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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 THEMICROWAVE SPECTRA OF SOME GASEOUS MOLECULESByKristine D. HenselB. Sc., Queen’s University, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESCHEMISTRYWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1993© Kristine D. Hensel, 1993In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.ChemistryThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1W5Date:AbstractThe microwave spectra of the following gaseous molecules have been observed and analyzed, using Stark-modulated microwave spectroscopy and cavity microwave Fouriertransform (MWFT) spectroscopy.Bromine‘80-isocyanate, BrNC’80 The microwave spectrum of BrNC18Ohas beenmeasured in the frequency region 23-52 GHz, using a Stark-modulated microwave spectrometer. Because the spectrum is that of a prolate near-symmetric rotor with stronga-type and weak b-type transitions, perturbations in the quadrupole hyperfine patternsof Br were used to improve the precision of A0. The geometry of the molecule has beendetermined; in particular, the NCO chain has been found to have a bend of-S.’ 8° awayfrom Br.Dichiorosilane, SiH2C1 The b-type rotational spectrum of 28SiF135C1 has been remeasured in the frequency region 10-16 GHz (J=1—10) using a cavity MWFT spectrometer. The MWFT technique has permitted resolution of the complex hyperfine patternsobserved for this molecule, which in turn has allowed the precise determination of theCl nuclear quadrupole coupling constants. In particular, perturbations in the 918-827transition have been analyzed to obtain a value for Xab.The quadrupole couplingtensor has been diagonalized to yield principal values, and the results are discussed interms of the bonding in SiH2C1.11Tetrolyl fluoride, CH3-CC-COF The microwave spectrum of the unstable moleculetetrolyl fluoride has been observed for the first time. The a-type rotational spectrum observed with a Stark-modulated microwave spectrometer is very dense, owing to internalrotation of the methyl group. The spectrum has also been measured in the frequencyrange 9-17 GHz using a pulsed jet cavity MWFT spectrometer. Cooling in the jet hasremoved all internal rotation states other than m =0 and m =1, permitting assignment of the microwave spectrum. The threefold barrier to internal rotation has beenconfirmed to be very low (V3=2.20(12) cm1.)Metal Halides: AgCI, Aid, CuC1, mCi, InBr, InF, YC1 An apparatus hasbeen constructed to produce metal compounds using laser ablation and to investigatetheir rotational spectra with a microwave Fourier transform (MWFT) cavity spectrometer. Metal halides have been produced by ablation of metal rods in the presence of ahalogen-containing gas, using a Q-switched Nd:YAG laser (532 nm). The first seven suchcompounds 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. Nuclearspin-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 coupling constants have been used to examine the electronic structures of the molecules, andeQq(Cl) of YC1 has been interpreted in terms of the ionicity of the Y-C1 bond. Valuesof the rotational and nuclear quadrupole coupling constants have also been improved forthe metal halides.111Table of ContentsAbstract iiList of Tables viiList of Figures ixAcknowledgement Xi1 Introduction 12 Theory 42.1 Energy levels of the rigid rotor 42.2 Selection rules for rotational transitions of a rigid rotor . . 72.3 Centrifugal distortion 102.4 Structure determination 142.5 Nuclear quadrupole hyperfine interactions 202.6 Nuclear spin-rotation interactions 312.7 Internal rotation 343 Experimental Methods 393.1 Stark-modulated microwave spectroscopy. . 393.1.1 Theory 393.1.2 Instrumentation 413.2 Microwave Fourier transform spectroscopy 43iv3.2.1 Theory . 433.2.2 Instrumentation 494 The Microwave Spectrum of BrNC’80 574.1 Introduction 574.2 Experimental methods 584.3 Observed spectrum and analysis 594.3.1 Harmonic force field and structure 674.3.2 Discussion and conclusions 71S The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, SiH2C1 755.1 Introduction 755.2 Experimental methods 765.3 Observed spectrum and analysis 775.4 Discussion 866 The Microwave Spectrum of Tetrolyl Fluoride 906.1 Introduction 906.2 Experimental methods 916.3 Results and discussion 916.3.1 Prediction of transition frequencies 926.3.2 Selection rules 946.3.3 Assignments 967 Microwave Spectra of Metal Halides Produced Using Laser Ablation 1037.1 Introduction 1037.2 Experimental methods 1047.3 Observed spectra 108v7.3.1 AgC1 1087.3.2 A1C1 1107.3.3 CuC1 1107.3.4 mCi 1137.3.5 InBr 1157.3.6 InF 1187.3.7 YC1 1227.4 Analysis and discussion 1257.4.1 Conclusions 147Appendices 149A Spherical Tensor Relations 149Bibliography 151viList of Tables2.1 Character table for the point group D2 and symmetry species of rotationaland direction cosine fullctions 102.2 Selection rules for rigid asymmetric top rotational transitions 114.1 Measured transitions of79BrNC18O(in MHz) 634.2 Measured transitions of81BrNC’0 (in MHz) 654.3 Spectroscopic constants of BrNCO 684.4 Structural parameters of bromine isocyanate 704.5 Centrifugal distortion constants of BrNCO 714.6 Spectroscopic constants of the ground state average structure of BrNCO 724.7 Principal values of the bromine quadrupole coupling tensor 735.1 Measured transitions of28SiH35C1 845.2 Chlorine nuclear quadrupole coupling constants of SiH2C1 875.3 Comparison of 35Cl principal quadrupole coupling constants and Si-Clbond lengths 896.1 Measured transitiolls of tetrolyl fluoride 996.2 Spectroscopic constants of tetrolyl fluoride 1006.3 Measured transitions of‘3C-tetrolyl fluoride 1017.1 Measured transitions of AgC1 1097.2 Measured hyperfine components of the J=1-0 transition of A1C1 1127.3 Measured transitions of CuCl 114vii7.4 Measured transitions of mCi. . 1167.5 Measured transitions of lnBr 1197.6 Measured hyperfine components of the J=l-0 transition of‘15n’9F . 1227.7 Measured transitions of YC1 1247.8 Spectroscopic Constants of AgC1 1277.9 Spectroscopic Constants of A1C1 1287.10 Spectroscopic Constants of CuC1 1297.11 Spectroscopic constants of mCi 1307.12 Spectroscopic constants of InBr 1307.13 Spectroscopic constants of 115n’9F 1317.14 Spectroscopic constants of YC1 1317.15 Ratio of Cl quadrupole coupling constants 1337.16 Ratios of nuclear spin-rotation coupling constants 136viiiList of Figures2.1 Correlation between free (V3=O) and hindered (V3 —* co) threefold internalrotor states 373.1 Schematic cross-section of a Stark cell 423.2 Schematic diagram of the Stark-modulated microwave spectrometer. . 443.3 Schematic circuit diagram for the cavity MWFT spectrometer 513.4 Schematic pulse sequence diagram for the cavity MWFT spectrometer 554.1 Rotational energy levels of79BrNC’80 614.2 The molecular structure of BrNCO, given in the principal inertial axissystem of79BrNC16O 745.1 Rotational energy level diagram of28SiH35C1 795.2 Schematic diagram of the 918 — 827 transition of28SiH35C1 815.3 Portion of the— 827 rotational transition of28SiH35C1 826.1 The m =0, 44-33 transition of tetrolyl fluoride 977.1 Schematic diagram of the arrangement of the cavity MWFT spectrometerand the Nd:YAG laser 1057.2 Schematic diagrams of the nozzle caps 1077.3 A portion of the J=1-0 rotational spectrum of 27A13C 1117.4 The J=2-1 rotational transition of 115n79Br and“51n81Br 117ix7.5 TheF1=3.5-4.5 hyperfine component of the J=1-O rotational transition of115n’9F 1217.6 The overlapped F=2.5-1.5,3.5-2.5 hyperfine components of the J=2-1 rotational transition of 89Y35C1 1267.7 Schematic diagram of the energies of the valence molecular orbitals of A1C1 1387.8 Schematic diagram of the energies of the valence molecular orbitals of CuC1 1417.9 Schematic diagram of the energies of the valence molecular orbitals of anindium (I) halide, InX 1437.10 Schematic diagram of the energies of the valence molecular orbitals of YC1. 146xAcknowledgementThis thesis would not have been possible without the guidance and encouragement of mysupervisor, Mike Gerry, who was occasionally more certain about the eventual productionof this thesis than I was. I would like to thank him for being a great supervisor and anice person.I’ve been fortunate to work in a lab constantly filled with experienced (and German-speaking) post-docs. I would like to thank all of them for their help: Wolfgang Jäger, forfixing 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. Iwould also like to thank Yunjie Xu for setting a good example, and Beth Gatehouse andKaley Walker for any times they asked me questions that I could answer, making me feellike 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 coupling constants come from; Mark Barnes and Photos Hajigeorgiou, for trying to explainthose electronic spectroscopy papers to a l+state spectroscopist; Chris Chan, for dealing with unhappy computers and overheated electronics; Ilona Merke, for trans-Atlanticfits of the dichlorosilane data in the midst of finishing her own thesis; Mike Pungente, formaking sure that the synthesis of tetrolyl fluoride produced more than a black tar; BillHenderson of the Mechanical Shop, for making things correctly even when my instructions were wrong; and Gordon Burton, for advice on thesis production from the frontlines. I would like to thank NSERC for financial support.xiI thank my family, for their mostly-unquestioning support and because this is theonly part of the thesis they will read. I thank Anne Dunlop, for living with me for fouryears, taking me dancing, and thrilling me with Fun Facts, linguistic and otherwise. Ithank my e-mail friends for providing a tenuous link to the outside world. I thank thegraduate 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.xiiChapter 1IntroductionIn molecular spectroscopy, discrete molecular energy levels are investigated via transitionsbetween levels, which occur when a molecule emits or absorbs electromagnetic radiation.Microwave spectroscopy is that branch of molecular spectroscopy which uses radiationwith frequencies between GHz and —lOO GHz. For molecules in the gas phase, mostof the transitions which fall within this frequency range are between rotational energylevels, and so microwave spectroscopy is also referred to as rotational spectroscopy. Sincethe positions of these levels depend on the masses of the atoms and their geometricalarrangements in a molecule, microwave spectroscopy is an important source of structuralinformation for gaseous molecules. Most microwave spectroscopic studies can probe theground electronic and vibrational states, and this structural information is very usefulfor theoretical calculations of potential energy surfaces near the potential minimum.The Hamiltonian operator which governs the positions of the energy levels is parameterized, and the ‘molecular constants’ are those values of the parameters which lead tothe best representation of the experimentally-determined positions of the energy levels.These constants are obtained by least-squares fitting of the observed transition frequencies to an appropriate Hamiltonian. Often a ‘bootstrap’ method is used, which involvesfitting to observed transitions and predicting other transitions; the predicted transitionscan then be measured and included in the fit to improve the precision of the molecularconstants.Rotational energy levels may also be split or shifted by interactions between the1Chapter 1. Introduction 2rotational angular momentum J and another angular momentum. Hyperfine effects arecommonly observed in rotational spectra, the result of interactions between the rotationof the electron ‘cloud’ and one or more nuclear spins I. ‘Coupling’ of J and I mayoccur via different mechanisms, to produce nuclear quadrupole hyperfine effects and/ormuch smaller nuclear spin-rotation hyperfine effects. Another possible interaction is thatbetween the overall rotation of the molecule and some rotation which is internal to themolecule. In the studies described in this thesis, both hyperfine and internal rotationeffects in rotational spectra have been investigated.In Chapter 4, a microwave spectroscopic investigation of the unstable molecule bromine‘80-isocyanate, BrNC’80, is reported. This study follows an earlier one from this laboratory [1, 2] in which the microwave spectrum of BrNC’60was investigated. The structureof 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 transitionswhich were perturbed by second order nuclear quadrupole hyperfine interactions [3].The study of BrNC’80was a fairly conventional microwave spectroscopic investigation, using the well-established technique of Stark modulation [4, 5] to improve thesignal-to-noise ratio of weak microwave absorption signals. The other investigations described here have been motivated, at least in part, by the development of other microwavetechniques, especially that of pulsed molecular beam cavity microwave Fourier transform(MWFT) spectroscopy. This technique, first described in the early 1980s [6], allows formuch greater resolution of spectral lines than is possible in Stark spectroscopy. As isdetailed in Chapter 5, the microwave spectrum of dichiorosilane, SiH2C1 has been reinvestigated in order to resolve the copious nuclear quadrupole hyperfine structure whichcould not be fully resolved in an earlier Stark-modulated spectroscopic study [7]. Thecomplete Cl nuclear quadrupole coupling tensor has been determined, which has in turnChapter 1. Introduction 3provided information regarding the nature of the Si-Cl bond in SiH2C1[8].In addition to improved resolution, the supersonic expansion of the molecular beamemployed in cavity MWFT spectroscopy allows the study of molecules at low rotationaland vibrational temperatures. As such, cavity MWFT spectroscopy is a very usefultechnique when a microwave spectrum is potentially very crowded because of rotationallyand vibrationally excited molecules which are undergoing rotational transitions. Thisproblem is commonly encountered when molecules with unhindered internal rotation ortorsion are studied. In Chapter 6, the microwave spectrum of tetrolyl fluoride, CR3-CC-COF, is reported. As is described in that chapter, the spectrum of tetrolyl fluorideis very dense when recorded at room temperature using a Stark-modulated microwavespectrometer, and analysis would have been virtually impossible without some methodof selectively removing some of the torsionally-excited rotational transitions. Using themolecular beam, only the lowest torsional levels were populated, and the few remaininglines in the microwave spectrum could be assigned with relative ease, permitting anapproximate determination of the barrier to internal rotation of the methyl group intetrolyl fluoride.Finally, laser ablation has been used to create metal-containing molecules in thegas phase, as described in Chapter 7. Seven diatomic metal halides were formed usingthis technique, from solid metal target rods and halogen-containing gas samples. Themolecules were studied using cavity MWFT spectroscopy, and so even unstable metalhalides 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 firsttime. Where possible, these latter constants have been related to perturbations of theelectronic ground state caused by excited electronic states [9].Chapter 2Theory2.1 Energy levels of the rigid rotorThe Hamiltonian for a rigid rotating molecule may be expressed asHrot (2.1)where J, J,, and J refer to components of the rotational angular momentum Jalong the x, y, and z principal inertial axes, respectively. The rotational constants B9(g=x,y,z) are related to the principal moments of inertia 1 of the molecule, and aregiven in frequency units1 asB9.(2.2)The principal axes are usually labelled a,b,c such that Ia I, I. In this case, therotational constants are rewritten as Ba = A, Bb = B, B = C. All of the symmetricrotors discussed in this work are prolate (cigar-shaped), where the principal inertial axisa has both the largest rotational constant and the highest symmetry; in the discussionwhich follows, x,y,z will be transformed to a,b,c as z —* a,x —* b, y —* c. In moleculeswhich 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 rotorbasis functions JKM >, where K is the quantum number associated with the projectionof J along the molecule-fixed z axis and M is associated with the projection of J along1Unless otherwise stated, all energies in this thesis will be given in frequency units.4Chapter 2. Theory 5the space-fixed Z axis. Both K and M may take integral values from —J to +J. In thesymmetric rotor basis, the non-zero matrix elements of the squared angular momentumoperators are given as:<JKM IJ 2 JKM> K2 (2.3)<JKMIJ2IJ > = <JKMIJJK >= [( +1) - K2] (2.4)<J,K+2,MIJ2I M> = —<JK+2MIJ2JKM>[J(J +1) — K(K+1)] [J(J +1) — (K±1)(K+2)]4.(2.5)The matrix elements of Hrot are thus given by<JKM Hrot JKM> = AK2 +B + C [J(J + 1) — K2] (2.6)<J,K±2,MHrotJKM> =BC [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 areindependent of M.For rigid linear molecules, there is no rotational angular momentum about the zasymmetry axis, and hence K = 0. Since B=C, the off-diagonal elements of Hrot disappear, and the energy levels may be given as a function of J:Ej<JIHrtIJ>B (J+l). (2.8)For a rigid prolate symmetric top, an A dependence is introduced to the rotational energysince K may now be non-zero. Again, Hrot is diagonal since B=C, and the energy levelsChapter 2. Theory 6depend on both J and K:EJK = <JK Hrot JK> (2.9)= BJ(J + 1) + (A — B)K2. (2.10)For an asymmetric rigid rotor, ABC and matrix elements of Hrot which areoff-diagonal by 2 in K are non-zero. Hrot may be simplified by applying a Wang transformation [11], which results in a new set of basis functions, related to the symmetrictop basis functions asI J0> = JO> (2.11)JK> = [ JK > + J, —K>] (2.12)IJK> HJK>—IJ,—K>]. (2.13)In this new basis, Hrot consists of four smaller block diagonals, which may more easilybe diagonalized. As in the original Hamiltonian, elements which are even and odd in Kdo not interact, and so the four submatrices which must be diagonalized may be denotedE+,E,0+, and 0-, where E and 0 stand for even and odd K, respectively; E+ and0+ involve symmetric combinations of symmetric top basis functions (see Eq. 2.12) andE and 0- involve antisymmetric combinations (Eq. 2.13).Since the Hrat matrix is not diagonal in K for an asymmetric rigid rotor, K isno longer a ‘good’ quantum number. For a near-symmetric prolate or oblate rotor,eigenfunctions of Hrot will strongly resemble the symmetric top basis functions. Thiscorrespondence is used to label the 2J+1 asymmetric top wavefunctions which exist foreach value of J. A common notation is JKaKc where Ka is the value of K in the limitingprolate case (z —* 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 invarious ways. The degree of asymmetry may be represented by the Wang asymmetryChapter 2. Theory 7parameter, b, where [11]C-Bbp=2ABC; (2.14)b=0 corresponds to a prolate symmetric rotor, and b= —1 corresponds to an oblatesymmetric rotor. For a prolate near-symmetric rotor (b0), the rotational Hamiltonianmay be rewritten in terms of b asHrot= B + CJ2+ [A — B H- Cj Hwang (br), (2.15)whereHwg (l)) = a 2 + b(J 2— b2). (2.16)The reduced energy W(b) corresponding to Hwg (br) may be expressed in terms ofan infinite series in b, which depends only on b and the quantum numbers for a givenasymmetric rotor energy level. Analytical expressions for and values of W(b) have beentabulated [10, 12]. For more asymmetric molecules, rotational energies must be calculatedby diagonalizing the Hrot matrix.2.2 Selection rules for rotational transitions of a rigid rotorThe molecular rotational transitions reported in this thesis are caused by a coupling ofthe electric component of an external electromagnetic radiation field with the electricdipole moment t of the molecule. When the external radiation oscillates at a frequency1mn, corresponding to the frequency of a molecular rotational transition from state m tostate n, the probability that the molecule in state m will undergo this transition is givenbyPrn-n = p(i’mn)Bm_*n, (2.17)Chapter 2. Theory 8where P(Vmrj is the density of the radiation at frequency mn Bm÷n is the Einsteincoefficient 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 ponto the space-fixed F axes, and are given in the basis of rotational eigenfunctions of themolecule. However, since the electric dipole moment is determined by the positions ofelectrons and protons in the molecule, it is a molecule-fixed quantity, and its space-fixedcomponents will change with rotation. It is thus necessary to transform matrix elementsdefined with respect to the space-fixed axis system to elements defined with respect tothe molecule-fixed axis system, using direction cosine matrix elements.If the components of p projected onto the principal inertial axes of the molecule areconsidered to be constant with rotation, then<fl lIFIm > FgI> g =x,y,z, (2.19)where ‘Fg is the direction cosine between the space-fixed F axis and the molecule-fixedg axis. Using symmetric top basis functions, the direction cosine matrix elements maybe factored as follows:<J’K’M’ JKM> = < J FgI J>< J’K’ 4F9 JK >< J’M’ JM>.(2.20)Analytical expressions for the values of the matrix elements on the right-hand side ofEq. 2.20 have been tabulated [13].Most microwave spectroscopic experiments use plane-polarized radiation, and thusthe molecule’s electric dipole moment will interact with radiation oscillating in only onespace-fixed direction. If this direction is chosen to be the Z axis, then the properties ofChapter 2. Theory 9the direction cosine matrix elements place the following constraints on allowed rotationaltransitions 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 withare 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 symmetryaxis.Selection rules for rotational transitions in asymmetric molecules are somewhat morecomplicated. J and M are still good quantum numbers, and the selection rules zSJ=0,±1and zM=0 still hold. However, the selection rule zK=0 has no meaning, since K is nolonger a good quantum number.As was stated in Section 2.1, the asymmetric rotor Hamiltonian may be factoredinto four submatrices upon application of the Wang transformation. The asymmetricrotor wavefunctions form an irreducible representation in the group D2, whose operationscorrespond to an identity operation (A) and rotations of 1800 about each of the threeprincipal inertial axes (C, C, and Cf). Upon application of a Wang transformation tothe Hrot matrix, each of the four submatrices is of a different symmetry species accordingto this group. The character table for D2 is given in Table 2.1, along with the symmetriesof the submatrices for both even and odd values of J. The symmetries of the rotationalstates 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, theChapter 2. Theory 10E ca rib riC Jeven Jodd Ka KC Fg2 ‘—2 ‘—‘2A 1 1 1 1 E E— e e —Ba 1 1 1 1 E C 0 ZaBb 1 —1 1 —1 0 0 ° ° ZbB 1 —1 —1 1 0 0 e ZcTable 2.1: Character table for the point group D2 and symmetry species of rotationaland direction cosine functionsmatrix element <J’KK I JKaKc > must be both non-zero and symmetric withrespect to all operations of the group D2. Since<J’IçK ILzI JKaKc> i’a< J’KK IZaI J’a’c > + [Lb< J’IcK IZbI JKaKc>+Itc< J’KK IzcI j1’a’c>, (2.24)this requires both that 1u be non-zero and that <J’KK I I jKaKc> be totallysymmetric for g=a, b, or c (note that all three components of it may be non-zero in anasyn-u-netric top.) For example, if a molecule has a component of its permanent electricdipole moment along its a principal axis, then the symmetry of Za matrix elementsmust be considered. Since Za is of symmetry Ba, if the product of the symmetries ofJKaKc> and I J’KK> is also Ba, then the direction cosine matrix element will benon-zero and a transition will be allowed. Because the intensity of this transition willdepend on the value of [ta, it is known as an ‘a-type’ transition. As can be seen fromTable 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 Table 2.2.2.3 Centrifugal distortionReal molecules are not rigid, and therefore the rigid rotor model for molecular rotation isonly an approximation. Interatomic bonds will stretch as the molecule rotates and anglesChapter 2. Theory 11Ka KctLaO ee—eo,oe-+oo1LbO ee—*oo,eo-+oej&LO ee+.-*oe,oo—eoTable 2.2: Selection rules for rigid asymmetric top rotational transitionsbetween the bonds will change, thereby distorting the molecule. This centrifugal distortion increases with increasing rotational angular momentum, and must be accounted forby 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 = Hrot + Ha (2.25)= BJ2+ BJ2+ BJ 2+ r5JJ JJ5, (2.26)where Br, B, and B are as given in Eq. 2.2, and c,/3,-y,6 = x,y, or z. In principle, thereare 81 possible rs; however, symmetry considerations reduce the number of meaningful rsto nine [14]. Commutation rules for angular momenta may then be applied to eliminateterms of the form Tag, folding their effects into the other coefficients. Hsemirigid maythen be written asHsernirigici = BJ 2 + 2 + BJ 2 + >TcrJcr 2j 2 (2.27)whereB = B + — — (2.28)B, = B + — — (2.29)= B + — — (2.30)andT = (2.31)Chapter 2. Theory 12Ta = + 2r) (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 willbe dropped from the effective rotational constants.As was pointed out by Watson [15, 16, 17], not all the coefficients in Hsemirigici maybe determined from observed energy levels. In order to decide which coefficients or combinations of coefficients are determinable, he proposed subjecting Hsem.jrjgjd to a unitarytransformation, which by definition leaves the eigenvalues of Hsejñgja unchanged. Thosecombinations of coefficients which are changed only to a very small extent by the transformation are eigenvalues of Hsemirigid and can hence be determined from the observedenergy levels. A ‘reduced’ Hamiltonian is obtained by choosing a unitary transformationwhich leaves only determinable combinations of coefficients. These combinations werefound to be [15]= B — 2T (2.33)= B— 2T (2.34)13,, = B,,— 2T (2.35)(2.36)(2.37)T,, (2.38)T1 = (2.39)T2 = (2.40)Since there are only eight determinable combinations of coefficients to fourth orderin J, only eight of the nine possible linear combinations of the rotational and centrifugalChapter 2. Theory 13distortion constants in Eq. 2.27 are independent. By choosing the unitary transformationcorrectly, one of these linear combinations can be set to zero. Watson’s first choice ofunitary transformation gave the following reduced Hamiltonian [15]:Hrot = (2.41)Hd = —LJJ ‘— /-JKJ 2jz 2 — LKJz ‘ 26J 2(JT 2 — j, 2)6K[Jz 2(J 2 — j 2) + (J 2 — j 2)J 2], (2.42)This is known as the Watson A (asymmetric) reduction, which is the most commonlyused reduction. However, the A reduction breaks down as the molecule becomes nearsymmetric and B — B approaches zero. By choosing a different unitary transformationwhich eliminates a different linear combination of Bas and Ts, One obtains the WatsonS reduction, which may be used for nearly symmetric molecules [18]:Hrot (S) = BJ2+ BJ 2 + BJ 2 (2.43)H,d. (S) = —DJ” — DJKJ 2J 2 — DKJ4+d1J2(J +J—)+d(J++J_, (2.44)where= Jx — iJ (2.45)J_ = Jx + jJ. (2.46)Centrifugal distortion constants are more than empirical parameters used to improvethe fitting of spectroscopic data. If the potential V governing intramolecular motions isassumed to be harmonic, V can be expressed as [10]V = (2.47)Chapter 2. Theory 14where fj is a harmonic force constant and R, and R, are internal displacement coordinates; 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 bendingmodes. For a non-linear molecule, there are 3N —6 internal displacement coordinates, aswell as 3N Cartesian displacement coordinates. R is related to the Cartesian coordinatesX3 by the B matrix, andR=B1X. (2.48)If the molecule is rotating but not vibrating, the only effects of V will be in therestoring potential forces which balance the centrifugal forces caused by rotation, and socentrifugal distortion constants are directly linked to the harmonic force constants. Ifis 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 obtainedfrom vibrational spectra to determine harmonic force fields [19].2.4 Structure determinationAs was shown in Eq. 2.2, the rotational constants obtained in microwave spectroscopy areinversely proportional to the principal moments of inertia of a molecule. These principalmoments are obtained by diagonalizing the inertia tensor, whose components are relatedto the centre-of-mass system coordinates (xj,yj,zj) of the atoms as follows:I = mj(y+z) (2.50)Chapter 2. Theory 15I, = m(x + z) (2.51)I, = > m(x + y) (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 atomiccoordinates. However, although rotational constants and hence moments of inertia aredetermined to high precision by microwave spectroscopy, molecular structures determinedfrom these constants are of much lower precision. This is a result of the corrections thatneed to be made for vibration-rotation interactions in the molecule.Since rotational transitions are measured for a particular vibrational state, the moments of inertia obtained represent effective values of r2, where r is an atomic position,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 structurewhich is generally of most interest, since it may be used to improve vibrational potentialfunctions which are obtained theoretically. However, the structure of the molecule in allvibrational states will differ from the re structure for two reasons: < r >re, since vibrational potentials are anharmonic, and < r >2L< r2 > 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 vibrationaleffects. For a diatomic molecule, the rotational constant as a function of the vibrationallevel 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 byA = Ae—>Za(vs +d8/2) +... (2.56)Chapter 2. Theory 16with corresponding expressions for B and G; summation is over all vibrational modes s,where d5 is the degeneracy of the mode. If enough vibrational levels are studied, a valuescan 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 rarelydone for polyatomics, although it is commonly done for diatomics [20].Instead, the effects of vibration are accounted for in various ways when a molecularstructure is reported. The simplest structure to obtain is the ‘effective’ or r0 structure(or r structure for a general vibrational state v.) This may be determined directly fromthe rotational constants if the number of structural parameters to be determined is lessthan or equal to the number of rotational constants obtained; for example, the bondlength r in a diatomic molecule may be determined from the rotational constant usingthe relationB= 2’ (2.57)8K ,irwhere t is the reduced mass of the molecule, given by= m12 (2.58)m1 + m2In general it is not possible to determine an r0 structure for an asymmetric moleculefrom the three rotational constants available from microwave spectroscopy, unless assumptions are made concerning some structural parameters. However, if rotational constants can be obtained for different isotopomers of the molecule, it may be possibleto determine a ‘substitution’ or r8 structure using a series of equations developed byKraitchman [21]. Bond distances and angles must first be assumed to be unchangedwhen isotopes are substituted at different atomic positions. Components of the inertiatensor for the substituted molecule (S) may then be calculated in terms of the principalinertial axes of the ‘parent’ molecule (F, usually the most abundant isotopomer). Ifsubstitution occurs at the atom with centre-of-mass coordinates x, y, and z in F, thenChapter 2. Theory 17the components are given by [21]I = I + p(y + z2) (2.59)I = I + p(x + z2) (2.60)= I + p(x2 + y2) (2.61)I = —pxy (2.62)I = —pxz (2.63)= —pyz, (2.64)where is a component of the inertia teilsor of S, ‘g is a principal moment of inertia ofF, and p is the reduced mass for isotopic substitution, given in terms of the total massesof the two molecules by= Mp(Ms— Mp) (2.65)The inertia tensor is then diagonalized to obtain principal moments of inertia for Sin 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 usedto 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) aregiven byI= [()(i+(2.66)I= [(Ii; ‘ (1+ . (2.67)Five moment equations exist for the parent planar asymmetric top:I = (2.68)Chapter 2. Theory 18I = (2.69)I,= —>Zmxy = 0 (2.70)= 0 (2.71)>mjyj = 0, (2.72)and hence x and y coordinates may be determined for two of the N atoms if the positionsof the remaining atoms are known. A minimum of (N — 2) isotopic substitutions atdifferent atomic sites is thus required to determine the ‘full’ substitution structure. Fora non-planar asymmetric top, nine moment equations exist (Eqs. 2.50-2.55 and threecentre-of-mass conditions), and so (N — 3) substitutions are required. The number ofrequired substitutions is also reduced if the positions of some atoms are constrained bymolecular symmetry.Substitution structures are based on the assumption that the molecule is rigid, anassumption that can be seen to break down when two different isotopes are substitutedat the same position, giving two slightly different substitution structures. Using theplanarity 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 substitute 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 +z0). It is not possible to determine a prioriwhich combination will give the most accurate substitution structure.Costain [22] has shown that most of the uncertainty in substitution structures arisesfrom forcing the structures to reproduce the measured moments of inertia (given for theChapter 2. Theory 19planar 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 eachatom in the molecule and the centre-of-mass coordinates for all atoms are determinedindependently. Substitution structures determined in this way have also been found tobe nearly independent of the choice of isotopic species used [22]. However, this is a time-consuming 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 thesubstitution method, and an r3 structure is usually closer to the equilibrium structurethan an r0 structure: for a diatomic molecule [22],r8Te + To. (2.74)However, the r3 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 groundvibrational state. If the molecule had a harmonic vibrational potential, the r structurewould be the same as the equilibrium structure, but would still differ from the r0 structurebecause of the harmonic vibration effects which make < r >2z4< r2 >. The o constantsin 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= A0 + d8ch, (2.76)Ae_d81 (2.77)Chapter 2. Theory 20An r structure may thus be obtained from an effective structure if harmonic force constants are known (see Section 2.3), from which values of ah are calculated [19].2.5 Nuclear quadrupole hyperfine interactionsThe electrostatic Hamiltonian for a system of interacting nuclear and electronic chargedensities is given classically as [23]Heiectrostatic= j pn(rn) [J dVe] dv. (2.78)where p(r) is the nuclear charge density of the nuclear volume element dv at positionr, and pe(re) is the electron charge density of volume element dye at position re; forconvenience, the centre of the nucleus under consideration is taken as the origin of thecoordinate system, and interactions of this nucleus with other nuclei in the molecule andwith the extramolecular surroundings are ignored. Eq. 2.78 may be expanded in termsof 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 kHeiectrostatic = 2k 1q(_) 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 interaction between point charges, separated by distance re. The k=1 term accounts forinteractions between electric dipoles in the nucleus and the surrounding electric field ofthe molecule; this term disappears for nuclei in their ground states, which do not possessa permanent electric dipole moment.Chapter 2. Theory 21In microwave spectra, only those effects due to the k=2 term of the electrostaticHamiltonian are observed, as a result of interactions between the electric field gradient surrounding the nucleus and the nuclear quadrupole moment. The Hamiltonian fornuclear quadrupole coupling is given byHQ = f p, r Y(O, ) dv, f Y2)(Oe, e) dVe. (2.81)q=—2 “ ereHQ may also be written as the scalar product of two second-rank spherical tensors,HQ = Q(2).V(= (2.82)whereQ2)= (2.83)andv2=Y(Oe, ) dv (2.84)represent components of the nuclear quadrupole moment tensor and the electric fieldgradient tensor, respectively.Neglecting centrifugal distortion effects, the microwave spectrum of a molecule withnuclear quadrupole coupling may be analyzed using a Hamiltonian which is the sum ofthe rotational Hamiltonian and the quadrupolar Hamiltonian,H Hrot +HQ. (2.85)In order to transform the classical expression for HQ given in Eq. 2.81 into a quantummechanical expression, it is necessary to derive expressions for Q (2) and V (2) whichdepend 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. Theory 22Consider first the nuclear quadrupole tensor, which will depend on the nuclear spinquantum number I. By the Wigner-Eckart theorem (Eq. A.3, Appendix A), it suffices to calculate only one component of Q (2), from which the rest may easily be derived. If the q=O component is considered, the spherical harmonic Y2(O, q) is simply(5/161r)h2(3cos2 O — 1), where O, is usually defined with respect to the symmetry axisof the (classical) nuclear charge distribution. The nuclear quadrupole moment operatorQ is conventionally defined asQ = jpn(rn)r(3cos2On_1)dvn, (2.86)with the corresponding expectation valueQ = <I,Mj=IQII,Mi=I> (2.87)= II Q2I II>. (2.88)Note that the nuclear quadrupole moment Q is defined for the state in which the (constant) 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 Q, which depends on the average value of (3 cos2 O, — 1); Q is positive for prolate charge distribution,zero for spherical charge distribution, and negative for a nucleus in which the charge isdistributed in an oblate fashion about the symmetry axis.Similarly, the appropriate operator may be chosen to correspond to the electric fieldgradient about the nucleus, which will depend on both the electronic structure and therotational state of the molecule. The electric field gradient coupling constant qj’j isdefined asqj’j = <r’, J’, M’ = J Vzzi r, J, M = J>, (2.89)where I r, J, M = J> is the rigid rotor basis function with maximum projection of Jalong a space-fixed Z axis. The electric field gradient operator Vzz is defined as theChapter 2. Theory 23second derivative of the potential at the nucleus, surrounded by a generalized electroncharge density distribution pe(re):= () = jPe) (3cos2O— 1)de. (2.90)Thus from Eq. 2.84 the electric field gradient coupling constant may be defined in termsof 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 arecalculated 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 asI+J =F,and MF is the projection of F on a space-fixed Z axis; r refers to all other relevantquantum numbers. Using Eq. A.5 given in Appendix A, matrix elements of HQ aregiven in this basis by<T’J’IF’M IHQ I TJIFMF>=J I F< Q (2) >< r’J’ y (2) rJ>.(I J 2J(2.92)The reduced matrix elements < Q (2) II I> and <r’J’ II V (2) II rJ> may be calculated using the Wigner-Eckart theorem as—1I I 2 I’\< II Q (2) = I I < II IQ2>I II> (2.93)0 I)Chapter 2. Theory 24/ —1if I 2 i\= —I I eQ (2.94)2_ o iI —1I J’ 2<r’J’ v (2) rJ> = (_)J’J f <r’J’J rJJ> (2.95)0 J)—1if J’ 2 J’\= (_Y ‘— I I qj’j. (2.96)2\—J 0 J)Matrix elements of HQ are thus given by<T’J’IFMF HQ I TJIFMF>/ —1 / —1 /ill 21\ fj’ 2J\ IJ’IF= 6FFlMFM(—) I I I I eQqj’j.—IOI) \\—JOJ) (I 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 whenJ=0 or 1/2.However, it must be recognized that qj’j is not a constant, but is instead dependenton the rotational state of the molecule, since qj’j is defined in terms of a space-fixed Zaxis: rotation will caused the electric field due to the electrons in the molecule to rotatein the space-fixed frame. Hence, qj’j must be transformed to the molecular frame, usingone of two approaches. When using symmetric top basis functions I JKM>, it is oftenconvenient to transform components defined in the space-fixed frame, to componentswhich are defined in the molecule-fixed frame and which are independent of therotational state, using the relation [23]= Dr (2.98)Chapter 2. Theory 25where D is a (2k+1)-dimensional matrix representation of the rotation operator, anddepends on the relative orientation of the space-fixed and molecule-fixed frames. Theelectric field gradient coupling ‘constant’ qj’j is then given in terms of expectation valuesof molecule-fixed components of V’ (2) [23]:IJ’2 J’\ , , IJ’ 2qj’j = 2 -K-v /(2J’ + l)(2J + 1 I I v’J 0 —J ) q’ —K’ q’ K /(2.99)where q’=K’ — K. In the molecular frame, with molecule-fixed axes x, y, z, the components of v’ (2) are given in a Cartesian representation as:= (2.100)V’ = + i14) (2.101)V’ = — V + 2iV), (2.102)where Vfg represents a component of the electric field gradient, evaluated at the nucleus:Vf9 ——1i’9fögJ(2.103)The molecule-fixed axes are usually taken to be the principal inertial axes. Terms of theform eQVf9 (f, g = x, y, z) will appear in HQ, and are often abbreviated to the nuclearquadrupole coupling constantsOften molecular symmetry permits elimination of certain Cartesian components ofv, (2) It is also helpful to recall that Laplace’s equation dictates that the field gradientvanish at the nucleus, i.e.V + V, + Vi,. = 0. (2.104)As a result of this relationship, only two of the three diagonal coupling constants areindependent; spectroscopic data are commonly fit to Xzz and X — x. For linearChapter 2. Theory 26molecules, or for symmetric tops with the coupling nucleus on the molecular symmetryaxis, the field gradient will be cylindrically symmetric about the molecular symmetryaxis, and so xxx— x,=O. The molecular constant q is defined asq = V (2.105)and eQq is the pertinent spectroscopic constant.If the electronic charge distribution surrounding a quadrupolar nucleus (i.e. a nucleuswith I1, QE0) is not symmetric about the x,y,z principal inertial axes, as in an asymmetric molecule or a symmetric molecule where the nucleus does not lie on the symmetryaxis, 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 chargedistribution 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 toonly 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 chargedistribution 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 qjjand q for linear molecules and symmetric tops: [25]qjj= 2j+ 3 [J(J 1) — i] for symmetric tops (2.106)=—for linear molecules:K = 0. (2.107)2J+3For a symmetric top with one quadrupolar nucleus on the symmetry axis, the first orderquadrupole coupling energy may be calculated for a state KJIFMF> asE)= <KJIFMFIHQIKJIFMF>Chapter 2. Theory 271 [c(c + 1) — J(J + 1)1(1 + 1)] J 31(2= eQqJ(2J — 1)1(21— 1) 2J +3 J(J + 1) — 1 , (2.108)whereC = F(F+ 1)—J(J+ 1) —1(1+1). (2.109)In an asymmetric rotor basis, it is instructive to use direction cosines to transformVzz, and hence qj’j, to the molecule-fixed frame. Using this approach,VZ = + Vyzy + Vzzyz+2Vxy4’z z + 2VyzzyFzz + (2.110)where Vjg is defined as in Eq. 2.103. Unlike the direction cosines, the VfgS are molecular constants and hence are unchanged by rotation, and calculation of qj’j amounts toevaluating direction cosine matrix elements for the appropriate rotational states.This calculation is greatly simplified if the symmetry properties of the direction cosinesand the asymmetric rotor basis functions JKaKcM > are considered, especially if firstorder perturbation theory is sufficient to account for nuclear quadrupole coupling ina given molecule. If this is the case, then only diagonal matrix elements of HQ arecalculated, and the relevant electric field gradient coupling constants are given byqjj = Vjg< JKaKcM ZfZgI JKaKcM> f,g =x,y,z. (2.111)fgOnly those matrix elements which are totally symmetric will be non-zero; since the matrixelements are diagonal, this amounts to requiring that the products of direction cosineoperators, ZfJZg, be totally symmetric. Since only and are of symmetryspecies A in the point group D2 (see Table 2.1), qjj reduces toqjj = Vccx < JKaKcM I JKaKcM> +V <JI(KM I zy I JKaKcM>+Vzz < JKaKcM I JKaKcM>. (2.112)Chapter 2. Theory 28The expectation values of the squared direction cosines in a given rotational state maybe related to the expectation values of squared components of the rotational angularmomentum operator J [10]:<g> (J+ 1)(2J+3)2>+2J+3 g = x,y,z. (2.113)Using Eq. 2.113, the reduced energy Wb corresponding to Hwg (Eq. 2.16), Laplace’sequation (Eq. 2.104), and <j 2 >= J(J + 1), one obtainsJ (3<J2> ‘qjj = VZZ 2J +3 J(J + 1) — + b V J(J + 1) (2.114)For a given state I TJIFMF>, the first order quadrupole coupling energy E) is givenby < 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 . Bynon-degenerate perturbation theory,E(JKaKcIFMF)<J’KK VxyZxZy + VyzZyZz + VzxzI JKaKc> 2E(J’IçKI’F’M) — E(JKaKc1FMF)+terms in g, (2.115)where E(JKaKcIFMF) is the rotational energy for the state JKaKc > (withoutnuclear quadrupole coupling) and E(JKaKcIFMF) is the second order correction.Summation is carried out over all states J’IçKI’F’M1> I J1aKcI1’MF>. Again,molecular symmetry may allow elimination of V, V, or V. The most significantcontributions 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 symmetry so that the matrix elements containing the products of direction cosines are non-zero.Chapter 2. Theory 29For a molecule whose plane of symmetry is the ab plane, this latter restriction places thefollowing constraints on interacting states I JKaKcIFMF> and J’K,KI’F’M >:J’—J = O,+l,+2IcK ÷- KK = ee 4-* oe, eo —* 00(see Table 2.1.) Another restriction arises upon consideration of matrix elements ofthe 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 predictedupon examination of the energy levels for a given asymmetric top molecule in order tofind near-degeneracies between levels of appropriate symmetry. These perturbations maybe 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 arewell-separated from the principal inertial axes. A large quadrupole moment Q will alsoamplify the effects of perturbations. Hrot must be augmented by a centrifugal distortionHamiltonian in the final analysis, as small shifts in the rotational energy levels can causelarge 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 torotation, as well as to each other. The coupling scheme used to determine matrix elementsof HQ depends on the nature of the two nuclei. For two nuclei having similar or equalnuclear quadrupole coupling constants, it is convenient to use the coupling schemeIi +12 = II+J =F,where I and 12 are the nuclear spins of the two nuclei, in order to create basis functionsof the formIJFMF>. The effect of the two quadrupolar nuclei is additive, andChapter 2. Theory 30HQ may be written asHQ = HQ (1) + HQ (2). (2.116)From application of Eqs. A.5 and A.6 of Appendix A, the matrix elements of HQ (1) aregiven by<T’I1I2’J’F M Q (2) (1). V (2) (1)1TII2JFMF>= FF,6MFM(_)21+J+ 11+12j+ l)(21’+ 1 { ‘ ‘ F’ } { i1 P i2 }•< r’J’ V (2) (1)W rJ >< I’ W Q (2) (1)W I >, (2.117)=+1+2+ 1)(21’ + { ‘ ‘ } { }eQ(1)qjij(1), (2.118)0 Ii Iwhere the reduced matrix elements have been evaluated as in the case of only one couplingnucleus. An equivalent formula applies to the matrix elements of HQ (2), with 1 and ‘2interchanged.For a molecule containing two nuclei with very different quadrupole coupling constants, it is more convenient to use the coupling scheme1+J = F1F1 +12 = F.In order to evaluate HQ (1) for the nucleus with much stronger coupling, basis functionsof the form r’ J I F M1 > may be used, leading to a simplified expression for matrixelements which determine the predominant part of the quadrupole interaction:<r’J’I1FM Q (2) (1). V (2) (1) TJIFMF >Chapter 2. Theory 31— J’+11+F’ I Ii J’ Ff— 6F1F6MF1M. (—) 1(J I 2—1,2 J’ (Ii 2 i\I I I I eQ(1)qjij(1). (2.119)0 J) ‘i 0 1)The smaller effect of coupling‘2 and F1 must be evaluated using basis functions of theform r’J’I1FI2’Mj >:<r’J’I1FI2’M Q (2) (2). V (2) (2) TJI1FIMF>Fi+12+F’F 12 F’= 6FF’8MFM()( 1 F1 2< r’J’I1F V (2) (1)11 rJIF >< ‘2 Q (2) (1)11 ‘2 >= 6FF,6MFM(_)2F1 2+F+J+I1(F 1)(2F + i){F‘2< r’J’ V (2) (1)I TJ >< 1 1 Q (2) (1)11 12 >= 8FF,MFM(_)2F1+I2+F+J+h1V(2F + 1)(2F + 1 {2.6 Nuclear spin-rotation interactionsThe Hamiltonian for the interaction of a magnetic dipole p with a magnetic field H isgiven byHmag = 1L H. (2.123)In a rotating molecule, such an interaction exists between the magnetic dipole momentsof the nuclei and the magnetic field induced by the rotating protons and electrons. The1 ( J’(2.120)12 F’ J’ F IF1 2 J(F1 J 2(2.121)‘2 F’ J’ F IF1 2 J(F1 J 2(2.122)—1‘2 2 12)eQ(2)qjij(2).12 0 ‘2Chapter 2. Theory 32nuclear magnetic moment depends on the spin I of the nucleus, as well as on the dimensionless gyromagnetic ratio gi for that nucleus:1tI= gI1NI, (2.124)where PN is the nuclear magneton. The precession of I is slow relative to the rotation ofthe molecule [10], so that u1interacts with an average magnetic field Heff in the directionof J. Heff may be expressed as [10]Heff < H> , (2.125)/J(J+ 1)where < H > 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 asHnuc.spin_rot = Mrjl J , (2.127)where the nuclear spin-rotation coupling constants MTJ depend on the rotational staterJ> under consideration, and are defined asMTJ (2.128)In addition to the rotation of positive and negative charges, contributions to the magneticfield about the nucleus arise from mixing of the ground electronic state and excited stateswith electronic angular momentum, a mixing which is brought about by rotation of themolecule. This effect may be represented as a second order perturbation of the electronicstates, 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; inChapter 2. Theory 33addition, symmetry about z makes h=h and <J,, 2 >=0. The nuclear spin-rotationcoupling constant for a nucleus with spin I is then given byM=J(J 1)kXX(< 2 > + <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 bedescribed by the vector equationI+J =F.Generally this coupling is weak, with splittings of IJF> levels often too small tobe resolved by microwave spectroscopy, and I and J may be considered to be goodquantum numbers. However, the coupling of I and J also occurs via much strongernuclear quadrupole interactions (see Sec. 2.5), and transitions between I IJF > levelshave the same selection rules for both mechanisms. Nuclear spin-rotation effects maythen be measured as small shifts of the expected nuclear quadrupole hyperfine structurepatterns.An even smaller hyperfine effect which is sometimes observed in microwave spectrais nuclear spin-spin coupling [27], which arises from a magnetic dipole-magnetic dipoleinteraction between two nuclei in a molecule. However, since this effect was not observedin any of the studies described in this thesis, it will not be discussed here.Chapter 2. Theory 342.7 Internal rotationIn addition to the overall rotation of a molecule, parts of the molecule may rotate relative to each other, a motion that is often described as a torsion. If a symmetric subgroup undergoes such a torsion relative to the rest of the molecule, interconverting structurally equivalent forms, this internal rotation may couple with the overall rotation ofthe molecule to affect noticeably the observed microwave spectrum.The effects of internal rotation on rotational energy levels depend on the potentialenergy function V(a) which governs the rotation of the subgroup through an angle arelative to the frame of the molecule. V(a) must be periodic in a, and the potentialfunction is usually expanded asV(a) = ajcos(kNa), (2.132)kwhere N is the number of equivalent configurations. As the internal rotor of interest inthis thesis is the threefold-symmetric methyl group, the remainder of this section will bedeveloped to treat the case where N=3. The reference energy level is usually shifted sothat V(a) may be written asV(a) = (1 — cos3a) + (1 — cos6a) +..., (2.133)where V3 is the threefold barrier to internal rotation and V6 is the sixfold barrier. UsuallyV6 << V3, and so the leading term in Eq. 2.133 is sufficient.The Schrödinger equation for internal rotation may then be written as—d2U(a)+ [(i — cos3a) — E} U(a) = 0, (2.134)where U(a) are the eigenfunctions and E is the total energy of internal rotation.‘r isthe reduced moment of inertia for the relative motion of the internal rotor (‘top’) andChapter 2. Theory 35the rest of the molecule (‘frame’), and is given by= ‘top’frame, (2.135)‘top + ‘framewhere 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 (V3=O), Eq. 2.134 has solutionsin ‘ ima— eE = m2, (2.137)where m=O, ±1, ±2 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 (V3 —* oo), a will be infinitelysmall, and cos 3a may be expanded in a Taylor series. V(a) is usually taken to be equalto the leading term in the expansion:V(a) = a2. (2.138)Eq. 2.134 then has solutions which resemble those for the harmonic oscillator (whereV(x)=kx/2), and the energy of internal rotation is given byE = h/3(v + )‘ (2.139)where v=O,1,2 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 V3 is finite, then the possibility of tunnelling exists and there are three types ofsolutions to Eq. 2.134, one of A symmetry and two of E symmetry. The A states are nondegenerate, and correspond to internal motions localized in one of the potential wells.The E states are degenerate, and correspond to internal motions with some tunnellingChapter 2. Theory 36character. The solutions for low and high barrier heights may be related, as is shown inFigure 2.1. Each v state splits into an A state and an E state as V3 falls, and each m statemust 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 torsionalbarrier is introduced. Transitions between A and E states are forbidden [28].The model usually taken to treat the coupling of internal and overall rotation isthat of a rigid symmetric top attached to a rigid frame; the top rotates about the bondconnecting it to the frame. In most cases, the remaining vibrational degrees of freedommay be ignored [28]. The Hamiltonian used to treat a molecule with both internal andoverall rotation is usually separated into three parts:H = Hrot + Htorsion + Hrot_torsion , (2.140)where Hrot resembles the usual rigid rotor Hamiltonian (see Section 2.1), describes 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 Htorsionare 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 aslightly asymmetric molecule with a light frame, the axis system is often chosen so thatthe symmetry axis of the top is one of the coordinate axes: this is known as the internalaxis method (lAM) [29]. Rotation-torsion coupling is then eliminated or minimized bycoordinate transformations. However, since the principal axis system is not used, termsinvolving products of inertia I (Eqs. 2.53-2.55) will remain, even in HrotAnother approach is the principal axis method (PAM [30, 31, 32]), in which theprincipal axes of the molecule are chosen as the axis system. Off-diagonal rotation-torsionterms are then larger than in the JAM, but the principal moments of inertia, which areunchanged by internal rotation, may be used. If j is the total angular momentum of theChapter 2. Theory±4±3±2±102fl V3V337rotor StatesFigure 2.1: Corre1atjo between free (J’O) and hindered (J threefold internal±5v3O2Chapter 2. Theory 38top about its symmetry axis, then H is given by [32]H = +J3J)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 asA’ = A+Fp (2.142)B’ = B+Fp (2.143)C’ = C+Fp (2.144)andh2F = (2.145)2rIt0j2——______— ,4x1top — ytop 2 146z yg top=g=x,y,z, (2.147)9with the direction cosine between the g principal axis and the symmetry axis of thetop. H may then be divided asHrot = A’J 2 + B’J2 + C’J 2 + Fpp(J1J + J J) (2.148)ijHtorsion = Fj 2 + V(a) (2.149)Hrot_torsion = —2F pJ j. (2.150)This division is particularly convenient for computational purposes, as symmetric rotorbasis functions may be used to calculate Hrot matrix elements, while the free rotor basisfunctions of Eq. 2.136 may be used to calculate matrix elements of Htorsjon.Chapter 3Experimental Methods3.1 Stark-modulated microwave spectroscopy3.1.1 TheoryWhen microwave radiation is passed through a gas sample and molecules in the sample undergo rotational transitions, only a small fraction of the radiation is absorbed.Microwave spectroscopy is an inherently low-sensitivity branch of spectroscopy for tworeasons: radiation densities p(v) vary as and microwave frequencies are relatively low;and rotational energy levels are closely spaced, resulting in a ‘smeared’ Boltzmann distribution of molecules over many rotational levels, leaving few molecules in any one levelto undergo a given transition. A method commonly used to increase the signal-to-noiseratio in microwave spectroscopy is Stark modulation, whereby only those signals whichare affected by an external electric field are recorded.The Stark effect is the result of interactions between and the permanent molecularelectric dipole moment pHstk =—it £. (3.1)The application of an external electric field breaks partially or fully the 2J+1-fold degeneracy of M, the projection of J onto a space-fixed axis Z; the result is that rotationalenergy levels are split by the Stark effect. Since has a fixed direction in space and jt isdefined in terms of the rotating molecular frame, p is conventionally given in terms of itsspace-fixed components, calculated using direction cosines as in Eq. 2.19. If e is assumed39Chapter 3. Experimental Methods 40to be of constant magnitude and pointing in the space-fixed Z direction, thenHStark = —e itLgzg. (3.2)g=x,y,zIn general, Stark effect energies are calculated to sufficient accuracy using non-degenerateperturbation theory, where H5tark is considered to perturb the eigenstates I JrM > ofHrotThe first order Stark effect energy is given byErk = —6 [tg< JrM zg1 JrM>. (3.3)gz,y,zFor asymmetric molecules, <JrM 4z9 JrM >=0 for all states, since this matrix element cannot be totally symmetric (i.e. of symmetry species A) in the group D2 (seeTable 2.1); a first order Stark effect is not observed. For linear molecules and symmetrictops, [L2f and z==O, soEk = jLE< JKM IzzI JKM> (3.4)— 1ueKM35— J(J+1)’where the direction cosine matrix element has been evaluated from Ref. [13]. Linearmolecules will have no first order Stark effect, since K=0.The second order Stark effect energy may be calculated as(2)—<JTMI11Stark J’r’M> 2Stark — (.)-‘-‘Jr —V’ V’ 2 21 < JrM Zg1 J’r’M> 12— Ld LdIg6 p p ‘ (.)JrIJT g -‘-‘Jr —where EJT is an eigenvalue of Hrot. Since EJ — is generally large relative to ,ue,Ek<E)k for a molecule in which both effects are present. In that case, relatively lowelectric fields are required to observe Stark splittings, and the magnitude of the splittingChapter 3. Experimental Methods 41varies linearly with . For asymmetric tops, where only Erk is non-zero, higher fieldsare required and splittings vary with E2. However, when a near-degeneracy exists betweenthe asymmetric top rotational energy levels, non-degenerate perturbation theory breaksdown, and Ek cannot be calculated as in Eq. 3.7; rather, the Stark splittings must beevaluated by diagonalization of the (Hrot +Hstk) matrix, or by transforming that matrixin such a way that the effects of near-degeneracies may be ignored in an approximateperturbation treatment [33].3.1.2 InstrumentationA schematic cross-section of a Stark cell is shown in Fig. 3.1. The electric field is introduced by applying a voltage to the electrode, which sits in the middle of the cell andwhich is insulated from the grounded cell walls by Teflon strips. The electric field direction Z is as indicated in the figure, although some distortion of this field occurs at theedges 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 cellwas used. The cell is sealed at both ends with mica windows, and so acts as both asample cell and a waveguide. Gas samples are introduced into the cell from an attachedvacuum line, to static gas pressures of 5-25 mTorr.The Stark-modulated microwave spectrometer is shown schematically in Fig. 3.2, andfollows the general design first given by Hughes and Wilson [4, 5]. Microwave radiationis provided by a microwave synthesizer, which can be computer controlled [34]. For theexperiments described in this work, a Watkins-Johnson 1291A synthesizer was used tocreate fundamental frequencies in the range 8-18 GHz, accurate to ‘1O Hz, which werethen doubled or tripled as required. The microwave radiation transmitted through theStark cell is detected by a point-contact diode detector and the resulting signals areChapter 3. Experimental Methods 42Teflontrode2lltzFFigure 3.1: Schematic cross-section of a Stark cell.Chapter 3. Experimental Methods 43pre-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. Thefrequency of the square wave is 100 kHz; this frequency is also sent to the LIA. The squarewave 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 rotationaltransition frequencies‘rot; when E=EStk, the sample absorbs at the frequencies 1/Stkof all the M components of the J’r’-Jr transition. The LIA detects only those signalswhich are modulated at 100 kHz, and so spectral noise which is not modulated is filteredout, improving the signal-to-noise ratio. The LIA is phase-sensitive, so the absorptionsoccurring at e=0 and6Stark are displayed on opposite sides of the baseline.The limiting contribution to the linewidths is modulation broadening, a result of finiteobservation periods as the molecules alternate rapidly between =0 and =Stark states.Wall broadening and pressure broadening also contribute, and linewidths observed inthis work using the Stark-modulated spectrometer were typically —250 kHz FWHM (fullwidth half maximum).3.2 Microwave Fourier transform spectroscopy3.2.1 TheoryIn MWFT spectroscopy, a short pulse of microwave radiation interacts with the dipolemoments of a large number of molecules, creating a macroscopic polarization of thesample. This quantum mechanical interaction may be treated macroscopically usingdensity matrix formalism [35]. For a molecule with only two rotational levels of energiesEa and Eb (Eb > Ea), the rotational wavefunction may be expanded in terms of theChapter 3. Experimental Methods 44to vacuumlineFigure 3.2: Schematic diagram of the Stark-modulated microwave spectrometer.Chapter 3. Experimental Methods 45rotational eigenstates ç and cb as= Caa + Cbbb. (3.8)The density matrix for this two-state system is defined asp = Ib><’b (3.9)= I + Cbb >< Ca/fa + Cbb I. (3.10)Since q!r. and q are orthonormal,CaC* CaC*a b (3.11)CbC CbCbandptj = 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 mechanical 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 Aand 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 byz ---=[H,pj. (3.15)If the two-level system described above is subjected to a microwave pulse of frequencyw, then the Hamiltonian may be given asH =Hrot +Hpert, (3.16)Chapter 3. Experimental Methods 46where Hrot is given in the {& b} basis asEaOHrot = . (3.17)0E6Hpert describes the interaction of the molecular dipole moment ,u with the electric fieldproduced by the microwave pulse. In the {g, qi,} basis, only the off-diagonal matrixelements [Lab of p are non-zero, as transitions between rotational eigenstates depend onL Hpert is thus given by0 2[tabE cos(.t + )Hpert = , (3.18)2[tab6 cos(wt + 4) 0where 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 isgiven by0PaaPaa = iX(PbaPab)cO5(Wt+) (3.19)Pab = ZWoPab + iX(pbb Paa) cos(wt + q) (3.20)Pba = ‘OPba + X(Paa pbb) cos(wt + q5) (3.21)pbb iX(Pba — Pab) cos(wt + 4), (3.22)where X=2[IabE/h and w0 is the angular frequency of the rotational transition, (EbEa)/h.The matrix elements pj are usually transformed to elements pj, usingPaa = Paa (3.23)Pbb = Pbb (3.24)Pab = pabe (3.25)Pba = pae_t (3.26)Chapter 3. Experimental Methods 47The elements 4 are defined in a frame of reference rotating at angular frequency w andare therefore without an explicit time dependence. The real quantities .s, u, v, and w arethen defined as8 = Paa+Pbb=’ (3.27)= pab + p’ (3.28)= (Pa— Pab) (3.29)W = Paa — Pbb (3.30)Neglecting terms in 2w (known as the rotating wave approximation [36]) and choosingthe phase of the microwave pulse such that çz=O,= 0 (3.31)= —viw (3.32)= u/sw — xw (3.33)th = xv (3.34)where Zw=w0— 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, andso Zw is small. The amplitude of the microwaves is then chosen such thatx= 2abE>> (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 givev(t) = —w0 sin(xt) (3.39)w(t) wocos(xt); (3.40)during the excitation pulse, v(t) and w(t) oscillate between —w0 and w0 at an angularfrequency x.The microwave field is then turned ofF at time t, whereupon x becomes zero. Neglecting relaxation process, Eqs. 3.32-3.34 becomeIL = —v/sw (3.41)= uZw (3.42)th = 0, (3.43)and so during the ‘observation’ period afteru(t) = —v sin(LSt) (3.44)v(t) = cos(L.ct) (3.45)w(t) wt1,. (3.46)The macroscopic polarization of the sample is given byP(t) = N < t> (3.47)= NTr(pp) (3.48)= NTr0 ab Paa Pab (349)ILab 0 Pba. Pbb= N,LLab(pab + Pba). (3.50)Chapter 3. Experimental Methods 49Using 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) = N,LtabVtp[Sfl(/Wt) cos(wt) + cos(Lwt) sin(wt)] (3.52)= N/JabVtp sin[(& + w)t] (3.53)= NIIabVtp sin(wot). (3.54)The macroscopic polarization will thus oscillate at the frequency of the rotational transition. The electric field produced by this oscillating polarization will also oscillate at w0,in quadrature to P(t). Since the amplitude of the field is proportional to v,, optimumsignals are obtained if the microwave field is turned off when v is at its maximum orminimum. This is known as the ‘ir/2’ condition: optimized microwave pulses satisfyXtp2[LabEtp/h = nir/2, where n is an integer. Assuming e to be constant, a moleculewith a smaller transition dipole moment will require a longer microwave pulse to producean optimized signal. However, the signal intensity is proportional to I-tab, rather thanto ,u as in conventional absorption spectroscopy (Eq. 2.18), and so time-domain microwave spectroscopy is well-suited to the study of molecules with small transition dipolemoments.3.2.2 InstrumentationMicrowave cavityThe pulsed molecular beam, pulsed microwave, cavity Fourier transform spectrometerused in this work is of the type first developed by Balle and Flygare [6]. The FabryPerot microwave cavity is formed by two spherical aluminum mirrors, of diameter 28 cmChapter 3. Experimental Methods 50and radius of curvature 38.4 cm. The mirrors sit within a vacuum chamber, separatedby -.30 cm. The cavity must be adjusted so that the exciting microwave radiation canform a standing wave pattern between the antennae centred in the two mirrors; this isaccomplished by moving one of the microwave mirrors with a micrometer screw until thecavity 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 frequencyis scanned in discrete steps no more than 1 MHz apart, with measurements performedat each frequency. The frequency range of the spectrometer is currently 4-26 GHz.ElectronicsA schematic circuit diagram for the spectrometer is shown in Fig. 3.3. The microwavesource, accurate to r”O.l Hz, is a Hewlett-Packard 8340A microwave synthesizer, whichis controlled by a personal computer via an IEEE bus. This synthesizer also produces a10 MHz reference signal, which is used for up- and down-conversion and which controlsthe timing of the experiment. At each excitation frequency Vexcjtatjon=VMW2OMHz, theoutput of the synthesizer is first swept over a frequency range centred at 11Mw• The radiation is coupled into the cavity through an antenna in the tuning mirror, and monitoredwith an oscilloscope connected to the antenna in the stationary mirror. The cavity istuned so that maximum transmission of radiation is centred at 1/MW-20 MHz.Once the cavity has been tuned, the synthesizer output frequency is locked at 1/MWThis frequency is then mixed with 20 MHz in a single side-band modulator to produceMW-20 MHz (the vMW +20 MHz side-band may also be used.) The microwave pulse iscreated by opening and closing a PIN switch (PIN switch 1). The pulse passes throughthe circulator to the microwave cavity, where it interacts with the gas sample whichhas been introduced into the cavity. After the microwave pulse, the gas sample emitsradiation at the frequencies of the transitions which are off-resonant from 1/excitation by aChapter 3. Experimental Methods 51tuningmirrorstationarymirrornozzleoscilloscope10 MHz‘MW 20 MHz+ tw—Figure 3.3: Schematic circuit diagram for the cavity MWFT spectrometer.Chapter 3. Experimental Methods 52small amount Lv and which have been excited by the microwave pulse. Those transitionfrequencies which fall within the bandwidth of the cavity are transmitted back throughthe circulator. This molecular signal is amplified, down-converted to 20 MHz-Zv, andthen to 5 MHz+tv. This last signal is fed through a 5 MHz bandpass filter to eliminateother signals leaving the RF mixer.Fourier transformationThe 5 MHz+LSi’ signal is collected by a transient recorder board in the personal computer. 4 K data points are collected at 50, 100, or 150 ns sample intervals (st), and arethen transferred to the computer, where signal averaging is accomplished by addition ofsuccessive time-domain signals. A frequency-domain spectrum F(zi), with both real andimaginary components, is computed from the time-domain signal f(nzt) using a discreteFourier transform [37]:F(v) = f(nt)e2. (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 ofF(v). The frequencies obtained in this spectrum are then added to vMw-2O MHz-5 MHzto obtain the molecular transition frequencies. When two or more frequencies in thespectrum are very close together, distortions of the peak positions in the power spectrummay lead to inaccurate frequencies. These distortions may be avoided by performing a‘decay fit’ [38], whereby amplitudes, relaxation times, frequencies, and phases of all ofthe signals are fit to the time-domain spectrum by least-squares.If the molecular signal were purely sinusoidal and could be observed for an infinitetime, the frequency-domain spectrum before down-conversion would consist of a series ofinfinitely sharp peaks, corresponding to a series of 6-functions centred at the transitionChapter 3. Experimental Methods 53frequencies. However, the molecular signal is observed for a finite time T; the uncertaintyprinciple dictates that the precision with which transition frequencies can be determinedis inversely proportional to T, and so the peaks in F(v) have a width which dependson 1/T. In addition, relaxation processes do occur in the gas sample and the molecularsignals 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 ratioby 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 eachexperiment.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 spectralrange is then 10 MHz. However, time-domain signals with v>10 MHz will appear inthe frequency-domain spectrum, as the frequency-domain spectrum is replicated every1/Lt=20 MHz [37], allowing peaks outside the spectral range to ‘fold in’. Care must betaken to filter out these frequencies before Fourier transformation, which is accomplishedwith a 5 MHz bandpass filter in the arrangement shown in Fig. 3.3.Gas sampleThe 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 andpressures of —1-2 atm, and is allowed to expand adiabatically through the orifice intothe vacuum chamber. Gas samples typically contain a high fraction of a rare gas, andso in front of the nozzle the molecule of interest undergoes collisions with mostly raregas atoms; rotational and vibrational energy is thus transferred to translational energyalong the jet axis. The velocity distribution narrows, producing a very low translationaltemperature. Rotational energy is lost rapidly, and low rotational temperatures (‘—.4 K)Chapter 3. Experimental Methods 54are obtained, while vibrational energy transfer is somewhat slower. Higher rotationaltemperatures 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 severaladvantages over static gas samples. Most of the population is collapsed down into thelowest electronic, vibrational, and rotational states; by increasing the populations in thelevels involved in rotational transitions, some of the sensitivity problems encountered inroom-temperature microwave spectroscopy are avoided. Supersonic cooling also permitsthe study of unstable species such as van der Waals molecules (e.g. Ar-OCS [40]), whichare 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 microwave cavity. However, sensitivity and resolution are improved dramatically if thenozzle is placed near the centre of one of the microwave mirrors, presumably because themolecules travel longer through the region of highest microwave field strength. Becausethe molecules travel parallel to the direction of microwave propagation, the Doppler effect splits lines in the frequency spectrum into doublets; the average frequency of the twocomponents corresponds to the transition frequency. With this arrangement, linewidthsas low as 6 kHz FWHM can be achieved, limited by the transit time of the moleculesthrough the microwave field. While the gas expansion is not collimated, and hence is nota true molecular beam, little if any Doppler broadening is observed.Pulse sequenceA schematic diagram of the pulse sequence used in a MWFT experiment is given inFig. 3.4. A pulse generator creates the transistor-transistor logic (TTL) pulses whichcontrol the PIN diode switches, the gas nozzle, and the transient recorder board. Thenozzle is opened and closed to allow the gas into the cavity, and then a microwave pulse isformed by opening and closing PIN switch ]. Acquisition of the molecular signal is startedChapter 3. Experimental Methods 55openPIN 1__________ ________closedopenPIN 2__________closedtrigger P________opennozzle_ ____ _ _ _ __ ____closeFigure 3.4: Schematic pulse sequence diagram for the cavity MWFT spectrometer.Chapter 3. Experimental Methods 56by sending a trigger pulse to the transient recorder. Microwave pulses of mW areused, and so while PIN 1 is open, PIN 2 is kept closed to protect the sensitive detectioncircuit from exposure to the full microwave power. The exciting microwave radiation isstored by the Fabry-Perot cavity, causing the cavity to ‘ring’ at the excitation frequencyafter the pulse is turned off. This ringing is often stronger than the molecular signal, sofor each experiment the exponential decay of the cavity ringing is recorded before the gassample is introduced. The cavity decay is then subtracted from the signal obtained in thepresence of gas to leave only the molecular signal. The repetition rate of the experimentsis generally limited by the pumping speed of the diffusion pump attached to the vacuumchamber, and typically ranges from 0.3-10 Hz.Chapter 4The Microwave Spectrum of BrNC’804.1 IntroductionBromine isocyanate, BrNCO, is one of several isocyanates whose gas phase structureshave been determined in the past 20 years, via both microwave spectroscopy and electron diffraction. Isocyanates contain the NCO moiety, giving this group of moleculessimilar chemical properties to the halides; along with azides, thiocyanates, and cyanides,isocyanates are known as pseudohalides [41]. Monomeric isocyanates are thermally unstable at room temperature [42]. Of the possible halogen isocyanates, a full substitutionstructure has been previously determined for C1NCO alone [43], while only one isotopiccombination of INCO(1274N’2C60)has been studied by microwave spectroscopy [44].Attempts to prepare FNCO in the gas phase have been unsuccessful [45], although JRspectra have been recorded for FNCO produced by photolysis of matrix-isolated FCON3and XCONF2 (X = H, NF2, CF3) [46].Bromine isocyanate lies between C1NCO and INCO in terms of thermal stability. Jtwas first prepared by Cottardi by reacting Br2 with silver cyanate at 150°C [47], as wellas by vacuum thermolysis of tribromoisocyanuric acid, (BrNCO)3 [48]. Evidence that themolecule is indeed an isocyanate, containing a Br-N bond, was obtained from the He(J)photoelectron spectrum [42], and from the ‘4N nuclear quadrupole hyperfine structureobserved in an initial microwave spectroscopic study [1, 2]. Jnfrared spectra of the normaland of isotopically labelled species have been reported [48, 49] and used to determine a57Chapter 4. The Microwave Spectrum of BrNC’80 58harmonic force field and some structural parameters.The structures of the halogen isocyanates are interesting, as they are expected fromsimple bonding theories to have C symmetry, with a bend of .‘ 1200 at N, and a linearNCO chain. However, C1NCO has been shown instead to have a bend in the NCOchain of i-.’ 90 [43] (as does HNCO [50]), and the infrared studies are consistent with asimilar bend for BrNCO [49]. This could not be confirmed from the initial microwavestudy, because spectra of only two isotopic species, 79Br’4N2C60and81Br’4N2C60,had been measured [2]; 79Br and 81Br are of approximately equal natural abundance,allowing both isotopomers to be observed. At least one more isotopic substitution wasrequired to determine the structure completely (see Section 2.4.)In the work described here, the microwave spectra of the substituted isotopomers79Br’4N2C180and 81Br14N’2C0have been measured using a Stark-modulated microwave spectrometer. The results of this study have been combined with those obtainedearlier [2] to permit an unambiguous determination of the geometry of bromine isocyanate. The initial measurements and a preliminary analysis were made by Ms. MimiLam [51]. The work carried out by this author expanded greatly the number and types ofmeasured transitions, and included a detailed analysis and structure determination [3].4.2 Experimental methodsIn the previous microwave study [2], fresh BrNCO was synthesized during measurementsby flowing Br2 over heated AgNCO. However, as the natural abundance of 180 is 0.20 %(as compared to 99.76 % for 160),‘80-enriched samples of BrNCO were necessary inthe 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 stateChapter 4. The Microwave Spectrum of BrNC18O 59(below —60°C), BrNCO forms yellow crystals, but at higher temperatures it melts to abrown liquid, and the molecules tend to dimerize to crystalline N,N-dibromocarbamoylisocyanate [47]. To maintain quantities of monomeric BrNC’80, samples were stored inliquid nitrogen.BrNCO vapour was introduced into the Stark cell by placing the sample on a vacuumline connected to the cell and allowing it to warm to room temperature until the entiresample had melted. The sample tube was then placed in liquid nitrogen, and most ofthe BrNCO vapour was collected back in the tube, leaving 15-30 mTorr in the microwavecell. The cell was surrounded by dry ice, and this cooling, combined with relativelylow pressures within the cell, discouraged dimerization. The sample remained stable forapproximately 20 minutes, after which point the cell was evacuated and fresh BrNCOwas added.The Stark-modulated spectrometer described in Section 3.1 was used to make measurements in the frequency region 23-52 GHz. Measurements are estimated to be accurateto better than +0.05 MHz.4.3 Observed spectrum and analysisInitial predictions of the rotational constants of79BrNC’80and81BrNC’0were based onthe three possible structures proposed by Jemson et al. [2]. Since the molecule is planar,with C3 symmetry, a- and b-type transitions were expected, with the former, followingRef. [2], much stronger than the latter. Transition frequencies were calculated using theserotational constants, combined with centrifugal distortion constants transferred from thenormal species.The transitions were all expected to show quadrupole hyperfine structure causedby the Br nucleus (IBr=3/2); the much smaller 14N quadrupole coupling (I14N=1) wasChapter 4. The Microwave Spectrum of BrNC’80 60not resolved in this study. In BrNC16O, the Br principal quadrupole axis z is essentially aligned with the Br-N bond, at an angle of to the a principal inertial axis,and QBr is relatively large; the only non-zero off-diagonal quadrupole coupling constantXab 500 MHz [2]. It was expected that any accidental rotational near-degeneraciesof appropriate symmetry would result in measurably perturbed hyperfine patterns (seeSection 2.5.)Obtaining an accurate value for Xab depends on observing such perturbations. Froman examination of Eq. 2.10 in the limiting case of a prolate symmetric rotor, it is alsoapparent that the A rotational constant cannot be determined precisely from rotationaltransition frequencies alone if only transitions with zSKa=0 (i.e. a-type transitions) aremeasured. However, as the positions of the rotational energy levels and hence the neardegeneracies are determined by all three rotational constants, the perturbations are asource of information, and perturbed hyperfine patterns may allow A to be determinedif b-type transitions are too weak to be found easily.The energy level diagram for79BrNC’80(Fig. 4.1) shows an important near-degeneracybetween the levels 1Ooo and 918, where 918 is 822.5 MHz above These levels willinteract 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 transitions 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 toimprove the precision of the derived constants. Ultimately lines of 22 a-type rotationaltransitions of each of79BrNC18Oand81BrNC18Owere measured and assigned. Individuallines were typically located within a scan frequency range of <40 MHz about the predicted frequencies; the assignments were confirmed by the hyperfine patterns. These lineswere fit to rotational, centrifugal distortion and Br quadrupole coupling constants usinga global least-squares fitting program [1, 2] employing Watson’s A reduction; the resultsChapter 4. The Microwave Spectrum of BrNC18O 61________________15114 2,1100 2031.9 MHz3212150,15 2,101,13141,14 t 122,11_________________2,9140,14 I I 2,101,12400 ‘31,132,8N ‘02,9130,131 227‘-‘2,81,10‘N 3 0 0 120,12 111,11____________2,6__ __ __ __ __ __2,5822.5 MHz1062,5100,10 1,9 2,352,4200 1,7 2342.29Q,9 81,8 342;9__ __ ___ ___1,68Q,9 71,7_1,561,6100 79,7 1,4695 41:5 31:2l:49,4 1 1:30,3o 1 0,1 000Figure 4.1: Rotational energy levels of79BrNC’80.Chapter 4. The Microwave Spectrum of BrNC’80 62are given in Table 4.3. Values for three rotational constants, three centrifugal distortionconstants and three independent Br quadrupole coupling constants were produced foreach isotopomer. In particular, precise values for both A0 and Xab were produced froma-type transitions alone. (Note that the sign of Xab cannot be determined, since thesecond order perturbation energy depends on Xb.)The rotational constants so determined were now sufficiently precise to make searchesfor the very weak b-type transitions practicable. The searches were only partially successful; even with frequency predictions accurate to better than ±5 MHz, many b-type transitions were too weak to be observed, or were overlapped by other lines. Thus only threesuch transitions were unambiguously assigned and included in the final fit for79BrNC’80(12111-12012, 15114-15015, and 19019-18118); two such transitions were included in the81BrNC’0fit (121 11-12012 and 151 14-15015).Chapter 4. The Microwave Spectrum of BrNC180 63obs.-calc. obs.-calc.F’- F” wt.a frequency no Xab Xab F’ - F” wt. frequency no Xab Xab*,b fl 0*V16- is ‘09 °o7.5 6.5 1.00 23609.528 0.335 0.025 9.5 8.5 1.00 35848.981 -1.343 -0.0056.5 5.5 1,00 23613.688 0.223 0.013 10.5 9.5 1.00 35849.905 0.002 -0.0134.5 3.5 1.00 23615.828 -0.013 0.008 7.5 6.5 1.00 35851.093 -1.609 -0.0345.5 4.5 1.00 23620.228 0.002 0.050 8.5 7.5 1.00 35852.929 -0.186 -0.018606 - 928 - 8276.5 5.5 0.50 23908.713 -0.177 -0.023 10.5 9.5 1.00 35880.482 0.084 0.0307.5 6.5 0.50 23908.713 0.085 0.063 7.5 6.5 1.00 35881.869 -0.034 -0.0284.5 3.5 1.00 23919.621 4.294 -0.024 9.5 8.5 1.00 35885.476 0.078 0.0355.5 4.5 0.10 23923.302 7.756 0.041 8.5 7.5 1.00 35887.014 0.000 0.008615-54 91s-877.5 6.5 1.00 24220.376 0.237 -0.036 9.5 8.5 0.50 36329.970 0.403 0.0534.5 3.5 1.00 24223.341 -0.064 -0.045 7.5 6.5 0.50 36329.970 -0.049 -0.0166.5 5.5 1.00 24224.321 -0.064 -0.061 10.5 9.5 1.00 36342.321 14.084 -0.0305.5 4.5 1.00 24227.981 0.189 -0.083 8.5 7.5 1.00 36343.155 11.776 -0.0067i 7 - 6 10010 — 9 98.5 7.5 1.00 27544.958 0.119 -0.029 10.5 9.5 1.00 39812.810 -14.037 0.0677.5 6.5 1.00 27547.632 0.083 -0.018 8.5 7.5 1.00 39816.782 -11.836 0.0295.5 4.5 0.75 27549.891 -0.035 -0.014 11.5 10.5 1.00 39826.436 0.059 0.0456.5 5.5 0.75 27552.650 -0.049 -0.010 9.5 8.5 1.00 39828.668 -0.416 0.017716—615 1029-9288.5 7.5 1.00 28257.665 0.623 0.111 11.5 10.5 1.00 39866.364 0.093 -0.0035.5 4.5 0.10 28259.828 0.117 0.141 8.5 7.5 1.00 39867.615 -0.065 -0.0597.5 6.5 0.10 28259.828 0.090 0.085 10.5 9.5 1.00 39870.054 0.116 0.0396.5 5.5 1.00 28263.098 0.613 0.064 9.5 8.5 1.00 39871.395 -0.025 -0.018808- 77 1028-9279.5 8.5 0.25 31871.232 -0.103 -0.120 11.5 10.5 1.00 39899.740 -0.035 -0.0838.5 7.5 0.25 31871.232 -0.473 0.026 8.5 7.5 1.00 39901.173 -0.047 -0.0416.5 5.5 1.00 31874.100 -0.828 -0.049 10.5 9.5 1.00 39903.010 0.046 0.0107.5 6.5 1.00 31874.969 -0.317 -0.013 9.5 8.5 1.00 39904.480 -0.003 0.000817-76 1019-9189.5 8.5 1.00 32294.387 1.331 -0.017 11.5 10.5 1.00 40341.072 -21.508 0.0418.5 7.5 0.50 32294.916 0.022 -0.007 9.5 8.5 1.00 40343.826 -21.265 -0.0076.5 5.5 0.10 32295.096 -0.130 -0.102 10.5 9.5 1.00 40363.014 -0.576 0.0287.5 6.5 1.00 32298.515 1.402 -0.052 8.5 7.5 1.00 40364.081 0.021 0.062a Measurements were weighted according to 1/o.2, where o is the uncertaintyin 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 79BrNC’80(in MHz)Chapter 4. The Microwave Spectrum of BrNC180 64obs.-calc. obs.-calc.F’ - F” wt. frequency no xab x F’ - F” wt. frequency no Xab Xabii - 12o 1212.5 12.5 1.00 42429.904 -1.93911.5 11.5 1.00 42434.845 -2.07913.5 13.5 1.00 42472.588 -1.11610.5 10.5 1.00 42475.983 -2.871111 ii - 1012.5 11.5 1.00 43279.046 -0.01111.5 10.5 1.00 43279.889 0.0349.5 8.5 1.00 43281.161 -0.07910.5 9.5 1.00 43282.007 -0.04211011-1001012.5 11.5 0.50 43800.531 -0.03010.5 9.5 1.00 43803.504 0.59311.5 10.5 1.00 43822,719 21.6409.5 8.5 1.00 43823.583 21.18711210-102912.5 11.5 0.50 43851.682 0.3229.5 8.5 0.50 43852.635 -0.00311.5 10.5 0.50 43854.480 0.34110.5 9.5 0.50 43855.500 0.03215i 14 - l5 1515.5 15.5 1.00 44781.782 -0.77914.5 14.5 1.00 44786.716 0.01616.5 16.5 1.00 44824.419 -0.46313.5 13.5 1.00 44828.725 -0.34419 19 - l8 1819.5 18.5 1.00 45042.956 0.62818.5 17.5 1.00 45044.982 -0.30220.5 19.5 1.00 45070.258 -0.03617.5 16.5 1.00 45073.506 0.2900.0390.041-0.040-0.05012 12 — lii ii13.5 12.5 1.00 47210.598 -0.00912.5 11.5 1.00 47211.264 0.00410.5 9.5 1.00 47212.323 -0.12811.5 10.5 1.00 47213.043 -0.07012012— lloii12.5 11.5 0.25 47770.270 -2.55010.5 9.5 0.25 47770.270 -3.51313.5 12.5 1.00 47772.256 0.00211.5 10.5 1.00 47774.287 -0.060l3 13 - 12 1214.5 13.5 1.00 51141.276 -0.01313.5 12.5 1.00 51141.894 0.05111.5 10.5 1.00 51142.758 -0.10912.5 11.5 1.00 51143.373 -0.054132 12 — 122 1114.5 13.5 1.00 51815.458 -3.74313.5 12.5 1.00 51817.977 -2.95611.5 10.5 1.00 51820.212 -0.01312.5 11.5 1.00 51822.035 0.05213211— 1221014.5 13.5 1.00 51893.253 0.59311.5 10.5 1.00 51893.718 -0.00113.5 12.5 1.00 51894.192 0.41612.5 11.5 1.00 51894.879 0.0170.029-0.0 640.0 240.036-0.0430.014-0.047-0.005-0.0420.0450.0840.0020.0810.0030.1430.038-0.015-0.0130.038-0.030-0.034-0.012-0.085-0.022-0.0610.010-0.0100.050-0.0340.039-0.0430.020-0.114-0.055-0.0060.0560.1020.0060.0280.014Table 4.1: Measured transitions of79BrNC’80(cont’d)Chapter 4. The Microwave Spectrum of BrNC18O 65*,b A 0*U16—‘09 - 0087.5 6.5 9.5 8.56.5 5.5 10.5 9.54.5 3.5 7.5 6.55.5 4.5 8.5 7.56o6-55 928-8276.5 5.5 10.5 9.57.5 6.5 7.5 6.54.5 3.5 9.5 8.55.5 4.5 8.5 7.5717-616 927-8268.5 7.5 10.5 9.57.5 6.5 7.5 6.5A 0*J) ‘±.) ?180176.5 5.5 10.5 9.5716- 6i 8.5 7.58.5 7.5 10 10 - 9 95.5 4.5 11.5 10.57.5 6.5 10.5 9.56.5 5.5 8.5 7.50 ‘7*008‘07A o in n*) 0.) IVO 10— ‘o 98.5 7.5 10.5 9.56.5 5.5 8.5 7.57.5 6.5 11.5 10.5826-725 9.5 8.59.5 8.5 1029-9288.5 7.5 11.5 10.5A *‘19 1810.5 9.5 10.5 9.59.5 8.5 9.5 8.57.5 6.58.5 7.5a Measurements were weighted according to 1/2, where o- is the uncertaintyin the measurements. Unit weight corresponds to an uncertainty of 0.05 MHz.b Transitions marked by * were measured by Ms. Mimi Lam [51].obs.-calc. obs.-calc.F’ - F” wt.a frequency no Xab Xab F’ - F” wt. frequency no Xab Xab1.00 23435.419 0.233 0.0081.00 23438.947 0.179 0.0261.00 23440.765 0.003 0.0181.00 23444.419 -0.005 0.0290.50 23730.320 -0.192 -0.0870.50 23730.320 0.027 0.0120.50 23738.906 2.996 0.1040.50 23741.056 4.957 -0.0541.00 27341.628 0.073 -0.0331.00 27343.944 0.117 0.0451.00 27345.864 0.044 0.0591.00 27348.120 -0.016 0.0111.00 28044.167 0.312 -0.0280.25 28046.075 -0.034 -0.0170.25 28046.075 -0.042 -0.0451.00 28048.727 0.301 -0.0660.50 31633.496 -0.013 -0.0250.50 31633.496 -0,319 0.0121.00 31635.959 -0.564 -0.0311.00 31636.562 -0.259 -0.0391.00 31672.443 0.034 0.0151.00 31678.060 0.016 0.0021.00 35151.923 0.036 -0.0051.00 35153.015 0.012 -0.0161.00 35154.515 -0.056 -0.0401.00 35155.625 -0.079 -0.0571.00 35581.887 -0.888 -0.0321.00 35582.397 -0.030 -0.0411.00 35583.669 -1.107 -0.0751.00 35584.929 -0.189 -0.0711.00 35612.557 0.041 0.0051.00 35613.781 -0.006 -0.0021.00 35616.805 0.100 0.0711.00 35618.005 -0.050 -0.0441.00 35636.304 0.041 0.0170.50 35637.492 -0.073 -0.0691.00 36060.920 6.940 -0.0061.00 36063.185 6.567 0.0691.00 39056.036 0.097 0.0681.00 39056.864 0.081 0.0611.00 39058.160 0.024 0.0421.00 39058.994 0.002 0.0241.00 39522.775 -6.927 0.0131.00 39524.611 -6.584 -0.0261.00 39529.345 0.031 0.0211.00 39531.344 -0.236 0.0011.00 39568.552 0.050 -0.0131.00 39569.646 -0.044 -0.0391.00 39571.686 0.113 0.0621.00 39572.804 -0.009 -0.004Table 4.2: Measured transitions of81BrNC’0 (in MHz)Chapter 4. The Microwave Spectrum of BrNC’80 66obs.-calc. obs.-calc.F’ - F” wt. frequency no Xab Xab F’ - F” wt. frequency no Xab Xab1028-927 11110- 101911.5 10.5 1.00 39601.062 -0.027 -0.059 12.5 11.5 1.00 44062.936 2.074 0.0548.5 7.5 1.00 39602.286 -0.020 -0.016 10.5 9.5 1.00 44065.426 2.835 0.07710.5 9.5 1.00 39603.786 0.019 -0.006 15i 14 - 150159.5 8.5 1.00 39605.046 0.009 0.012 15.5 15.5 1.00 44659.172 -0.542 0.001101 9 - 91 8 14.5 14.5 1.00 44663.228 0.068 -0.02911.5 10.5 0.75 40045.583 -12.263 0.113 16.5 16.5 1.00 44694.596 -0.380 -0.0239.5 8.5 1.00 40046.536 -13.418 0.024 13.5 13.5 1.00 44698.316 -0.139 0.01310.5 9.5 0.75 40058.348 -0.344 0.018 12 12 - 111 118.5 7.5 1.00 40059.078 -0.016 0.012 13.5 12.5 1.00 46861.740 0.014 -0.00312 - 12 12 12.5 11.5 1.00 46862.340 0.068 0.05712.5 12.5 1.00 42342.248 -1.452 -0.009 10.5 9.5 1.00 46863.228 -0.043 -0.01411.5 11.5 0.50 42346.579 -1.354 0.082 11.5 10.5 1.00 46863.822 -0.001 0.03113.5 13.5 1.00 42377.787 -0.810 0.015 12o 12 - 1101110.5 10.5 1.00 42380.770 -2.108 -0.008 13.5 12.5 1.00 47416.264 0.053 0.045111 fl - 101 10 11.5 10.5 1.00 47417.902 -0.057 0.02812.5 11.5 1.00 42959.217 -0.014 -0.036 13113 - 121 1211.5 10.5 1.00 42959.943 0.046 0.031 14.5 13.5 1.00 50763.333 -0.046 -0.0619.5 8.5 1.00 42960.990 -0.070 -0.048 13.5 12.5 0.50 50763.843 0.001 -0.00710.5 9.5 1.00 42961.728 -0.007 0.018 11.5 10.5 1.00 50764.599 -0.102 -0.058110 11 - 10 10 12.5 11.5 0.50 50765.091 -0.077 -0.02812.5 11.5 1.00 43473.915 -0.059 -0.06810.5 9.5 1.00 43476.189 0.249 -0.07311.5 10.5 1.00 43486.780 12.378 -0.0039.5 8.5 1.00 43488.946 13.432 0.040Table 4.2: Measured transitions of81BrNC’0 (cont’d)The measured transitions included in the fits are given in Tables 4.1 and 4.2, alongwith the residual (observed—calculated) frequencies. Note that when Xab was set tozero in the fit, the residuals (marked ‘no Xab’) were noticeably larger than those obtainedwhen Xab was released into the fit (marked ‘Xab’). As expected, frequencies for transitionsinvolving the 918 and 10010 levels depend strongly on Xab• In particular, the F=8.5 andF=lO.5 hyperfine components of these levels are shifted by as much as 21 MHz by theperturbation. Inspection of Fig. 4.1 also shows that the level 12211 is only 2031.9 MHzabove 13 in 79BrNC’80,and so transitions involving these levels should be perturbed.Chapter 4. The Microwave Spectrum of BrNC’80 67Only one such transition was measured (13212-12211), but the F=12.5 and F=13.5 levelsof 12211 seem to be shifted by ‘-‘3 MHz.The final data set was analyzed using the global least-squares program to producerotational, centrifugal distortion, and Br nuclear quadrupole coupling constants. Theresults are in the upper part of Table 4.3, in the columns labelled ‘all transitions’. It isclear that inclusion of the b-type transitions has greatly improved the precision of severalconstants, notably A0, (Xbb — Xcc), and Xab.Table 4.3 also includes the ground stateinertial defect z0.Only three of the five quartic centrifugal distortion constants (zJ, ZJK, and 5j) couldbe determined in the fit. As the extents of the 918-10010 and 12211-141 13 perturbationsdepend on the relative positions of the Ka=0, iç=i and Ka=2 energy level stacks, thiscould allow /K to be separated from A0 [34]. Evidently the latter perturbation is notsufficiently large, as a fit carried out with /K released produced an indeterminate valuefor that constant.4.3.1 Harmonic force field and structureWith rotational constants available for four isotopic combinations of BrNCO, the molecular geometry could now be determined without assuming a value for any bond length orangle. The inertial defects of all four isotopic species are small and positive, with littledependence on isotopic species, supporting the assumption that BrNCO is planar. Thisassumption is also confirmed by the values of Xcc, obtained from xaa and XbbXcc usingEq. 2.104. If BrNCO were non-planar, the angle of the c inertial axis relative to the yprincipal axis of the quadrupole coupling tensor would change upon isotopic substitution,and so the relative values of Xcc for 79BrNC’80and 81BrNC’0would depend on thatchange, as well as on the ratio of the 79Br and 81Br quadrupole moments. However,is 0.8359 for BrNC’80, in excellent agreement with the ratio of theChapter 4. The Microwave Spectrum of BrNC’80 6879BrNC18O 81BrNC’0a-type all a-type alltransitions transitions transitions transitionsA0 (MHz) 40311 .7(24)a 40308.326(72) 40258.8(44) 40260.088(94)B0 (MHz) 2044.0895(24) 2044.0919(18) 2028.4733(39) 2028.4840(23)Co (MHz) 1942.4052(23) 1942.4047(17) 1928.1979(24) 1928.1931(20)zij (kllz) 1.0377(38) 1.0442(27) 1.0114(62) 1.0263(46)JK (kllz) -162.00(17) -162.17(14) -160.20(20) -160.04(21)6j (kHz) 0.1445(50) 0.1530(26) 0.1268(70) 0.1480(29)Xaa (MHz) 598.6(14) 598.3(14) 500.6(15) 500.2(17)Xbb — Xcc (MHz) 295.6(36) 289.99(20) 242.3(42) 242.29(25)I Xab I (MHz) 557.04(68) 556.06(25) 464.09(71) 464.28(38)Lo (a.m.u. A2) 0.4055(4) 0.4057(4)no. of rot. trans. 22 25 22 24fit (MHz) 0.042 0.044 0.041 0.04779BrNC16Ob 81BrNC16Oall alltransitions transitionsA0 (MHz) 41189.506(25) 41141.914(29)B0 (MHz) 2175.63391(52) 2 159.50429(53)C0 (MHz) 2063.09857(53) 2048.47014(55)j (kllz) 1.1370(19) 1.1281(18)LJK (kllz) -173.97(14) -172.23(14)6j (kllz) 0.17367(26) 0.17004(31)Xaa (MHz) 608.41(52) 508.49(52)Xbb — Xcc (MHz) 280.086(64) 233.311(70)Xab I (MHz) 549.67(12) 458.60(12)L0 (a.m.u. A2) 0.401 0.401no. of rot. trans. 43 46alIt (MHz) 0.032 0.034a Numbers in parentheses are one standard deviation in units of the last significant figure.b Spectroscopic constants for 79BrNC16Oand81BrNC16Oare taken from Ref. [2].Table 4.3: Spectroscopic constants of BrNCOChapter 4. The Microwave Spectrum of BrNC’80 69quadrupole moments of the two nuclei (0.8353).The planarity of the molecule is thus confirmed, with the result that only two of A0,B0and C0 are independent. The positions of Br and 0 were calculated using Kraitchman’sequations [21] for a rigid molecule (Eqs. 2.66 and 2.67.) Four sets of calculations weremade, using each of 79BrNC’60,81BrNC’60,79BrNC’80, and 81BrNC’0 as the basismolecule. For each basis molecule three calculations were possible, depending on whichpair of rotational constants was considered to be independent. Since the force field (seebelow and Table 4.6) suggested that B0 and C0 had the smallest vibrational contributions, this pair was chosen. The remaining two atoms were located using the productof 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 r0 values. Note that isotopicsubstitution at the carbon or nitrogen atoms would remove the necessity of reproducingI, 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 observedvibrational frequencies was used to estimate a ground state average (re) structure forBrNCO. The only adjustment to the force field was a minor one: the out-of-plane forceconstant f66 was changed to correspond to a different definition of this coordinate. Inthe present work, f66 = 0.0142 mdyn A—’. The coordinate was defined as a torsion, withB-matrix elements (rad A’): Br, -0.603; N, -4.419; C, 10.791; 0, -5.770. Since thisforce field also reproduced the observed centrifugal distortion constants to within a fewpercent (see Table 4.5), it was reasonable to leave the rest unchanged.The harmonic contributions to the force field were subtracted from the observedrotational constants for all four isotopomers to give ground state average values (seeEq. 2.76.) These are given in Table 4.6, along with the resulting calculated inertialdefects L, which are nearly zero, indicating that they are essentially all accounted forby the force field. The structural parameters were calculated again using Kraitchman’sChapter 4. The Microwave Spectrum of BrNC’80 70Effective (ro) Average (ri) Values in related moleculesvalues valuesr(Br-N) (A) 1.8617(23)a 1.85565(73) 1.82 C6H5ONHBr [52]1.82 n-CH3CONHBr [53]1.84 N-bromosuccinimide [52]1.84 sum of single bond radii [54]1.88 C6H5SO2NBr [52]r(N-C) (A) 1.2166(40) 1.2224(15) 1.199 SiH3NCO [55]1.214 HNCO [50]1.226 C1NCO [43]1.234 SF5NCO [56]r(C-O) (A) 1.1692(13) 1.16496(53) 1.162 C1NCO [43]1.166 HNCO [50]1.174 SiH3NCO [55]1.179 SF5NCO [56]L(XNC)b 117.38(42)° 117.99(14)° 118.83° C1NCO [43]123.9° HNCO [50]124.9° SF5NCO [56]127.1° C1CONCO [57]L(NCO) 172.33(44)° 173.13(15)° 170.87° C1NCO [43]172.6° HNCO [50]173.4° C1CONCO [57]173.8° SF5NCO [56]L(Br — N_a)D 28.66(31)° 28.68(24)°a Numbers in parentheses are one standard deviation in the mean taken over the four choices ofbasis molecule, in units of the last significant figure.b E.g. X=Br for BrNCO.L(Br — N—a) is the angle between the BrN bond and the a inertial axis for 79BrNC16O.Table 4.4: Structural parameters of bromine isocyanateChapter 4, The Microwave Spectrum of BrNC’80 71obs. calc.a obs. calc.79BrNC18O 79BrNC16Obj (kllz) 1.0442 1.1212 1.1370 1.2555LJK (kHz) -162.17 -153.77 -173.97 -164.52j (kHz) 0.1530 0.1487 0.1737 0.172681BrNC18O 81BrNC16OL.j (kllz) 1.0263 1.1050 1.1281 1.2378ZJK (kHz) -160.04 -152.43 -172.23 -163.156j (kHz) 0.1480 0.1456 0.1700 0.1691a Calculated using the force field of Ref. [49], adjusted as describedin Section 4.3.1.b Distortion constants for 79BrNC16Oand 81BrNC16Oare takenfrom Ref. [2].Table 4.5: Centrifugal distortion constants of BrNCOequations, using a method analogous to that described above. The resulting values arethe r,, parameters in Table 4.4. No attempt was made, however, to account for changesin these parameters with isotopomer; these changes should be approximately the sameorder of magnitude as the uncertainties given.4.3.2 Discussion and conclusionsThe measured rotational constants of BrNC18O indicate conclusively that the moleculeis the isocyanate, with a Br-N bond, rather than the cyanate, with a Br-O bond. Thederived 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 thoseof 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 structuressuggested previously for BrNCO [2], number III most closely approximates the one determined here. The atoms of 79BrNC16Oin its principal inertial axis system are shownChapter 4. The Microwave Spectrum of BrNC’80 7279BrNC’60 81BrNC’60 79BrNC’80 81BrNC’0Rotational Constants (MHz)A0 41189.506 41141.914 40308.326 40260.088aa/2 -642.934 -641.213 -623.193 -621.327A 40546.572 40500.701 39685.133 39638.761B0 2175.634 2159.504 2044.092 2028.484ab/2 -2.839 -2.808 -2.583 -2.554B 2172.795 2156.696 2041.509 2025.930C0 2063.099 2048.470 1942.405 1928.193c/2 -1.114 -1.108 -1.049 -1.042C 2061.985 2047.362 1941.356 1927.151Inertial defects (a.m.u. A2)& 0.0353 0.0355 0.0363 0.0366a From Ref. [2] and Table 4.3 of this workTable 4.6: Spectroscopic constants of the ground state average structure of BrNCOChapter 4. The Microwave Spectrum of BrNC’80 7379BrNC’80 79BrNC16OaXzz (MHz) 8934(18)b 893.95(68)x (MHz) -449.3(15) -449.70(56)Xyy (MHz) -444.14(35) -444.25(13)8 27.96(1O)° 27.451(37)°81BrNC’0 81BrNC16OXzz (MHz) 746.4(23) 746.40(68)x (MHz) -375.2(19) -375.50(56)Xyy (MHz) -371.24(43) -370.90(13)Oza 27.94(15)° 27.419(44)°a Values for BrNC16O are taken from Ref. [2].6Numbers in parentheses are one standarddeviation in units of the last significant figure.Table 4.7: Principal values of the bromine quadrupole coupling tensorin Fig. 4.2.The principal values of the 79Br and 81Br quadrupole coupling constants of BrNC’80,obtained by diagonalizing the quadrupole tensor, are given in Table 4.7, along with theangle °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 BrNC’60:as expected,isotopic substitution at 0 has no effect on the electronic structure at Br. There is a slightdifference (‘..sl°) between the alignment of the z principal axis and the bond containing thequadrupolar 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 74c0Figure 4.2: The molecular structure of BrNCO, given in the principal inertial axis systemof79BrNC’60.bChapter 5The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, S1H2C15.1 IntroductionThe 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 correspondinghalomethanes. The electronic structures of the halosilanes are therefore of interest, andmay be investigated via nuclear quadrupole coupling effects when the bonded halogenatom has a quadrupolar nucleus (as is the case for Cl, Br, and I, with nuclear spins ofI=3/2, 3/2, and 5/2 respectively.) Values of the halogen nuclear quadrupole couplingconstant eQq have been reported for the silyl halides SiH3X (X=C1 [60], Br [61, 62],and I [63]) and the trifluorohalosilanes SiF3X (X = Cl [64], Br [65], and I [64]). Thehalosilanes in which the halogen does not lie on a symmetry axis have been studiedless extensively by microwave spectroscopy, although partial sets of quadrupole couplingconstants have been determined for SiHC13 [66, 67], CH3SiC1 [66, 68], and SiH32I[69], and the complete coupling tensor x has been determined in the case of CH3Si2I[70, 71]. Electronic information derived from incomplete quadrupole tensors is usuallybased on the assumption that the z principal axis of the tensor is coincident with thesilicon-halogen bond. In CH3Si2I, this assumption was confirmed: the z axis and theSi-I bond were found to be separated by only ‘-‘ 41’ [71].It has traditionally been difficult to measure the complete Cl nuclear quadrupolecoupling tensors of asymmetric top molecules where the Cl nucleus is not on a symmetry75Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, S1H2Cl2 76axis. The quadrupole moments of the 35C1 and 37C1 nuclei are relatively small, and effectsof off-diagonal coupling terms such as Xab are difficult to see with conventional Starkmodulated spectrometers. Usually an accidental rotational near-degeneracy is requiredfor their effects to be seen (see, for example, Ref. [58].) However, with the adventof MWFT spectrometers these effects are now more readily observed, and Cl couplingtensors can be measured easily [72].The first measurement of the microwave spectrum of dichlorosilane, SiH2C1 wasmade 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 coupling nuclei was not possible, and not all the quadrupole coupling constants could bedetermined accurately. In particular, Xab was unavailable. In contrast, despite the existence of similar complications in the microwave spectra of CH21 [73, 74], CH2Br[75, 76, 77, 78], and CH2BrC1 [79], complete sets of coupling constants have been determined for these species. In light of evidence of (p —* d)?r back-bonding and asymmetric Clfield gradients in both SiCl2 [80] and the SiC1 radical [81], a more conclusive examinationof the bonding in dichiorosilane was warranted. The MWFT spectrometer described inSection 3.2 has facilitated the remeasurement of the microwave spectrum of dichlorosilane with much higher resolution. All values of the quadrupole coupling tensor havebeen determined accurately, allowing examination of the electrostatic potential aroundthe chlorine nuclei without prior assumptions about the nature of the Si-Cl bond [8].5.2 Experimental methodsA 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 thisChapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, SiH2Cl2 77percentage was increased for some higher-J transitions which required higher rotationaltemperatures. For the 918-82 7 transition, discussed extensively below, the SiH2C1:Armixing ratio was 1:2.The cavity MWFT spectrometer was used to make measurements in the frequencyregion 10-16 GHz. Measurements are estimated to be accurate to better than ±1 kllz.5.3 Observed spectrum and analysisFour of the previously reported [7] rotational transitions of dichlorosilane were re-examinedin this study, and seven transitions were examined for the first time. Most of the hyperfine components of the rotational transitions, which were often overlapped in the earlierwork, could now be resolved because of the small linewidths obtainable with the cavityMWFT spectrometer.Dichlorosilane has C2 symmetry, with the SiC12 moiety in the ab inertial plane. Themolecular symmetry axis is the b inertial axis. Two diagonal Cl quadrupole couplingconstants, Xbb and Xcc (or Xaa and Xbb— Xcc) may be determined, as may the off-diagonalcoupling constant Xab, although only the diagonal constants contribute to first order.Since most of the hyperfine structure may be accounted for without including Xab, initialassignments were made using first order patterns generated by a computer program written by the author for this purpose. The coupling of the spins of two identical quadrupolarCl nuclei to rotation was treated using the coupling scheme‘Cli + ‘Cl2 = II+J =F.To first order, only those matrix elements which are diagonal in J are included in thequadrupolar Hamiltonian; the matrix is still off-diagonal in I, and so must be diagonalized. The form of the matrix elements, which may be derived from Eq. 2.118, was takenChapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, Sill2Cl2 78from Ref. [74]. Relative transition probabilities P were calculated using [82]P cc <JIF’MF IitI JIFMF> 2 (5.1)I j F’ I Icc /(2F+1)(2F’+1) . (5.2)(FJ1JSince the HQ matrix is not diagonal in I, the matrix element in Eq. 5.1 must be transformed to the basis of eigenfunctions of HQ.Upon examination of relatively low-lying energy levels of the most abundant isotopomers of dichlorosilane, two near-degeneracies were found of the symmetry requiredfor second order perturbations via Xab• Both of these were for 28SiH35C1: 413 — 221134 MHz) and 99— 827 (‘ 64 MHz), as indicated on the energy level diagram givenin Fig. 5.1. It was hoped that measurement of transitions involving these levels wouldreveal 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 fordichlorosilane and fell within the frequency range of the spectrometer. The more easilyobserved transition, 4 3-322, was measured first. Surprisingly, the measured splittingpattern corresponded well with that predicted by a first order calculation, using the previously determined values of Xaa and Xbb— x [7]. Deviations were found, however, forthe second potentially perturbed transition, 918-827.As these perturbations could not be accounted for using the first order program, afitting program which could calculate exactly the effects of two coupling quadrupolarnuclei was required. At the author’s request, the exact nuclear quadrupole hyperfinepatterns were predicted by Ilona Merke of Universität Kiel, using her program Q2DIAG[83]. With values of rotational, centrifugal distortion, and diagonal quadrupole couplingconstants from the previous microwave study [7], and using a value of Xab consistent withplacing the z axis of the quadrupole tensor along the Si-Cl bond, the exact programChapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, SiH2Cl2 79300-,—‘ 1 ,9I U110___________2,71 00,10 2,865.4 MHz1,8‘• 1 ,9 2,69Q9 82,7N20°817182,52,68Q,31,6 2,41,7 6257Q,72,31,5 52,4100 661,62,242,31 ,40,515 341: 2?40,4 3J:3Q,3 21:_______134.2 MHz20,2_ ___1 1:0 1 0,1 0o,Figure 5.1: Rotational energy level diagram of28SiH35C1.Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, SiH2Cl2 80confirmed that neither 4 nor 221 should be noticeably perturbed by a Xab-type interaction. However, comparison of first order and exact predictions of energy levels indicatedthat hyperfine components of 827 were shifted by as much as 800 kHz. The effect of thisshifting on the 91 8-827 transition is shown in Fig. 5.2. The bottom spectrum of this figureshows transitions calculated by the first order program, the middle spectrum show transitions calculated by the exact program, and the top spectrum shows transitions whichwere measured, unambiguously assigned, and included in the final fit. The perturbationof 827 was confirmed as being due to an interaction with 919 via Xab by first reducing Xabslightly 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 firstorder splitting pattern.Measurements of the 9 8-827 transition agreed well with the exact prediction, permitting assignment of the perturbed hyperfine components. A portion of this transitionis displayed in Fig. 5.3. The Doppler-split spectrum predicted using Q2DIAG is shownbelow the observed spectrum, with dashed lines indicating transitions which were notincluded 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 transition frequencies with Q2DIAG. Besides 413—322 and 9 8—827, nine relatively unperturbedtransitions were also measured, primarily in order to refine the values of the diagonal nuclear quadrupole coupling constants for Cl. The measured transition frequencies andtheir assignments are in Table 5.1, aild the derived quadrupole coupling constants are inTable 5.2. Note that hyperfine levels are labelled by I and F, although I is not strictlya good quantum number. The Hamiltonian used in the fitting program, which includedChapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, Sill2Cl2 8115310.44 MHz15305.5 MHz 15313.4 MHz1MHz $I I I I. i HI. . iI.fittedexactcalculationfirst ordercalculationFigure 5.2: Schematic diagram of the 918 — 827 transition of28SiH35C1.Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, Sill2Cl2 82IF=2 10—2 915310.8 MHz1 10—1 915311.1 MHz3 11—3 10Figure 5.3: Portion of the 918 — 827 rotational transition of 28SiH35Cl2. Experimental conditions: gas sample 33% SiHC1/Ar; excitation frequency 15310.9 MHz; 150 nssample interval; 4 K FT; 200 averaging cycles.27—262 11—2 10Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, SiH2Cl2 83rotational, quartic centrifugal distortion, and nuclear quadrupole coupling terms, was sufficient to account for the observed spectra, and it was not necessary to consider nuclearspin-rotation coupling.Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, SiH2Cl2 84F’ - P’ F” frequency obs.-calc. I’ F’ - I” F” frequency obs.-calc.(MHz) (kHz) (MHz) (kHz)2 8 10281.76002 4 10281.77763 8 10283.56421 5 10285.64833 6 10286.16062 6 10286.72221 10 11528.64963 13 11528.86543 8 11529.76122 11 11530.13922 9 11530.15211 11 11530.23602 12 11530.25803 12 11530.57161 9 11531.52133 10 11531.69162 10 11531.76223 11 11531.95960 1 11889.29421 1 11890.67463 2 11890.68841 1 11894.04892 2 11895.75440 1 11895.83961 2 11898.11963 4 11898.16583 3 11900.88003 2 11901.07572 1 11902.20482 3 11902.31763 3 11906.33363 4 11906.47442 1 11908.75082 2 11908.86241 2 11909.04443 3 11911.80701 2 11911.87883 3 11914.64162 1 11915.31102i i-2o 231123212133211220235341213343311351334023433322424233331 2-303311232221435130333363123143 1 12243.44323 1 12243.45763 1 12243.54841 2 12244.71941 2 12244.74241 2 12244.80571 2 12244.84200 2 12245.10440 2 12245.27293 5 12247.69053 5 12247.80441 3 12249.14221 3 12249.16201 3 12249.28621 3 12249.31821 1 12255.00583 4 12255.09223 4 12255.07503 4 12255.20402 2 12256.34643 3 12256.71723 3 12256.75043 3 12256.66022 4 12250.68142 3 12250.69722 3 12250.71183 2 12250.80583 0 12780.12701 3 12780.76813 1 12781.54020 3 12781.71261 3 12781.82633 6 12782.86721 3 12782.93400 3 12782.97963 2 12783.45983 6 12784.49293 2 12785.55242 2 12785.93021 4 12786.5992-0.6-1.22.3-0.21.2-1.2-1.0-0.80.8-0.0-1.10.5-1.41.10.00.14.0-6.40.92.10.60.7-0.40.3-2.2-1.83.0-0.80.3-0.1-2.71.0-0.3-0.5-0.60.90.5-2.60.70.3523-6162 72 33 71 4352 5936-10291 93 123 72 10281 102 113 111 83 92 93 10110-1010 13 21 11 10 12 23 23 432330 12 33 4332 22 11 21 23 33 32 1-5.96.1-0.80.2-0.60.9-0.8-0.5-0.8-2.01.3-3.60.51.0-0.72.21.21.80.6-2.51.6-0.2-0.5-0.4-0.11.2-2.21.0-0.44.62.4-1.1-0.80.2-0.3-0.2-0.7-0.4-1.5Table 5.1: Measured transitions of28SiH35C1Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, S1H2Cl2 85I’ F’ - I” F” frequency obs.-calc. I’ F’ - I” F” frequency obs.-calc.(MHz) (kllz) (MHz) (kllz)312-303 cont’d2 5 2 5 12787.1825 -1.13 4 3 3 12788.4262 3.60 3 2 4 12788.4522 -1.83 4 3 5 12789.2548 0.73 3 3 3 12789.2793 -1.03 5 3 5 12789.9220 1.11 2 1 2 12790.1254 0.53 3 1 2 12790.3907 0.53 2 3 3 12790.6298 -1.61 4 3 5 12790.9026 -1.52 3 2 3 12791.3831 -0.23 6 3 5 12791.5480 2.23 4 3 4 12792.1068 0.82 4 2 3 12792.6398 2.53 5 3 4 12792.7747 1.83 3 3 4 12792.9619 -1.91 4 3 4 12793.7578 1.70 3 2 3 12793.9062 -1.94i 34043 2 3 1 13521.8690 0.21 5 1 4 13522.6848 0.33 1 3 1 13522.8550 2.03 3 1 4 13523.1022 -0.53 6 3 7 13523.1746 -2.63 3 3 2 13524.0053 2.71 4 1 4 13524.3476 -0.30 4 2 4 13524.3650 3.73 2 3 2 13524.9911 0.53 4 3 3 13525.2148 2,93 7 3 7 13525.4257 0.23 1 3 2 13525.9742 -0.62 4 2 5 13526.3024 -0.53 4 1 5 13526.3778 -1.43 6 1 5 13526.6222 0.63 3 3 3 13526.9980 0.81 5 1 5 13527.7466 0.43 2 3 3 13527.9844 -0.92 6 2 6 13528.0853 0.62 3 2 3 13528.1259 -1.72 5 2 5 13528.1418 1.71 4 3 3 13528.2454 3.0413-404 con’d3514353624373533343615325o 54143632332534381627351437053635322-413311331363214242535331323343 6 13528.9206 -0.51 5 13529.4086 -1.03 4 13529.8196 3.03 6 13530.0992 0.50 4 13531.8437 1.73 6 13532.3498 2.93 5 13532.5264 0.73 4 13532.5368 -0.23 5 13533,4594 -1.43 5 13533.7040 0.83 6 13531.2230 -0.31 3 13533.5902 0.73 6 13760.6478 -1.23 1 13760.9994 -0.03 2 13761.2558 -0.02 4 13762.0649 1.03 3 13762.3934 0.23 7 13762.7406 0.21 5 13763.2374 0.22 6 13763.5292 0.53 4 13763.9754 0.91 3 13764.4083 -0.63 6 13764.7618 0.50 4 13764.9974 0.13 5 13765.3228 0.33 5 13767.4826 -2.63 1 14282.1064 -1.71 4 14282.8344 3.03 2 14283.0912 -1.13 7 14283.3716 2.03 3 14284.0828 -0.91 5 14284.4990 1.02 5 14284.6088 1.62 6 14284.6634 0.43 6 14285.6217 0.23 4 14285.8635 -2.71 3 14285.9060 -0.72 4 14286.4452 0.53 5 14286.7954 -1.2Table 5.1: Measured transitions of28SiH35C1 (cont’d)Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichlorosilane, S1H2Cl2 86I’ F’ - I” F” frequency obs.-calc. J F’ - F” frequency obs.-calc.(MHz) (kHz) (MHz) (kllz)5i 4-505 9 8-827 coni ‘d3 3 3 2 14488.3368 1.0 2 8 2 7 15310.4385 -2.12 4 2 5 14488.7585 -1.4 2 10 2 9 15310.8915 -0.81 6 1 5 14488.8877 1.9 2 7 2 6 15310.9850 0.93 4 1 5 14489.1314 -3.0 2 11 2 10 15311.0162 -3.33 7 3 8 14489.1760 -3.7 3 9 3 8 15312.1648 -3.43 2 3 2 14489.7120 -0,7 1 8 1 7 15312.1876 3.20 5 2 5 14490.8588 0.1 615-6061 5 1 5 14490.8588 5.2 3 4 3 3 15704.9720 0.63 8 3 8 14491.7273 -0.9 3 5 1 6 15705.5386 0.73 3 3 3 14492.0914 -0.3 3 8 3 9 15705.5812 -4.22 5 2 6 14492.3554 -0.9 3 3 3 3 15706.6134 0.83 4 3 4 14493.8091 -0.0 3 9 3 9 15708.3178 -0.31 6 1 6 14494.2205 -0.5 3 4 3 4 15709.2231 1.32 6 2 6 14494.4538 -0.9 3 5 3 5 15710.7345 -0.11 5 3 4 14495.5312 2.9 1 7 1 7 15710.9864 0.23 7 3 7 14496.0984 -0.0 2 7 2 7 15711.1594 -1.21 4 1 4 14497.1055 0.1 3 8 3 8 15712.5884 -0.83 5 3 5 14497.3277 0.2 3 7 3 6 15713.5162 3.82 5 0 5 14498.0477 0.8 1 5 1 5 15713.9140 0.991 8-827 3 6 3 6 15714.2344 0.23 6 3 5 15308.3096 0.3 2 6 2 6 15714.7439 0.21 9 1 8 15308.8930 8.8 3 7 3 7 15715.3350 0.53 12 3 11 15309.4208 -0.8 3 6 3 7 15716.0546 -1.83 7 3 6 15309.5866 -1.4 3 4 1 5 15716.9212 -3.53 8 3 7 15310.2523 -1.6 3 8 3 7 15717.1211 3.0Table 5.1: Measured transitions of28SiH35C1 (cont’d)5.4 DiscussionThe 35C1 quadrupole coupling constants are compared in Table 5.2 with those fromthe earlier study of SiH2C1 [7]. Xbb and Xcc are the same for both 35C1 atoms, andtheir absolute signs have been determined. Not oniy is there an improvement in theirprecision, but also the new values are slightly outside the estimated uncertainties of theearlier work. The off-diagonal constants, Xab , have been measured directly for thefirst time. In this case the absolute signs have not been determined as they enter toChapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, SiH2C1 87previous work’ this workInertial axis systemXbb (MHz) 013(23)b -0.3095(19)Xcc (MHz) 21.00(23) 20.7245(19)I Xab I (MHz) — 26.099(74)Principal axis systemXzz (MHz) 42.0(7)c -38.33(13)x (MHz) 21.0(6) 17.60(13)Xyy (MHz) 21.00(23) 20.7245(19)za 35.140c 34.467(27)°a Values are taken from Ref. [7]. Uncertainties wereestimated outside limits.b Numbers in parentheses are one standard deviationin units of the last significant figure.c Note that previous principal values of the Cl nuclearquadrupole coupling tensor were calculated assumingcoincidence of the z principal axis of the tensorand the Si-Cl bond [7], and thus Oza is the anglebetween the a inertial axis and the Si-Cl bond.Table 5.2: Chlorine nuclear quadrupole coupling constants of SiH2C1Chapter 5. The Cl Nuclear Quadrupole Coupling Tensor of Dichiorosilane, SiH2Cl2 88second and higher order in the nuclear quadrupole coupling energy, although they mustbe opposite for the two atoms [74].The results of diagonalizing the 35C1 quadrupole coupling tensor are also shown inTable 5.2. The principal values of the tensor are compared with those of the previouswork [7], where it was assumed that the z principal axis of the tensor lay along the Si-Clbond. It is clear that the assumption was not unreasonable, although the present resultshave produced slightly different constants. The diagonalization produces Oza, the anglebetween the z principal axis of the tensor and the a principal inertial axis. The valueobtained 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 —Xyy)/Xzz) is very nearly zero in SiH2C1, implying that the Si-Cl bond is essentiallycylindrically symmetric; this would seem to justify the assumption made in estimatingXzz in SiHC13. The value for Xzz of SiH2Cl2 is now found to lie between those of SiH3C1and SiHC13, in contrast to what was found earlier [7]; there is now a logical sequence,paralleled by the trend in the Si-Cl bond length. The coupling constant Xzz is similarto those of CH3SiHC12 and (CH3)2SiC1,although for those molecules Xzz is relativelypoorly determined. For the molecules containing tetravalent Si there does not seem tobe a general correlation of Xzz with Si-Cl bond distance, with the exception of the onenoted above. Only for SiCl2 and SiC1 are the coupling constants markedly different. Thelarge value for in SiC12 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 SiH2C1,however, suggest thatthe 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, SiH2Cl2 89molecule Xzz (MHz) method r(Si-Cl, A) Ref.SiH2C1 -38.33(13) 0.082(3) b 2.O33f this workSIH3C1 -39.70(7) 0 d 2048 [60]SiHC13 -37.2 0 c,e 2.012 [7, 84]CHSiHC12 -41.2(20) 0.16(3) c 2.040 [85](CH3)SiC1 -38.0(16) 0.22(8) c 2.055 [86]SiCl2 -36.08(35) 0.859(18) c 2.070 [80]SiC1 -23.13(96) 0 d 2.061 [81]SiF3C1 -39.83(16) 0 d 1.996 [64]CH21 -76.2(40) 0.0(21) b — [74]CH21 -76.92(2) 0.0356(3) c — [74]a‘i=(x — Xyy)/Xzz.b Determination of Xab and diagonalization of the 35C1 quadrupole tensor.Assuming Si-Cl bond is a principal axis of the 35C1 quadrupole tensor.d In linear and symmetric top molecules where the quadrupolar nucleus is on thesymmetry axis, the measured quadrupole coupling constant is Xzz and the bondis cylindrically symmetric.Cylindrical symmetry assumed for the Si-Cl bond.Ref. [7].‘ Ref. [87].Table 5.3: Comparison of 35C1 principal quadrupole coupling constants and Si-Cl bondlengthsChapter 6The Microwave Spectrum of Tetrolyl Fluoride6.1 IntroductionAccording to basic concepts of chemical bonding, a methyl group attached to a nonlinear acetylene fragment -CC-R should experience essentially free rotation. Microwavespectroscopic studies of several molecules of the form CH3-CC-M (where M = CD3[88], CF3 [28], SiR3 [89], and CH21 [90]) have confirmed that the threefold barrier tointernal rotation of the methyl group is indeed low, and where V3 has been determinedit has been found to be no greater than 10 cm’. The infrared spectrum of tetrolylfluoride (2-butynoyl fluoride, CH3-CC-COF) also gave evidence of free rotation of themethyl group [91].Molecules with low barriers to internal rotation exhibit very crowded microwavespectra, as lines due to different torsional states may be very closely spaced. Assignment 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 overcome the problem of spectral congestion in the case of CH3-CC-CH2l [90]. In thework described here, the microwave spectrum of tetrolyl fluoride has been obtained usinga cavity MWFT spectrometer. As an alternative to MW-MW DR, the efficient coolingin the molecular beam of the spectrometer has been used ‘freeze out’ all but the lowest torsional states of tetrolyl fluoride. This has vastly reduced the congestion in themicrowave spectrum and simplified the assignment of lines.90Chapter 6. The Microwave Spectrum of Tetrolyl Fluoride 916.2 Experimental methodsTetrolyl 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 ‘9F nmr, as well as by comparison of thegas phase infrared spectrum to that obtained by Balfour et al.[91]. As the compound isunstable at room temperature and turns brown over time, the sample was kept at liquidnitrogen temperatures when not in use and remained colourless over the period of thisstudy.Preliminary microwave studies were performed on the Stark-modulated spectrometerdescribed in Section 3.1. For the most part, scans were performed in the frequency region8-18 GHz, as this was the region available on the cavity MWFT spectrometer at the timeof these experiments. Spectra were recorded both with and without dry ice cooling ofthe Stark cell. The tetrolyl fluoride was warmed to room temperature in order to fillthe cell, and spectra were recorded at pressures of 30-50 mtorr. At these pressures, thesample remained stable in the cell for at least an hour, independent of the temperatureof the cell.High resolution microwave spectra of tetrolyl fluoride were recorded with the cavityMWFT spectrometer (see Section 3.2), using gas mixtures of 1-3% tetrolyl fluoride inargon.6.3 Results and discussionGiven the anticipated structure of tetrolyl fluoride, it was expected that the dipole moment would lie very nearly along the a inertial axis, resulting in strong a-type rotationaltransitions. The initial microwave spectra recorded with the Stark spectrometer revealedthe typical a-type R-branch pattern, with groups of strong, equally spaced, unresolvedChapter 6. The Microwave Spectrum of Tetrolyl Fluoride 92lines. 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 frequencyregion of the cavity MWFT spectrometer. Since their frequencies were roughly in theratio 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 rotormodel.6.3.1 Prediction of transition frequenciesA computer program was written to predict rotational transition frequencies for a moleculewith one internal rotor and a planar framework. The Principal Axis Method (PAM) waschosen, as the symmetry axis of the methyl group in tetrolyl fluoride was expected to lienearly along the a principal axis. When the principal axes of a molecule are chosen suchthat the frame lies in the xz plane, the Hamiltonians given in Eqs. 2.148-2.150 may berewritten as [28]Hrot = A’J 2 + B’J 2 + C’J, 2 + Fpp(J J + J J) (6.1)Htorsion Fj 2 + V(o) (6.2)Hrot_torsion = —2F(pJj + pJ2j). (6.3)The computer program has been designed to solve the Hamiltonian in steps, after themethod of Anderson and Gwinn [93]. The rigid rotor part of the Hamiltonian, Hrot, istreated first, using symmetric rotor basis functions. The matrix elements of Hrot in thisbasis are given by<JK IHrot I J1> = A’K2 +(B’ + C’) [J(J + 1) — K2] (6.4)<j K + 1 IHrot JK>= (2K + 1)[J(J + 1) — K(K + 1)]h/2 (6.5)Chapter 6. The Microwave Spectrum of Tetrolyl Fluoride 93<J K±2HrotjJK>= (B’ — C’) [J(J + 1) — K(K + 1)]’/2[J(J + 1) — (K ± 1)(K + 2)11/2. (6.6)Diagonalization of this part of the Hamiltonian leads to eigenvalues and eigenvectorssimilar to those of an asymmetric rigid rotor; a Wang transformation does not producea 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,using free rotor basis functions of the form given in Eq. 2.136. Assuming that V6 issufficiently small that V(c) may be truncated after the V3 terms (Eq. 2.133), the matrixelements of the torsional Hamiltonian in the free rotor basis are given by<m IHtorsjon m> = Fm2 + (6.7)<m ±3 Htorsion m> = —. (6.8)The introduction of a non-zero threefold barrier to internal rotation thus introducesoff-diagonal elements in Htorsion. Free rotor states with m40 are doubly degenerate,but the barrier acts to remove the degeneracy for states with m a multiple of 3, asshown in Fig. 2.1. H010 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 highm has little effect on small values of m , and only small values of m were expectedto be seen for tetrolyl fluoride with the cavity MWFT spectrometer, it was decided todiagonalize 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 ; jChapter 6. The Microwave Spectrum of Tetrolyl Fluoride 94is also transformed to the basis of eigenfunctions of Htorsion, with A and F submatricesconsidered separately throughout. If the final Hamiltonian matrix is written in terms ofbasis 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 ) and (—2FJ xj ), where the matrix elements of A x B =C are given byC,kl = AkBl. (6.9)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 treatthe torsional part of the Hamiltonian separately using free rotor basis functions hingeson the assumption that internal rotation is essentially unhindered, an assumption thatwas made early in this study of tetrolyl fluoride.6.3.2 Selection rulesThe selection rules for rotational transitions may be determined by considering the permutation inversion group of a molecule such as tetrolyl fluoride, which has a planar frameand for which the possibility of torsional tunnelling exists. The character table for therelevant permutation inversion group, G6, is given as follows:G6 E 2.(123) 3.(23)*A1 1 1 1A2 1 1 -lF 2 -1 0where F is the identity operation, (123) is an operation which is equivalent to rotation ofthe methyl group by 2nir/3 radians (n = 1, 2,...) about its symmetry axis, and (23)* isequivalent to a combined rotation of the methyl group by o= 2n7r/3 radians and rotationof the entire molecule by ir radians about the out-of-plane axis y. In this permutationChapter 6. The Microwave Spectrum of Tetrolyl Fluoride 95inversion group, both the a and b components of the dipole moment are of symmetryA2. In order to determine the selection rules for rotational transitions, it is necessary todetermine the symmetries of the eigenfunctions of Hrot_toion.The extent to which these eigenfunctions are properly labelled by the KaKc asymmetric rotor labels and the m torsional quantum number depends on both V3 and 0, theangle between the top symmetry axis and the z principal axis. A large barrier to internalrotation results in greater mixing of the free rotor states in Htorsjon, while 0 determinesthe size of F, which in turn determines both the deviation of Hrot from a pure asymmetric rigid rotor model and the contribution of cross terms in the final Hamiltonian.V3 was expected to be small for tetrolyl fluoride, in which case m is nearly a goodquantum number for torsional states of E symmetry, while states of A symmetry maybe approximated as the following linear combinations:+m > + —m>): A1 symmetry (6.10)+m >——m >) : A2 symmetry. (6.11)The angle 0 was also expected to be small (i.e. p—O, pz*l), making the factorFpp2 in Hrot almost zero. Diagonalization of Hrot will therefore give asymmetric rigidrotor wavefunctions for which the labels KaKe may still be used. The 2FpJ,j crossterm in Hrot_torsion will also be small, but the 2FpJ j term will be significant. As aconsequence, considerable mixing of asymmetric rigid rotor states with the same value ofKa occurs. The K label is thus meaningless, although it has been retained in order todistinguish between the resulting Ka states. The symmetry species of the rotation-torsionstates are preserved in this mixing.The computer program described above also calculates approximate relative intensities for transitions with low values of m.Using the direction cosine matrix elements ofCross, Hainer, and King [13], the dipole moment matrix is calculated in the symmetricChapter 6. The Microwave Spectrum of Tetrolyl Fluoride 96rotor 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 intensitiesof 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 havem = 0. However, transitions such as 303— 3 , which are nominally c-types but whichdepend 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, thetransitions should be vanishingly weak in tetrolyl fluoride, where 0 is expected to be lessthan 506.3.3 AssignmentsThe cavity MWFT spectrometer was used to search for some of the strongest lines observed 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 spectra, leaving a small number of lines. Some of these could be fit to a rigid rotor modelas Ka=O and Ka2 transitions, with initial rotational constants calculated from a trialstructure. This made it possible to search for Kal satellites, which had not been identified in the Stark spectra; these were soon found.Searching was facilitated by the large a component of the dipole moment of tetrolylfluoride, estimated to be —3 D by comparison with similar molecules. Thus, despitethe small bandwidth of the spectrometer, searches for a-type lines could confidently beperformed in 1 MHz steps, with short exciting pulse lengths (0.1 its). Very strong signalswere obtained for many transitions. An example is the line at 12598.412 MHz shownin Fig. 6.1, later assigned to the m =0, 414-313 transition; this transition could beobserved with no signal averaging.With reasonable initial values for the rotational constants, 0, and V3, all of the linesChapter 6. The Microwave Spectrum of Tetrolyl Fluoride 9748 kHz125983 1298. MHzFigure 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 sampleinterval; 8 K FT; 1 averaging cycle.MHzChapter 6. The Microwave Spectrum of Tetrolyl Fluoride 98observed thus far using the cavity MWFT spectrometer could be assigned. Those lineswhich corresponded to an asymmetric rigid rotor spectrum could be assigiled to stateswith 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 subsequentlyobserved with the cavity spectrometer. The measured transition frequellcies and assignments are given in Table 6.1. In order to obtain spectroscopic constants, the programVC3IAM [94] was used to fit the data. Although this program uses the internal axismethod, frequencies predicted by the PAM prediction program described above agreedvery well with those predicted by VC3IAM.While many attempts were made to locate b-type transitions, no such transitions wereobserved. 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 a-type lines. The b component of the dipole momeilt is due entirely to the COF group, andshould be relatively small. For the related molecule CH3OF, Pb=°•88 D, compared to4Ua2.83 D [10].As only a-type transitions were observed, the A rotational constant cannot be determined. In addition, A and I, the moment of inertia of the methyl group about itssymmetry 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 thedata, ‘c’€ was assumed to have the value 3.18 a.m.u.A2 [95] and A was released into thefit. However, since the a-type transition frequencies are insensitive to the value of A, itwould remain near its initial value. The lowest standard deviation was obtained withA ‘.‘11050 MHz, and so this was used for the initial value in the final fits. The centrifugaldistortion constants were set to zero in the fits, as attempts to determine D arid DJKChapter 6. The Microwave Spectrum of Tetrolyl Fluoride 99ImI=0 ImI=lJ4’ J(’ j(’ J” K K’ frequency obs.-calc. frequency obs.-calc.(MHz) (kllz) (MHz) (kHz)3 1 3 2 1 2 9452031b 0.466 9778.413 0.5363 0 3 2 0 2 9777.556 -0.535 9838.027 0.3033 2 2 2 2 1 9793.147*,c 0.089 9746.914* 0.4853 2 1 2 2 0 9808.165* 0.139 9796.063* -0.5333 1 2 2 1 1 10129.989 0.118 9798.918 -0.0894 1 4 3 1 3 12598.412 0.603 13020.406 0.7404 0 4 3 0 3 13019.263 -0.773 13165.910 0.4124 2 3 3 2 2 13054.608 0.109 12938.029 0.5124 3 2 3 3 1 13064.459* -0.325 13075.727* 0.3454 3 1 3 3 0 13064.689* -0.327 13061.567* -0.6394 2 2 3 2 1 13092.091 0.217 13063.362 -0.7164 1 3 3 1 2 13502.172 0.136 13070.045 -0.1135 1 5 4 1 4 15741.226 0.726 16246.069 -0.4715 0 5 4 0 4 16247.566 0.367 16526.493 0.4845 2 4 4 2 3 16313.559 0.115 16088.505 0.4645 4 2 4 4 1 16328.843* -1.240 16335.322* -0.1275 4 1 4 4 0 16328.843* -1.243 16327.630* -0.1775 3 3 4 3 2 16333.820 -0.411 16356.106 0.4695 3 2 4 3 1 16334.636 -0.407 — —5 2 3 4 2 2 16388.336 0.329 16332.337 -0.8965 1 4 4 1 3 16870.396 0.134 16345.380 -0.127a 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 anaverage.Table 6.1: Measured transitions of tetrolyl fluorideChapter 6. The Microwave Spectrum of Tetrolyl Fluoride 100normal isotopomer ‘3C substitutedA (MHz) 11049.705(90)a 11049705bB (MHz) 1745.221(17) 1744.601(54)C (MHz) 1519.119(19) 1518.056(93)I (a.m.u. A2) 318b9 1.5092(14)° 0.9994(25)°V3 (cm’) 2.20(12)fit (MHz) 0.545 1.118a Numbers in parentheses are one standard deviation in units ofthe last significant figure.b Held constant in the fit.Table 6.2: Spectroscopic constants of tetrolyl fluoridewere unsuccessful. Determined in the fit were the B and C rotational constants, 9, and V3,as given in Table 6.2. As expected, 0 is small; the threefold barrier to internal rotationV3 is small (.—‘2.2 cm’) and very comparable to those determined for CH3-CC-CD(5.6 cm1 [88]) and CH3-CC-SiH (3.8 cm’ [89]).In the course of searching for b-type transitions, several weak lines were found whichcould be assigned to m =zO and m =1 a-type transitions of a ‘3C isotopomer of tetrolylfluoride. The measured transition frequencies and assignments are given in Table 6.3.In order to fit the limited number of lines, A, ‘a and V3 were constrained to the valuesobtained in the fit for the normal isotopomer; the spectroscopic constants obtained aregiven in Table 6.2. The isotopic shifts in B and C are quite small, indicating that isotopicsubstitution has taken place very near the centre of mass. From a rough calculation ofthe expected structure of tetrolyl fluoride, the isotopomer is probablyCH3-C’-C0F.No lines were observed which could be assigned to m =2 or 3 transitions. Theabsence of transitions with high values of m is not unreasonable considering the energiesof such states (which increase rapidly because of the Fj 2 term in Htorsjon ) and the efficientChapter 6. The Microwave Spectrum of Tetrolyl Fluoride 101jmI=0 ImI=1j! J( j(a J” I(j K’ frequency obs.-calc. frequency obs.-calc.(MHz) (kllz) (MHz) (kllz)3 1 3 2 1 2 — — 9772721b -1.3583 0 3 2 0 2 9773.601 0.636 9833.142 0.7723 1 2 2 1 1 10124.843 -0.604 — —4 1 4 3 1 3 12592.392 2.240 13014.021 -0.3044 0 4 3 0 3 13012.853 -0.285 13159.336 0.9174 2 2 3 2 1 13085.581c 0.337 — —4 1 3 3 1 2 13495.317 -0.803 — —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‘3C-tetrolyl fluoridecooling in the molecular beam. While collisional relaxation of m =1 states to rn =0is symmetry-forbidden, depopulation of torsional states with higher values of m ispermitted in the beam. With only m =0 and m =1 transitions, it is impossible totest the validity of the assumption that V3 is the dominant term of V(a), as V6 cannotbe determined [96].It is important to note that the standard deviations in the fits were considerablylarger than the uncertainty in the measurements; for the larger data set of the normalisotopomer, Ufit is roughly 500 times larger than the measurement uncertainty. Largediscrepancies between observed and calculated transition frequencies have been foundin other internal rotation studies (e.g. CH3ONO [95]). A probably source of thesediscrepancies is the use of rigid top - rigid frame models, which neglects interactionsbetween 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 intoChapter 6. The Microwave Spectrum of Tetrolyl Fluoride 102two and/or three sublevels, with the splittings on the order of 10-30 kHz. As multipletswere observed for both m =0 and m =1 states in all cases, it seems unlikely that themechanism of the splitting is related to internal rotation. However, none of the likelymechanisms (‘9F and/or ‘H spin-rotation or spin-spin coupling) split the J=tKa levelspreferentially, but rather affect all JKaK > states to comparable extents. It is possiblethat terms which .contribute to splittings of the other levels cancel out accidentally, butthis is difficult to determine, especially without a fitting program which will simultaneously fit the hyperfine effects of four coupling nuclei in an asymmetric rotor. Even withsuch a program, the large residual frequencies mentioned above could make it difficult tofit very small hyperfine parameters.Chapter 7Microwave Spectra of Metal Halides Produced Using Laser Ablation7.1 IntroductionRecent experiments in other laboratories have demonstrated the feasibility of coupling alaser vaporization source to a cavity MWFT spectrometer in order to produce and studydiatomic metal-containing species in the gas phase [98, 99, 100, 101]. Rapid vaporizationof a metal or metal oxide target produces transient species which may be reacted withother species that are present in small concentrations in a rare gas. Stabilization ofthe product is then achieved via the supersonic expansion of the mixture through anozzle into the microwave cavity. By this method, molecules which would otherwise beproduced in highly excited states in an oven or electrical discharge are generated at verylow rotational temperatures.In the work described here, laser ablation has been used to prepare seven diatomicmetal halides: AgC1, A1C1, CuC1, InCl, InBr, InF’, and YC1. The rotational spectra ofthese molecules have been measured using a cavity MWFT spectrometer. A major aimof this work was the optimization of the experimental conditions necessary to prepareboth stable and unstable metal-containing species in the gas phase. As a result of thisoptimization, 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, manypreviously known spectroscopic constants of the metal halides have been determined103Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 104considerably more precisely in this study. The effects of nuclear quadrupole couplinghave been observed in the spectra of all the metal halides studied here, and nuclearquadrupole coupling constants have been obtained which are a major improvement overearlier values. Nuclear spin-rotation coupling constants, which are unattainable at lowerresolution, have also been determined with relative ease. For some of the diatomic metalhalides discussed here, interpretation of these coupling constants provides a glimpse ofthe structures of the excited electronic states of the molecules [9].7.2 Experimental methodsThe experimental arrangement for the laser ablation experiments described here is shownschematically in Fig. 7.1. The output of a Q-switched Nd:YAG laser is focused outside themicrowave cavity of the MWFT spectrometer by a 50 cm focal-length lens. The radiationthen passes through a quartz window and a small hole in the tuning microwave mirrorbefore traversing the cavity to hit the target metal rod, which is fixed in front of the gasnozzle in the stationary microwave mirror. Both Bosch and General Valve nozzles wereused for these experiments, with no noticeable difference in signals obtained. However,the General Valve nozzles were much more reliable when corrosive gas mixtures were inuse.A brass cap fits on the end of the nozzle and holds the target metal rod directlyin front of it, as is shown in Fig. 7.2. The laser radiation enters the cap through theexit channel, and the laser is aligned visually using a low power setting. The targetrod is typically 1 cm long and 2 mm in diameter, while the exit channel is 1 mm indiameter; the rod thus serves to protect the end of the nozzle from damage caused bydirect 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 aChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 105windownozzlestationarymirrorlenstuningFigure 7.1: Schematic diagram of the arrangement of the cavity MWFT spectrometerand the Nd:YAG laser used in these experiments.to pump mirrorChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 106few percent in neon. Both the laser and the nozzle are pulsed, and the delay betweenthe pulse of the gas mixture and the laser pulse is adjusted to optimize the productionof metal halides. A small chamber, approximately 0.04 cm3 in volume, surrounds therod, giving sufficient time for reaction before the backing gas carries the metal halidemolecules through the exit channel. The molecules are stabilized by collisions in thechannel 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. AgC1is a stable molecule, and was successfully produced using a variety of chlorine/neongas mixtures with a wide range of Cl2 concentrations. The other six metal monohalidesproduced here are less stable: for example, A1C1 is less stable than other, more chlorinatedpotential reaction products such as A1C13/A12Cl6. Low concentrations of the reactant gas(‘—0.1%) were found to favour production of the monohalides. Total static pressures ofthe gas mixtures were typically 1-2 atm. The delay between the nozzle pulse and thelaser 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 rodwithout opening the vacuum chamber, and consequently the laser pulse hit the samespot 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 rodwould sometimes give weak signals which improved as more laser pulses were fired. Theenergy of the laser pulses was kept near the threshold value for the production of metalchlorides. 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 severalhours of operation; ‘softer’ metals such as indium are ablated more rapidly.Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 107cap532 nmrod532 nmFigure 7.2: Schematic diagrams of the nozzle caps which hold the sample rod in front ofthe nozzle (Bosch or General Valve.)Generalvalvesample rodChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 1087.3 Observed spectra7.3.1 AgC1The rotational spectrum of silver chloride, AgC1, has been studied previously using high-temperature microwave [102, 103] and millimeter-wave [104] spectroscopy. More recently,the cavity MWFT spectrometer used in this work was coupled to an electric dischargesource, and AgC1 was produced by applying a pulsed high voltage to Ag electrodes inthe presence of Cl2 [105].The laser ablation technique has now been employed to produce AgC1, using a silvertarget rod and C12/Ne gas mixtures of varying concentrations (—0.05%—2%). The rotational transitions J=1-0 and J=2-1 have been observed for four isotopomers of silverchloride, ‘°7Ag35Cl (38.7% natural abundance), ‘°9Ag35C1 (36.7%), 107Ag37C1 (12.6%),and ‘°9Ag37C1 (12.0%). These rotational transitions are split by nuclear quadrupolecoupling of the Cl nucleus (I =3/2); the measured transition frequencies and their assignments for the four isotopomers are given in Table 7.1, where frequencies from boththe electric discharge [105] and laser ablation studies are given. Each hyperfine transitionis labelled by the quantum number F, whereTci +J =F.Many hyperfine components could be seen for the 35C1 isotopomers with virtually nonoise even without any signal averaging, while four hyperfine components of the J=2-1transition have been observed for 107Ag37C1 and 109Ag37C1 which could not be observedusing the electric discharge method [105]. Clearly, the laser vaporization technique is amore efficient method for producing AgC1 in the gas phase. By comparison of transitionsobserved using both methods, the laser vaporization method is estimated to improve thesignal-to-noise ratio by one to two orders of magnitude for this molecule.Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 109‘°7Ag35C1 ‘°9Ag35C1F’ - F” frequency obs.-calc. frequency obs.-calc.(MHz) (kllz) (MHz) (kllz)J=1_01.5 1.5 7348.7934a 0.0 7315.5410 0.02.5 1.5 7357.9022 0.0 7324.6502 0.40.5 1.5 7365.1925 0.0 7331.9397 -0.4J=2-11.5 0.5 147030059D,b -0.4 146365008D -0.42.5 2.5 147037851 0.4 146372800 0.40.5 0.5 14712.1132 0.5 14645.6068 -0.7} 14712.8936’ 14646.3892D {1.5 1.5 147194055D 0.1 146529004D 0.2‘°7Ag37C1 ‘°9Ag37C1F’ - F” frequency obs.-calc. frequency obs.-calc.(MHz) (kllz) (MHz) (kllz)J=1-01.5 1.5 7051.2379 -0.1 7017.9832 0.82.5 1.5 7058.4171 0.3 7025.1606 -0.80.5 1.5 7064.1617 -0.2 7030.9068 0.0J=2-11.5 0.5 14106.7389 -0.3 14040.2300 0.02.5 2.5 14107.3536 0.5 14040.8427 -1.20.5 0.5 14113.9155 -0.9 14047.4096 2.1} 14114.5326 { 14048.0229 {:1.5 1.5 14119.6632 0.2 14053.1540 -0.4a Measurement accuracy is estimated to be better than ±1 kllz.b Frequencies denoted by D are taken from the electric discharge study [105].Table 7.1: Measured transitions of AgC1Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 1107.3.2 A1C1Aluminum (I) chloride has been the subject of both microwave [106, 107] and millimeter-wave [108] spectroscopic studies, both performed at high temperatures. Recently, ahigh-resolution infrared emission spectrum of A1C1 was recorded to refine the Dunhamcoefficients used to parameterize rovibrational transition frequencies [109]. The rotationalspectrum of A1C1 is of particular interest, since A1C1 has been detected in the interstellargas 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 aC12/Ne gas mixture of low chlorine percentage (-.-0.1%). A1C1 exists as two isotopomers,27Al35C (75.5% natural abundance) and 27A13C (24.5%). Previous analyses of thehyperfine structure of A1C1 (IA1 =5/2, Ici =3/2) were limited to the more abundant isotopomer [106, 107]. In this work, hyperfine components of the J=1-0 rotational transitionhave been observed for both isotopomers.The complex hyperfine patterns for A1C1 consist of groups of overlapped hyperfinecomponents. A section of the spectrum of 27A13C is given in Fig. 7.3; this shows threeDoppler doublets corresponding to eight hyperfine transitions. Each hyperfine transitionis labelled by the quantum numbers F1 and F, corresponding to the coupling schemeIAi+J = F1Ici+Fi = F.The measured transition frequencies for A1C1 are given in Table 7.2.7.3.3 CuC1In the present study, copper (I) chloride was formed by ablation of a copper rod in thepresence of a Cl2/Ne gas mixture. Formation of CuC1 was particularly sensitive to theChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 1111.5,1 — 2.5,2? 2.5,114239.2 MHzç 2.5,32.5,22.5,114239.9 MHzFigure 7.3: A portion of the J=1-0 rotational spectrum of 27A13C1, showing three setsof overlapped hyperfine components. Experimental conditions: Al rod; 0.05% C12/Negas sample; 14239.397 MHz excitation frequency; 50 ns sample interval; 4 K FT; 200averaging cycles.P,F= 1.5,22.5,41.5 — 2.5,385 kHz 2.5,2___I IChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 11227A135C 27A135CF F’ F’ F” frequency obs.-calc. frequency obs.-calc.(MHz) (kllz) (MHz) (kllz)a Observed frequency for overlapped hyperfine components. Measurementaccuracy is estimated to be better than ±1 kllz.2.5 1.0 2.5 14571.0739a 14228.97132.5 4.0 2.5 14571.8527 14229.60862.5 2.0 2.5 2.0 14572.4124 0.9 14230.0720 1.21.0 J 0.4 0.83.0 1-0.3 1-0.32,5 3.0 2.5 2.0 14573.5586 -0.6 14230.9376 -0.54.0 J (-0.8 (-0.73.0 -0.8 0.03.5 3.0 2.5 2.0 14577.9643 -1.0 14235.8264 -0.23.5 4.0 2.5 14578.3261 14236.00303.5 5.0 2.5 4.0 14579.4010 0.3 14236.8830 0.53.0 0.5 0.43.5 2.0 2.5 2.0 14579.6313 0.3 14237.1183 0.21.5 1.0 2.5 14581.8121 14239.34761.5 2.0 2.5 2.0 14581.9938 0.0 14239.39251.0 J (-o.s (-0.61.5 0.0 2.5 1.0 -0.3 —3.0 0.5 0.31.5 3.0 2.5 2.0 0.24.0 0.014582.104414582.5121 14239.8170Table 7.2: Measured hyperfine components of the J=1-0 transition of A1C1Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 113concentration of Cl2 in the mixture, and so low concentrations (O.1%) were necessaryin order to preclude formation of such products as CuC12. CuC1 has been studied beforein both the microwave [111] and millimeter-wave [112] regions, but nuclear quadrupolecoupling constants for Cu and Cl (‘Cu =Tci =3/2) had been determined for only the mostabundant isotopomer, 63Cu351 (52.2% natural abundance). In this work, the J=1-o rotational transition has been observed for 63Cu35l, 65Cu31 (23.3%), and 63Cu371(16.9%). As in the case of A1C1, the J=1-O transition is composed of overlapped hyperfinecomponents. Several hyperfine components of the J=2-1 transition were also measuredfor Cu35l and 65Cu31. The observed frequencies and their assignments are given inTable 7.3, where the transitions are labelled according to the coupling schemeIci+J = F11C+F1 = F.Very recently, a laser ablation/cavity MWFT study of the J=1-O transition in CuC1 wasreported [113], with all of the observed frequencies of the hyperfine components fallingwithin experimental uncertainty of the frequencies reported in this work.7.3.4 mCIThe pure rotational spectrum of indium (I) chloride has been observed previously viamicrowave [114, 115, 116, 117] and millimeter-wave [118] spectroscopy. The large nuclearspin of indium (I =9/2), coupled to the nuclear spin of Cl as well as to the rotationof the molecule, results in a large number of hyperfine components for even the J=1-Orotational transition. In the present work, InCl has been produced by ablation of In inthe presence of a dilute Cl2/Ne gas mixture (‘.O.1% to avoid formation of the more stableInCl3.)Indium is easily ablated from the ‘soft’ In target rod, and InCl is stable enoughChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 11463Cu351 65Cu31 63Cu371F F’ F’ F” frequency obs.-calc. frequency obs.-calc. frequency obs.-calc.(MHz) (kllz) (MHz) (kllz) (MHz) (kllz)J=1-03.0 ‘ (1.8 ( 2.0 ( 2.71.5 2.0 1.5 1.0 10648.3416a 1.8 10531.5532 2.0 10280.0472 2.72.0 J ( 0.2 ( 0.4 (1.30.0 2.9 2.8 0.31.5 1.0 1.5 1.0 10650.0642 1.3 10533.2112 1.3 10281.5242 -1.01.5 3.0 1.5 10650.9242 10533.9448 10282.63341.5 0.0 1.5 1.0 10653.8852 0.2 10536.6590 0.1 — —2.5 1.0 1.5 } 10657.2448 {j) 10540.3060 {‘ 10287.5529 {:2.5 4.0 1.5 3.0 10657.8984 0.0 10540.9769 -0.1 10287.9860 0.13.0 -0.7 -0.7 -0.72.5 2.0 1.5 1.0 10658.2404 -0.7 10541.3286 -0.7 10288.1966 -0.72.5 3.0 1.5 10660.8588 10543.6902 10291.02700.5 1.0 1.5 1.0 10665.5242 1.3 10548.4833 1.5 10294.3592 0.90.5 2.0 1.5 10666.2516 10549.1200 10295.1952J=2-12.5 3.0 2.5 3.0 21304.9936 -2.32.5 4.0 2.5 4.0 21306.9156 1,31.5 3.0 2.5 2.0 21312.9254 1.42.5 4.0 1.5 3.0 21313.8877 -0.8 21079.9820 0.93.5 5.0 2.5 4.0 21314.4335 0.9 21080.4937 -1.23.5 4.0 2.5 3.0 21314.9672 1.13.5 3.0 2.5 2.0 21316.5075 -1.22.5 3.0 1.5 2.0 21317.5143 -0.4 21083.3542 0.91.5 3.0 1.5 3.0 21086.2723 -0.6a Observed frequency for overlapped hyperfine components. Measurement accuracy is estimatedto be better than ±1 kllz.Table 7.3: Measured transitions of CuC1Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 115to be commercially available. This, combined with the large dipole moment of mCi( to(115mn35C1)I=3.79 10) D [117]), resulted in very strong signals. Both the J=1-0and J=2-1 rotational transitions were measured for the two most abundant isotopomers,‘15In3C1 (72.5% natural abundance) and“5In37C1 (23.2%). The measured transitionsare given in Table 7.4, labelled according to the coupling scheme= F1= F.7.3.5 InBrIndium (I) bromide was the first non-chloride produced in this work using the laser ablation technique. However, simply substituting Br2 for Cl2 in the gas mixture was sufficientto produce InBr; again, low concentrations (‘-.‘O.05%) of Br2 in Ne were used to inhibitformation of InBr3. The microwave [114, 119] and millimeter-wave [120] spectra of InBrare known; the microwave spectrum of InBr exhibits complex and widespread hyperfinepatterns, the result of two coupling nuclei 9/2, ‘Br =3/2) with large quadrupolemoments. In this work, the J=2-1 rotational transition has been measured for‘5In79Br(48.5% natural abundance) and5In81Br (47.2%). The hyperfine components of thistransition are overlapped for the two isotopomers, and extend over a frequency rangeof 300 MHz, as is shown in Fig. 7.4. The measured transition frequencies and theirassignments are given in Table 7.5, labelled according to the coupling scheme= F1TBr+F1 = F.Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 116“5In35C1 115In37C1frequency obs.-calc. frequency obs.-calc.F F’ F’ F” (MHz) (kllz) (MHz) (kllz)J=1-04.5 3.0 4.5 3.0 6434.1926a 0.84.5 6.0 4.5 6.0 6435.0110 -0.8 6165.3614 -0.34.5 4.0 4.5 4.0 6436.8213 0.84.5 5.0 4.5 5.0 6437.6728 -0.5 6167.4547 -1.55.5 6.0 4.5 5.0 6555.6403 0.1 6285.9940 0.65.5 5.0 4.5 4.0 6555.8488 0.2 6286.1986 0.45.5 7.0 4.5 6.0 6557.3901 -0.3 6287.3774 0.35.5 4.0 4.5 3.0 6557.7150 -0.6 6287.6571 0.73.5 4.0 4.5 5.0 6583.6770 -0.2 6313.8886 -0.73.5 3.0 4.5 4.0 6583.9029 0.2 6314.0930 -0.13.5 5.0 4.5 6.0 6584.6868 0.7 6314.6761 0.53.5 2.0 4.5 3.0 6584.8274 -0.4J=2-14.5 6.0 3.5 5.0 12930.0468 -1.9 12390.7558 -1.24.5 3.0 3.5 2.0 12930.4296 1.8 12391.0807 1.44.5 5.0 3.5 4.0 12933.7853 -1.0 12393.7491 1.54.5 4.0 3.5 3.0 12933.8542 0.65.5 4.0 5.5 4.0 12957.2001 1.3 12417.8908 0.14.5 6.0 5.5 7.0 12957.3449 0.4 12418.0560 0.45.5 7.0 5.5 7.0 12958.4523 -0.04.5 6.0 5.5 6.0 12959.0634 -0.35.5 5.0 5.5 5.0 12959.9006 0.8 12419.9792 -0.55.5 6.0 5.5 6.0 12961.3212 -1.04.5 5.0 5.5 6.0 12961.8234 0.04.5 4.0 5.5 5.0 12961.9076 -0.13.5 2.0 3.5 2.0 12988.4497 0.63.5 5.0 3.5 5.0 12989.3760 -0.23.5 5.0 3.5 4.0 12990.3551 1.03.5 3.0 3.5 3.0 12991.1514 -0.73.5 4.0 3.5 4.0 12992.1008 -1.36.5 7.0 5.5 7.0 13059.0279 0.86.5 7.0 5.5 6.0 13060.7454 -1.0 12521.0767 -0.46.5 6.0 5.5 5.0 13061.0634 -0.7 12521.2880 -1.06.5 8.0 5.5 7.0 13061.4214 -0.3 12521.6088 -0.46.5 5.0 5.5 4.0 13061.6656 -0.8 12521.7724 -0.86.5 5.0 5.5 5.0 13063.5033 0.9a Measurement accuracy is estimated to be better than ±1 kllz.Table 7.4: Measured transitions of InClChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 117I d115 81 111111In Br 1ii 111111111 III ill “51n79BrI I6470.6 MHz 6764.2 MHzFigure 7.4: A composite spectrum, showing the hyperfine components of the J=2-1rotational transition measured for“51n79Br and“51n81Br, scaled according to predictedtransition intensities.Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 118“5InC1 ‘15In37C1frequency obs.-calc. frequency obs.-calc.F F’ F’ F” (MHz) (kllz) (MHz) (kHz)J=2-1 (cont’d)2.5 3.0 3.5 4.0 13072.4186 -1.2 12532.7690 -0.02.5 4.0 3.5 5.0 13072.5800 0.6 12532.9003 -0.92.5 2.0 3.5 3.0 13072.8460 -1.5 12533.0786 0.42.5 1.0 3.5 2.0 13073.0441 0.1 12533.2370 1.24.5 6.0 4.5 5.0 13077.0284 -2.25.5 5.0 4.5 4.0 13078.9285 0.6 12539.3888 -0.55.5 6.0 4.5 5.0 13079.2904 1.4 12539.7208 0.24.5 5.0 4.5 5.0 13079.7908 0.65.5 4.0 4.5 3.0 12540.8329 0.15.5 7.0 4.5 6.0 13080.8314 0.5 12540.8774 0.95.5 6.0 4.5 6.0 13081.9835 2.04.5 4.0 4.5 4.0 12541.0476 -0.34.5 3.0 4.5 3.0 12541.1898 -0.4Table 7.4: Measured transitions of InCl (cont’d)7.3.6 InFPrevious studies [121, 122] of the rotational spectrum of indium (I) fluoride have requiredhigh 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 rodand a SF6/Ne (.-..‘O.l%) gas mixture. Only the J=1-0 rotational transition of InF fallswithin the frequency range of the cavity MWFT spectrometer; this transition has beenmeasured for“5In19F(95.7% natural abundance); transitions of the much less abundant“31n’9F (4.3%) were not observed. In addition to the “51n nuclear quadrupole splitting,19F nuclear spin-rotation splittings (IF =1/2) have been resolved, as is shown in the upperspectrum 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 usedChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 119‘15n3Br “51n37Brfrequency obs.-calc. frequency obs.-calc.F F’ F’ F” (MHz) (kllz) (MHz) (kllz)J= 2-14.5 5.0 5.5 6.0 6470.3476a 195.5 7.0 5.5 7.0 6491.0736 0.93.5 4.0 3.5 3.0 6604.1039 0.43.5 3.0 3.5 3.0 6607.9857 -2.4 6511.7997 2.23.5 5.0 3.5 5.0 6612.0374 0.8 6515.8834 -1.63.5 2.0 3.5 2.0 6628.7970 -0.56.5 8.0 5.5 7.0 6680.5504 -1.4 6583.4056 -0.22.5 4.0 3.5 5.0 6683.4523 -0.1 6588.4788 -1.26.5 7.0 5.5 6.0 6587.8420 -0.76.5 5.0 5.5 4.0 6687.2419 1.5 6588.2380 -0.45.5 6.0 4.5 6.0 6687.6085 5.32.5 3.0 3.5 4.0 6690.4708 0.42.5 1.0 3.5 2.0 6691.7346 -2.8 6594.5962 -1.05.5 7.0 4.5 6.0 6693.3651 -0.8 6597.0359 0.55.5 4.0 4.5 3.0 6694.5730 -0.7 6598.3234 1.32.5 2.0 3.5 3.0 6694.6634 -3.94.5 4.0 4.5 4.0 6696.0571 1.4 6599.3748 0.14.5 3.0 4.5 3.0 6696.2324 1.2 6599.5942 1.74.5 5.0 4.5 5.0 6606.6898 0.86.5 6.0 5.5 5.0 6698.7110 0.0 6597.3860 1.84.5 5.0 4.5 5.0 6704.4096 1.14.5 6.0 4.5 6.0 6705.5634 0.1 6607.2960 0.15.5 5.0 4.5 4.0 6707.7118 -2.0 6609.1366 -2.15.5 6.0 4.5 5.0 6708.5354 -2.8 6609.6932 -0.35.5 5.0 4.5 5.0 6713.0034 3.25.5 4.0 4.5 4.0 6717.4856 1.84.5 3.0 4.5 4.0 6719.1435 2.33.5 5.0 4.5 6.0 6758.8904 0.8 6660.6886 3.03.5 3.0 4.5 4.0 6762.8429 -1.6 6664.1811 -3.03.5 4.0 4.5 5.0 6764.2450 -1.3a Measurement accuracy is estimated to be better than ±1 kllz.Table 7.5: Measured transitions of InBrChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 1201’5n35Br 5In37Brfrequency obs.-calc. frequency obs.-calc.F F’ F’ F” (MHz) (kllz) (MHz) (kllz)J=3-25.5 7.0 5.5 7.0 9996.2778 2.6 9850.6363 2.33.5 5.0 3.5 5.0 10003.6364 1.5 9857.2374 0.27.5 6.0 6.5 5.0 10009.2192 0.8 9863.3901 0.37.5 9.0 6.5 8.0 10010.0396 -1.6 9863.9498 -1.17.5 7.0 6.5 6.0 10012.0460 -0.2 9865.7220 -0.17.5 8.0 6.5 7.0 10012.4468 -0.9 9866.0036 -0.06.5 6.0 5.5 5.0 10015.9650 -4.31.5 2.0 2.5 3.0 10017.6481 1.16.5 8.0 5.5 7.0 10017.6940 -1.1 9871.9570 -0.01.5 3.0 2.5 4.0 10019.1348 2.6 9872.5022 1.06.5 7.0 5.5 6.0 10037.5968 -0.7 9888.2025 -2.7Table 7.5: Measured transitions of InBr (cont’d)to assign the hyperfine components, with the coupling scheme= F1IF+F1 = F.In addition, theF1=3.5-4.5 hyperfine component of the J=1-O rotational transitionin the first excited vibrational state (v=1) has been observed, as is shown in the lowerspectrum of Fig. 7.5; with the much lower signal-to-noise ratio, ‘9F nuclear spin-rotationsplittings cannot be determined accurately. By comparing the signal-to-noise ratios forthe two spectra of Fig. 7.5, and assuming a Boltzmann distribution over vibrationallevels (which may not be the case in the non-equilibrium environment of the molecularexpansion), the vibrational temperature in the expansion may be estimated very roughlyas 380 K. This temperature is sufficiently high that the other two F-F’ components werenot observed for v=1. Recently, a vibrational temperature of .-..i500 K was estimated forChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 12115738.5 MHz15625.8 MHzv015738.9 MHzvO\15626.2 MHzFigure 7.5: TheF1=3.5-4.5 hyperfine component of the J=1-O rotational transition of“51n’9F, observed for both the ground vibrational state (upper spectrum) and the firstexcited vibrational state (lower spectrum). Experimental conditions (upper spectrum):In rod; 0.1% SF6/Ne gas sample; 15738.701 MHz excitation frequency; 50 us sampleinterval; 8 K FT; 80 averaging cycles. Experimental conditions (lower spectrum): In rod;0.1% SF6/Ne gas sample; 15625.980 MHz excitation frequency; 50 ns sample interval;2 K FT; 687 averaging cycles.F1,F= 3.5,4—4.5,43.5,3—4.5,5Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 122frequency obs.-calc.F F’ F’ F” (MHz) (kllz)v=O5.5 5.0 4.5 4.0 15708.6416 -2.75.5 6.0 4.5 5.0 15708.6669 2.73.5 4.0 4.5 4.0 15738.7072 -3.33.5 3.0 4.5 5.0 15738.6975 3.34.5 4.0 4.5 4.0 15575.8635 1.84.5 5.0 4.5 5.0 15575.8635 -1.8v=13.5 4.5 15625.9667a Measurement accuracy is estimated to bebetter than ±1 kllz.Table 7.6: Measured hyperfine components of the J=1-0 transition of‘15In9FMgCl produced using a similar laser ablation/cavity MWFT arrangement [101].7.3.7 YC1Having demonstrated the capabilities of the cavity MWFT-laser ablation system, attempts were made to produce and observe metal-containing species whose pure rotational spectra were previously unknown. The first rotational spectrum to be successfullyobserved for the first time with this system was yttrium (I) chloride. Relatively littleis known about the spectroscopy of YC1. Even its electronic spectrum, of interest totheoreticians because of the presence of partially filled d orbitals, has been explored littlesince the discovery of the C’1 — X’E band system of YC1 in 1966 [124]. In addition tothe ‘ electronic ground state, the A1/ [125], B11 [126], C’D [124, 127], D’H [128], andJ’L[ [128] states have been investigated, with much of this work having been done veryrecently.In their laser-induced fluorescence (LIF) study [127] of the C’E — X’E (0,0) band ofChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 123YC1, Simard et al. used supersonic cooling to reduce spectral congestion near the bandorigin. With sub-Doppler resolution of 120 MHz, they were able to resolve rotationalstructure and determine rough rotational constants, in addition to permanent electricdipole moments for the X and C states. YC1 was prepared in that study by laserablation of an yttrium target rod in the presence of a CC14/He gas mixture.In the present work, YC1 has been produced by ablation of an yttrium target rod inthe presence of a 0.05% C12/Ne gas mixture. The J=1-0, J=2-1, and J=3-2 rotationaltransitions of 89Y35C1 (75.8% natural abundance) and 89Y37C1 (24.2%) have been measured. Initial searches were based on the rotational constants produced by the LIF study[127]. Once the rotational transitions had been located, other hyperfine components ofthe transitions were sought, as eQq(Cl) was unknown. The hyperfine patterns were discovered to spread over very narrow frequency regions, as can be seen in Table 7.7 wherethe measured transitions of YC1 are given with their assignments. Labelling is accordingto the coupling schemeTci +J =F.No 89Y nuclear spin-rotation splitting (ly =1/2) was observed.The signals obtained for YC1 were frustratingly weak, even after much care was takento optimize experimental parameters such as gas mixture composition, microwave pulselength, nozzle pulse/laser pulse delay, and laser power. Both Cl2 and CC14 were usedas chlorinating agents, with comparable results. The predicted ground state dissociationenergy of YC1 is ‘-‘5.4 eV [129], which is greater than that determined experimentally forA1C1 (‘—‘5.2 eV [130]), and so the problem seems to lie in either poor formation of YC1 orits rapid conversion to another form (e.g. YC13), rather than in premature dissociation.A sample spectrum is given in Fig. 7.6, showing the overlapped hyperfine componentsF=2.5-1.5 and F=3.5-2.5 of the J=2-1 transition for 89Y35C1. This is the most intenseChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 12489y35c 89Y37c1F’ - F” frequency obs.-calc. frequency obs-ca1c.(MHz) (kllz) (MHz) (kllz)J=1-01.52.50.5J=2-12.53.51.5 7080.1021a1.5 7080.31421.5 7080.4678a Measurement accuracy is estimated to be better than ±1 kllz.f 2.80.66806.0640 -1.70.3-0.20.6f 1.71.52.5 J 14160.5252 13612.0474J=3-21.52.53.54.50.51.52.53.521240.6288 1.4 20417.930621240.6863 2.30.620417.9750 2.0Table 7.7: Measured transitions of YC1Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 125transition observed for YC1, and yet 320 averaging cycles were required to obtain thissignal-to-noise ratio.7.4 Analysis and discussionAnalysis of the data obtained for all seven metal halides was performed using the least-squares global fitting program SPFIT [131], which treats nuclear quadrupole and nuclearspin-rotation coupling exactly for up to four nuclei by diagonalizing the appropriateHamiltonian matrices. For two coupling nuclei, SPFIT uses the coupling scheme1+J = F12+F = F.Overlapped hyperfine components were weighted according to their theoretical intensities,and the observed frequency was treated as a blend of these components. The Hamiltonianemployed was of the formHspFIT = Hrot + Hc.ci. + HQ (1) + HQ (2) + Hnuc.spin_rot (1)+Hnuc.spin_rot (2) (7.1)= B0J 2— D0J “ + v (2) (1). Q (2) (1) + V (2) (2) . Q (2) (2)+M±(1)11 J +M±(2)12 .J (7.2)Where no hyperfine splittings were observed for the metal nucleus (i.e. AgCI, YCI), hyperftne 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 observeddata; any effects of nuclear spin-spin coupling were evidently negligible.The spectroscopic constants obtained for the metal halides studied are given in Table 7.8-7.14, along with the standard deviations fit of the fits. For comparison, valuesChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 12681 kHz14160.275 MHz 14160.800 MHzFigure 7.6: The overlapped F=2.5-1.5,3.5-2.5 hyperfine components of the J=2-1 rotational transition of 89Y35C1. Experimental conditions: Y rod; 0.05% C12/Ne gas sample;14160.3 MHz excitation frequency; 50 ns sample interval; 4 K FT; 320 averaging cycles.Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 127‘°TAg35CI ‘°9Ag35C1this work lit, value Ref. this work lit, value Ref.B0 (MHz) 3678.04299(39)L 3678043712)b[10] 3661.41682(39) 3661.41815(62)[104]D0 (MHz) 1.851(53) 1.89057(40) [104] 1.869(53) 1.87416(50)[104]eQq(Cl)(kllz) -36.4408(20) -36.44089(95) [105] -36.4404(20) -36.44113(95)[105]fit (kHz) 0.3 0.4‘°7Ag37C1 ‘°9Ag37C1this work lit, value Ref. this work lit, value Ref.B0 (MHz) 3528.49352(39) 3528.49242(82) [104] 3511.86565(39) 3511.8629(10) [104]D0 (MHz) 1.802(53) 1.73846(80) [104] 1.720(53) 1.7195(11) [104]eQq(Cl)(kllz) -28.7184(20) 28.7213(43)c [105] -28.7197(20) -28.7213(43) [105]7fit (kllz) 0.4 1.0a Numbers in parentheses are one standard deviation in units of the last significant figure.b Calculated from the constants given in Ref. [104]. Numbers in parentheses reflect the errorlimits given there.C Values of eQq(37C1) predicted in Ref. [105] using experimental values of eQq(35C1) andthe ratio of (5C1)/Q(71)=1.26878(15), given in Ref. [10].Table 7.8: Spectroscopic Constants of AgClof rotational, centrifugal distortion, and nuclear quadrupole coupling constants obtainedfrom previous microwave, millimeter-wave, and infrared studies are also presented. Incases where only one rotational transition has been studied in this work, D0 was set tothe literature value in the fits.For the molecules whose pure rotational spectra were previously known, B0 has generally been determined to a precision comparable to or better than that obtained frommuch more extensive millimeter-wave studies. One exception is A1C1, where the recentinfrared study [109] determined B0 one order of magnitude more precisely; note thatdata from the present work was included in their analysis [109]. D0 does not fare asChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 12827A135C 27A13Cthis work lit, value Ref. this work lit, value Ref.B (MHz) 7288.2462(1)b 7288.72504(57)’ [109] 7117.51219(16) 7117.51235(59)[109]eQq(Al) (MHz) -30.4081(27) -29.8(50) [107] -30.4112(28) —eQq(C1) (MHz) -8.8290(35) -8.6(10) [107] -6.9586(36) —M±(Al) (kllz) 5.54(16) — 5.44(16) —M(Cl) (kllz) 3.52(30) — 2.62(32) —cTfit (kHz) 0.8 0.6a B0 is obtained by using the known value [108] of D0 in the least-squares fitfor the J=1-0 transition.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 reflectthe error limits given there.Table 7.9: Spectroscopic Constants of A1C1well because of the limited number of transitions measured for each molecule, althoughwhere D0 has been released in the fits to the laser ablation data it compares quite well tomillimeter-wave results. The precision of D0 values so obtained is also remarkably good,given that the constants have been obtained from two or three rotational transitions withthe smallest observable distortion effects, and is a result of the high resolution availablewith the spectrometer. The precision of eQq values has generally been improved considerably over earlier results. In some cases, eQqs have been determined for the firsttime for less abundant isotopomers; although eQq(35C1) was determined for AgC1 withcomparable precision in the electric discharge study [105], measurement of the remaining hyperfine components has allowed determination of eQq(37C1). Nuclear spin-rotationcoupling constants have also been determined for the first time in most cases. A notableexception is InF, whose hyperfine parameters were determined by molecular beam electricresonance to a precision impossible to match using today’s cavity MWFT spectrometers.Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 12963Cu351 65Cu31this work lit. value Ref. this work lit. value Ref.B0 (MHz) 5328.54998(23 5328.512(ii)b [1121 5270.05804(23) 5270.0598(12J112]D0 (kllz) 3.911(36) 3.89315(80) [112] 3.836(40) 3.8084(80{112]eQq(C1) (MHz) -32.1257(19) -32.25(15) [111] -32.1247(20) —eQq(Cu) (MHz) 16.1712(24) 16.08(20) [111] 14.9635(27) —M(Cu) (kllz) 10.39(28) — 11.59(30) —fit (kllz) 0.9 0.863Cu371this work lit. value Ref.B0 (MHz) 5143.76368(18) 5143.7647(12) [112]D0 (kllz) 3.62747c 3.62747(80) [112]eQq(Cl) (MHz) -25.3181(31) —eQq(Cu) (MHz) 16.1667(42) —M±(Cu) (kllz) 9.96(36) —gfit (kHz) 0.9a Numbers in parentheses are one standard deviation lit units of the last significant figure.b Calculated from the constants given in Ref. [112]. Numbers in parentheses reflect the errorlimits given there.C D0 fixed at known value [112] in order to determine B0.Table 7.10: Spectroscopic Constants of CuClChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 130“5In35C1 “5In37Clthis work lit, value Ref. this work lit, value Ref.B0 (MHz) 3261.73321(2oy 3261 73242(53)b [118] 3126.79399(23) 3126.79430(67)[118]D0 (kllz) 1.548(26) 1.54373(14) [118] 1.386(30) 1.41872(17)[118]eQq(In)(MHz) -657.8487(18) -657.52(50) [132] -657.8913(23) -657.20(45) [115]eQq(Cl)(MHz) -13.7575(15) -13.63(20) [132] -10.8399(26) -10.11(45) [115]M(In) (kllz) 10.454(38) — 9.980(46) —M(Cl) (kllz) 1.71(11) — 1.19(14) —iit (kHz) 0.9 0.7a Numbers in parentheses are one standard deviation in units of the last significant figure.b Calculated from the constants given in Ref. [118]. Numbers in parentheses reflect the errorlimits given there.Table 7.11: Spectroscopic constants of InCl115n79Br “51n81Brthis work lit. value Ref. this work lit. value Ref.B0 (MHz) 1667.29199(11)a 166729062(9)b[120] 1642.90726(12) 1642.90394(90)[120]D0 (kllz) 0.4092(76) 0.41713(13) [120] 0.3941(83) 0.40472(12)[120]eQq(In) (MHz) -633.5756(35) -633.50(26) [119] -633.5731(34) -633.20(26) [119]eQq(Br)(MHz) 110.6501(22) 110.63(27) [119] 92.4367(28) 92.33(26) [119]M(In) (kllz) 6.168(44) — 5.924(52) —M±(Br) (kllz) 5.30(10) — 5.36(11) —Ufi (kllz) 2.0 1.4a Numbers in parentheses are one standard deviation in units of the last significant figure.b Calculated from the constants given in Ref. [120]. Numbers in parentheses reflect the errorlimits given there.Table 7.12: Spectroscopic constants of InBrChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 131this work lit. value Ref.B0 (MHz) 7836l049(88)ab 7836.1392(57)c [122]eQq(In) (MHz) -723.794(10) -723.7996(2) [123]M(In) (kllz) 17.46(11) 17.50(1) [123]M±(F) (kllz) 18.3(10) 18.77(10) [123]aiit (kllz) 2.6a B0 is obtained by using the known value [122] of D0 in theleast-squares fit for the J=1-0 transition.b Numbers in parentheses are one startdard deviation in unitsof the last significant figure.c Calculated from the constants given in Ref. [122]. Numbers inparentheses reflect the error limits given there.Table 7.13: Spectroscopic constants of 115n’9F89y35c 89Y37c1this work lit. value Ref. this work lit, value Ref.B0 (MHz) 3540.13730(30)a 354020(12)b [126] 3403.01828(52) 3405.6(13)C [127]D0 (kllz) 1.398(19) 1.6(3) [127] 1.345(33) 2.1(4) [127]eQq(Cl)(MHz) -0.8216(43) — -0.621(20) —M(Cl) (kHz) 2.86(39) — 229d —‘fit (kllz) 0.4 1.2a 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 errorlimits given there.Calculated from the constants given in Ref. [127]. Numbers in parentheses reflect the errorlimits given there.d M±(37C1) fixed at value calculated using M±(35C1) and the relation M1 cx Bo I_ti.Table 7.14: Spectroscopic constants of YC1Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 132All of the spectroscopic constants obtained for YC1 are significant: B0 and D0 aremuch more precisely determined than from the LIF study [127], as expected given thevery large improvement in resolution. The values of eQq(Cl) in YC1 have been determinedfor the first time for both 35C1 and 37Cl, as has M(35Cl). eQq(Cl) may be interpreted asa measure of the ionic character i of YC1. If screening effects and orbital hybridizationare neglected, the ionic character of a bond is related to the nuclear quadrupole couplingconstant along that bond (Xz, or eQq for a diatomic molecule) by [10]Xz (73)eQqiowhere Q is the quadrupole moment of the coupling atom and nio is the contributionof an electron in a p atomic orbital to the field gradient at the atomic nucleus. For35C1, eQqio=109.74 MHz [10], and so i=0.9925 for YC1. This highly ionic bond isconsistent with the large electronegativity difference for the two atoms (‘—4.8 [10]); in thisrespect, YC1 is much more similar to the alkali halides (e.g. KC1: i=1, electronegativitydifference 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 quadrupolecoupling constants, eQq(35C1)/eQq(71). The values for this ratio in ‘°7AgC1, 109AgC1,27A1C1, 63CuCl,“5InCl, and 89YC1 are given in Table 7.15, along with a weighted averageof the six ratios, where weighting was according to the squared inverse of the standarddeviations. In the case of 89YC1, where eQq(Cl) was determined from a very limitednumber of hyperfine splittings, the ratio of the Cl quadrupole coupling constants is withintwo standard deviations of the known value; for the other six molecules, the ratios agreewithin one standard deviation with the known ratio of the Cl quadrupole moments,Q(35Cl)/Q(71) [133], as does the more precise weighted average. While it was hopedinitially that the high precision of the eQq values obtained in this work would result inChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 133molecule eQq(35C1)/eQq(71)‘°7AgC1 1.26890(11)a‘°9AgC1 1.26883(11)27AlCl 1.26879(83)63CuCl 1.26889(17)“5IiCl 1.26915(33)89YC1 1.323(43)weighted average 1.268881(49)Q(35C1)/Q(71) [133] 1.2688773(15)a Numbers in parentheses are one standarddeviation in units of the last significant figure.Table 7.15: Ratio of Cl quadrupole coupling constantsan improvement of theQ(35C1)/Q(71) ratio, this was not the case.Nuclear spin-rotation coupling constantsIn a diatomic molecule with one coupling nucleus, nuclear spin-rotation coupling is described by a term of the form MIJ in the molecular Hamiltonian, a term which isdiagonal in electronic state. This Hamiltonian may be derived considering effects of second order perturbations of the electronic states. Following the procedure described byMiller [134], the electron and nuclear spin dependent terms of the total Hamiltonian maybe 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)whereHrot = B(J — L — S )2 (75)Chapter 7. Microwave Spectra of Metal Halides ProdNced Using Laser Ablation 134Heiectron spin—orbit = a1, s (7.6)Heiectron spin-spin 2/3A(3s — S 2) (77)Heiectron spin—rotation = — L — S ) S (7.8)Hmagnetic hyperfine structure aI L + bFI S + c/3(I S — 3IS)+d/2(exp(2iq)I_S_ + exp(—2i)IS)+e[exp(ic/)(S_I + I_S2) + exp(—iq)(SI + I+S)1.(7.9)L is the total electronic orbital angular momentum, S is the total electron spin, andR =J -L -s has replaced J as the nuclear rotational angular momentum. Then, usingdegenerate perturbation theory, the second order term of the effective Hamiltonian isgiven by(2) lok>< lokIH ‘Ilk>< lkH ‘Ilok>< lokIHeff L.lLd E E, (7.10)k llo 0 1where summation is carried out over all electronic states 1 (where lo is the state of interest)and all rotational and spin quantum numbers k; Eo and E1 are the unperturbed energiesof the 1 and 1 electronic levels, respectively.Using the perturbing Hamiltonian given in Eq. 7.4, there exists a second order crossterm between the rotational and nuclear spin-orbit parts of H’, of the form<lokaI.LIlk’><lk’I—2BJ.Llok”> (711)k’,l — .This term is equivalent to an I J interaction, acting within the electronic state 10. (Theelectron spin analogue, involving bFI .5 and -2BJ S , vanishes for a linear diatomicmolecule when the electronic state of interest is 1E+.) Treating the interaction describedin Eq. 7.11 as the origin of the nuclear spin-rotation coupling described by MI J, oneChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 135obtainsM1 = 2B <l0aL Il>< lL lb>• (7.12)—Since the a parameter is given by 2/BPI < r3 >11, where ,UB is the Bohr magneton,M± can also be expressed as1uj <loITfl>< lILt ho>M1 =41uBB—- . (7.13)Eq. 7.13 is given by Schlier [27] as the dominant electronic contribution to M1.. Whenthe electronic state of interest is Xl+, as is the case in this work for all of the metalhalides studied, only those electronic states of 1fl symmetry will interact with the groundstate via the L operator to produce nuclear spin-rotation coupling.Nuclear spin-rotation coupling constants M1 have been determined for the Cu nucleusin CuCI, the Cl nucleus in YC1, and both nuclei in Aid, mdl, InBr, and InF. Where Mis known for a pair of isotopomers, the values may compared by observing in Eq. 7.13that M is proportional to the product of and the rotational constant [136]. Theratios of the values of M± obtained for pairs of metal halide isotopomers are given inTable 7.16, along with the ratios calculated using 1u and B0. In general, the agreementbetween experimental and predicted ratios is quite good. Small differences are expected,as variations in effective ground state structures will also affect the ratios via the < r3 >dependence of M1 (Eq. 7.13.); in most cases, these differences fall within the experimentaluncertainties of the ratios.A1C1 The ratio of the spin-rotation constants for the two nuclei in A1C1, M(Al)/M1(Cl), may be used to gain some insight into the electronic structure of A1C1. Relativeenergies of the pertinent molecular orbitals (MOs) of A1C1 are shown schematically inFig. 7.7, following the MO configuration proposed by Ram et al. [137]. The 37r and 8uMOs may be correlated to the 3pir and 3pu atomic orbitals (AOs) of Cl respectively, whileChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 136Ratioa exptl. value pred. value Ratio exptl. value pred. value27 cl 1.018(42)c 1.0240 I 115J 1.0475(62) 1.043227A1 7C1I”InCl27A 35Cl I 1.34(20) 1.2301151fl1 35C1 1.44(19) 1.253227A]j 37C1115j 37C1163Cu5Cl0.986(34) 0,9438 I”InBr 1.041(12) 1.014865CCl1115111 ‘Br6Cu Ci 1.043(47) 1.0359“‘I 79Br 0.988(28) 0.941463Cu 7C11151111 81Br65 cul 35Cl I 1.164(53) 1.097563cu1 37C1 Ia Ratio of nuclear spin-rotation coupling constants for the two nuclei indicated by boxes.b Calculated as the ratio of the products of the nuclear magnetic moment pi and therotational constant B0 for the two isotopomers.C Numbers in parentheses are one standard deviation in units of the last significant figure.Table 7.16: Ratios of nuclear spin-rotation coupling constantsChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 137the 9cr and 47r MOs correspond to the 3s and 3pTr AOs of Al. The greatest interactionbetween Al and Cl AOs is assumed to occur between 3pcr(Cl) and 3scr(Al), resulting inthe 8cr bonding MO and the 9cr antibonding MO.In the X’ state, the 9cr orbital is the highest occupied MO (HOMO). The low-lying A11 state is then formed by promoting an electron from 9cr to 4ir. Assuming thatcontributions from the A’ll state dominate, M1 is given byt< X’jL A’ll>< A’ll-IX’>M1 = 4pBBo-- (7.14)-A1fl —p < 9cr1 4K >< 47rI-9cr>= 4ILBB0— . (7.15)I 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 < 9crI1 4r > not vanish for agiven nucleus, 9cr and 4K must be comprised at least partly of AOs which are centred onthat nucleus and which have the same values of both the principal quantum number nand the azimuthal quantum number 1, and values of the magnetic quantum number m1which differ by one. This condition is satisfied by assuming the following: (i) 9cr is madeup of a large contribution from 3.scr(Al) and a lesser contribution from 3pcr(Cl), but alsocontains small contributions from 3pcr(Al) and 3scr(Cl); (ii) 4K is composed primarilyof 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 onlythose terms of M which differ for the two nuclei need be considered when comparingM±(Al) and M±(Cl):M cx <r3 >3p< 3pcr9cr >< 3pK4K>. (7.16)Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 138_____UU3s 3su’9u3p U\____ 3p8 /37T3sAl ClFigure 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 139Here < 3pa9o > represents the coefficient of 3pu in a linear-combination-of-atomicorbitals (LCAO) expansion of 9a, and < 3p7r47r > the coefficient of 3p7r in 47r. Substituting values for 1uj, I, and < r3 >3p [138] for (atomic) 27A1 and 35C1,M(Al) 3.64150 1.5 1.493 <3po(Al)9u> < 3pr(Al)4’r> (7 17)M(Cl) — 0.82187 2.5 8.389 < 3po(Cl)9u > < 3pir(Cl)4ir >When this ratio is compared to the experimental valueM(Al) ——1 57 7 18M(Cl) . , ( . )one obtains<3p(Al)9u> <3p7r(Al)47r>3.33. (7.19)<3pu(Cl)I9cr> <3pK(Cl)4K>Although it is difficult to draw further quantitative conclusions from this result, it isconsistent 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) byproviding configuration interaction (CI) coefficients for electronic states of CuCl. Thelow-lying electron configurations of CuCl which could contribute to M (Cu) areCu(3d543d7rc2)Cl(3spu : config.(a)Cu(3d53dira14s’)Cl(3spu : config.(b)Cu(3d63thrdu24s’)Cl (3s23pa7r4) config.(c).The dominant configuration of the X+ electronic ground state of CuC1 is config. (a),while that of the lowest-lying ‘H state, D’H at 22959 cm1 [140], is config. (c). However,X1> does contain a finite contribution from config. (b), and therefore we can write<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 140where CI(b,X1)and CI(c,D11) are the CI coefficients representing the amount ofthose configurations in the given electronic states. The nuclear spin-rotation couplingconstant for Cu is then given byM = 41LBBo<3dI I3d >2< r3 >3d (CI(b,X1E+)CI(c,Dhll))2 (7.21)I ED1H—Ex1E+= 6BBo1<r3 >3d (CI(b,X’E+)CI(c,D’fl)). (7.22)I Ejin— Exi+Substituting in the appropriate values for63Cu351 [138, 139], one obtains M±(Cu)=1.22 Hz,which is approximately 85 times too small. It is important to note that this large discrepancy does not arise from using an atomic value of < r3 > for Cu, as the value ofa = 2pLI < r3 > /1 determined experimentally by Burghardt et al. [140] differs byonly about 7% from that calculated for Cu using <r3 >(atomic) [138].One may also use this method to predict M±(Cl), assuming that the configurations of X’Y2 and D11 which interact are Cu(3d843d7ru2s’)Cl(3s23pa’3p7r4)andCu(3d643thrdu2s’) Cl(3s23pcr27) respectively, corresponding to promotion of anelectron from 3p7r(C1) to 3p(Cl). Using the CI coefficients given in Ref. [139], M±(Cl) iscalculated 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 ofA1C1. 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 consideredto be formed from by promoting an electron from 57r to l2ci. If it is then assumedthat the valence AOs of Cu and Cl are sufficiently separated in energy that only the4so- and 3do- AOs of Cu interact to form llu and 12cr (an assumption supported bythe conclusion of Ramirez-Solis and Daudey [139] that the wavefunctions of CuCl areChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 1414s 4su12licT3d___5Tv1ô3p4 rr—‘v” TvCu ClFigure 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 142strongly ionic,) then the degree of this interaction may be evaluated quantitatively usingM±(Cu). Let112u >= ciI4sa(Cu) > +c2I3d(Cu) > . (7.23)Then<5I1 I12>< l2uI5 >= 3c <r3 >3d. (7.24)Using this result in an expression analogous to Eq. 7.15, together with < r >3d derivedfrom the experimental a constant of Ref. [140] and the value of M(Cu) determinedexperimentally for63Cu351, we find c2 0.34. Normalization of 12a then gives c1 0.94.In this picture, 4scr(Cu) and 3da(Cu) are mixed to an appreciable extent. This is notindicated by the ab initio CI coefficients [139], which therefore give a much smallerprediction for M(Cu): it seems that the small value of CI(b,X’), which is 0.007, isthe 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 halidesare less attractive than lighter molecules as subjects of ab initio calculations; as a consequence, much less information is available regarding their electron configurations. Inaddition, 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 theindium 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, 3for X=C1, and 4 for X=Br (it will be assumed that the 3d and 4d AOs of Cl and Brare not involved in the MOs to be discussed.) In the X’ electronic ground state, Dais the HOMO; in the single-configuration approximation, the lowest perturbing ‘II stateChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 143_____/5p ciE cip5s4d_4d Cu4 d B\4c16Tl_____ArrA ci — flS U flSIn XF,Ci,BrFigure 7.9: Schematic diagram of the energies of the valence molecular orbitals of anindium (I) halide, InX.Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 144corresponds to promotion of an electron from Do to C’r. In general, Do and Cir maybe written as composed of In and X AOs asDo = d1 nso(X) > + d2 npcr(X) > + d3 4do(In) > + d4 5so(In)>+d51 5po(In) > (7.25)C7r = cii npir(X) > + c2 4d7r(In) > + c3 5pir(In) >. (7.26)Perturbations contributing to M (In) will involve matrix elements of the form<Do I I C >< C Do> = < T >4d +cd <r3 >, (7.27)and thus< Do 1 C >< C I Do>M±(In) = 4tLBBo-- (7.28).1 JIlH ——2jtBBOuI(3cd <r3 >4d +cd <r3> (7 29)— I(EaH—E1E+)M (X) will involve matrix elements of the form<Do 1 C>< C I Do> = <r3 (7.30)andC) D 2J2 —3/ LUBJJO[LICiU2<r >,1VIjj/k)— 1/ 1—’Without a priori knowledge of the composition of Do and C?r, two nuclear spin-rotation 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 theseries InF—InCl---InBr. In particular, since the ionization potential (IP) of X decreaseswith increasing size of X (IP(F)=17.422 eV, IP(Cl)=12.967 eV, IP(Br)=11.814 eV), theIn and X AOs become closer in energy and Do and C7r should acquire more halogencharacter. The magnitude of the coefficients d1, d2, and c1 should thus increase, and d3,d4, d5, c2, and c3 should decrease in magnitude.Chapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 145This trend should be reflected in the M(X) values, since2d — M±(X)I(E1n—EIE+) 732C1 2— 2[LBBOILI < r3 >np— — (7 33)— 2gIN[LBBO <r3 >,Using M±(X) and B0 values from this work, <r3 > values from Ref. [138], and energiesof the C’l1-XD electronic transitions [141] (InF: 42809.2 cm; InCl: 37483.6 cm’;InBr: —34000 cm’), cd is 0.68 for5In’9F, 1.34 for5In35Cl, and 1.58 for5In79Br.While the increase in cd with increasing size of X is as expected, the fact that cd isgreater than unity is troubling. In principle, Dcr and C7r are normalized functions, andso c1 and d2 should never be greater than unity. This discrepancy points out a definitelimitation of this very approximate interpretation of the spin-rotation constants; as aresult, the quantitative mixing ratios determined above for the CuCl MOs should betaken 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 thisscheme, 3p7r(Cl) interacts with a hybridization of 4d7r(Y) and 5p7r(Y) to create the 6irand 7ir MOs, and 3pcr(Cl) interacts with 4da(Y) to form 13cr and 15cr. Both the 14crand 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 theground state, the HOMO is 14o-, and the B’ll-X’> transition is assumed to correspondto promotion of an electron from 14cr to 7r, although two other electron configurationswere 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 thisstudy to the MO picture, one must consider matrix elements linking the 14cr and 77r MOsChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 146_4d4d/5p 4d/5p4c15 255s/5pu 5su/5pa l4ci3pY ClFigure 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 147via the ‘x operator. If 14a and 7ir are considered to be composed of Y and Cl AOs as14u> = cjj 5su(Y) > + c2l 5pa(Y) > (7.34)7ir> = d1 3pir(Cl) > + d2 4dir(Y) > + dl 5pir(Y)>, (7.35)then M(Cl) would be expected to be vanishingly small, as there is no way to obtainthe necessary matrix elements <3pa(Cl) lix I 3p2r(Cl) >. However, since M(Cl) is nonzero, 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 dueto 89Y was too small to be resolved, and so M±(Y) could not be determined. With onlyone coupling constant, it is impossible to determine all of the coefficients of Eqs. 7.34and 7.35, especially considering that Eq. 7.34 will also require a term of the formc31 3p(Cl) >, and so the degree to which the Y and Cl AOs interact cannot be quantifiedfrom M1(Cl).7.4.1 ConclusionsThis work has confirmed that laser ablation techniques may be applied to generate a variety of metal compounds, which may then be studied by MWFT spectroscopy. The simpleablation system described here clearly gives much stronger spectra of metal-containingcompounds than the pulsed electric discharge method described earlier [105], and haspermitted 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 ofmetal-metal dimers, trimers, and larger clusters.The efficiency of the laser ablation technique, combined with the precision and sensitivity available with the cavity MWFT spectrometer, have led to the improvementChapter 7. Microwave Spectra of Metal Halides Produced Using Laser Ablation 148of many spectroscopic constants for the metal halides, as well as allowing the first determination of nuclear spin-rotation coupling constants. As these latter constants areparticularly sensitive to the composition of certain MO functions, they should be considered along with larger spectroscopic parameters in the optimization of future theoreticalcalculations for the metal halides.Appendix ASpherical Tensor RelationsA spherical .tensor T (k) is an operator whose components (q=—k, —k + 1,. .. ,k,where k is the ‘rank’ of T) are proportional to the spherical harmonics y) and transformin the same way under rotation. may also be defined according to the commutationrelations,T)] = qT) (A.l){Jx + iJy , T)] [k(k + 1) — q(q ± 1)]1’2 T1. (A.2)Spherical tensor components may be separated into two parts, a geometric part dependingon the coordinate system in which the spherical tensor operates, and a ‘physical’ partwhich is independent of coordinate system. When acts on basis functions I cxjm>,where in is the projection of j onto some axis, this separation may be accomplished usingthe Wigner-Eckart theorem:<a’j’m’ ITI jm> =(1)i’_m’ ( ‘ k j <‘j’ T (k) 1 >. (A.3)—m’ q m )< cVj’ II T (ic) cj > is a ‘reduced matrix element’; since it has no in dependence, it doesnot depend on the coordinate system used. If T (‘a) operates on a molecular system,the reduced matrix element will contain all of the ‘inherent’ physical information andcan often be related to molecular constants. The components can then be calculatedfrom the molecular constants, using Eq. A.3.149Appendix A. Spherical Tensor Relations 150The dot product of two spherical tensors of equal rank, T (‘a) and u is givenbyT (“) U (k) = (_1)T)U. (A.4)If T (k) and U (k) act on different parts of a couple basis function, the basis functionmust be ‘decoupled’. 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