steps no more than 1 MHz apart, with measurements performed at each frequency. The frequency range of the spectrometer is currently 4-26 GHz. Electronics A schematic circuit diagram for the spectrometer is shown in Fig. 3.3. The microwave source, accurate to r”O.l Hz, is a Hewlett-Packard 8340A microwave synthesizer, which is controlled by a personal computer via an IEEE bus. This synthesizer also produces a 10 MHz reference signal, which is used for up- and down-conversion and which controls the timing of the experiment. At each excitation frequency Vexcjtatjon=VMW O MHz, the 2 output of the synthesizer is first swept over a frequency range centred at 11 Mw• The radi ation is coupled into the cavity through an antenna in the tuning mirror, and monitored with an oscilloscope connected to the antenna in the stationary mirror. The cavity is tuned so that maximum transmission of radiation is centred at  20 1/MW-  MHz.  Once the cavity has been tuned, the synthesizer output frequency is locked at  MW 1/  This frequency is then mixed with 20 MHz in a single side-band modulator to produce 20 MW-  MHz (the  vMW +20  MHz side-band may also be used.) The microwave pulse is  created by opening and closing a PIN switch (PIN switch 1). The pulse passes through the circulator to the microwave cavity, where it interacts with the gas sample which has been introduced into the cavity. After the microwave pulse, the gas sample emits radiation at the frequencies of the transitions which are off-resonant from  excitation 1/  by a  Chapter 3. Experimental Methods  stationary mirror  51  tuning mirror  20 MHz  ‘MW  nozzle  +  tw  —  oscilloscope 10 MHz  Figure 3.3: Schematic circuit diagram for the cavity MWFT spectrometer.  52  Chapter 3. Experimental Methods  small amount Lv and which have been excited by the microwave pulse. Those transition frequencies which fall within the bandwidth of the cavity are transmitted back through the circulator. This molecular signal is amplified, down-converted to 20 MHz-Zv, and then to 5 MHz+tv. This last signal is fed through a 5 MHz bandpass filter to eliminate other signals leaving the RF mixer. Fourier transformation The 5 MHz+LSi’ signal is collected by a transient recorder board in the personal com puter. 4 K data points are collected at 50, 100, or 150 ns sample intervals (st), and are then transferred to the computer, where signal averaging is accomplished by addition of successive time-domain signals. A frequency-domain spectrum F(zi), with both real and imaginary components, is computed from the time-domain signal f(nzt) using a discrete Fourier transform [37]: F(v)  =  . 2t f(nt)e  (3.55)  For all of the studies outlined in this work, the frequency-domain spectra were displayed as ‘power’ spectra, calculated by adding the squares of the real and imaginary components of F(v). The frequencies obtained in this spectrum are then added to vMwO MHz-5 MHz 2 to obtain the molecular transition frequencies. When two or more frequencies in the spectrum are very close together, distortions of the peak positions in the power spectrum may lead to inaccurate frequencies. These distortions may be avoided by performing a ‘decay fit’ [38], whereby amplitudes, relaxation times, frequencies, and phases of all of the signals are fit to the time-domain spectrum by least-squares. If the molecular signal were purely sinusoidal and could be observed for an infinite time, the frequency-domain spectrum before down-conversion would consist of a series of infinitely sharp peaks, corresponding to a series of 6-functions centred at the transition  Chapter 3. Experimental Methods  53  frequencies. However, the molecular signal is observed for a finite time T; the uncertainty principle dictates that the precision with which transition frequencies can be determined is inversely proportional to T, and so the peaks in F(v) have a width which depends on 1/T. In addition, relaxation processes do occur in the gas sample and the molecular signals decay exponentially, resulting in Lorentzian line shapes in the frequency spectrum. The resolution can be improved artificially and at the expense of the signal-to-noise ratio by adding N’ zeros at the end of f(n/t) so that T becomes (N + N’)Lt. For example, an 8 K FT is the result of adding 4 K zeros to the 4 K data points collected in each experiment. In order to determine unambiguously the frequencies v present in the time domain, f(nLt) must be sampled twice per period [39], i.e. 2Lt1/v. If /t=50 ns, the spectral range is then 10 MHz. However, time-domain signals with v>10 MHz will appear in the frequency-domain spectrum, as the frequency-domain spectrum is replicated every 1/Lt=20 MHz [37], allowing peaks outside the spectral range to ‘fold in’. Care must be taken to filter out these frequencies before Fourier transformation, which is accomplished with a 5 MHz bandpass filter in the arrangement shown in Fig. 3.3. Gas sample The gas sample is injected into the cavity as a pulsed jet, using a nozzle with a small (‘—.4 mm) orifice diameter. The gas is held behind the nozzle at room temperature and pressures of —1-2 atm, and is allowed to expand adiabatically through the orifice into the vacuum chamber. Gas samples typically contain a high fraction of a rare gas, and so in front of the nozzle the molecule of interest undergoes collisions with mostly rare gas atoms; rotational and vibrational energy is thus transferred to translational energy along the jet axis. The velocity distribution narrows, producing a very low translational temperature. Rotational energy is lost rapidly, and low rotational temperatures (‘—.4 K)  Chapter 3. Experimental Methods  54  are obtained, while vibrational energy transfer is somewhat slower. Higher rotational temperatures may be achieved by simply decreasing the amount of rare gas in the sample. In terms of microwave spectroscopy, this supersonically cooled gas sample has several advantages over static gas samples. Most of the population is collapsed down into the lowest electronic, vibrational, and rotational states; by increasing the populations in the levels involved in rotational transitions, some of the sensitivity problems encountered in room-temperature microwave spectroscopy are avoided. Supersonic cooling also permits the study of unstable species such as van der Waals molecules (e.g. Ar-OCS [40]), which are formed by collisions in the jet but which lack sufficient energy to dissociate. Balle and Flygare [6] positioned the gas nozzle perpendicular to the axis of the mi crowave cavity. However, sensitivity and resolution are improved dramatically if the nozzle is placed near the centre of one of the microwave mirrors, presumably because the molecules travel longer through the region of highest microwave field strength. Because the molecules travel parallel to the direction of microwave propagation, the Doppler ef fect splits lines in the frequency spectrum into doublets; the average frequency of the two components corresponds to the transition frequency. With this arrangement, linewidths as low as 6 kHz FWHM can be achieved, limited by the transit time of the molecules through the microwave field. While the gas expansion is not collimated, and hence is not a true molecular beam, little if any Doppler broadening is observed. Pulse sequence A schematic diagram of the pulse sequence used in a MWFT experiment is given in Fig. 3.4. A pulse generator creates the transistor-transistor logic (TTL) pulses which control the PIN diode switches, the gas nozzle, and the transient recorder board. The nozzle is opened and closed to allow the gas into the cavity, and then a microwave pulse is formed by opening and closing PIN switch ]. Acquisition of the molecular signal is started  55  Chapter 3. Experimental Methods  open  PIN 1 open  PIN 2 trigger  closed  P_______ open  nozzle  Figure 3.4: Schematic pulse sequence diagram for the cavity MWFT spectrometer.  Chapter 3. Experimental Methods  by sending a trigger pulse to the transient recorder. Microwave pulses of  56  mW are  used, and so while PIN 1 is open, PIN 2 is kept closed to protect the sensitive detection circuit from exposure to the full microwave power. The exciting microwave radiation is stored by the Fabry-Perot cavity, causing the cavity to ‘ring’ at the excitation frequency after the pulse is turned off. This ringing is often stronger than the molecular signal, so for each experiment the exponential decay of the cavity ringing is recorded before the gas sample is introduced. The cavity decay is then subtracted from the signal obtained in the presence of gas to leave only the molecular signal. The repetition rate of the experiments is generally limited by the pumping speed of the diffusion pump attached to the vacuum chamber, and typically ranges from 0.3-10 Hz.  Chapter 4  The Microwave Spectrum of BrNC’ 0 8  4.1  Introduction  Bromine isocyanate, BrNCO, is one of several isocyanates whose gas phase structures have been determined in the past 20 years, via both microwave spectroscopy and elec tron diffraction. Isocyanates contain the NCO moiety, giving this group of molecules similar chemical properties to the halides; along with azides, thiocyanates, and cyanides, isocyanates are known as pseudohalides [41]. Monomeric isocyanates are thermally un stable at room temperature [42]. Of the possible halogen isocyanates, a full substitution structure has been previously determined for C1NCO alone [43], while only one isotopic combination of INCO 0 6 1 ( 1 1 C 2 N’ 27 4 ) ’ has been studied by microwave spectroscopy [44]. Attempts to prepare FNCO in the gas phase have been unsuccessful [45], although JR spectra have been recorded for FNCO produced by photolysis of matrix-isolated FCON 3 and XCONF 2 (X  =  H, NF , CF 2 ) [46]. 3  Bromine isocyanate lies between C1NCO and INCO in terms of thermal stability. Jt was first prepared by Cottardi by reacting Br 2 with silver cyanate at 150°C [47], as well as by vacuum thermolysis of tribromoisocyanuric acid, (BrNCO) 3 [48]. Evidence that the molecule is indeed an isocyanate, containing a Br-N bond, was obtained from the He(J) photoelectron spectrum [42], and from the ‘ N nuclear quadrupole hyperfine structure 4 observed in an initial microwave spectroscopic study [1, 2]. Jnfrared spectra of the normal and of isotopically labelled species have been reported [48, 49] and used to determine a  57  Chapter 4. The Microwave Spectrum of BrNC’ 80  58  harmonic force field and some structural parameters. The structures of the halogen isocyanates are interesting, as they are expected from simple bonding theories to have C symmetry, with a bend of NCO chain. chain of  i-.’  90  .‘  1200  at N, and a linear  However, C1NCO has been shown instead to have a bend in the NCO [43] (as does HNCO [50]), and the infrared studies are consistent with a  similar bend for BrNCO [49]. This could not be confirmed from the initial microwave study, because spectra of only two isotopic species, 0 6 N 4 Br’ 79 C 2 ’ ’ and 0 6 N 4 Br’ 81 C 2 , ’ ’ had been measured [2]; 79 Br and 81 Br are of approximately equal natural abundance, allowing both isotopomers to be observed. At least one more isotopic substitution was required to determine the structure completely (see Section 2.4.) In the work described here, the microwave spectra of the substituted isotopomers 0 N 4 Br’ 79 1 C 2 8 ’ and 0 8 C 2 N’ 14 Br 81 ’ have been measured using a Stark-modulated mi crowave spectrometer. The results of this study have been combined with those obtained earlier [2] to permit an unambiguous determination of the geometry of bromine iso cyanate. The initial measurements and a preliminary analysis were made by Ms. Mimi Lam [51]. The work carried out by this author expanded greatly the number and types of measured transitions, and included a detailed analysis and structure determination [3]. 4.2  Experimental methods  In the previous microwave study [2], fresh BrNCO was synthesized during measurements by flowing Br 2 over heated AgNCO. However, as the natural abundance of (as compared to 99.76 % for  160),  180  is 0.20 %  0-enriched samples of BrNCO were necessary in 8 ‘  the present study, and hence the flow system was impractical. Instead, the sample, provided by Prof. H. Willner of Universitãt Hannover, was prepared by pyrolysis of  18o.  enriched tribromoisocyanuric acid, following the method of Ref. [49]. In the solid state  59  O 18 Chapter 4. The Microwave Spectrum of BrNC  (below —60°C), BrNCO forms yellow crystals, but at higher temperatures it melts to a brown liquid, and the molecules tend to dimerize to crystalline N,N-dibromocarbamoyl , samples were stored in BrNC’ 0 isocyanate [47]. To maintain quantities of monomeric 8 liquid nitrogen. BrNCO vapour was introduced into the Stark cell by placing the sample on a vacuum line connected to the cell and allowing it to warm to room temperature until the entire sample had melted. The sample tube was then placed in liquid nitrogen, and most of the BrNCO vapour was collected back in the tube, leaving 15-30 mTorr in the microwave cell. The cell was surrounded by dry ice, and this cooling, combined with relatively low pressures within the cell, discouraged dimerization. The sample remained stable for approximately 20 minutes, after which point the cell was evacuated and fresh BrNCO was added. The Stark-modulated spectrometer described in Section 3.1 was used to make mea surements in the frequency region 23-52 GHz. Measurements are estimated to be accurate to better than +0.05 MHz.  4.3  Observed spectrum and analysis  0 were based on 8 BrNC’ 0 and 81 8 BrNC’ Initial predictions of the rotational constants of 79 the three possible structures proposed by Jemson et al. [2]. Since the molecule is planar, with C 3 symmetry, a- and b-type transitions were expected, with the former, following Ref. [2], much stronger than the latter. Transition frequencies were calculated using these rotational constants, combined with centrifugal distortion constants transferred from the normal species. The transitions were all expected to show quadrupole hyperfine structure caused N quadrupole coupling by the Br nucleus (IBr=3/2); the much smaller 14  (I14N=1)  was  Chapter 4.  60  80 The Microwave Spectrum of BrNC’  O, the Br principal quadrupole axis z is essen 16 not resolved in this study. In BrNC tially aligned with the Br-N bond, at an angle of  to the a principal inertial axis,  is relatively large; the only non-zero off-diagonal quadrupole coupling constant  and  QBr  Xab  500 MHz [2]. It was expected that any accidental rotational near-degeneracies  of appropriate symmetry would result in measurably perturbed hyperfine patterns (see Section 2.5.) Obtaining an accurate value for  Xab  depends on observing such perturbations. From  an examination of Eq. 2.10 in the limiting case of a prolate symmetric rotor, it is also apparent that the A rotational constant cannot be determined precisely from rotational transition frequencies alone if only transitions with zSKa=0 (i.e. a-type transitions) are measured. However, as the positions of the rotational energy levels and hence the near degeneracies are determined by all three rotational constants, the perturbations are a source of information, and perturbed hyperfine patterns may allow A to be determined if b-type transitions are too weak to be found easily. 0 (Fig. 4.1) shows an important near-degeneracy 8 BrNC’ The energy level diagram for 79 between the levels 1 Ooo and  918,  where  918  is 822.5 MHz above  These levels will  interact via Xab, as they satisfy the requirements LiF=0, LJ=0, +1, +2, and iK,= eo—*oo, oe—*ee. The earliest searches, performed by Ms. Lam [51], included a-type tran sitions involving these levels, as they were expected to produce the most information. In the present work, many more transitions of the same type were assigned in order to improve the precision of the derived constants. Ultimately lines of 22 a-type rotational transitions of each of 79 O and 81 18 BrNC O were measured and assigned. Individual 18 BrNC lines were typically located within a scan frequency range of <40 MHz about the pre dicted frequencies; the assignments were confirmed by the hyperfine patterns. These lines were fit to rotational, centrifugal distortion and Br quadrupole coupling constants using a global least-squares fitting program [1, 2] employing Watson’s A reduction; the results  61  O 18 Chapter 4. The Microwave Spectrum of BrNC  15114  00  2,11 3212  2031.9 MHz 150,15  2,10 1,13  141,14  400  t  140,14  122,11  I I  2,9 2,10  1,12 ‘31,13 2,8  N  ‘02,9  130,13  12 27 ‘-‘2,8  ‘N  300  120,12  1,10 111,11  2,6  2,5  822.5 MHz  10 62,5  100,10  1,9  9Q,9  81,8  8Q,9  71,7  200  1,7  1,6  1,5  61,6  100  79,7  1,4  695  41: 31: 2l:  5  1 1:  49,4  o  30,3  1 0,1  000  Figure 4.1: Rotational energy levels of 79 0. 8 BrNC’  2,3 52,4 42.2 23  34 2;9  62  0 8 Chapter 4. The Microwave Spectrum of BrNC’  are given in Table 4.3. Values for three rotational constants, three centrifugal distortion constants and three independent Br quadrupole coupling constants were produced for 0 and each isotopomer. In particular, precise values for both A a-type transitions alone. (Note that the sign of  Xab  Xab  were produced from  cannot be determined, since the  second order perturbation energy depends on Xb.) The rotational constants so determined were now sufficiently precise to make searches for the very weak b-type transitions practicable. The searches were only partially success ful; even with frequency predictions accurate to better than ±5 MHz, many b-type tran sitions were too weak to be observed, or were overlapped by other lines. Thus only three 0 8 BrNC’ such transitions were unambiguously assigned and included in the final fit for 79 (12111-12012, 15114-15015, 0 fit (121 11-12012 8 BrNC’ 81  and 19019-18118); two such transitions were included in the and  151 14-15015).  Chapter 4. The Microwave Spectrum of BrNC 18 0  F’  -  F”  wt.a  frequency  obs.-calc. no Xab Xab  *,b  V16  -  is 6.5 5.5 3.5 4.5  7.5 6.5 4.5 5.5 606 6.5 5.5 7.5 6.5 4.5 3.5 4.5 5.5 615-54 7.5 6.5 4.5 3.5 6.5 5.5 4.5 5.5 8.5 7.5 5.5 6.5  -  1.00 1,00 1.00 1.00  23609.528 23613.688 23615.828 23620.228  0.335 0.223 -0.013 0.002  0.025 0.013 0.008 0.050  0.50 0.50 1.00 0.10  23908.713 23908.713 23919.621 23923.302  -0.177 0.085 4.294 7.756  -0.023 0.063 -0.024 0.041  1.00 1.00 1.00 1.00  24220.376 24223.341 24224.321 24227.981  0.237 -0.064 -0.064 0.189  -0.036 -0.045 -0.061 -0.083  1.00 1.00 0.75 0.75  27544.958 27547.632 27549.891 27552.650  0.119 0.083 -0.035 -0.049  -0.029 -0.018 -0.014 -0.010  7.5 4.5 6.5 5.5  1.00 0.10 0.10 1.00  28257.665 28259.828 28259.828 28263.098  0.623 0.117 0.090 0.613  0.111 0.141 0.085 0.064  8.5 7.5 5.5 6.5  0.25 0.25 1.00 1.00  31871.232 31871.232 31874.100 31874.969  -0.103 -0.473 -0.828 -0.317  -0.120 0.026 -0.049 -0.013  8.5 7.5 5.5 6.5  1.00 0.50 0.10 1.00  32294.387 32294.916 32295.096 32298.515  1.331 0.022 -0.130 1.402  -0.017 -0.007 -0.102 -0.052  1.00 1.00 1.00 1.00  35848.981 35849.905 35851.093 35852.929  -1.343 0.002 -1.609 -0.186  1.00 1.00 1.00 1.00  35880.482 35881.869 35885.476 35887.014  0.084 0.030 -0.034 -0.028 0.078 0.035 0.000 0.008  0.50 0.50 1.00 1.00  36329.970 36329.970 36342.321 36343.155  0.403 0.053 -0.049 -0.016 14.084 -0.030 11.776 -0.006  1.00 1.00 1.00 1.00  39812.810 39816.782 39826.436 39828.668  1.00 1.00 1.00 1.00  39866.364 39867.615 39870.054 39871.395  0.093 -0.003 -0.065 -0.059 0.116 0.039 -0.025 -0.018  1.00 1.00 1.00 1.00  39899.740 39901.173 39903.010 39904.480  -0.035 -0.083 -0.047 -0.041 0.046 0.010 -0.003 0.000  1.00 1.00 1.00 1.00  40341.072 40343.826 40363.014 40364.081  -21.508 0.041 -21.265 -0.007 -0.576 0.028 0.021 0.062  0*  9.5 10.5 7.5 8.5  8.5 9.5 6.5 7.5 827 928 10.5 9.5 7.5 6.5 9.5 8.5 8.5 7.5 91s-87 9.5 8.5 7.5 6.5 10.5 9.5 8.5 7.5 10010 9 9 10.5 9.5 8.5 7.5 11.5 10.5 9.5 8.5 1029-928 11.5 10.5 8.5 7.5 10.5 9.5 9.5 8.5  -0.005 -0.013 -0.034 -0.018  -14.037 -11.836 0.059 -0.416  0.067 0.029 0.045 0.017  1028-927  817-76  9.5 8.5 6.5 7.5  frequency  —  808- 77  9.5 8.5 6.5 7.5  obs.-calc. no Xab Xab  wt.  -  716—615  8.5 5.5 7.5 6.5  F”  °o  ‘09  6  7.5 6.5 4.5 5.5  -  fl  -  i7 7  F’  63  11.5 10.5 8.5 7.5 10.5 9.5 9.5 8.5 1019-918 11.5 10.5 9.5 8.5 10.5 9.5 8.5 7.5  a  Measurements were weighted according to 1/o.2, where o is the uncertainty in the measurements. Unit weight corresponds to an uncertainty of 0.05 MHz. b Transitions marked by * were measured by Ms. Mimi Lam [51].  Table 4.1: Measured transitions of 79 0 (in MHz) 8 BrNC’  Chapter 4.  F’  -  ii  12.5 11.5 13.5 10.5 111 ii 12.5 11.5 9.5 10.5  -  -  F” o 12 12 12.5 11.5 13.5 10.5 10 11.5 10.5 8.5 9.5 11.5 9.5 10.5 8.5  frequency  obs.-calc. no xab x  F’  1.00 1.00 1.00 1.00  42429.904 42434.845 42472.588 42475.983  -1.939 -2.079 -1.116 -2.871  0.029 -0.0 64 0.0 24 0.036  12 12 13.5 12.5 10.5 11.5  1.00 1.00 1.00 1.00  43279.046 43279.889 43281.161 43282.007  -0.011 0.034 -0.079 -0.042  -0.043 0.014 -0.047 -0.005  0.50 1.00 1.00 1.00  43800.531 43803.504 43822,719 43823.583  -0.030 0.593 21.640 21.187  -0.042 0.045 0.084 0.002  -  -  11.5 8.5 10.5 9.5 l5 15 15.5 14.5 16.5 13.5 l8 18 18.5 17.5 19.5 16.5  0.50 0.50 0.50 0.50  43851.682 43852.635 43854.480 43855.500  0.322 -0.003 0.341 0.032  0.081 0.003 0.143 0.038  1.00 1.00 1.00 1.00  44781.782 44786.716 44824.419 44828.725  -0.779 0.016 -0.463 -0.344  -0.015 -0.013 0.038 -0.030  1.00 1.00 1.00 1.00  45042.956 45044.982 45070.258 45073.506  0.628 -0.302 -0.036 0.290  0.039 0.041 -0.040 -0.050  -  —  12012—  11210-1029  12.5 9.5 11.5 10.5 15i 14 15.5 14.5 16.5 13.5 19 19 19.5 18.5 20.5 17.5  obs.-calc.  wt.  11011-10010  12.5 10.5 11.5 9.5  64  18 0 The Microwave Spectrum of BrNC  12.5 10.5 13.5 11.5 l3 13 14.5 13.5 11.5 12.5 132 12 14.5 13.5 11.5 12.5  -  —  F”  wt.  frequency  no  1.00 1.00 1.00 1.00  47210.598 47211.264 47212.323 47213.043  -0.009 0.004 -0.128 -0.070  -0.034 -0.012 -0.085 -0.022  0.25 0.25 1.00 1.00  47770.270 47770.270 47772.256 47774.287  -2.550 -3.513 0.002 -0.060  -0.061 0.010 -0.010 0.050  1.00 1.00 1.00 1.00  51141.276 51141.894 51142.758 51143.373  -0.013 0.051 -0.109 -0.054  -0.034 0.039 -0.043 0.020  1.00 1.00 1.00 1.00  51815.458 51817.977 51820.212 51822.035  -3.743 -2.956 -0.013 0.052  -0.114 -0.055 -0.006 0.056  1.00 1.00 1.00 1.00  51893.253 51893.718 51894.192 51894.879  0.593 -0.001 0.416 0.017  0.102 0.006 0.028 0.014  Xab  Xab  lii ii  12.5 11.5 9.5 10.5 lloii 11.5 9.5 12.5 10.5 12 12 13.5 12.5 10.5 11.5 122 11 13.5 12.5 10.5 11.5  13211— 12210  14.5 11.5 13.5 12.5  13.5 10.5 12.5 11.5  Table 4.1: Measured transitions of 79 0 (cont’d) 8 BrNC’  O 18 Chapter 4. The Microwave Spectrum of BrNC  F’ U16  F”  -  wt.a  frequency  obs.-calc. no Xab Xab  *,b  6.5 5.5 3.5 4.5  1.00 1.00 1.00 1.00  23435.419 23438.947 23440.765 23444.419  0.233 0.179 0.003 -0.005  0.008 0.026 0.018 0.029  0.50 0.50 0.50 0.50  23730.320 23730.320 23738.906 23741.056  -0.192 0.027 2.996 4.957  -0.087 0.012 0.104 -0.054  5.5 6.5 3.5 4.5  1.00 1.00 1.00 1.00  27341.628 27343.944 27345.864 27348.120  0.073 0.117 0.044 -0.016  -0.033 0.045 0.059 0.011  1.00 0.25 0.25 1.00  28044.167 28046.075 28046.075 28048.727  0.312 -0.034 -0.042 0.301  -0.028 -0.017 -0.045 -0.066  0.50 0.50 1.00 1.00  31633.496 31633.496 31635.959 31636.562  -0.013 -0,319 -0.564 -0.259  -0.025 0.012 -0.031 -0.039  1.00 1.00  31672.443 31678.060  0.034 0.016  0.015 0.002  1.00 1.00 1.00 1.00  35151.923 35153.015 35154.515 35155.625  0.036 0.012 -0.056 -0.079  -0.005 -0.016 -0.040 -0.057  717-616  8.5 7.5  7.5 6.5  J)  ‘±.)  6.5  5.5 6i 7.5 4.5 6.5 5.5  8.5 5.5 7.5 6.5 0 008  o  0.)  8.5 7.5 6.5 5.5 7.5 6.5 826-725 9.5 8.5 8.5 7.5 ‘19  10.5 9.5 7.5 8.5  9.5 10.5 7.5 8.5  8.5 9.5 6.5 7.5  10.5 7.5 9.5 8.5  9.5 6.5 8.5 7.5  1.00 1.00 1.00 1.00  35581.887 35582.397 35583.669 35584.929  -0.888 -0.030 -1.107 -0.189  -0.032 -0.041 -0.075 -0.071  1.00 1.00 1.00 1.00  35612.557 35613.781 35616.805 35618.005  0.041 -0.006 0.100 -0.050  0.005 -0.002 0.071 -0.044  1.00 0.50  35636.304 35637.492  0.041 -0.073  0.017 -0.069  1.00 1.00  36060.920 36063.185  6.940 6.567  -0.006 0.069  1.00 1.00 1.00 1.00  39056.036 39056.864 39058.160 39058.994  0.097 0.081 0.024 0.002  0.068 0.061 0.042 0.024  1.00 1.00 1.00 1.00  39522.775 39524.611 39529.345 39531.344  -6.927 -6.584 0.031 -0.236  0.013 -0.026 0.021 0.001  1.00 1.00 1.00 1.00  39568.552 39569.646 39571.686 39572.804  0.050 -0.044 0.113 -0.009  -0.013 -0.039 0.062 -0.004  927-826  10.5 7.5  9.5 6.5 0*  A  ?18017  10.5 8.5 10 10 11.5 10.5 8.5  -  9.5 7.5 99 10.5 9.5 7.5  in  IVO  10  n* —  ‘o 9  10.5 9.5 8.5 7.5 10.5 11.5 9.5 8.5 1029-928 11.5 10.5  *  18  9.5 8.5 6.5 7.5  obs.-calc. no Xab Xab  008  ‘07  )  A  -  ‘7*  A  frequency  928-827  6o6-55  6.5 7.5 4.5 5.5  wt.  0*  ‘09  7.5 6.5 4.5 5.5  F”  -  A  —  716-  F’  65  10.5 9.5  9.5 8.5  a  Measurements were weighted according to 1/2, where o- is the uncertainty in the measurements. Unit weight corresponds to an uncertainty of 0.05 MHz. b Transitions marked by * were measured by Ms. Mimi Lam [51].  Table 4.2: Measured transitions of 81 0 (in MHz) 8 BrNC’  0 8 The Microwave Spectrum of BrNC’  Chapter 4.  F’  F”  -  obs.-calc. no Xab Xab  wt.  frequency  1.00 1.00 1.00 1.00  39601.062 39602.286 39603.786 39605.046  0.75 1.00 0.75 1.00  40045.583 40046.536 40058.348 40059.078  1.00 0.50 1.00 1.00  42342.248 42346.579 42377.787 42380.770  -1.452 -0.009 -1.354 0.082 -0.810 0.015 -2.108 -0.008  1.00 1.00 1.00 1.00  42959.217 42959.943 42960.990 42961.728  -0.014 -0.036 0.046 0.031 -0.070 -0.048 -0.007 0.018  1.00 1.00 1.00 1.00  43473.915 43476.189 43486.780 43488.946  -0.059 0.249 12.378 13.432  F’  10.5 7.5 9.5 8.5  -0.027 -0.059 -0.020 -0.016 0.019 -0.006 0.009 0.012  91 8  -  -  -  -  -  F”  obs.-calc. no Xab Xab  wt.  frequency  1.00 1.00  44062.936 44065.426  2.074 2.835  0.054 0.077  1.00 1.00 1.00 1.00  44659.172 44663.228 44694.596 44698.316  -0.542 0.068 -0.380 -0.139  0.001 -0.029 -0.023 0.013  1.00 1.00 1.00 1.00  46861.740 46862.340 46863.228 46863.822  0.014 0.068 -0.043 -0.001  -0.003 0.057 -0.014 0.031  1.00 1.00  47416.264 47417.902  0.053 -0.057  0.045 0.028  1.00 0.50 1.00 0.50  50763.333 50763.843 50764.599 50765.091  -0.046 0.001 -0.102 -0.077  -0.061 -0.007 -0.058 -0.028  11110- 1019  1028-927  11.5 8.5 10.5 9.5 101 9 11.5 9.5 10.5 8.5 12 12.5 11.5 13.5 10.5 111 fl 12.5 11.5 9.5 10.5 110 11 12.5 10.5 11.5 9.5  66  10.5 8.5 9.5 7.5 12 12 12.5 11.5 13.5 10.5 101 10 11.5 10.5 8.5 9.5 10 10 11.5 9.5 10.5 8.5  -12.263 -13.418 -0.344 -0.016  0.113 0.024 0.018 0.012  12.5 10.5 15i 14 15.5 14.5 16.5 13.5 12 12 13.5 12.5 10.5 11.5 12o 12 13.5 11.5 13113  14.5 13.5 11.5 12.5  -  -  11.5 9.5 15015 15.5 14.5 16.5 13.5 111 11  12.5 11.5 9.5 10.5 -  -  11011  12.5 10.5 121 12 13.5 12.5 10.5 11.5  -0.068 -0.073 -0.003 0.040  Table 4.2: Measured transitions of 81 0 (cont’d) 8 BrNC’ The measured transitions included in the fits are given in Tables 4.1 and 4.2, along with the residual (observed—calculated) frequencies. Note that when zero in the fit, the residuals (marked ‘no when  Xab  Xab’)  was released into the fit (marked  involving the  918  Xab  was set to  were noticeably larger than those obtained  ‘Xab’).  As expected, frequencies for transitions  and 10010 levels depend strongly on Xab• In particular, the F=8.5 and  F=lO.5 hyperfine components of these levels are shifted by as much as 21 MHz by the perturbation. Inspection of Fig. 4.1 also shows that the level 12211 is only 2031.9 MHz above  13  in 79 0, and so transitions involving these levels should be perturbed. 8 BrNC’  Chapter 4.  67  0 8 The Microwave Spectrum of BrNC’  Only one such transition was measured (13212-12211), but the F=12.5 and F=13.5 levels of  12211  seem to be shifted by ‘-‘3 MHz.  The final data set was analyzed using the global least-squares program to produce rotational, centrifugal distortion, and Br nuclear quadrupole coupling constants. The results are in the upper part of Table 4.3, in the columns labelled ‘all transitions’. It is clear that inclusion of the b-type transitions has greatly improved the precision of several , 0 constants, notably A  (Xbb  —  Xcc), and  Xab  .  Table 4.3 also includes the ground state  inertial defect z . 0 Only three of the five quartic centrifugal distortion constants  (zJ,  ZJK,  and  j) 5  could  be determined in the fit. As the extents of the 918-10010 and 12211-141 13 perturbations depend on the relative positions of the Ka=0, iç=i and Ka=2 energy level stacks, this could allow  /K  to be separated from A 0 [34]. Evidently the latter perturbation is not  sufficiently large, as a fit carried out with  /K  released produced an indeterminate value  for that constant. 4.3.1  Harmonic force field and structure  With rotational constants available for four isotopic combinations of BrNCO, the molec ular geometry could now be determined without assuming a value for any bond length or angle. The inertial defects of all four isotopic species are small and positive, with little dependence on isotopic species, supporting the assumption that BrNCO is planar. This assumption is also confirmed by the values of Xcc, obtained from  xaa  and  XbbXcc  using  Eq. 2.104. If BrNCO were non-planar, the angle of the c inertial axis relative to the y principal axis of the quadrupole coupling tensor would change upon isotopic substitution, and so the relative values of  Xcc  0 and 81 8 BrNC’ 0 would depend on that 8 BrNC’ for 79  change, as well as on the ratio of the 79 Br and 81 Br quadrupole moments. However, is 0.8359 for BrNC’ 0, in excellent agreement with the ratio of the 8  Chapter 4.  The Microwave Spectrum of BrNC’ 80  A 0 0 B Co  (MHz) (MHz) (MHz) zij (kllz) (kllz) JK (kHz) j 6 (MHz) Xaa Xbb Xcc (MHz) (MHz) Xab I I Lo (a.m.u. A ) 2 no. of rot. trans. (MHz) fit —  79 O 18 BrNC a-type all transitions transitions  81 0 8 BrNC’ a-type all transitions transitions  40311 .7(24)a 40308.326(72) 2044.0895(24) 2044.0919(18) 1942.4052(23) 1942.4047(17) 1.0377(38) 1.0442(27) -162.00(17) -162.17(14) 0.1445(50) 0.1530(26) 598.6(14) 598.3(14) 295.6(36) 289.99(20) 557.04(68) 556.06(25) 0.4055(4) 22 25 0.042 0.044  40258.8(44) 40260.088(94) 2028.4733(39) 2028.4840(23) 1928.1979(24) 1928.1931(20) 1.0114(62) 1.0263(46) -160.20(20) -160.04(21) 0.1480(29) 0.1268(70) 500.6(15) 500.2(17) 242.3(42) 242.29(25) 464.09(71) 464.28(38) 0.4057(4) 22 24 0.041 0.047  79 Ob 16 BrNC all transitions A 0 0 B 0 C  (MHz) (MHz) (MHz) (kllz) j (kllz) LJK (kllz) j 6 (MHz) Xaa Xbb Xcc (MHz) (MHz) Xab I 0 L (a.m.u. A ) 2 no. of rot. trans. (MHz) alIt —  68  41189.506(25) 2175.63391(52) 2063.09857(53) 1.1370(19) -173.97(14) 0.17367(26) 608.41(52) 280.086(64) 549.67(12) 0.401 43 0.032  O 16 BrNC 81  all transitions 41141.914(29) 2 159.50429(53) 2048.47014(55) 1.1281(18) -172.23(14) 0.17004(31) 508.49(52) 233.311(70) 458.60(12) 0.401 46 0.034  a Numbers in parentheses are one standard deviation in units of the last significant figure. b  Spectroscopic constants for 79 O and 81 16 BrNC O are taken from Ref. [2]. 16 BrNC Table 4.3: Spectroscopic constants of BrNCO  Chapter 4.  The Microwave Spectrum of BrNC’ 80  69  quadrupole moments of the two nuclei (0.8353). ,B 0 0 The planarity of the molecule is thus confirmed, with the result that only two of A and C 0 are independent. The positions of Br and 0 were calculated using Kraitchman’s equations [21] for a rigid molecule (Eqs. 2.66 and 2.67.) Four sets of calculations were made, using each of 79 0, 81 6 BrNC’ 0, 79 6 BrNC’ 0, and 81 8 BrNC’ 0 as the basis 8 BrNC’ molecule. For each basis molecule three calculations were possible, depending on which pair of rotational constants was considered to be independent. Since the force field (see below and Table 4.6) suggested that B 0 and C 0 had the smallest vibrational contribu tions, this pair was chosen. The remaining two atoms were located using the product of inertia and centre-of-mass conditions (Eqs. 2.70, 2.71, 2.72), and by reproducing I. The resulting parameters are given in Table 4.4 as the r 0 values. Note that isotopic substitution at the carbon or nitrogen atoms would remove the necessity of reproducing I, and would improve the precision of the substitution structure (see Section 2.4.) The harmonic force field calculated by Gerke ci al.  [49] by fitting the observed  vibrational frequencies was used to estimate a ground state average (re) structure for BrNCO. The only adjustment to the force field was a minor one: the out-of-plane force constant  f66  was changed to correspond to a different definition of this coordinate. In  the present work,  f66 =  B-matrix elements (rad  0.0142 mdyn  A’):  A—’.  The coordinate was defined as a torsion, with  Br, -0.603; N, -4.419; C, 10.791; 0, -5.770. Since this  force field also reproduced the observed centrifugal distortion constants to within a few percent (see Table 4.5), it was reasonable to leave the rest unchanged. The harmonic contributions to the force field were subtracted from the observed rotational constants for all four isotopomers to give ground state average values (see Eq. 2.76.)  These are given in Table 4.6, along with the resulting calculated inertial  defects L, which are nearly zero, indicating that they are essentially all accounted for by the force field. The structural parameters were calculated again using Kraitchman’s  80 Chapter 4. The Microwave Spectrum of BrNC’  Effective (ro) values  Average (ri) values  70  Values in related molecules  r(Br-N)  (A)  1.8617(23)a  1.85565(73)  1.82 1.82 1.84 1.84 1.88  C C 5 H 6 ONHBr [52] CONHBr [53] 3 n-CH N-bromosuccinimide [52] sum of single bond radii [54] O [52] SBr 5 H 6 C N 2  r(N-C)  (A)  1.2166(40)  1.2224(15)  1.199 1.214 1.226 1.234  NCO [55] 3 SiH HNCO [50] C1NCO [43] NCO [56] 5 SF  r(C-O)  (A)  1.1692(13)  1.16496(53)  1.162 1.166 1.174 1.179  C1NCO [43] HNCO [50] NCO [55] 3 SiH NCO [56] 5 SF  L(XNC)b  117.38(42)°  117.99(14)°  118.83° 123.9° 124.9° 127.1°  C1NCO [43] HNCO [50] NCO [56] 5 SF C1CONCO [57]  L(NCO)  172.33(44)°  173.13(15)°  170.87° 172.6° 173.4° 173.8°  C1NCO [43] HNCO [50] C1CONCO [57] NCO [56] 5 SF  28.66(31)°  28.68(24)°  L(Br  —  N_a)D  a  Numbers in parentheses are one standard deviation in the mean taken over the four choices of basis molecule, in units of the last significant figure. b E.g. X=Br for BrNCO. O. 16 BrNC L(Br N—a) is the angle between the BrN bond and the a inertial axis for 79 —  Table 4.4: Structural parameters of bromine isocyanate  Chapter 4,  The Microwave Spectrum of BrNC’ 80  obs.  j  LJK j  L.j  ZJK j 6  calc.a  71  obs.  calc.  (kllz) (kHz) (kHz)  79 O 18 BrNC 1.0442 1.1212 -162.17 -153.77 0.1530 0.1487  79 Ob 16 BrNC 1.1370 1.2555 -173.97 -164.52 0.1737 0.1726  (kllz) (kHz) (kHz)  O 18 BrNC 81 1.0263 1.1050 -160.04 -152.43 0.1480 0.1456  O 16 BrNC 81 1.1281 1.2378 -172.23 -163.15 0.1700 0.1691  a  Calculated using the force field of Ref. [49], adjusted as described in Section 4.3.1. b Distortion constants for 79 O and 81 16 BrNC O are taken 16 BrNC from Ref. [2].  Table 4.5: Centrifugal distortion constants of BrNCO equations, using a method analogous to that described above. The resulting values are the r,, parameters in Table 4.4. No attempt was made, however, to account for changes in these parameters with isotopomer; these changes should be approximately the same order of magnitude as the uncertainties given.  4.3.2  Discussion and conclusions  The measured rotational constants of BrNC O indicate conclusively that the molecule 18 is the isocyanate, with a Br-N bond, rather than the cyanate, with a Br-O bond. The derived structural parameters are compared with those of related molecules in Table 4.4. There seem to be some minor differences between the bond lengths of BrNCO and those of related species. The most interesting feature, however, is the bend in the NCO chain, which is very comparable to the one found in both C1NCO and HNCO. Of the structures suggested previously for BrNCO [2], number III most closely approximates the one de termined here. The atoms of 79 O in its principal inertial axis system are shown 16 BrNC  Chapter 4. The Microwave Spectrum of BrNC’ 0 8  0 6 BrNC’ 79  0 6 BrNC’ 81  72  0 8 BrNC’ 79  0 8 BrNC’ 81  Rotational Constants (MHz)  0 A  41189.506 -642.934 40546.572  41141.914 -641.213 40500.701  40308.326 -623.193 39685.133  40260.088 -621.327 39638.761  B  2175.634 -2.839 2172.795  2159.504 -2.808 2156.696  2044.092 -2.583 2041.509  2028.484 -2.554 2025.930  C 0 c/2 C  2063.099 -1.114 2061.985  2048.470 -1.108 2047.362  1942.405 -1.049 1941.356  1928.193 -1.042 1927.151  aa/2 A 0 B ab/2  Inertial defects (a.m.u.  & a  0.0353  0.0355  ) 2 A 0.0363  0.0366  From Ref. [2] and Table 4.3 of this work  Table 4.6: Spectroscopic constants of the ground state average structure of BrNCO  Chapter 4.  The Microwave Spectrum of BrNC’ 0 8  0 8 BrNC’ 79 Xzz  x Xyy  (MHz) (MHz) (MHz)  )b 1 ( 8934 8  -449.3(15) -444.14(35) 27.96(1O)°  8  0 8 BrNC’ 81 Xzz  x Xyy Oza  (MHz) (MHz) (MHz)  746.4(23) -375.2(19) -371.24(43) 27.94(15)°  73  Oa 16 BrNC 79 893.95(68) -449.70(56) -444.25(13) 27.451(37)°  O 16 BrNC 81 746.40(68) -375.50(56) -370.90(13) 27.419(44)°  a Values for BrNC O are taken from Ref. [2]. 16  Numbers in parentheses are one standard 6 deviation in units of the last significant figure. Table 4.7: Principal values of the bromine quadrupole coupling tensor in Fig. 4.2. The principal values of the 79 Br and 81 Br quadrupole coupling constants of BrNC’ 0, 8 obtained by diagonalizing the quadrupole tensor, are given in Table 4.7, along with the angle °za between the z principal axis of the tensor and the a principal inertial axis. They are in excellent agreement with the corresponding values for 6 BrNC’ 0 : as expected, isotopic substitution at 0 has no effect on the electronic structure at Br. There is a slight difference (‘..sl°) between the alignment of the z principal axis and the bond containing the quadrupolar atom, as indicated by the difference between L (Br  —  N—a) (Table 4.4) and  °za (Table 4.7.) This difference was also noted in the structures of INCO [44], C1HCCO  [58], and BrSCN [59].  Chapter 4. The Microwave Spectrum of BrNC’ 80  74  b  0  c  Figure 4.2: The molecular structure of BrNCO, given in the principal inertial axis system of 79 0. 6 BrNC’  Chapter 5  C1 S1H The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, 2  5.1  Introduction  The presence of low-lying d orbitals in the silicon atom allows for the possibility of (p  —÷  d)ir back-bonding in halosilanes, a mechanism which is not available to the corresponding halomethanes. The electronic structures of the halosilanes are therefore of interest, and may be investigated via nuclear quadrupole coupling effects when the bonded halogen atom has a quadrupolar nucleus (as is the case for Cl, Br, and I, with nuclear spins of I=3/2, 3/2, and 5/2 respectively.) Values of the halogen nuclear quadrupole coupling X (X=C1 [60], Br [61, 62], 3 constant eQq have been reported for the silyl halides SiH X (X 3 and I [63]) and the trifluorohalosilanes SiF  =  Cl [64], Br [65], and I [64]). The  halosilanes in which the halogen does not lie on a symmetry axis have been studied less extensively by microwave spectroscopy, although partial sets of quadrupole coupling SiC1 [66, 68], and 2 3 SiH 3 SiH I constants have been determined for SiHC1 3 [66, 67], CH [69], and the complete coupling tensor  x  has been determined in the case of 2 SiH 3 CH I  [70, 71]. Electronic information derived from incomplete quadrupole tensors is usually based on the assumption that the z principal axis of the tensor is coincident with the silicon-halogen bond. In 2 SiH 3 CH I , this assumption was confirmed: the z axis and the Si-I bond were found to be separated by only  ‘-‘  41’ [71].  It has traditionally been difficult to measure the complete Cl nuclear quadrupole coupling tensors of asymmetric top molecules where the Cl nucleus is not on a symmetry  75  2 Cl 2 76 Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, S1H  C1 nuclei are relatively small, and effects C1 and 37 axis. The quadrupole moments of the 35 of off-diagonal coupling terms such as Xab are difficult to see with conventional Stark modulated spectrometers. Usually an accidental rotational near-degeneracy is required for their effects to be seen (see, for example, Ref. [58].)  However, with the advent  of MWFT spectrometers these effects are now more readily observed, and Cl coupling tensors can be measured easily [72]. C1 was SiH , The first measurement of the microwave spectrum of dichlorosilane, 2 made with a Stark-modulated spectrometer [7]. Since the linewidths were large (>250 kllz), full resolution of the complex hyperfine patterns resulting from the presence of two cou pling nuclei was not possible, and not all the quadrupole coupling constants could be determined accurately. In particular, Xab was unavailable. In contrast, despite the ex Br CH C1 [73, 74], 2 CH istence of similar complications in the microwave spectra of 2 BrC1 [79], complete sets of coupling constants have been deter 2 [75, 76, 77, 78], and CH mined for these species. In light of evidence of (p  —*  d)?r back-bonding and asymmetric Cl  2 [80] and the SiC1 radical [81], a more conclusive examination field gradients in both SiCl of the bonding in dichiorosilane was warranted. The MWFT spectrometer described in Section 3.2 has facilitated the remeasurement of the microwave spectrum of dichlorosi lane with much higher resolution. All values of the quadrupole coupling tensor have been determined accurately, allowing examination of the electrostatic potential around the chlorine nuclei without prior assumptions about the nature of the Si-Cl bond [8].  5.2  Experimental methods  A dichlorosilane sample of stated purity 97% was purchased from Pfalz and Bauer, Inc., and used without further purification. For most experiments a gas mixture of 5% dichlorosilane in argon, at a total backing pressure of 1-2 atm, was used, although this  Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, SiH 2 Cl 2 77  percentage was increased for some higher-J transitions which required higher rotational temperatures. For the  918-82 7  transition, discussed extensively below, the 2 C SiH : Ar 1  mixing ratio was 1:2. The cavity MWFT spectrometer was used to make measurements in the frequency region 10-16 GHz. Measurements are estimated to be accurate to better than ±1 kllz. 5.3  Observed spectrum and analysis  Four of the previously reported [7] rotational transitions of dichlorosilane were re-examined in this study, and seven transitions were examined for the first time. Most of the hyper fine components of the rotational transitions, which were often overlapped in the earlier work, could now be resolved because of the small linewidths obtainable with the cavity MWFT spectrometer. Dichlorosilane has C 2 symmetry, with the SiC1 2 moiety in the ab inertial plane. The molecular symmetry axis is the b inertial axis. Two diagonal Cl quadrupole coupling constants,  Xbb  and Xcc (or Xaa and  coupling constant  Xab,  Xbb  —  Xcc) may be determined, as may the off-diagonal  although only the diagonal constants contribute to first order.  Since most of the hyperfine structure may be accounted for without including  Xab,  initial  assignments were made using first order patterns generated by a computer program writ ten by the author for this purpose. The coupling of the spins of two identical quadrupolar Cl nuclei to rotation was treated using the coupling scheme ‘Cli  I  + ‘Cl 2  =  I+J  =F.  To first order, only those matrix elements which are diagonal in J are included in the quadrupolar Hamiltonian; the matrix is still off-diagonal in I, and so must be diagonal ized. The form of the matrix elements, which may be derived from Eq. 2.118, was taken  Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, Sill 2 Cl 2 78  from Ref. [74]. Relative transition probabilities P were calculated using [82] P  cc cc  <JIF’MF  IitI JIFMF>  I  2  j F’ I I /(2F+1)(2F’+1) (FJ1J  (5.1)  .  (5.2)  Since the HQ matrix is not diagonal in I, the matrix element in Eq. 5.1 must be trans formed to the basis of eigenfunctions of HQ. Upon examination of relatively low-lying energy levels of the most abundant iso topomers of dichlorosilane, two near-degeneracies were found of the symmetry required for second order perturbations via Xab• Both of these were for C1 35 SiH 28 : 2 134 MHz) and 99  —  413  —  221  827 (‘ 64 MHz), as indicated on the energy level diagram given  in Fig. 5.1. It was hoped that measurement of transitions involving these levels would reveal sufficient perturbations from first order behaviour that Xab could be determined. Only one transition for each near-degeneracy both satisfied the b-type selection rules for dichlorosilane and fell within the frequency range of the spectrometer. The more easily observed transition, 4 3-322, was measured first. Surprisingly, the measured splitting pattern corresponded well with that predicted by a first order calculation, using the pre viously determined values of Xaa and Xbb  —  x  [7]. Deviations were found, however, for  the second potentially perturbed transition, 918-827. As these perturbations could not be accounted for using the first order program, a fitting program which could calculate exactly the effects of two coupling quadrupolar nuclei was required. At the author’s request, the exact nuclear quadrupole hyperfine patterns were predicted by Ilona Merke of Universität Kiel, using her program Q2DIAG [83]. With values of rotational, centrifugal distortion, and diagonal quadrupole coupling constants from the previous microwave study [7], and using a value of Xab consistent with placing the z axis of the quadrupole tensor along the Si-Cl bond, the exact program  2 79 2 Cl Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, SiH  300 1 ,9 I 1 U 10  -  1  ,—‘  2,7 2,8  00,10  65.4 MHz ‘•  1,8 1 ,9  2,6  82,7  9Q9 0 ° 2 N  817  8Q,3  18  2,5 2,6  1,6 1,7  625  1,5  2,3 52,4  2,4  7Q,7  100  6  2,2 42,3  1 ,4 15 0,5 40,4  3Q,3  0  61,6  20,2 1 0,1  41: 3J: 21: 1 1: 0o,  3 2? 134.2 MHz  Figure 5.1: Rotational energy level diagram of C1 35 SiH 28 . 2  2 Cl 2 80 Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, SiH  confirmed that neither 4 nor 221 should be noticeably perturbed by a Xab-type interac tion. However, comparison of first order and exact predictions of energy levels indicated that hyperfine components of shifting on the  91  8-827  827  were shifted by as much as 800 kHz. The effect of this  transition is shown in Fig. 5.2. The bottom spectrum of this figure  shows transitions calculated by the first order program, the middle spectrum show tran sitions calculated by the exact program, and the top spectrum shows transitions which were measured, unambiguously assigned, and included in the final fit. The perturbation of 827 was confirmed as being due to an interaction with  919  via  Xab  by first reducing  Xab  slightly in the exact calculation, which lessened the effect of the perturbation somewhat. Xab  was then set to zero, which removed the perturbation altogether and produced a first  order splitting pattern. Measurements of the 9 8-827 transition agreed well with the exact prediction, per mitting assignment of the perturbed hyperfine components. A portion of this transition is displayed in Fig. 5.3. The Doppler-split spectrum predicted using Q2DIAG is shown below the observed spectrum, with dashed lines indicating transitions which were not included in the fit. Given that the width of this portion of the spectrum is only 300 kHz, it is clear that conventional Stark spectroscopy, with typical linewidths of ‘250 kHz, would not be sufficient to resolve hyperfine patterns of this complexity. The final constants were obtained by doing a least-squares fit to the measured transi tion frequencies with Q2DIAG. Besides 413—322 and 9 8—827, nine relatively unperturbed transitions were also measured, primarily in order to refine the values of the diagonal nu clear quadrupole coupling constants for Cl. The measured transition frequencies and their assignments are in Table 5.1, aild the derived quadrupole coupling constants are in Table 5.2. Note that hyperfine levels are labelled by I and F, although I is not strictly a good quantum number. The Hamiltonian used in the fitting program, which included  Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, Sill 2 Cl 2 81  15310.44 MHz  $  1MHz  I .  I  II  HI.  i  fitted  .  iI.  exact calculation  first order calculation 15305.5 MHz  Figure 5.2: Schematic diagram of the  15313.4 MHz  918  —  827  transition of C1 35 SiH 28 . 2  Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, Sill 2 Cl 2 82  IF=2 10—2 9 27—26 2 11—2 10  15310.8 MHz  15311.1 MHz  1  10—1 9  3 11—3 10  Figure 5.3: Portion of the 918 827 rotational transition of 28 35 Cl 2 SiH . Experimen 2 tal conditions: gas sample 33% 2 C SiH / Ar; 1 excitation frequency 15310.9 MHz; 150 ns sample interval; 4 K FT; 200 averaging cycles. —  Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, SiH 2 Cl 2 83  rotational, quartic centrifugal distortion, and nuclear quadrupole coupling terms, was suf ficient to account for the observed spectra, and it was not necessary to consider nuclear spin-rotation coupling.  2 84 SiH Cl Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, 2  F’  -  frequency (MHz)  obs.-calc. (kHz)  8 4 8 5 6 6  10281.7600 10281.7776 10283.5642 10285.6483 10286.1606 10286.7222  -5.9 6.1 -0.8 0.2 -0.6 0.9  1 3 3 2 2 1 2 3 1 3 2 3  10 13 8 11 9 11 12 12 9 10 10 11  11528.6496 11528.8654 11529.7612 11530.1392 11530.1521 11530.2360 11530.2580 11530.5716 11531.5213 11531.6916 11531.7622 11531.9596  -0.8 -0.5 -0.8 -2.0 1.3 -3.6 0.5 1.0 -0.7 2.2 1.2 1.8  0 1 3 1 2 0 1 3 3 3 2 2 3 3 2 2 1 3 1 3 2  1 1 2 1 2 1 2 4 3 2 1 3 3 4 1 2 2 3 2 3 1  11889.2942 11890.6746 11890.6884 11894.0489 11895.7544 11895.8396 11898.1196 11898.1658 11900.8800 11901.0757 11902.2048 11902.3176 11906.3336 11906.4744 11908.7508 11908.8624 11909.0444 11911.8070 11911.8788 11914.6416 11915.3110  0.6 -2.5 1.6 -0.2 -0.5 -0.4 -0.1 1.2 -2.2 1.0 -0.4 4.6 2.4 -1.1 -0.8 0.2 -0.3 -0.2 -0.7 -0.4 -1.5  P’  F”  2 2 3 1 3 2  523-616  2 7 2 3 3 7 1 4 35 2 5  110-101  1 0 3 2 1 1 1 1 1 0 2 2 2 3 4 3 32 33 1 0 2 3 4 3 33 2 2 2 1 1 2 1 2 3 3 3 3 2 1  F’  o2 2 i31 12 32 12 13 32 11 22 02 35 34 12 13 34 33 11 35 13 34 02 34 33 32 24 24 23 33  -  frequency (MHz)  obs.-calc. (kHz)  1 1 1 2 2 2 2 2 2 5 5 3 3 3 3 1 4 4 4 2 3 3 3 4 3 3 2  12243.4432 12243.4576 12243.5484 12244.7194 12244.7424 12244.8057 12244.8420 12245.1044 12245.2729 12247.6905 12247.8044 12249.1422 12249.1620 12249.2862 12249.3182 12255.0058 12255.0922 12255.0750 12255.2040 12256.3464 12256.7172 12256.7504 12256.6602 12250.6814 12250.6972 12250.7118 12250.8058  -0.6 -1.2 2.3 -0.2 1.2 -1.2 -1.0 -0.8 0.8 -0.0 -1.1 0.5 -1.4 1.1 0.0 0.1 4.0 -6.4 0.9 2.1 0.6 0.7 -0.4 0.3 -2.2 -1.8 3.0  0 3 1 3 3 6 3 3 2 6 2 2 4  12780.1270 12780.7681 12781.5402 12781.7126 12781.8263 12782.8672 12782.9340 12782.9796 12783.4598 12784.4929 12785.5524 12785.9302 12786.5992  -0.8 0.3 -0.1 -2.7 1.0 -0.3 -0.5 -0.6 0.9 0.5 -2.6 0.7 0.3  I”  F”  3 3 3 1 1 1 1 0 0 3 3 1 1 1 1 1 3 3 3 2 3 3 3 2 2 2 3 3 1 3 0 1 3 1 0 3 3 3 2 1  i 2  936-1029  1 9 3 12 3 7 2 10 28 1 10 2 11 3 11 1 8 3 9 2 9 3 10  I’  31 2-303  31 12 32 22 14 35 13 03 33 36 31 23 14  Table 5.1: Measured transitions of C1 35 SiH 28 2  Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, S1H 2 Cl 2 85  I’  F’  312-303  2 5 3 4 0 3 3 4 3 3 3 5 1 2 3 3 3 2 1 4 2 3 3 6 3 4 2 4 3 5 3 3 1 4 0 3 4i 3404 3 2 1 5 3 1 3 3 3 6 3 3 1 4 0 4 3 2 3 4 3 7 3 1 2 4 3 4 3 6 3 3 1 5 3 2 2 6 2 3 2 5 1 4  -  I”  F”  frequency (MHz)  obs.-calc. (kllz)  cont’d  I’  F’  413-404  2 3 2 3 3 3 1 1 3 3 2 3 3 2 3 3 3 2  5 3 4 5 3 5 2 2 3 5 3 5 4 3 4 4 4 3  12787.1825 12788.4262 12788.4522 12789.2548 12789.2793 12789.9220 12790.1254 12790.3907 12790.6298 12790.9026 12791.3831 12791.5480 12792.1068 12792.6398 12792.7747 12792.9619 12793.7578 12793.9062  -1.1 3.6 -1.8 0.7 -1.0 1.1 0.5 0.5 -1.6 -1.5 -0.2 2.2 0.8 2.5 1.8 -1.9 1.7 -1.9  3 1 3 1 3 3 1 2 3 3 3 3 2 1 1 3 1 3 2 2 2 3  1 4 1 4 7 2 4 4 2 3 7 2 5 5 5 3 5 3 6 3 5 3  13521.8690 13522.6848 13522.8550 13523.1022 13523.1746 13524.0053 13524.3476 13524.3650 13524.9911 13525.2148 13525.4257 13525.9742 13526.3024 13526.3778 13526.6222 13526.9980 13527.7466 13527.9844 13528.0853 13528.1259 13528.1418 13528.2454  0.2 0.3 2.0 -0.5 -2.6 2.7 -0.3 3.7 0.5 2,9 0.2 -0.6 -0.5 -1.4 0.6 0.8 0.4 -0.9 0.6 -1.7 1.7 3.0  35 14 35 36 24 37 35 33 34 36 15 32  -  I”  frequency (MHz)  obs.-calc. (kllz)  6 5 4 6 4 6 5 4 5 5 6 3  13528.9206 13529.4086 13529.8196 13530.0992 13531.8437 13532.3498 13532.5264 13532.5368 13533,4594 13533.7040 13531.2230 13533.5902  -0.5 -1.0 3.0 0.5 1.7 2.9 0.7 -0.2 -1.4 0.8 -0.3 0.7  3 3 3 2 3 3 1 2 3 1 3 0 3 3  6 1 2 4 3 7 5 6 4 3 6 4 5 5  13760.6478 13760.9994 13761.2558 13762.0649 13762.3934 13762.7406 13763.2374 13763.5292 13763.9754 13764.4083 13764.7618 13764.9974 13765.3228 13767.4826  -1.2 -0.0 -0.0 1.0 0.2 0.2 0.2 0.5 0.9 -0.6 0.5 0.1 0.3 -2.6  3 1 3 3 3 1 2 2 3 3 1 2 3  1 4 2 7 3 5 5 6 6 4 3 4 5  14282.1064 14282.8344 14283.0912 14283.3716 14284.0828 14284.4990 14284.6088 14284.6634 14285.6217 14285.8635 14285.9060 14286.4452 14286.7954  -1.7 3.0 -1.1 2.0 -0.9 1.0 1.6 0.4 0.2 -2.7 -0.7 0.5 -1.2  con’d 3 1 3 3 0 3 3 3 3 3 3 1  F”  5o 5414  36 32 33 25 34 38 16 27 35 14 37 05 36 35 322-413  31 13 31 36 32 14 24 25 35 33 13 23 34  Table 5.1: Measured transitions of C1 35 (cont’d) SiH 28 2  Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, S1H 2 Cl 2 86  I’  F’  -  frequency (MHz)  obs.-calc. (kHz)  I”  F”  3 2 1 1 3 3 2 1 3 3 2 3 1 2 3 3 1 3 0  2 5 5 5 8 2 5 5 8 3 6 4 6 6 4 7 4 5 5  14488.3368 14488.7585 14488.8877 14489.1314 14489.1760 14489.7120 14490.8588 14490.8588 14491.7273 14492.0914 14492.3554 14493.8091 14494.2205 14494.4538 14495.5312 14496.0984 14497.1055 14497.3277 14498.0477  1.0 -1.4 1.9 -3.0 -3.7 -0,7 0.1 5.2 -0.9 -0.3 -0.9 -0.0 -0.5 -0.9 2.9 -0.0 0.1 0.2 0.8  3 1 3 3 3  5 8 11 6 7  15308.3096 15308.8930 15309.4208 15309.5866 15310.2523  0.3 8.8 -0.8 -1.4 -1.6  5i 4-505  3 2 1 3 3 3 0 1 3 3 2 3 1 2 1 3 1 3 2  3 4 6 4 7 2 5 5 8 3 5 4 6 6 5 7 4 5 5  91 8-827  3 1 3 3 3  6 9 12 7 8  J  F’  -  9 8-827 coni ‘d 2 8 2 2 10 2 2 2 7 2 11 2 3 9 3 1 1 8 615-606 4 3 3 1 3 5 3 8 3 3 3 3 3 9 3 4 3 3 3 5 3 1 7 1 2 7 2 3 8 3 3 7 3 1 1 5 3 6 3 2 6 2 3 7 3 6 3 3 4 1 3 3 8 3  F”  frequency (MHz)  obs.-calc. (kllz)  7 9 6 10 8 7  15310.4385 15310.8915 15310.9850 15311.0162 15312.1648 15312.1876  -2.1 -0.8 0.9 -3.3 -3.4 3.2  3 6 9 3 9 4 5 7 7 8 6 5 6 6 7 7 5 7  15704.9720 15705.5386 15705.5812 15706.6134 15708.3178 15709.2231 15710.7345 15710.9864 15711.1594 15712.5884 15713.5162 15713.9140 15714.2344 15714.7439 15715.3350 15716.0546 15716.9212 15717.1211  0.6 0.7 -4.2 0.8 -0.3 1.3 -0.1 0.2 -1.2 -0.8 3.8 0.9 0.2 0.2 0.5 -1.8 -3.5 3.0  Table 5.1: Measured transitions of 2 C1 3 SiH 28 5 (cont’d) 5.4  Discussion  The 35 C1 quadrupole coupling constants are compared in Table 5.2 with those from the earlier study of 2 C1 [7]. Xbb and SiH  Xcc  are the same for both 35 C1 atoms, and  their absolute signs have been determined. Not oniy is there an improvement in their precision, but also the new values are slightly outside the estimated uncertainties of the earlier work. The off-diagonal constants,  Xab  , have been measured directly for the  first time. In this case the absolute signs have not been determined as they enter to  Chapter 5.  The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, 2 C1 87 SiH  previous work’ Inertial axis system (MHz) Xbb Xcc Xab  I  (MHz)  I  (MHz)  23 ( 013 )b  21.00(23) —  this work  -0.3095(19) 20.7245(19) 26.099(74)  Principal axis system Xzz x Xyy za  (MHz) (MHz)  (MHz)  42.0(7)c 21.0(6)  -38.33(13) 17.60(13)  21.00(23)  20.7245(19)  35.140c  34.467(27)°  a Values are taken from Ref. [7]. Uncertainties were estimated outside limits. b Numbers in parentheses are one standard deviation in units of the last significant figure. c Note that previous principal values of the Cl nuclear quadrupole coupling tensor were calculated assuming coincidence of the z principal axis of the tensor and the Si-Cl bond [7], and thus Oza is the angle between the a inertial axis and the Si-Cl bond.  Table 5.2: Chlorine nuclear quadrupole coupling constants of 2 C1 SiH  2 Cl 2 88 Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, SiH  second and higher order in the nuclear quadrupole coupling energy, although they must be opposite for the two atoms [74]. The results of diagonalizing the 35 C1 quadrupole coupling tensor are also shown in Table 5.2. The principal values of the tensor are compared with those of the previous work [7], where it was assumed that the z principal axis of the tensor lay along the Si-Cl bond. It is clear that the assumption was not unreasonable, although the present results have produced slightly different constants. The diagonalization produces Oza, the angle between the z principal axis of the tensor and the a principal inertial axis. The value obtained places the z axis of the tensor essentially along the Si-Cl bond, as expected. However, as has been found for other molecules [58], including BrNCO (see Chapter 4), there seems to be a slight (1°) difference between the two. The present results are compared with those of several related molecules in Table 5.3, along with a comparison of the Si-Cl bond distances. Note that the value for  ((Xzx  —  C1 implying that the Si-Cl bond is essentially SiH , Xyy)/Xzz) is very nearly zero in 2 cylindrically symmetric; this would seem to justify the assumption made in estimating Xzz  in SiHC1 . The value for Xzz of 2 3 SiH 2 Cl is now found to lie between those of SiH C1 3  and SiHC1 , in contrast to what was found earlier [7]; there is now a logical sequence, 3 paralleled by the trend in the Si-Cl bond length. The coupling constant Xzz is similar to those of 2 SiHC1 and 2 3 CH SiC1 although for those molecules Xzz is relatively ) 3 (CH , poorly determined. For the molecules containing tetravalent Si there does not seem to be a general correlation of Xzz with Si-Cl bond distance, with the exception of the one noted above. Only for SiCl 2 and SiC1 are the coupling constants markedly different. The large value for  in SiC1 2 has been rationalized in terms of (p  —*  d)7r back-donation [80],  as has the low value of Xzz in SiC1 [81]. The results for 2 C1 however, suggest that SiH , the bonding between the Si and Cl atoms can be treated as primarily a in character.  Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, SiH 2 Cl 2 89  molecule  Xzz (MHz)  C1 2 SiH C1 3 SIH 3 SiHC1 SiHC1 3 CH 2 SiC1 ) 3 (CH 2 2 SiCl SiC1 C1 3 SiF C1 CH 2 C1 CH 2  -38.33(13) -39.70(7) -37.2 -41.2(20) -38.0(16) -36.08(35) -23.13(96) -39.83(16) -76.2(40) -76.92(2)  a b  ‘i=(x  —  method 0.082(3) 0 0 0.16(3) 0.22(8) 0.859(18) 0 0 0.0(21) 0.0356(3)  b d c,e c c  c d d b c  r(Si-Cl, .O 2 f 33 2048 2.012 2.040 2.055 2.070 2.061 1.996 —  —  A)  Ref. this work [60] [7, 84] [85] [86] [80] [81] [64] [74] [74]  Xyy)/Xzz.  Determination of Xab and diagonalization of the 35 C1 quadrupole tensor. Assuming Si-Cl bond is a principal axis of the 35 C1 quadrupole tensor. d In linear and symmetric top molecules where the quadrupolar nucleus is on the symmetry axis, the measured quadrupole coupling constant is Xzz and the bond is cylindrically symmetric. Cylindrical symmetry assumed for the Si-Cl bond. Ref. [7]. Ref. [87].  ‘  Table 5.3: Comparison of 35 C1 principal quadrupole coupling constants and Si-Cl bond lengths  Chapter 6  The Microwave Spectrum of Tetrolyl Fluoride  6.1  Introduction  According to basic concepts of chemical bonding, a methyl group attached to a non linear acetylene fragment -CC-R should experience essentially free rotation. Microwave spectroscopic studies of several molecules of the form 3 CH CC-M (where M  =  3 CD  [88], CF 3 [28], SiR 3 [89], and CH C1 [90]) have confirmed that the threefold barrier to 2 internal rotation of the methyl group is indeed low, and where V 3 has been determined it has been found to be no greater than  10 cm’. The infrared spectrum of tetrolyl  fluoride (2-butynoyl fluoride, CH -CC-COF) also gave evidence of free rotation of the 3 methyl group [91]. Molecules with low barriers to internal rotation exhibit very crowded microwave spectra, as lines due to different torsional states may be very closely spaced. Assign ment is therefore quite difficult unless some sort of state-selective technique is employed. Microwave-microwave double resonance (MW-MW DR) has recently been used to over come the problem of spectral congestion in the case of C 2 3 CH CC-CH l [90]. In the work described here, the microwave spectrum of tetrolyl fluoride has been obtained using a cavity MWFT spectrometer. As an alternative to MW-MW DR, the efficient cooling in the molecular beam of the spectrometer has been used ‘freeze out’ all but the low est torsional states of tetrolyl fluoride. This has vastly reduced the congestion in the microwave spectrum and simplified the assignment of lines.  90  Chapter 6.  6.2  The Microwave Spectrum of Tetrolyl Fluoride  91  Experimental methods  Tetrolyl fluoride was prepared by heating a mixture of tetrolic acid and benzoyl fluoride, following the method of Olah et al. [92]. The resulting clear distillate (h.p. 77-78°) was identified as tetrolyl fluoride by  111  and ‘ F nmr, as well as by comparison of the 9  gas phase infrared spectrum to that obtained by Balfour et al.[91]. As the compound is unstable at room temperature and turns brown over time, the sample was kept at liquid nitrogen temperatures when not in use and remained colourless over the period of this study. Preliminary microwave studies were performed on the Stark-modulated spectrometer described in Section 3.1. For the most part, scans were performed in the frequency region 8-18 GHz, as this was the region available on the cavity MWFT spectrometer at the time of these experiments. Spectra were recorded both with and without dry ice cooling of the Stark cell. The tetrolyl fluoride was warmed to room temperature in order to fill the cell, and spectra were recorded at pressures of 30-50 mtorr. At these pressures, the sample remained stable in the cell for at least an hour, independent of the temperature of the cell. High resolution microwave spectra of tetrolyl fluoride were recorded with the cavity MWFT spectrometer (see Section 3.2), using gas mixtures of 1-3% tetrolyl fluoride in argon.  6.3  Results and discussion  Given the anticipated structure of tetrolyl fluoride, it was expected that the dipole mo ment would lie very nearly along the a inertial axis, resulting in strong a-type rotational transitions. The initial microwave spectra recorded with the Stark spectrometer revealed the typical a-type R-branch pattern, with groups of strong, equally spaced, unresolved  92  Chapter 6. The Microwave Spectrum of Tetrolyl Fluoride  lines. From this pattern, a rough value for B+C of 32O0 MHz was calculated (Eq. 2.15.) Very few lines were found in between these groups. Three groups fell within the frequency region of the cavity MWFT spectrometer. Since their frequencies were roughly in the ratio 3:4:5, they were assigned to the J=3-2, J=4-3, and J=5-4 transitions. However, these groups contained far more lines than could be accounted for by a simple rigid rotor model. 6.3.1  Prediction of transition frequencies  A computer program was written to predict rotational transition frequencies for a molecule with one internal rotor and a planar framework. The Principal Axis Method (PAM) was chosen, as the symmetry axis of the methyl group in tetrolyl fluoride was expected to lie nearly along the a principal axis. When the principal axes of a molecule are chosen such that the frame lies in the xz plane, the Hamiltonians given in Eqs. 2.148-2.150 may be rewritten as [28]  Hrot  = A’J 2 + B’J 2 + C’J, 2 + Fpp(J J + J J) Fj  Htorsion Hrot_torsion  2  (6.1) (6.2)  + V(o)  (6.3)  j). 2 = —2F(pJj + pJ  The computer program has been designed to solve the Hamiltonian in steps, after the method of Anderson and Gwinn [93]. The rigid rotor part of the Hamiltonian, Hrot, is treated first, using symmetric rotor basis functions. The matrix elements of Hrot in this basis are given by <JK <j  IHrot I  J1>  K + 1 IHrot JK>  = 2 A’K + =  (B’ + C’)  [J(J + 1)  (2K + 1)[J(J + 1)  —  —  (6.4)  ] 2 K  K(K +  1)]h/2  (6.5)  Chapter 6.  The Microwave Spectrum of Tetrolyl Fluoride  <J K±2HrotjJK> = (B’ C’) [J(J + 1) —  —  K(K + 2 1)]’/ [ J(J + 1)  93  —  (K ± 1)(K + 2)11/2.  (6.6)  Diagonalization of this part of the Hamiltonian leads to eigenvalues and eigenvectors similar to those of an asymmetric rigid rotor; a Wang transformation does not produce a useful simplification because of the terms off-diagonal by 1 in K. An analogous procedure is used for the torsional part of the Hamiltonian, Htorsion, 6 is using free rotor basis functions of the form given in Eq. 2.136. Assuming that V sufficiently small that V(c) may be truncated after the V 3 terms (Eq. 2.133), the matrix elements of the torsional Hamiltonian in the free rotor basis are given by <m  Fm + IHtorsjon m> = 2  <m ±3 Htorsion m> =  (6.7) (6.8)  —.  The introduction of a non-zero threefold barrier to internal rotation thus introduces off-diagonal elements in Htorsion. Free rotor states with m 0 are doubly degenerate, 4 but the barrier acts to remove the degeneracy for states with m a multiple of 3, as shown in Fig. 2.1. H 010 may then be divided into submatrices of A symmetry (m = —  6, —3,0, 3,6,...) and E symmetry. Since the E states are still doubly degenerate,  it suffices to consider only half of the F submatrix, i.e. m =  ...  5, +2, ±1, ±4, ±7,...,  where the choice of phase is arbitrary. Htorsion thus extends from m = —oo to m = +oo, and numerical solutions are necessarily approximate. As truncation at sufficiently high m has little effect on small values of m  , and only small values of  m were expected  to be seen for tetrolyl fluoride with the cavity MWFT spectrometer, it was decided to diagonalize a truncated Htorsjon directly. The ‘cross terms’ of Hrot_torsion are treated by transforming representations of J, and J,. in the rigid rotor basis to representations in the basis of eigenfunctions of Hrot ;  j  Chapter 6. The Microwave Spectrum of Tetrolyl Fluoride  94  is also transformed to the basis of eigenfunctions of Htorsion, with A and F submatrices considered separately throughout. If the final Hamiltonian matrix is written in terms of basis functions which are the direct product of the eigenfunctions of Hrot and Htorsjon, the cross terms are easily calculated as the direct products (—2FJ x  j ),  j)  and (—2FJ x  where the matrix elements of A x B =C are given by C,kl  (6.9)  = AkBl.  As most of the diagonalization of Hrot_torsjon has been achieved in the first two steps, the matrix is nearly diagonal and diagonalization proceeds rapidly. The decision to treat the torsional part of the Hamiltonian separately using free rotor basis functions hinges on the assumption that internal rotation is essentially unhindered, an assumption that was made early in this study of tetrolyl fluoride. 6.3.2  Selection rules  The selection rules for rotational transitions may be determined by considering the per mutation inversion group of a molecule such as tetrolyl fluoride, which has a planar frame and for which the possibility of torsional tunnelling exists. The character table for the relevant permutation inversion group, G , is given as follows: 6 6 G  E  2.(123)  3.(23)*  1 A  1  1  1  2 A  1  1  -l  F  2  -1  0  where F is the identity operation, (123) is an operation which is equivalent to rotation of the methyl group by 2nir/3 radians (n = 1, 2,...) about its symmetry axis, and (23)* is equivalent to a combined rotation of the methyl group by o= 2n7r/3 radians and rotation of the entire molecule by ir radians about the out-of-plane axis y. In this permutation  Chapter 6. The Microwave Spectrum of Tetrolyl Fluoride  95  inversion group, both the a and b components of the dipole moment are of symmetry  . In order to determine the selection rules for rotational transitions, it is necessary to 2 A determine the symmetries of the eigenfunctions of Hrot_toion. The extent to which these eigenfunctions are properly labelled by the KaKc asym metric rotor labels and the m torsional quantum number depends on both V 3 and 0, the angle between the top symmetry axis and the z principal axis. A large barrier to internal rotation results in greater mixing of the free rotor states in Htorsjon, while 0 determines the size of F, which in turn determines both the deviation of Hrot from a pure asym metric rigid rotor model and the contribution of cross terms in the final Hamiltonian. 3 was expected to be small for tetrolyl fluoride, in which case V  m is nearly a good  quantum number for torsional states of E symmetry, while states of A symmetry may be approximated as the following linear combinations:  +m > + —m>):  1 symmetry A  (6.10)  —m >) :  2 symmetry. A  (6.11)  +m >  —  The angle 0 was also expected to be small (i.e.  p—O,  pz*l), making the factor  2 in Hrot almost zero. Diagonalization of Hrot will therefore give asymmetric rigid Fpp rotor wavefunctions for which the labels KaKe may still be used. The 2FpJ,j cross term in Hrot_torsion will also be small, but the 2FpJ j term will be significant. As a consequence, considerable mixing of asymmetric rigid rotor states with the same value of Ka occurs. The K label is thus meaningless, although it has been retained in order to distinguish between the resulting Ka states. The symmetry species of the rotation-torsion states are preserved in this mixing. The computer program described above also calculates approximate relative intensi ties for transitions with low values of m  .  Using the direction cosine matrix elements of  Cross, Hainer, and King [13], the dipole moment matrix is calculated in the symmetric  Chapter 6.  The Microwave Spectrum of Tetrolyl Fluoride  96  rotor basis. This matrix is then transformed to the basis of eigenfunctions of Hrot_torsjon, neglecting intensity effects due to pure torsion, and is then used to calculate intensities of transitions. For the most part, the standard a- and b-type selection rules are retained, in spite of the fact that K is no longer a good label; allowed transitions must also have m = 0. However, transitions such as  303—  3 , which are nominally c-types but which  depend on the b component of the dipole moment, are allowed for states of E symmetry. Although these transitions are predicted to have finite intensity for large values of 0, the transitions should be vanishingly weak in tetrolyl fluoride, where 0 is expected to be less than  50  6.3.3  Assignments  The cavity MWFT spectrometer was used to search for some of the strongest lines ob served in the Stark spectra. As expected, because of the rotational and torsional cooling, most of the lines observed using the Stark spectrometer disappeared in the MWFT spec tra, leaving a small number of lines. Some of these could be fit to a rigid rotor model as Ka=O and Ka 2 transitions, with initial rotational constants calculated from a trial structure. This made it possible to search for Kal satellites, which had not been iden tified in the Stark spectra; these were soon found. Searching was facilitated by the large a component of the dipole moment of tetrolyl fluoride, estimated to be —3 D by comparison with similar molecules. Thus, despite the small bandwidth of the spectrometer, searches for a-type lines could confidently be performed in 1 MHz steps, with short exciting pulse lengths (0.1 its). Very strong signals were obtained for many transitions. An example is the line at 12598.412 MHz shown in Fig. 6.1, later assigned to the  m =0,  414-313  transition; this transition could be  observed with no signal averaging. With reasonable initial values for the rotational constants, 0, and V , all of the lines 3  Chapter 6. The Microwave Spectrum of Tetrolyl Fluoride  97  48 kHz  125983 MHz  1298. MHz  Figure 6.1: The m =0, 44-33 transition of tetrolyl fluoride. Experimental conditions: 2.3% tetrolyl fluoride/Ar gas sample; 12598.4 MHz excitation frequency; 50 ns sample interval; 8 K FT; 1 averaging cycle.  Chapter 6. The Microwave Spectrum of Tetrolyl Fluoride  98  observed thus far using the cavity MWFT spectrometer could be assigned. Those lines which corresponded to an asymmetric rigid rotor spectrum could be assigiled to states with m =0, while the remaining unassigned lines were assigned to states with m =1. It was then possible to predict the remaining  m =1 lines, which were subsequently  observed with the cavity spectrometer. The measured transition frequellcies and assign ments are given in Table 6.1. In order to obtain spectroscopic constants, the program VC3IAM [94] was used to fit the data. Although this program uses the internal axis method, frequencies predicted by the PAM prediction program described above agreed very well with those predicted by VC3IAM. While many attempts were made to locate b-type transitions, no such transitions were observed. This is consistent with the spectra recorded using the Stark spectrometer, where no lilles were observed in the portions of the spectrum between the groups of atype lines. The b component of the dipole momeilt is due entirely to the COF group, and should be relatively small. For the related molecule 3 CH C OF, 88 Pb=°• D, compared to Ua2.83 D [10]. 4 As only a-type transitions were observed, the A rotational constant cannot be de termined. In addition, A and I, the moment of inertia of the methyl group about its symmetry axis, are highly correlated. If the inertial defect is assumed to be zero, then (6.12) since the only out-of-plane atoms are the hydrogens of the methyl group. In fitting the data, ‘c’€ was assumed to have the value 3.18 a.m.u.A2 [95] and A was released into the fit. However, since the a-type transition frequencies are insensitive to the value of A, it would remain near its initial value. The lowest standard deviation was obtained with A ‘.‘11050 MHz, and so this was used for the initial value in the final fits. The centrifugal distortion constants were set to zero in the fits, as attempts to determine D arid DJK  Chapter 6. The Microwave Spectrum of Tetrolyl Fluoride  99  ImI=0  ’ 4 J  J(’  j(’  J”  K  K’  3 3 3 3 3  1 0 2 2 1  3 3 2 1 2  2 2 2 2 2  1 0 2 2 1  2 2 1 0 1  0.466 9777.556 -0.535 9793.147*,c 0.089 9808.165* 0.139 10129.989 0.118  4 4 4 4 4 4 4  1 0 2 3 3 2 1  4 4 3 2 1 2 3  3 3 3 3 3 3 3  1 0 2 3 3 2 1  3 3 2 1 0 1 2  12598.412 13019.263 13054.608 13064.459* 13064.689* 13092.091 13502.172  5 5 5 5 5 5 5 5 5  1 0 2 4 4 3 3 2 1  5 5 4 2 1 3 2 3 4  4 4 4 4 4 4 4 4 4  1 0 2 4 4 3 3 2 1  4 4 3 1 0 2 1 2 3  15741.226 16247.566 16313.559 16328.843* 16328.843* 16333.820 16334.636 16388.336 16870.396  frequency (MHz)  obs.-calc. (kllz)  b 9452031  ImI=l frequency (MHz)  obs.-calc. (kHz)  9778.413 9838.027 9746.914* 9796.063* 9798.918  0.536 0.303 0.485 -0.533 -0.089  0.603 -0.773 0.109 -0.325 -0.327 0.217 0.136  13020.406 13165.910 12938.029 13075.727* 13061.567* 13063.362 13070.045  0.740 0.412 0.512 0.345 -0.639 -0.716 -0.113  0.726 0.367 0.115 -1.240 -1.243 -0.411 -0.407 0.329 0.134  16246.069 16526.493 16088.505 16335.322* 16327.630* 16356.106  -0.471 0.484 0.464 -0.127 -0.177 0.469  —  16332.337 16345.380  —  -0.896 -0.127  a K is retained to distinguish between states with the same values of J and Ka. b Measurement accuracy is estimated to be better than ±1 kllz. L * Transitions marked by were observed as multiplets, and the frequency given is an average.  Table 6.1: Measured transitions of tetrolyl fluoride  Chapter 6.  The Microwave Spectrum of Tetrolyl Fluoride  normal isotopomer A B C I  (MHz) (MHz) (MHz) (a.m.u. A ) 2  3 V fit  C substituted 3 ‘  11049.705(90)a  b 11049705  1745.221(17) 1519.119(19)  9  (cm’) (MHz)  100  1744.601(54) 1518.056(93)  b 318  1.5092(14)° 2.20(12) 0.545  0.9994(25)° 1.118  a Numbers in parentheses are one standard deviation in units of  the last significant figure. b Held constant in the fit. Table 6.2: Spectroscopic constants of tetrolyl fluoride were unsuccessful. Determined in the fit were the B and C rotational constants, 9, and V , 3 as given in Table 6.2. As expected, 0 is small; the threefold barrier to internal rotation 3 is small (.—‘2.2 cm’) and very comparable to those determined for 3 V -CC-CD CH 1 [88]) and 3 (5.6 cm -CC-SiH (3.8 cm’ [89]). CH In the course of searching for b-type transitions, several weak lines were found which could be assigned to m =zO and m =1 a-type transitions of a ‘ C isotopomer of tetrolyl 3 fluoride. The measured transition frequencies and assignments are given in Table 6.3. In order to fit the limited number of lines, A, ‘a and V 3 were constrained to the values obtained in the fit for the normal isotopomer; the spectroscopic constants obtained are given in Table 6.2. The isotopic shifts in B and C are quite small, indicating that isotopic substitution has taken place very near the centre of mass. From a rough calculation of the expected structure of tetrolyl fluoride, the isotopomer is probably 3 --C0F. CH C C’ No lines were observed which could be assigned to  m =2 or 3 transitions. The  absence of transitions with high values of m is not unreasonable considering the energies of such states (which increase rapidly because of the Fj  2  term in Htorsjon ) and the efficient  Chapter 6.  j!  J(  j(a  jmI=0 J”  3  1 0  3 3  2 2  3  1  2  4  1 0 2 1  4 4 2 3  3  4 4 4  101  The Microwave Spectrum of Tetrolyl Fluoride  I(j  K’  ImI=1  frequency  obs.-calc.  frequency  obs.-calc.  (MHz)  (kllz)  (MHz)  (kllz)  2  1 0 1  2 2 1  9773.601 10124.843  3 3 3 3  1 0 2 1  3 3 1 2  12592.392 13012.853 13085.581c 13495.317  b 9772721  -1.358  0.636 -0.604  9833.142  0.772  2.240 -0.285 0.337 -0.803  13014.021 13159.336  —  —  —  —  -0.304 0.917  —  —  —  —  a K is retained to distinguish between states with the same value of J and Ka. b  Measurement accuracy is estimated to be better than ±1 kHz. Observed as a multiplet; the frequency given is an average.  Table 6.3: Measured transitions of ‘ C-tetrolyl fluoride 3 cooling in the molecular beam. While collisional relaxation of m =1 states to rn =0 is symmetry-forbidden, depopulation of torsional states with higher values of  m is  permitted in the beam. With only m =0 and m =1 transitions, it is impossible to test the validity of the assumption that V 3 is the dominant term of V(a), as V 6 cannot be determined [96]. It is important to note that the standard deviations in the fits were considerably larger than the uncertainty in the measurements; for the larger data set of the normal isotopomer,  Ufit  is roughly 500 times larger than the measurement uncertainty. Large  discrepancies between observed and calculated transition frequencies have been found in other internal rotation studies (e.g. discrepancies is the use of rigid top  -  ONO [95]). 3 CH  A probably source of these  rigid frame models, which neglects interactions  between internal and overall rotation and other vibrational modes [97]. As is noted in Table 6.1, several rotational transitions were observed as multiplets. Rotational energy levels with JKa  (221, 220, 331, 330, 441, 440)  seem to be split into  Chapter 6.  The Microwave Spectrum of Tetrolyl Fluoride  102  two and/or three sublevels, with the splittings on the order of 10-30 kHz. As multiplets were observed for both m =0 and m =1 states in all cases, it seems unlikely that the mechanism of the splitting is related to internal rotation. However, none of the likely F and/or ‘H spin-rotation or spin-spin coupling) split the J=tKa levels 9 mechanisms (‘ preferentially, but rather affect all JKaK > states to comparable extents. It is possible that terms which .contribute to splittings of the other levels cancel out accidentally, but this is difficult to determine, especially without a fitting program which will simultane ously fit the hyperfine effects of four coupling nuclei in an asymmetric rotor. Even with such a program, the large residual frequencies mentioned above could make it difficult to fit very small hyperfine parameters.  Chapter 7  Microwave Spectra of Metal Halides Produced Using Laser Ablation  7.1  Introduction  Recent experiments in other laboratories have demonstrated the feasibility of coupling a laser vaporization source to a cavity MWFT spectrometer in order to produce and study diatomic metal-containing species in the gas phase [98, 99, 100, 101]. Rapid vaporization of a metal or metal oxide target produces transient species which may be reacted with other species that are present in small concentrations in a rare gas. Stabilization of the product is then achieved via the supersonic expansion of the mixture through a nozzle into the microwave cavity. By this method, molecules which would otherwise be produced in highly excited states in an oven or electrical discharge are generated at very low rotational temperatures. In the work described here, laser ablation has been used to prepare seven diatomic metal halides: AgC1, A1C1, CuC1, InCl, InBr, InF’, and YC1. The rotational spectra of these molecules have been measured using a cavity MWFT spectrometer. A major aim of this work was the optimization of the experimental conditions necessary to prepare both stable and unstable metal-containing species in the gas phase. As a result of this optimization, it has been possible to observe the pure rotational spectrum of yttrium (I) chloride (YC1) for the first time. As a consequence of the high resolution of the cavity MWFT spectrometer, many previously known spectroscopic constants of the metal halides have been determined  103  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  104  considerably more precisely in this study. The effects of nuclear quadrupole coupling have been observed in the spectra of all the metal halides studied here, and nuclear quadrupole coupling constants have been obtained which are a major improvement over earlier values. Nuclear spin-rotation coupling constants, which are unattainable at lower resolution, have also been determined with relative ease. For some of the diatomic metal halides discussed here, interpretation of these coupling constants provides a glimpse of the structures of the excited electronic states of the molecules [9].  7.2  Experimental methods  The experimental arrangement for the laser ablation experiments described here is shown schematically in Fig. 7.1. The output of a Q-switched Nd:YAG laser is focused outside the microwave cavity of the MWFT spectrometer by a 50 cm focal-length lens. The radiation then passes through a quartz window and a small hole in the tuning microwave mirror before traversing the cavity to hit the target metal rod, which is fixed in front of the gas nozzle in the stationary microwave mirror. Both Bosch and General Valve nozzles were used for these experiments, with no noticeable difference in signals obtained. However, the General Valve nozzles were much more reliable when corrosive gas mixtures were in use. A brass cap fits on the end of the nozzle and holds the target metal rod directly in front of it, as is shown in Fig. 7.2. The laser radiation enters the cap through the exit channel, and the laser is aligned visually using a low power setting. The target rod is typically 1 cm long and 2 mm in diameter, while the exit channel is 1 mm in diameter; the rod thus serves to protect the end of the nozzle from damage caused by direct laser radiation. The laser is used to vaporize metal atoms from the target rod. The vapour is then allowed to react with a halogen-containing species, contained as a  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  105  window nozzle lens stationary mirror  to pump  tuning mirror  Figure 7.1: Schematic diagram of the arrangement of the cavity MWFT spectrometer and the Nd:YAG laser used in these experiments.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  106  few percent in neon. Both the laser and the nozzle are pulsed, and the delay between the pulse of the gas mixture and the laser pulse is adjusted to optimize the production 3 in volume, surrounds the of metal halides. A small chamber, approximately 0.04 cm rod, giving sufficient time for reaction before the backing gas carries the metal halide molecules through the exit channel. The molecules are stabilized by collisions in the channel before being cooled by supersonic expansion of the gas mixture into the cavity. Operation of the cavity MWFT spectrometer is otherwise as described in Section 3.2. Successful production of metal halides is very sensitive to several parameters. AgC1 is a stable molecule, and was successfully produced using a variety of chlorine/neon 2 concentrations. The other six metal monohalides gas mixtures with a wide range of Cl produced here are less stable: for example, A1C1 is less stable than other, more chlorinated /A1 Cl 3 A1C1 . Low concentrations of the reactant gas 6 potential reaction products such as 2 (‘—0.1%) were found to favour production of the monohalides. Total static pressures of the gas mixtures were typically 1-2 atm. The delay between the nozzle pulse and the laser pulse is crucial; the optimized delay was ‘500 s in the experiments described here, varying little for the range of molecules studied. It should be noted that no provision has been made here for moving the target rod without opening the vacuum chamber, and consequently the laser pulse hit the same spot on the rod until the chamber was opened. However, the ‘freshness’ (or lack thereof) of the spot was not found to have a significant effect, and in fact a well-polished rod would sometimes give weak signals which improved as more laser pulses were fired. The energy of the laser pulses was kept near the threshold value for the production of metal chlorides. For the ‘harder’ metals used in these experiments (silver, aluminum, copper, yttrium), pulses of this energy produced only a small indentation in the rod after several hours of operation; ‘softer’ metals such as indium are ablated more rapidly.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  107  cap  532 nm  rod  General valve  532 nm  sample rod  Figure 7.2: Schematic diagrams of the nozzle caps which hold the sample rod in front of the nozzle (Bosch or General Valve.)  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  108  Observed spectra  7.3 7.3.1  AgC1  The rotational spectrum of silver chloride, AgC1, has been studied previously using hightemperature microwave [102, 103] and millimeter-wave [104] spectroscopy. More recently, the cavity MWFT spectrometer used in this work was coupled to an electric discharge source, and AgC1 was produced by applying a pulsed high voltage to Ag electrodes in the presence of Cl 2 [105]. The laser ablation technique has now been employed to produce AgC1, using a silver target rod and C1 /Ne gas mixtures of varying concentrations (—0.05%—2%). The ro 2 tational transitions J=1-0 and J=2-1 have been observed for four isotopomers of silver chloride, 35 Ag (38.7% natural abundance), 35 7 ‘° Cl Ag (36.7%), 107 9 ‘° C1 C1 (12.6%), 37 Ag and 37 Ag (12.0%). 9 ‘° C1  These rotational transitions are split by nuclear quadrupole  coupling of the Cl nucleus (I =3/2); the measured transition frequencies and their as signments for the four isotopomers are given in Table 7.1, where frequencies from both the electric discharge [105] and laser ablation studies are given. Each hyperfine transition is labelled by the quantum number F, where ci +J =F. T Many hyperfine components could be seen for the 35 C1 isotopomers with virtually no noise even without any signal averaging, while four hyperfine components of the J=2-1 transition have been observed for 107 C1 and 109 37 Ag C1 which could not be observed 37 Ag using the electric discharge method [105]. Clearly, the laser vaporization technique is a more efficient method for producing AgC1 in the gas phase. By comparison of transitions observed using both methods, the laser vaporization method is estimated to improve the signal-to-noise ratio by one to two orders of magnitude for this molecule.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  F’  -  J=1_0 1.5 2.5 0.5  35 A 7 ‘° C1 g frequency obs.-calc. (MHz) (kllz)  F”  1.5 1.5 1.5  35 A 9 ‘° C1 g obs.-calc. frequency (kllz) (MHz)  7348.7934a 7357.9022 7365.1925  0.0 0.0 0.0  7315.5410 7324.6502 7331.9397  0.0 0.4 -0.4  D,b 147030059  -0.4 0.4 0.5  D 146365008  -0.4 0.4 -0.7  J=2-1 1.5  0.5  2.5  2.5  D 147037851  0.5  0.5  14712.1132  1.5  F’  1.5  -  J=1-0 1.5 2.5 0.5  }  14645.6068 14646.3892D  14712.8936’ D 147194055  0.1  Ag 7 ‘° C1 37 frequency obs.-calc. (MHz) (kllz)  F”  D 146372800  D 146529004  {  0.2  Ag 9 ‘° C1 37 frequency obs.-calc. (MHz) (kllz)  1.5 1.5 1.5  7051.2379 7058.4171 7064.1617  -0.1 0.3 -0.2  7017.9832 7025.1606 7030.9068  0.8 -0.8 0.0  0.5 2.5 0.5  14106.7389 14107.3536 14113.9155  -0.3 0.5 -0.9  14040.2300 14040.8427 14047.4096  0.0 -1.2 2.1  J=2-1 1.5 2.5 0.5  1.5 a  b  1.5  }  14114.5326 14119.6632  {  14048.0229 0.2  14053.1540  {:  -0.4  Measurement accuracy is estimated to be better than ±1 kllz. Frequencies denoted by D are taken from the electric discharge study [105]. Table 7.1: Measured transitions of AgC1  109  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  7.3.2  110  A1C1  Aluminum (I) chloride has been the subject of both microwave [106, 107] and millimeterwave [108] spectroscopic studies, both performed at high temperatures.  Recently, a  high-resolution infrared emission spectrum of A1C1 was recorded to refine the Dunham coefficients used to parameterize rovibrational transition frequencies [109]. The rotational spectrum of A1C1 is of particular interest, since A1C1 has been detected in the interstellar gas cloud IRC+10216 using fairly low-J rotational transitions [110]. A1C1 has been prepared here by ablation of an aluminum rod in the presence of a /Ne gas mixture of low chlorine percentage (-.-0.1%). A1C1 exists as two isotopomers, 2 C1 Cl (75.5% natural abundance) and 27 35 Al 27 C1 (24.5%). 37 A1 hyperfine structure of A1C1  (IA1  Previous analyses of the  =5/2, Ici =3/2) were limited to the more abundant iso  topomer [106, 107]. In this work, hyperfine components of the J=1-0 rotational transition have been observed for both isotopomers. The complex hyperfine patterns for A1C1 consist of groups of overlapped hyperfine components. A section of the spectrum of 27 C1 is given in Fig. 7.3; this shows three 37 A1 Doppler doublets corresponding to eight hyperfine transitions. Each hyperfine transition is labelled by the quantum numbers F 1 and F, corresponding to the coupling scheme IAi+J  =  1 F  Ici+Fi  =  F.  The measured transition frequencies for A1C1 are given in Table 7.2. 7.3.3  CuC1  In the present study, copper (I) chloride was formed by ablation of a copper rod in the presence of a Cl /Ne gas mixture. Formation of CuC1 was particularly sensitive to the 2  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  ç P,F= 1.5,2  2.5,3 2.5,2 2.5,1  1.5 1.5,1  —  ?  2.5,2 2.5,1  14239.2 MHz  111  2.5,4 2.5,3 2.5,2  —  85 kHz I  I  14239.9 MHz  C1, showing three sets 37 A1 Figure 7.3: A portion of the J=1-0 rotational spectrum of 27 /Ne 2 of overlapped hyperfine components. Experimental conditions: Al rod; 0.05% C1 gas sample; 14239.397 MHz excitation frequency; 50 ns sample interval; 4 K FT; 200 averaging cycles.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  27 C1 35 A1 frequency obs.-calc. (kllz) (MHz)  F  F’  F’  2.5  1.0  2.5  14571.0739a  14228.9713  2.5  4.0  2.5  14571.8527  14229.6086  2.5  2.0  2.5  2,5  3.0  2.5  3.5  3.0  2.5  3.5  4.0  2.5  3.5  5.0  2.5  3.5  2.0  2.5  1.5  1.0  2.5  1.5  2.0  2.5  1.5 1.5  0.0 3.0  2.5 2.5  F”  27 C1 35 A1 frequency obs.-calc. (MHz) (kllz)  2.0 1.0 3.0 2.0 4.0 3.0 2.0  14572.4124  J J  4.0 3.0 2.0  0.9 0.4  1-0.3  1-0.3  14573.5586 -0.6 (-0.8 -0.8 14577.9643 -1.0  14230.9376 -0.5 (-0.7 0.0 14235.8264 -0.2  14578.3261  14236.0030  14579.4010 14579.6313  0.3 0.5 0.3  14581.9938  J  14236.8830 14237.1183  0.0  14239.3925  (-o.s 14582.1044 14582.5121  0.5 0.4 0.2  14239.3476  14581.8121 2.0 1.0 1.0 3.0 2.0 4.0  1.2 0.8  14230.0720  -0.3 0.5 0.2 0.0  (-0.6 —  0.3 14239.8170  a  Observed frequency for overlapped hyperfine components. Measurement accuracy is estimated to be better than ±1 kllz. Table 7.2: Measured hyperfine components of the J=1-0 transition of A1C1  112  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  113  concentration of Cl 2 in the mixture, and so low concentrations (O.1%) were necessary in order to preclude formation of such products as CuC1 . CuC1 has been studied before 2 in both the microwave [111] and millimeter-wave [112] regions, but nuclear quadrupole coupling constants for Cu and Cl  (‘Cu  ci =3/2) had been determined for only the most T =  abundant isotopomer, 63 C1 (52.2% natural abundance). 35 Cu  o  In this work, the J=1-  rotational transition has been observed for 63 Cl, 65 35 Cu C1 (23.3%), and 63 35 Cu C1 37 Cu  (16.9%). As in the case of A1C1, the J=1-O transition is composed of overlapped hyperfine components. Several hyperfine components of the J=2-1 transition were also measured for Cu Cl and 65 35 C1. The observed frequencies and their assignments are given in 35 Cu Table 7.3, where the transitions are labelled according to the coupling scheme Ici+J  =  1 F  1C+F1  =  F.  Very recently, a laser ablation/cavity MWFT study of the J=1-O transition in CuC1 was reported [113], with all of the observed frequencies of the hyperfine components falling within experimental uncertainty of the frequencies reported in this work. 7.3.4  mCI  The pure rotational spectrum of indium (I) chloride has been observed previously via microwave [114, 115, 116, 117] and millimeter-wave [118] spectroscopy. The large nuclear spin of indium (I =9/2), coupled to the nuclear spin of Cl as well as to the rotation of the molecule, results in a large number of hyperfine components for even the J=1-O rotational transition. In the present work, InCl has been produced by ablation of In in the presence of a dilute Cl /Ne gas mixture (‘.O.1% to avoid formation of the more stable 2 .) 3 InCl Indium is easily ablated from the ‘soft’ In target rod, and InCl is stable enough  114  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  F  F’  F’  63 C1 35 Cu frequency obs.-calc. (kllz) (MHz)  F”  65 C1 35 Cu obs.-calc. (MHz) (kllz)  frequency  63 C1 37 Cu obs.-calc. (kllz) (MHz)  frequency  J=1-0 1.5  2.0  1.5  1.5  1.0  1.5  1.5  3.0  1.5  1.5  0.0  1.5  2.5  1.0  1.5  2.5  4.0  1.5  3.0 1.0 2.0 0.0 1.0  ‘  10648.3416a  (1.8 1.8 0.2 2.9 1.3  10650.0642  10653.8852  1.0  }  10657.2448 10657.8984 10658.2404  10536.6590  0.1  10540.3060  0.0 -0.7 -0.7  10540.9769  {‘ -0.1  10541.3286  -0.7 -0.7  0.2  1.5  2.5  3.0  1.5  0.5  1.0  1.5  0.5  2.0  1.5  J=2-1 2.5 3.0 2.5 4.0 1.5 3.0 4.0 2.5 5.0 3.5  2.5 2.5 2.5 1.5 2.5  3.0 4.0 2.0 3.0 4.0  21304.9936 21306.9156 21312.9254 21313.8877 21314.4335  -2.3 1,3 1.4 -0.8 0.9  4.0 3.0 3.0 3.0  2.5 2.5 1.5 1.5  3.0 2.0 2.0 3.0  21314.9672 21316.5075 21317.5143  -1.2 -0.4  3.5 3.5 2.5 1.5  1.3  10548.4833  1.5  2.7 (1.3 0.3 -1.0  —  10287.5529 10287.9860 10288.1966  —  {:  0.1 -0.7 -0.7  10294.3592  0.9  10295.1952  10549.1200  10666.2516  10281.5242  ( 2.7  10291.0270  10543.6902  10660.8588  10280.0472  10282.6334  {j)  2.0  10665.5242  10533.2112  2.0 0.4 2.8 1.3  10533.9448  2.5  1.0  (  (  J  10650.9242  3.0 3.0 1.0  10531.5532  ( 2.0  21079.9820 21080.4937  0.9 -1.2  21083.3542 21086.2723  0.9 -0.6  1.1  a  Observed frequency for overlapped hyperfine components. Measurement accuracy is estimated to be better than ±1 kllz.  Table 7.3: Measured transitions of CuC1  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  115  to be commercially available. This, combined with the large dipole moment of mCi  (  C1)I=3.79(10) D [117]), resulted in very strong signals. Both the J=1-0 35 mn 115 to(  and J=2-1 rotational transitions were measured for the two most abundant isotopomers, In (23.2%). The measured transitions 5 “ C1 15 (72.5% natural abundance) and 37 ‘ C1 35 In are given in Table 7.4, labelled according to the coupling scheme  7.3.5  =  1 F  =  F.  InBr  Indium (I) bromide was the first non-chloride produced in this work using the laser abla tion technique. However, simply substituting Br 2 for Cl 2 in the gas mixture was sufficient to produce InBr; again, low concentrations (‘-.‘O.05%) of Br 2 in Ne were used to inhibit formation of InBr . The microwave [114, 119] and millimeter-wave [120] spectra of InBr 3 are known; the microwave spectrum of InBr exhibits complex and widespread hyperfine 9/2, ‘Br =3/2) with large quadrupole  patterns, the result of two coupling nuclei  In 5 ‘ Br moments. In this work, the J=2-1 rotational transition has been measured for 79 In (47.2%). The hyperfine components of this 5 Br (48.5% natural abundance) and 81 transition are overlapped for the two isotopomers, and extend over a frequency range of 300 MHz, as is shown in Fig. 7.4. The measured transition frequencies and their assignments are given in Table 7.5, labelled according to the coupling scheme  TBr+F1  =  1 F  =  F.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  F’  F’  F”  J=1-0 4.5 3.0 6.0 4.5 4.5 4.0 4.5 5.0 6.0 5.5 5.5 5.0 7.0 5.5 4.0 5.5 4.0 3.5 3.5 3.0 5.0 3.5 2.0 3.5  4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5  3.0 6.0 4.0 5.0 5.0 4.0 6.0 3.0 5.0 4.0 6.0 3.0  J=2-1 4.5 6.0 4.5 3.0 4.5 5.0 4.0 4.5 4.0 5.5 4.5 6.0 7.0 5.5 4.5 6.0 5.0 5.5 5.5 6.0 4.5 5.0 4.0 4.5 2.0 3.5 5.0 3.5 5.0 3.5 3.5 3.0 4.0 3.5 6.5 7.0 7.0 6.5 6.5 6.0 6.5 8.0 5.0 6.5 6.5 5.0  3.5 3.5 3.5 3.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 5.5 3.5 3.5 3.5 3.5 3.5 5.5 5.5 5.5 5.5 5.5 5.5  5.0 2.0 4.0 3.0 4.0 7.0 7.0 6.0 5.0 6.0 6.0 5.0 2.0 5.0 4.0 3.0 4.0 7.0 6.0 5.0 7.0 4.0 5.0  F  a  In 5 “ C1 35 frequency obs.-calc. (kllz) (MHz)  6434.1926a 6435.0110 6436.8213 6437.6728 6555.6403 6555.8488 6557.3901 6557.7150 6583.6770 6583.9029 6584.6868 6584.8274  12930.0468 12930.4296 12933.7853 12933.8542 12957.2001 12957.3449 12958.4523 12959.0634 12959.9006 12961.3212 12961.8234 12961.9076 12988.4497 12989.3760 12990.3551 12991.1514 12992.1008 13059.0279 13060.7454 13061.0634 13061.4214 13061.6656 13063.5033  0.8 -0.8 0.8 -0.5 0.1 0.2 -0.3 -0.6 -0.2 0.2 0.7 -0.4  -1.9 1.8 -1.0 0.6 1.3 0.4 -0.0 -0.3 0.8 -1.0 0.0 -0.1 0.6 -0.2 1.0 -0.7 -1.3 0.8 -1.0 -0.7 -0.3 -0.8 0.9  C1 37 In 115 frequency obs.-calc. (kllz) (MHz)  6165.3614  -0.3  6167.4547 -1.5 6285.9940 0.6 6286.1986 0.4 6287.3774 0.3 6287.6571 0.7 6313.8886 -0.7 6314.0930 -0.1 6314.6761 0.5  12390.7558 12391.0807 12393.7491  -1.2 1.4 1.5  12417.8908 12418.0560  0.1 0.4  12419.9792  -0.5  12521.0767 -0.4 12521.2880 -1.0 12521.6088 -0.4 12521.7724 -0.8  Measurement accuracy is estimated to be better than ±1 kllz. Table 7.4: Measured transitions of InCl  116  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  115  Id  81  In Br  1  111111 ii  I  6470.6 MHz  117  111111111  III  ill  79 1 5 “ Br n  I  6764.2 MHz  Figure 7.4: A composite spectrum, showing the hyperfine components of the J=2-1 rotational transition measured for 79 1n and 81 5 “ Br 1n scaled according to predicted 5 “ Br, transition intensities.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  In 1 ‘ C1 37 5 frequency obs.-calc. (MHz) (kHz)  12532.7690 -0.0 12532.9003 -0.9 12533.0786 0.4 12533.2370 1.2  F’  F”  “ I 5 nC1 frequency obs.-calc. (MHz) (kllz)  J=2-1 (cont’d) 2.5 3.0 3.5 2.5 4.0 3.5 2.5 2.0 3.5 1.0 3.5 2.5 4.5 6.0 4.5 5.0 5.5 4.5  4.0 5.0 3.0 2.0 5.0 4.0  13072.4186 -1.2 13072.5800 0.6 13072.8460 -1.5 13073.0441 0.1 13077.0284 -2.2 13078.9285 0.6  12539.3888 -0.5 12539.7208  0.2  12540.8329  0.1  12540.8774  0.9  F  F’  5.5  6.0  4.5  5.0  13079.2904  1.4  4.5 5.5  5.0 4.0  4.5 4.5  5.0 3.0  13079.7908  0.6  5.5  7.0  4.5  6.0  13080.8314  0.5  5.5 4.5 4.5  6.0 4.0 3.0  4.5 4.5 4.5  6.0 4.0 3.0  13081.9835  2.0  118  12541.0476 -0.3 12541.1898 -0.4  Table 7.4: Measured transitions of InCl (cont’d) 7.3.6  InF  Previous studies [121, 122] of the rotational spectrum of indium (I) fluoride have required high temperatures, as InF does not exist as a monomer at room temperature. However, InF is easily produced using the laser ablation technique, with an indium target rod and a 6 SF / Ne (.-..‘O.l%) gas mixture. Only the J=1-0 rotational transition of InF falls within the frequency range of the cavity MWFT spectrometer; this transition has been measured for F(95.7% 19 I 5 “ n natural abundance); transitions of the much less abundant 9 1 3 “ F n’ (4.3%) were not observed. In addition to the 5 “ 1 n nuclear quadrupole splitting, F nuclear spin-rotation splittings (IF =1/2) have been resolved, as is shown in the upper 19 spectrum given in Fig. 7.5. The measured hyperfine components are given in Table 7.6, where M±(F) as determined by molecular beam electric resonance [123] has been used  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  F  F’  F’  F”  1n 1 ‘ Br 35 5 frequency obs.-calc. (MHz) (kllz)  37 1 5 “ Br n frequency obs.-calc. (MHz) (kllz)  J= 2-1 6470.3476a 4.5 5.0 6.0 5.5 5.5 7.0 7.0 6491.0736 5.5 3.5 4.0 6604.1039 0.4 3.5 3.0 3.5 3.0 3.5 3.0 6607.9857 -2.4 6511.7997 5.0 6612.0374 0.8 6515.8834 3.5 5.0 3.5 2.0 2.0 3.5 3.5 6628.7970 -0.5 6.5 8.0 7.0 6680.5504 -1.4 6583.4056 5.5 2.5 4.0 6683.4523 -0.1 6588.4788 3.5 5.0 6.5 7.0 5.5 6.0 6587.8420 4.0 6687.2419 1.5 6588.2380 6.5 5.0 5.5 4.5 5.5 6.0 6.0 6687.6085 5.3 2.5 4.0 3.0 3.5 6690.4708 0.4 2.5 1.0 2.0 3.5 6691.7346 -2.8 6594.5962 4.5 5.5 7.0 6.0 6693.3651 -0.8 6597.0359 4.0 4.5 6694.5730 -0.7 6598.3234 5.5 3.0 2.5 2.0 6694.6634 -3.9 3.5 3.0 4.5 4.0 4.5 4.0 1.4 6696.0571 6599.3748 4.5 4.5 1.2 3.0 3.0 6696.2324 6599.5942 4.5 5.0 4.5 5.0 6606.6898 6.5 6.0 5.5 5.0 6698.7110 0.0 6597.3860 4.5 4.5 6704.4096 1.1 5.0 5.0 4.5 4.5 6.0 6705.5634 0.1 6607.2960 6.0 4.5 5.5 5.0 4.0 6707.7118 -2.0 6609.1366 5.5 4.5 6708.5354 -2.8 6609.6932 6.0 5.0 4.5 5.5 5.0 5.0 6713.0034 3.2 4.0 4.5 5.5 4.0 6717.4856 1.8 4.5 4.5 3.0 4.0 6719.1435 2.3 4.5 3.5 5.0 6758.8904 0.8 6.0 6660.6886 4.5 3.5 3.0 4.0 6762.8429 -1.6 6664.1811 4.5 3.5 4.0 6764.2450 -1.3 5.0 a Measurement accuracy is estimated to be better than ±1 kllz. Table 7.5: Measured transitions of InBr  19 0.9 2.2 -1.6 -0.2 -1.2 -0.7 -0.4  -1.0 0.5 1.3 0.1 1.7 0.8 1.8 0.1 -2.1 -0.3  3.0 -3.0  119  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  F  F”  1n 5 ’ 1 Br 35 frequency obs.-calc. (MHz) (kllz)  120  In 5 Br 37 frequency obs.-calc. (MHz) (kllz)  F’  F’  J=3-2 5.5 7.0  5.5  7.0  9996.2778 2.6  9850.6363  2.3  3.5 7.5  5.0 6.0  3.5 6.5  5.0 5.0  10003.6364 1.5 10009.2192 0.8  9857.2374 9863.3901  0.2 0.3  7.5 7.5 7.5  9.0 7.0 8.0  6.5 6.5 6.5  8.0 6.0 7.0  10010.0396 -1.6 10012.0460 -0.2 10012.4468 -0.9  9863.9498 9865.7220 9866.0036  -1.1 -0.1 -0.0  6.5 1.5 6.5  6.0 2.0 8.0  5.5 2.5 5.5  5.0 3.0 7.0  10015.9650 -4.3 10017.6481 1.1 10017.6940 -1.1  9871.9570  -0.0  1.5  3.0  2.5  4.0  10019.1348 2.6  9872.5022  1.0  6.5  7.0  5.5  6.0  10037.5968 -0.7  9888.2025  -2.7  Table 7.5: Measured transitions of InBr (cont’d) to assign the hyperfine components, with the coupling scheme  IF+F1  =  1 F  =  F.  =3.5-4.5 hyperfine component of the J=1-O rotational transition 1 In addition, the F in the first excited vibrational state (v=1) has been observed, as is shown in the lower spectrum of Fig. 7.5; with the much lower signal-to-noise ratio, ‘ F nuclear spin-rotation 9 splittings cannot be determined accurately. By comparing the signal-to-noise ratios for the two spectra of Fig. 7.5, and assuming a Boltzmann distribution over vibrational levels (which may not be the case in the non-equilibrium environment of the molecular expansion), the vibrational temperature in the expansion may be estimated very roughly as 380 K. This temperature is sufficiently high that the other two F-F’ components were not observed for v=1. Recently, a vibrational temperature of .-..i500 K was estimated for  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  121  ,F= 3.5,4—4.5,4 1 F 3.5,3—4.5,5  v0  15738.5 MHz  15738.9 MHz  vO  \  15625.8 MHz  15626.2 MHz  Figure 7.5: The F =3.5-4.5 hyperfine component of the J=1-O rotational transition of 1 1n’ 5 “ F 9 , observed for both the ground vibrational state (upper spectrum) and the first excited vibrational state (lower spectrum). Experimental conditions (upper spectrum): In rod; 0.1% SF /Ne gas sample; 15738.701 MHz excitation frequency; 50 us sample 6 interval; 8 K FT; 80 averaging cycles. Experimental conditions (lower spectrum): In rod; /Ne gas sample; 15625.980 MHz excitation frequency; 50 ns sample interval; 6 0.1% SF 2 K FT; 687 averaging cycles.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  frequency  F  F’  F’  F”  5.0 6.0 4.0 3.0 4.0 5.0  4.5 4.5 4.5 4.5 4.5 4.5  4.0 5.0 4.0 5.0 4.0 5.0  (MHz)  122  obs.-calc. (kllz)  v=O 5.5 5.5 3.5 3.5 4.5 4.5 v=1 3.5  4.5  15708.6416 15708.6669 15738.7072 15738.6975 15575.8635 15575.8635  -2.7 2.7 -3.3 3.3 1.8 -1.8  15625.9667  a  Measurement accuracy is estimated to be better than ±1 kllz.  Table 7.6: Measured hyperfine components of the J=1-0 transition of In 15 ‘ F 19 MgCl produced using a similar laser ablation/cavity MWFT arrangement [101].  7.3.7  YC1  Having demonstrated the capabilities of the cavity MWFT-laser ablation system, at tempts were made to produce and observe metal-containing species whose pure rota tional spectra were previously unknown. The first rotational spectrum to be successfully observed for the first time with this system was yttrium (I) chloride. Relatively little is known about the spectroscopy of YC1. Even its electronic spectrum, of interest to theoreticians because of the presence of partially filled d orbitals, has been explored little  since the discovery of the C’1 the  ‘  —  X’E band system of YC1 in 1966 [124]. In addition to  electronic ground state, the A / [125], B 1 11 [126], C’D [124, 127], D’H [128], and 1  J’L[ [128] states have been investigated, with much of this work having been done very recently. In their laser-induced fluorescence (LIF) study [127] of the C’E  —  X’E (0,0) band of  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  123  YC1, Simard et al. used supersonic cooling to reduce spectral congestion near the band origin. With sub-Doppler resolution of 120 MHz, they were able to resolve rotational structure and determine rough rotational constants, in addition to permanent electric dipole moments for the X and C states.  YC1 was prepared in that study by laser  ablation of an yttrium target rod in the presence of a CC1 /He gas mixture. 4 In the present work, YC1 has been produced by ablation of an yttrium target rod in the presence of a 0.05% C1 /Ne gas mixture. The J=1-0, J=2-1, and J=3-2 rotational 2 37 (24.2%) have been mea Y 89 transitions of C1 35 (75.8% natural abundance) and C1 Y 89 sured. Initial searches were based on the rotational constants produced by the LIF study [127]. Once the rotational transitions had been located, other hyperfine components of the transitions were sought, as eQq(Cl) was unknown. The hyperfine patterns were dis covered to spread over very narrow frequency regions, as can be seen in Table 7.7 where the measured transitions of YC1 are given with their assignments. Labelling is according to the coupling scheme ci +J =F. T No 89 Y nuclear spin-rotation splitting (ly =1/2) was observed. The signals obtained for YC1 were frustratingly weak, even after much care was taken to optimize experimental parameters such as gas mixture composition, microwave pulse length, nozzle pulse/laser pulse delay, and laser power. Both Cl 4 were used 2 and CC1 as chlorinating agents, with comparable results. The predicted ground state dissociation energy of YC1 is ‘-‘5.4 eV [129], which is greater than that determined experimentally for A1C1 (‘—‘5.2 eV [130]), and so the problem seems to lie in either poor formation of YC1 or its rapid conversion to another form (e.g. YC1 ), rather than in premature dissociation. 3 A sample spectrum is given in Fig. 7.6, showing the overlapped hyperfine components 35 This is the most intense Y 89 F=2.5-1.5 and F=3.5-2.5 of the J=2-1 transition for C1.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  F’  -  F”  35 y 89 c frequency obs.-calc. (MHz) (kllz)  37 Y 89 c1 frequency obs-ca1c. (MHz) (kllz)  7080.1021a 7080.3142 7080.4678  6806.0640  -1.7  13612.0474  f 2.8  J=1-0 1.5 2.5 0.5  1.5 1.5 1.5  J=2-1 2.5 3.5  1.5 2.5  J=3-2 1.5 2.5  0.5 1.5  21240.6288  3.5 4.5  2.5 3.5  21240.6863  a  J  14160.5252  0.3 -0.2 0.6  f 1.7 1.4 2.3  20417.9306 20417.9750  Measurement accuracy is estimated to be better than ±1 kllz. Table 7.7: Measured transitions of YC1  0.6  0.6 2.0  124  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  125  transition observed for YC1, and yet 320 averaging cycles were required to obtain this signal-to-noise ratio.  7.4  Analysis and discussion  Analysis of the data obtained for all seven metal halides was performed using the leastsquares global fitting program SPFIT [131], which treats nuclear quadrupole and nuclear spin-rotation coupling exactly for up to four nuclei by diagonalizing the appropriate Hamiltonian matrices. For two coupling nuclei, SPFIT uses the coupling scheme +J 1  = F  +F 2 1  = F.  Overlapped hyperfine components were weighted according to their theoretical intensities, and the observed frequency was treated as a blend of these components. The Hamiltonian employed was of the form HspFIT  = Hrot + Hc.ci. + HQ (1) + HQ (2) + Hnuc.spin_rot (1) +Hnuc.spin_rot (2) = B J 0  2 —  J 0 D  1 +M±(1)1  “  +  (7.1)  v  (2)  (1)  .  Q  (2)  (1) + V  (2)  (2)  .  J +M±(2)1 2 .J  Q  (2)  (2) (7.2)  Where no hyperfine splittings were observed for the metal nucleus (i.e. AgCI, YCI), hy perftne effects of the halogen nucleus alone were analyzed. Where M± was indeterminate, it was set to zero in the final fits. HSpFIT was sufficient to account for all the observed data; any effects of nuclear spin-spin coupling were evidently negligible. The spectroscopic constants obtained for the metal halides studied are given in Ta ble 7.8-7.14, along with the standard deviations  fit  of the fits. For comparison, values  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  126  81 kHz  14160.275 MHz  14160.800 MHz  Figure 7.6: The overlapped F=2.5-1.5,3.5-2.5 hyperfine components of the J=2-1 rota /Ne gas sample; 2 35 Experimental conditions: Y rod; 0.05% C1 Y 89 tional transition of C1. averaging cycles. 320 4 K FT; 14160.3 MHz excitation frequency; 50 ns sample interval;  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  g 35 A T ‘° CI lit, value this work B 0 (MHz) (MHz) 0 D eQq(Cl)(kllz) (kHz) fit  62 ( 367804371 ] 104 3678.04299(39)L )b[ 1.89057(40) [104] 1.851(53) -36.44089(95) [105] -36.4408(20) 0.3  g 37 A 7 ‘° C1 lit, value this work (MHz) B 0 (MHz) 0 D eQq(Cl)(kllz) (kllz) fit 7  Ref.  Ref.  3528.49352(39) 3528.49242(82) [104] 1.73846(80) [104] 1.802(53) 28.7213(43)c [105] -28.7184(20) 0.4  g 35 A 9 ‘° C1 lit, value this work  127  Ref.  3661.41682(39) 3661.41815(62)[104] 1.87416(50)[104] 1.869(53) -36.44113(95)[105] -36.4404(20) 0.4  g 37 A 9 ‘° C1 lit, value this work  Ref.  3511.86565(39) 3511.8629(10) [104] 1.7195(11) [104] 1.720(53) -28.7213(43) [105] -28.7197(20) 1.0  a  Numbers in parentheses are one standard deviation in units of the last significant figure. Calculated from the constants given in Ref. [104]. Numbers in parentheses reflect the error limits given there. C C1) and 35 C1) predicted in Ref. [105] using experimental values of eQq( 37 Values of eQq( C1)=1.26878(15), given in Ref. [10]. 37 C1)/Q( 35 the ratio of Q( b  Table 7.8: Spectroscopic Constants of AgCl of rotational, centrifugal distortion, and nuclear quadrupole coupling constants obtained from previous microwave, millimeter-wave, and infrared studies are also presented. In  0 was set to cases where only one rotational transition has been studied in this work, D the literature value in the fits.  0 has gen For the molecules whose pure rotational spectra were previously known, B erally been determined to a precision comparable to or better than that obtained from much more extensive millimeter-wave studies. One exception is A1C1, where the recent infrared study [109] determined B 0 one order of magnitude more precisely; note that  0 does not fare as data from the present work was included in their analysis [109]. D  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  this work B eQq(Al) eQq(C1) M±(Al) M(Cl) cTfit  (MHz) (MHz) (MHz) (kllz) (kllz) (kHz)  6 )b 2 . 7288 1 ( 462  -30.4081(27) -8.8290(35) 5.54(16) 3.52(30) 0.8  27 C1 35 A1 lit, value  Ref.  7288.72504(57)’ [109] [107] -29.8(50) [107] -8.6(10) —  —  27 C1 37 A1 lit, value this work  128  Ref.  7117.51219(16) 7117.51235(59)[109] -30.4112(28) -6.9586(36) 5.44(16) 2.62(32) 0.6 —  —  —  —  0 in the least-squares fit B is obtained by using the known value [108] of D 0 transition. for the J=1-0 b Numbers in parentheses are one standard deviation in units of the last significant figure. Calculated from the constants given in Ref. [109]. Numbers in parentheses reflect the error limits given there. a  Table 7.9: Spectroscopic Constants of A1C1 well because of the limited number of transitions measured for each molecule, although where D 0 has been released in the fits to the laser ablation data it compares quite well to  0 values so obtained is also remarkably good, millimeter-wave results. The precision of D given that the constants have been obtained from two or three rotational transitions with the smallest observable distortion effects, and is a result of the high resolution available with the spectrometer. The precision of eQq values has generally been improved con siderably over earlier results. In some cases, eQqs have been determined for the first C1) was determined for AgC1 with 35 time for less abundant isotopomers; although eQq(  comparable precision in the electric discharge study [105], measurement of the remain C1). Nuclear spin-rotation 37 ing hyperfine components has allowed determination of eQq(  coupling constants have also been determined for the first time in most cases. A notable exception is InF, whose hyperfine parameters were determined by molecular beam electric resonance to a precision impossible to match using today’s cavity MWFT spectrometers.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  C1 35 Cu 65  C1 35 Cu 63 lit. value  this work (MHz) B 0 (kllz) 0 D eQq(C1) (MHz) eQq(Cu) (MHz) M(Cu) (kllz) (kllz) fit  5328.54998(23 3.911(36) -32.1257(19) 16.1712(24) 10.39(28) 0.9  Ref.  512 (ii)b 5 . 5328  [1121 3.89315(80) [112] [111] -32.25(15) [111] 16.08(20)  this work  lit. value  —  —  —  —  0 B D 0 eQq(Cl) eQq(Cu) M±(Cu) gfit  (MHz) (kllz) (MHz) (MHz)  5143.76368(18) 3.62747c -25.3181(31) 16.1667(42)  (kllz)  9.96(36)  (kHz)  0.9  lit. value  Ref.  5270.05804(23) 5270.0598(12J112] 3.8084(80{112] 3.836(40) -32.1247(20) 14.9635(27) 11.59(30) 0.8  C1 37 Cu 63 this work  129  Ref.  5143.7647(12) [112] 3.62747(80) [112] —  —  —  Numbers in parentheses are one standard deviation lit units of the last significant figure. Calculated from the constants given in Ref. [112]. Numbers in parentheses reflect the error limits given there. C . 0 0 fixed at known value [112] in order to determine B D a  b  Table 7.10: Spectroscopic Constants of CuCl  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  this work  B 0 0 D  (MHz) (kllz) eQq(In)(MHz) eQq(Cl)(MHz) M(In) (kllz) M(Cl) (kllz) (kHz) iit  n 35 I 5 “ C1 lit, value  53 ( 73242 3261.73321(2oy 3261 )b 1.54373(14) 1.548(26) -657.8487(18) -657.52(50) -13.63(20) -13.7575(15) 10.454(38) 1.71(11) 0.9  Ref. [118] [118] [132] [132]  this work  n 37 I 5 “ Cl lit, value  130  Ref.  3126.79399(23) 3126.79430(67)[118] 1.41872(17)[118] 1.386(30) [115] -657.8913(23) -657.20(45) [115] -10.11(45) -10.8399(26) 9.980(46) 1.19(14) 0.7 —  —  —  —  Numbers in parentheses are one standard deviation in units of the last significant figure. Calculated from the constants given in Ref. [118]. Numbers in parentheses reflect the error limits given there. a  b  Table 7.11: Spectroscopic constants of InCl  this work  B 0 0 D  (MHz) (kllz) eQq(In) (MHz) eQq(Br)(MHz) M(In) (kllz) M±(Br) (kllz) (kllz) Ufi  115 Br 79 1n lit. value  Ref.  0 [120] 9062 )b 2 . 1667 9 1667.29199(11)a ( 0.41713(13) [120] 0.4092(76) [119] -633.5756(35) -633.50(26) [119] 110.63(27) 110.6501(22) 6.168(44) 5.30(10) 2.0 —  —  this work  n 81 1 5 “ Br lit. value  Ref.  1642.90726(12) 1642.90394(90)[120] 0.40472(12)[120] 0.3941(83) [119] -633.5731(34) -633.20(26) [119] 92.33(26) 92.4367(28) 5.924(52) 5.36(11)  —  —  1.4  Numbers in parentheses are one standard deviation in units of the last significant figure. Calculated from the constants given in Ref. [120]. Numbers in parentheses reflect the error limits given there. a  b  Table 7.12: Spectroscopic constants of InBr  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  this work  lit. value  Ref. [122] [123] [123] [123]  0 B  (MHz)  0496 l 7836 8 ( )ab 8  7836.1392(57)c  eQq(In) M(In) M±(F)  (MHz) (kllz) (kllz) (kllz)  -723.794(10) 17.46(11) 18.3(10) 2.6  -723.7996(2) 17.50(1) 18.77(10)  aiit  131  a  0 in the 0 is obtained by using the known value [122] of D B least-squares fit for the J=1-0 transition. b  Numbers in parentheses are one startdard deviation in units of the last significant figure. c Calculated from the constants given in Ref. [122]. Numbers in parentheses reflect the error limits given there.  F 9 1n’ Table 7.13: Spectroscopic constants of 115  35 y 89 c this work  B 0 0 D  (MHz)  (kllz) eQq(Cl)(MHz) M(Cl) (kHz) (kllz) ‘fit  lit. value  Ref.  this work  c1 3 Y 89 7 lit, value  3540.13730(30)a  12 ( 354020 )b  [126]  3403.01828(52)  1.398(19) -0.8216(43) 2.86(39) 0.4  1.6(3)  [127]  1.345(33) -0.621(20)  —  —  d 229  3405.6(13)C  2.1(4) —  —  1.2  a  Numbers in parentheses are one standard deviation in units of the last significant figure. Calculated from the constants given in Ref. [126]. Numbers in parentheses reflect the error limits given there. Calculated from the constants given in Ref. [127]. Numbers in parentheses reflect the error limits given there. d C1) fixed at value calculated using M±( 37 M±( C1) and the relation M 35 1 cx Bo I_ti. ‘  Table 7.14: Spectroscopic constants of YC1  Ref. [127]  [127]  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  132  0 are 0 and D All of the spectroscopic constants obtained for YC1 are significant: B much more precisely determined than from the LIF study [127], as expected given the very large improvement in resolution. The values of eQq(Cl) in YC1 have been determined C1 and 37 Cl, as has M( Cl). eQq(Cl) may be interpreted as 35 for the first time for both 35 a measure of the ionic character i of YC1. If screening effects and orbital hybridization are neglected, the ionic character of a bond is related to the nuclear quadrupole coupling constant along that bond (Xz, or eQq for a diatomic molecule) by [10] Xz  eQqio where  Q  (73)  is the quadrupole moment of the coupling atom and nio is the contribution  of an electron in a p atomic orbital to the field gradient at the atomic nucleus. For C1, eQqio=109.74 MHz [10], and so i=0.9925 for YC1. This highly ionic bond is 35 consistent with the large electronegativity difference for the two atoms (‘—4.8 [10]); in this respect, YC1 is much more similar to the alkali halides (e.g. KC1: i=1, electronegativity difference 2.2) than to the other metal halides studied in this work (e.g. AgCl: i=0.66, electronegativity difference 1.2.) Relatively precise values have also been determined for the ratio of the Cl quadrupole AgC1, C1). The values for this ratio in ‘° 37 C1)/eQq( 35 AgC1, 109 7 coupling constants, eQq( A1C1, 63 27 CuCl, “ InCl, and 89 5 YC1 are given in Table 7.15, along with a weighted average of the six ratios, where weighting was according to the squared inverse of the standard deviations. In the case of 89 YC1, where eQq(Cl) was determined from a very limited number of hyperfine splittings, the ratio of the Cl quadrupole coupling constants is within two standard deviations of the known value; for the other six molecules, the ratios agree within one standard deviation with the known ratio of the Cl quadrupole moments, C1) [133], as does the more precise weighted average. While it was hoped 37 Cl)/Q( 35 Q( initially that the high precision of the eQq values obtained in this work would result in  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  molecule  C1) 37 C1)/eQq( 35 eQq(  ‘° A 7 gC1 AgC1 9 ‘° AlCl 27 CuCl 63 IiCl 5 “ YC1 89  1.26890(11)a 1.26883(11) 1.26879(83) 1.26889(17) 1.26915(33) 1.323(43)  weighted average C1) [133] 37 C1)/Q( 35 Q(  1.268881(49) 1.2688773(15)  133  a Numbers in parentheses are one standard deviation in units of the last significant figure.  Table 7.15: Ratio of Cl quadrupole coupling constants an improvement of the Q( C1) ratio, this was not the case. 37 C1)/Q( 35 Nuclear spin-rotation coupling constants In a diatomic molecule with one coupling nucleus, nuclear spin-rotation coupling is de scribed by a term of the form MIJ in the molecular Hamiltonian, a term which is diagonal in electronic state. This Hamiltonian may be derived considering effects of sec ond order perturbations of the electronic states. Following the procedure described by Miller [134], the electron and nuclear spin dependent terms of the total Hamiltonian may be grouped together as a perturbing Hamiltonian H’ as follows [135]: H  ‘  =  Hrot + Heiectron spin—orbit + Heiectron spin—spin +Heiectron spin—rotation + Hmagnetic hyperfine structure  (7.4)  )2  (75)  where Hrot  =  B(J  —  L  —  S  Chapter 7. Microwave Spectra of Metal Halides ProdNced Using Laser Ablation  Heiectron spin—orbit  Hmagnetic hyperfine  =  —  aI  structure  (7.6)  s  2/3A(3s  Heiectron spin-spin Heiectron spin—rotation  a1,  =  L  —  —  134  S  2)  S  )  L + bFI  (77) (7.8)  S S + c/3(I  S  —  3IS)  +d/2(exp(2iq)I_S_ + exp(—2i)IS) ) + exp(—iq)(SI + 2 +e[exp(ic/)(S_I + I_S  I+S)1. (7.9)  L is the total electronic orbital angular momentum, S is the total electron spin, and R =J -L -s has replaced J as the nuclear rotational angular momentum. Then, using degenerate perturbation theory, the second order term of the effective Hamiltonian is given by Heff  (2)  L.lLd k  llo  lok>< lokIH ‘Ilk>< lkH ‘Ilok>< lokI , E0 E1  (7.10)  where summation is carried out over all electronic states 1 (where lo is the state of interest)  1 are the unperturbed energies and all rotational and spin quantum numbers k; Eo and E of the 1 and 1 electronic levels, respectively. Using the perturbing Hamiltonian given in Eq. 7.4, there exists a second order cross term between the rotational and nuclear spin-orbit parts of H’, of the form  <lokaI.LIlk’><lk’I—2BJ.Llok”> k’,l  (711) .  —  This term is equivalent to an I J interaction, acting within the electronic state 10. (The electron spin analogue, involving bFI .5 and -2BJ S  ,  vanishes for a linear diatomic  E+.) Treating the interaction described molecule when the electronic state of interest is 1  in Eq. 7.11 as the origin of the nuclear spin-rotation coupling described by MI J, one  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  135  obtains  1 M  =  2B  <  L Il>< lL l a 0  lb  (7.12)  >•  —  3 Since the a parameter is given by 2 /BPI < r  >11,  where  ,UB  is the Bohr magneton,  M± can also be expressed as 1 M  =  uj 1 4 u 1 BB—-  <loITfl>< lILt ho>  .  (7.13)  . When M . Eq. 7.13 is given by Schlier [27] as the dominant electronic contribution to 1 the electronic state of interest is Xl+, as is the case in this work for all of the metal halides studied, only those electronic states of 1fl symmetry will interact with the ground state via the L operator to produce nuclear spin-rotation coupling.  1 have been determined for the Cu nucleus Nuclear spin-rotation coupling constants M in CuCI, the Cl nucleus in YC1, and both nuclei in Aid, mdl, InBr, and InF. Where M is known for a pair of isotopomers, the values may compared by observing in Eq. 7.13 that M is proportional to the product of  and the rotational constant [136]. The  ratios of the values of M± obtained for pairs of metal halide isotopomers are given in  . In general, the agreement 0 u and B Table 7.16, along with the ratios calculated using 1 between experimental and predicted ratios is quite good. Small differences are expected,  3 > as variations in effective ground state structures will also affect the ratios via the < r 1 (Eq. 7.13.); in most cases, these differences fall within the experimental dependence of M uncertainties of the ratios. A1C1  The ratio of the spin-rotation constants for the two nuclei in A1C1, M(Al)/  (Cl), may be used to gain some insight into the electronic structure of A1C1. Relative 1 M energies of the pertinent molecular orbitals (MOs) of A1C1 are shown schematically in Fig. 7.7, following the MO configuration proposed by Ram et al. [137]. The 37r and 8u MOs may be correlated to the 3pir and 3pu atomic orbitals (AOs) of Cl respectively, while  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  Ratioa  27  cl  exptl. value  pred. value  Ratio  exptl. value  1.018(42)c  1.0240  J I 115  1.0475(62)  1.0432  1.44(19)  1.2532  1.041(12)  1.0148  0.988(28)  0.9414  A1 7 27 C1  pred. value  I”InCl 27A 35 Cl 27A]j  I  1.34(20)  1.230  C1 37  163Cu5Cl  0.986(34)  0,9438  C Cl 65  1151fl1  C1 35  j 115  C1 37  I”InBr 1115111 ‘Br  Cu Ci 6  1.043(47)  1.0359  Cu 7 63 C1  Br “‘I 79 1151111 81 Br  Cl I cul 35 63cu1 37 C1 I 65  1.164(53)  1.0975  a  Ratio of nuclear spin-rotation coupling constants for the two nuclei indicated by boxes. Calculated as the ratio of the products of the nuclear magnetic moment pi and the rotational constant B 0 for the two isotopomers. C Numbers in parentheses are one standard deviation in units of the last significant figure. b  Table 7.16: Ratios of nuclear spin-rotation coupling constants  136  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  137  the 9cr and 47r MOs correspond to the 3s and 3pTr AOs of Al. The greatest interaction between Al and Cl AOs is assumed to occur between 3pcr(Cl) and 3scr(Al), resulting in the 8cr bonding MO and the 9cr antibonding MO. In the X’ state, the 9cr orbital is the highest occupied MO (HOMO). The lowlying A 11 state is then formed by promoting an electron from 9cr to 4ir. Assuming that 1 contributions from the A’ll state dominate, M 1 is given by 1 M  t<  4pBBo--  =  X’jL A’ll>< A’ll-IX’> fl 1 -A  p < 9cr1  — 0 4ILBB  =  4K ><  47rI-9cr>  (7.15)  .  I  (7.14)  —  EA1H—Ex1E+  In Eq. 7.15, only one-electron integrals are considered, and the one-electron operator l, has been written instead of L. In order that the integral given nucleus, 9cr and  4K  <  9crI1 4r  >  not vanish for a  must be comprised at least partly of AOs which are centred on  that nucleus and which have the same values of both the principal quantum number n and the azimuthal quantum number 1, and values of the magnetic quantum number m 1 which differ by one. This condition is satisfied by assuming the following: (i) 9cr is made up of a large contribution from 3.scr(Al) and a lesser contribution from 3pcr(Cl), but also contains small contributions from 3pcr(Al) and 3scr(Cl); (ii)  4K  is composed primarily  of 3pK(Al) but contains a small contribution from 3pir(Cl). In that case, the integrals <  3pcrI1  I3pK  >  and  <  3pKI-I3pcr  >  will be non-vanishing for both nuclei, and only  those terms of M which differ for the two nuclei need be considered when comparing M±(Al) and M±(Cl): M cx  3 <r  >3p<  3pcr9cr  ><  3pK4K>.  (7.16)  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  138  U U  9u 3s  3su’  p 3  U\____  8  /  p 3  37T  3s  Al  Cl  Figure 7.7: Schematic diagram of the energies of the valence molecular orbitals of Aid.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  139  Here < 3pa9o > represents the coefficient of 3pu in a linear-combination-of-atomic orbitals (LCAO) expansion of 9a, and < 3p7r47r > the coefficient of 3p7r in 47r. Substi tuting values for M(Al) M(Cl)  uj, 1  —  3 I, and < r  3.64150 0.82187  1.5 2.5  >3p  C1, A1 and 35 [138] for (atomic) 27  1.493 8.389  <3po(Al)9u> < 3po(Cl)9u >  3pr(Al)4’r > < 3pir(Cl)4ir >  <  (7 17)  When this ratio is compared to the experimental value M(Al) M(Cl)  —  —  ( 7 18 )  1 57 ,  .  .  one obtains <3p(Al)9u>  <3pu(Cl)I9cr>  <3p7r(Al)47r> <3pK(Cl)4K>  3.33.  (7.19)  Although it is difficult to draw further quantitative conclusions from this result, it is consistent with 4ir being ‘mostly’ 3pK(Al) with ‘some’ 3p7r(Cl), and with 9a being ‘mostly’ 3sa(Al) with ‘some’ 3pu(Al) and ‘some’ 3pa(Cl).  CuCI  A recent theoretical study [139] permits the direct calculation of M±(Cu) by  providing configuration interaction (CI) coefficients for electronic states of CuCl. The low-lying electron configurations of CuCl which could contribute to M (Cu) are d7r p7r dcr pu : config.(a) Cl(3s Cu(3d5 3 4 ) 3 3 2 )  pu da : config.(b) dir pir s’)Cl(3s Cu(3d5 3 4 4 1 3 3 2 ) p7r pa 4 3 (3s 3 2 thr du ) s’)Cl 2 3 4 Cu(3d6 3 3 4  config.(c).  The dominant configuration of the X+ electronic ground state of CuC1 is config. (a), 1 [140], is config. (c). However, while that of the lowest-lying ‘H state, D’H at 22959 cm > does contain a finite contribution from config. (b), and therefore we can write 1 X <X’EL D’ll >=< 3d7rll,r 3dcr> CI(b,X’Y)CI(c,D’H),  (7.20)  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  140  where CI(b,X 11) are the CI coefficients representing the amount of 1 ) and CI(c,D 1 those configurations in the given electronic states. The nuclear spin-rotation coupling constant for Cu is then given by M  =  =  41LBBo<  3dI  I  1 6BBo  I3d  >2<  3 r  >3d  2 (7.21) (CI(b,X1E+)CI(c,Dhll))  ED1H—Ex1E+  3 <r  I Ejin  —  >3d  Exi+  . 2 (CI(b,X’E+)CI(c,D’fl))  (7.22)  Substituting in the appropriate values for 63 C1 [138, 139], one obtains M±(Cu)=1.22 Hz, 35 Cu  which is approximately 85 times too small. It is important to note that this large dis crepancy does not arise from using an atomic value of < r 3 > for Cu, as the value of a  pLI = 2  3 > < r  /1 determined experimentally by Burghardt et al. [140] differs by  only about 7% from that calculated for Cu using <r 3 >(atomic) [138]. One may also use this method to predict M±(Cl), assuming that the configura 3pa’3p7r and 2 Cl(3s ) tions of X’Y2 and D 11 which interact are 2 1 3d7r Cu(3d8 3 4 4 s’) du 4 3p7r respectively, corresponding to promotion of an 2 3pcr ) 3thr Cu(3d6 3 4 4 2 s’) du Cl(3s 23 electron from 3p7r(C1) to 3p(Cl). Using the CI coefficients given in Ref. [139], M±(Cl) is calculated to be ‘-0.02 kflz. While this small value is supported by the fact that M(Cl) could not be determined in this study, no conclusions can be drawn about its accuracy, especially given the difficulty in predicting M(Cu). M±(Cu) may also be treated in a manner similar to that used above in the case of A1C1. A schematic diagram of the CuC1 MO energies is given in Fig. 7.8. In the X’> electronic ground state, the llo MO, which corresponds primarily to the 3d AO of Cu, is the HOMO. Using the single-configuration approximation, the D’ll state is considered to be formed from  by promoting an electron from 57r to l2ci. If it is then assumed  that the valence AOs of Cu and Cl are sufficiently separated in energy that only the 4so- and 3do- AOs of Cu interact to form llu and 12cr (an assumption supported by the conclusion of Ramirez-Solis and Daudey [139] that the wavefunctions of CuCl are  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  4s  141  12 4su licT 5Tv  3d___ 1ô  —‘v”  Tv  p 3  4 rr  Cu  Cl  Figure 7.8: Schematic diagram of the energies of the valence molecular orbitals of CuC1.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  142  strongly ionic,) then the degree of this interaction may be evaluated quantitatively using M±(Cu). Let 112u >= ciI4sa(Cu) > +c I3d(Cu) > 2  (7.23)  .  Then  <5I1 I12>< l2uI5  >=  3c <r 3  (7.24)  >3d.  Using this result in an expression analogous to Eq. 7.15, together with < r  >3d  derived  from the experimental a constant of Ref. [140] and the value of M(Cu) determined experimentally for 63 C1, we find c 35 Cu 2  0.34. Normalization of 12a then gives c 1  0.94.  In this picture, 4scr(Cu) and 3da(Cu) are mixed to an appreciable extent. This is not indicated by the ab initio CI coefficients [139], which therefore give a much smaller prediction for M(Cu): it seems that the small value of CI(b,X’), which is 0.007, is the most likely source of the disagreement. mCi, InBr, InF  The nuclear spin-rotation coupling constants obtained for the indium  (I) halides are more difficult to interpret than those of A1C1 and CuCl. Indium halides are less attractive than lighter molecules as subjects of ab initio calculations; as a con sequence, much less information is available regarding their electron configurations. In addition, M± of the halogen nucleus is less obviously linked to the electron configurations, as the differences in configurations between  and 111 states tend to be localized on the  indium nucleus. A generalized schematic MO diagram for an indium (I) halide is given in Fig. 7.9. The valence AOs of the halogen atom are ns and np orbitals, where n=2 for X=F, 3 for X=C1, and 4 for X=Br (it will be assumed that the 3d and 4d AOs of Cl and Br are not involved in the MOs to be discussed.) In the X’ electronic ground state, Da is the HOMO; in the single-configuration approximation, the lowest perturbing ‘II state  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  p  5p  E  ci  143  ci  5s  4d  Cu B  4d 4d  \4c16  Tl  Arr A  In  —  ci  flS U  flS  XF,Ci,Br  Figure 7.9: Schematic diagram of the energies of the valence molecular orbitals of an indium (I) halide, InX.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  144  corresponds to promotion of an electron from Do to C’r. In general, Do and Cir may be written as composed of In and X AOs as Do  =  1 nso(X) > + d d 2 npcr(X) > + d 3 4do(In) > + d 4 5so(In)>  1 5po(In) 5 +d C7r  =  cii  >  (7.25)  npir(X) > + c 2 4d7r(In) > + c 3 5pir(In) >.  (7.26)  Perturbations contributing to M (In) will involve matrix elements of the form <Do I  IC  ><  Do>  C  <  =  T  >4d  3 >, +cd <r  (7.27)  and thus M±(In)  < Do =  4tLBBo--  1  .1 —  I Do>  C >< C JIlH  2jtBBOuI(3cd 3 <r >4d +cd 3 <r >  I(EaH—E1E+)  —  (7.28)  —  (7 29)  M (X) will involve matrix elements of the form <Do 1 C>< C  I Do>  =  3 <r  (7.30)  and C)  D  2J2  2 LUBJJO[LICiU  /  1VIjj/k) —  1/  <r —3 >, 1—’  Without a priori knowledge of the composition of Do and C?r, two nuclear spinrotation coupling constants are not sufficient to determine all of the coefficients d and c, for InX. However, one would expect certain trends to exist in these coefficients over the series InF—InCl---InBr. In particular, since the ionization potential (IP) of X decreases with increasing size of X (IP(F)=17.422 eV, IP(Cl)=12.967 eV, IP(Br)=11.814 eV), the In and X AOs become closer in energy and Do and C7r should acquire more halogen character. The magnitude of the coefficients d , d 1 , and c 2 1 should thus increase, and d , 3 , d 4 d , c 5 , and c 2 3 should decrease in magnitude.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  145  This trend should be reflected in the M(X) values, since 12 C d2  —  —  —  —  M±(X)I(E1n—EIE+) 3 2[LBBOILI < r  732  >np  —  gIN[LBBO <r 2 3 >,  (7 33)  Using M±(X) and B 0 values from this work, <r 3 > values from Ref. [138], and energies of the C’l1-XD electronic transitions [141] (InF: 42809.2 cm; InCl: 37483.6 cm’; InBr: —34000 cm’), cd is 0.68 for 9 In’ 5 F , 1.34 for 35 In and 1.58 for 79 5 Cl, In 5 Br. While the increase in cd with increasing size of X is as expected, the fact that cd is greater than unity is troubling. In principle, Dcr and C7r are normalized functions, and so c 1 and d 2 should never be greater than unity. This discrepancy points out a definite limitation of this very approximate interpretation of the spin-rotation constants; as a result, the quantitative mixing ratios determined above for the CuCl MOs should be taken with a grain of salt. YC1  In interpreting the ab initio calculations of Langhoff et al. [129], Simard et al.  proposed the MO scheme for YC1 which is shown schematically in Fig. 7.10. In this scheme, 3p7r(Cl) interacts with a hybridization of 4d7r(Y) and 5p7r(Y) to create the 6ir and 7ir MOs, and 3pcr(Cl) interacts with 4da(Y) to form 13cr and 15cr. Both the 14cr and 25 MOs are considered to be non-bonding. Hybridization of the 5s(Y) and 5pcr(Y) AOs is assumed to occur such that the larger lobe of the 14cr MO points away from Cl, leaving little electron density between the nuclei with which 3pcr(Cl) can interact. In the ground state, the HOMO is 14o-, and the B’ll-X’> transition is assumed to correspond to promotion of an electron from 14cr to 7r, although two other electron configurations were needed to describe the B’ll state in the ab initio calculation [129]. In order to relate the nuclear spin-rotation coupling constant obtained for Cl in this study to the MO picture, one must consider matrix elements linking the 14cr and 77r MOs  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  4d/5p  5s/5pu  146  4d/5p 4c15  25  5su/5pa  l4ci 3p  Y  Cl  Figure 7.10: Schematic diagram of the energies of the valence molecular orbitals of YC1.  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  via the  ‘x operator.  147  If 14a and 7ir are considered to be composed of Y and Cl AOs as  14u> 7ir>  =  cjj 5su(Y) > +  = d 1  l 2 c  (7.34)  5pa(Y) >  3pir(Cl) > + d 2 4dir(Y) > + dl 5pir(Y)>,  (7.35)  then M(Cl) would be expected to be vanishingly small, as there is no way to obtain the necessary matrix elements <3pa(Cl)  lix I 3p2r(Cl)  >.  However, since M(Cl) is non  zero, 14o must have some 3pu(Cl) character, assuming that the ab initio calculations [129] predicting only one electron configuration in the ground state are correct. Although < 5pa(Y)  li  5pir(Y) > is non-zero, any nuclear spin-rotation splitting due  to 89 Y was too small to be resolved, and so M±(Y) could not be determined. With only one coupling constant, it is impossible to determine all of the coefficients of Eqs. 7.34 and 7.35, especially considering that Eq. 7.34 will also require a term of the form  c 3p(Cl) 1 3  >,  and so the degree to which the Y and Cl AOs interact cannot be quantified  (Cl). 1 from M 7.4.1  Conclusions  This work has confirmed that laser ablation techniques may be applied to generate a vari ety of metal compounds, which may then be studied by MWFT spectroscopy. The simple ablation system described here clearly gives much stronger spectra of metal-containing compounds than the pulsed electric discharge method described earlier [105], and has permitted the first observation of the pure rotational spectrum of yttrium (I) chloride. In the near future, improvements to this system will hopefully lead to the production of metal-metal dimers, trimers, and larger clusters. The efficiency of the laser ablation technique, combined with the precision and sen sitivity available with the cavity MWFT spectrometer, have led to the improvement  Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation  148  of many spectroscopic constants for the metal halides, as well as allowing the first de termination of nuclear spin-rotation coupling constants. As these latter constants are particularly sensitive to the composition of certain MO functions, they should be consid ered along with larger spectroscopic parameters in the optimization of future theoretical calculations for the metal halides.  Appendix A  Spherical Tensor Relations  A spherical .tensor T  (k)  (q=—k, —k + 1,.  is an operator whose components  where k is the ‘rank’ of T) are proportional to the spherical harmonics in the same way under rotation.  y)  ..  ,k,  and transform  may also be defined according to the commutation  relations ,T)] {Jx  +  iJy  ,  =  T)]  (A.l)  qT) [k(k + 1)  —  q(q ±  1)]1’2  (A.2)  . 1 T  Spherical tensor components may be separated into two parts, a geometric part depending on the coordinate system in which the spherical tensor operates, and a ‘physical’ part acts on basis functions  which is independent of coordinate system. When where  in  I cxjm>,  is the projection of j onto some axis, this separation may be accomplished using  the Wigner-Eckart theorem:  <a’j’m’  <  cVj’ I T  (ic)  ITI jm>  =  )i’_m’ 1 (  (  ‘  —m’  k  j  q m  <‘j’  (‘a)  (k)  1  >.  (A.3)  )  cj > is a ‘reduced matrix element’; since it has no  not depend on the coordinate system used. If T  T  in dependence, it does  operates on a molecular system,  the reduced matrix element will contain all of the ‘inherent’ physical information and can often be related to molecular constants. The components from the molecular constants, using Eq. A.3.  149  can then be calculated  150  Appendix A. Spherical Tensor Relations  u  The dot product of two spherical tensors of equal rank, T (‘a) and  is given  by T (“)  If T  (k)  and U  (k)  U  j,  1 j  then matrix elements of T T  <jjj’m’  (_1)T)U.  =  (A.4)  act on different parts of a couple basis function, the basis function  must be ‘decoupled’. 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