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Pulsed molecular beam cavity microwave Fourier transform spectroscopy of some fundamental van der Waals… Xu, Yunjie 1993

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PULSED MOLECULAR BEAM CAVITY MICROWAVE FOURIERTRANSFORM SPECTROSCOPY OF SOME FUNDAMENTAL VAN DER WAALSDIMERS AND TRIMERSBYYunjie XuB.Sc., Xiamen University, China, 1988A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Chemistry)We accept this thesis as conforming to therequired standardTHE UNIVERSITY OF BRITISH COLUMBIACANADASeptember, 1993© Yunjie Xu, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) CIV-1/1 (3-7Department ofThe University of British Colum1'aVancouver, CanadaDateDE-6 (2/88)ii-ABSTRACTThe pure rotational spectra of several van der Waalsdimers and trimers, namely Ar-C12, Ne-Kr, Ne-Xe, Ar-Xe,Kr-Xe, Ne2-Kr and Ne2-Xe, as well as Ar2-0CS and Ar2-0O2,have been investigated using a pulsed molecular beamcavity microwave Fourier transform spectrometer.Ar-C12 has been found to have a T-shaped geometrydominated by pairwise additive forces, in contrast to thepreviously studied linear complex Ar-C1F. An effectivevan der Waals bond distance has been obtained, as well asan average vibrational angle for C12. This study has notonly confirmed the structure obtained from electronicspectroscopy, but has also provided a reliable estimateof the 35t1 nuclear quadrupole coupling constant in thefree chlorine molecule.In the microwave investigation of four mixed raregas dimers, the newest available mixed rare gas pairpotentials have been adjusted to obtain the equilibriumdistances (re) between the rare gas atoms. The resultshave been compared with those obtained from isotopicdata. It has been found that the present literature revalues for these dimers had been underestimated in theorder of 1-2%. The quadrupole coupling constants of 83Krand 131Xe have been determined for several differentspecies containing these nuclei. The magnitudes of theseconstants are in the order of several hundreds of kHz,which are surprisingly large. The induced dipole momentsiiiare estimated to be in the order of 0.01 D.Transitions of four isotopomers of 20Ne2-Kr and ofsix isotopomers of 20Ne2-Xe, all with Cn symmetry, as wellas two mixed isotopom^NenNeers of^-Kr, have beenmeasured. A recent theoretical model derived for suchfloppy systems has been used to obtain structuralparameters. The structures, harmonic force constants,83 Kr and 131Xe nuclear quadrupole coupling constants anddipole moments have been compared with the correspondingparameters of the dimers. Possible nonpairwise additivecontributions to these properties have been deduced fromdeviations of the experimental values from those obtainedfrom corresponding dimers assuming pairwise additivity.The complexes Ar2-0CS and Ar2-0O2 have been found tohave distorted tetrahedral geometries (Cs and C2vsymmetry, respectively), with the linear molecule lyingin the symmetry plane. Harmonic force field analyses havebeen performed for Ar2-0CS and Ar2-0O2. The structures andforce constants obtained indicate a dominance of pairwiseadditivity. The basic geometric trend in Ar2-moleculetrimers is discussed.ivTABLE OF CONTENTSABSTRACT ^  iiTABLE OF CONTENTS ^  ivLIST OF TABLES  viiiLIST OF FIGURES ^  xiACKNOWLEDGEMENT  xiii1. INTRODUCTION  ^1Bibliography ^  112. THEORY ^  142.1 Energy Levels of Asymmetric Rotors ^ 152.2 Nuclear Quadrupole Hyperfine Structure ^ 202.3 Structural Information From Rotational Spectra ^ 262.4 Harmonic Force Field Analysis ^  332.5 Theoretical Description of PulsedExcitation Experiments ^  36Bibliography ^  443. EXPERIMENTAL METHODS ^  473.1 Pulsed Molecular Beam Cavity MicrowaveFourier Transform Spectrometer ^ 493.1.1 Microwave Radiation Source and Generationof Microwave Pulses ^  513.1.2 Fabry-Perot Resonator and Pulsed MolecularBeam Source ^  523.1.3 Signal Detection System and DataAcquisition System ^  543.1.4 Experiment Control  553.2 Optimization of the Experimental Conditions ^ 57Bibliography ^  644. THE ROTATIONAL SPECTRUM OF ARGON-CHLORINE DINER ^ 664.1 Introduction ^  664.2 Search and Spectrum Assignment ^ 684.3 Analysis of the Argon-Chlorine Dimer Spectrum ^ 714.4 Estimation of the 35C1 Nuclear Quadrupole CouplingConstant in the Free Chlorine Molecule ^ 744.5 Harmonic Force Field and Structure  764.6 Comments on Argon-Halogen Dimers andExperimental Observations ^  79Bibliography ^  825. PURE ROTATIONAL SPECTROSCOPY OF THE MIXEDRARE GAS DINERS ^  915.1 Introduction  915.2 Experimental Considerations ^  955.3 Search and Spectral Assignments  965.3.1 Search and Spectral Assignment of Ne-Kr ^ 965.3.2 Ne-Xe Spectra and Assignment ^ 985.3.3 Ar-Xe Spectra and Assignment  1005.3.4 Kr-Xe Spectra and Assignment ^ 102vi5.4 Estimation of the Induced Dipole Momentsof the Mixed Rare Gas Dimers ^  1045.5 Interpretation of the Spectroscopic Constants ^ 1065.5.1 Estimations of the Equilibrium Distancesfrom the Isotopic Data ^  1065.5.2 Comments on the Newest Potentials availablefor Ne-Kr, Ne-Xe, Ar-Xe and Kr-Xe ^ 1105.5.3 Manual Adjustment of Equilibrium Distancesand Dissociation Energies ^  1135.6 Comments on the Dipole Moments of the MixedRare Gas Dimers ^  1165.7 Discussion of the Nuclear Quadrupole CouplingConstants of °Kr and 131Xe ^  1185.8 Conclusions and Some Future Prospects ^ 124Bibliography ^  1276. MICROWAVE SPECTROSCOPIC INVESTIGATION OF THE MIXEDRARE GAS VAN DER WAALS TRIMERS Ne 2-Kr AND Ne2-Xe .... 1546.1 Introduction ^  1546.2 Search and Assignments ^  1576.3 Spectra Analysis ^  1636.4 Estimation of the Dipole Moments ofNe2-Kr and Ne2-Xe ^  1666.5 Possible Evidence of Three-Body Effects ^ 1666.5.1 Structure of the Ne2-Kr and Ne2-Xe Trimers .. 1676.5.2 Harmonic Force Field Approximationof Ne2-Kr ^  1756.5.3 Induced Dipole Moments of the Trimers ^ 177vii6.5.4 Nuclear Quadrupole Hyperfine Structures ^ 1786.6 Conclusion and Future Prospect ^  181Bibliography ^  1827. ROTATIONAL SPECTROSCOPY OF THE VAN DER WAALS TRIMERSAr2 -OCS AND Ar2-CO2   1957.1 Introduction ^  1957.2 Experimental Methods ^  1997.3 Search and Rotational Assignment ^ 2007.4 Analysis of the Ar2-0CS and Ar2-0O2 Spectra ^ 2057.5 Geometry and Structure of Ar2-0CS and Ar2-0O2 ^ 2067.6 Harmonic Force Field Approximation ^ 2137.7 Comments on the Effects of Three-bodyInteraction and the General GeometricTrends for Ar 2-Molecule Trimers   218Bibliography ^  223APPENDIX ^  242viiiLIST OF TABLES4.1 Observed Frequencies of Ar-35C12 and Ar-35C137C1 ^ 844.2 Spectroscopic Constants of Ar-35C12 and Ar-33C137C1 ^ 884.3 The Structure and Harmonic Force Field of Ar-35C12 ^ 894.4 Comparison of the Properties of Ar-C12with Those of Ar-C1F ^  905.1 Observed Frequencies of Ne-Kr ^  1315.2 Observed Frequencies of 20Ne-0Kr  1335.3 Spectroscopic Constants of Ne-kr ^  1345.4 Observed Frequencies of Ne-Xe  1355.5 Observed Frequencies of Ne-131Xe ^  1375.6 Spectroscopic Constants of Ne-Xe  1385.7 Observed Frequencies of Ar-Xe ^  1395.8 Observed Frequencies of Ar-131Xe  1415.9 Spectroscopic Constants of Ar-Xe ^  1435.10 Observed Frequencies of Kr-Xe  1445.11 Observed Frequencies of 84Kr-131Xe ^ 1455.12 Spectroscopic Constants of Kr-Xe  1465.13 Estimated Electric Dipole Moments in Debye ofthe Mixed Rare Gas Dimers ^  1475.14 Nuclear Quadrupole Coupling Constants of 83Krand 131Xe in RG -83Kr and RG-131Xe Dimers ^ 1475.15 Coordinates and Bond Lengths (I)of Rare Gas Dimers ^  148ix5.16 Comparison of the Observed Rotational Frequencieswith Those Predicted From HFD-B Potentialsfor Ne-Kr and Ne-Xe ^  1495.17 Comparison of the Observed Rotational Frequencieswith Those Predicted From HFD-C Potentialsfor Ar-Xe and Kr-Xe ^  1515.18 Equilibrium Distances (A) of Rare Gas Dimers ... 1536.1 Observed Frequencies of 28Ne2-Kr ^ 1846.2 Observed Frequencies of NenNe -Kr  1856.3 Observed Frequencies of 20Ne-83Kr ^ 1866.4 Observed Frequencies of 20Ne2-Xe Trimer ^ 1876.5 Observed Frequencies of 20Ne2_131xe ^  1886.6 Spectroscopic Constants of Ne2-Kr Trimer ^ 1896.7 Spectroscopic Constants of 20Ne2-Xe Trimer ^ 1906.8 Estimates Electric Dipole Moments of Ne2-RGTrimers and the Corresponding Dimers ^ 1916.9 Quadrupole Coupling Constants (MHz)of 83Kr and 131Xe in Ne2-RG Trimersand in the Corresponding Dimers ^  1916.10 Structural Parameters of Ne2-Kr and Ne2-Xe andthe Corresponding Dimers ^  1926.11 The Harmonic Force Field Analysis of Ne2-Kr ^ 1937.1 Observed Transitions of Ar2-0CS ^  2257.2 Measured Line Frequencies of SubstitutedIsotopomers of Ar2-0CS ^  2287.3 Observed Transitions of Ar2-0O2 ^  230x7.4 Spectroscopic Constants of Ar2-0CS ^ 2317.5 Spectroscopic Constants of Ar2-0O2  2327.6 Structure Parameters of Ar2-0CS, Ar-OCS and Ar2 ^ 2337.7 Structure Parameters of Ar2-0O2, Ar-0O2 and Ar2 ^ 2347.8 The Harmonic Force Field of Ar2-0CS ^ 2357.9 Comparison of Observed Centrifugal DistortionConstants with Those Obtained From the^Harmonic Force Field of Ar2-0CS ^  2377.10 The Harmonic Force Field of Ar2-0O2 ^ 2387.11 Comparison of Structures and Forces ConstantsDerived for the Ar and Ar2 complexes ofHX (X=F,C1,CN) and OCS and CO2 ^  240xiLIST OF FIGURES3.1 Schematic Circuit Diagram of the Pulsed MolecularBeam Cavity Fourier Transform Spectrometer ^ 503.2 A Train of Pulse Sequences forControlling One Experiment ^  563.3 Hyperfine Components due to 79Br and 14N NuclearQuadrupole Coupling of Rotational Transition3 03 -202 of 79BrCH 2CMs1 ^  584.1 The Chlorine Nuclear Quadrupole Hyperfine Splittingof the Transition 31,3-21,2 of Ar-C12 ^ 725.1 Hyperfine Pattern of the J=2-1 RotationalTransition of 28Ne-83Kr ^  995.2 Hyperfine Pattern of the J=2-1 RotationalTransition of 28Ne-131Xe ^  1015.3 x(83Kr) Plotted as a Function of the ElectricField Gradient qo ^  1225.4 x (13i Xe) Plotted as a Function of the ElectricField Gradient qo ^  1235.5 List of All Isotopomers of the Four Rare GasDimers Observed in Natural Abundance ^ 1266.1 Hyperfine Splitting due to 83Kr in the RotationalTransition 30,3-20,2 of 20Ne2-83Kr ^  1606.2 Nuclear Quadrupole Hyperfine Components ofTransition 30,3-20,2 of 20Ne2_131xe ^  1626.3 Nonpairwise Additivity of x (131xe)-ccin the Ne2-131Xe Trimer ^  180xii7.1 Example Transitions of Ar2OCS and Ar2-0O2 ^ 2017.2 Ar2-0CS in Its Principal Axes ^  2077.3 Ar2-0O2 in Its Principal Axes  212xiiiAcknowledgementsThis thesis is completed at the University of BritishColumbia under the supervision of Dr.M.C.L.Gerry. It givesme great pleasure to thank him for his guidance, hisencouragement and his approachability. I am also verygrateful for the considerable latitude he has allowed me inmy choices of research subjects.Furthermore, I would like to express my gratitude toDr.W.Jager for teaching me to use the spectrometer when Iwas a junior graduate student, and for numerous help andvaluable discussions throughout this thesis work on van derWaals complexes. I would also like to further extend myappreciation to Dr.N.Heineking for stimulating discussionson the search for Ar2-0CS, and to Dr.I.Ozier for manyhelpful discussions on the Ar-C12 project.Specific thank goes to Dr.D.Clouthier for kindlylending us the General Valve nozzle and the nozzle driverat the early development stage of the spectrometer, and toChris Chan for his electronic expertise, without which thethesis work would not have progressed so smoothly.My gratitude also goes to my fellow colleagues at theUniversity of British Columbia, for their support andfriendship which make my stay in Vancouver a very enjoyableone.Finally, I would like to thank my family for theirsupport during all my years of study.1CHAPTER 1IntroductionThis thesis is concerned with the measurement andanalysis of microwave pure rotational spectra of weaklybound van der Waals complexes. The spectra of thecomplexes have been measured using a cavity pulsedmolecular beam microwave Fourier transform (MWFT)spectrometer. The underlying aim is to provideinformation on the weak interactions between the atomsand the molecules comprising these complexes.Weak intermolecular interactions have receivedconsiderable attention in recent years and their study isa field of growing experimental and theoreticalactivity [1,2]. Such weak interactions are responsiblefor various physical and biological phenomena, rangingfrom gas imperfections, and condensation andsolidification of matter [3], to the unique foldedstructures of polypeptide chains in proteins [4] in muchmore sophisticated in vivo molecular systems.Furthermore, computer simulation of molecular dynamics,widely used in bio-science and chemistry [5], is arapidly advancing field. Such simulations are importantto increase our knowledge of the nature of complexchemical and biochemical processes as well as of complexmolecules, especially for systems which are not amenableto direct measurements. However, the success of thesimulation approach relies on our knowledge of weak2intermolecular interactions and on the correctness of thepotentials used.Speculation concerning the origin and the form ofintermolecular interactions goes back to as early as thebeginning of the twentieth century [6]. But it was notuntil the establishment of the principles of quantummechanics that a basic understanding of the nature ofintermolecular forces was developed in the 1930s [3]. Ittook another 40 years for a reasonably precisedescription of the forces, i.e. potential energy curves,to be obtained for even the simplest systems, namelypairs of rare gas atoms [7].Weak intermolecular interactions, in general,include both long-range attractive forces, such aselectrostatic forces (between permanent multipoles),induction and dispersion forces, and short-rangerepulsive forces [3]. The hydrogen bond, as a specialcase, is now generally accepted as being due mainly toelectrostatic interactions with significant overlapeffects, after many years of controversy [8]. It may seemsurprising at first, yet is important to note, that thedispersion energy usually makes the major contribution tothe attractive energy, with the only exception being forsmall, highly polar molecules such as water, where it isthe hydrogen bond which makes the major contribution [9].The induction energy on the other hand, is almost alwayssmall, unless one or more of the interacting species is3charged [9].A decade ago, information about the potential energysurfaces of van der Waals complexes was dominated byresults from "bulk" phase experiments such as secondvirial coefficients, gas transport properties, andenergies of crystallization of rare gases, and by resultsfrom molecular beam scattering experiments [3]. Becauseproperties of dense systems are influenced by multi-bodyinteractions, approximations had to be made to extractinformation about the binary systems from these "bulk"experiments. This was difficult, because currentknowledge about many-body nonpairwise additivecontributions [10] is far from satisfactory [11]. On theother hand, although molecular beam scatteringexperiments seemed to show promise for success indetermining potential energy surfaces, these experimentsare in general very difficult for heavier systems, andthe majority of experiments has been done on very lightsystems such as He, H2 and Ne [12,13].Recent developments in high resolution spectroscopyin the microwave (MW), infrared (IR) and far infrared(FIR) regions, combined with the use of seeded molecularbeams, have led to a breakthrough in the experimentalstudy of weak intermolecular interactions [14,15,16]. Oneadvantage is that such studies provide detailedinformation about intrinsic properties of the isolatedgas phase complexes, such as van der Waals dimers,4trimers, and larger clusters.* Perturbations caused bysolvent and lattice interactions, which are unavoidablein dense systems, are absent. This makes it possible touse spectroscopic data directly, to construct thepotential energy surfaces and to compare the experimentaldata with theoretical calculations.The recent rapid progress in this field, resultingto a large degree from cooperation betweenexperimentalists and theoreticians, is reflected by thefact that the number of publications related to van derWaals complexes has doubled every 2.1 years since1980 [15]. The experimental work in the MW and IR rangesprior to 1988 has been summarized by Novick, Leopold andKlemperer [17], followed by a review by Klemperer andYaron on the current problems and the future prospects ofthe experimental aspects of this field [18]. Highresolution infrared spectroscopy of weakly boundcomplexes has been reviewed by Nesbitt [15]. Thecorresponding theoretical work has been reviewed byBuckingham, Fowler, and Hutson [19] and by Hutson [20].Most recently, some prospects for far infrared probing ofvan der Waals complexes have been reviewed by Saykallyand Blake [16].* It should be noted that binary (e.g. Ar-HC1) andternary (e.g. Ar2-HC1) systems are sometimes referred toas "dimers" and "trimers" in this thesis, as is commonlydone in publications of this field.5Current knowledge of potential energy surfaces ofinteratomic and intermolecular interactions in dimers isgrowing rapidly, even though we are still far away fromthe ultimate goal of predicting the properties of anysystem simply by solving the Schredinger equation. Forexample, homonuclear rare gas dimer potentials are fairlyprecisely known: any properties of these systems can bederived from their potential energy curves with anuncertainty of up to only a few percent [21]. For morecomplicated systems, atom-diatom interactions are thebest understood. For example, Ar-HC1 has been extensivelystudied using high resolution microwave [22], farinfrared [23] and near infrared [24] spectroscopy toprovide sufficient data for a full dimensional potentialenergy surface to be determined [25]. The potentialenergy surface for Ar-HC1 has been repeatedlyrevised [26] and improved [25] to become one of the bestquantitatively determined surfaces. It has survived somerigorous tests, mainly imposed by new far infraredmeasurements, although the part of the potential energysurface concerning the secondary minimum is still notquantitatively well represented [16]. Detailedinformation on dimer potential energy surfaces dwindlesdramatically, however, from rare gas dimer systems tolarger molecular systems. For example, for slightlylarger binary systems such as Ar-H20 and Ar-NH3, onlyqualitative details of the potential energy surfaces are6well established [16].The current goal in this field is to understand notonly binary interactions, but also three and more-bodyinteractions [3]. Knowledge of many-body interactions isessential for understanding properties of condensed phasesystems in terms of pairwise additive binary potentials,by considering three and more-body interactions as aperturbation [3]. The possibility of spectroscopicdetermination of many-body contributions to clusterproperties depends on the small discrepancies between theexperimentally measured properties from those predictedusing a pairwise additive calculations. Clearly, in orderto attribute the small discrepancies to many-bodyeffects, not only binary interaction potentials must beknown to a very high degree of accuracy, but alsopairwise additive calculations on larger systems need tobe as rigorous as possible; furthermore, accuratequantitative experimental data on isolated trimers andlarger clusters, provided by high resolutionspectroscopy, are unequivocally necessary.It was only five years ago that high resolutionspectroscopy was first applied to larger van der Waalsclusters such as trimers and tetramers [17]. Pioneeringwork on microwave rotational spectroscopy of largerclusters was carried out by Gutowsky and co-workers [27],followed by the recent near IR and FIR investigations onsuch systems by Nesbitt and co-workers [28], and by7Saykally and co-workers [29]. One of the experimentallyand theoretically best studied trimer systems so far isAr2-HC1. A series of FIR investigations byElrod et al. [29], along with a microwave study [27],has provided much spectroscopic information. In aparallel theoretical treatment, Cooper and Hutson [30]pointed out the possibility of three-body nonpairwiseadditive contributions, and attempted to calculate someof these contributions theoretically. However, theyencountered several problems. First, intensive computingis needed for the pairwise calculation using afull-dimensional treatment [30], since the cheaperadiabatic separation [31] has been found to be inadequatefor Ar 2-HC1. Secondly, the theory dealing with many-bodyinteractions has been developed mostly for atomicsystems [32] and has not yet been well established formolecular systems [30]. Therefore approximations wereneeded in the case of Ar 2-HC1 [30].Theoreticians have shown considerable interest forseveral decades in simpler and more fundamental systemssuch as dimers and trimers containing only rare gasatoms [33], as well as atom-homonuclear diatom complexes[34]. The problem with using these species to investigateinteratomic and intermolecular forces has been a lack ofexperimental spectroscopic data, largely because therequired transition moments are nearly prohibitivelysmall.8This thesis presents data which overcome part ofthis problem, in the form of microwave rotational spectraof several very fundamental systems. These systemsinclude both van der Waals dimers and trimers, namelyAr-C12, Ne-Kr, Ne-Xe, Ar-Xe, Kr-Xe, Ne2-Kr, Ne2-Xe,Ar2-0CS and Ar2-0021 which have been investigated using acavity pulsed molecular beam MWFT spectrometer. None ofthese complexes had been previously studied in themicrowave region; no high resolution spectra of any kindhad been investigated previously for the four trimers.Most of these complexes have extremely low electricdipole moments, making them a challenging experimentalproblem in microwave spectroscopy.Discussion of these complexes is presented inChapters 4-7. Although the complexes in each individualchapter are somewhat independent, with the specificinteresting points discussed in the introduction of eachchapter, they are also connected in a general way.Chapter 4 describes the microwave investigation of thelow dipole moment atom-homonuclear diatom dimer Ar-C12.The geometry of Ar-C12 has been found to be T-shapeddominated by pairwise additivity, which is in contrast tothe previously studied linear complex Ar-C1F [35]. It ispresented as the first complex in this thesis because theexperimental experience from the study of this dimer wasessential to the later investigation of other low dipolemoment complexes, particularly those containing only rare9gas atoms.The following three chapters present a series ofinvestigations of complexes ranging from the simplestrare gas dimers (Ne-Kr, Ne-Xe, Ar-Xe and Kr-Xe) to tworare gas trimers (Ne2-Kr and Ne2-Xe), and eventually tomore complicated trimers containing molecular subunits(Ar2-0CS and Ar2-0O2). Chapter 5 focuses on the propertiesof the mixed rare gas dimers, especially theirequilibrium internuclear distances, as well as suchproperties as the quadrupole coupling constants of °Krand 131Xe nuclei and the induced dipole moments. It wasfound that the literature equilibrium distances for thesedimers, which had been obtained from "bulk" properties,had been underestimated in the order of 1-2%. Suchdifferences, although small, would have significantimpact in any subsequent many-body nonpairwise additivestudy, where a nonpairwise additive contribution isusually a very small fraction of the pairwisecontribution.The microwave spectroscopy of the mixed rare gas vander Waals trimers Ne2-Kr and Ne2-Xe is described inChapter 6. The geometries of these two trimers have beenestablished using various isotopic data. Informationabout their structures and dipole moments, as well asnuclear quadrupole coupling constants of °Kr and 1251Xehave been obtained. This study provides high resolutiondata, which are essential in the determination of the10three-body nonpairwise additive effects.Chapter 7 describes the spectra of a new series ofrare gas-small molecule van der Waals trimers, namelyAr2-0CS and Ar2-0O2. The basic geometric trend inAr2-molecule trimers is discussed. This work providesmore complicated and challenging problems for futuretheoretical calculations.Each study involved a large amount of spectralmeasurement and analysis. To assist the reader inlocating these results, the tables in this thesis arepresented at the end of each chapter. It should be noted,however, that the figures are placed in the text near thepoint where they are first cited.11Bibliography1. J.Michl, ed., van der Waals interactions. Chem.Rev.88, 815-988 (1988).2. D.A.Young, ed., Faraday Discuss.Chem.Soc. 86, 1982.3. G.C.Maitland, M.Rigby, E.B.Smith, W.A.Wakeham,"Intermolecular Forces", Oxford:Clarendon, 1981. Seealso M.Rigby, E.B.Smith, W.A.Wakeham, G.C.Maitland,"The Forces Between Molecules", Oxford:Clarendon,1986.4. M.F.Perutz, in "The Chemical Bond: Structure andDynamics", ed A.Zewail, Academic Press, 17 (1992).5. See for example, E.Velasco, S.Toxvaerd,Phys.Rev.Lett. 71, 388 (1993).6. J.D.van der Waals, Ph.D. thesis, The University ofLeiden, (1873).7. R.A.Aziz, in "Inert Gases", ed. M.L.Klein, pp.5-86.Berlin:Springer-Verlag (1984).8. A.D.Buckingham, J.Mol.Struct. 250, 111 (1991).9. M.Rigby, E.B.Smith, W.A.Wakeham, G.C.Maitland, "TheForces Between Molecules", p.12 Table 1.1, Oxford:Clarendon, 1986.10. J.A.Baker, "Rare Gas Solids" Vol.1, ed. M.L.Klein andJ.A.Venables, Chapter 4, Academic Press, 1976. Seealso R.J.Bell and I.J.Zucker, ibid. Chapter 2.11. W.J.Meath and R.A.Aziz, Mol.Phys. 52, 225 (1984).12. R.B.Gerber, V.Buch, and U.Buck, J.Chem.Phys. 72 3596(1980).13. J.Andres, U.Buck, F.Huisken, J.Schleusener, andF.Torello, J.Chem.Phys. 73, 5620 (1980).14. A.C.Legon and D.J.Millen, Chem.Rev. 86, 635 (1986).15. D.J.Nesbitt, Chem.Phys. 88, 843 (1988).16. R.J.Saykally and G.A.Blake, Science 259, 1570 (1993).17. S.E.Novick, K.R.Leopold and W.Klemperer, in "Atomicand Molecular Clusters", ed. E.R.Bernstein, Elsevier,1990.1218. W.Klemperer and D.Yaron, in "Dynamics of Polyatomicvan der Waals Complexes", ed. N.Halberstadt andK.C.Janda, Plenum Press, 1 (1990).19. A.D.Buckingham, P.W.Fowler and J.M.Hutson, Chem.Rev.88, 963 (1988).20. J.M.Hutson, Annu.Rev.Phys.Chem. 41, 123 (1990).21. R.A.Aziz and M.J.Slaman, J.Chem.Phys. 94, 8047(1991); Chem.Phys. 130 187 (1989); Mol.Phys. 58, 679(1986); Mol.Phys. 57 825 (1986); See also G.Scoles,Ann.Rev.Phys.Chem. 31, 81 (1980).22. S.E.Novick, P.Davies, S.J.Harris, and W.Klemperer,J.Chem.Phys. 59, 2273 (1973).23. K.L.Busarow, G.A.Blake, K.B.Laughlin, R.C.Cohen,Y.T.Lee, and R.J.Saykally, Chem.Phys.Lett. 141, 289(1987).24. B.J.Howard and A.S.Pine, Chem.Phys.Lett. 122, 1(1985); C.M.Lovejoy and D.J.Nesbitt, Chem.Phys.Lett.146, 582 (1988).25. J.M.Hutson, J.Chem.Phys. 89, 4550 (1988).26. J.M.Hutson and B.J.Howard, Mol.Phys. 43, 493 (1981);J.M.Hutson and B.J.Howard, Mol.Phys. 45, 791 (1982).27. H.S.Gutowsky, T.D.Klots, C.Chuang, C.A.Schmuttenmaer,and T.Emilsson, J.Chem.Phys. 86, 569 (1987);H.S.Gutowsky, T.D.Klots, C.Chuang, J.D,Keen,C.A.Schmuttenmaer, and T.Emilsson, J.Am.Chem.Soc.109, 5633 (1987); H.S.Gutowsky, C.Chuang, T.D.Klots,T.Emilsson, R.S.Ruoff, and K.R.Krause, J.Chem.Phys.88, 2919 (1988).28. A.McIlory and D.J.Nesbitt, J.Chem.Phys. 97, 6044(1992).29. M.J.Elrod, D.W.Steyert and R.J.Saykally, J.Chem.Phys.94, 58 (1991), 95, 3182 (1991); M.J.Elrod,J.G.Loeser, and R.J.Saykally, 98, 5352 (1993).30. A.R.Cooper and J.M.Hutson, J.Chem.Phys. 98, 5337(1993).31. J.M.Hutson, J.A.Beswick, and N.Halberstadt,J.Chem.Phys. 90, 1337 (1989).32. W.J.Meath and M.Koulis, J.Mol.Structure 266, 1(1991).1333. "Inert Gases", ed. M.L.Klein, Berlin:Springer-Verlag(1984).34. See for example, P.McCabe, J.N.L.Connor, andK.-E.Thylwe, J.Chem.Phys. 98, 2947 (1993);K.-E.Htylwe and J.N.L.Connor, J.Chem.Phys. 91, 1668(1989); F.A.Gianturco and M.Venanzi, J.Chem.Phys. 91,2525 (1989).35. S.J.Harris, S.E.Novick, W.Klemperer, andW.E.Falconer, J.Chem.Phys. 61, 193 (1974).14CHAPTER 2TheoryThis chapter summarizes briefly some of the basictheory used to analyze the spectra and to obtain themolecular parameters reported in this thesis. Althoughthis theory has not been developed as part of the thesisresearch, it is presented to familiarize the reader withthe notation used in the following chapters.First, some theory relevant to the assignment andanalysis of pure rotational spectra is outlined. Thetopics include: (1) the semirigid rotor model; (2)nuclear quadrupole hyperfine structure; (3) structuralinformation from rotational constants; (4) harmonic forcefield analysis. They are treated in detail in severaltext books [1,2]. Although these treatments have beendeveloped for "normal" chemically bonded systems, theyhave also been applied to van der Waals complexes in themajority of recent publications [3].Secondly, some basic theoretical aspects of pulsedexcitation experiments are also briefly summarized inthis chapter.Theoretical aspects of the dynamics of van der Waalscomplexes in terms of their intermolecular potentialenergy surfaces are a rapidly developing field [4]. Someof the most recent theoretical results will be comparedwith the experimental results presented in the following15chapters. However, because the theories underlying thosecalculations are not tightly connected with the thesiswork presented here, and are out of the scope of thiswork, they will not be summarized here. The reader isreferred to some introductory books on this field [5] andto some review articles [6].2.1 Energy levels of asymmetric rotorsFor a rigid rotor in its principal inertial axessystem, the Hamiltonian for the rotational energy levelscan be written as:9-CR - A Pa2 + B Pi; +C PC^ (2.1)where Pa,^and Pc are the components of the totalrotational angular momentum vector P with respect to themolecule fixed axes. A, B and C are the rotationalconstants, which are related to the principal moments ofinertia 'g by G (A, B or C)=h/(87r2I) with g=a, b or c.Here a, b and c are the principal inertial axes, chosenso that A>B>C. The rotational constants are in frequencyunits.Rigid rotors are generally classified into severaldifferent types:(1) linear molecules^la=0, Ib=Ic (A-+00, B=C);(2) spherical tops Ia=Ib=Ic (A=B=C);16(3) prolate symmetric tops Ia<Ib=Ic (A>B=C);(4) oblate symmetric tops Ia=Ib<Ic (A=B>C);(5) asymmetric tops Ia<Ib<Ic (A>B>C);The range of B values between A and C corresponds tovarious conditions of asymmetry, which can be denoted byasymmetry parameters. For example, Ray's asymmetryparameter, K, is defined as [7]:2B -A -CK A -C (2.2)ranging from -1 to 1, corresponding to the prolate andoblate symmetric top limits, respectively.By solving the SchrOdinger equation, one can obtainthe corresponding rotational energy levels of a rigidrotor. For example, a linear rigid rotor has rotationalenergies ER=BJ(J+1); while prolate symmetric topscorrespond to ER=BJ(J+1)+(A-B)K2 and oblate symmetrictops to ER=BJ(J+1)+(C-B)K2 [8]. Here J and K are quantumnumbers, with J denoting the total rotational angularmomentum, and K representing the projection of J onto themolecular axis; K can take integer values from -J to J.Any state of an isolated rigid symmetric top can bedescribed by the quantum numbers J and K, and thecorresponding wavefunction may be denoted by IJ,K> inbra-ket notation. There is a third quantum number, Mj,representing the component of J along the space fixed Z17axis; it has no effect on rotational energies in theabsence of external fields.However, for an asymmetric top, such closed formexpressions for rotational energy levels no longerexist [8]. An asymmetry splits the levels IJ,±K>, whichare degenerate for symmetric tops. Although the totalangular momentum P is a constant of motion and J is stilla "good" quantum number which can be used to specify thestate of the rotor, K is no longer a "good" quantumnumber and cannot specify the rotational state very well.Instead IJ,Ka,Kc> or IJ,T> are used to denote thewavefunctions of asymmetric rotors, with Ka and Kc beingthe IKI values in the limiting prolate and oblatesymmetric top cases, and r=-Ka-Kc• The common notation isto designate the levels as Jo, Kc (e.g. 10,1, 73,5, -). Foran asymmetric top, the energies of rotational levels canbe obtained by directly diagonalizing the rigid rotorenergy matrix expressed in a basis set of symmetric topwavefunctions IJ,K>. The non-zero matrix elements, in theIr representation [9], which is usually chosen forprolate asymmetric tops, are:<J,K 1 XRIJ,K> -1 (B+C) J (J+1) + [A-1 (B+C) 1K22^ 2(2.3)18<J,K±2 I 9-CRIJ,K> = 1 (B-C) { [J (J+1 ) -K (}C±1 ) ] x41[J (J+1) - (K±1 ) (K±2) ] j 2(2.4)For very flexible systems such as the van der Waalscomplexes studied in this thesis, the simple rigid rotormodel cannot accurately describe even low J transitions.Extra terms such as quartic and sextic centrifugaldistortion terms [10,11] are needed for an adequateanalysis of the rotational spectra of these systems. Oneof the most commonly used rotational Hamiltonians,including both quartic and sextic distortion terms, inWatson's A-reduction, 1r representation can be writtenas [11]:5( -9{R+1-CD+XD19-CR = Al2,2+81 +C/3c29-CD - -Aa P4 - A j K P2 Pa2 — A K Pa4 - 26j P2 (121, - P2)_ 8 K { [ pa2 ( pk2 p: ) + ( pk2 ... p: ) pa2 i }^(2.5)}C D / ' 11 j P5 + H JK P4 Pa2 I- liK j P2 Pa4 + HK P: + hj PI (le, - Pc2)+ hal( p2 [pa2 [ pk2 _ P2) + (p2 _ p ) pa2]+ hK[pa4 (pi:: .....e) 4. ( pk2 _ p: ) pa4 iThe matrix elements of RD and RD, can be evaluated in arigid symmetric top basis in a similar fashion as thoseof R The non-zero elements are, in the IrR 'representation [11]:19<J, KO-CD +5-CD/1J,K> - -AJJ2 (J+1)2-AJKJ(J+1) K2-AKK4^(2.6)+1-/JJ3 (J+1) 3 + HJKJ2 (J+1) 2 K2 + HKJJ(J+1) K4 + HK K6<J,K±219-CD+9{Di1J,K>={-8JJ(J+1)-18K[(K±2)2+K2]+hjJ2(J+1)2+-1-hjK(J+1) [(K±2)2+K2] +-12:- h ^(K±2)4+K4]]1x{[J(J+1)-K(K±1)] [J(J+1)-(K±1) (K±2)1} 2(2.7)A complete diagonalization to obtain the semirigid rotorenergy levels can easily be done on a personal computer.Several such programs have been used in this thesis work,and their sources have been documented in each specificcase.Transitions between the rotational energy levels1J,Ka,Kc> and 1J',K;,K> require the presence of anelectric dipole moment. For any rotor, these transitionsare governed by the selection rules AJ=0, ±1.Furthermore, for an asymmetric rotor, the dipole momentmay in general lie in any arbitrary direction withrespect to the principal inertial axes. Additionalselection rules connected with its nonzero componentsalong the three axes result:a-type transitions: Ka , Kc-K; K = ee-eo, oo-oe;b-type transitions: Ka Kc--K; , K = ee-oo, eo-oe;c-type transitions: Ka , Kc-K; , K = ee-oe, oo-eo.with e and o denoting even and odd quantum numbers,20respectively.2.2 Nuclear quadrupole hyperfine structure A nucleus with a spin I greater than 1/2 possesses anuclear electric quadrupole moment eQ, which can interactwith the electric charge distribution surrounding thenucleus. The interaction energy will be zero in case of aspherically symmetric electric charge distribution aroundthe nucleus, for example in case of an isolated rare gasatom, since there is no preferred orientation for thequadrupolar nucleus. However, when such a sphericalsymmetric charge distribution is distorted uponchemically bonding with other atoms, or through weak vander Waals interactions, different energies connected withdifferent orientations of the quadrupolar nucleus result.Quantum mechanically, such phenomena can bedescribed as the coupling between the rotational angularmomentum J with the nuclear spin angular momentum I togive a total angular momentum F: F=J+I, where the quantumnumber F ranges from J+I to IJ-II . Each rotational energylevel is split into 21+1 (if J>1) or 2J+1 (if J5J)hyperfine levels. Transitions between these hyperfinelevels are governed by the additional selection ruleAF=0, ±1. This results in a splitting of a rotationaltransition into a hyperfine pattern.21The Hamiltonian describing such interaction can bewritten as the scalar product of two second rank tensors,the electric nuclear quadrupole moment Q and the electricfield gradient Vt at the site of the quadrupolarnucleus [12]:^9-CQ — — 2- co: vz— 1^E Qi. v..6^6 3 ij1,J-x,y,z (2.8)where V11=-VEii. For a rotating molecule, this can berearranged to give as the classical interactionenergy [12]:1^A217Equad = —4 eQ ( ^ ,aZ 2 ° (2.9)where eQ is the quadrupole moment of the nucleus definedby jp(3Z2-r2)dr, where p is the charge density of thenucleus, and (32vi3z2)0 is the field gradient along aspace-fixed Z axis at the nucleus. The classical energyexpression is transformed to quantum mechanics via theWigner-Eckart theorem [13]. The matrix elements of thequadrupolar Hamiltonian Xcl are written in terms of thebasis set IJ,i,I,F,MF> with i denoting any inner quantumnumbers such as Ka,1<c or T. For linear or symmetric topmolecules, both the diagonal matrix elements<J,K,MJ=JINQIJ,K,Mj=J> [14] and the off-diagonal matrixelements <J,K,MJ=JIICIJ+1,K,MJ=J> and<J,K,I4J=JIXQIJ+2,K,MJ=J> [15], can be evaluated explicitly22and can be expressed in a closed form [12]. Matrixelements off diagonal in K exist for symmetric tops, butwill not be discussed here. For the more general case ofasymmetric rotors, the non-vanishing matrix elements oflic) have been evaluated by Bragg [16]:<J,i,I,F,MF I X0 IJ,V,I,F,MF> -C0eQ<J,i,M3-J I Vzz IJ, ii,MJ-J><J,i,I,F,MF I 9-00 IJ+1, V, I,F,MF> -(2.10)C1eQ<J,i,14J-J I Vzz IJ+1,V,MJ-J><J,i,I,F,MF I 9-CQ1J+2,ii,I,F,MF> —C2 eQ<J,i,MJ-J I Vzz 1J+2,ii,MJ-J>with Vzz =(32viaz2) o . m=J indicates the maximum projectionJof J along a space-fixed axis. Co, C1 and C2 are constantsand are functions of I, J and F. Since the matrix isHermitian, the matrix elements for J'=J-1 and J'=J-2 mayalso be obtained from these expressions.Vz is the ZZ-component of the field gradientzexpressed in terms of space-fixed axes. This quantity canbe related to the molecule-fixed components Vgg, along theprincipal inertial axes with g,g'=a,b,c:vzz - E1 Vogt (1)Zg 4)Zglg, g-a,b,c—VaaSeZ2a+Vbb4)Z2b+Vcc 4z2c + 2V d Zao dog^• Zb+2Vac 4)Za 4)Zc + 2Vbc 4)Zb 4)Zc(2.11)23where cpzg and cpw are the direction cosines which relatethe Z axis of the space fixed system to the a,b and caxes of the molecule fixed system; Vgg, are the fieldgradients with reference to the principal inertial axes,and are to a high order of approximation, independent ofmolecular rotational state.Conventionally, the nuclear quadrupole couplingconstants are designated by xgw with xggi=eQVgg,. It shouldbe noted that for the three diagonal terms xgg, Laplace'sequation holds, and Xaa+Xbb+X cc=0. Thus there are only twoindependent variables; the final constants for this partare usually expressed in terms ofxu and (,Xbb-Xcc) •The first order perturbation values of thequadrupole coupling constants can be evaluated in thebasis set of the unperturbed rotor IJ,i,Mj>. As a result,the off-diagonal elements, involving OzgOzg, ((Jog') areidentically zero, leaving the quadrupole matrix diagonalwith only xgg in the diagonal terms [12]. The hyperfinesplitting thus generated is called a "first orderpattern".However, deviations from first order behaviour havebeen observed in some cases, for example for a nucleuswith a large nuclear quadrupole moment Q. Furthermore,with the much higher resolution (FWHH-7kHz [17]) andsensitivity of MWFT spectrometers, compared with those ofconventional Stark modulated instruments (FWHH typically24-100kHz), even small deviations from "first orderpatterns" for a moderate eQ nucleus can easily bedetected and need to be accounted for [Chapter 4]. Oneapproach is to use higher order perturbation theory [12].Another approach is to include all terms in the matrixelements and to employ a complete diagonalizationprocedure for a matrix which is diagonal only in F andMF. The second approach has been used in this thesis workusing Dr.H.M.Pickett's SPCAT and SPFIT programs [18].Symmetry considerations can be used to decide whichterms of the matrix elements are non-vanishing.Nonvanishing contributions to IC will appear whenever thedirect product of the symmetries of IJ,Ka,Kc>,Ozg and Ozg, belong to the representation Aof the Four-group, D2, whose character table is thefollowing:E Ca Cb Cc KK1 1 1 1^ee1 1 -1 -1^eo1-1 1-1^oo1 -1 -1 1^oeFor an asymmetric rotor, the non-zero elements arerelated to the parities of Ka and Kc as follows:25non-vanishing terms inXaa Xbb XccabXacbcKaKc—K;Kee-ee, oo-oo, eo-eo, oe-oeee-oe, eo-ooee-oo, eo-oeee-eo, oe-ooIt can be seen from the above that off-diagonal elementsin r(or Ka,Kd will appear for symmetry reasons, withadditional selection rule: AF=0 and AJ=0, ±1, ±2.There are many cases where there are more than onequadrupolar nucleus in a molecule or a complex. Forexample, Ar-C12, studied in this thesis (Chapter 4), hastwo quadrupolar chlorine nuclei. In this case, twocoupling schemes are possible:(1) Fi=J+Ii; F=F1l-I2; with F1 taking integers or halfintegers from J+Il to 1J-I11, and F taking integers orhalf integers from F1+12 to 1F1-121;or (2) 1=11+12; F=J+1; with I ranging from 11+12 to111-121, and F from J+I to 1J-I1.Since a complete diagonalization program was employed inthis work, the final result is the same for these twocoupling schemes. The latter one has some advantage foridentical nuclei, such as in case of Ar-C12 described inChapter 4, since it allows one to consider the symmetryof the nuclear spin wavefunctions. The problem for anynumber of coupling nuclei was studied in detail by26Thaddeus et al. [19], who expressed the matrix elementsof Ifo in terms of 6j Wigner coefficients [20], which aretabulated [21].2.3 Structural information from rotational spectra Microwave spectroscopy is one of the most widelyused methods to provide accurate structural informationof small to medium sized molecules in the gas phase.However, there are limitations. For example, it isusually necessary to observe spectra of severalisotopomers in order to determine complete moleculargeometries. Also, molecular vibrational effects are oftendifficult to account for in the determination ofstructural parameters.In general, four types of geometrical parameters,designated re, re, ray (or rd and re, are most oftenencountered in microwave spectroscopy. These are theeffective ground vibrational state, substitution, groundstate average, and equilibrium structures, respectively,and have different physical meanings. Detailed discussionon how to extract structural information from microwavespectra for chemically bonded systems can be found inRef.[2]. There have also been several reviews on therelationships between the different structuralparameters [22].27However, it should be pointed out that for the veryweakly bonded and flexible van der Waals complexesinvestigated in this thesis work such structural conceptsdo not always apply in a straightforward fashion. This isbecause of the very low dissociation energies, which arein general about 100 times smaller than those of normalchemical bonds, and because of the very large vibrationalamplitudes involved [5]. The structures derived byapplying some conventional microwave structuralapproximations should therefore be treated with caution.The details are discussed in later chapters separatelyfor each individual complex studied here. It should alsobe pointed out that van der Waals complexes areconsidered dynamic entities, which are perhaps moreappropriately characterized by their potential energysurfaces rather than by application of conventionalstructural concepts [23]. Determination of potentialenergy surfaces, however, would require additional dataabout vibrational transitions which are not yet availablefor the systems studied here.The following considerations apply to mostchemically bonded systems.In a rigid rotor approximation, the principalmoments of inertia are defined as:where Ia, Ib and Ic are the principal moments of inertiaas defined before; mi is the mass of the ith atom; ai, bi28Ia - E mi ( bf +c))iib = E mi ( cf + af )1lc - E mi ( al + bt )1(2.12)and c. are the coordinates of the ith atom along the1principal moment of inertia axes.One way to extract structural information is toutilise directly the rotational constants obtained fromrotational spectra in the ground vibrational state. Thestructure thus obtained is called an effective, or ro,structure. For a simple diatomic molecule, there isenough information to calculate the structure for eachisotopomer. However, for more complicated molecules,extra isotopic data are usually necessary to determinethe structure. In these cases, a least squares procedurecan be used to fit the ro structural parameters to theobserved principal moments of inertia of a sufficientnumber of isotopic species [24]. Such a procedureneglects zero-point vibrational effects by assuming thatthe r0 parameters are the same for different isotopicspecies.Neglect of zero-point vibrational effects severelylimits the accuracy of the ro structural parametersobtained. For example, for a planar molecule, theinertial defect is defined as:where v denotes the specific vibrational state. A. would29(2.13)be automatically zero if the molecule were rigid. Butthis is not the case for a real molecule. Ao is usually asmall positive value ranging from 0.05 to 0.2 amu A2 fora planar molecule in its ground vibrational state [25].Although this residual is relatively small compared tothe moments of inertia, it can cause variations in thebond lengths as large as 0.01 A when different pairs ofprincipal moments of inertia are used [24]. Thisdiscrepancy would be expected to be even larger in thecase of van der Waals complexes, such as in the case ofNe2-Kr (described in Chapter 6), where A. is in the orderof 6.8 amu A2.The contributions to A can be written as [26]:- Avib Acent Aelec^ (2.14)where Aviv kern' and Aelec are contributions of vibration,centrifugal distortion and electronic effects to theinertial defect A. For a normal molecule Avib is usuallythe largest contribution. It has been shown that for aplanar molecule, Avib comes solely from the harmonic partsof the vibrations [27], and can be calculated from aharmonic force field analysis [26]. The degree to whichAo can be predicted by a particular harmonic force fieldis sometimes used as a measure of the quality of the30force field analysis.Another type of structural parameter often used inmicrowave spectroscopy is the so called substitution, orrs, structure obtainable using Kraitchman'sequations [28]. In this method, the coordinate of acertain atom is determined by the changes in theprincipal moments of inertia upon isotopic substitutionat this atom. The formulae for the determination ofsubstitution coordinates in a general asymmetric topmolecule are tabulated in Chapter 7, where they have beenapplied to the case of Ar2-0CS.The rstructure is considered to have somesadvantage over the ro structure since the vibrationalcontributions to the principal moments of inertia arelargely cancelled [29]. In practice, one difficulty is tosynthesize the isotopic species with isotopicsubstitution at a specific atom. However, with the greatsensitivity improvement of MWFT spectrometers over theearlier Stark cell spectrometer, spectra of increasednumbers of substituted isotopomers can be observed intheir natural abundances [30]. Some atoms, e.g. F, haveonly one stable isotopic species. In this case, the firstmoment equation (the centre of mass condition) is used tolocate it. However, the rs procedure has notoriousproblems in dealing with atoms close to the principalinertial axes. In these cases, vibrational effects are31significantly magnified and the coordinates of such atomscannot be obtained reliably [31], without consideringcorrections for isotopic shrinkage [32].Although the above two structures are most oftenencountered in the literature, they suffer from having nospecific physical meaning. One of the physically betterdefined parameters is the ground state average, ray (orrz) structure. ray can be calculated in the same fashionas ro by replacing I0 with Iay• The average rotationalconstants are obtained by removing the harmonic part ofthe vibrational contributions to the vibration-rotationconstant ai, calculated from a harmonic force fieldanalysis from the effective rotational constants [33]:.Gay - G° 3N-6 d. a+ E  " ( harmonic), i-i^2G-A,B,C. (2.15)Gay and Go denote the ground state average rotationalconstants and effective rotational constants,respectively; ai is the vibration-rotation interactionconstant and di is the degeneracy of the ith normal mode.The equilibrium, or re, structure defines thecoordinates of a molecule at the hypotheticalvibrationless state (i.e. at the potential minimum). Itis normally obtained by measuring the rotationalconstants for a sufficient number of isotopomers of amolecule in its ground vibrational state and in at least32one excited state of each normal mode. The equilibriumrotational constants are obtained by applying theequation [24]:31k6^d.Gn .• Ge - L ai (v. +^) +...i-i^i^2(2.16)where Gn is the rotational constant in the nthvibrational state; ai now includes both anharmonic andharmonic contributions; vi are the vibrational quantumnumbers for the 3N-6 normal modes which specify the nthvibrational state; di has been defined previously. Foranything more than a triatomic molecule, the number ofrequired measurements is normally prohibitively high.If an ray structure is obtainable , then one can usethe following equation derived by Kuchitsu andco-workers [34,35] to obtain an approximation for theequilibrium structure:3^2ray re + au - K (2.17)where u2 is the zero-point, mean-square amplitude of thebond in question and K the corresponding mean-squareperpendicular amplitude correction. These quantities canbe evaluated from a harmonic force field [36]. The Morseanharmonicity parameter a may be estimated from similardiatomic molecules. The derivative of the above equationcan also be used to estimate isotopic changes in the bondlengths with isotopic substitution [34,35]:33Or^—32 a 81.12 - 8K^ (2.18)This equation is used in determining r" using severalisotopomers, because values for ray are isotopomerdependent.2.4 Harmonic force field analysisA harmonic force field analysis is of interest toobtain an ray structure. The quartic centrifugaldistortion constants from the analysis of the rotationalspectra can be used in such an analysis. The quarticdistortion constants are related to the T'S of Wilson andHoward [10]. For Watson's A-reduction constants in the Irrepresentation, the r's are obtained from [11]:Taaaa — -4 (AJ AJK A K )Tbbbb '" — 4 (Aj + 28j)tcccc - - 4 (Aj - 28j)jtaabb + 2 abab + taacc + tbcbc = - 4 ( 3A + A JOaabb Atbbcc Btccaa 2Ctabab-4 [ (A + B + c) Aj +^(B C) AJK (B C) (8j. 8 K )(2.19)These T'S are related to the elements of the inverseharmonic force constant matrix [37]. In the approximationof a harmonic force field and small vibrationalamplitudes, one can write [37]:34Too - - [1/2 (iaaippiyyi,o) ] E (Jaip)0 (f -1) ii (J,)8) .^(2.20)i, 3For adaptation to the units commonly used in microwavespectroscopy, the above equation can be rewritten [38]:AA 3Tors ='^2h X 1017 E (J 1ap) a (f -1)13 (Jy8)0 (2.21)I IPP YY^1,Jwith h in erg-sec. A,A are the rotational constants inMHz; 'BB ^In are the equilibrium principal moments ofinertia in amu A2; and (IB), and (J4)0 are derivatives ofthe inertia tensor with respect to the ith internalcoordinate evaluated at the equilibrium configuration, inamu A and amu K2 rad-1 for a stretch and a bend,respectively; (f-1)1i are elements of the inverse forceconstant matrix, with f in mdyn A-1, mdyn A rad-2 andmdyn rad-1 for a stretch, a bend, and a stretch bendinteraction; the r's thus obtained are in MHz.(3100 can be evaluated using Polo's vectorsei [39] • If e1=(07). C), Q1.1), 4). . •• t QI;(1)) is any set ofCartesian displacements in the principal inertial axeswhich produce a unit increment 6R1 in the ith internalcoordinate and leave the other 3N-7 internal coordinatesunchanged, then [39]:Jai. = 2^Mk { 13k QIIP,) + yk Qit 1Jalp = -2 1 Mh 4, mk ok Qac) + mh ig mk ak Qg) VI YY^(2.22)3 5Here the kth atom has mass mk and equilibrium coordinatesakf BV ykf•I is an equilibrium moment of inertia.YYStrictly speaking, only equilibrium values should beused in such a force field analysis. However, equilibriumgeometries are very rarely available, and approximationshave to be made by using re instead of re values in theanalyses. A fitting program is available to fit the forceconstants to the corresponding vibration frequencies andthe quartic centrifugal distortion constants. For amoderately large molecule, the inversion of Eq.(2.21) toallow the determination of a molecular force field fromthe microwave data is in general not possible, becausethe number of force constants usually exceeds the numberof determinable centrifugal distortion constants.Instead, the quartic distortion constants are usedtogether with known vibrational frequency information torefine the harmonic force field of the molecule. For acomplex, it has been shown that the vibrations of therigidly bonded subunits make a negligible contribution tothe centrifugal distortion constants since the van derWaals modes are much lower in energy compared thevibrations of covalent bonds [40]. This assumption helpsto simplify a harmonic force field analysis forcomplexes, although for such flexible systems, the degreeof validity of a harmonic force field is still underdiscussion [41].3 62.5 Theoretical description of pulsedexcitation experiments This section is concerned with the interaction of anensemble of molecular dipoles with a microwave pulse, andwith the transient effects which arise immediately aftera sudden radiation-induced change in the system. In suchexperiments, the radiation power levels not only causerapidly oscillating superposition states, but alsoconsiderably change the thermal equilibrium populationsof the stationary states [42]. In an experiment, amacroscopic dipole moment is created by a microwave pulseof appropriate strength and duration. This dipole momentevolves in time and emits microwave radiation at themolecular transition frequencies. The intensity of theemitted microwave radiation decreases in time as themacroscopic dipole moment decreases, as a result ofincreasing disorder or relaxation due to molecularcollisions and other effects.These phenomena can be described by analogy with theBloch equations used in nuclear magnetic resonance(NMR) [40]. Details of the derivation have been given byFlygare [42], Shoemaker [43], and Dreizler andco-workers [44,45]. Only some important results and somesimple derivations are discussed here in order toacquaint the reader with some phrases and notation used37in the following chapters of this thesis.An appropriate frame of description is provided bytime dependent SchrOdinger theory in a density matrixformalism (44]. The complete time dependent Hamiltonian Xcan be expressed as: X=1(0-FIC1, where Ho is the timeindependent Hamiltonian describing the stationary statesof the system, and Ifi describes the time dependentperturbation introduced by the interaction with amicrowave field. For a two-level particle, the space (q)and time (t) dependent wave function can be written as:Tv (q, t) = cva(t)(1)a(q) + cvb (t) (0)D (q)^(2.23)where v denotes the vth particle; cva(t) and cvb(t) arecomplex time-dependent coefficients; 0a(q)and Ob(q) arethe solutions of the time independent SchrOdingerequation with the Hamiltonian X0 of a two-level particle:9-1:0 4 1(q ) - E , (1)i ( q) ;^i -a,b^(2.24)The perturbation Hamiltonian for the interaction with anexternal field is: 1(1=-A-E(t), with E(t)=ecos(wt-w), theexternal microwave field. Here A is the molecular dipolemoment operator, with the matrix elements Aaa=Abb=0 andab=1/ba" f corresponding to an electric dipole allowedtransition between states (a) and (b); e is the externalmicrowave field amplitude; w is the angular microwaveradiation frequency, not necessarily resonant with the38rotational transition frequency co:b=(Eb-Ea)/1); p is anarbitrary phase factor. The matrix elements of R in thebasis set {Oi} are:Haa=Ea Hblo=Eb and Hab=Hba=— Aabe cos (wt-cp ) .The expectation value <A> of an operator A of thevth particle in the state Tv(q,t) can be written as:<A> <Tv(q,t) IA layq,t)>= E^E Cm(t) cs,n(t) Amnn-a,b m-a,b(2.25)with Amr,=<Om I A I On> • Now if one considers an ensemble of Nparticles, the mean expectation value of the sameoperator A is:<A>wv--gA01C(ci,t) IA li(q,t)>E Ef *n-a,b m-a,b^v-1 Cm(t) c(t) Aran(2.26)The density matrix Q(t) of an ensemble of N particles isgiven by its elements:^1^ *Qnm^ik= — C (t) c(t) N^vm^vnm,n a,b^(2.27)and can be further separated into two parts, namely theamplitude and phase factors:ICvm(t) l'ICvn(t)^{1-^exP [i (yvn(t) -yv,(t)11-^N (2.28)39At thermal equilibrium, the density matrix is diagonalsince the phase relation between individual particles israndom. The diagonal elements describe the populationdistributions of the stationary states. However, when theparticles interact with external coherent radiation, aphase coherence between the wavefunctions of theindividual particles is created and the off-diagonalelements are no longer necessarily zero. From the aboveequations, the quantity <A>m, can also be written in asimple matrix expression: <A>m,---TrfA-el, where Trindicates the trace of the matrix. For example, theinduced macroscopic dipole moment P of the ensemble of Nparticles can be expressed as P=N.Trfg-el.The time dependence of e(t) is described byitl-k = [ 9-C , Q] -X Q — Q9-Cat (2.29)where X is the matrix representation of the Hamiltonoperator in the cpi(q) basis. By substituting the matrixexpressions of X and e(t) into Eq.(2.29), a set of lineardifferential equations for the elements of the densitymatrix results:Oaa —^i Xab COS (COt -9) ceba - eadQbb — —i Xab COS (Cat — (P) (Qba — Qab)Qab •• i WoQab + i Xab COS (4)t — 9) (ebb - Qaa)+Oba —i oQba i Xab COS (G)t — (p) (Qaa — Qbb) '.` 0:b— w(2.30)40where xab=e-ab,u^by analogy to the Rabi frequencies in, NMR spectroscopy. By introducing a new set of matrixelements Om (n,m=a,b):Qaa gaaQbb gbbQab -gab exp[i(cat-9) ]Qba b-ba exp [-i (cot - 9) ](2.31)and by neglecting terms in 2o (the "rotating waveapproximation" [46]), Eq.(2.30) can be transformed intothe simpler form:gaa^ Xab (gba — gab)gbb Xab (gab — gba)gab^(Wo — (a) gab + xab (ebb - eaa)Oba^((0. - (0) Oba - I xab (Qbb Qaa)(2.32)In order to keep the notation simple for laterdiscussion, the following real variables connected withthe above quantities are introduced:Uab — gab + gba^Val)^gba gab)Wab^gaa gbbSab^gaa gbb(2.33)where uab and vab are connected with the polarization P bythe following relationship which can be derived fromEqs.(2.31) and (2.33):P N^[uab (t) cos (c)t -^- vab (t) sin(cot - 9)]^(2.34)41uab is called the real part of the macroscopicpolarization P, which is in-phase with the microwaveperturbation, and vab the imaginary part of P, which is900 out-of-phase. wab=AN/N, where AN is the populationdifference, and sab is the sum of populationprobabilities, which is a constant in the two levelsystem and needs no longer to be considered. Eq.(2.32)can now be written in a simple form as Bloch Equations:1:lab — — A COVab^Vab — A (Lit-lab — XabWab^ (2.35)Wab —^XabVabmo^ 0with AO= .,ab_m'''" wab and o are defined as before.The Bloch equations will now be applied to adescription of a pulse excitation experiment. Theexperiment will be described in two phases: theexcitation and the observation periods. The initialcondition is such that the polarization is zero i.e.and ^population difference s Nuao (ti ) =yob (ti )=0^  that th i^i^i A 0i.e. wab(ti)=- ANo/N. Under the assumption of a nearresonant, strong, short excitation pulse Aw<xab, Aw canbe neglected in Eq.(2.35):flab — 01.1 — — X Wab^ab abWab ••• XabVab(2.36)The solution is:Uab (t)^Uab (t.)Vab (t) -Wab (ti) sin (xabte)^ (2.37)Wab (t) " Wab (t.) COS (Xabte)42with ra=t-ti denoting the microwave excitation pulselength. The condition of a "11/2 pulse" is fulfilled when:XabTe —2Rab. e • *re —712(2.38)and it corresponds to a maximum macroscopic polarization.The maximum signal is connected to AN(t712)=0.The observation period starts at the completion ofthe "v/2 pulse"; its initial conditions are:u (ab • tit/2) =Wab tir/2) =0 and vab(t)=-AN0/N. Since the microwavepulse is switched off, i.e. the system no longerinteracts with the radiation field, all terms involvingxab can be dropped. Eq.(2.35) becomes:laab -A COVab1:Tab AGM-lab^ (2.39)Wab - 0with the solution:uab (t) = +w(ti) sin(At)v(t) = -w(ti) cos (AG)t)w (t)ab(2.40)43The variables uab and vab oscillate with the off-resonantfrequency AG). The observed polarization becomes:P(t) -Ngabw(ti) sin(oal,t-(p)^(2.41)and oscillates with the rotational transition frequency(6:b. Notice that P is proportional to 4th'Eq.(2.39) pictures the situation after a pulsewithout considering the relaxation effects. In a realcase, AN (i.e. wab) relaxes to AN() (ANo/N), and P (i.e. uaband vth) to zero with relaxation times T 1 and T2,respectively. The exponential terms exp(-t/TO andexp(-t/T2) are then introduced to describephenomenologically the relaxation of the particularquantities to their equilibrium values.44Bibliography1. C.H.Townes and A.L.Schawlow, "MicrowaveSpectroscopy", McGraw-Hill, New York, 1955.2. W.Gordy and R.L.Cook, "Microwave Molecular Spectra",Interscience Publishers, New York, 1970.3. For example, S.E.Noyick, K.R.Leopold and W.Klemperer,"Atomic and Molecular Clusters", ed. E.R.Bernstein,Elsevier, New York, 1990. And references thereincited.4. A.D.Buckingham, P.W.Fowler, and J.M.Hutson, Chem.Rey.88, 963 (1988).5. G.C.Maitland, M.Rigby, E.B.Smith, and W.A.Wakeham,"Intermolecular Forces", Oxford: Clarendon, 1981;Also M.Rigby, E.B.Smith, W.A.Wakeham, andG.C.Maitland, "The Forces Between Molecules", Oxford:Clarendon, 1986.6. J.M.Hutson, Ann.Rev.Phys.Chem., 41, 123 (1990).7. B.S.Ray, Z.Physik. 78, 74 (1932).8. See for example, W.Gordy and R.L.Cook, "MicrowaveMolecular Spectra", Chapter 3. IntersciencePublishers, New York, 1970.9. G.W.King, R.M.Hainer and P.C.Cross, J.Chem.Phys. 11,27 (1943).10. E.B.Wilson, Jr. and J.B.Howard, J.Chem.Phys. 4, 260(1936); E.B.Wilson, Jr. and J.Chem.Phys. 4, 526(1936); also D.Kivelson and E.B.Wilson, Jr.,J.Chem.Phys. 21, 1229 (1953).11. J.K.G.Watson, J.Chem.Phys. 45, 1360 (1966). ibid, 46,1935 (1967).12. W.Gordy and R.L.Cook, "Microwave Molecular Spectra",Chapter 9. Interscience Publishers, New York, 1970.13. R.N.Zare, "Angular Momentum", Chapter 5, p.180,Interscience Publishers, New York, 1988.14. J.K.Bragg and S.Golden, Phys.Rev. 75, 735 (1949).15. J.Bardeen and C.H.Townes, Phys.Rey. 73, 97 (1948).16. J.K.Bragg, Phys.Rev., 74, 533 (1948).4517. Y.Xu, W.Jager, and M.C.L.Gerry, J.Mol.Spectrosc.,151, 206 (1992).18. Dr.Pickett, private communication.19. P.Thaddeus, L.C.Krisher, and J.H.Loubser,J.Chem.Phys. 40, 257 (1964).20. B.R.Judd, "Operator Techniques in AtomicSpectroscopy", McGraw-Hill, New York, 1963; AlsoA.R.Edmonds, "Angular Momentum in Quantum Mechanics"2nd ed., Princeton Univ. Press, Princeton, 1960.21. M.Rotenberg, R.Bivens, N.Metropolis and J.K.Wooten,"The 3j and 6j Symbols", The Technology Press,Massachusetts Inst. Tech., Cambridge, 1959.22. For example, K.Kushitsu and S.J.Cyvin, in "MolecularVibrations and Structure Studies", Ed. S.J.Cyvin,Elsevier, Amsterdam, 1972.23. R.J.Saykally and G.A.Blake, Science 259, 1570 (1993).24. W.Gordy and R.L.Cook, "Microwave Molecular Spectra",Chapter 13. Interscience Publishers, New York, 1970.25. W.Gordy and R.L.Cook, "Microwave Molecular Spectra",Chapter 13, Table 13.11, Interscience Publishers, NewYork, 1970.26. T.Oka and Y.Morino, J.Mol.Spectrosc. 6, 472 (1961).27. B.T.Darling and D.M.Dennison, Phys.Rev. 57, 128(1940).28. J.Kraitchman, Amer.J.Phys. 21, 17 (1953).29. C.C.Costain, J.Chem.Phys. 29, 864 (1958).30. See for example, N.Heineking, W.Jager, andM.C.L.Gerry, J.Mol.Spectrosc. 158, 69 (1993).31. C.C.Costain, Trans.Am.Crystallographic Assoc. 2, 157(1966).32. D.R.Herschbach and V.W.Laurie, J.Chem.Phys. 37, 1687(1962).33. T.Oka, J.Phys.Soc.Japan 15, 2274 (1960).34. K.Kuchitsu, J.Chem.Phys. 49, 4456 (1968).35. K.Kuchitsu, T.Fukuyama and Y.Morino, J.Mol.Struct. 4,41 (1969).4636. R.Stolevik, H.M.Seip and S.J.Cyvin, Chem.Phys.Lett.15, 263 (1972).37. D.Kivelson and E.B.Wilson, Jr., J.Chem.Phys. 21, 1229(1953).38. C.R.Parent and M.C.L.Gerry, J.Mol.Spectrosc. 49, 343(1974).39. S.R.Polo, J.Chem.Phys. 24, 1133 (1956).40. J.A.Shea, W.G.Read, and E.J.Campbell, J.Chem.Phys. 792559 (1983).41. S.W.Sharpe, D.Reifschneider, C.Wittig, andR.A.Beaudet, J.Chem.Phys. 94, 233 (1991).42. W.H.Flygare, in "Molecular Structure and Dynamics"p.423, Prentice-Hall, 1978. See also J.C.McGurk,T.G.Schmalz, and W.H.Flygare, Adv.Chem.Phys., 25, 1(1974).43. R.L.Shoemaker, in "Laser and Coherent Spectroscopy",p.197, Plenum Press, 1978.44. H.Dreizler, Mol.Phys. 59, 1 (1986).45. W.Jäger, Ph.D. thesis, Kiel University, (1989).46. R.L.Shoemaker, in "Laser and Coherent Spectroscopy",209-210, Plenum Press, 1978.47CHAPTER 3Experimental Methods The pulsed molecular beam cavity microwave Fouriertransform spectrometer used in this work is similar tothat of the original design of Halle and Flygare [1]. Thecombination of the Fourier transform technique and pulsedmolecular expansions has sparked a small revolution inmicrowave spectroscopy, and has improved the sensitivity,resolution and precision of the measurement of transitionfrequencies by orders of magnitude. Since its appearancein 1981 [1], this type of spectrometer has been widelyused in the study of transient species [2-5] such asweakly bound van der Waals complexes, ions, radicals, andunstable molecules. Today, there are about a dozenmicrowave groups worldwide using these instruments.The advantages of the pulsed excitation Fouriertransform technique over a conventional Stark modulationexperiment, have been discussed in detail by Flygare [6]and by Dreizler [7]. First, the signal-to-noise (S/N)improvement achieved by the pulsed excitation Fouriertransform technique, compared to a Stark cell experimentis estimated to be more than an order of magnitude [6].This great improvement can be rationalized as follows. Ina pulsed experiment, a number of transitions within agiven excitation bandwidth can be excited simultaneously.The time domain signal contains, besides noise,information about all these transitions. A Fourier48transform then yields a display of the transitions in thefrequency domain. In a Stark cell experiment, on theother hand, at any instant of time, only one point of thefrequency spectrum under investigation is recovered.Because microwave spectra in general are not very dense,much of the sweeping time is spent in recording the baseline between the spectral lines.Secondly, a pinhole jet expansion is the mostcommonly used method to introduce sample into themicrowave cavity in this type of spectrometer. Themolecular beam thus generated can have extremely lowtranslational (well below 1K) and rotational (-1K to 2K)temperatures as well as low vibrational temperatures [8],which vary according to the sample mixture, nozzlestructure and backing pressure. This significantlyreduces the number of quantum states being populated, andsqueezes the broad Boltzmann distribution at roomtemperature into a very sharp distribution with only thefew lowest lying states being populated. This not onlysimplifies the spectra greatly, but also enhances theintensities of lower lying rotational transitions.Thirdly, the pulsed technique has an advantage inthe case of molecules with very small dipole moment g.The observed signal is proportional to g, once the "v/2pulse" [Chapter 2] condition is achieved, as compared tothe square of the dipole moment in the case of Stark cellexperiments [6]. The pulse technique thus allows49detection of transitions with very low transitionmoments.The resolution is also greatly enhanced compared toconventional Stark modulated spectroscopy, because of theabsence of several line broadening mechanisms. The linesare essentially free of pressure and wall collisionbroadening since a molecular jet expansion is used. Alsomodulation broadening is absent because there is nomodulation; and power broadening is absent since signalsare observed in emission. The full line width at halfheight (FWHH) in the present case was typically -7 kHz.However, it is in general not possible to obtainreliable intensity information with such spectrometers.The actual intensity observed is affected by severalfactors such as the off-resonance from the excitationfrequency, microwave pulse length and cavity mode.Although the basic design of the spectrometer hasbeen given by Halle and Flygare [1], with the subsequentmodifications described elsewhere [9,10], a briefdescription of the instrument used is given here. Theemphasis is on the experimental conditions, which had tobe carefully chosen and optimized for the weak van derWaals complexes studied in this thesis.3.1 Pulsed molecular beam cavity microwaveFourier transform spectrometer: circulatordetectorprotectiveMW-switchMW-amplifier 5010MHz HP MW-25 MHzSynthesizersinglesidebandmodulator,20MHz<^vMWI"- RF-^tRF-mixer^amplifier^mixer+ 20 MHz\ MW-switch20 MHz + Av5 MHzband passpower dividerV kov + 20 MHz + Av5 MHz + AvRF-amplifiertransientrecorder0 >^personalcomputerivkm-Fig.3.1 Schematic circuit diagram of the pulsed molecularbeam cavity Fourier transform spectrometerThe numbers 1,2,3,4, here represent a train ofpulses which control the experiment. The details areillustrated in Fig.3.2.51A schematic circuit diagram of the spectrometer isgiven in Figure 3.1. Following this diagram, thespectrometer is divided into five main parts: (1)microwave radiation source and generation of microwavepulses; (2) Fabry-Perot resonator and molecular beamsource; (3) signal detection system; (4) data acquisitionsystem (computer data processing); (5) experimentalcontrol.3.1.1 Microwave radiation source and generation of microwave pulses The microwave source used is a HP 8341A synthesizedsweeper which is controlled via an IEEE-bus with a 286personal computer. With use of a single sidebandmodulator the upper sideband 20 MHz from the carrierfrequency vm is obtained. The microwave radiation ismodulated by two fast microwave PIN diode switches togenerate the excitation pulses. Because the switchesreflect the microwave power when closed, isolators areused to reduce the influence on the signal source. Themicrowave pulse is then fed through a circulator into thecavity via a simple "wire hook" antenna (of length -1/4).The microwave radiation power from the HPsynthesizer is 13.0 dBm (20 mw). A considerable amount ofpower (-18 dB) is lost before the microwave radiation iscoupled into the cavity. For the measurement of lowdipole moment complexes with g in the order of or less52than 0.1 D, more power is needed in order to achieve the"71/2 pulse" condition [Chapter 2]. The microwaveexcitation pulses in these cases are amplified with solidstate power amplifiers with a specified maximum microwaveoutput power of 30 dBm (1000 mW) in the frequency rangeof 8-18 GHz, and 27 dBm (500 mW) in the range 4-8 GHz.The present operating frequency region of thespectrometer is from 4.0 GHz to 24.0 GHz3.1.2 Fabry-Perot resonator and pulsedmolecular beam source The heart of the spectrometer is a Fabry-Perotcavity, where the molecules interact with the microwaveradiation. It is situated in a stainless steel vacuumchamber and consists of two spherical aluminum mirrorswith diameters of 28 cm and radii of curvature of38.4 cm. The separation of the mirrors is in the range of30 cm. One of the mirrors is movable, allowing manualtuning of the cavity with a micrometer screw; the othermirror is held at a fixed position. The mirrors aresupported by three 3/4-inch diameter aluminum rods whichare fastened to the wall of the vacuum chamber. Threeteflon rings riding on the rods are attached to themovable mirror to allow smooth linear motion foradjusting the mirror separation. The cavity is tuned intoresonance with the microwave excitation frequency byfeeding a frequency sweep into the cavity and monitoring53the microwave power in transmission. The tuning mirror isthen moved with a micrometer screw until a cavity modeappears at the excitation frequency. For example, afrequency step of 100 kHz at 10 GHz and 30 cm mirrorseparation requires the cavity to be tuned by -3 gm.The quality factor of the cavity is -10,000: thebandwidth of the cavity (FWHH) is about 1 MHz at afrequency of 10 GHz. This comparatively broadbandedmicrowave cavity makes searching for unknown lines lesstedious than in the case of higher Q cavities [11]because a single experiment can cover a wider frequencyrange. The excitation frequency can be changed in stepsof up to 1 MHz in order to scan a wide frequency region.Another important parameter is the beam waist radiuscoo, corresponding to the radial distance from the centreof the cavity to the 1/e field strength point. It can berelated to the geometric parameters of the cavity by thefollowing equation [12]:1^11(0° == [ 7---.7c [ d ( 2R-d ) ] 2 1 2 (3.1)where d is the distance between mirrors, R is the radiusof curvature and 1 is the radiation wave length. Theresulting value for coo is 4.2 cm at 1=3.0 cm (10 GHz).Two different types of commercially availablenozzles have been used as the molecular beam source:54Bosch fuel injector nozzles with an orifice diameter of1 mm and General Valve series 9 nozzles with 0.8 mmorifices. In the normal operation mode, a nozzle isplaced parallel to the input-output radiation axis,mounted near the centre of one of the mirrors.3.1.3 Signal detection system and data acquisition systemThe radiation signal emitted by the excitedmolecules is picked up by the same antenna that was usedto broadcast the excitation pulse, and then fed through acirculator to couple out of the cavity. The signal, whichcontains the frequencies corresponding to rotationaltransitions, is amplified by a high quality low-noisemicrowave amplifier. High quality amplifiers and mixersin the detection circuit, especially a microwaveamplifier with a very low noise figure, seem to becrucial for the high sensitivity of the spectrometer.Since further data processing, likeanalog-to-digital (A/D) conversion, is much easier in theradio frequency range than in the microwave frequencyrange, the microwave signal is downconverted to in twosteps to frequencies -5 MHz. The data then undergoanalog-to-digital conversion and are stored with acommercially available 'plug-in' transient recorder board(obtained from Dr. Strauss GMBH) for the personalcomputer. This board features a programmablepreamplifier, an 8-bit A/D-converter and a memory depth55of 32k 8-bit words. The board can be operated at samplerates of up to 25 MHz. In our instrument 4k data pointsat 50, 100 or 150 ns sample intervals are transferred tothe computer, where the signal averaging of successiveexperiments is done.3.1.4. Experiment control A home built programmable pulse generator createsthe TTL pulses for controlling the sample injectionnozzle and the microwave PIN diode switches and providesthe trigger signal for the start of the data acquisition.The timing of the pulse sequences is qualitatively shownin Figure 3.2. It is essential for signal averaging thatthe master clock of the pulse sequence generator and the5 MHz intermediate frequency after the seconddownconverting step be phase synchronous. The timing ofthe experiment is based on the internal 10 MHz referenceoscillator of the HP synthesized sweeper.In a given experiment, the microwave cavity is firsttuned into resonance with the excitation frequency. Aburst of sample gas is then injected into the cavitybetween the aluminum mirrors. After a short delay, amicrowave pulse, usually in the order of 1 to 10 As, iscoupled into the cavity via the antenna, and a standingwave pattern results. The microwave radiation interactswith the molecular dipoles of the molecules. A short time(in the order of 10 As) after the pulse is turned off,^ mol. pulse widthmol. MW delayMW pulse width---HH^protective switchH MW pulse width +adjustable width^§^microwave pulsemolecular pulse delayIitrigger signalsampling time * number of sample pointscavity, molecular signalbackground signal^ background signal + molecular signalFig.3.2 Control pulse sequence for one experiment cycle.57the microwave switch to the detection circuit is opened.This third PIN diode switch protects the sensitivedetection circuit from the strong microwave pulse duringthe excitation period. The molecular emission signal,coupled out of this cavity, is further amplified,downconverted and then stored in a personal computer as atime domain signal. The cavity background signal,obtained by applying only a MW-pulse and no molecularpulse, is subtracted in each experiment. A Fouriertransform then yields the frequency spectrum.The repetition rate of the experiments wasrestricted to ca. 1 Hz by the pumping speed of thediffusion pump.3.2 Optimization of the experimental conditions The pulsed nozzle is commonly mounted at a positionwhere the axis of the molecular beam is perpendicular tothe input-output microwave radiation axis [13]. Thisarrangement was also used during the development stage ofthe spectrometer used here. A parallel nozzleconfiguration was originally mentioned by Halle andFlygare back in 1981 [1]. The first test experiment waspublished by Grabow and Stahl [10] in 1990. This parallelarrangement results in a longer observation time for thecoherent emission, compared to the perpendicular set up(a)12644.35 MHz 12644.75 MHz58FiL F Fi=^4.5,4.5-3.5,3.5 4.5,5.5-3.5,4.53.5,3.5-2.5,2.5^4.5,3.5-^I3.5,2.513.5,4.5-2.5,3.5(b) 12644.35 MHz^ 12644.75 MHzFig.3.3 Hyperfine components due to 79Br and "N nuclearquadrupole coupling of rotational transition 3(0-20.2 of79BrCH2CEN. (a) measured with the parallel nozzleposition, 10 experimental cycles. Each component isdoubled by the Doppler effect. (b) measured with theperpendicular nozzle position, 800 experimental cycles.I 3.5,2.5-2.5,1.559and therefore in a higher sensitivity and a higherresolution. A great improvement of sensitivity andresolution was also found in this thesis work by changingthe nozzle from the perpendicular to the parallelposition. An example is illustrated in Figure 3.3. Thespectrum shown is a 0.5 MHz section of hyperfinesplitting due to both 7913r and 14N, of the 30,3-20,2rotational transition of 7913rCH2CN. It can be seen thatthe spectrum obtained with the parallel nozzle positionnot only has a better S/N ratio but also narrower linewidths. Furthermore, it has been obtained with only 10experimental cycles, compared to 800 cycles with theperpendicular nozzle position. Each line is split into adoublet due to the Doppler effect [14]. The linefrequencies are calculated by taking the average of thetwo Doppler components. It should be noted that theexcitation frequencies applied in these two experimentsare different, resulting in different intensity patterns.One of the original concerns about mounting a nozzlein one of mirrors was possible interference with themicrowave radiation. However, the diameters of the holesin the mirror are in the order of 0.5 cm, which issmaller than the microwave wavelength (7.5 cm at 4GHz and1.7 cm at 18 GHz), and no perturbations have beendetected. Another question was the position of the nozzlein the mirror. As was mentioned before, two types ofcommercial nozzles have been used as the molecular beam60source: Bosch fuel injectors and General Valve (GV)Series 9 nozzles. It was interesting to note that the GVnozzles were more sensitive to the position in the mirrorthan the Bosch nozzles. A severe intensity decrease (anorder of magnitude) has been observed by moving the GVnozzle from its current position (3cm off centre) by only2cm in either direction, close to or further away fromthe centre. A similar effect was not observed for theBosch nozzles. This seems to suggest that the samplegenerated by the GV nozzle is more confined, thus moresensitive to the microwave field strengths at differentpositions, while that from the Bosch nozzle is more of acloud shape.Furthermore, it was found that the GV nozzle wasmechanically more reliable compared to the Bosch fuelinjector. The GV nozzle can also work against highbacking pressures (-5 to 6 atm), which is essential forachieving the very low translational and rotationaltemperatures needed to stabilize some very weakly boundvan der Waals complexes such as the mixed rare gas dimersand trimers studied in this work. The high backingpressure would, on the other hand, affect the performanceof the Bosch nozzles.One of the most important problems was how toprepare a sufficient amount of the transient species inorder to permit the observation of a microwave spectrum.In a static gas mixture of Ar and HF at a pressure of 161Torr at room temperature only one molecule in 109 of themolecules is bound as a complex with a very short lifetime [8]. This number is greatly increased in the jetexpansion [8]. The high number of collisions inside thenozzle orifice provides a high yield of complexes, whichare subsequently stabilized through the extremely lowtranslational and rotational temperature achieved in thejet expansion.A further question was how to enhance the yield ofthe complex to be investigated. The molecular beamexpansion used is a very complex dynamical process and isfar from being understood in every detail [15]. Theoptimization of the expansion condition is therefore atedious, time consuming procedure with many variableslike the nature of the carrier gas, sample composition,backing pressure, opening time of the nozzle,modification of the nozzle, and tubing material of samplesystem, to be considered. The importance of thisoptimization procedure cannot be overemphasized; it wascrucial for the detection of any of the complexesreported in this thesis. Most of the parameters have tobe adjusted empirically. The choice of carrier (backing)gas, however, warrants some special consideration.The most commonly used carrier gases for beamexperiments are Ar, Ne and He. Helium is the atom withthe smallest polarizability in this series and istherefore least likely to form clusters, either with62itself or other substances. It has been established thatthe rotational temperature in a He beam is usually higherthan in an Ar beam, and higher energetic isomers arequite often observed, for example in infraredexperiments [16]. A possible explanation is that He isvery light and moves very fast, so that the formation ofclusters might block the beam, which would result in ahigher beam temperature. Attempts were made to use He ascarrier gas to observe some complexes such as Ne-Xe/Kr,but no signal could be observed. This is possibly becauseof the very low dissociation energy of these very weaklybound complexes and because of a relatively higherrotational temperature in the He beam.Ar, on the other hand, is the most commonly usedcarrier gas for economic reasons. However, the use of Aras carrier gas resulted in some cases in only very weaklines, even in the case of Ar-containing van der Waalscomplexes such as Ar-N 2 [17] or Ar-CO [18]. Theexplanation is that Ar, in contrast to He, has a strongtendency to form clusters among itself, resulting in amuch lower yield of the desired complexes.The properties of Ne are somewhat between those ofHe and Ar, and it was found that use of Ne as a carriergas significantly improved the S/N ratio in the case ofthe van der Waals complexes studied in this thesis.The ratio of the mixture plays an important role forthe rotational temperature achieved after the beam63expansion. For example, while 1-2% of F2CS in 0.5 atm Aris quite suitable for measuring lower J transitions suchas 1m-0of it is not possible to observe higher Jtransitions such as 71,3-62,4. On the other hand, a mixturewith 30% F2CS nicely improved the sensitivity for71,3-62,4; the observed intensity for 10,1-00,0 decreased,indicating a much higher rotational temperature [19].This can also serve as an extra piece of information forassignment purposes.64Bilioaraphv1. T.J.Balle and W.H.Flygare, Rev.Sci.Instruments 52,33 (1981).2. See for example, M.R.Keenan, D.B.Wozniak, andW.H.Flygare, J.Chem.Phys. 75, 631-640 (1981).3. R.D.Suenram, F.J.Lovas, G.T.Fraser, and K.Matsumura,J.Chem.Phys. 92, 4724 (1990).4. Y.Ohshima and Y.Endo, J.Mol.Spectrosc. 159, 458(1993); Y.Ohshima, M.Iida, and Y. Endo, J.Chem.Phys.95, 7001 (1991).5. N.W.Howard, A.C.Legon, and G.J.Luscombe,J.Chem.Faraday Trans. 87, 507 (1991).6. T.G.Schmalz and W.H.Flygare, in "Laser and CoherentSpectroscopy",125, Plenum Press, 1978.7. H.Dreizler, Mol.Phys. 59, 1 (1986).8. D.J.Nesbitt, Chem.Rev. 88, 843 (1988).9. U.Andresen, H.Dreizler, J.-U.Grabow, and W.Stahl,Rev.Sci.Instruments 61, 3694-3699 (1990).10. J.-U. Grabow and W.Stahl, Z.Naturforsch. 45a, 1043(1990).11. J.-U. Grabow, thesis, Kiel University (1992).12. H.Kogelnik and T.Li, Applied Optics 5, 1550 (1966).13. F.J.Lovas and R.D.Suenram, J.Chem.Phys. 87, 2010(1987).14. C.H.Townes and A.L.Schawlow, in "MicrowaveSpectroscopy", Dover Publication, New York, p337(1955).15. D.R.Miller, in "Atomic and Molecular Beam Methods,Volume 1", Ed. G.Scoles, Oxford University Press,Oxford, 1988, p.14-54.16. W.Klemperer and D.Yaron, in "Dynamics of PolyatomicVan der Waals Complexes", ed. N.Halberstadt andK.C.Janda, Plenum Press, 1(1990).17. W.Jager, and M.C.L.Gerry, Chem.Phys.Lett. 196, 274(1992).6518. T.Ogata, W.Jäger, I.Ozier, and M.C.L.Gerry,J.Chem.Phys. 98, 9399 (1993).19. Y.Xu, M.C.L.Gerry, D.L.Joo, and D.J.Clouthier,J.Chem.Phys. 97, 3931 (1992).66CHAPTER 4Rotational Spectrum of Ar-C124.1 IntroductionVan der Waals complexes between an atom and ahomonuclear diatomic molecule, such as Ar-C12, are amongthe simplest possible which can undergo large amplitudemotions due to both stretching and bending. The bindingforces here are very weak, and are mainly due todispersion forces; only a minor role is played by theinduction part arising from the molecular electricquadrupole moment of the diatomic monomer [1].There is special interest in the Ar-C12 van derWaals dimer. Since it was established that two differentforms of rare gas-halogen complexes exist, a great dealof attention has been paid to these systems. One form islinear, which can be ascribed to an "incipient chargetransfer" [2,3] from the rare gas atom to the halogen.Examples are Ar-C1F [2] and Kr-C1F [3]. The other has aT-shaped equilibrium structure which is consistent withthe atom-atom additive van der Waals model [4]. Examplesare the He- [5], Ne- [6] and Ar-C12 [7] complexes. Thedifference provides a challenge for both experimentalistsand theoreticians to try to explain its origin in detail.A recent theoretical investigation [8] has suggestedthat a linear conformer of Ar-C12 should also exist,close in energy to the T-shaped one. There is also recent67evidence for the existence of both T-shaped and linearAr-I2, provided by the optical spectrum of thiscomplex [10], where the observed continuum fluorescencehas been interpreted as being due to a linear Ar-I2 form.It was thus of considerable interest to search for thepure rotational spectra of both T-shaped and linearAr-C12 complexes.There is further reason to investigate the microwavespectrum of Ar-C12. Over the last twenty years,Ar-C1F [2] and Kr-C1F [3] have been the only tworare gas-halogen complexes studied by microwavespectroscopy, and the only two rare gas-halogen complexesfound to be linear. All other rare gas-halogen complexesstudied by electronic spectroscopy, such as Ar-C12 [7],Ne-IBr [10], have been found to be T-shaped. A furtherpuzzle came when electronic spectroscopy failed to revealspectra of Ar-C1F and Kr-C1F, possibly because ofunfavourable Frank-Condon factors [7].The anticipated difficulties in observing therotational spectrum of Ar-C12 were twofold. The dipolemoment for Ar-C12, which is essential for the observationof a pure rotational spectrum, is due entirely to theweak van der Waals interaction between Ar and C12. Theasymmetric top (in case of T-shaped Ar-C12) spectrum isirregular and is difficult to search for. However, Ar-N2[11] and Kr-N2 [12] have been recently measured with a68high S/N ratio. This success stimulated the search forthe microwave spectrum of Ar-C12 with the hope of solvingthis puzzle.In this chapter, a microwave spectroscopicinvestigation of the T-shaped Ar-C12 complex isdescribed. Accurate rotational constants and quarticcentrifugal distortion constants have been determined andhave been used to estimate the structure of Ar-C121 aswell as its harmonic force field. These results arecompared with the structure obtained from electronicspectroscopy [7] and with the van der Waals frequenciesfrom the theoretical study [8]. In addition, it has alsobeen possible to resolve the hyperfine structure due tothe quadrupole coupling of the Cl nuclei. The quadrupolecoupling constants in both Ar-35C12 and Ar-35C137C1 havebeen determined, from which an estimate of the previouslyunmeasured Cl nuclear quadrupole coupling constant forthe free diatomic chlorine molecule has been obtained.4.2 Search and spectrum assignment The Ar-Cl2 complex was formed by the expansion of agas mixture consisting of 0.4% C12 and 0.8% Ar in Ne at abacking pressure of 5.0 atm. Because of the anticipatedlow dipole moment of the Ar-C12 complex(g<0.01 Debye [7]), the solid state amplifier was used to69increase the MW excitation pulse power.To make an initial prediction of the rotationalspectrum of Ar-35C12, the complex was taken to beT-shaped, with Ar on the a-inertial axis. Rotationalconstants were calculated using the geometric parametersderived from the electronic spectrum [7] and were in turnused to predict the frequencies of the anticipated a-typetransitions. Because the chlorine nucleus has a spin 3/2,all rotational transitions were expected to show nuclearquadrupole hyperfine structure. Since no accurate nuclearquadrupole coupling constants for free C12 werepreviously available, a value of 115 MHz estimated fromthe rotational spectrum of HF-C12 [13] was used topredict the hyperfine patterns.In the expected T-shaped equilibrium structure ofAr-35C12 the two chlorine nuclei are equivalent. The twoequivalent fermions require the total wavefunction to beantisymmetric with respect to a C2 operation. Withsymmetric electronic and vibrational functions, theproduct of the rotational and nuclear spin functions hasto be antisymmetric. The coupling scheme used is:1=11+121 F=J+1, where II and 12 are the individualchlorine nuclear spins. The quantum number I can take thevalues 0,1,2,3. As a result, rotational levels with Ka=0have antisymmetric nuclear spin functions with I= 0 and2, and Ka=1 rotational levels have symmetric spin70functions with I= 1 and 3. The spin statistical weightsof Ka=0 and Ka=1 transitions are in the ratio 6:10.The initial search was carried out for the 30,3-20,2transition, as it has the lowest lying rotational energylevel accessible within the specified frequency range ofthe power amplifier and a relatively simple hyperfinepattern, namely a symmetric triplet in a first orderapproximation. From the experience with Ar-N2 [11] andKr-N2 [12], the line intensities were expected to befairly high. The first line found, however, at 7903.06MHz, was rather weak. It was checked by removing Ar orCl2 and was confirmed to be due to Ar+C12' Using thepredicted hyperfine patterns, two weaker satellite linescould be located after a very careful search, thusconfirming the assignment of the 30,3-20,2 transition ofAr-35C12. Three other transitions, 20,2-10,2, 40,4-30,3 and5 -40,5 0,41 were soon found and their assignments were againconfirmed with their hyperfine triplets.Because spin conversion between spin functions ofdifferent symmetries in Ar-35C12 is not allowed in thebeam expansion, molecules in energy level stacks withKa=0 and Ka=1 cool separately. Levels with Ka=1 aremetastable, and it is possible to observe K8=1transitions with similar intensity as in correspondingKa=0 transitions, even though these levels are higher inenergy by -0.2 cm-1. Six such transitions were found, one71of which, 31,3-21,2, is depicted in Fig.4.1 to illustratethe signal-to-noise ratio and resolution achieved.Searches were also carried out for the mixedisotopomer, Ar-35C137C1, (37% natural abundance) using thestructure calculated for the main isotopomer, Ar-35C12(57%). This time, although three Ka=0 transitions wereeasily found, only a few components from the two lowestKa=1 rotational transitions could be observed, since thechlorine nuclei were no longer equivalent and separatecooling for the two different Ka stacks no longeroccurred.All measured frequencies for Ar-35C12 and Ar-37C135C1are listed with their assignments in Table Analysis of the argon-chlorine dimer spectrumThe Hamiltonian used to account for the observedspectrum is given by: 1{4(R-14(Q, where 'CR is the semirigidrotor Hamiltonian, including rotational constants andquartic centrifugal distortion constants. The quadrupoleHamiltonian Ifcl includes nuclear quadrupole coupling dueto two chlorine nuclei: IVI1Q(1)+1C(2). The programs SPCATand SPFIT [14], which employ complete diagonalization ofthe Hamiltonian matrix, were used. The programs wereapplied in the coupling scheme: Fi=J+Ii, F2=F1i-I2. However,the quantum numbers in Table 4.1 are given in the1MHz1---1723,6-3,5l',F'- I",F"=1,4-1,31,3-1,27556.0 MHz^ 7568.0 MHzFig. 4.1 The chlorine nuclear quadrupole hyperfinesplitting of the rotational transition 31,3-21,2 of Ar-C12.Each doublet (due to Doppler splitting) has beenrecorded separately using 200 to 400 averaging cycles.The polarization frequency was always near resonant tothe observed line. The Doppler splitting in this regionis -41 kHz.3,4-3,33,5-3,473coupling scheme 1=11+121 F=J+I, which is more appropriatefor a molecule with two equivalent or near equivalentnuclei.A global least squares analysis was used to fit themeasured frequencies simultaneously to the rotationalconstants, centrifugal distortion constants of Watson'sA-reduction Hamiltonian in Ir representation [15], andnuclear quadrupole coupling constants. Since only AKa=0transitions were observed, not all distortion constantscould be obtained from the spectra, and an iterativeprocedure was used. In the first fit for Ar-35C12 theconstants AK and 6K were set to zero, and the onlydistortion constants included were AJ, AJK and 6j. Thesewere sufficient to evaluate the harmonic force field (asdescribed in the next section), from which values for AKand 6K were estimated and fixed in the next fit. Thisiterative procedure converged rapidly and the inclusionof AK and 6K improved the fit slightly. The results arein Table 4.2. Although only a-type transitions wereobserved, the A rotational constant could be determinedfairly accurately, because of the relatively high degreeof asymmetry in the complex (K=-0.921). For Ar-35C137C1,since many fewer transitions were observed, all thequartic distortion constants were fixed at values derivedfrom the force field analysis of the most abundantisotopomer.74The differences between measured frequencies and thefrequencies calculated from the derived constants arealso in Table 4.1 for Ar-35C12 and Ar-37C135C1. Thestandard deviation of the fit for Ar-35C12 is about 2.3kHz, while that for Ar-35C137C1 is about 1.3 kHz. Thedifferences between observed and calculated frequenciesof Ar-35C12 show small systematic deviations in thetransitions 21,2-110 and 210-11,0: the same hyperfinecomponents (I,F= 3,4-3,4 and 3,4-3,3) in both transitionsshow similar differences, considerably larger than themeasurement uncertainty. Inclusion of the off diagonalelement xoo or spin rotation interaction did not removethese deviations. Similar but more pronounced effectshave been observed for Ar-N2 [11], Kr-N2 [12], and Ar-HCN[16] and Kr-HCN [17], where the nuclear quadrupolecoupling constants seem to be functions of rotationalstates. However, the effect for Ar-35C12 is small and nofurther attempt was made to account for these deviations.4.4 Estimation of the 35C1 nuclear quadrupole couplingconstant in the free chlorine molecule The Cl quadrupole coupling constants xo of Ar-C12along the g-principal inertial axes can be related to xo,the value for free C12, through the following equation,75Xc,Xg^<3COS21399 —1> ( 4 . 1)where 9 gg is the instantaneous angle between the C1-C1bond and the g-axis. Evaluation of ON requires priorknowledge of xo, the 35C1 coupling constant in free C12.Since it had not been previously evaluated, however, itwas determined using x m of Ar-35C12, because theout-of-plane component xm is free of van der Waalsvibrational averaging (cosOm=0). The value of xo thusobtained is -111.7902(38) MHz.This determination of xo assumes that there is nosevere perturbation of the electronic structure of thesubunits on complex formation. In some complexes, such aslinear HF-C12 [13], this is not entirely the case. Thefield gradients at the Cl nuclei differ by about 3%; inthis case the coupling constants for the inner and theouter 35C1 nuclei are -111.530(17) MHz and-108.161(17) MHz, respectively. However, previous studieson the rare gas-small molecule complexes have not yetshown such large perturbations in the field gradients ofthe small molecules on complex formation. In T-shapedAr-C1CN [18], for example, it was found that theperturbation along the c-axis is only -0.15% at the Clnucleus. Assuming a similar effect in Ar-C12, theestimated xo in free diatomic C12 should be accurate towithin -100 kHz.76From the above equation an average value for theangle O., defined by arccos<cos2eaa>Y2 can be calculated.The result is 85.40(1)°. This represents an averagedeviation of -4.6° from the T-shaped configuration,arising essentially from an internal rotation of the Cl 2unit about its centre of mass. This value is to becompared with 21.7° in Ar-N2 [11] and 21.3° inKr-N2 [12]; evidently Ar-C12 is much more rigid thanthese two complexes. This observation is consistent withthe much lower centrifugal distortion constants in Ar-C12than in the N2 complexes.The xm values for the 35C1 nuclei in bothisotopomers agree within twice the standard deviation.Also, the ratio of the xcc values of 35C1 and 37.C1 in themixed isotopomer is 1.2681(6), which agrees with theratio of the quadrupole moments of the two nuclei1.2688773(15) [19]. It is unusual to be able to determinethis ratio by measurement on 35C1 and 37C1 in a singleisotopomer.4.5 Harmonic force field analysis and structure The distortion constants which could be determinedfrom the spectrum made it possible to perform a harmonicforce field analysis on Ar-C12. There are only threevibrational modes, namely the C1-C1 stretch (v1) and the77van der Waals stretch (v2), both of Al symmetry, and thevan der Waals bend (v3), of B2 symmetry. The two van derWaals modes describe, respectively, the change indistance between the Ar and C12 subunits, and anessential internal rotation of C12 about its centre ofmass. The modes can be approximately characterized by thesymmetry coordinates in Table 4.3. The B2 mode is treatedformally as an asymmetric ClArC1 stretch, without losingits essential internal rotation character.The measured distortion constants of Ar-35C12 werefit by least squares to the diagonal force constants f22and f33' The C1-C1 stretching constant f 111 was held at3.2882 mdynkl, estimated from the C12 vibrationfrequency [20], with the interaction constant, f121 setto zero. The fitting was done in two iterations, inconjunction with the fits to the microwave spectrum, asdescribed in section 4.3. The results of the final forcefield fit are in Table 4.3. Evidently the force constantsare well determined, and reproduce the distortionconstants well. The wavenumbers of the van der Waalsmodes, estimated from the force constants, are also inTable 4.3.The inertial defects A were estimated using thederived force field. The values obtained are 2.395 amukand 2.424 amull2 for Ar-35C12 and Ar-35C137C1, respectively.These are in moderate agreement with the experimental78values in Table 4.2. An alternative approach is to assumethat A arises mainly from the low-lying van der Waalsbending mode, and to use the formula of Herschbach andLaurie [22] to estimate the wavenumbers of this mode. Theequation is Az4K/03, where K=16.85763 amu A2 cm-1, for Ain amu A2 and 03 in cm-1. The result for Ar-35C12 is03=26.6 cm-1, which is rather less than 29.9 cm-1, obtainedin the force field analysis. This conclusion is verysimilar to those obtained for several other complexes,including Ar-OCS [23,24], Ar-0O2 [25] and Ar-C1CN [18],where the approximate method consistently underestimatedthe bending wavenumbers compared to those from a forcefield analysis.The equivalence of the two 35C1 sites in Ar-35C12 hasbeen confirmed by the details of the hyperfine structurein the rotational transitions and by the fact that theKa=0 and Ka=1 stacks cooled separately in the molecularexpansion, as shown by the similar intensities of thecorresponding transitions. Consequently, with the angle 0between the C1-C1 bond and the a-axis fixed at 900, thereremained only one parameter to be determined, namely thevan der Waals distance R between the Ar atom and thecentre of mass of the C12 subunit. The C rotationalconstant here is independent of the average vibrationalangle, since Icc=gR2+I0, where g is the reduced mass ofAr-C12 and It) is the moment of inertia of free C12. In79similar cases such as in Ar-OCS [23,24] and Ar-0O2 [25],it has been shown that the C rotational constant is theone least influenced by vibrational effects. From the Crotational constant values of 3.7190 A and 3.7184 A werecalculated for Ar-35C12 and Ar-35C137C1, respectively. Thevalue 3.72±0.1 A from electronic spectroscopy [7] is inexcellent agreement with the one presented here.4.6 Comments on the argon-halogen dimers and experimental observations From the MW spectrum it has been found that theAr-Cl2 complex is T-shaped and can be fairly welldescribed by a semirigid rotor model. The possibility ofthe existence of a linear conformer was discussed by Taoand Klemperer in their recent ab initio study of Ar-C1Fand Ar-C12 [8]. In a later ab initio calculation with alarger basis set, Tao and Klemperer [26] found that thelinear conformer should be the higher energy conformer.Following their prediction a search was carried out forthis conformer, but without success. Because of theuncertainty in the predicted structural parameters of thelinear conformer and the very low line intensities of theT-shaped Ar-C12, no definite conclusion could be drawnfrom the search. It is very likely that there is a verylow probability for structural degeneracy at the very low80rotational temperature (<5 K) achieved by the supersonicjet expansion. In this sense, the observation of T-shapedAr-Cl2 would rule out the existence of a linear conformerwith lower zero point energy, which is in agreement withthe later ab initio calculation.The experimental and theoretical parameters ofAr-Cl2 are compared in Table 4.4 to those of Ar-C1F.While the ab initio studies [8,26] and the realisticmodel intermolecular potential energy surface [27]produced only slightly higher or lower van der Waalsstretch frequencies compared to the value from the forcefield for Ar-Cl21 both calculations overestimate theflexibility of the van der Waals bending motion. This isshown by the much lower predicted bending frequencies.The chlorine quadrupole coupling constant estimatedfor the free molecule fits into the trend x(freemolecule)=-111.7902 MHz > x(atom)=-109.746 MHz [28] >x(crystal)=-108.95 MHz [29], that was previously observedfor iodine and bromine [13].The surprisingly low intensity of the transitionsobserved, as compared to the case of e.g. Ar-N2 [11], isworth some consideration. The slightly higher number ofhyperfine components in the rotational transitions ofAr-C12 cannot exclusively account for this. Three factorscould cause the weak lines. These are a small dipolemoment, a low abundance of the complex in the molecular81expansion, and a high temperature. The last point can beruled out, however, since a considerable decrease in lineintensity has been observed in going from lower to higherJ transitions, thus verifying a low rotationaltemperature. The length of the applied MW excitationpulses, optimized for maximum polarization of themolecular sample, indicated that the dipole moment isslightly higher than that of Ar-N2 (-0.01 D) [11]. Arelatively low abundance of Ar-C12 might be caused by theformation of C12 dimers and larger clusters in theexpansion. The intensity of the transitions decreased ifthe chlorine percentage in the sample (0.4% C12, 0.8% Arin Ne) was only slightly raised. This picture issupported also by the higher boiling point of chlorine,as compared to nitrogen, indicating relatively strong vander Waals forces between the C12 molecules.82Bibliography1. A.D.Buckingham, P.W.Fowler, and J.M.Hutson, Chem.Rev.88, 963 (1988).2. S.J.Harris, S.E.Novick, W.Klemperer, andW.E.Falconer, J.Chem.Phys. 61, 193 (1974).3. S.E.Novick, S.J.Harris, K.C.Janda, and W.Klemperer,Can.J.Phys. 53, 2007 (1975).4. G. Delgado-Barrio, J. Campos-Martinez, S.Miret-Artês,and P.Villarreal in: "Dynamics of Polyatomic Van derWaals Complexes", edited by N.Halberstadt andK.C.Janda (Plenum Press, New York, 1990), P. 409.5. J.I.Cline, B.P.Reid, D.D.Evard, N.Sivakumar,N.Halberstadt, and K.C.Janda, J.Chem.Phys. 89, 3535(1988).6. J.I.Cline, N.Sivakumar, D.D.Evard, C.R.Bieler,B.P.Reid, N.Halberstadt, S.R.Hair, and K.C.Janda,J.Chem.Phys. 90, 2605 (1989).7. D.D.Evard, J.I.Cline, and K.C.Janda, J.Chem.Phys. 88,5433 (1988).8. F.-M.Tao and W.Klemperer, J.Chem.Phys. 97, 440(1992).9. M.L.Burke and W.Klemperer, J.Chem.Phys. 98, 1797(1993).10. S.A.Walter and T.A.Stephenson, J.Chem.Phys. 96, 3536(1992).11. W.Jager and M.C.L.Gerry, Chem.Phys.Lett. 196, 274(1992).12. W.Jager, Y.Xu, N.Heineking and M.C.L.Gerry,J.Chem.Phys., 1993 (in press).13. F.A.Baiocchi, T.A.Dixon, and W.Klemperer,J.Chem.Phys. 77, 1632 (1982).14. H.M.Pickett, private communication.15. J.K.G.Watson, in "Vibrational Spectra and Structure:A Series of Advances", edited by J.R.Durig (Elsevier,Amsterdam, 1977).8316. T.D.Klots, C.E.Dykstra, and H.S.Gutowsky,J.Chem.Phys. 90, 30 (1989). See also K.R.Leopold,G.T.Fraser, F.J.Lin, D.D.Nelson, Jr., andW.Klemperer, J.Chem.Phys. 81, 4922 (1984).17. T.C.Germann, T.Emilsson, and H.S.Gutowsky,J.Chem.Phys. 95, 6302 (1991). See also E.J.Campbell,L.W.Buxton, and A.C. Legon, J.Chem.Phys. 78, 3483(1983).18. M.R.Keenan, D.B.Wozniak, and W.H.Flygare,J.Chem.Phys. 75, 631 (1981).19. F.J.Lovas and E.Tiemann, J.Phys.Chem. Ref.Data 3, 609(1974) (see p.661).20. G.Herzberg, "Molecular Spectra and MolecularStructure" (Van Nostrand, New York, 1966), Vol I,p. 519.21. J.A.Coxon, J.Mol.Spectrosc. 82, 264 (1980).22. D.R.Herschbach and V.W.Laurie, J.Chem.Phys. 40, 3142(1964).23. Y.Xu, W.Jager and M.C.L.Gerry, J.Mol.Spectrosc.151,206 (1992).24. F.J.Lovas and R.D.Suenram, J.Chem.Phys. 87, 2010(1987).25. G.T.Fraser, A.S.Pine, and R.D.Suenram, J.Chem.Phys.88, 6157 (1988).26. F.Tao and W.Klemperer, private communication (1992).27. B.P.Reid, K.C.Janda, and N.Halberstadt, J.Phys.Chem.92, 587 (1988).28. V.Jaccarino and J.G.King, Phys.Rev. 83, 471 (1951).29. R.Livingston, J.Chem.Phys. 19, 803 (1951).84Table 4.1Observed Frequencies of Ar-35C12 and Ar-35C137C1TransitionJ'^._ JI1^I' F' - I" F"K:1‹^K1E:1(Observed^Obs.-Frequency^Calc.(MHz)^(kHz)Ar-35C1 22 02 -^101 2 2 - 0 1 5284.8003 -0.32 4 - 2 3 5281.2187 1.60 2 - 2 1 5277.6249 -1.22 1.1 -^1 1,o 3 4 - 3 4 5560.9627 -8.43 3 - 1 2 5554.1780 -1.11 3 - 3 2 5544.0945 0.43 4 - 3 3 5538.6083 3.33 5 - 3 4 5521.3749 -0.72 1.2 -11,1 3 4 - 3 3 5066.0652 4.13 3 - 1 2 5046.9235 -0.83 5 - 3 4 5041.5892 -0.53 4 - 3 4 5021.7926 -8.71 3 - 1 3 5008.4428 1.230,3 -^20,2 2 3 - 2 2 7904.8239 -1.02 4 - 2 3 7903.0635 0.52 5 - 2 4 7903.0635 1.50 3 - 0 2 7901.2865 0.385Table 4.1 (continued)TransitionV K:^- J;IIKu I' F" - I" FI‹ flObserved^Obs.-Frequency^Calc.(MHz)^(kHz)3 10 -^2 1,2 3 5 - 3 4 7566.8362 3.11 2 - 1 1 7565.6719 0.53 4 - 3 3 7565.0001 -0.43 6 - 3 5 7561.3425 1.01 4 - 1 3 7559.6287 0.21 3 - 1 2 7557.3367 0.93 1,2 -^2 1,1 3 4 - 3 3 8297.9554 0.31 4 - 1 3 8295.5233 1.11 2 - 1 1 8291.6951 0.81 3 - 1 2 8291.5650 2.53 5 - 3 4 8288.9675 1.63 6 - 3 5 8287.4075 1.440,4 -^3 0,3 2 4 - 2 3 10504.5169 -0.92 5 - 2 4 10502.6732 0.52 6 - 2 5 10502.6732 1.50 4 - 0 3 10500.8139 0.64 1,4 -^3 1,3 3 6 - 3 5 10075.7144 -0.63 5 - 3 4 10075.1238 -0.63 7 - 3 6 10073.8349 1.01 5 - 1 4 10073.2055 -1.486Table 4.1 (continued)Transition^Observed^Obs.-Frequency^Calc.J' 1<p‹ - j7KK^I' F' - I" F"^(MHz)^(kHz)4 1.3 -^3 1.2 1 5 - 1 4 11047.8160 -1.23 5 - 3 4 11047.3538 -1.63 7 - 3 6 11043.7242 0.03 6 - 3 5 11043.0576 0.4505  4-^04 2 5 - 2 4 13075.5617 0.62 6 - 2 5 13073.4716 -0.62 7 - 2 6 13073.4716 0.40 5 - 0 4 13071.3164 -0.5Ar-35c137c120,2 -^10,1 3 4 - 3 4 5237.0880 0.73 3 - 1 2 5226.9762 -0.42 3 - 2 2 5222.4106 -1.62 4 - 2 3 5222.1248 -1.33 5 - 3 4 5220.5347 -1.83 4 - 3 3 5217.3670 -2.32 1.1 -^11.0 4 3 - 3 3 5480.4015 -0.44 2 - 3 2 5473.6246 0.85 3 - 4 3 5464.7621 2.287Table 4.1 (continued)TransitionJ' 1‹^- J;ftio I' F' - I" Fu:1<Observed^Obs.-Frequency^Calc.(MHz)^(kHz)21,2 -^11,1 4 2 - 3 2 4984.8948 2.15 3 - 4 3 4982.1541 0.43 -^2020,2 1 4 - 1 3 7815.3355 -1.02 3 - 2 2 7815.2765 0.62 4 - 2 3 7814.0443 0.82 5 - 2 4 7813.7577 0.13 6 - 3 5 7813.4824 -0.53 4 - 3 3 7813.1372 0.10 3 - 0 2 7812.1597 0.43 5 - 3 4 7810.8633 -0.440,4 -^30,3 1 4 - 1 3 10384.2970 1.12 4 - 2 3 10383.8730 -0.71 5 - 1 4 10382.8805 -0.93 7 - 3 6 10382.8289 1.32 6 - 2 5 10382.3182 0.23 5 - 3 4 10380.4238 -0.23 6 - 3 5 10380.2508 0.188Table 4.2Spectroscopic Constants of Ar-35C12 and Ar-35C137C1Parameter^Ar-35C12^ Ar-35C137C1Rotational constants IMIlzaA^ 7373.50(12)^7173.929(34)1444.08802(23)^1429.91722(19)1200.31288(20)^1185.11335(19)Centrifugal distortion constants /kHzaAj^ 8.2522(47) 8.048bAR 108.21(14) 105.4bAK —111.1b —107.8b6 j 1.4155(36) 1.407bs K 73.69b 71.69b35C1 and 37C1 nuclear quadrupole coupling constants /MHzXaa (mC1) 54.8180(16)^54.8407(27)-‘43b(5C1) -110.7131(19) -110.706(20)'Coo (mC1) 55.8951(19) 55.866(20)aa37C1)X( 43.2111(29)(37CiXbb -87.267(12)2(cc (37Ci) - 44.056(12)Inertial defect /amu 12Ao^ 2.5354(11)^2.5604(3)Standard deviation /kHz2.3^ 1.1a 1r representation, Watson's A-reduction Hamiltonianwas used [16].bFixed at the values obtained from the force field analysis.89Table 4.3The structure and harmonic force field of Ar-35C12Structural parameters: r(C1(1)-C1(2))=r=1.991 liar(Ar-c.m. of C12)=R=3.7190 Ar(Ar-C1 (1) )=r1=3.8499 Ar(Ar-C1(2) )=r2=3 .8499 ASymmetry coordinates: A • S---Ar1 :^1S2=ARB1: S3=(1/20 (Ar1 -Ar2)Harmonic force constants andvibrational frequencies: fil=f (C12) (mdyn A-1)^3.2882f22=fs (mdyn 11-1)w (cm-1)sf33=fb (mdyn A-1)4/13 (cm-1)0.0178234.50.0165129.9Comparison of Observed and Calculatedcentrifugal distortion constants (kHz):Observed'^CalculatedcA j 8.2522(47) 8.251Ax 108.21(14) 108.2AK -111.1^(fixed) -111.16 J 1.4155(36) 1.4226K 73.69^(fixed) 73.69aRef. 21.bThe values are those in Table 4.2.`Calculated from the derived force constants.Table 4.4 Comparison of the properties of Ar-C12 with those of Ar-C1F.Parameter^ Ar-35c1 2^ Ar-mciFExp.^Theo.^Theo.^Exp.^Theo.Structure type^T-shaped^T-shaped^T-shaped^linear^linearR(Ar-C1)/ A 3.8499^3•9^3.97 3.33^3.38Bond energy/ cm-1^188±1a^197.9^180.8^230^233.5ks/ mdyn V^0.01782^- - 0.0301^-/^-1cm 34.5^33.1^41^47.2^51.461skb/ mdyn V^0.01651^- - 0.0215^--1ob/ cm^ 29.9^20.7^12^ 41.0^48.600 4.6 - - 11.1 -References^This work^26^27^ 2^8a Ref. 791CHAPTER 5Pure Rotational Spectra of the Mixed Rare Gasvan der Waals Dimers 5.1 IntroductionRare gas (RG) pair potentials were among the firstvan der Waals potentials studied, since the van der Waalsinteractions between two rare gas atoms are the simplesttype possible and exhibit no angular dependence of thepair potentials. Since the 1970s some major advances havebeen made in the determination of rare gas pairpotentials, as more precise experimental and theoreticalinformation as well as new techniques of analysis havebecome available [1,2]. Research involving the rare gaspair potentials is an active field. This is shown by thecontinuing close interaction between experimentalists andtheoreticians, for these pair potentials have beenrepeatedly revised whenever new experimental data havebecome available [3].The homonuclear dimers in particular have beenextensively studied, and accurate interaction potentialshave been derived [4]. Results from spectroscopicinvestigations have been especially valuable, whencombined with data from 'bulk' and scatteringexperiments [2]. Absorption and laser inducedfluorescence spectra [5,6] obtained in the vacuumultraviolet (VUV) region were of comparatively high92resolution, and vibrational and rotational structurescould be resolved. They have led in some cases to fairlyaccurate rotational and centrifugal distortion constantsfor several vibrational levels in the ground and excitedelectronic states, as well as to equilibrium bond lengthsand harmonic vibration frequencies.The situation is far less satisfactory in the caseof the heteronuclear PG dimers. The newest availablepotentials for He-Ne, He-Ar, He-Kr, He-Xe [7], Ne-Ar [8],Ne-Kr, Ne-Xe [9], Ar-Kr [10], and Ar-Xe, Kr-Xe [11] arebased on scattering and 'bulk' experimental data only,and even this is not very accurate, because homonuclearinteractions must also be accounted for [2]. VUVabsorption spectra have been obtained for Xe- and Kr-RGmixtures [12], but no conclusive assignment of theobserved bands could be made. Collision inducedabsorption (CIA) spectra, which probe the repulsive wallof the potential curve near the collision diameter of RGmixtures, have yielded information about the induceddipole moments of RG-RG' collision pairs [13].On the other hand, information from high resolutioninfrared and microwave spectroscopy has proven to be veryuseful for the investigation of many van der Waalsmolecules [14]. For example, information about theinteraction potential energy surfaces, especially neartheir minima, can be extracted from microwave93(rotational) spectra of such systems [15]. Furthermore,information can be obtained about the molecular electriccharge distribution through the determination of electricdipole moments and, if quadrupolar nuclei (e.g. 131xe,°Kr, 21Ne) are involved, of nuclear quadrupole couplingconstants. These parameters may then be test cases forsophisticated ab initio calculations.However, high resolution IR and MW spectroscopieshave never previously been applied to the heteronuclearRG dimer systems, although these spectroscopies are inprinciple possible for the systems mentioned, because oftheir small, but nonvanishing, induced dipole moments.The search for the rotational spectra of these systemsfaces some serious unknowns, such as the yield of thecomplexes, the rough magnitude of the dipole moments, anddrawbacks, such as the large uncertainty in thefrequencies.The success with some very low dipole moment van derWaals complexes, such as Ar-N2 [16], Kr-N2 [17], andespecially Ar-C12 [Chapter 4], have shown that the methodof pulsed molecular beam MW Fourier Transform (FT)spectroscopy is particularly powerful when only a smalltransition dipole moment is involved. Furthermore, fromthe study of Ar-C12, as discussed in Chapter 4 of thisthesis, it seemed that the abundance of the species ofinterest is more critical for an observation than the94magnitude of the dipole moment. In addition, another partof this thesis work, namely the investigations of the vander Waals trimers Ar2-0CS and Ar2-0O2 [Chapter 7], withAr2 subunits, has suggested that there is an abundance ofrare gas dimers in the molecular expansion. These resultssuggested that microwave spectroscopic investigation ofRG dimers might be feasible. Rotational and vibrationaltransition frequencies of all possible heteronuclear RGdimers have recently been predicted [18], based on thenewest available potentials, and these were expected toprovide valuable starting points for spectroscopicsearches.In this chapter the measurements of rotationaltransitions of various isotopomers of the RG dimersNe-Kr, Ne-Xe, Ar-Xe and Kr-Xe with low values of J (up toJ=9) are described. The analyses have yielded rotationalconstants, along with quartic and, in some cases, sexticcentrifugal distortion constants. The experimentalfrequencies and constants are compared to those predictedfrom the newest mixed rare gas potentials. Furthermore,some parameters in the potentials, e.g. the equilibriumdistances re and (or) the dissociation energies e, havebeen manually adjusted in order to bring the experimentaldata and the predictions based on the potentials intoagreement. The high resolution capability of MWFTspectroscopy has made it possible to resolve the nuclear95hyperfine structure of rotational transitions ofisotopomers containing quadrupolar RG nuclei, and toderive the nuclear quadrupole coupling constants of thesenuclei. Values for the electric dipole moments have beenestimated for all four dimers from the MW excitationpulse lengths, optimized for maximum signal strength("v/2 condition" [19], see Chapter 2).5.2 Experimental considerations In the present work the following gas mixtures wereused: (i) for Ne-Kr: 1.5% Kr in Ne; (ii) for Ne-Xe: 1% Xein Ne; (iii) for Ar-Xe: 2% Ar and 1% Xe in Ne; (iv) forKr-Xe: 1.5% Kr and 1% Xe in Ne. The backing pressureswere up to 5 atm. All gases were of research gradepurity, purchased from Linde, Canada. The MW poweramplifier was used here because of the very low dipolemoments anticipated for all these dimers.Since the molecular electric multipole moments ofthe isolated dimers usually play a minor role in theattractive forces [20], the yields of rare gas dimersformed in the molecular expansion were probably similarto those of van der Waals complexes containing polarsubunits.For well separated lines, the frequency informationis obtained after Fourier transformation, using a three96point interpolation formalism. On the other hand, wherelines were closely spaced (as was found for complexescontaining 83Kr and 131Xe in the present work), thedisplayed spectra were often slightly distorted. In thiscase accurate frequencies were obtained by directanalysis of the time-domain signals [21].5.3 Search and spectral assignments5.3.1 Search and spectral assignment of Ne-KrThe prediction from Ref.18 provided the startingpoint for the search. It was done in 0.2 MHz incrementsfor the excitation frequency, using a pulse length of2 As and 10 averaging cycles at each frequency.There are several possible isotopic combinations inNe-Kr, since Ne has two major isotopes mNe (90.92%) and22Ne (8.28%); and Kr has five: &Kr (56.90%), 86Kr(17.37%), 82Kr (11.56%), 83Kr (11.55%), and mKr (2.27%).The first transition was found at 8188.6913 MHz and waslater assigned to the complex 22Ne-86Kr by comparingisotopic data and intensity. It could be seen easily withone experimental cycle. Following this, it was relativelystraightforward to find transitions at other values of J,as well as lines of other isotopomers.97The intensities of the lines decreased dramaticallywith increasing J values, indicating an extremely lowrotational temperature achieved in the expansion.Altogether spectra of nine isotopomers have beenobserved, with J values ranging from 0 to 4, constrainedby the frequency range of the microwave power amplifier(4-18 GHz) or by the line intensities. The complicatednuclear quadrupole hyperfine structures due to mKr(nuclear spin 9/2) have been resolved forone transition, namely J=2-1, has been observed for21^84Ne--Kr because of its extremely low natural abundance(-0.1%); no hyperfine splitting due to 21Ne (1=3/2) couldbe observed.All measured rotational transition frequencies andtheir assignments are given in Table 5.1. The measuredfrequencies of the hyperfine components of mNe-mKr arein Table 5.2; for this isotopomer the frequencies inTable 5.1 are those of the hypothetical unsplit lines.Fits of the rotational constants and quartic andsextic centrifugal distortion constants to the measuredfrequencies gave the spectroscopic constants inTable 5.3, which reproduce the experimental frequenciesto within 1 kHz (see Table 5.1). The quadrupole couplingconstant for 20Ne-83Kr was obtained from a fit of thisconstant and the hypothetical unsplit line frequencies tothe observed hyperfine components. The standard deviationn- _Ne 83Kr. Only98of this fit was again within the measurement accuracy(see Av in Table 5.2); the constant is given inTable 5.13.The J=2-1 rotational transition of 20Ne-83Kr isdepicted in Fig. 5.1, showing its resolved hyperfinecomponents. The illustration is a composite one, obtainedfrom three experiments. This was necessary because anexperiment at a certain frequency is relatively narrowbanded, covering roughly 200 kHz around the excitationfrequency for this particular transition. For some weakcomponents such as components 7/2-9/2 in Fig. 5.1, theexcitation bandwidths are even narrower. This is theresult of the quality factor Q=10000 of the MW cavity andthe fairly long MW excitation pulse length of 2 gs.5.3.2. Ne-Xe spectra and assignments Essentially the same search procedure was used forNe-Xe as for Ne-Kr. Xe has even more major isotopes thanKr: 128xe (2%),-129Ae_ (26%), 130xe (4%) ,131Xe (21%), 132xe(27%) , 134Xe (10%), and 136Xe (9%) . This time the searchwas started for the J=2-1 transition and the line ofn- _ _128Xe at 7326.1225 MHz was the first one found.Subsequently spectra of ten other isotopomers wereassigned. Four transitions, ranging from J=2-1 to J=5-4,could be measured for most of the isotopomers, and the99^11/2-9/29/2-9/213/2-11/2 ^7/2 - 9/21 1/2-1 1/29/2 - 1 1/2I^ I7/2-7/29/2-7/2F"-r===lt6PAsys..........wm0ftworssow8877.35 MHz 8877.85 MHzFig.5.1. Hyperfine pattern of the J=2-1 rotationaltransition of Ne-mKr.The spectrum is a composite one, obtained from threeexperiments with different excitation frequencies. 200averaging cycles were used. Each hyperfine component issplit into a Doppler doublet because the molecularexpansion is injected parallel to the cavity axis.100measured frequencies are given in Table 5.4. Thefrequencies of the hyperfine components due to thequadrupolar 131Xe nucleus in nNe- 13.1Xe and 22Ne- 131Xe are inTable 5.5; for these two isotopomers the frequencies inTable 5.4 are those of the hypothetical unsplit lines.The derived rotational and centrifugal distortionconstants are in Table 5.6, while the quadrupole couplingconstants obtained are in Table 5.13. Again, thespectroscopic constants obtained here reproduce theexperimental frequencies well, as can be seen from Av inTables 5.4 and 5.5.Spectra of isotopomers containing the quadrupolarVNe nucleus (0.21% natural abundance) were too weak tobe observed.The hyperfine splitting of J=2-1 rotationaltransition of 20Ne-131Xe is depicted in Fig.5.2. Theillustration is again a composite one, obtained fromthree experiments.5.3.3 Ar-Xe spectra and assignment Essentially the same search procedure was used forAr-Xe as for Ne-Kr. The first transition was found at7702.13 MHz and was assigned to the complex Ar-131Xe,because of its 131Xe nuclear quadrupole hyperfine pattern.Altogether spectra of seven isotopomers have beenobserved, with J values ranging from 2 to 9. The effects3/2-3/21 I1101r±-1 7/2-5/21 I 5/2-3/25/2-5/23/2-1/2vi^17303.7 MHzI I I 1.7304.3 MHz Fig.5.2. Hyperfine pattern of the J=2-1 rotationaltransition of 20Ne-131Xe.The spectrum is a composite one, obtained from threeexperiments with different excitation frequencies. Forthe F"-F'=3/2-1/2 component 400 averaging cycles wereused; 20 cycles were needed for the strongest components.102of spin rotation coupling in the case of the 129Xeisotopomer (spin 1/2) were too small to be observed. Noattempt was made to measure transitions of isotopomerscontaining mAr because of the very low intensityexpected.All rotational transition frequencies are given inTable 5.7. The measured frequencies of the hyperfinecomponents of Ar-131Xe are in Table 5.8; for thisisotopomer the frequencies in Table 5.7 are those of thehypothetical unsplit lines. The derived rotationalconstants and quartic and sextic centrifugal distortionconstants are in Table 5.9. The quadrupole couplingconstant for Ar-131Xe is given in Table 5.13. The fits areof similar quality as those of Ne-Kr and Ne-Xe.5.3.4 Kr-Xe spectra and assignments Although Kr-Xe was anticipated to be the most stableof the complexes among these four mixed rare gas dimers,it was considered the most unfavourable case. Both raregases occur as several isotopes, each with considerablenatural abundance. The lowest transition within theoperating range of the spectrometer was J=4-3, near4.4 GHz, well below the specified operating range of thethen available power amplifier (8-18 GHz). The J=5-4transition at roughly 5.5 GHz was expected for searchpurposes to be the best compromise between available MW103power and population. The search was started slightlybelow the frequency estimated from the predictions inRef.18, following the trend observed in the other threedimers. The first line found was at 5496.05 MHz; it wasassigned to 84Kr-132Xe. Measurements for furtherisotopomers were straightforward. The measuredfrequencies for five different isotopomers are given inTable 5.10, together with the values of the deviationsAv, obtained from the rotational and centrifugaldistortion constants in Table 5.12. The fits are ofsimilar quality to those of the other three dimers.The measured frequencies of the quadrupole hyperfinepatterns of 84Kr-131Xe are in Table 5.11. The derivedquadrupole coupling constant in Table 5.13 is the leastprecise of those reported here. The reason is that onlytwo transitions with relatively high J quantum numberscould be measured. Furthermore the sign of the quadrupolecoupling constant could not be unambiguously determinedbecause only two hyperfine components could be resolvedfor each rotational transition. The sign in Table 5.11was inferred from the intensity ratios of the doubletsand from the signs determined for the other two complexescontaining 131xe.No attempt was made to observe quadrupole hyperfinestructure due to 83Kr. The intensity of isotopomerscontaining 83Kr was expected to be much too small because104of its low natural abundance (11.5%), its relatively highnuclear spin (1=9/2) and the relatively high J of thelowest accessible transition (J=4-3).5.4 Estimation of the induced dipole moments of the mixed rare gas dimers Rough estimates for the dipole moments of the dimerswere obtained using the dependence of the signal strengthon the duration of the MW excitation pulse. The productof transition dipole moment and MW excitation pulselength is a constant at a given field strength for thecondition of maximum signal strength, i.e. "v/2condition" [19, see also Chapter 2]:Tcilab•^e (5.1)The permanent dipole moment g of a linear molecule isconnected with the transition dipole moment ^by [22]:'jab12J +1% 7"^) gabJ + 1(5.2)with J denoting the higher level in a transition.The spectrometer was calibrated by finding the "7T/2condition" for the 20,2-1m transition of Ar-0O2 at7317.288 MHz [23] using its known dipole moment of1050.06793(20) D [24]. The method was checked using Ne-0O2,by measuring the ratio of the optimized microwave pulselengths of Ne-0O2 and Ar-0O2. The value obtained forNe-0O2 is 0.026 D, in good agreement with the known valueg=0.0244(13) D [25] measured with the Stark effect.The J=2-1 transition of 22Ne_86 Kr at 8188.6913 MHz,the J=2-1 transition of n- __Die 132Xe at 7296.9286 MHz, andthe J=4-3 transition of Ar-129Xe at 7729.7703 MHz arequite close in frequency to the Ar-0O2 calibrationtransition and were used to estimate the induced dipolemoments in Ne-Kr, Ne-Xe and Ar-Xe, respectively. Theestimated values of the dipole moments are given inTable 5.14. Their uncertainties are difficult toevaluate. The values of the dipole moments obtained arebelieved to be of the correct order of magnitude.Unfortunately, applying the above estimation methodwas much more difficult in the case of Kr-Xe than in theother three cases. The J=7-6 transition, which is theclosest in frequency to the calibration transition ofAr-002/ was too weak for a reasonable intensityinvestigation because the J=6 energy level had too low apopulation. Instead, the J=5-4 transition of 84Kr-129Xe at5545.3940 kHz was used. The result in Table 5.14 is muchless reliable than for the other two complexes.1065.5 Interpretation of the spectroscopic constants 5.5.1 Estimations of the equilibrium distancesfrom the isotopic data With the rotational constants obtained, two types ofdistances, namely the ground state effective (rd and thesubstitution (re) distances have been calculated. ForNe-Kr, Ne-Xe, and Kr-Xe, in which isotopic substitutionswere made at both nuclei, complete rs structures havebeen obtained. The resulting values, obtained using20Ne_84Kr , 20.,e_132--Xe and 84Kr-132Xe as the "parent" species,are given in Table 5.15. These distances were crosschecked using several different isotopic combinations,and were found to be the same within 0.001 A,irrespective of the isotopomers used. For Ar-Xe, whereisotopic substitutions could be made only at Xe, the Arnucleus was located using the first moment condition.This procedure was checked by using the same procedurefor the other three complexes: the resulting distanceswere within 0.001 A of the true rs values. The resulting"re" distance for 40Ar-132Xe is given in Table 5.15.Table 5.15 also contains the ro values for theisotopomers given above, along with the ro values for theisotopomers which have the biggest ro value changes withrespect to their "parent" species.However, one important parameter used to107characterize the potential energy curve is theequilibrium distance re Even though all the measuredspectra were of the complexes in their ground vibrationalstates, it has nevertheless been possible to obtainvalues of the equilibrium (re) distances, because spectraof so many isotopomers have been observed. It was shownby Costain [26] that for diatomic molecules, the rsdistances are, to a good approximation, the averages ofthe ground state effective (rd and equilibrium (re)distances:re - 2r, - ro^ (5.3)The re values thus calculated are in Table 5.15.Since re should in principle be independent ofisotopomer, uncertainties in the re values will arisefrom the variations in r and r0 derived above. Thesoutside error limit from this source is -±0.005 A forNe-Kr, -±0.003 A for Ne-Xe, and -±0.001 A for Ar-Xe andKr-Xe.On the other hand, the accuracy of these re valuesdepends on the validity of the approximation used inderiving Eq.(5.3). This is a particularly importantconsideration for these dimers because they are so weaklybound, and their internuclear potentials can be expectedto be quite anharmonic. Eq.(5.3) was derived [26] usingthe equation:108h% rBt 18n^) _ ( 1 ) ]AI); — l ----) Lk^/I^Bo2 o (5.4)with the approximationa,2Bo — Be _ (5.5)This gives the equationhAIkc: — AI]: + (-87;2 ) [ a ei^a,]^,^+ • . •2 (Bei) 2^2B,`(5.6)where the primed terms refer to the isotopicallysubstituted species. In Eq.(5.6) terms of the formare ignored; this approximation is tantamount to sayingthat (ae/ae')=(Be/Be')Y with y=1.5.To test whether Eqs.(5.4)-(5.6) are valid in thepresent case, it is assumed that a e has the formB2a, - - (6.--) (1+ al)we(5.7)where al is a cubic potential constant. Eq.(5.7) givesthe first approximation to ae in Dunham's [27]expressions, given by Townes and Schawlow [28]; ae alsotakes this form if the Morse potential is adequate todescribe rare gas pairs around the potentialro - re2 re^2re - re38 e ( 1 + a )4er - r0^s (5.8)109minimum [29]. In this case y=1.5, and the terms in cc:/B3ebecome independent of isotopomer, so that theirdifferences in Eq.(5.6) vanish. Any further approximationto Bo, by adding a term -ye/4, would be ignored.There are enough data in Ref.[6] to check whetherEq.(5.7) is a reasonable approximation for the case of40Az'2' The cubic coefficient a1 was calculated using thevalues of Be, ae and we given there: it came to a1=-6.23.This was then used to estimate ro with the equationwhich can be obtained [26] from the derivation ofEq.(5.3). Using the re value of Ref. [6], this givesr0=3.818 A, which is within 0.004 A of the value 3.822 Aderived from Bo [6]. Evidently Eq.(5.7) is a goodapproximation for Ar2. Uncertainties in re from breakdownof Eq.(5.3) are probably roughly ±0.005 A. In the casesof the heavier pairs such as Ar-Xe and Kr-Xe, theuncertainties is probably less than stated. However, inthe cases of the lighter pairs such as Ne-Kr and Ne-Xe,whose potentials support only three bound vibrationalstates [18], it is likely that there is bigger deviationthan -±0.005 A. The total uncertainties in re are thusconservatively estimated to be -±0.01 A for Ne-Kr,-±0.008 A for Ne-Xe, and -±0.005 A for Ar-Xe and Kr-Xe.110The above method has the advantage of using solelythe microwave data with a simple assumption. The Morsepotential approximation for the bottom of the well hasbeen used in the construction of several other rare gaspair potentials [30]. However, the Morse potential hasproven to be inadequate for a characterization of theentire potential [31], because it has only two adjustableparameters. For example, in the case of Ar 2 , the Morsepotential failed to predict the rotational andvibrational spacings measured [31] by Colbourn andDouglas [5].However, the relatively large error bounds attachedto the re values do not reflect the accuracy of themicrowave data presented here. Furthermore, several newtypes of potentials have been developed for these raregas pairs in the last ten years [9,11,32], and it washighly desirable to test these new potentials with thehigh resolution microwave data presented and also to tryto incorporate these data into the potential parameters.5.5.2 Comments on the newest potentials available for Ne-Kr, Ne-Xe, Ar-Xe and Kr-XeOne motivation of this work was to provide accuratespectroscopic data for these mixed rare gas dimers nearthe bottom of the potential well. Highly accuratespectroscopic data would help to refine the existing pair111potentials of these dimers. It might also act as adiscriminator between different potentials proposed forsome pairs.The HFD-B [32,9] and HFD-C [11] type potentials fromAziz and co-workers for these four dimers were used forthis purpose. These potential functions have been widelyused in the theoretical research work involving rare gassystems, such as dipole moment studies of rare gassystems [33] and three-body nonadditive studies [34]. Theanalytical form of HFD-B type potential V(r) is given byAziz [32,9]:V(r) -eV*(x),V*(x) -A*exp (-a*x+ I3*x2) -F(x) i C2i +6 /X 2j +6 ,j-0(5.9)with the damping functionF(x) - exp [- (D/x-1) 2] , x<D-1,^ x121,andX = //r e(5.10)c6, c8, and cw are the dispersion coefficients, usuallyfixed at the values from ab initio calculations. Thispotential has five fully adjustable parameters: e, re, D,B, and in some cases cu [32,9] to add some moreflexibility to the potential. D is a damping parameter; 6112is the dissociation energy from the potential minimum;and re is the equilibrium distance. A*, a* and B* are thereduced parameters, which can be calculated from theabove input. Here a* is positive and B* is negative.The analytical form of HFD-C type potential V(r)is [11]:V (r ) - e V* (x) ,V* (x) - A*xY exp (-a*x) -F (x) i C2i +6 / X2i "Ij -0(5.11)with F(x) the same as defined in Eq.(5.10). The potentialhas four adjustable parameters: e, re, y, and D [11]; e,re, D have the same meaning as before, and y is arepulsive parameter. The rest of parameters are the sameas defined before.All these potentials have been fitted to certainscattering and/or bulk experimental data for various raregas pairs, and have then successfully predicted themajority of all other existing experimental data withinexperimental uncertainties [32,9,11]. However, themeasurement accuracies of these experiments are ingeneral not as high as those of the microwavemeasurements presented here. It was very interesting tosee how well these potentials would predict the measuredfrequencies of the rotational transitions.For this purpose, the program LEVEL, kindly providedby Dr.R.J.LeRoy, was used [35]. This program calculates113vibration-rotation energy levels and B rotationalconstants for diatomic molecules for a wide choice ofpotential functions, including HFD-B and HFD-C. For Ne-Krand Ne-Xe, the absolute differences between the measuredand predicted frequencies and the correspondingpercentage differences are tabulated in Table 5.16, andin Table 5.17 for Ar-Xe and Kr-Xe. The deviations are inthe order of 1.0% to 1.5%, with predicted valuessystematically larger than the measured ones. Thisindicates that the re values of all these potentials wereunderestimated. This observation is consistent with thenew re values presented in Section Manual adjustment of re and c There is no doubt that the microwave measurementspresented are by far more sensitive to the re values thanthe scattering and the "bulk" experimental data, whichhad been incorporated in the original fitting of the raregas dimer potentials [32,9,11]. It was very interestingto see how much one needs to shift re in order to bringthe predictions into agreement with the experimentalfrequencies.This was done by manually adjusting the re valuesuntil agreement between the predicted B, and experimentalrotational constants was achieved. The best results areshown in Table 5.16 for Ne-Kr and Ne-Xe and in Table 5.17114for Ar-Xe and Kr-Xe. The new re values are given inTables 5.16 and 5.17, as well as Table 5.18. Theagreements are much better now as shown by the muchsmaller deviations in Tables 5.16 and 5.17.However, the frequencies of higher J transitionswere still not in good agreement. In each case, theabsolute deviations increase systematically with increaseof the rotational quantum number J. Such discrepanciespossibly depend on the dissociation energy and/or theshape of the potentials, which are connected with theexperimental centrifugal distortion constants.In a second attempt, both re and e were adjustedmanually. It was possible to bring the predictedfrequencies into excellent agreement with theexperimental data. The deviations are in the ranges of0.1 kHz to 20 kHz, close to the experimental uncertainty(1 kHz). The best results are listed in Table 5.16 forNe-Kr and Ne-Xe, and in Table 5.17 for Ar-Xe and Kr-Xe.The adjustments described above have been done forthe most abundant isotopomer of each species. To estimatethe uncertainty in the adjustment procedure, the finalpotentials with the adjusted e and r e were checked onother isotopomers. The result for 22Ne-84Kr, which has thebiggest relative mass change in all these pairs, is givenhere. The potential adjusted for 2°Ne-84Kr predicted theexperimental frequency measurements of 22Ne-84Kr to better115than 0.01%, as compared to 0.00004% in the case ofnNe-m-Kr. In order to predict the 22Ne-114Kr frequencieswith the same quality as for 20Ne-84Kr, one would need tochange re in the order of less than 0.0002 A, with e heldconstant. For other isotopomers, the relative masschanges are smaller, and the necessary changes in re arealso smaller. From the above consideration, theuncertainty for re is estimated to be ±0.0002 A. The revalues from the adjustments are listed in Table 5.18,along with the re values obtained in section 5.5.1 fromthe isotopic data and the literature values. Theuncertainties in the e values are more difficult toestimate.From Table 5.18, it is clear that the re valuesobtained from the present work, despite differentapproaches used in the data analyses, are all larger thanthe literature values obtained from "bulk" experiments.For the heavier pairs Ar-Xe and Kr-Xe, the re valuesobtained using conventional microwave structuralapproaches are very similar to those obtained usingnewest available potentials, with deviations in the orderof 0.001-0.002 A. However, for the lighter pairs Ne-Xeand Ne-Kr, especially Ne-Kr, the deviations are muchlarger, in the order of 0.01 A, indicating the possiblebreak down of the conventional approaches.It should also be pointed out that the rare gas116potentials are still an active research area, and theresults presented here are by no means the final answers.The uncertainties quoted for the re values, obtained fromthe microwave data by manually adjusting re and e,reflect rather the reproducibility of the measuredfrequencies by the adjusted potentials than the absoluteuncertainties. For example, even though the adjustedpotentials can predict the microwave measurements to avery high accuracy, it is still outside the experimentaluncertainty (1 in 107). Moreover, Aziz's HFD-B potentialshave difficulty in being consistent with high energy beamscattering data without distorting the well [34]. Aninitial test run using the newly adjusted re and E valuesshifted the predicted the total collision cross sectionexperiments even further away from those fromexperiments [36]. Clearly, the rare gas potentials haveto be reexamined. The microwave data have to be includedin the fitting procedure of all adjustable parameters inthe analytical forms [36]. Furthermore, there is alsotheoretical interest in developing more flexible rare gaspair potentials [36].5.6 Comments on the dipole moments of themixed rare gas dimers It has been possible in this work to estimate values117for the dipole moments of the dimers, using theoptimization of the MW pulse excitation condition. Theyrepresent the first experimental values available at theseparation of the potential minimum. The induced dipolemoments are somewhat surprisingly large, in the order of0.01 D, comparable to that of Ar-N2 [16]. It was notpossible to obtain the direction of the dipole momentwith the technique applied.Certain difficulties were encountered in theexperiments anAR '*+AU* +(he accuracy of the finalresults. The difficulties arose from the relatively smallavailable MW power, especially in the lower frequencyrange. The problem was severe for Kr-Xe, where pulselengths > 20gs were necessary. The molecules travel inthis time distances in the order of 1 cm and variation ofthe field strength in the MW cavity may well not benegligible. In addition, the spectrometer performancevaries with frequency range and the conditions mightchange in tuning from the transition frequency of thecomparison molecule (Ar-0O2) to the correspondingfrequencies of the mixed rare gas dimers.The only other available experimental source ofinformation about the dipole moments are the CIAspectra [37]. Bar-Ziv et a/. [38] derived a "reduced"dipole moment function from these spectra which isapplicable to a number of mixed RG pairs. The118'meaningful' range of this function is, according tothese authors, from 0.6*re to 0.85*re, thus reflectingthat the CIA spectra probe the repulsive wall of theinteraction potentials. Nonetheless, an extrapolation tothe separation at the potential minimum was done [38] andthe magnitudes of the values in Table 5.14 are inrelatively good agreement with those obtained in thiswork. The sign of their dipole moments is such that thelighter atom is the negative end.However, theoretical calculations of some lightermixed RG dimers suggest a different sign [39]. The dipolemoment is usually decomposed into two contributions [40]:i) a short range part, resulting from electronic overlapand distortion of the electron cloud, making the morepolarizable atom the negative end; ii) a long range partdue to electron correlation effects with opposite sign.Future theoretical work might clarify which is thedominant contribution at potential minimum separation.5.7 Discussion of the nuclear quadrupole couplingconstants of °Kr and 121XeThe observed nuclear quadrupole hyperfine structuresdue to °Kr and 121Xe arise through the coupling of theirnuclear spins (1=9/2 for °Kr, and 1=3/2 for 131Xe) withthe overall rotation of the complexes. Figure 5.1 shows119the hyperfine splitting of the J=2-1 rotationaltransition due to 83Kr nucleus of n e-N 83Kr, whileFigure 5.2 shows the hyperfine splitting of the J=2-1rotational transition due to 131Xe nucleus of 20Ne-131Xe.The mechanism requires a non-zero electric field gradientat the site of the 83Kr or 131Xe nuclei. This fieldgradient is zero in a free atom in a 1S state. Clearlythe observed structures are due to a perturbing effect ofthe attached RG atom.Several van der Waals complexes consisting of amolecule and a RG atom with a quadrupolar nucleus such as0Kr or 131Xe have been investigated earlier, using cavityMWFT or molecular beam electric resonance techniques (seefor example Refs.[41-44]). In these cases, the hyperfineeffects are expressed in terms of the permanent electricmultipole moments of the polar molecule. The fieldgradients qo at the sites of RG nuclei due to theseelectric multipole moments can be calculated using [42]:Pi (cosi))^P, (cos())^ > - 12Q <^4 5P,(cos0)- 20La <^ > -6(5.12)where g, Q, and n are the electric dipole, quadrupole,and higher order multipoles of the polar molecule;Pi(cose), P2(cose), and P3(cose) etc. are the Legendrepolynomials; and the angular brackets indicate theexpectation values in the vibrational ground state.120However, the actual field gradient q, experienced bythe quadrupolar nucleus is not directly the externalfield gradient go. The reason is that the electron cloudsurrounding the nucleus is distorted and polarized by theexternal field, and that changes the field gradientsensed by the nucleus. Such effects have been analyzed bySternheimer and co-workers [45-47], and thephenomenological outcome of their analyses was that q canbe taken to be a "shield" external fieldgradient [45-47]:q^(1 - y„)^ (5.13)where y, is the Sternheimer shielding factor, which is aconstant for a certain nucleus. It should be pointed outthat the value of ly,1 is not necessarily smaller than1.0. For example, in case of 153Kr, y,=-77.5±15 [42]. Thenuclear quadrupole coupling constants, x, can then beobtained using the relation x=eQq.However, since RG atoms in a 1S state have nopermanent electric moments, such a consideration wouldresult in a zero x value. Yet the observed °Kr and 121Xecoupling constants (Table 5.13) are remarkably large:their absolute values are about 10% of those of theconstants measured in Refs.[41-44]. For example, forKr-HF, x=10.23(8) MHz [41]; 83Kr-HC1, x=5.2(1) MHz [42];Kr-HCN, x=7.46(6) MHz [43]; 131Xe-HF,121x=-8.54(4) MHz [44]; for 131Xe-HC1, x=-4•64(5) MHz [42].Furthermore, the coupling constants in the mixed rare gasdimers have opposite sign of those of the RG-smallmolecule complexes. It is clear that formation of a vander Waals bond causes a measurable distortion of theelectron clouds of the rare gas atoms, even though thereare no permanent multipoles involved.There is an earlier indication that these effectsmight not be negligible. Campbell et al. [43] haveplotted the observed °Kr quadrupole coupling constantsof 83Kr-HX and 83Kr-DX (X=F [41], Cl [42], CN [43]) as afunction of the electric field gradients qo at the Krnuclei, calculated from the electric multipole moments ofthe RG binding partner (see Fig. 5.3). A similar plot(Fig. 5.4) had also been done for the 121Xe-HX and 131Xe-DX(X=F [44], C1[42]) complexes. The x axis intercept (q0=0)is at -0.44±1.7 MHz in case of °Kr, and at+0.243 ±1.6 MHz in case of 121Xe. Both observations areconsistent in sign and order of magnitude with thecoupling constants obtained in this work, where bydefinition q0=0. However, because of the attached error1^183Kr—DF 83 Kr—HF ^83Kr—DC 14N1_83 Kr—HC 14 N83 Kr—D 350183 K r —H 35C11 61 21220^0.2^0.4^0.6^0.8—eq (calc.)/h (MHz/b)Fig.5.3 Experimentally determined values of the 83Krnuclear quadrupole coupling constants plotted as afunction of the electric field gradient qo at the site ofKr nucleus calculated from Eq.(5.12). The solid line isobtained from a weighted least squares fit to the data.The horizontal bar indicates the uncertainty in thecalculated qo [43].I 1 I 1 1I-I131 Xe—DF-131 Xe—HF-131 Xe—D 35CI131 Xe—H 35 CI---—12—101230^0.2^0.4^0.6—eq. (calc.)/h (MHz/b)Fig.5.4 Experimentally determined values of the 131Xenuclear quadrupole coupling constants plotted as afunction of the electric field gradient qo at the site ofXe nucleus calculated from Eq.(5.12) [43].124limits of ±1.7 MHz and ±1.6 MHz, the agreement must beregarded as fortuitous.It is interesting to note that no quadrupolecoupling effects were observed in the investigation ofthe rotational spectra of the an Hg-Ar dimer [48]. Fromthe lack of those effects, an upper limit for the 201Hgcoupling constant of 0.5 MHz was estimated in Ref.[48].From a comparison of the 131Xe and amH-g coupling constantsin the pairs 131Xe-HC1 (x=-4•64(5) MHz) [42], 201.-ny HC1(x=6.0(2) MHz) [49] and 131Xe-0O2 (x=-3.05(6) MHz) [50],201 Hg-0O2 (x=4.80(6) MHz) [51], respectively, one wouldexpect the magnitude of the 201 Hg quadrupole couplingconstant to be slightly larger than that of Ar-131Xe.5.8 Conclusions and some future prospects The spectra observed were of quite high intensityand it has been possible to observe a multitude ofdifferent isotopomers in natural abundance. An overviewof all the isotopomers observed is given in Fig.5.5. Therarest one is 21Ne-84Kr with a natural abundance of-0.12%, which is about 500 times less abundant than themain isotopomer 20Ne-84Kr.It has been possible to incorporate the highresolution microwave data presented here into the newestrare gas pair potentials of Ne-Kr, Ne-Xe, Ar-Xe and Kr-Xe125to obtain r e values. It was found that the presentliterature r e values for these dimers, had beenunderestimated in the order of 1-2%. Clearly, these pairpotentials need to be reexamined. Furthermore, suchdifferences, although small, have significant impact onstudies concerned with nonpairwise additive contributionsin many-body systems.The values for the dipole moments and nuclearquadrupole coupling constants are surprisingly large. Themagnitudes of these parameters suggest that electroncorrelation and/or overlap effects must be considered inattempts to account for these properties in RG-moleculecomplexes. It might be feasible to measure the FIRspectra of these dimers to obtain vibrationalfrequencies, which are important for characterizing thepair potentials.126Ar_128xe 1.9% 84Kr_129xe20 Ne_80Kr20Ne_82Kr20Ne_ 83Kr20Ne_128xe20Ne_129xe20Ne_130xe84Kr_131xe84Kr_132xe 15%86Kr_129xe 4.6%Ar-129XeAr-130xe20ik Te_IN 84Kr 51.6% 20Ne_131xe Ar-131xe20Ne_86Kr22Ne_ 80Kr22Ne_82Kr22Ne_84Kr22Ne_86Kr20Ne_134xe20Ne_136xe22Ne_129xe22Ne_131xe20Ne_132xe 24%^Ar-132Xe 27% 86Kr_132xeAr-134XeAr-138Xe21Ne_84Kr 0.12% 22Ne_132xe22Ne_134xe 0.96%Fig.5.5 List of all isotopomers of the four mixed raregas dimers observed in natural abundance.The highest and the lowest natural abundance of eachdimer are indicated.127Bibliography1. J.A.Barker, in "Rare gas solids", Vol.1, ed.M.L.Klein and J.A.Venables, Academic, London,New York, San Francisco, 1976, p.212.2. R.A.Aziz, in "Inert Gases, Potentials, Dynamics andEnergy Transfer in Doped Crystals", Ed. M.L.Klein,Springer Verlag, Berlin, 1984, p.5.3. R.A.Aziz and H.H.Chen, J.Chem.Phys. 67, 5719 (1977);R.A.Aziz and M.J.Slaman, Mol.Phys. 58, 679 (1986).4. R.A.Aziz and M.J.Slaman, J.Chem.Phys. 94, 8047(1991); Chem.Phys. 130, 187 (1989); Mol.Phys. 58, 679(1986); Mol.Phys. 57, 825 (1986).5. E.A.Colbourn and A.E.Douglas, J.Chem.Phys. 65, 1741(1976). R.H.Lipson, P.E.LaRocque, and B.P.Stoicheff,J.Chem.Phys. 82, 4470 (1985). P.E.LaRocque,R.H.Lipson, P.R.Herman, and B.P.Stoicheff,J.Chem.Phys. 84, 6627 (1986).6. P.R.Herman, P.E.LaRocque, and B.P.Stoicheff,J.Chem.Phys. 89, 4535 (1988).7. M.Keil, L.J.Danielson, and P.J.Dunlop, J.Chem.Phys.94, 296 (1991).8. D.A.Barrow and R.A.Aziz, J.Chem.Phys. 89, 6189,(1988).9. D.A.Barrow, M.J.Slaman, and R.A.Aziz, J.Chem.Phys.91, 6348 (1989). J.Chem.Phys. 96, 5555 (1992) (E).10. R.A.Aziz and A. van Dalen, J.Chem.Phys. 78, 2413(1983).11. R.A.Aziz and A. van Dalen, J.Chem.Phys. 78, 2402(1983).12. D.E.Freeman, K.Yoshino, and Y.Tanaka, J.Chem.Phys.67, 3462 (1977). M.C.Castex, J.Chem.Phys. 66, 3854(1977).13. Z.J.Kiss and H.L.Welsh, Phys.Rev.Lett. 2, 166 (1959).D.R.Bosomworth and H.P.Gush, Can.J.Phys. 43, 729(1965).12814. S.E.Novick, K.R.Leopold, and W.Klemperer, in "Atomicand Molecular Clusters", Ed. E.R.Bernstein, Elsevier,Amsterdam, 1990, p.359.15. J.M.Hutson, Annu.Rev.Phys.Chem. 41, 123 (1990).A.E.Thornley and J.M.Hutson, Chem.Phys.Lett. 198, 1(1992).16. W.Jager and M.C.L.Gerry, Chem.Phys.Lett. 196, 274(1992).17. W.Jager, Y.Xu, N.Heineking, and M.C.L.Gerry,J.Chem.Phys., 1993 (in press).18. J.F.Ogilvie and F.Y.H.Wang, J.Mol.Structure. 291, 313(1993).19. H.Dreizler, Mol.Phys. 59, 1 (1986).20. G.C.Maitland, M.Rigby, E.B.Smith, and W.A.Wakeham,"Intermolecular Forces, Their Origin andDetermination", Clarendon Press, Oxford, 1981.21. J.Haekel and H.Mader, Z.Naturforsch. 43a, 203 (1988).22. W.Gordy and R.L.Cook, "Microwave Molecular Spectra",Interscience Publishers, New York, 1970.23. G.T.Fraser, A.S.Pine, and R.D.Suenram, J.Chem.Phys.88, 6157 (1988).24. J.M.Steed, T.A.Dixon, and W.Klemperer, J.Chem.Phys.70, 4095 (1979).25. M.Iida, Y.Ohshima, and Y.Endo, J.Phys.Chem. 97, 357(1993).26. C.C.Costain, J.Chem.Phys. 29, 864 (1958).27. J.L.Dunham, Phys.Rev. 41, 721 (1932).28. C.H.Townes and A.L.Shawlow, "Microwave Spectroscopy",McGraw-Hill, New York, 1953; W.Gordy and R.L.Cook,"Microwave Molecular Spectra", IntersciencePublishers, New York, 1970, p.692.29. L.Pauling and E.B.Wilson, Jr., "Intoduction toQuantum Mechanics", McGraw-Hill, New York, 1935,P.274.30. See for example, J.M.Farrar, Y.T.Lee, V.V.Gldman,M.L.Klein, Chem.Phys.lett. 19, 359 (1973).12931. J.M.Hutson, Annu.Rev.Phys.Chem. 41, 123 (1990).32. R.A.Aziz, private communication.33. B.Guillot, R.D.Mountain, and G.Birnbaum, J.Chem.Phys.90, 650 (1989); B.Guillot, J.Chem.Phys. 91, 3456(1989).34. A.R.Cooper and J.M.Hutson, J.Chem.Phys. 98, 5337(1993); A.R.Cooper, S.Jain, and J.M.Hutson,J.Chem.Phys. 98, 2160 (1993).35. R.J. Le Roy, private communication.36. R.A.Aziz, private communication.37. I.R.Dagg, G.E.Reesor, and M.Wong, Can.J.Phys. 56,1046 (1978).38. E.Bar-Ziv and S.Weiss, J.Chem.Phys. 64, 2417 (1976).39. M.Krauss and B.Guillot, Chem.Phys.Lett. 158, 142(1989).40. E.W.Pearson, M.Waldman, and R.G.Gordon, J.Chem.Phys.80, 1543 (1984).41. L.W.Buxton, E.J.Campbell, M.R.Keenan, T.J.Balle, andW.H.Flygare, Chem.Phys. 54, 173 (1981).42. E.J.Campbell, L.W.Buxton, M.R.Keenan, andW.H.Flygare, Phys.Rev. A 24, 812 (1981).43. E.J.Campbell, L.W.Buxton, and A.C.Legon, J.Chem.Phys.78, 3483 (1983).44. F.A.Baiocchi, T.A.Dixon, C.H.Joyner, and W.Klemperer,J.Chem.Phys. 75, 2041 (1981).45. R.M.Sternheimer, Phys.Rev. 80, 102 (1950); 84, 244(1951); 86, 316 (1952); 95, 736 (1954); 102, 731(1956); 146, 140 (1966).46. R.M.Sternheimer and H.M.Foley, Phys.Rev. 92, 1460(1953).47. H.M.Foley, R.M.Sternheimer, and D.Tycko, Phys.Rev.93, 734 (1954).48. Y.Ohshima, M.Iida, and Y.Endo, J.Chem.Phys. 92, 3990(1990).13049. J.A.Shea and E.J.Campbell, J.Chem.Phys. 81, 5326(1984).50. M.Iida, Y.Ohshima, and Y.Endo, J.Phys.Chem. 97, 357(1993).51. M.Iida, Y.Ohshima, and Y.Endo, J.Chem.Phys. 95, 4772(1991).131Table 5.1 Observed frequencies of Ne-Krmolecule J'- J"^V OBS(MHz)^(kHz)^2 °Ne-86Kr 1 - 0^4411.0577^0.1^2 - 1^8819.3663 -0.13 - 2 13222.1597^0.04 - 3 17616.6345^0.020Ne-84 Kr 1 - 0^4430.2661 -0.22 - 1^8857.7571^0.23 - 2 13279.6785 -0.14 - 3 17693.2023^0.020Ne_83Kr 1 - 0^4440.1936 -0.52 - 1^8877.5987^0.3^20Ne-82Kr 1 - 0^4450.3964^0.32 - 1^8897.9882 -0.33 - 2 13339.9561^0.14 - 3 17773.4422^0.0Ne-80Kr 1 - 0^4471.5179 -0.22 - 1^8940.2024^0.13 - 2 13403.2012^0.0Table 5.1 (continued)molecule J'- J"^v^AvOBS(MHz)^(kHz)13222Ne-84Kr 1 - 0 4114.7964^0.82 - 1^8227.2299 -0.83 - 2 12334.9322^0.34 - 3 16435.5011 -0.122N e _86K r 1 _ 0 4095.5134^0.02 - 1^8188.6913 -0.03 - 2 12277.1850^0.04 - 3 16358.6186 -0.022Ne-82Kr 1 - 0^4135.0015^0.12 - 1^8267.6170 -0.13 - 2 12395.4490^0.0Table 5.2Observed frequencies of nNe-mKrJI _Fljfil-^F"V obs(MHz)Av(kHz)1 - 03.5 - 4.5 4440.2408 -0.55.5 - 4.5 4440.2197 0.14.5 - 4.5 4440.1247 0.52 - 13.5 - 4.5 8877.6709 -0.35.5 - 4.5 8877.6250 0.24.5 - 4.5 8877.6250 0.26.5 - 5.5 8877.6112 1.33.5 - 3.5 8877.5533 -0.85.5 - 5.5 8877.5286 -0.84.5 - 5.5 8877.5286 -0.84.5 - 3.5 8877.5086 0.9133134Table 5.3 Spectroscopic constants of Ne-KrParameter B (MHz) D^(kHz)J H^(Hz)J20Ne-84Kr 2215.36414(12) 115.453(13) -24.68(42)2 °Ne-mKr 2205.75756(73) 114.3322(83) -25.02(26)20Ne-0Kr 2220.32925(13) 116.045(fixed)a -24.68(fixed)b20Ne-82Kr 2225.43140(20) 116.636(23) -24.35(73)20te-80Kr 2235.99489(8) 117.88(51) -24.35(fixed)b22Ne_84Kr 2057.59433(51) 98.221(58) -17.4(18)22Ne_86-Kr 2047.95108(2) 97.1598(20) -18.675(63)22Ne _82Kr 2067.69929(6) 99.2636(37) -17.4(fixed)c0^84a fixed at the value interpolated from 2-Ne- Kr and28Ne- 82Kr.84b fixed at the value of 2-0Ne- Kr.C fixed at the value of 22Ne-84kr.135Table 5.4 Observed frequencies of Ne-Xemolecule J'- J"^V obs^Av(MHz) (kHz)^20Ne- 128Xe 2 - 1^7326.1225 -0.1^3 - 2^10984.7648^0.14 - 3^14638.0801^0.0^20Ne_129xe 2 _ 1^7318.6471^0.03 - 2^10973.5605^0.04 - 3^14623.1584^0.05 - 4^18265.6300^0.0^20Ne- 130Xe 2 - 1^7311.3041^0.33 - 2^10962.5552 -0.34 - 3^14608.5033^0.1^20Ne_131xe 2 _ 1^7304.0521^0.13 - 2^10951.6875 -0.24 - 3^14594.0307^0.15 - 4^18229.2759^0.0^20Ne_132xe 2 _ 1^7296.9286^0.13 - 2^10941.0109 -0.24 - 3^14579.8121^0.15 - 4^18211.5310^0.020Ne_134xe 2 _ 1 7282.9757 -0.13 - 2^10920.1015^0.1136Table 5.4 (continued)molecule J'- J"^yobs^Av(MHz) (kHz)^20Ne- 134Xe 4 - 3^14551.9672 -0.1^5 - 4^18176.7790^0.0^20Ne_136xe 2 - 1^7269.4304 -0.13 - 2^10899.8006^0.24 - 3^14524.9324 -0.15 - 4^18143.0415^0.0^22Ne_129xe 2 _ 1^6757.9258 -0.13 - 2^10133.1820^0.24 - 3^13503.9703 -0.15 - 4^16868.7708^0.0^22Ne_131xe 2 _ 1^6743.2790 -0.23 - 2^10111.2289^0.24 - 3^13474.7320 -0.1^221419-132Xe 2 - 1^6736.1296^0.23 - 2^10100.5123 -0.24 - 3^13460.4597^0.15 - 4^16814.4617^0.0^22Ne_134xe 2 _ 1^6722.1277 -0.23 - 2^10079.5267^0.24 - 3^13432.5084 -0.1137Table 5.5 Observed frequencies of Ne-131Xe20Ne- 131Xe^22Ne- 131Xej” _ jv^v dn^Av v obs^AvF" - F'^(MHz) (kHz)^(MHz) (kHz)2- 10.5 - 1.5 7303.8778 0.1 6743.1050 0.41.5 - 1.5 7303.9747 0.1 6743.2018 0.33.5^- 2.5 7304.0435 -0.3 6743.2701 -0.62.5 - 1.5 7304.0435 -0.3 6743.2701 -0.62.5 - 2.5 7304.1408 0.0 6743.3678 0.21.5 - 0.5 7304.1492 0.1 6743.3763 0.43- 22.5^-^1.5 10951.7070 0.1 10111.2488 0.61.5 - 0.5 10951.7070 0.1 10111.2488 0.63.5 - 2.5 10951.6828 -0.1 10111.2237 -0.64.5 - 3.5 10951.6828 -0.1 10111.2237 -0.64- 33.5^- 2.5 14594.0389 -0.1 13474.7400 -0.32.5^-^1.5 14594.0389 -0.1 13474.7400 -0.34.5^- 3.5 14594.0278 0.1 13474.7294 0.35.5 - 4.5 14594.0278 0.1 13474.7294 0.35- 44.5^- 3.5 18229.2814 0.93.5^- 2.5 18229.2814 0.95.5^- 4.5 18229.2729 -0.96.5 - 5.5 18229.2729 -0.9138Table 5.6^Spectroscopic constants of Ne-XeParameter B0 (MHz) D^(kHz)J H^(Hz)J20Ne- 128Xe 1832.11859(3) 73.4167(12) -11.79(fixed)a20Ne- 129Xe 1830.24854(3) 73.27059(22) -11.5850(45)20Ne- 130Xe 1828.41138(12) 73.1011(45) -11.79(fixed)a20Ne- 131Xe 1826.59707(9) 72.9300(61) -12.09(12)20Ne- 132Xe 1824.81506(10) 72.7913(68) -11.79(14)20Ne- 134Xe 1821.32426(7) 72.4594(52) -12.31(10)20Ne- 136Xe 1817.93574(8) 72.1886(56) -11.66(11)22Ne_129xe 1689.97473(8) 61.5953(57) -9.41(10)22Ne- 131 Xe 1686.31083(73) 61.3215(28) -8.97(fixed)b22Ne- 132Xe 1684.52228(12) 61.1811(84) -8.97(17)22Ne_134xe 1681.01973(8) 60.9104(32) -8.97(fixed)ba fixed at the value of 20Ne- 132Xe.b fixed at the value of 22Ne- 132Xe.139Table 5.7Observed frequencies of Ar-Xemolecule J'- J"^V dn^Av(MHz) (kHz)^A r _ 1 28 x e 3 - 2^5808.5116 -0.2^4 - 3^7743.9259 -0.15 - 4^9678.6913^0.56 - 5^11612.6445 -0.2^Ar-129Xe 3 - 2^5797.8927^0.34 - 3^7729.7703^0.05 - 4^9661.0015 -0.36 - 5^11591.4235^0.07 - 6^13520.8730 -0.28 - 7^15449.1858^0.49 - 8^17376.1968 -0.1^Ar-130Xe 3 - 2^5787.4639^0.14 - 3^7715.8674 -0.15 - 4^9643.6265 -0.16 - 5^11570.5796^0.27 - 6^13496.5644^0.0^Ar-131Xe 3 - 2^5777.1636 -0.14 - 3^7702.1367^0.15 - 4^9626.4681^0.16 - 5^11549.9965 -0.27 - 6^13472.5600^0.0Table 5.7^(continued)molecule J'- J"^V obs^Av(MHz) (kHz)^Ar- 132Xe 3 - 2^5767.0446^0.04 - 3^7688.6476^0.15 - 4^9609.6112 -0.16 - 5^11529.7746^0.07 - 6^13448.9764 -0.38 - 7^15367.0544^0.39 - 8^17283.8456 -0.1^Ar- 134Xe 3 - 2^5747.2292 -0.34 - 3^7662.2322^0.45 - 4^9576.6003 -0.26 - 5^11490.1742^0.07 - 6^13402.7943^0.0^Ar- 136Xe 3 - 2^5727.9905 -0.24 - 3^7636.5861^0.35 - 4^9544.5513 -0.26 - 5^11451.7274^0.17 - 6^13357.9551^0.0140141Table 5.8Observed frequencies of Ar-131Xejfl^_F"jf-^F'V obs(MHz)Av(kHz)3 - 23.5 - 3.5 5777.3362 0.52.5 - 1.5 5777.1990 -0.81.5 - 0.5 5777.1990 -0.84.5 - 3.5 5777.1556 0.63.5 - 2.5 5777.1556 0.62.5 - 2.5 5777.0699 -0.81.5 - 1.5 5777.0199 0.84 - 33.5 - 2.5 7702.1522 0.02.5 - 1.5 7702.1522 0.04.5 - 3.5 7702.1312 -0.05.5 - 4.5 7702.1312 -0.05 - 44.5 - 3.5 9626.4767 0.03.5 - 2.5 9626.4767 0.05.5 - 4.5 9626.4643 0.06.5 - 5.5 9626.4643 0.0142Table 5.8 (continued)ju _FHjil-^F'Vdn(MHz)Av(kHz)6 - 55.5 - 4.5 11550.0016 -0.34.5 - 3.5 11550.0016 -0.36.5 - 5.5 11549.9940 0.37.5 - 6.5 11549.9940 0.37 - 66.5 - 5.5 13472.5648 1.05.5 - 4.5 13472.5648 1.07.5 - 6.5 13472.5568 -1.08.5 - 7.5 13472.5568 -1.08 - 77.5 - 6.5 15394.0011 2.46.5 - 5.5 15394.0011 2.48.5 - 7.5 15393.9919 -2.49.5 - 8.5 15393.9919 -2.4143Table 5.9 Spectroscopic constants of Ar-XeParameter Bo (MHz) Dj^(kHz) H^(Hz)JAr_izaxe 968.20670(6) 6.7444(11) -0.1922(fixed)Ar-129Xe 966.43643(5) 6.7182(10) -0.1922(66)Ar-130Xe 964.69798(5) 6.7018(18) -0.122(18)Ar-131Xe 962.98073(1) 6.6723(4) -0.1558(30)Ar-132Xe 961.29377(4) 6.64655(78) -0.1714(51)Ar-134Xe 957.99043(10) 6.6037(36) -0.113(37)Ar-136Xe 954.78304(8) 6.5507(26) -0.196(27)144Table 5.10Observed frequencies of Kr-Xemolecule J'- JH AvOBS(MHz)^(kHz)^84Kr- 129Xe 4 - 3^4436.5638 -0.1^5 - 4^5545.3947^0.76 - 5^6654.0189 -0.97 - 6^7762.3951^0.3^AL- -Xe 4 - 3^4397.0832 -0.15 - 4^5496.0492 -0.16 - 5^6594.8119^0.47 - 6^7693.3307 -0.284Kr-181 Xe 4 - 3^4410.0332^0.65 - 4^5512.2346 -0.586Kr-129-Xe 4 - 3^4374.4932^0.55 - 4^5467.8167 -0.86 - 5^6560.9403^0.3132AL" --Xe 4 - 3^4335.0076 -0.15 - 4^5418.4642^0.16 - 5^6501.7240 -0.0Table 5.11Observed frequencies of 84Kr-131Xe_F" -^F'V dn(MHz)Av(kHz)4 - 33.5 - 2.5 4410.0482 -0.22.5 - 1.5 4410.0482 -0.24.5 -^3.5 4410.0280 0.25.5 - 4.5 4410.0280 0.25 - 44.5 - 3.5 5512.2433 0.33.5 - 2.5 5512.2433 0.35.5 - 4.5 5512.2306 -0.36.5 - 5.5 5512.2306 -0.3145Table 5.12Spectroscopic constants of Kr-XeParameter Bo^(MHz) Dj^(kHz)84Kr- 129Xe 554.62559(12) 1.7224(16)84Kr- 132Xe 549.68960(5) 1.6936(6)84Kr- 131Xe 551.30842(6) 1.6936a86Kr_129xe 546.86496(21) 1.6658(36)86Kr- 132Xe 541.92845(3) 1.6406(5)a Fixed at the value of 84Kr- 132Xe.146147Table 5.13Nuclear Quadrupole couplingconstants of mKr and 131Xe inRG-83Kr and RG-131Xe dimers.complex^X20Ne-83Kr20Ne- 131 Xe22Ne- 131 XeAr-131Xe84Kr- 131Xe-0.5205(23) MHz0.3877(9) MHz0.3875(9) MHz0.7228(36) MHz0.7079(86) MHzTable 5.14Estimated electric dipole moments inDebye of the mixed rare gas dimers.molecule^this work^Bar-ZivaNe-Kr^0•01113Ne-Xe 0. 0 12 b^0.00696Ar-Xe^0.014b^0.0124Kr-Xe 0.007" -a The values of go in Table I of Ref.38.b The number of digits given does notreflect the uncertainty. See textfor evaluation procedure.Table 5.15. Coordinates and bond lengths (A) of rare gas dimers.complexRG-RG1subst. coordinates r +c.m.a r0 reb reZN ZN,20Ne_84Kr 2.9865 -0.7117 3.6982 3 "" 3.7589 3.63753.639c22Ne_84 Kr 2.9301 -0.7677 3.6978 - 3.7544 3.641220Ne_132xe 3.4191 -0.5184 3.9375 3.9385 3.9940 3.88103.883c22Ne_132xe 3.3820 -0.5553 3.9373 - 3.9896 3.885040Ar_132xe - -0.9573 - 4.1171 4.1402 4.09404.094c40Ar_136xe - -0.9355 - 4.1170 4.1400 4.094084Kr-132Xe 2.5783 -1.6401 4.2184 4.2184 4.2340 4.20284.203c86Kr-132Xe 2.5546 -1.6637 4.2182 4.2183 4.2338 4.2026a The substitution procedure was used for the Xe or Kr coordinate. The remainingatom was located with the first moment condition.ID The equilibrium distance re=2rs-ro.C Estimated uncertainties are ±0.01 A for Ne-Kr and ±0.008 A for Ne-Xe, with thosefor Ar-Xe and Kr-Xe estimated as ±0.005 A.Table 5.16 Direct comparison of the observed rotational frequencies with those predicted fromHFD-B potentials for Ne-Kr and for Ne-Xe respectively. Not all figures displayed for re and eare significant. Some are displayed only to avoid roundoff errors. "%" here denotes(obs-calc)/obs %.observedMHzNe-Kr^original re and E^adjust re^adjust re and ere=3.631, re=3.65058 re=3.6480165 H^e=50.28 cm-1^e=50.28 cm-1 e=48.20 cm-1^.p.obs-calc.^% obs-calc.^%^obs-calc.^%1-0 4430.2261 -46.3869^-1.047 -0.0171^-3.9E-4 -0.0012^-0.3E-42-1 8857.7571 -92.8501^-1.048 -0.1729^-19.5E-4 -0.0017^-0.2E-43-2 13279.6785 -139.4696^-1.050 -0.6072^-45.7E-4 -0.0026^-0.2E-44-3 17693.2023 -186.3213^-1.053 -1.4614^-82.6E-4 0.0006^0.0E-4B/MHz 2215.36414(12) 2238.5510 2215.36113 2215.3646D/kHz 115.453(13) 112.229 109.682 115.434H/Hz -24.68(42) -24.455 -23.408 -26.111Table 5.16 (continued)Ne-Xeobserved^original re and e^adjust re^adjust re and eMHzre=3.8610,e=51.57 cm-1obs-calc.^%re=3.889818e=51.57 cm-1obs-calc.^%re=3.88691e=49.00 cm-1obs-calc.^%I-.2-1 7296.9286 -106.3415^-1.457 -0.1373^-18.8E-4 -0.0105^-1.4E-4 ulo3-2 10941.0109 -159.6380^-1.459 -0.4641^-42.4E-4 -0.0165^-1.5E-44-3 14579.8121 -213.0846^-1.462 -1.1010^-75.5E-4 -0.0271^-1.9E-45-4 18211.5310 -266.7318^-1.465 -2.1541 -118.3E-4 0.0451^2.5E-4B/MHz 1824.81506(10) 1851.38376 1824.81506 1824.8125D/kHz 72.7913(68) 70.7007 68.5024 72.7928H/Hz -11.79(14) -11.81 -11.28 -12.79Table 5.17 Direct comparison of the observed rotational frequencies with those predicted fromHFD-C potentials for Ar-Xe and for Kr-Xe respectively. Not all figures displayed for re and Eare significant. Some are displayed only to avoid roundoff errors. H%" here denotes(obs-calc)/obs %.observedMHzAr-Xeoriginal re and Ere=4.0668,E=131.10 cm-1obs-calc.^%adjust rere=4.093187E=131.10 cm-1obs-calc.^%adjust re and Ere=4.0922456=126.19 cm-1obs-calc.^% I-I-flI-.3-2 5767.0446 -74.2055^-1.287 -0.0300^-5.2E-4 -0.0006^-0.1E-44-3 7688.6476 -98.9514^-1.287 -0.0701^-9.1E-4 -0.0004^-0.1E-45-4 9609.6112 -123.7065^-1.287 -0.1358^-14.1E-4 -0.0004^-0.0E-46-5 11529.7746 -148.4730^-1.288 -0.2335^-20.3E-4 0.0001^0.1E-47-6 13448.9764 -173.2529^-1.288 -0.3697^-27.5E-4 0.0010^0.1E-48-7 15367.0544 -198.0485^-1.289 -0.5506^-35.8E-4 0.0027^0.2E-49-8 17283.8456 -222.8618^-1.289 -0.7824^-45.3E-4 0.0058^0.3E-4B/MHz 916.29377(4) 973.65962 961.29395 961.29384D/kHz 6.64655(78) 6.54993 6.37820 6.64640H/Hz -0.1714(51) -0.1861 -0.1806 -0.1924Table 5.17 (continued)observedMHzKr-Xeoriginal re and ere=4.174e=162.275 cm-1obs-calc.^%adjust rere=4.20160e=162.275 cm-1obs-calc.^%adjust re and Ere=4.200903e=155.062 cm-1obs-calc.^%4-3 4397.0832 -106.3415^-1.457 -0.1373^-18.8E-4 -0.0105 -1.4E-45-4 5496.0492 -159.6380^-1.459 -0.4641^-42.4E-4 -0.0165 -1.5E-46-5 6594.8119 -213.0846^-1.462 -1.1010^-75.5E-4 -0.0271 -1.9E-47-6 7693.3307 -266.7318^-1.465 -2.1541 -118.3E-4 0.0451 2.5E-4B/MHz 549.68960(5) 556.94261 549.68576 549.68953D/kHz 1.6939(6) 1.6572 1.6133 1.6941153Table 5.18 Equlibrium distances (A) of rare gas dimers.complex^rea reb reb re^(Ref.)isotopic adjusted adjusted literaturedata re only re and 6 valueNe-Kr 3.639 3.6506 3.6480 3.631 (32)Ne-Xe 3.883 3.8898 3.8869 3.861 (9)Ar-Xe 4.094 4.0932 4.0922 4.067 (11)Kr-Xe 4.203 4.2016 4.2009 4.174 (11)a The values are those in Table 5.15.b Obtained from the potentials by adjusting only re.See text for the procedure.C Obtained from the potentials by adjusting both re and E.See text.154CHAPTER 6.Microwave Spectroscopic Investigation of the Mixed Rarevan der Waals Trimers Ne2-Kr and Ne2-Xe6.1 IntroductionMany-body nonpairwise additive effects haveattracted much attention for several decades [1]. Anunderstanding of these effects is essential for adescription of bulk phase properties in terms ofmicroscopic scale properties. The most common correctionterm to the pairwise additive calculation of three-bodysystems is the triple-dipole term given by Axilrod andTeller (AT term) [2], which was derived directly from thestudy of liquid and solid phase matter. Although thiscorrection term has proven to work fairly well inpractice, recent theoretical investigation has shown thatits success implies fortuitous cancellation involving anegative contribution from short-range nonpairwiseadditive repulsive forces and a positive contributionfrom higher order dispersion corrections [3]. Meath andAziz have shown that inclusion of first order nonpairwiseadditive three-body exchange energies and the AT term atthe same time would result in a significant cancellation,and thus be in disagreement with experiments [4]. On theother hand, the accuracy of "bulk" experiments is usually155not high enough to separate the AT term and higher ordercorrection terms. Furthermore, sophisticated computersimulations are usually needed in order to extractinformation from those experiments. In some complicatedsystems, even pairwise additive calculations aredifficult to perform [3].Recent success in high resolution microwave (MW),infrared (IR) and far infrared (FIR) spectroscopicinvestigations of van der Waals trimers Ar-HX(X=F,C1) [5-10] has shown great promise in providingdirect information on the nonpairwise additivity ofintermolecular forces. Several papers concerned with thetheoretical description of such systems have beenpublished [11,12]. Furthermore, recent FIRinvestigation [10] has indicated that a fully dynamicaltheoretical treatment of such systems is necessary inorder to obtain quantitative information about three-bodynonpairwise additive contributions. Additional challengewas faced in theoretically modelling such complicateddynamic problems. Meanwhile, hope has also been given tomeasure spectra of simpler rare gas (RG) van der Waalstrimers, without the aid of polar monomers to provide alarge transition moment [13].Rare gas trimers are among the simplest typespossible for the investigation of three-body nonpairwiseadditive effects, since the respective pair potentialsare isotropic, exhibiting no orientational dependence on156the individual constituents (atoms). Furthermore, therare gas dimers have been extensively studied, and theirpair interaction potentials are known with very highprecision [14]. This makes them excellent prototypes forthe investigation of nonpairwise additive effects.Two potential difficulties are expected in thespectroscopic (MW, FIR) investigation of rare gastrimers: production of enough trimers in the molecularbeam expansion, and a small magnitude of the induceddipole moments for providing observable intensity for theexperimental technique being used. A recent theoreticalcalculation on the Ar3 trimer has evaluated thetransition dipole moment for the lowest infrared-allowedtransitions to be -7x10-5 D, making it unlikely to beobservable using current FIR techniques [13].The search for the mixed rare gas van der Waalstrimers in this work was inspired by the success of theMW investigations of the rare gas dimers [Chapter 5] andof two trimers Ar2-0CS and Ar2-0O2 [Chapter 7]. Theinduced dipole moments in the rare gas dimers wereestimated to be about 0.01 D and were found to besufficiently large for their rotational spectra to beobserved. Furthermore, a fairly high intensity has beenobserved for the rotational spectra of the van der Waalstrimer Ar2-0O2 [Chapter 7], despite its low dipolemoment; this observation suggested that there is anabundance of trimers in the molecular beam expansion.157In this chapter, the pure rotational spectra ofvarious isotopomers of the rare gas trimers Ne2-Kr andNe2-Xe are described. Rotational constants andcentrifugal distortion constants have been determined andhave been used to obtain structural parameters. Thenarrow nuclear hyperfine splitting of rotationaltransitions of °Kr- and 131Xe-containing isotopomers havebeen resolved and have yielded accurate nuclearquadrupole coupling constants. Estimates of the inducedelectric dipole moments in these two trimers have beenobtained from values of excitation pulse lengths,optimized for maximum signal strength ("v/2 condition")[15, see also Chapter 2]. Observation of the spectra ofthese fundamental trimers makes it possible to comparetheir properties to those of their constituent rare gasdimers.6.2 Search and assignments In the present work gas mixtures of 1% Kr or Xe inNe were used for Ne2-Kr and Ne 2-Xe, respectively, withbacking pressures of up to 5 atm.A. Ne2-KrThe search for the spectrum of Ne 2-Kr was carriedout assuming that the structure of the trimer isdominated by the influence of pairwise additive158interatomic forces. Using the bond lengths of theNe-Ne [16] and Ne-Kr [Chapter 5] dimers, the complex waspredicted to be a highly asymmetric prolate top(K--0.748) with C2v symmetry, with Kr lying on thea-principal axis of inertia. Because of the largeuncertainty in the bond lengths, especially for the Ne2dimer, the uncertainty in the estimated rotationalconstants was very large.The predicted frequency for the 30,3-20,2 transitionwas near 8.5 GHz. A systematic search was carried out inthis region, varying the excitation frequency in 0.2 MHzsteps, using 10 averaging cycles at each step. The firstline found was at 8379.928 MHz. It clearly arose from aspecies with a very low dipole moment because it requiredvery long excitation pulses to optimize the signalstrength (i.e. to reach the "71/2 condition") [15, seealso Chapter 2]. It was later identified as being due toNe2-84Kr by finding the corresponding lines of Ne2-86Kr andNe2-82Kr. Because the trimer is highly asymmetric, it wasvery difficult to predict further transitions. Inpractice, searches were carried out for some of thelowest J transitions of each isotopomer, and theassignments were supported utilizing the lengths of theexcitation pulse, isotopic data, the relative intensitiesobserved, and the analysis described below. The measuredfrequencies and assignments are in Table 6.1.The assignment difficulties were compounded by the159effects of nuclear spin statistics. A complex with Cnsymmetry would have two equivalent nNe atoms, with I=0,and Bose-Einstein statistics would apply. The total wavefunction would thus be symmetric with respect to exchangeof these two nuclei (i.e. to the C2 operation). Theresult would be that only Ka=even levels are allowed, andtransitions involving levels with Ka=1 are missing. Onthe other hand, failure to observe these transitions wasstrong evidence that the observed lines were due to thedesired complex Ne2-Kr.This evidence was strengthened further by thediscovery of lines due to the mixed isotopomers20Ne22Ne-84 Kr and 20Ne22Ne-88 Kr. In this case there are nolonger two equivalent nuclei, and it has been possible tomeasure both Ka=0 and Ka=1 a-type R branch transitions.Their frequencies and assignments are in Table 6.2.Final confirmation of the assignments was made withtransitions due to 20Ne2-83Kr. In this case nuclearquadrupole hyperfine structure due to °Kr was observed.Because of its high nuclear spin (1=9/2), and because ofthe relatively small field gradient at the °Kr nucleusin the complex, the hyperfine structure is verycongested. Computer simulation of the hyperfine splittingpatterns with different assumed coupling constants werenecessary to match the experimental spectrum and toassign quantum numbers. An example, showing thetransition 30,5-20,2, is given in Figure 6.1. The spectrum1609/2 - 7/2 11/2- 11/211/2 - 9/215/2-13/2F" - F'=13/2 - 13/218406.15 MHz 8406.45 MHz IFig.6.1 Hyperfine splitting due to ftr in therotational transition 30,3-20,2 of 20Ne2-831CrThe spectrum is composed from three experiments,with 200 up to 1000 experimental cycles.161is composed from three experiments. The frequencies andassignments of the individual hyperfine components of thetransitions of 20N 2_e °Kr are in Table 6.3.B. Ne2-XeThe search for the spectrum of Ne2-Xe followedessentially the same procedure as for Ne2-Kr. However,because Ne2-Xe is heavier, transitions with the same Jvalues as in Ne2-Kr occur at lower frequencies (7 GHz).Unfortunately the sensitivity of the spectrometer is muchlower here than in the higher frequency range(-8-18 GHz). The search was therefore carried out for thetransition 40,4-30,3 at 8.8 GHz. Furthermore, Xe hasseveral major isotopes with similar abundances. Theobserved signal-to-noise ratios were much lower for thiscomplex. As a result, only a few transitions could bemeasured for 20Ne2-Xe, and lines due to the mixedisotopomers nNenNe -Xe were too weak to be observed.The measured frequencies for the isotopomers20Ne 2 _129xe 20Ne 2 _132xe, 20Ne 2_134Xe and 20Ne2-136Xe are listedin Table 6.4. Once again only lines having Ka=0 could beobserved, consistent with On symmetry for the complex.132_1Three transitions were measured for 20Ne -Xe; allshowed nuclear quadrupole hyperfine structure due to 131Xe(1=3/2). Their frequencies and assignments are inTable 6.5 with the corresponding hypothetical unsplitline frequencies in Table 6.4. No hyperfine splitting due162ri, 7/2-5/211 5/2-3/23/2-3/2IIIV I5/2-5/2ril 3/2-1/21 I11\1IL„..,„4,i \#,A1 koAILA1^14450.7MHzFig.6.2 Nuclear quadrupole hyperfine components oftransition 2 0,2 -10,1 of 20Ne2- 131XeThe spectrum is composed from three experiments,with 200 experimental cycles each.I^I^IevkAAWAA11 4451.3 MHz163to 129Xe (I=1/2) was observed for Ne2-129Xe. An exampletransition of 20Ne 2_131Xe showing hyperfine splitting isdepicted in Fig. Spectral analyses For Ne2-Kr, all the measured rotational transitionfrequencies were fit to Watson's semirigid rotorHamiltonian, using the A-reduction in its Irrepresentation [17]. Because the observed lines arecomprised of only a limited number of a-type R branchtransitions, there is not enough information to determineexperimentally all the rotational and quartic centrifugaldistortion constants.In the cases of the mixed isotopomers 20Ne22Ne-24Kr2oNe22Ne_86and^—Kr, three quartic centrifugal distortionconstants, namely Aj, Ajic, and 6j could be determined,with AK and 61( fixed at the values predicted by the forcefield analysis, as discussed in Section 6.5.2. In thecases of 20Ne2-Kr, all quartic distortion constants werenecessarily constrained at the values predicted by theharmonic force field analysis. Nevertheless, this hasproduced reasonable fits for the 20Ne2-Kr isotopomers.More significantly, this has allowed values of Ao to bedetermined for 20Ne2-84Kr and 20Ne2-86Kr, with 5 and 4measured lines, respectively. The asymmetry of thesecomplexes is evidently high enough that this could be164done, even though only rotational transitions having Ka=0were measured. There is even a small, but real,measurable isotopic variation in Ao. Furthermore, Bo andCo both show a roughly linear variation with Kr mass2_number in these two isotopomers and in 20Ne 83Kr and20Ne2-82Kr. Consequently, an isotopic scale factor was usedto determine the fixed values of Ao in the fits for2ome _2 83Kr and 20Ne 2_82Kr.The derived spectroscopic constants for all theisotopomers of Ne2-Kr are in Table 6.6. The residuals,designated Av obs-calc in Table 6.1, are all less than 6 kHzfor the fitted lines. These residuals could probably bereduced if there were enough data to include sexticdistortion constants in the analysis; this was found forAr2-HF [5] and Ar2-HC1 [6], where inclusion of theseparameters reduced the standard deviations of the fitsfrom 10-20 kHz down to -2 kHz.For Ne2-Xe, the rotational analysis followed thesame procedure as for Ne2-Kr. However, there were notenough experimental data to evaluate both the rotationaland the quartic centrifugal distortion constants. Allquartic centrifugal distortion constants were fixed at0.0 in the fitting procedures. Furthermore, in theanalyses, the rotational constant Ao were necessarilyfixed at the Ao value of 28Ne2-84Kr for all the isotopomersbecause of high correlations (>0.999) between the Ao, Bo,and C0 rotational constants; the lines were fit to the165rotational constants Bo and Co only for all theisotopomers. The resulting spectroscopic constants are inTable 6.7.Because the nuclear quadrupole hyperfine splittingsdue to 83Kr in the 28Ne2-83Kr complex were closely spaced,the displayed line shapes were often distorted. Accuratefrequencies were obtained using the DECAYFIT program [18]which directly analyses the time-domain signals. The 83Krnuclear quadrupole coupling constants and hypotheticalunsplit line frequencies were fit to the observedhyperfine components. The standard deviation of the fitis reasonable (-1.4 kHz), considering the congestedpatterns observed. The coupling constants are inTable 6.8, and the unsplit line frequencies with theirassignments are in Table 6.1.For 20Ne2_131xe, the analysis followed the sameprocedure as for 20Ne2_83Kr, and the resulting 131Xe nuclearquadrupole coupling constants are in Table Estimation of the dipole moments of Ne2-Kr and Ne2-XeAgain, the "v/2 condition" [15, see also Chapter 2]method was used for the estimation of the induced dipolemoments of the trimers, as already described in Chapter 5for the mixed rare gas dimers. Again, the spectrometerwas calibrated using Ar-0O2 and the method was checkedfor consistency against Ne-0O2. For Ne2-Kr, the pulse166length optimization for the transition 30,3-20,2 of2om_ 2 _e 84Kr at 8379.928 MHz produced a value of 0.015 Debye.For Ne2-Xe, the dipole moment has been estimated usingthe transition 3m-20,2 of 20Ne2-132Xe at 6626.259 MHz and avalue of 0.011 Debye has been obtained. These two valuesare listed in Table 6.9, along with the correspondingdimer values of Ne-Kr and Ne-Xe. However, the uncertaintyof this method is large, as has been discussed already inSection 5.4. The measured dipole moments are believed tobe of the right order of magnitude.6.5 Possible evidence of three-body effects A primary interest of this work was to detect anydeviations from pairwise additive behaviour. Possibleinformation about three-body nonpairwise additivecontributions will be discussed in the following sectionsin terms of three different types of measurableproperties: the rotational constants i.e. the structures,the induced dipole moments, and the nuclear quadrupolecoupling constants of these two trimers.6.5.1 Structures of Ne2-Kr and Ne2-XeThe observed spectra of several isotopomers ofNe2-Kr and Ne2-Xe indicate that the assumed geometriesare essentially correct. The strongest evidence comesfrom the effect of spin statistics in 20Ne2-Kr and 20Ne2-Xe167described earlier: these complexes have Cn symmetry,with the symmetry axis being the a-inertial axis.However, because the trimers observed here areextremely flexible, as indicated by the relatively largequartic distortion constants as well as by their largeinertial defects, it is expected that the usualstructural concepts do not apply in a straightforwardfashion.There have been some different approaches toevaluate the structural information for this type offloppy complex in the literature. The microwave data forthe Ar2-HX (X=F [5], Cl [6]) trimers have beeninterpreted in terms of semirigid models, an approachwhich has often been taken in microwave spectroscopicinvestigations. However, Hutson and co-workers haverecently taken a slightly different approach for thefloppy systems in their theoretical treatments ofAr2-HC1 [11,12] and Ar 3 [13]. The reason for such anapproach is that the separation of rotation and vibrationmay not be entirely complete because of the very lowvibrational modes with large vibrational amplitudes inthese complexes.By analogy with the theoretical treatment ofAr2-HC1 [11], the instantaneous elements of the inertialtensors of Ne2-RG (RG=Kr or Xe) can be written as [11]:168Iaa — INe2 S1112X7.. v 2^2'bb — I-4 "c.m. -r- J-Ne2 COS Xn 2'cc — r pp,-c.m. + INe2Iab — INe2 COS X SiriXIbc — lac — 0(6.1)where INe2 (=2MNe rNe-Ne2 ) is the moment of inertia of thediatomic subunit Ne2; Rcm. is the line connecting thecentres of mass of Ne2 and the third RG atom; x is theinstantaneous angle between Rcm. and the Ne-Ne bond; A isthe pseudodiatomic reduced mass:MNe2MRGp, = m "Ne2 + MRG(6.2)with RG=Kr or Xe. In the semirigid model treatment ofAr2-HX (X=F [5], Cl [6]), the observed rotationalconstants A, B, and C were related to the expectationvalues of the inverse of the inertial tensor elementsIaal Ibb, and I directly. However, in Hutson's treatment,ccthe A, B, and C rotational constants are related to theexpectation values of the diagonal elements of theinverse ternsor of I (i.e. I-1) [11]:2 152A .. <^ + ^ >2 INe2 Sirl2X^2 p, R ^tan2Xti2B ,. <^ >2 A Rc2.m.1.12C P. <^ >2 il Rc2.,. + 2 Ille2(6.3)169where I is the inertial tensor with its elements definedas in Eq.(6.1). The angular brackets indicate averageover the ground state vibrational wavefunction. Evidentlythe equations for the A and B rotational constants arenot the same as would be obtained by simply inverting Iaaand Ith• This model is used here since it approximatesthe large amplitude motions more properly than thesemirigid model.The above equations can be rewritten to calculatethe structures from the experimental rotationalconstants:2^505379p, R.c.m. =^B. 2^505379Fhl-‘D 4. Tc.m. . -'-Ne2^C (6.4)sin2x ^B2 (B-C) (A+B)It should be noted that the bracket expressions are nolonger used in Eq.(6.4); and the structural parameterscalculated from Eq.(6.4) have slightly different physicalmeanings when different types of rotational constants(e.g. effective, ground state average) are used. When theeffective rotational constants (in MHz) are used, Rcam.and rNe-Ne calculated from Eq.(6.4) denote the effectivedistances. In order to understand the physical meaning ofx calculated from Eq.(6.4) and to compare the result here170with other trimer studies, an angle B defined as (90°-x)is introduced, which is referred to as the averagevibrational bending amplitude. When the averagerotational constants are used, the Rc.m. and rNe-Ne denotethe average bond distances; and x denotes approximatelythe average structural angle between Ne-Ne bond and Rc.m..For Ne2-Kr, there are enough data to carry out areasonably complete structural determination. With Cnsymmetry it is possible to obtain complete geometriesindependently for each isotopomer utilizing Eq.(6.4). Theeffective structures calculated by using the effectiverotational constants in Table 6.6 are listed inTable 6.10 for the two most abundant isotopomers 20Ne2-84Krand 20Ne2-86Kr. The structural parameters calculated for3s2oNe_the other two isotopomers 2^-Kr and 20Ne2-82Kr showedless than 0.001 A differences in the bond lengths ofNe-Ne and Ne-Kr, and less than 0.10 difference in theangles, compared to 20Ne2-84Kr. The angle B here indicatesthe average vibrational amplitude of the Ne-Ne bend, asdefined in the previous paragraph.Another approach has been to evaluate a ground stateaverage (rd structure. A harmonic force field isrequired for this purpose. An approximation was obtainedusing the measured quartic centrifugal distortionconstants of 20Ne22Ne-Kr. The detailed analysis is insection 6.5.2. To evaluate the rz parameters the harmoniccontributions to the a-constants were first subtracted171from the ground state rotational constants for eachisotopomer. The resulting values are listed as Az, Bz andCz in Table 6.6, as are the corresponding inertialdefects Az. The rz parameters were evaluated separatelyfor each symmetric isotopic species. The resultingstructural parameters for the two most abundantisotopomers 2oN- 2_e 84Kr and 28Ne2-86Kr are listed inTable 6.10. It should be emphasized that the angle xcalculated here is defined as approximately the averageangle between Roal. and Ne-Ne bond, different from theones calculated from effective rotational constants.Since rotational constants for the mixed isotopomers2°Nen-Ne-Kr have also been obtained it has been possibleto obtain a full substitution (re) structure [19].Kraitchman's equations for a planar molecule [19] havebeen applied in two separate calculations using Bo and Corotational constants taking 2oN-e 2_84Kr and 20Ne2-86Kr asparent molecules. The a, b and c coordinates of asubstituted atom can be evaluated by:lal=(1+ ^Aib 114lcAIibl- [^- Alb (1+JAI,^1IL^) 1 22Ib -^jIcl-(6.5)with 111)=1'13 and AIo:=I-Io the differences between themoments of inertia of the different isotopomers. Theresults are also given in Table 6.10.172Similar structural analysis procedures as above havebeen applied for Ne2-Xe, using Eq.(6.4). However, sinceit has not been possible to perform a harmonic forcefield analysis, because of the lack of centrifugaldistortion data, an rz structure could not be evaluatedfor Ne2-Xe. The calculated ro structures for two mostabundant isotopomers Ne2- 129Xe and Ne2-132Xe using theeffective rotational constants in Table 6.7 are listed inTable 6.10.The structural parameters of the correspondingdimers, Ne-Kr, Ne-Xe and Ne-Ne are also summarized inTable 6.10 to allow comparison with the trimers. For theNe-Ne distance, since there is no accurate experimentalvalue available [16], an ro value predicted from the mostrecent Ne-Ne potential [20] is given in Table 6.10. Theprogram LEVEL [21] was used to calculate ro from theoriginal potential [20]. The Ne-Ne potential has anuncertainty in the bond length of better than 1%.Apparently there is a lengthening of the Ne-Ne distanceof -0.07 A in the trimers compared to that of the dimer.Such lengthening has also been observed in the bondlengths of Ar-Ar in trimers, such as Ar2-HX (X=F [5], Cl[6]), Ar2-0CS and Ar2-0O2 [Chapter 7]. This can possiblybe attributed to the existence of an overall three-bodyrepulsive force in the trimers [5,6,11]. There seems tobe also a lengthening in the order of 0.005 A, althoughless obvious, in the Ne-RG (RG=Kr and Xe) distances.173The Ne-Ne vibrational bending amplitude (8) is about11.8° in Ne2-Kr and 13.6° in Ne 2-Xe, although the lattervalue is probably not very accurate since the A0rotational constants could not be determined for theNe2-Xe complexes. These two values can be compared withthe bending amplitude of 4.2° for Ar-Ar in Ar 2-HC1 [6],calculated using Eq.(6.4) and the effective rotationalconstants in Ref.[6]. The larger amplitudes observed forthe Ne2 motion reflect the higher floppiness due tosmaller mass and weaker bonding of Ne.Although the above structural analyses allowqualitative comparison of the effective structures ofdifferent complexes, the large uncertainties in thestructural parameters do not reflect the accuracy of themicrowave measurements. In order to extract three-bodynonpairwise additive contributions, it is highlydesirable to compare these experimental rotationalconstants directly with those predicted by rigorouspairwise additive potentials for Ne 2-Kr and Ne2-Xe,respectively. Such a pairwise additive potential hasrecently been constructed for Ar 3 [13]. The inclusion ofthree-body nonpairwise additive contributions showed thatthere is a significant cancellation between the leadingthree-body dispersion term described by the triple-dipolemodel and the short-range nonpairwise additive term,leaving the final rotational constants almostunchanged [13]. Similar cancellation phenomena have also174been observed in the theoretical investigation ofthree-body nonpairwise additive effects inAr2-HC1 [11,12], where inclusion of these three-bodynonpairwise additive contributions still cannot accountfor the discrepancy between theoretical and experimentalrotational constants. From the rough comparison discussedabove, it seems that there would be detectable deviationfrom pairwise behaviour for Ne2-Kr and Ne2-Xe. Thesetriatomic trimers should be easier to deal withtheoretically, as opposed to those containing molecularsubunits, especially since the theory for nonpairwiseadditive effects in atomic systems is much moreadvanced [4,12].6.5.2 Harmonic force field approximation of Ne2-KrFor the triatomic Ne2-Kr complex, there are threenormal modes of vibration. The basis set was defined withthe following symmetry coordinates: S1=(1/1/2) (Ad1+Ad2),S2=Ar, and S3=(1/V2) (Ad1-Ad2) with Adi and Ar the NeKr andNeNe bond stretches, respectively. In the C2v pointgroup, S1 and S2 have Al symmetry and S3 has B2 symmetry;there are four force constants in this case, designatedf", fn and f 33 ^off-diagonal constant fu.Because only three quartic centrifugal distortionconstants were obtained for 20lie22Ne-134Kr and 20Ne22Ne-86Kr,respectively, there is not enough information to fit allfour force constants. Instead, four trial fits were175carried out, each with one of the force constants heldfixed. The structure used was the ro structure. When adiagonal force constant was fixed at its value for thecorresponding dimer (a reasonable approximation if thepotential were all pairwise additive), the off diagonalforce constant fu was found to be at least an order ofmagnitude smaller than the diagonal elements.Furthermore, fu could not be well determined, eventhough the Jacobian in the least squares fit indicatedthat the measured distortion constants were sensitive toit. When fu was set to zero the diagonal constants werelittle changed from those from the fits when fu wasincluded. Consequently, the final fit assumed f,2=0; theresulting force field, as well as the correspondingvibrational frequencies, is given in Table 6.11. Theestimated values of the strength force constants of thetwo corresponding dimers Ne-Kr and Ne-Ne are also listedin Table 6.11 for comparison purpose.It should be pointed out that the force fielddetermination was part of an iterative process whichincluded the fitting of the observed spectra to thespectroscopic constants. Preliminary distortion constantsAj, AjK and 6j were obtained for 28NONe-84Kr andNeuNe -86Kr from the spectra and were used in the firstforce field calculation. Distortion constants AK and 6Kwere then predicted from this force field and fixed inthe next fit of the spectra, which produced new values176for the other three constants. The procedure was repeateduntil it converged. The final values of the distortionconstants obtained from semirigid rotor fits are comparedwith those predicted from the force field analysis for20Ne22Ne-Kr in Table 6.11. Distortion constants predictedfrom the final force field were used in the fits to therotational constants of the 20Ne2-Kr complexes.Because of the constraint (f12=0) in the harmonicforce field analysis, the very large vibrationalamplitudes involved, and the expected high anharmonicityof the vibrations, the derived force field must be viewedwith caution. Inclusion of the quartic distortionconstants predicted from the force field reduced thestandard deviations in the fits from -10 kHz to 1-3 kHzfor-nNenNe -Kr and from -50 kHz to less than 6 kHz for20Ne2-Kr. More importantly, the force field can accountfor over 97% of the inertial defects.6.5.3 Induced dipole moments of the trimers Much attention has been given to the study ofcollision induced absorption spectra which are due tocolliding rare gas pairs [22,23]. In recent years, suchattention has been extended to rare gas triatomic systemsin connection with the observed features in FIRabsorption spectra [24], in particular to explorepossible three-body effects on their dipolemoments [25,26]. At present the theoretical treatment for177such triatomic systems is still aiming at a qualitativeunderstanding. However, it is established that the signand the magnitude of the three-body contribution to theinduced dipole moment depend on the actual properties ofthe constituents and the geometry of trimers [25,26].Moreover, theoretical treatments near potential minimahave been extremely difficult, since both long and shortrange interactions contribute significantly in thisregion. The estimated induced dipole moments presentedhere are important additional experimental informationfor theoreticians in order to describe such systems.The dipole moments of the trimers can be describedin terms of pairwise and nonpairwise additivecontributions [25,26]. The pairwise contribution issimply the vector sum of the dipole moments of all thepairs in the trimers [25,26]. However, there are possiblenonpairwise additive contributions. For example, when twoNe atoms approach each other, an electric quadrupolemoment is induced. This quadrupole moment can in turninduce a dipole moment in Kr or Xe, which would not existin the corresponding dimers [26]. In a similar fashion,the Ne-Kr or Ne-Xe dipole in these two trimers can inducea dipole moment in the other Ne atom [26].The estimates of the dipole moments of the twotrimers studied here are listed in Table 6.9. The orderof magnitude of the induced dipole moments of the trimersis about the same as that of the corresponding178dimers [Chapter 5], in the order of 0.01 D. Consideringonly the pairwise additive contribution from the vectorsum of the dimer constituents and taking into account thestructures of the trimers, the magnitudes of the dipolemoments for Ne2-Kr and Ne2-Xe are expected to be about1.8 times the values of Ne-Kr and Ne-Xe, respectively.The experimental estimated values in Table 6.9 areslightly smaller than the expected pairwise additivevalues, and the discrepancies might be attributable tothree-body effects. Unfortunately, since theuncertainties involved in the experimental determinationsare large, as discussed before [Chapter 5], no explicitconclusion can be drawn.6.5.4 Nuclear quadrupole hyperfine structures In both Ne2-Kr and Ne 2-Xe trimers, the nuclearquadrupole coupling constants have been determined for°Kr and 1251Xe containing complexes. Such information isvery important as it provides detailed information aboutthe electronic charge distribution at the sites of thequadrupolar rare gas nuclei.However, because only a-type transitions wereobserved, xbb-y-ccX"- Nonetheless, the highly asymmetric structures ofthese two trimers made it possible to obtain also valuesfor 7"bia-1", and thus values for 7-cc by using Laplace'sequation [see Chapter 2]. This provides a uniqueis in general not as well determined as179opportunity to compare them with the magnitudes in thecorresponding dimers in the light of three-bodyinteractions. xcc in the trimers, which is not influencedby the large van der Waals vibrational motions, iscompared with the xi values in the corresponding dimersin Table 6.8. If one assumes pairwise additivity for thisproperty, one would expect xi values of the dimers to bedoubled in the trimers(x) upon approaching of the„second Ne atom, as illustrated in Fig.6.3 for Ne2-131Xe.In both cases, Ne2-83Kr and Ne2-131Xe, the simplecombination sum rule does not apply here, i.e. Xcc*2X,,although the considerable large uncertainty attached, incase of Ne2-Kr, makes it not possible to draw definiteconclusion for this trimer. The directions and thepercentages of the deviations from pairwise additivityare of opposite sign and different magnitudes, i.e. 4.7%(±7.4%) for 83Kr and -20.8%(±4.8%) for 131Xe. The values inbrackets indicate the error limits arising from theuncertainties in xcc. The different signs and sizes ofthe deviationsseem to suggest that three-body effects on the fieldgradient at the site of the quadrupolar nuclei depend onthe actual properties of the constituents and thestructures of the trimers, paralleling the theoreticalobservations of the many-body effects on dipolemoments [25,26].180 N e XCCXeVN eFig.6.3 Nonpairwise additivity of xcc(131Xe) in theNe2-131Xe trimer.2y1 indicates twice the y, value of the Ne-131Xedimer, which is the value expected for y (131Xe) in-ccNe2 -131Xe assuming pairwise additivity. y (131Xe) is theccexperimental determined value for Ne2-131Xe1816.6 Conclusion and future prospectIn this chapter, the first high resolution spectraof the mixed rare gas van der Waals trimers Ne-Kr andNe2-Xe have been described. The observed rotationalconstants, the dipole moments and the quadrupole couplingconstants indicate possible three-body nonpairwiseadditive contributions to these quantities.The magnitude of the measured induced dipole momentsuggests that it might be feasible to observe these mixedtrimers in the FIR region where complementary informationon nonpairwise additivity of interatomic (molecular)forces could be obtained.182Bibliography1. R.J.Saykally and G.A.Blake, Science 259, 1570 (1993).2. B.M.Axilrod and E.Teller, J.Chem.Phys. 11, 299(1943).3. G.C.Maitland, M.Rigby, E.B.Smith, and W.A.Wakeham,"Intermolecular Forces, Their Origin andDetermination", Clarendon Press, Oxford, 1981.4. W.J.Meath and R.A.Aziz, Mol.Phys. 52, 225 (1984).5. H.S.Gutowsky, T.D.Klots, C.Chuang, C.A.Schmuttenmaer,and T.Emilsson, J.Chem.Phys. 86, 569 (1987).6. T.D.Klots, C.Chuang, R.S.Ruoff, T.Emilsson, andH.S.Gutowsky, J.Chem.Phys. 86, 5315 (1987).7. A.Mcllroy and D.J.Nesbitt, J.Chem.Phys. 97, 6044(1992).8. M.J.Elrod, D.W.Steyert, and R.J.Saykally,J.Chem.Phys. 94, 58 (1991).9. M.J.Elrod, D.W.Steyert, and R.J.Saykally,J.Chem.Phys. 95, 3182 (1991).10. M.J.Elrod, J.G.Loeser, and R.J.Saykally, J.Chem.Phys.98, 5352 (1993).11. J.M.Hutson, J.Beswick, and N.Halberstadt,J.Chem.Phys. 90, 1337 (1989).12. A.R.Cooper and J.M.Hutson, J.Chem.Phys. 98, 5337(1993).13. A.R.Cooper, S.Jain, and J.M.Hutson, J.Chem.Phys. 98,2160 (1993).14. G.Scoles, Ann.Rev.Phys.Chem. 31, 81 (1981), and thereferences therefore cited.15. T.G.Schmalz and W.H.Flygare, in "Laser and CoherenceSpectroscopy" (J.I.Steinfeld, Ed.), pp.125-195,Plenum Press, New York and London, 1978.16. Y.Tanaka and K.Yoshino, J.Chem.Phys. 57, 2964 (1972).18317. J.K.G.Watson, in "Vibrational Spectra and Structure:A Series of Advances" (J.R.Durig,Ed.), Vol.6,pp.1-89,Elsevier, New York, 1977.18. J.Haeckel and H.Mader, Z.Naturforsch. 43a, 203(1988).19. J.Kraitchman, Am.J.Phys. 21, 17 (1953).20. R.A.Aziz and M.J.Slaman, Chem.Phys. 130, 187 (1989).21. R.J.Le Roy, private communication.22. E.Bar-Ziv and S.Weiss, J.Chem.Phys. 64, 2417 (1976).23. E.W.Pearson, W.Waldman, and R.G.Gordon, J.Chem.Phys.80, 1543 (1984).24. U.Buontempo, S.Cunsolo, P.Dore, and P.Maselli,J.Chem.Phys. 66, 1278 (1977).25. B.Guillot, R.D.Mountain, and G.Birnbaum, J.Chem.Phys.90, 650 (1989).26. B.Guillot, J.Chem.Phys. 91, 3456 (1989).Table 6.1 Observed frequencies of 28/2-KrTransition^20Ne2-84Kr^20Ne2-soKr^20Ne2-83Kr^20Ne2-82KrJ"^— J' v bsa^Av obs-calc^Vobsa AvotnM-cc^V dn^Av -catc^Vdn^AVdn-calcKaKe^KaKc^o Hc02 0,2 - 10,1^5660.3974 4.4^5624.3018 1.1^5679•0228b 1.8^5698.1425 3.4 .p.3 3 - 2 0,2^8379.9282 -2.8^8328.7513 -1.6^8406•3113b -2.3^8433.3762 -4.24 4 - 3 03^10982.7663 -3.7^10919.2181 0.8 11015.5066c 19.3^11049.0451 1.45 ,5 -^0,44 13469.0183 4.4^13394.9018 -0.20 606 - 5 5^15870.7364 -1.3,a , oln in MHz and Av otn-cec in kHz.Hypothetical unsplit line frequencies, free of the mKr hyperfine structure.C Estimated frequency with hyperfine structure due to 83Kr unresolved.185Table 6.2 Observed frequencies of 20Ne22Ne -KrTransition^20Ne22Ne-84Kr^20Ne22Ne-86KrJ" tl If^ji g IKaKc KaKc^V obsa^A V aobs-calc^V obs^"obs-calc202 -^101 5469.9686 0.6 5433.9719 -0.22 1.2 -^1 1.1 5082.1565 -0.9 5050.5493 -1.42 1.1 -^1 1,0 5946.0218 2.8 5903.0997 0.33 3 - 202 8095.4319 -0.4 8044.4685 0.33 1.3 -^2 1.2 7595.5323 1.0 7548.8785 1.53 1.2 -^2 1.1 8887.1467 -3.4 8823.6693 -0.240,4 - 3 3 10605.9192 0.1 10542.7305 -0.14 1,4 -^3 1.3 10079.6175 -0.3 10018.6534 -0.441,3 -^3 1.2 11786.1746 1.1 11703.3900 4.6a vdn in MHz and Avan-cec in kHz.186Table 6.3 Observed frequencies of 20Ne2-83Kr_KaKc^KaKcF" - F'^(MHz)^(kHz)V obs Av obs - calc2^-^1010,13.5^- 4.5 5679.1194 -2.55.5 - 4.5 5679.0578 1.04.5 - 4.5 5679.0578 1.06.5 - 5.5 5679.0401 0.33.5^- 3.5 5678.9618 -0.84.5^- 3.5 5678.8984 0.930,3^-^20,22.5^-^3.5 8406.3744 2.26.5 - 5.5 8406.3322 0.37.5 - 6.5 8406.3222 -2.35.5 - 4.5 8406.3049 0.55.5 - 5.5 8406.3049 0.54.5^- 3.5 8406.2634 0.86.5 - 6.5 8406.2171 -1.9Table 6.4 Observed frequencies of 20Ne2-XeTransition^20Ne2 -129XeVI 11^I.— VI V obsa^AVaobs-calcKaKc^KaKc20Ne 2 —131xeV obs^Av obs-calc20Ne2-132XeV obs^A vobs -calc20Ne2-134XeV obs_136xeV obs202,3034 45 5606,-^10,1- 202- 3 3- 4 4- 5 54465.22556657.23798802.690010892.125012924.245348.7-5.6- 6586.9693 oaa Vdn in MHz and Av obs-calc in kHz.188Table 6.5 Observed frequencies of 20Ne2-131Xe_KaKc^KaKcF" -^F'V obs(MHz)AV obs-ca Lc(kHz)20,2^-^10,13.5^- 2.5 4450.9796 -0.72.5^- 1.5 4450.9796 -0.81.5 - 1.5 4450.8804 0.82.5 - 2.5 4451.1217 0.21.5 - 0.5 4451.1339 0.53 0,3^-^2024.5 - 3.5 6636.4194 0.93.5 - 2.5 6636.4194 0.62.5 - 2.5 6636.3522 -0.62.5 - 1.5 6636.4531 -0.61.5 - 0.5 6636.4531 -0.340,4^-^30,35.5 - 4.5 8776.8833 1.14.5 -^3.5 8776.8833 0.83.5 - 2.5 8776.8977 -1.12.5 -^1.5 8776.8977 -0.8Table 6.6^Spectroscopic Constants of Ne2-KrParameter^20N_e 2_ 84Kr^20Ne 2_86Kr^20Ne2 -83Kr 20Ne2-82Kr 20Ne22Ne-84Kr 20Ne22Ne_86KrGround state effective rotational constants /MHzAo^4728.39(32)^4727.00(38)^4729.09a 4729.78a 4529.258(79) 4529.249(37)Bo^1648.9885(66)^1636.793(10)^1655.3001(91) 1661.7762(54) 1595.2281(13) 1583.07491(71)Co^1204.1318(82)^1197.653(11)^1207.4657(58) 1210.8914(126 1162.4545(14) 1155.96567(51)Ground state average rotational constants /MHzAz^4578.2220^4576.975^4578.849 4579.464 4388.584 4388.713Bz^1623.4785 1611.575 1629.638 1635.958 1571.0071 1559.1409^HCz^1197.9508^1191.521^1201.260 1204.660 1156.6255 W1150.1837^kaCentrifugal distortion constants /kHzAj^38.58b^38.05b^38.86b 39.15b 35.713(32) 35.290(21)AJK^30.91b 29•93b 31.41b 31.93b 33.54(93) 31.71(38)AK^1473•0b^1474.0b^1472.0b6j 11.92" 11.72b 12.02b1471.0b 1339•16b 1340.63b12.13b 11.046(20) 10.910(16)6K^131.5b^130.5b^132.0b 132.5b 122.35" 121.43bInertial defect /amu A2Ao^6.3440^6.2995^6.3696 6.3909 6.3639 6.3720Az 0.1880 0.1352 0.2174 0.2431 0.0939 0.0962Standard deviation /kHz5.6 2.8 1.2a^5.6^2.1^5.9a Fixed at the values extrapolated from 28Ne 2-84Kr and 28Ne2-86Kr.b Fixed at the values from the force field analysis.190Table 6.7 Spectroscopic Constants of 20Ne2-XeParameter^20Ne2- 129xe 20Ne 2_131xe 20Ne2- 132XeRotational constants /MHzAo^4728.39a^4728•398 4728.398Bo^1260.844(30) 1256.236(52) 1254.163(29)Co^979.967(13) 977.296(34) 975.891(12)Inertial defect /amu A2Ao^8.00 7.94 8.02Standard deviation /kHz38.0 60.0a^63.6a Fixed at the value of 20Ne2-84Kr.191Table 6.8Quadrupole coupling constants (MHz) of °Kr and 131Xein Ne2-RG trimers and in the corresponding dimerscomplex^20Ne2-83Kr^20Ne-83Kr^20Ne2- 131 Xe^20Ne- 131 XeXaa -0.7080(63) -0.5205(23) 0.5641(23) 0.3877(9)Zbb 0.163(35) - -0.257(18) -Xm or xi- 0.545(36) 0.2602(12) -0.307(18) -0.1939(5)Ref. this work Chapter 5 this work Chapter 5Table 6.9Estimated electric dipole moments ofNe2-RG trimers and the corresponding dimersmolecule^dipole momenta^Ref.(Debye)Ne2-Kr^0.015^this workNe-Kr 0.011^Chapter 5Ne2-Xe^0.011^this workNe-Xe 0.012^Chapter 5aThe number of digits given does not reflect theuncertainty. See text for evaluation procedure.192Table 6.10 Structural parametersa of Ne2-Kr, Ne2-Xeand the corresponding dimersbparameter Roxi.X^BrNe-Kr^rNe-Ne^ rNe-Kr^rNe-Ne20Ne 2_84Kr 20Ne_84Kr 20Ne_20Nero^3.364^3.762^3.366^78.2 11.8^3.759^3.290rz^3.390^3.776^3.326^88.0^-^-^-r -3.700^3.179^-^- -s 3.69820Ne2-86Kr^ 20Ne_86Kr 20Ne-20Nero^3.364^3.761^3.365^78.3 11.7^3.759^3.290rz^3.390^3.776^3.326^88.3^-^-^-r -^3.700^3.186^-^-^3.698^-sparameter Roal.X^BrNe-Xe^rNe-Ne^ rNe-Xe^rNe-Ne20mm. 2_132xe^ 20Ne_132xe 20Ne_20Ne'"'"ro 3.624 4.001^3.390 76.4 13.6 3.994 3.29020Ne2- 129Xe 20Ne_129xe 20Ne_20Nero 3.624 4.001^3.390 76.4 13.6 3.994 3.290aBond length in (A), angle in (°).bThe values for Ne-Kr and Ne-Xe were those in Table 5.15; thevalue for Ne-Ne distance was calculated using the potential inRef.20 (see text).193Table 6.11The harmonic force field analysis of Ne2-KrStructural parameters: r(Ne(1)-Ne(2))=r=3.366 Ar(Ne(1) -Kr)=ri=r(Ne(2) -Kr)=r2=3 .762 ASymmetry coordinates:Al: S1=(1/2)Y2 (Ari+Ar2)S2=ArB2 : S3=(1/2)1/2 (Ari-Ar2)Harmonic force constants of Ne2-Kr andthose of the corresponding dimers:Ne2-Kr Dimerf" ^(mdyn A-1) 0.00408(3) 0.00397bfu (mdyn rad-1) 0.0^(fixed)afn (mdyn A rad-2) 0.00146(4) 0.00221cfm(mdyn A-1) 0.00362(5) 0.00397bPredicted van der Waals vibrationalfrequencies /cm-1 V1 23.1V 2 13.3V 3 18.4Table 6.11 (continued)Comparison of observed and calculatedcentrifugal distortion constants (kHz): 20Ne22Ne-114KrObserved^Calculated20Ne22Ne-86KrObserved CalculatedAj 35.713(32) 35.77 35.290(21) 35.26Ao 33.54(93) 32.82 31.71(38) 31.86AK 1339.16(fixed) 1339.16 1340.63(fixed) 1340.636j 11.046(20) 11.08 10.910(16) 10.89(51( 122.35(fixed) 122.35 121.43(fixed) 121.43a See discussion in the text.Ne-Kr stretch force constant calculated from thespectroscopic constants of 20Ne-84Kr in Chapter 5Table 5.3, using formulae fs=167r2gwe-KrBO/DJ194cCaluclated from Ne-Ne stretch frequency in Ref.16.195CHAPTER 7The Microwave Spectra of the van der Waals trimersAr2-0CS and Ar 2-0027.1 IntroductionIn the previous chapter, the microwave investigationof two of the simplest possible van der Waals trimers wasdescribed. In this chapter, another important, but morecomplicated, type of van der Waals trimer, namely Ar2-0CSand Ar2-002, is discussed.The first observed pure rotational spectrum of aneutral trimer was that of Ar2-HF, studied and analyzedby Gutowsky et al. [1] in 1987. Subsequently, this grouphas succeeded in obtaining the microwave spectra of otherAr2-small molecule complexes, such as: Ar2-HF, DF [1],HC1 [2], DC1 [3], and HCN [4], as well as some largerclusters. Their work provided the first accurate groundstate structural information of larger clusters, andraised the possibility of determining accurate potentialsfor three- and four-body interactions and of obtaining inparticular direct information on the nonpairwiseadditivity of intermolecular forces [5]. Following theseobservations on Ar2-HC1, Hutson et al. [5] have been ableto develop and test a theory for treating the dynamics ofpolymeric van der Waals clusters. They calculatedpairwise additive and "adjusted" potential energy196surfaces (adjusted empirically to account for nonpairwiseadditive effects in the ground state) for Ar2-HC1, todetermine whether high resolution spectroscopy could, inprinciple, characterize the nature of three-body forces.It is very encouraging that, guided by this theoreticalprediction, Elrod et al. [6] have been successful inobserving the first far infrared intermolecularvibration-rotation spectrum of Ar2-HC1.Although these trimers, especially Ar2-HCN, havetheir own peculiarities, they share some generalproperties: (1) Ar-HX (X=F [1], Cl [2,3], CN [4])trimers have C2v symmetry, and are T-shaped; (2) TheAr-Ar distances are very similar in these differenttrimers (see Table 7.11 for details), and is also verysimilar to the reported value of Ar-Ar distance,3.821(10) A, in the free Ar2 dimer [7]; (3) The distancefrom Ar to the center of mass (c.m.) of HX is almost thesame as that in the corresponding linear dimersAr-HF [8,9] and Ar-HC1 [10], although this distance is,surprisingly, rather shorter in Ar2-HCN [4,11,12] (seealso Table 7.11); (4) All these trimers have largeinertial defects, ranging from 3.379 amu K2 for Ar-HF to13.9 amu K2 for Ar2-HCN, which is enormous for anominally planar species.The distinctive behaviour of Ar2-HCN, asGutowsky et al. pointed out [4], is caused by the197cylindrical shape of HCN. Because of this shape the HCNcan get closer to a spherical species with a side-onapproach, rather than by an end-on one. To maintain thiscondition, it undergoes an in-plane internal rotationwith respect to Ar2. In contrast, both HF and HC1 arevirtually spherical, and can easily maintain a T-shapedstructure at close contact without undergoing complexinternal rotation. On the other hand, some othercylindrically shaped small molecules, such as OCS andCO2, with the most positive atom in the middle, preferside-on approaches while bonding weakly to rare gasatoms. It was thus of considerable interest to studyother types of Ar2-small molecule trimers such as Ar2-0CSand Ar2-0O2, for a better understanding of the nature ofthe intermolecular forces. How do the structures and thestrength of the forces change upon replacing the smallmolecules? How do the properties of the Ar-small moleculedimers change as the second rare gas atom approaches? Ifone classifies the dimers according to the essentialshapes of the small molecules, spherical or cylindrical,a trend is easily seen: Ar-HF, Ar-HC1 (semirigid, linear)-* Ar-HCN (highly nonrigid, linear) -+ Ar-OCS,Ar-0O2(semirigid, planar T-shaped). Here Ar-HCN can beviewed as intermediate between the two differentstructure types. This may be the reason for its highnonrigidity and its peculiarities, such as its unusually198large centrifugal distortion constants and its abnormalsensitivity to isotopic substitution, as well as itsunexpectedly large bending amplitude [11,12]. The aim wasthen to learn whether the trend of the trimers would besimilar.The only previous study on Ar2-0CS appears to bethat of Leopold [13]. Six very weak radio frequency andmicrowave transitions were found by molecular beamelectric resonance (MBER), and were tentatively ascribedto the trimer, but were insufficient to determine itsstructure or to lead to other detectable transitions. Asa result of the present work, five of the lines have beenassigned and are listed in Table 7.1.In the work reported here, the search for Ar2-0CSwas triggered by the observation of two "mystery" linesat the frequencies 7516.1554 MHz and 9930.3470 MHz duringthe search for lines of the Ar-"OCS [14] complex.Following the elimination of transitions due to Ar-OCSisotopomers [14], complexes due to clusters of argon withOCS, other than the dimer Ar-OCS, appeared most likely.In fact, these two lines turned out to be very helpfulfor the entire search procedure, and were eventuallyidentified as two transitions of the Ar2-0CS trimer.For Ar2-0O2, there is no previous study reported, tothe best of my knowledge. The search for Ar2-0O2 was moreor less straightforward though tedious, following the199study of Ar2-0CS. These two trimers were the first twotrimers studied in this work.7.2 Experimental Conditions The Ar2-0CS trimer was formed by the expansion of Arwith 0.2% OCS at a backing pressure of 1.5 atm. It wasfound later that use of mixtures of 15% Ar and 85% Ne asthe carrier gas with 0.2% OCS would increase thesigal-to-noise (S/N) ratio by a factor of four to five,especially for the low J(<4) transitions. Thissignificant improvement made it possible to search forlines of the Ar2-0C34S isotopomer in natural abundance.The line strength of Ar2-0CS was about six or seven timessmaller than that of Ar-OCS under the same conditions.For Ar2-0021 the trimer was formed by the expansionof a gas mixture consisting of 0.5% CO2 and 1% Ar in Neat a backing pressure of 3.5 atm. Because of theanticipated low dipole moment of the trimer Ar2-0O2(4-0.068 Debye for Ar-0O2 [15]), the microwave excitationpulses were amplified using the microwave poweramplifier.When this project was almost complete and a draftfor a publication had been written, I learned of aparallel study by Connelly and Howard at the Universityof Oxford, who had measured transitions of the normal and20034S-containing isotopomers of Ar2-0CS. The data werecombined in the final analyses in the following way. Onlymeasurements made with a parallel nozzle at theUniversity of British Columbia were included, along withthose made with a perpendicular nozzle at Oxford. Thelatter, indicated by "*" in Table 7.1, had measurementaccuracies of ±4 kHz. In the final fit, made by thisauthor, the data were weighted according to the(measurement accuracy) 2•The %CS-enriched sample was synthesized by themethod of Ref. 16, using isotopically enriched water. Thepercentage of 180CS was estimated to be about 4% bycomparing the line strength of Ar-0C34S in naturalabundance with that of Ar-180CS using the enrichedsample.Finally, examples of transitions in Ar2-0CS andAr2-0O2, are shown in Fig.7.1 to give an indication ofthe signal-to-noise ratio and the resolution achieved.7.3 Search and rotational assignment A. Ar2-OCSThe possible sources of the aforementioned two"mystery" transitions were first carefully checked. Theselines had very similar properties: (1) They could be seeneasily with 0.2% OCS in Ar. The optimized microwave20130 kHzcc1,2^=3 ,2 - 2027516.06 MHz^7516.26 MHzFig.7.1 (a) The rotational transition 31,2-20,2 of Ar2-0CSrecorded with 1 experimental cycle. The transition issplit into a Doppler doublet.I130 kHz,1J Kc =3 3,1 - 2209879.44 MHz 9879.64 MHz'Fig.7.1 (b) The rotational transition 330-22,0 of Ar2-0O2recorded with 5 experimental cycles.202excitation pulse length indicated that the dipole momentof the unknown substance was similar to that of OCS.These lines were weaker than those of Ar-OCS but strongerthan those of Ar-004S in natural abundance under thesame conditions. (2) They could not be seen with pure Ar.(3) They could not be seen with 0.2% OCS in Ne. Thesecond and the third observations suggested that thesource was Ar+OCS. The first, from a comparison with theline strengths observed for the Ar2-HX [1,2] and Ar-HX(X=F [8,9] and Cl [10,11]) systems, suggested that it wasmost likely to be Ar2-0CS.Following Ar2-HX (X=F [1], Cl [2,3], CN [4]) asexamples, a Cn T-shaped structure for Ar2-0CS was firstassumed. However, to keep a planar structure and a ;dm.similar to Ar-OCS would result in an Ar-O distance of2.5 A, which is far too short compared with the sum ofthe van der Waals radii of argon (1.91 A) and oxygen(1.4 A). A reasonable structure to meet the generalproperties observed for Ar2-small molecule complexes, asdescribed in the introduction, was a Cs, distortedtetrahedral shape with both Ar-OCS [14] and Ar-Ar [7]moieties remaining similar to their corresponding freedimers. With these dimensions the trimer would be anoblate, highly asymmetric top (Fig.7.2). Since the a andc axes would be in the Cs symmetry plane, making theprojection of the OCS dipole moment on the c axis about203twice as large as that on the a axis, strong c and weak atype transitions were expected.The initial calculation based on this assumedstructure predicted a series of c-type transitions ofJ=3-2 and J=4-3 with frequencies around 7.5 GHz and10.0 GHz respectively. If the assumed structure were moreor less correct, the two "mystery" lines should probablybelong to these two series respectively. The assignmentof the quantum numbers was made by comparing thefrequency ratio of each pair of predicted frequencies inJ=3-2 and J=4-3 to that of the two lines observed. Twopairs of possible candidates were found, namely : 31,2-20,2142,2-3 1,2 and 3 2 2 21 21 4 -3^For the second case, search,,^2,3^1,3'for other lines belonging to the same Al<c=0 stack wasunsuccessful. However for the first possibility, twoother lines, namely 32,2-21,2 andWith these four lines, an approximate set of rotationalconstants was obtained. Ultimately 65 lines were observedfor the main isotopomer; their frequencies are given inTable 7.1. The a-type transitions were observed withabout three times less intensity than the c-type, aspredicted.The spectra of Ar2 -0e4S and Ar2-180CS isotopomerswere also measured to confirm the assignments and toprovide more information on the structure and the forcefield of the trimer. Transitions for these two32,2 were observed.204isotopomers were easily located using rotationalconstants calculated from the preliminary structure anddistortion constants derived for the main isotopomer. Thetransitions measured for these two isotopomers are listedin Table 7.2.The line intensities of these two isotopomers wereabout four or five times smaller than those of theircorresponding Ar-OCS dimers under the same experimentalconditions. This observation was consistent with what hadbeen observed in the case of the main isotopomer.B. Ar2-002The search for Ar2-002 followed essentially theprocedure used in Ar2-0CS. An initial structure wasproposed for Ar2-0O2. It is a Cn, distorted tetrahedralshape, with both Ar-0O2 [15] and Ar-Ar [7] moietiesremaining similar to their corresponding free dimers.With these dimensions, the trimer is an oblate, highlyasymmetric top (K=-0.360, see Fig.7.3). Because of theC2V symmetry of the complex, the only non-zero induceddipole moment is along the axis connecting the centres ofmass of the Ar2 and CO2 subunits, coinciding with theb-inertial axis; only b-type transitions are allowed. Theinitial rotational constants were predicted from theassumed structure, and were used to predict b-typetransition frequencies. Searches were carried out for low205J transitions because of the expected low rotationaltemperature in the jet expansion. Most low J (4) lineswere located within ±50 MHz of the predicted frequencies.Further lines were easily found by bootstrapping.Eventually twenty-one lines, all b-type, with Ka+Kc=even,were found (See Table 7.3). The intensities of theselines were about five to six times less than those of theAr-0O2 dimer under the same conditions. A rough estimatefrom the optimized microwave excitation pulse lengths forboth Ar-0O2 and Ar2-0O2 suggested that the induced dipolemoment of Ar2-0O2 is very similar to that of Ar-0O2.7.4 Analyses of the Ar2-0CS and Ar2-0O2  spectra The observed frequencies of Ar2-0CS were fit to aIIIr representation, Watson A-reduction, semirigid rotormodel [18] consisting of 3 rotational constants and 5quartic centrifugal distortion constants. Thespectroscopic constants for all three isotopomers ofAr2 -0CS are listed in Table 7.4. The differences betweenthem and those calculated using the constants tabulatedin Table 7.4, are listed in Tables 7.1 and 7.2. Table 7.1also contains the MBER measurements [13], and theirassignments. Although they were not included in the fit,they are very well predicted by our constants.Similarly, all observed frequencies of Ar2-0O2 were206fit to the same semirigid rotor model used above,consisting of three rotational constants, five quarticdistortion constants, as well as three sextic distortionconstants. The spectroscopic constants for Ar2-0O2 arelisted in Table 7.5. The observed frequencies, along withthe differences between the observed and calculatedfrequencies using the constants tabulated in Table 7.5,are listed in Table 7.3.The standard deviations for all these fits arecomparable to or less than the estimated measurementaccuracies, demonstrating that the data are fittedadequately by the models employed. For Ar2-0O2, a fitwithout the sextic distortion constants gave essentiallythe same quartic distortion constants with slightlylarger standard deviations and a 2.4 kHz standarddeviation for the fit.7.5 Geometry and structure of Ar2-0CS and Ar2-0O2A. Ar2-0CSAll the observed transitions listed in Tables 7.1and 7.2 are either a- or c-type. A search for b typetransitions was unsuccessful, thus suggesting stronglythat the b axis is perpendicular to the OCS axis, andthat OCS lies in the ac plane [Fig.7.2]. This is alsosupported by the very small change in the planar momentC. M207Fig.7.2 Ar2-0CS in its principal inertial axis system.R^connects the centres of mass of the Ar 2 and thec.m.OCS subunits. Rc.m., r and 0 are coordinates used indetermining the structure of Ar2-0CS.208Pb upon isotopic substitution, as shown in the nextparagraph.The inertial defect, defined as(7.1)is expected to be zero for a rigid planar molecule. InAr2-0CS, A0 is about -140 amu K2 for all threeisotopomers, clearly indicating that the complex isnonplanar. The planar moment Pb is given by^P - E^- (^+^- I °)^bi 2 b (7.2)where mi is the mass of atom i, and bi is its coordinatealong the b-inertial axis. This quantity describes themass located out of the ac plane. The Pb values for allthree Ar2-0CS isotopomers are tabulated in Table 7.4. Ifthe OCS moiety were not lying in the ac plane, one wouldexpect significant increases in the Pb values uponsubstitutions of 180 and 34S for 160 and 32S, respectively.However, the changes are very small, with differentsigns, for the two different substitutions:-0.0101 amu K2 for 160 to 180 and +0.0073 amu K2 for 32S toUS. The OCS molecule is clearly in the symmetry plane ofthe complex, and the sole contributions to Pb are fromthe two Ar atoms, giving209Pb- 2 MArbr^ (7.3)the bAr values obtained in this manner from the normalisotopomer are ±1.921 A, corresponding to an Ar-Ardistance of 3.842 A, which is very close to its value inthe Ar2 dimer, and in the Ar2-HX (X=F [1], Cl [2,3],CN [4]) trimers.From the 180 and 34S substitutions one can get thesubstitution coordinates of these two atoms using theusual formulae for nonplanar asymmetric topmolecules [18],A P^APb^lal - [ — ` (1 +^) (1 +APIbI = [ b^ (i. +^A Pc  ) ( 1 +11AP A PaICI = [ --E ( 1 + ^ ) ( 1 +11IC() _ .rao21711(7.4)where1A Pa= —2 ( - A Ia° + A Ib° + A I?)1A Pb ''^(- A -Tb° + AI: + A _Ta°)1A Pc= —2 ( - A /: + A If ÷ A Ib° )(7.5)210and where Ars(=n1-I), and so on, are the changes in theprincipal moments of inertia due to isotopicsubstitution. The a and c coordinates for S arecalculated to be, respectively: 2.187 A and -0.803 A, andfor 0: 1.101 A and 1.694 A. This gives an OCS length of2.723 A assuming it to be linear, which agrees very wellwith the rs value in the free OCS molecule, 2.721(2) A[19]. It seems that there is no significant change in theOCS structure upon weakly bonding to the Ar atoms, as isthe case for most small molecules in their van der Waalscomplexes with rare gases [14].Assuming that the two Ar atoms are equivalent, andusing the known structure of OCS [19], there are threestructural parameters to be determined for Ar2-0CS: theAr-Ar distance (r), the Ar-Ar c.m. to OCS c.m. distance(Rcan,), and the angle between and OCS (0). By fittingall these three parameters to the rotational constantsAo, Bo and Co of all three isotopomers, the ro structurefor Ar2-005 was obtained, which is listed in Table 7.6.The values are effective ground state values, and thequoted uncertainties are those from the least squaresfit. For comparison, the structural parameters of Ar-OCS,as well as of the Ar dimer are also in Table 7.6.B. Ar2-002In the case of Ar2-0O2, all of the observed211transitions listed in Table 7.4 are b-type, Ka+Kb=even.Efforts to measure b-type, Ka+Kb=odd transitions turnedout to be unsuccessful. This is consistent with thenuclear spin statistics associated with the Cnequilibrium structure for Ar2-0O2. The rotational wavefunctions are symmetric or antisymmetric with respect toa 1800 rotation about the b-axis if Ka+Kc=even or oddrespectively, while the nuclear spin functions are alwayssymmetric for the spinless nuclei. Since the two sets ofequivalent spinless Bosons, namely the two oxygen nucleiand the two argon nuclei, require a totally symmetricwave function, only Ka+Kc=even rotational levels arepossible.With the Cn symmetric structure for Ar2-0O2, andusing the known structure of CO2 [20], there are only twostructural parameters to determine for Ar2-0O2: the Ar-Ardistance (r), and the Ar-Ar c.m. to CO2 c.m. distance(Rc.r,L)• The Ar-Ar distance can be calculated in a similarfashion as in case of Ar2-0CS from the planar moment Pa,given by Pa=2mAra2= (1/ 2 ) Ii3)+I°c-I). The am, values obtainedin this manner are ±1.922 A, which corresponds to aneffective Ar-Ar distance of 3.844 A. This value is, ofcourse, subject to the vibrational effects in the trimer.Nonetheless, it is consistent with the value of the Ar-Ardistance given earlier for Ar2-0CS.Because of the higher symmetry of Ar2-0O2 (Cn)ArIIR c.m.212Fig.7.3 Ar2-0O2 in its principal inertial axes system.connects the centres of mass of the Ar 2 and thec.m.CO2 subunits, respectively. Rc.m. and r are coordinatesused in determining the structure of Ar2-0O2.213compared to Ar2-0CS (Cs), the Rcah can be simplycalculated with either Iaa=gRLL+I(CO2), using the knownvalue of I(CO2) from free CO2, or Icc=g12c2.m.+I(Ar2), usingthe I(Ar2) from the value of aAr derived earlier. Theresult from these two procedures yields values for 12c.rn.of 2.923 A and 2.936 A respectively, which in turn givethe Ar-C distance as 3.498 A and 3.509 A. It is importantto note that all three rotational constants are affectedby the large amplitudes of the van der Waals vibrationalmotions, and that the structures calculated usingdifferent procedures will therefore differ. The rostructure obtained for Ar2-0O2 using the second procedureis listed in Table 7.7. The structural parameters ofAr-0O2F as well as those of the Ar2 dimer are also givenin Table 7.7. Fig.7.3 shows the structure of Ar2-0O2 inits principal inertial axis system.7.6 Harmonic Force Field ApproximationA. Ar 2 -OCSThere is a significant fluctuation in thecorresponding centrifugal distortion constants for thethree isotopomers, as is shown in Table 7.4. A similarphenomenon observed in the earlier study of variousisotopomers of Ar-OCS [14] turned out to be mainly massand geometry dependent. A harmonic force field analysiswould certainly help to rationalize these fluctuations214and to confirm the assignments. At the same time, itwould also provide force constants for the van der Waalsmodes in the trimer, giving some insight into the natureof the van der Waals interactions.In principle, information about the force constantsof Ar2-0CS can be obtained from an analysis of thecentrifugal distortion constants of the complex. Such ananalysis has been done for the Ar-OCS dimer [14]. Forceconstants are related to the asymmetric-top TIS by theEq.(2.22) in Chapter 2 [see also Ref.21]:AA 2hx1017E (j,(1))^(f-1)- (J-(-1))010^/ I^P 0 /.7 y8 0PP YY j (2.22)The variables are the same as defined before. The TIS arerelated to the asymmetric-top distortion constants inTable 7.4 by the well-known equations in Ref. 22.The nine vibrational modes of the Ar2-0CS complextransform as 6A1-F3A2. Those of Al symmetry areapproximately: the C-0 stretch (v1), the C-S stretch(v2), the in-plane OCS linear bend (v3), the symmetricAr-C-Ar stretch (v4), the Ar-Ar stretch (v5), and theAr-Ar "wag" (v6) where two argon atoms bend towards OCSin the same direction; those of A2 symmetry are: theout-of-plane OCS linear bend (v7), the Ar-C-Ar asymmetricstretch (v8), and the Ar-Ar "torsion" (v9), where twoargon atoms bend towards OCS in opposite directions.These modes can be approximately characterized by the215symmetry coordinates shown in Table 7.8. Shea et al. [23]showed that, in the case of Ar-OCS, the values of theforce constants for the van der Waals modes were notsignificantly altered by neglect of the internal modes ofOCS. This insensitivity of van der Waals force constantsto the internal modes of the monomers has been confirmedby our previous study of various isotopomers ofAr-OCS [14], where the force constants for OCS were leftthe same as in the free monomer [24]. The same assumptionwill be made here.There is only enough information to fit the fivediagonal force constants. Some of the off-diagonalcoupling terms, namely those between Al and A2 modes, areidentically zero because of symmetry. In addition, in theabsence of three-body contributions to the intermolecularpotential, the coupling terms fo and fm should be zero.However, rather than neglecting the remainingintermolecular coupling terms, fi,o and fn, their valueshave been estimated from the related Ar-OCS bend-stretchcoupling term fro [14]. From the definition of thesymmetry coordinates in Table 7.8, and assumingnegligible three-body contributions, one would expectboth fi,o and fn to take the value fr8/V2, withfre-0.0015(2) mdyn rad-1 [14]. Inclusion of these crossterms improves the fit to the centrifugal distortionconstants. The five diagonal force constants, obtained by216fitting simultaneously to the centrifugal distortionconstants of all three isotopomers, are shown inTable 7.8, along with the corresponding vibrationalfrequencies predicted for the five van der Waals modes.The distortion constants were weighted according to thesquares of their standard deviations from the semirigidrotor fits.The centrifugal distortion constants calculated fromthe force constants in Table 7.8 are compared with theexperimental data in Table 7.9. As one can see fromTable 7.9, the set of the force constants reproduces theobserved distortion constants rather well, andparticularly the observed variations with isotopomer. Ithas also been noticed that although small changes in theforce constants give quite different values for thecentrifugal distortion constants, the relative valuesbetween different isotopomers are basically the same andare rather insensitive to variations in the forceconstants. This confirms that the observed variations aremainly mass and geometry dependent as in the case ofAr-OCS [14], and that the assignment is consistent withthe assumption of a distorted tetrahedral structure forthe trimer.B. Ar2-0O2Similarly, a harmonic force field estimate was217performed for Ar 2-0O2' There are nine vibrational modesof the Ar2-0O2 complex, which transform as 4A1+A2+2B1+2B2.Those of Al symmetry are: the symmetric O-C-0 stretch(v1), the CO2 in-plane linear bend (v2), the symmetricAr-C-Ar stretch (v3), and the Ar-Ar stretch (v4); that ofA2 symmetry is the Ar-Ar "torsion" (v5), where two argonatoms bend towards CO2 in opposite directions; those ofB1 symmetry are the O-C-0 asymmetric stretch (v6), andthe Ar-Ar "wag" (v7), where two argon atoms bend towardsCO2 in the same direction; those of B2 symmetry are: theCO2 out-of-plane linear bend (v8) and the Ar-C-Arasymmetric stretch (v9). These modes can be approximatelycharacterized by the symmetry coordinates shown inTable 7.10.The force constants of the internal modes of CO2 areconstrained to the values obtained for the free CO2monomer [20]. Because of the symmetry, the only possibleinteraction between the van der Waals modes is f34, theinteraction between Ar-C-Ar symmetric stretch and Ar-Arsymmetric stretch. Unfortunately, since there are onlyfive centrifugal distortion constants available, it isimpossible to fit all five diagonal force constants andone off-diagonal force constant. Several fits withdifferent combinations of four out of six force constantshave been performed to see how the value of each constantand the quality of the fit vary. It was found that the218interaction force constant f34 is more than an order ofmagnitude smaller than other constants and is basicallyindeterminate. Thus f34 was fixed at 0.0 in the furtherfits, as might be expected if pairwise interactions werepredominant. f55 and fn are highly correlated andinsensitive to the distortion constants. Since these twomodes originate from the van der Waals bending motion ofAr-0O2 (fb=0•009428 mdyn^[24]), a value of0.01 mdyn A-1 has been used as an initial guess for thesetwo constants. The Ar-C-Ar symmetric stretching constantfm and Ar-C-Ar asymmetric stretching constant f99, varyequally about 5-10% in opposite senses upon differentchoices, but the mean of fm and f99, which corresponds tothe single Ar-C stretch constant, is largely unaffected.The Ar-Ar symmetric stretch force constant, f441 isinsensitive to the choices of fitting parameters. As longas the fit converges, the value of f44 varies less than1% around 0.00816 mdyn P. One of the force field fitsis shown in table 7.10, along with the comparison of theobserved and calculated centrifugal distortion constantsfrom the fit.7.7 Comments on the Effects of Three-body Interaction andthe General Geometric Trends for Ar2-Molecule Trimers The geometries of both trimers Ar2-0CS and Ar2-0O2219are found to be dominated by the pairwise additivity. Thetrimer Ar2-0CS has been found to have a distortedtetrahedral geometry, evidently determined mainly bypairwise van der Waals interactions between the two Aratoms and between each individual Ar atom and OCS, takingthese interactions to be comparable to those in the Ar2and Ar-OCS dimers. The trimer Ar2-0O2 has a C2v distortedtetrahedral geometry, with its properties also mainlydetermined by pair interactions between the two argonatoms and between each argon and the CO2, which arecomparable with those in the Ar2 and Ar-0O2 dimers.However, a structural prediction based on theparameters for Ar-OCS and Ar2 overestimates therotational constants by 1 to 2 %. A similar phenomenonhas also observed for Ar2-0O2. This correlates with theincrease in Ar-Ar distance in both trimers when comparedwith Ar2 (3.821 A.), and might be expected if there were asmall induced dipole repulsion [5). In addition, the Ar-Cdistance is also greater in Ar2-0CS than in Ar-OCS, atrend also observed with the Ar-X distance in Ar2-HX andAr-HX. A similar lengthening in Ar2-0O2 is not obvious,3.509 A vs 3.505 A, comparable to the uncertainty(0.01 A) in the structural determination.A comparison of the structures of Ar2-0CS and Ar2-0O2with those of some other Ar2-small molecule trimers isgiven in Table 7.11. It is clear that the general trends220for the trimers parallel those of the dimers, namely:Ar2-HF, Ar2-HC1 (semirigid, planar)^Ar2-HCN (highlynonrigid planar)^Ar2-0CS, Ar2-0O2(semirigid, distortedtetrahedral).The force field analyses allow for some interestingdeductions, which must, however, be made cautiously. Inparticular, some care must be exercised in attributingsignificance to small changes in the force constants. Incarrying out the force field analyses, the 1'0 geometriesused were only approximations to the equilibriumgeometries. The harmonic force field for the van derWaals modes is similarly very approximate.Nevertheless, the derived diagonal force constantsfor Ar2-0CS are reasonable. This is in spite of theneglect in the analysis of two off-diagonal constants,f45 and f56, which may be significant, especially theformer. It is interesting that fR5fes, suggesting thatthe interaction between the two Ar-C stretches isnegligible. Similarly, f6ef99, suggesting that theinteraction between the two Ar-OCS bends is likewisenegligible. Both observations are consistent with thepairwise additive interaction model.The constants fia, and f ^Ar2-0CS, which correspondto the symmetric and asymmetric Ar-C-Ar stretches,respectively, are directly comparable to fR, the Ar-Cstretch in Ar-OCS, which has the value2210.0222(10) mdyn V [14]. Both f44 and f88 are smaller thanfR, suggesting a weakening of the van der Waalsinteraction on truer formation, which is the reverse ofthat observed for Ar n-HX. Similarly, the bend forceconstants f ^f99 are comparable tof0/2-0.0127 mdyn A rad-2, where f0 is the van der Waalsbend force constant in Ar-OCS [14]. This time the valuesare quite similar, suggesting that there is little changein the angular dependence between dimer and truer.The Ar-Ar stretch constant, f85, has been found tobe 0.008422(20) mdyn A-% This value is very comparableto 0.0078 mdyn V, estimated for free Ar2 from the v=1-0energy difference from vibronic spectra [7]. It contrastswith the decrease of about 10% found in going from Ar2 toAr2-HF/DF [1] and Ar2-HC1 [2]. In Ar2-0CS the wavenumbersof the nominal Ar-Ar stretch cover the small range 23.3to 24.6 cm-1 for the three isotopomers, which is veryclose to the corresponding values for Ar2, 25.7 cm-1 [7].There is, however, a significant contribution of the OCSmoiety to the normal mode of the trimer, so thiscomparison is perhaps not entirely valid.In the case of Ar 2 -002, there is not enoughinformation to obtain a complete force field. The forcefield analysis was performed with the goal of checkingits consistency with the assumption of pairwiseadditivity rather than to detect effects of nonpairwise222additive contributions. The individual Ar-C stretchconstants, calculated using the mean of the symmetric andasymmetric Ar-C-Ar stretch force constants, is0.01673 mdyn V, similar to that of the Ar-0O2 dimer0.01738 mdyn V [24]. The value of f44 is quitereasonable compared with the value 0.0078 mdyn V forthe free Ar2 dimer. The estimated vibrational frequencyof the Ar-Ar stretch is about 23.5 cm-1 in Ar2-0O2 , whichcompares favourably with 25.69 cm-1, the experimentalvalue reported for the fundamental vibrational frequencyin the free Ar2 dimer [7].223Bibliography1. H.S.Gutowsky, T.D.Klots, C.Chuang, C.A.Schmuttenmaer,and T.Emilsson, J.Chem.Phys. 86, 569 (1987).2. T.D.Klots, C.Chuang, R. S.Ruoff, T.Emilsson, andH.S.Gutowsky, J.Chem.Phys. 86, 5315 (1987).3. T.D.Klots and H.S.Gutowsky, J.Chem.Phys. 91, 63(1989).4. H.S.Gutowsky, T.D.Klots, and C.E.Dykstra,J.Chem.Phys. 93, 6216 (1990).5. J.M.Hutson, J.Beswick and N.Halberstadt, J.Chem.Phys.90, 1337 (1989). See also, A.R.Cooper and J.M.Hutson,J.Chem.Phys. 98, 5337 (1993).6. M.J.Elrod, D.W.Steyert, and R.J.Saykally,J.Chem.Phys. 94, 58 (1991).7. E.A.Colbourn and A.E.Douglas, J.Chem.Phys. 65, 1741(1976). Also the recent work of P.R.Herman,P.E.LaRocque, and B.P.Stoicheff, J.Chem.Phys. 89,4535 (1988).8. T.A.Dixon, C.H.Joyner, F.A.Baiocchi, and W.Klemperer,J.Chem.Phys. 74, 6539 (1981).9. M.R.Keenan, L.W.Buxton, E.J.Campbell, A.C.Legon, andW.H.Flygare, J.Chem.Phys. 74, 2133 (1981).10. (a) S.E.Novick, P.Davies, S.J.Harris, andW. Klemperer, J.Chem.Phys • 59, 2273 (1973). (b)S.E.Novick, K.C.Janda, S .L.Holmgren, W.Waldmam, andW. Klemperer, J.Chem.Phys. 65, 1114 (1976).11. K.R.Leopold, G.T.Fraser, F.J.Lin, D.D.Nelson, Jr.,and W.Klemperer, J.Chem.Phys. 81, 4922 (1984).12. T.D.Klots, C.E.Dykstra and H.S.Gutowsky, J.Chem.Phys.90, 30 (1989). See also G.T.Fraser and A.S.Pine,J.Chem.Phys. 91, 3319 (1989); and R.E.Bumgarner andG.A.Blake, Chem.Phys.Lett. 161, 308 (1989).13. K.R.Leopold, Ph.D. thesis, Harvard University, 1983.14. Y.Xu, W.Jager and M.C.L.Gerry, J.Mol.Spectrosc. 151,206 (1992).22415. J.M.Steed, T.A.Dixon and W.Klemperer, J.Chem.Phys.70, 4095-4100 (1979); 75, 5977 (E) (1981). See also,G.T.Fraser, A.S.Pine, and R.D.Suenram, J.Chem.Phys.88, 6157 (1988).16. G.Brauer, Ed., "Handbuch der Praparativen Chemie",p.496-497, Ferdinand Enke Verlag, Stuttgart, 1954.17. J.K.G.Watson, in " Vibrational Spectra and Structure:A series of Advances" (J. R. Dung, Ed. ), Vol. 6,p.1-89, Elservier, New York, 1977.18. W.Gordy and R.L.Cook, "Microwave Molecular Spectra",P. 511, Interscience Publishers, 1970.19. J.K.G.Watson, J.Mol.Spectrosc. 48, 479 (1973).20. G.Herzberg, "Infrared and Raman Spectra", VanNostrand-Reinhold, New York, 1945.21. D.Kivelson and E.B.Wilson, Jr., J.Chem.Phys. 21,1229(1953).22. J.K.G.Watson, J.Chem.Phys. 46, 1935 (1967).23. J.A.Shea, W.G.Read, and E.J.Campbell, J.Chem.Phys.,79, 2559 (1983).24. Y.Morino and T.Nakagawa, J.Mol.Spectrosc. 26, 496(1968).225Table 7.1^Observed transitions of Ar2-0CSTransitionK::[<^KUObservedfrequency(MHz)Distortionacontribution(MHz)Obs.-Calc.(kHz)3 1.2^-^20,2 7516.1554 -0.7324 -1.03 2.2^-^2 1.2 7708.9044 -0.6231 -1.73 2.1^-^2 1.1 7458.8154 -0.7540 -1.53 3.1^-^2 2.1 8028.4369 -0.9705 -0.63 3,0^-^22,0 7901.2192 -0.9503 0.041,3^-^30,3 10137.6417 -1.4875 -0.834 1,3^-^- 1,2 7813.1040 -0.9793 0.24 2,3^-^3 1,3 10193.4432 -1.3757 0.14 2,2^-^3 1,2 9930.3470 -1.7363 0.34 2,2^-^3 2,1 8704.8027 -1.5570 -0.64 2,3^-^3 2,2 7632.0945 -1.0022 -1.043,1^-^33,0 8738.9823 -1.5440 1.843.2^-^3 3.1 8192.1509 -1.2107 -0.843.1^-^3 2.1 10031.4064 -1.7592 0.043,2^-^32,2 10320.3771 -1.7079 -0.34^ 34,0^-^3,0 10668.3791 -2.4749 0.54 4.1^-^3 3.1 10747.8599 -2.5141 -0.65 0,5^-^4Q,4 8282.9612 -0.9736 0.25 1.5^-^4 1.4 8281.6753 -0.9799 -1.05 1.4^-^4 1.3 9308.8645 -1.6199 0.05 1.4^-^40,4 12716.3144 -2.5931 1.252.4^-^4 1.4 12727.7795 -2.5382 -0.852.3^-^4 1.3 12568.4400 -3.2540 1.352,3^-^42,2 10451.1967 -2.4970 0.95 2,4^-^4 2,3 9256.1581 -1.6940 1.353,2^-^4 2,2 12373.0635 -3.3122 0.7226Table 7.1 (continued)TransitionK:,1(^KU':Observedfrequency(MHz)Distortionacontribution(MHz)Obs.-Calc.(kHz)5 3,3 12739.9605 -3.0386 -0.553,2^-^43,1 11046.4592 -3.1100 -0.553,3 10051.6779 -2.3329 -1.254.1 12692.5764 -3.5517 0.654,2 12964.4787 -3.6352 1.85^- 45,0^4,0 13447.1916 -5.1738 0.755,1^-^44,1 13485.5450 -5.1975 0.36^- 50,6^0,5 9838.7579 -1.6323 0.56 1,6^-^5 1,5 9838.5809 -1.6340 0.36 1,5^-^5 1,4 10840.7131 -2.5616 0.361,5^-^50,5 15274.0665 -4.1811 1.462,4^-^51,4 15197.0802 -5.1928 0.862,5^-^51,5 15276.0347 -4.1626 -0.862,5^-^5 2,4 10829.9318 -2.6043 0.363,3^-^5 2,3 14945. 4885Th -5.7369 -0.663,4^-^5 2,4 15245.1664 -5.0117 -1.263,4^-^53,3 11761.3632 -3.6671 -0.264,2^-^53,2 14883.3312 -5.6715 0.464,3^-^53,3 15307.8218 -5.5926 0.665,1^-^54,1 15441.4746* -6.5583 3.46 5,2^-^54,2 15642.6191* -6.7208 2.266,0^-^55,0 16217.9809 -9.3448 -0.766,1^-^5 5,1 16233.8828* -9.3510 -2.870,7 -^60,6 11394.8760 -2.5282 0.371,6^-^6 1,5 12390.6382* -3.7855 -0.771,6^-^6 0,6 17825.9445* -6.3343 -2.1227Table 7.1 (continued)TransitionJ'^_ireK:,1‹^KI:,ICObservedfrequency(MHz)Distortionacontribution(MHz)Obs.-Calc.(kHz)71,7 -^61,6 11394.8526 -2.5286 -0.472,6 -^62,5 12388.7951* -3.8008 -0.972,5 -^61,5 17776.4260* -7.7083 -1.372,6 -^61,6 17826.2502* -6.3293 -0.773,4 -^62,4 17622.6738* -8.8419 0.473,5 -^6 2.5 17786.7559* -7.6255 -4.174,3 -^63,3 17323.8361* -8.9994 0.57 4.4 -^63.4 17763.7970* -8.4792 -3.57 5.2 -^64.2 17486.7628 -9.1096 -0.97 5.3 -^64.3 17907.8822* -9.3030 2.27 6.2 -^6 5.2 18350.5324* -11.3140 -1.380,8 -^70,7 12950.8181* -3.6985 -3.28 1.8 -^7 1.7 12950.8181* -3.6985 -0.441,4 -^40,4 1.4851.c 0.008 -2.06 2.5 -^6 1.5 2. 173t 0.021 0.083.6 -^8 2.6 2.2331 0.033 -3.05 5, 0 -^5 5.1 25.237t 0.005 1.065,1 -^6 5,2 216.855t -0.145 -4.0a This is the fourth-degree centrifugal distortioncontribution calculated from the constants of Table 7.4.b All frequencies indicated with * were measured with aperpendicular nozzle arrangement at Oxford by Connellyand Howard, with an accuracy of -±4 kHz as compared to-±1 kHz for the remainder.c All frequencies indicated with t were MBER measurementsin ref.13 and were not included in the fit.228Table 7.2Measured Line Frequencies of Substituted Isotopomers of Ar2-0CSTransition^Observed^obs.-^Observed^obs.-Frequency^calc.^Frequency^calc.Ar2-0C345 Ar2 -180CSJ' ;^' -^JIZ''^K"K K,^c ,^c (MHz) (kHz) (MHz) (kHz)2 - 20 231,, 7368.4543 -0.5 7379.4749 -0.41^-^21 132,, 7333.1594 -0.4 7330.3773 0.322^-^212 7607.6196 -0.9 7580.0018 0.933,0^-^22,0 7846.3588 -1.1 7780.1205 0.03 3,1^-^22,1 7971.6663 -1.1 7902.7897 -0.913^-^303 9958.3463 -0.4 9957.3371 -0.413^-^3 12 7711.1272 0.2 7733.7695 0.022^-^312 9729.8940 0.1 9749.5700 -0.04 2,2^-^32,1 8521.3158 0.3 8570.7407 -0.042,3^-^31,3 10038.2629 -0.4 10018.2534 0.043 1^-^32 1 9891.7127 -0.6 9865.5555 -0.443,2^-^3 2,2 10197.3571 -0.6 10150.7406 0.2o 44,^-^33,0 10611.7984 0.7 10507.9703 0.544,1^-^33,1 10681.9300 0.1 10581.9255 0.1505 -^404 8163.2458 0.351,4^-^40,4 12507.8003 0.751,4^-^41,3 9183.7733 -0.75 1.5^-^4 1.4 8161.1506 0.15 2,3^-^41,3 12315.8059 1.8 12337.7260 0.452,4^-^41,4 12526.7095 -0.7Table 7.2 (continued)Transition^Observed^obs.-^Observed^obs.-Frequency^calc.^Frequency^calc.Ar2 -0C34S Ar2 -180CSJ'Kl<^-^ 1Z1< (MHz) (kHz) (MHz) (kHz)52,4^-^4 2,3 9110.8585 -0.153,3^-^4 2,3 12551.8326 0.5 12522.6727 -0.453,2^-^4 2,2 12140.4679 0.254,1^-^4 3,1 12561.6967 1.4 12493.5511 -0.454,2^-^43,2 12828.0263 0.9 12755.5817 0.55 5,0^-^4 4,Q 13382.0607 -1.05 5,1^-^4 4,1 13412.3111 -0.7 13279.3742 -0.215096.3750 1.214643.1138 2.015323.9251 -1.914625.9306 -0.315034.1204 0.715030.4000 -0.714923.6413 -1.116141.9729 0.516153.2541 1.315001.0768 1.3229230Table 7.3 Observed transitions of Ar2-0O2TransitionJ'Observedfrequency(MHz)Obs.-Calc.(kHz)22.0^-^11,1 7182.2833 -0.333,1^-^22,0 9879.5340 0.131,3^-^20,2 6351.0264 0.532,2^-^21,1 8115.4690 -0.544,0^-^33,1 13818.4074 0.342,2^-^33,1 9327.4863 0.04 2.2^-^3 1.3 15800.2512 -0.04 0,4^-^3 1.3 8171.3919 0.24 3.1^-^3 2.2 14094.8706 0.44 1.3^-^3 2.2 9340.0529 -1.15 5.1^-^44,Q 17144.5090 0.453.3^-^4 2.2 13371.3746 1.851,5^-^40,4 10052.2285 -0.154,2^-^43,1 15560.6216 -2.052,4^-^41,3 11485.8343 0.462,4^-^53,3 14492.2462 0.060,6^-^51,5 11922.7823 -0.263,3^-^54.2 14624.3221 -0.06 1.5^-^5 2.4 13288.1001 -0.27 1.7^-^60,6 13794.5770 0.172.6^-^6 1.5 15169.2508 -0.0231Table 7.4 Spectroscopic Constants of Ar2-0CSParemeter Ar2- 1601202s^Ar2-16012c34s^Ar2-18012c32sRotational constants /MHzaAo 1381.53375(09) 1376.89177(14) 1360.66909(13)Bo 1188.30931(08) 1159.18449(14) 1166.19528(15)Co 778.58576(06) 767.39568(20) 775.67736(42)Centrifugal distortion constants /kHzaAj 9.8113(14) 9.5797(16) 9.2427(21)AJK -18.1894(66) -17.7630(75) -16.917(11)AK 9.8837(56) 9.6613(82) 9.252(42)6j 1.7345(08) 1.7715(11) 1.6470(16)6K -15.310(10) -13.219(11) -14.588(13)Inertial defect and planar moment Pb /amu A2ob -142.00388 -144.45765 -153.24408Pbb 294.80816 294.81455 294.79743Standard deviation of the fit /kHz 1.0^1.0^0.5a IIIr representation, A-reduction.b^ Pb= ( I:MI 4)/2; 4(amuA)=505379/B0 (MHz),etc.232Table 7.5Spectroscopic Constants of Ar2-0O2aRotational ConstantsA (MHz)°^1768.75772(18)B (MHz)0^1502.63155(17)C (MHz)0^936.57347(20)Ouartic Distortion ConstantsAj (kHz)AJK (kHz)AK (kHz)6j (kHz)6K (kHz)18.9904(43)-40.019(20)23.378(23)0.2672(20)-34.538(21)Sextic Distortion ConstantsHJK (Hz)^-5.71(86)HKJ (Hz)^15.1(18)HK (Hz)^-9.47(99)Standard Deviation of the Fita (kHz)^1.0a IIIr representation, A-reduction.233Table 7.6 Structural parameters of Ar2-0CS, Ar-OCS and Ar2parametere Ar2-0CS Ar-OCS Ar2r(CO) 1.1561(12)b 1.1561(12)b _r(CS) 1.5651(9)b 1.5651(9)b -rc 3.8412(6) _ 3.821(10)dRco.m. 3.1597(3) - _C0 _109.81(8) -r(Ar-c.m.of OCS) 3.698e 3.651f _r(Ar-C) 3.581e 3.579f _r(Ar-0) 3.591e 3.652f -6_(Ar-c.m.of OCS-C) 73.2e 73.6f _a Bond lengths in A; bond angles in degrees (°).b Fit to Bo values of various isotopomers of OCSreported in ref.19.C Fit parameters for Ar2-0CS as described in the text;uncertainties are one standard deviation in units ofthe last quoted digits.d Ref. 7.e Structural parameters calculated from Ro.m and 0.f Fitted to Ao, Bo and Co values of various isotopomersof Ar-OCS (except 170) in ref.14.Table 7.7Structural parameters of Ar2-0O2, Ar-0O2 and Ar2Parametera Ar2—002 Ar-CO2 Ar2r(CO) 1.1632b 1.1632b _rc 3.843 _ 3.821(10)d2.935Rril. _ -r(Ar-C) 3.509 3.5048(1)e _a Bond lengths in A.b Ref.20.C R^connects the center of mass of the Ar 2 subunitc.m.and the C atom of CO2; r is Ar-Ar distance.d Ref.7.e234Ref 15.235Table 7.8 The harmonic force field of Ar2-0CSStructural parameters:r(Ar(1)-Ar(2))=r=3.841 Ar (Ar (1) -C)=ri=r (Ar (2) -C)=r2=3.581r (CO) =r3=1.1561r (CS) =r4=1.5651 A(Ar (1) -C-0) =a1=6_ (Ar (2) -C-0)=131=81.2°(Ar (1) -C-S) =a2=6_ (Ar (2) -C-S) =B2=98.8 °Symmetry coordinates:: Si=Ar3S2=Ar4S3=iri-plane OCS linear bendS4= (1/2)1/2 (Ari-I-Ar2)S5=Ar56= (1/2) (Acri-Aa2+A1i-AB2)A2: S7=out-of-plane OCS linear bendS8= (1/2)1/2 (Ar1-Ar2)S9= (1/2) (Acti-Aa2-A131+AB2)Harmonic forcef"/mdynfv/mdynf22/mdynn/mdyn A rad-2f"/mdyn A-155/mdynconstants:16.14a1.040a7.443a0•6513a0.01901(15)0.008422(20)236Table 7.8 (continued)f46/mdyn rad-1 -0.00110bf66/mdyn A rad-2 0.01127(57)ffl/mdyn A rad-2 0.6513afas/mdyn V 0.01933(17)fn/mdyn rad-1 _0.00110bfn/mdyn A rad-2 0.01238(15)Predicted van der Waals vibrational frequencies 1cm-1: Ar2-0CS^Ar2-0C34S^Ar2-180CSV4 43.2 43.1 42.4V5 28.5 28.1 27.9v6 23.5 23.3 23.4V 8 34.5 34.5 34.1V 9 19.3 19.1 18.9a Constrained at the values of ref. 24.b Constrained. See discussion in text.Table 7.9 Comparison of observed centrifugal distortion constants (in kHz)with those obtained from the harmonic force field of Ar2-0CSAr2-0CS^Ar2 -0C34S^Ar2 -180CS^parameter Valuea Obs.-^Valuea^Obs.-^Valuea^Obs.-^Calc. Calc. Calc.wwAj^9.8113(14)^0.003^9.5797(16) -0.007^9.2427(21)^0.001^-...]Ax^-18.1894(66) -0.015 -17.7630(75) -0.031 -16.917(11) -0.036AK^9.8837(56) -0.040^9.6613(82) -0.015^9.252(42)^0.0926^1.7345(08) -0.001^1.7715(11) 0.001^1.6470(16) -0.003J-15.310(10)^0.049 -13.219(11) -0.118 -14.588(13)^0.034s Ka The values are those of Table 7.4.A •l•A •2 'B •l•B •2 '238Table 7.10 The harmonic force field of Ar2-0O2aStructural parameters:r(Ar(1)-Ar(2))=r=3.843 Ar(Ar(1)-C)=r1=r(Ar(2)-C)=r2=3.509 Ar(C-0(1))=r3=r(C-0(2))=r4=1.1632 AA.(Ar(1)-C-0(1))=a1=6_(Ar(2)-C-0(1))=B1=90.0°L(Ar(1)-C-0(2))=a2=L(Ar(2)-C-0(2))=B2=90.0°Symmetry coordinates:S =(1/2)1/2 (Ar 3+Ar4 )1S2=in-plane CO2 linear bendS3=(1/2)1/2 (Ar1+Ar2)S4=ArS5= (1/2) (Act1-Aa2-AB1+AB2)S6= (1/2) 1/2 (Ar3-Ar4)S7= (1/2) (Acr1-Aa2+A131-A132)S8=out-of-plane CO2 linear bendS9=(1/2) (Ar 1 -Ar2 )Harmonic force constants:f 11^(mdyn A-1)f22 (mdyn A rad-2)16.87b0.77bf33 (mdyn A) 0.01726(7)f44^(mdyn A-1) 0.00816(2)f55^(mdyn A rad-2) 0.00985(11)f66^(mdyn A-1) 14.2bfn (mdyn A rad-2) 0.01cf88 (mdyn A rad-2) 0.77bf99^(mdyn A-1) 0.01619(3)239Table 7.10 (continued)Predicted van der Waals vibrationalfrequencies /cm-1Ar2—002V 3 42.3v 4 23.5V 5 22.6V 7 35.9v 9 32.7Comparison of Observed and Calculatedcentrifugal distortion constants (kHz): Observed CalculatedAj 18.9904(43) 19.001klic -40.019(20) -40.002AK 23.378(23) 23.3886j 0.2672(20) 0.2666 K -34.538(21) -34.531a The number of digits quoted in thistable do not reflect the uncertaintiesof the values, see text.b Constrained at the values of ref. 20.C See discussion in the text.Table 7.11 Comparison of structures and force constants derived for the Arand Ar2 complexes of HX(X=F,C1,CN) and OCS.Complex Structure^bond length(A)Ar-c.m.HX/OCS^Ar-Arforce constanta(x10-5 dyn A-1)f(Rc.m)^f(Ar-Ar)Vibrationalc^ReferenceFreguency(cm-1)w(Rc.m)^w(Ar-Ar)Ar-HF linear^3.510 - 1.47 - 43.3 - 9Ar-DF linear^3.461 - 1.79 - 47.0 - 9Ar-H35C1 linear^3.980 - 1.17 - 32.4 - 10Ar-D35C1 linear^3.967 - 1.34 - 34.4 - 10Ar-HCN linear^4.343 - 0.099 - 10.2 - 11,12Ar-OCS planar,T-shaped 3.651 - 2•22b - 41.4 - 14Ar-OCO planar,T-shaped 3. 306 - 1.74 - 37.5 - 15Ar2 linear^- 3.821 _ 0.78 - 25.69 7Table 7.11 (continued)Complex^Structure^bond length(A) force constanta Vibrationalc ReferenceAr-c.m.^(x10-5 dyn A-1)^Frequency(cm-1)HX/OCS Ar-Ar f(Rem) f(Ar-Ar) o(Rc.m.) o(Ar-Ar)Ar2-HF^C2, T-shaped^3.541^3.825Ar2-DF^C2, T-shaped^3.510^3.819Ar2-H35C1 C2Vi T-shaped^3.988^3.832Ar2-D35C1 C2, T-shaped^3.970^3.828Ar2-HCN C2, T-shaped^4.16^3.850Ar2-0CSCs,Tetrahedral 3.698^3.841Ar2-0C0C2v, Tetrahedral 3.509^3.843^1.86^0.66^55.3^22.31.97^0.68^55.3^23.01.66^0.68^44.6^21.8- 45.1^22.1- -^-0.84^43.2^23.5^1.7 b^0.82^42.3^23.5112,334This workThis worka f(Rc.m) is the stretch force constant between Ar and HX c.m. in the case of Ar-HX,and it is the symmetric stretch constant of r1(Ar(1)-HX c.m.) and r2(Ar(2)-HX c.m.)in the case of Ar2-HX. f(Ar-Ar) is the stretch force constant between the two Ar atoms.b f(Rc.m) is the stretch force constant between the Ar and C atoms in the case of Ar-OCS;and it is the symmetric stretch constant of Ar-C-Ar in the case of Ar2-0CS.c The predicted van der Waals vibrational frequencies corresponding to the vander Waals modes in footnotes a and b.242AppendixIn addition to the work on this thesis, the authorhas also been involved in other investigations which haveresulted in the following publications:1. Yunjie Xu, Wolfgang Jager, and M.C.L.Gerry, "Therotational spectrum of the isotopically substitutedvan der Waals complex Ar-OCS, obtained using a pulsedbeam microwave Fourier transform spectrometer",J.Mol.Spectrosc. 151, 206-216 (1992).2. Yunjie Xu, M.C.L.Gerry, D.L.Joo and D.J.Clouthier,"The microwave spectrum, spin-rotation couplingconstants, and structure of thiocarbonyl fluoride,SCF2, observed with a cavity microwave Fouriertransform spectrometer", J.Chem.Phys. 97, 3931-3939(1992).3. Wolfgang Jager, Yunjie Xu and M.C.L.Gerry, "Amicrowave spectroscopic investigation of the weaklybound dimer DC-CV', J.Phys.Chem. 97, 3685 (1993).4. Yunjie Xu, Wolfgang Jager, M.C.L.Gerry andIlona Merke, "The rotational spectrum ofbromoacetonitrile, measured with Stark modulated andmicrowave Fourier transform spectrometers",J.Mol.Spectrosc. 160, 258 (1993).5. Wolfgang Jager, Yunjie Xu, Nils Heineking, andM.C.L.Gerry, "The microwave rotational spectrum of thevan der Waals complex Kr-by', J.Chem.Phys., 1993(in press).


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