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In all cases we have filmed the best available copy. University Microfilms International 300 N 2EEBRD , ANN ARBOR Ml 18106 8203501 KEENAN, MICHAEL ROBERT STRUCTURE AND DYNAMICS OF VAN DER WAALS MOLECULES FROM MICROWAVE SPECTROSCOPY University of Illinois at Urbana-Chatnpaign University Microfilms I n t e r n £ t t J O n a I 300 N Zeeb Road. Ann Arbor, MI 48106 PH.D. 1981 STRUCTURE AND DYNAMICS OF VAN DER WAALS MOLECULES FROM MICROWAVE SPECTROSCOPY BY MICHAEL ROBERT KEENAN B.S., State University of New York, 1976 THESIS Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemistry in the Graduate College of the University of Illinois at Urbana-Champaign, 1981 Urbana, Illinois UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN THE GRADUATE COLLEGE JUNE 1981 W E HEREBY RECOMMEND T H A T T H E THESIS BY MICHAEL ROBERT KEENAN ENTITLED STRUCTURE AND DYNAMICS OF VAN DER WAALS MOLECULES FROM MICROWAVE SPECTROSCOPY BE ACCEPTED IN PARTIAL F U L F I L L M E N T OF T H E REOUIREMENTS FOR T H E DEGREE O F DOCTOR OF PHILOSOPHY MJL2_Q^. Director of Thesis Research Head of DepaY Department Committee on Final Examination! 2J^t^ '/>vrX^^ t Required for doctor's degree but not for master's Chairman ACKNOWLEDGEMENTS I greatly appreciate having had the opportunity to work with Dr. W.H. Flygare. He was a tremendous source of ideas and insight and his enthusiasm was inspiring. Without him, the work described here would have been impossible. I would like to thank Terry Balle for building the microwave spectrometer, and Ed Campbell and Bill Buxton for keeping it in top working condition. In addition, assistance from all of the members of the research group is appreciated. The support of the University of Illinois through teaching and research assistantships, the E.I. Dupont Co. for a fellowship and the National Science Foundation for general support of the research is gratefully acknowledged. Most of all, I wish to thank my wife, Ginny, for her support and encouragement through the years and for her help in preparing this thesis. iv TABLE OF CONTENTS CHAPTER I. Page INTRODUCTION A. B. II. D. E. F. 10 11 21 35 52 61 Introduction Hindered Rotor Model Results and Discussion 70 71 72 XENON NUCLEAR QUADRUPOLE COUPLING IN XENON HYDROGEN CHLORIDE A. B. C. V. Introduction General Analytical Methods Krypton Hydrogen Chloride and Xenon Hydrogen Chloride Argon Hydrogen Bromide and Krypton Hydrogen Bromide Argon Deuterium Fluoride Summary of Results and Conclusion A HINDERED ROTOR MODEL FOR ESTIMATING ANGULAR EXPECTATION VALUES IN THE RARE GAS-HYDROGEN HALIDES A. B. C. IV. 1 4 STRUCTURE AND DYNAMICS OF RARE GAS-HYDROGEN HALIDE VAN DER WAALS MOLECULES A. B. C. III. Background and Review Experimental Method Introduction Spectra and Results Analysis of the Xenon Nuclear Quadrupole Coupling Constant 82 83 89 ARGON CYANOGEN CHLORIDE A. B. C. D. E. Introduction Spectra and Results Molecular Structure Intramolecular Force Field Nuclear Quadrupole Coupling 97 99 103 115 125 v TABLE OF CONTENTS CHAPTER APPENDIX I . Page THE ROTATIONAL CONSTANTS FOR AN ATOM-LINEAR MOLECULE COMPLEX 128 APPENDIX I I . RELATION OF MONOMER TENSOR PROPERTIES TO ATOM-LINEAR MOLECULE COMPLEXES 135 REFERENCES 138 VITA 143 1 CHAPTER I INTRODUCTION A. Background and Review Intermolecular forces have been under intensive invesitgation for several decades. These forces play a major role in the understanding of the structure and properties of condensed phases as well as the deviations of gases from ideal behavior. On the molecular scale, intermolecular forces can be described in terms of the intermolecular potential energy surface. Numerous theoretical and experimental techniques have been used m attempts to obtain potential surfaces 2-4 and many reviews are available. Theoretical understanding of the long range electrostatic, induction and dispersion exchange forces is well advanced. and short range However, the detailed shape of the potential near the potential minimum is currently difficult to obtain from first principles. From an experimental standpoint, nearly all phenomena of chemical interest bear relation to the potential energy surface. Bulk phase phenomena such as virial and transport properties are intimately related to the intermolecular potential but are complicated in that averaging over a large number of interactions may have to be performed. Individual intermolecular interactions are probed directly in various molecular beam scattering experiments (elastic, 3 6 7 inelastic and reactive) ' and rotational line shape analyses. All of 2 the above methods suffer, however, from not being equally sensitive to all parts of the potential. Indeed, the same experiment may probe different parts of the potential as an experimental condition such as temperature or molecular beam energy is changed. Perhaps the best potential surface information results from spectroscopic studies, in the gas phase, of bound states of molecular complexes or van der Waals molecules. Van der Waals molecules are weakly bound molecular complexes which may be characterized as having binding energies on the order of kT at room temperature, where k is the Boltzmann constant and T is the temperature. From simple equilibrium considerations, production of van der Waals molecules is favored by high pressures and/or low temperature. Infrared spectra were originally observed in highly compressed rare gaso hydrogen h a l i d e mixtures and were l a t e r i n t e r p r e t e d i n terms of van der 9 Waals molecules. More r e c e n t l y , spectroscopy has been performed on 10-12 supersonic gas expansions which takes advantage of the low i n t e r n a l temperature generated in these jets. Spectroscopy of van der Waals molecules has included a l l frequency ranges and the l i t e r a t u r e has been 13-15 r e c e n t l y reviewed. Studies of the r o t a t i o n a l s p e c t r a of van der Waals molecules present the g r e a t e s t i n t e r e s t h e r e . Klemperer and coworkers f i r s t found micro16 wave and radiofrequency s p e c t r a for ArHCl and have since s t u d i e d the v i b r a t i o n a l ground s t a t e of s e v e r a l complexes by employing the molecular 10,17 beam e l e c t r i c resonance t e c h n i q u e . Low r e s o l u t i o n microwave s p e c t r a including v i b r a t i o n a l s a t e l l i t e s have been observed f o r hydrogen 3 bonded species in relatively high pressure s t a t i c gas mixtures. With the recent development, in our laboratory, of a method for obtaining 12 high resolution rotational spectra of weakly bound molecular complexes, the body of information concerning van der Waals molecules has considerably increased. Rotational spectra yield a wealth of information about van der Waals molecules and their potential surfaces in the region of the potent i a l minimum. The extraction of this information for several molecular systems i s the topic of this d i s s e r t a t i o n . A rotational spectrum r e s u l t s from the absorption or emission of radiation at frequencies corresponding to changes in the s t a t e of molecular rotational angular momentum. Since angular momentum i s proportional to the molecular moment of i n e r t i a , which in turn i s a function of the atomic coordinates, analysis of the rotational spectrum serves to locate the region of the potential energy minimum. Hyperfine interactions, which appear as perturbations in the pure rotational spectrum, give further structural and orientational information directly related to the anisotropy of the potential surface. Since van der Waals molecules are not rigid r o t o r s , centrifugal d i s t o r tions of spectroscopic properties yield information about the shape of the potential near i t s minimum and if coupled with assumed forms for the potential, properties such as the binding energy can be estimated. Finally, certain spectroscopic features such as rare gas nuclear quadrupole coupling result solely from formation of the weak van der Waals bond. Analyses of these features lend a direct insight into the basic nature of the van der Waals binding interaction. 4 B. Experimental Method The spectroscopic method employed in the studies presented here involves pulsed Fourier transform microwave spectroscopy carried out in a Fabry-Perot cavity with a synchronized pulsed supersonic nozzle as the molecular source. The operational theory and details of the apparatus 12,19 have been given before. spectrometer. Figure 1-1 shoxre a block diagram of the Briefly, microwave power at frequency v is supplied to the Fabry-Perot cavity by the master oscillator (MO) which is phase locked to a frequency standard. The local oscillator (LO), which is used in the superheterodyne detection, is phase locked to the MO at v-30 MHz. A gas mixture used to generate the molecular complexes is pulsed into the cavity and, after a suitable delay, is polarized by a microwave pulse formed by opening the PIN diode 1 switch for 3-7 us. The micro- wave pulse dies away with the cavity relaxation time after which PIN diode 2 is opened and the coherent molecular emission at the resonant frequency, v^, is detected in the superheterodyne receiver. The result- ing time domain signal is digitized and subsequent signals averaged. An equal number of signals are taken with the gas absent to subtract out any coherent noise. The final time domain record is Fourier transformed to yield the frequency spectrum. In practice, 256 point time domain records are obtained by digitizing the molecular emission signals at rates of 1-2 MHz. After 256 zeros are added, this record is Fourier transformed to give a 256 point spectrum having 3;9-1.9 kHz/point resolution. Figure 1-2 shows the time domain record of the 1=5/2, F=5/2 to 1=5/2, F=7/2 LO v -30 MHz Frequency Stabilizer <Xf xer 30 MHz Monitor Detector a Pin Diode MO Frequency Standard v Pulsed no Nozzle ffie Pin Tuner .JUL Diode H3--Q Circulator Mixer <S) Block diagram of the microwave spectrometer. A±30MHz I [Diffusion '—' Pump Mixer • Frequency Stabilizer Figure I-l. Detector iMixer $ A t T Display Display 6 hyperfine component of the O Q Q - 1,, transition in ArClCN and Figure 1-3 shows the corresponding frequency spectrum. The spectral lines show a characteristic doubling which is a Doppler effect involving the natures of the gas expansion and the standing wave electric field in the cavity. A detailed theory explaining this effect is available in the literature. 2 As far as spectroscopic information is concerned, the true molecular resonance frequency lies at the midpoint of the Doppler pairs. The spectral lines have half widths at half height of about 5 kHz and the frequency uncertainty is estimated to be about 2 kHz. The key experimental feature that permits the study of van der Waals molecules is the use of a pulsed supersonic nozzle expansion as the molecular source. The adiabatic expansion rapidly cools the internal degrees of freedom and the large stagnation pressures available give the conditions of low temperature and high pressure needed to form significant quantities of molecular complexes. In addition, since the molecules are observed in the free molecular flow region of the beam, collisional line broadening effects are minimal. Dimer formation proceeds by 3 body collisions which are proportional to p^D where p is the stagnation pressure and D is the nozzle diameter. Use of a pulsed nozzle is advan- tageous since it permits a greater p and D than a continuous nozzle for given pumping speeds. The nozzle used in the present studies is a t h m plate flat orifice, approximately 0.5-1 mm in diameter, bolted onto a solenoid valve. For complexes involving rare gases, the gas mixtures consisted of 1-4% of the dipolar specie seeded in the rare gas and were held at room temperature and at 0.5-2.5 atra pressure. The estimated TIME (/JLS) Figure 1-2. Time domain record of the 1=5/2, F=5/2 -+ I'=5/2, F*=7/2 hyperfine component of the OQQ -*- 1 ^ transition in ArClCN. 7403.18 7403.10 7403.02 MHz Figure 1-3. The frequency spectrum corresponding to the time domain in Figure 1-2. The resolution is 3.9 kHz per point. <» 9 21 rotational temperature of the expanding gas is 1-10 K. This low tem- perature greatly facilitates the observation of the rotational spectra for low lying rotational levels. In general, the signal strength is proportional to the number density difference, Ap(J,K)/p0, between the J and K levels. Table I-l compares Ap/pQ for several transitions in KrHCl at temperatures of 0.5 K and 300 K. Table I-l. For J=0-M=1 and J=3"M=4 Comparison of the number density differences between rotational levels of KrHCl at temperatures of 0.5 K and 300 K. 0.5 K 300 K Ap(0,l)/pQ 2.3xl0~2 7.4xl0"8 Ap(3,4)/pQ 1.2xl0_1 2.1xl0~6 Ap(9,10)/p 5.9xl0"5 1.4xl0-5 there are large gains in Ap/p over a 300 K gas. J=9, however, there is a fall off in gain. For J transitions over 10 CHAPTER II STRUCTURE AND DYNAMICS OF RARE GAS-HYDROGEN HALIDE VAN DER WAALS MOLECULES A. Introduction The simplest, strongly anisotropic, van der Waals interaction is represented by the class of van der Waals molecules that involve the binding between a rare gas atom (X) and a hydrogen halide (HY). As such, the X-HY complexes, in general, and ArHCl, in particular, have been intensively investigated. Gas phase spectroscopic studies of X-HY 8 have ranged from the infrared radio frequencies. ' ' 22 and far infrared to the microwave and X-HY has several characteristics which permit a great deal of information to be gained about it from relatively little data. X can be treated, for the most part, as a structureless particle probing HY in the region of the potential minimum. X-HY can also be treated, in certain situations, as a pseudodiatomic molecule, since the mass of H is small compared to the mass of the halogen. only Finally, 3 modes of vibration are added to the atom-molecule system upon complexation. These are easily visualized in terms of bending and stretching of the van der Waals bond. Due to this simplicity, there is relatively little ambiguity in interpreting the spectroscopic findings in terms of the structure and potential of X-HY". The studies of several X-HY complexes presented here serve a twofold 11 purpose. The primary aim is to increase the body of knowledge concerning van der Waals molecules. In particular, the X-HY interaction is studied over a series of complexes to see how the interaction may correlate with the properties of the X and HY monomers. The second goal is to investigate spectroscopic features peculiar to particular complexes such as the centrifugal distortion of the Br nuclear quadrupole coupling constant in the HBr complexes and the HF nuclear spin-spin interaction in ArHF. These interactions serve to give more insight into the intermolecular potential and to test some assumptions made in the spectral and structural analyses. B. General Analytical Methods Due to the small mass of H as compared to the rare gas and halogen atoms, the pure rotational energies of X-HY can be described by the symmetric top Hamiltonian for all orientations of HY with respect to X. The extreme cooling caused by the nozzle expansion places essentially all molecules in the ground vibrational and K=0 rotational states, thus, the rotational Hamiltonian, H R , is given by % 33 = B 0 J 2 - DjJ4 , (II-l) where ~BQ = (B0+C0)/2 is the rotational constant, Dj is the centrifugal distortion constant and J is the rotational angular momentum. In addition, nuclei with spin angular momenta, I, may couple with the rotational angular momentum, J, to form a total angular momentum, F. This latter coupling appears as a perturbation on the pure rotational state and results in the observed rotational spectrum hyperfine structure. Three 12 types of coupling have been observed in the X-HY complexes, and the hyperfine Hamiltonian is given by H HF=HQ+HSR+HSS' (II "2> H Q describes the nuclear quadrupole coupling for nuclei have 1 ^ 1 and involves the interaction between a nuclear quadrupole moment and the electric field gradient at that nucleus. Hg^ is the spin-rotation inter- action which couples a nuclear magnetic moment to the rotational molecular magnetic moment and H g s , the nuclear spin-spm coupling, is the classical interaction between two magnetic dipoles. For all cases considered here, the rare gas nuclei do not couple and a first order perturbation treatment for the resulting two spin system gives H = i x1 [(2h'&2 Q i=l a + hij->i) -4*2) 2I1(2IjL-l)(2J-l)(2J+3) (H-3) H SR " c l<&.-# H« SS = ~ (2J-l)(2J+3) &K.'V<h'$ + c 2<fc'# + 3( l2'V(lvJ) - 2(l!^ 2 )^] where x 1 , c and d a are the appropriate coupling constants. Matrix elements of the coupling operators in Eq. (II-3) are readily computed in the coupled basis given by Aj -, 'tl = F ol (H-4) J, + I, = F 13 where I refers to the nucleus that is the most strongly coupled. The matrix elements are composed from 35—37 <JW2*V£2| " i W ' V = J(J+1)6 (j^F^FM^IJI^F'Mp) = I±(I1+l)«PiP,6Fpl (jI^^FMpl^.JlJI^^F'Mp) FlFi6FF' = [F 1 (F 1 +1)-J(J+1)-I 1 (I 1 + 1)J /26 F F,5FF, (JI 1 F 1 I 2 FM F |I 2 .J|JI 1 F'I 2 F'M F ) = (-l) 1 + I l + I 2 + J + 2 F i + F \, x[(2F1+l)(2F[+l)I2(I2+l)(2I2+l)J(J+l)(2J+l)] 1 F; Z 1 Fl n 2 A «*• i 2 i 2 FJ ( J J 1^ <JI1F1I2FMF|l1.I2|jI1FiI2F'MF> = 1+I +I +J+F +F'+F ( - Dn ^0 l ^ n ^ l x [(2F 1 +1) ( 2 F ' + 1 ) I 2 ( I 2 + 1 ) (212+1)1^^(^+1) (21^+1)] *2 \ Fi l] (h F[ i) J i2 i 2 F] (I X I L 6 FF" jj where 6 . is the Kronecker delta and the quantities in braces are 6-j symbols. The Hamiltonian matrices in Eqs. (II-2) and (II-3) are then readily constructed by matrix multiplication and addition and diagonalized 14 to yield the energy levels. In the special (and most common) case that only a single nucleus couples with J, energy levels are given by H = IL, + H„ F is diagonal and the 33 -E(J,I,F)=B o J(J+l)-D J J 2 (J+l) 2 - x a (3/4)G(G+l)-1(1+1)J(J+1) 21(21-1)(2J-1)(2J+3) 2 + —a. gxlO -3 + cG/2 where F = I + J , G = 2 I - J = (II-6) F(F+1)-I(I+1)-J(J+1), and g is a second order quadrupole energy correction taken from Appendix II in Townes and Schawlow. 33 To obtain the spectroscopic constants for X-HY, the experimental spectral frequencies were fit to B , D T and the appropriate coupling constants by a nonlinear least squares procedure involving diagonalization of H m spin case. the two spin case or Eq. (II-6) in the single In all cases, the Hamiltonian in Eq. (II-2) was sufficient to fit the data within the experimental uncertainty. The molecular constants of X-HY derived from the spectral data reflect vibrationally averaged rather than equilibrium properties of the complex. Considering the weak nature of the van der Waals bond, the vibrational amplitudes are expected to be large. As a starting point for determining the structure of X-HY, we assume that the complex is held rigidly fixed in its vibrationally averaged configuration and that all properties of free HY are preserved unchanged upon complexation. particular, the H-Y distance is fixed at the free HY bond length. In Under 15 these assumptions, the structure of X-HY can be described in terms of two parameters, a length and an angle. Figure II-l dinate system used in the X-HY structure analysis. shows the coor- The structural para- meters have been chosen, for convenience, to be the rare gas-halogen distance, R , and the angle, 9, between HY and R . The angle, a, between HY and the a-inertial axis is not a free parameter but is fixed once R and 8 are known. An alternative parameterization in which the length is taken between X and the HY center of mass is discussed in Appendix I. It suffers, however, from the fact that given a fixed atomic arrangement, the structural parameters vary with isotopic substitution. In any case, the two parameterizations are easily interconverted by simple geometry. The rotational constant of X-HY is very insensitive to the location of H, and thus 9, since H is much lighter than X or Y (this will be discussed at length in the next section). Fortunately, the hyperfine constants can be used to derive angular information. The measured nuclear quadrupole and nuclear spin-spin coupling constants, x a and d„ , c * EL * respectively, are simply those components of the respective tensors along the a-axis of the complex. Under the constancy of HY properties assumption discussed above, the coupling constant tensors for X-HY can be obtained by rotating the free HY tensors to the principal axes of the complex according to (see Appendix II ) P ^ ( c o s ^ a -1)P 0 , (II-7) where P represents either X„ or d 'and P. is the corresponding property of free HY. The angle a is shown in Figure (II-l). In practice, a and ® a-axis R Figure II-l. Coordinates used to describe the structure of rare gas (X)-hydrogen halide (HY) van der Waals molecules. a* 17 the structural angle, 9, differ by only a couple tenths of a degree and can often be used interchangeably. Once a has been determined by using Eq. (II-7), the evaluation of RQ is straightforward. An iterative technique is used whereby the rotational constant B = (B+C)/2 is calcu38 lated as a function of R and 9 until B 0 and a are reproduced. It is noted that the sign of cosa cannot be determined from Eq. (II-7). The question of whether a is acute or obtuse can be settled, however, by resort to isotopic substitution in the HY subunit since, as a first approximation, the structure is invariant to isotopic substitution. In all X-HY complexes, a has been found to be acute. The internal dynamics of van der Waals molecules which undergo large amplitude vibrations are not totally understood. Information about the intermolcular potential may be obtained, however, by considering the centrifugal distortion constant, D,, and the structures of X-HY determined above, along with simple assumptions about the form of the potential. In these comlexes, D, arises from stretching of the van der Waals bond, taken to be along the X-Y axis, as well as small effects due to the change in the bond angle, 9, with rotation. Neglecting the effects due to changes in 9, the van der Waals stretching motion can be treated by considering X-HY to be a pseudodiatomic molecule. In this model, the radial interaction between X and HY is described by an effective radial potential averaged over the bending mode. Dj is related to the van der Waals stretching frequency, v g , and force constant, k , by the usual diatomic expression 39 18 4B_ (H-8) where (H-9) and the reduced mass, u , is given by (11-10) X HY By assuming a functional form for the angle averaged effective radial potential, k and the molecular structure can be used to estimate the potential parameters. The choice, made here, is to describe the radial interaction in terms of the Lennard-Jones 6/12 potential given by V(R) = e Re\ 1 2 _2{Re (11-11) where e is the well depth and R e is the equilibrium mternuclear separation. Other choices could be made for the form of V(R). However, it has been found that the final results do not depend very much on the specific form chosen. Expanding V(R) in a Taylor series about R=Rg gives 36 (R-RJ 2 - 252 2 R' R. e "e V(R) = -e + ^ (R-P)3 + . (11-12) The harmonic force constant is related to the quadratic term of this potential by 19 k s = 36e/Re2 \ (H-13) and, through second order, the rotation-vibration interaction constant, 4 uby 39,40 <xe, Ais given a e = 36(B e 2 /v s ) . (11-14) In the ground vibrational state, a e relates the vibrationally averaged rotational constant,BQ, to the equilibrium rotational constant, B e , according to B o = Be " K = B e " 18(B e 2 /v s ). (H-15) Taking B0 2 ' Eq. (11-15) is solved by the quadratic formula to obtain B (H-16) and the equilibrium separation is calculated from Rg=| 8*2y B e \ * S_] # (II_17) Using this value for R e and the value of k s derived from Dj, Eq. (11-13) Is solved for the Lennard-Jones estimate of the effective radial well depth, e. The treatment of the van der Waals bending motion poses more severe problems. In contrast to the stretching motion which Is simply described in terras of the stretching between two heavy particles, the bending modes involve complicated, large amplitude motion by a light atom in two dimen- 20 sions. It is still useful, however, to consider the bending in terms of a harmonic oscillator model for at least qualitative and comparative purposes. By analogy with ArHCl ' and from evidence to be presented later, it appears that the equilibrium geometry of X-HY is linear. The measured bond angle results from averaging over the large amplitude bending vibrations. The two dimensional isotropic harmonic oscillator is the model appropriate to the bending motion of a linear system. The bending poten- tial is given by V(9) = f k ^ 2 (11-18) where the bending force constant, k., is related to the bending frequency, v D , and bending reduced mass, u^, by (11-19) The energy levels of the two dimensional oscillator are given by (E) = <T>+<V) = hvb(n+l) (11-20) where T and V are the kinetic and potential energies, respectively, and n is the vibrational quantum number. From the virial theorem <T) = (V) thus, from Eqs. (11-18) - (11-21) and for n=0 (11-21) 21 In other words, an estimate of the bending potential can be gained simply from the structure of X-HY and a value of u^. ,. Mfc, is taken to be The bending reduced mass, 42 where r is the bond length of HY. The choice of bending angle and re- duced mass is not unique and the choice made here is not the one conventionally made. However, with this choice lVf~ + — I" *" = 1 - ' (11-24) MY; where I is the moment of inertia of free HY and the bending problem becomes couched m the more physically meaningful picture of a hindered internal rotation. C. Krypton Hydrogen Chloride and Xenon Hydrogen Chloride KrHCl was the first molecular complex studied using the microwave spectrometer described in Section I-B. As such, it served to demonstrate the high resolution and sensitivity achievable with this spectroscopic technique. The J=3->J=4 and J=4-*J=5 rotational transitions in the ground vibrational state were readily observed and recorded for the 12 isotopes consisting of all combinations of 82 84 86 35 37 Kr, Kr, Kr, H, D, CI and CI. The measured KrHCl transition frequencies are listed in Table II-l. The spectrometer's high sensitivity is established by noting that the 22 Table II-l. Observed transition frequencies for KrHCl and those calculated with Eq. (II-6) and the constants in Table II-3. Isotope J,F J',F' Observed (MHz) Calculated (MHz) Residue (kHz) 82 35 , KrH CI 3,9/2 3,3/2,5/2 3,7/2,9/2 3,7/2 3,5/2 4,5/2,7/2 4,9/2,11/2 4,9/2 4,7/2 4,9/2 4,5/2,7/2 4,9/2,11/2 4,7/2 4,5/2 5,7/2,9/2 5,11/2,13/2 5,9/2 5,7/2 9665.464 9671.933 9672.779 9675.345 9679.246 12089.008 12089.514 12091.586 12096.326 9665.464 9671.930 9672.780 9675.349 9679.245 12089.011 12089.516 12091.580 12096.327 0 3 -1 -4 1 -3 -2 6 -1 84 3,9/2 3,9/2 3,3/2,5/2 3,7/2,9/2 3,7/2 3,5/2 3,7/2 4,11/2 4,5/2,7/2 4,9/2,11/2 4,9/2 4,7/2 4,9/2 4,7/2 4,5/2,7/2 4,9/2,11/2 4,7/2 4,5/2 4,5/2 5,11/2 5,7/2,9/2 5,11/2,13/2 5,9/2 5,7/2 9596.022 9598.592 9602.486 9603.333 9605.902 9609.801 9613.208 11995.399 12002.213 12002.723 12004.774 12009.524 9596.022 9598.591 9602.485 9603.334 9605.902 9609.795 9613.214 11995.403 12002.212 12002.717 12004.780 12009.523 0 1 1 -1 0 6 -6 -4 1 6 -6 1 3,7/2,9/2 3,7/2 3,5/2 4,11/2 4,5/2,7/2 4,9/2,11/2 4,9/2 4,7/2 4,9/2 4,5/2,7/2 4,9/2,11/2 4,7/2 4,5/2 5,11/2 5,7/2,9/2 5,11/2,13/2 5,9/2 5,7/2 9529.757 9536.223 9537.064 9539.637 9543.534 11912.592 11919.403 11919.903 11921.969 11926.713 9529.756 9536.220 9537.069 9539.638 9543.531 11912.591 11919.401 11919.906 11921.969 11926.713 1 3 -5 -1 3 1 2 -3 0 0 3,3/2,5/2 3,7/2,9/2 4,5/2,7/2 4,9/2,11/2 4,5/2,7/2 4,9/2,11/2 5,7/2,9/2 5,11/2,13/2 9315.690 9316.364 11643.709 11644.112 9315.690 9316.364 11643.710 11644.111 0 0 -1 1 KrH 35 C1 86 35 KrII Kru Cl UL 82 KrH 37 C1 3 9/2 ' 3)3/2j5/2 23 Table I I - l , Continued Isotope J,F J',F» Observed (MHz) Calculated (MHz) Residue (kHz) 8 3,9/2 3,3/2,5/2 3,7/2,9/2 3,7/2 3,5/2 4,11/2 4,5/2,7/2 4,9/2,11/2 4,9/2 4,7/2 4,9/2 4,5/2,7/2 4,9/2,11/2 4,7/2 4,5/2 5,11/2 5,7/2,9/2 5,11/2,13/2 5,9/2 5,7/2 9241.135 9246.243 9246.915 9248.938 9252.017 11551.541 11556.929 11557.322 11558.952 11562.696 9241.137 9246.242 9246.913 9248.941 9252.017 11551.544 11556.923 11557.322 11558.951 11562.699 -2 1 2 -3 0 -3 6 0 1 -3 KrH 3 7 Cl 3,3/2,5/2 3,7/2,9/2 3,7/2 4,5/2,7/2 4,9/2,11/2 4,5/2,7/2 4,9/2,11/2 4,7/2 5,7/2,9/2 5,11/2,13/2 9179.979 9180.650 9182.681 11474.113 11474.514 9179.979 9180.650 9182.681 11474.113 11474.513 0 0 0 0 1 ™. 35 r-i KrD CI 3,3/2,5/2 3,7/2,9/2 3,7/2 4,5/2,7/2 4,9/2,11/2 4,9/2 4,5/2,7/2 4,9/2,11/2 4,7/2 5,7/2,9/2 5,11/2,13/2 5,9/2 9537.198 9538.382 9541.973 11920.916 11921.622 11924.502 9537.197 9538.384 9541.972 11920.915 11921.621 11924.503 1 -2 1 1 1 -1 84 35 KrD CI 3,9/2 3,3/2,5/2 3,7/2,9/2 3,7/2 3,5/2 4,11/2 4,5/2,7/2 4,9/2,11/2 4,9/2 4,7/2 4,9/2 4,5/2,7/2 4,9/2,11/2 4,7/2 4,5/2 5,11/2 5,7/2,9/2 5,11/2,13/2 5,9/2 5,7/2 9458.627 9467.644 9468.825 9472.413 9477.832 11824.482 11833.983 11834.688 11837.569 11844.185 9458.626 9467.641 9468.826 9472.411 9477.838 11824.484 11833.983 11834.688 11837.568 11844.182 1 3 -1 2 -6 -2 0 0 1 3 V.H 3 7 P-. KrH CI 86 82 24 Table I I - l , Continued Isotope J,F J',F» Observed (MHz) 86 35 KrD CI 3,3/2,5/2 3,7/2,9/2 3,7/2 4,11/2 4,5/2,7/2 4,9/2,11/2 4,9/2 4,7/2 4,5/2,7/2 4,9/2,11/2 4,7/2 5,11/2 5,7/2,9/2 5,11/2,13/2 5,9/2 5,7/2 9401.276 9402.457 9406.046 11741.517 11751.038 11751.740 11754.616 11761.239 9401.272 9402.459 9406.048 11741.522 11751.031 11751.737 11754.620 11761.241 4 -2 -2 -5 7 3 -4 -2 82 37 KrD CI 3,3/2,5/2 3,7/2,9/2 4,5/2,7/2 4,9/2,11/2 4,5/2,7/2 4,9/2,11/2 5,7/2,9/2 5,11/2,13/2 9191.338 9192.276 11488.553 11489.106 9191.339 9192.275 11488.551 11489.108 -1 1 2 -2 84 v 37_. KrD CI 3,9/2 3,3/2,5/2 3,7/2,9/2 3,7/2 3,5/2 4,11/2 4,5/2,7/2 4,9/2,11/2 4,9/2 4,7/2 4,9/2 4,5/2,7/2 4,9/2,11/2 4,7/2 4,5/2 5,11/2 5,7/2,9/2 5,11/2,13/2 5,9/2 5,7/2 9114.707 9121.819 9122.756 9125.591 9129.867 11394.174 11401.668 11402.228 11404.497 11409.730 9114.705 9121.821 9122.757 9125.586 9129.872 11394.173 11401.672 11402.228 11404.501 11409.723 2 -2 -1 5 -5 1 -4 0 -4 7 86 37 , KrD CI 3,3/2,5/2 3,7/2,9/2 4,5/2,7/2 4,9/2,11/2 4,5/2,7/2 4,9/2,11/2 5,7/2,9/2 5,11/2,13/2 9055.487 9056.421 11318.772 11319.333 9055.486 9056.422 11318.774 11319.331 1 -1 -2 2 Calculated (MHz) Residue (kHz) 25 AF=-1 hyperfine components were observed in the J=3->-J=4 transition in KrH AF—1 CI. KrH CI is present at 43.0% isotopic abundance and the components comprise only 0.07% of the total transition intensity. on In addition, the spectra of easily assigned. o7 KrH CI (2.8% isotopic abundance) were Table II-l also exhibits the high spectral resolution inherent in this technique. within a couple of kHz. All transition frequencies are good to Subsequent to the KrHCl study, the resolution was further increased by a fuller understanding of the instrumental effects on the line shape and a slightly more sophisticated data handling procedure. 132 35 XeH CI was ini28 tially assigned using the molecular beam electric resonance technique. The rotational spectrum of the single isotope The microwave spectrum of 132 35 XeH CI was re-examined in this lab to obtain a more precise value of its centrifugal distortion constant which was useful in determining the 131 Xe nuclear quadrupole coupling constant in 131 XeHCl (see Chapter IV). In addition, the microwave spectra of 129 35 129 37 129 35 132 35 XeH CI, XeH CI, XeD CI and XeD CI were assigned and the measured frequencies are listed in Table II-2. Using the rotational constants, centrifugal distortion constants and CI nuclear quadrupole coupling constants derived from the spectra, a more complete structure was determined for XeHCl and greater information about the intermolecular potential was obtained by referring to the results of isotopic substitution. The analyses of the spectra, molecular structures and potentials of KrHCl and XeHCl proceeded in an identical fashion for both complexes. 26 Table II-2. Observed transition frequencies for XeHCl and those calculated with Eq. (II-6) and the constants in Table II-3. Isotope 129 v „35 XeH Cl 129 v TT37 , XeH Cl 129 v „35 , XeD Cl 132 XeH 3 5 Cl 132 J 35 XeD JJ Cl J,F J',F* Observed (MHz) Calculated (MHz) Residue (kHz) 0.6 3,3/2,5/2 3,7/2,9/2 4,5/2,7/2 4,9/2,11/2 5,7/2,9/2 5,11/2,13/2 6,9/2,11/2 6,13/2,15/2 4,5/2,7/2 4,9/2,11/2 5,7/2,9/2 5,11/2,13/2 6,9/2,11/2 6,13/2,15/2 7,11/2,13/2 7,15/2,17/2 7951.4371 7952.4462 9939.1278 9939.7248 11926.1776 11926.5781 13912.6138 13912.8994 7951.4365 7952.4464 9939.1273 9939.7246 11926.1798 11926.5776 13912.6137 13912.8988 3,3/2,5/2 3,7/2,9/2 4,5/2,7/2 4,9/2,11/2 5,7/2,9/2 5,11/2,13/2 4,5/2,7/2 4,9/2,11/2 5,7/2,9/2 5,11/2,13/2 6,9/2,11/2 6,13/2,15/2 7621.9843 7622.7796 9527.2647 9527.7326 11431.9749 11432.2877 7621.9861 7622.7807 9527.2615 9527.7315 11431.9756 11432.2886 3,3/2,5/2 3,7/2,9/2 4,5/2,7/2 4,9/2,11/2 5,7/2,9/2 5,11/2,13/2 6,9/2,11/2 6,13/2,15/2 4,5/2,7/2 4,9/2,11/2 5,7/2,9/2 5,11/2,13/2 6,9/2,11/2 6,13/2,15/2 7,11/2,13/2 7,15/2,17/2 7833.4321 7834.7333 9791.8363 9792.6065 11749.5909 11750.1065 13706.7637 13707.1307 7833.4320 7834.7338 9791.8355 9792.6060 11749.5927 11750.1059 13706.7631 13707.1309 4,5/2,7/2 4,9/2,11/2 6,9/2,11/2 6,13,2,15/2 7,11/2,13/2 7,15/2,17/2 5,7/2,9/2 5,11/2,13/2 7,11/2,13/2 7,15/2,17/2 8,13/2,15/2 8,17/2,19/2 9890.3078 9890.9044 13844.2962 13844.5786 15820.3100 15820.5230 9890.3096 9890.9054 13844.2926 13844.5770 15820.3107 15820.5251 -1.8 -1.0 6,9/2,11/2 6,13/2,15/2 7,11/2,13/2 7,15/2,17/2 8,13/2,15/2 8,17/2,19/2 7,11/2,13/2 7,15/2,17/2 8,13/2,15/2 8,17/2,19/2 9,15/2,17/2 9,19/2,21/2 13638.1782 13638.5446 15584.9436 15585.2190 17531.0282 17531.2446 13638.1784 13638.5447 15584.9431 15585.2191 17531.0286 17531.2444 -0.2 -0.1 -0.2 0.5 0.2 -2.2 0.5 0.1 0.6 -1.8 -1.1 3.2 1.1 -0.7 -0.9 0.1 -0.5 0.8 0.5 -1.7 0.6 0.6 -0.2 3.6 1.6 -0.7 -2.1 0.5 -0.1 -0.4 0.2 27 Table II-3. Isotope Spectroscopic Constants for KrHCl and XeHCl. B (MHz) Dj(kHz) X (MHz) 82 KrH 3 5 Cl 1209.3075(6) 7.431(15) -29.268(9) 84 KrH 3 5 Cl 1200.6255(7) 7.389(16) -29.250(7) 86 KrH 3 5 ci 1192.3386(5) 7.273(12) -29.253(6) 82 KrH 3 7 Cl 1164.7454(2) 6.935(5) -23.235(33) 84 KrH 3 7 Cl 1156.0596(5) 6.795(12) -23.106(6) 86 KrH 3 7 Cl 1147.7735(2) 6.691(4) -23.139(8) 82 KrD 3 5 Cl 1192.4708(4) 6.612(8) -40.842(14) 84 KrD 3 5 Cl 1183.7735(5) 6.530(12) -40.805(6) 86 KrD 3 5 Cl 1175.4763(11) 6.489(23) -40.849(14) 82„ „37„, KrD Cl 1149.1999(8) 6.126(19) -32.237(122) 84T, „37„, KrD Cl 1140.5069(8) 6.026(17) -32.211(9) 1132.2113(8) 5.909(18) -32.232(119) XeH 3 5 Cl 994.14484(12) 3.8130(15) -34.761(42) 129 37 ^yXeHJ/Cl 952.92241(38) 3.4927(63) -27.359(94) 129 v „35„, XeD Cl 979.40964(11) 3.4520(14) -44.816(38) 86 KrD 3 7 Cl 129 132 XeH 3 5 Cl 989.26113(34) 3.7762(31) -34.670(20) 132 XeD 3 5 Cl 974.50773(6) 3.4221(4) -44.629(54) U n c e r t a i n t i e s represent the standard deviation of the least squares f i t . 28 Good fits of the observed data were obtained by least squares fitting of the spectra to the single nuclear spin energy levels given in Eq. (II-6). Although deuterium nuclear quadrupole coupling gave rise to some line broadening in the deuterated species, its effects on the energy levels were ignored in this analysis as were the Cl spin-rotation interactions. The spectroscopic constants for all isotopes of KrHCl and XeHCl are listed in Table II-3. The frequencies calculated with Eq. (II-6) and the con- stants in Table II-3 are shown in Tables II-l and II-2 for KrHCl and XeHCl, respectively, and, in general, the fits are very good. The structures of KrHCl and XeHCl were derived from the measured rotational and Cl nuclear quadrupole coupling constants using the method outlined in Section II-A and known properties of HC1. The appropriate properties of free HC1 are listed in Table II-4. Table II-4. Isotope H 3 5 C1 H 35C1 37C1 D^'Cl D a Rotational constants, bond lengths and Cl nuclear quadrupole coupling constants for 4 isotopic species of HC1 B 0 (MHz) a r 312989.297 312519.121 161656.238 161183.122 1.28387 1.28386 1.28124 1.28123 0 (A)b See Ref. 43. Calculated from B . c See Ref. 44; X Q ( 37 C1) = x Q ( 35 Cl)/1.2688. X Q (MHz) C -67.61893 -53.294 -67.39338 -53.115 29 The structures of KrHCl and XeHCl are shown in Table II-5. All angles are acute as will be shown below. The structures of KrHCl and XeHCl show little change upon isotopic subsitution at the rare gas and halogen positions. A dramatic deuterium isotope effect is observed, however, and merits some comment. The deuterated complexes show a much smaller bond angle, 6, and an increased bond length, R , as compared to the hydrogen species. Due to a fractional mass change of near unity, there is a large difference in the manner that the angle is averaged over the van der Waals bending vibration with the result that H and D sample significantly different portions of the potential surface. Qualitatively, the lighter H is, on the average, farther away from the heavy atom axis than D and, consequently, there is less repulsion between the heavy atoms and they are allowed to approach more closely. The deuterium isotope effect in the structures of KrHCl and XeHCl can be used, to some advantage, in testing the assumption that these complexes are linear at equilibrium. For a linear two dimensional iso- tropic harmonic oscillator, Eq. (11-22) predicts that (9 2^ should scale as li^"'2 for a given force constant. Since u^ approximately doubles in going from the H to D isotopes, the deuterated complexes should have angles given by e(XDCl) *fe (j^eOEHCl). (II-25) Using Eq. (11-25), 9 is predicted to be 31.7° for KrDCl and 29.0° for XeDCl. These values compare well with the measured values of 30.5° and 30 Table II-5. Structures of KrHCl and XeHCl where 9 is determined from x a - Isotope R 0( A ) 9(Deg.) 82 KrH 35 Cl 4.1107 37.74 84 KrH 35 Cl 4.1106 37.75 86 35,,, KrH Cl 4.1105 37.75 82 KrH 37 Cl 4.1101 37.63 84 KrH 37 Cl 4.1100 37.73 86 KrH 37 Cl 4.1099 37.70 82 KrD 3 5 C l 4.1253 30.52 84 KrD 35 Cl 4.1251 30.54 86 KrD 35 ci 4.1250 30.51 82 KrD 37 Cl 4.1248 30.50 84 KrD 37 Cl 4.1246 30.52 86 KrD 37 Cl 4.1245 30.50 W5C1 4.2753 34.50 129 XeH 3 7 Cl 4.2747 34.54 129 XeD 3 5 ci 4.2875 27.91 132 XeH 3 5 Cl 4.2752 34.56 132 XeD 3 5 Cl 4.2873 28.03 12 31 27.9 , respectively, considering the level of approximation. It appears that the most likely equilibrium configuration is linear and that the measured angle results solely from vibrational averaging. While the structures of KrHCl and XeHCl have been determined above by resort to an analysis of the Cl nuclear quadrupole coupling, it is possible, in principle, to determine the structures from rotational constants alone if the structure is unchanged upon isotopic substitution. The problem encountered here is that the rotational constants are fairly insensitive to the location of the hydrogen. Since a large amount of isotopic data was available for KrHCl, an attempt was made to derive the structure of KrHCl from its rotational constants alone. In this analysis, it is assumed that the six hydrogen isotopes have identical structures as do the six deuterium isotopes. differ from each other, however. The hydrogen and deuterium isotopes may It is also assumed that the H-Cl dis- tance is rigidly fixed at the bond length of free HC1. To estimate the 45 molecular geometry, a grid search method is employed whereby the rota- tional constants are calculated for various assumed structures and the minimum in f is sought, where f = i ( I oi- ¥ ci ) 2 1=1 ' (II "26) and where Bo0 and Bc are the observed and calculated rotational constants, respectively. Table II-6 lists typical results from this calculation where the minimum of f and the corresponding bond length is given as a function of 9. The absolute minima were found at Ro=4.100A and 9=63.7 o o for KrHCl and at R =4.116A and 9=45.7 for KrDCl. While there are a range 32 Table II-6. Structure of KrHCl as from rotational constants alone. 6 0 ° \c-0.& ^CB^-B^CMHz ) iW-nw rv.r ui o z p B < c-r ol B •) osaz ) ci 0 4.1163 .472 4.1326 .484 10 4.1159 .449 4.1317 .440 20 4.1146 .384 4.1293 .322 30 4.1125 .288 4.1252 .171 40 4.1096 .181 4.1195 .051 50 4.1059 .086 4.1123 .045 60 4.1015 .031 4.1037 .241 70 4.0965 .042 4.0939 .730 80 4.0909 .144 4.0830 1.581 90 4.0849 .355 4.0713 2.837 100 4.0787 .678 4.0593 4.496 110 4.0725 1.106 4.0472 6.507 120 4.0664 1.617 4.0355 8.767 130 4.0608 2.173 4.0247 11.13 140 4.0559 2.728 4.0151 13.42 150 4.0518 3.231 4.0072 15.44 160 4.0487 3.632 4.0013 17.03 170 4.0468 3.891 3.9977 18.05 180 4.0462 3.980 3.9965 18.40 a r = 1.28387A\ H-Cl VCI = 1,28124i ' p 33 of angles that will adequately fit the rotational constants, it is clear that H(D) is located between the heavy atoms. It is also noted that the large deuterium isotope effect is reproduced. The relatively large values of 9 are expected since an increasing angle will compensate for the expected shortening of the Kr-Cl bond as substitution with heavier isotopes occurs. The fact that 9 is acute in XeHCl can also be established by referring to the results of Cl isotopic substitution. Table II-7 shows the 129 Xe-Cl distances, R Q , calculated using both acute and obtuse choices 35 37 for 9. Since the fractional mass change in going from Cl to Cl is very small, the expected isotope effect is small and since R over the van der Waals stretch, R the Cl mass increases. is averaged is expected to decrease slightly as Clearly, as demonstrated in Table II-7, only the acute choice for 9 fulfills both these conditions. 129 Table II-7. The bond lengths of XeHCl for both acute and obtuse choices for 6. 9 is acute Isotope R0(A) 129 4.2753 XeH 3 5 Cl 8 is obtuse R 0 ( 35 C1)-R 0 ( 37 C1)A R Q (R) 4.2162 0.0006 129 X e H 37 c l 4.2747 R o ( 35 Cl)-R 0 ( 37 Cl) (A) -0.0025 4.2187 The harmonic bending and stretching force constants and frequencies for KrHCl and XeHCl are shown in Table II-8. Also shown in Table II-8 are the Lennard-Jones estimates of the effective radial potential well OClCnCTiOOOOICOOOHOCOCTiCTiCTi oo(~.t^r-»oooooo>crvooofricocotnin H r l r l H H H N r H N N W N N N N N tncooN<f<rsr<rn^coo>NN>j>jN COrO<r«a--<rv>-HiHHHrHrHr^f^f^HH vovovovovovDeooooooooooofOcocomm O O O O O O O O O O O O C M C M C M C M C M r^-i~-i^.ooococricoooo\oHa\oo>srHrHtHCMHrHinmvOvDUOvOHOi-ll^vO CMCMCMCMCMCMcMCMCMCMCMCNCOcOcOCOcO vOvf<rHl^rOCMvD-*cMvOi-<OCMCO<rcO ^j-^vrr^^fLnocriOocriOoovo^ocMoo r~.r~r~rNp^r^i^vor^r~>£ii^.vovOvo<tco •<r-*-<r-d-<r<rcocococococoininm<rvt CTiOOHcMCMOOOH^mr-sOOCMCPiO vor-.r^oococoorHHinminr».r^oooH COCOCOCOCOCOHrHrHHHHCOCOCOHH vOvOvOvOvOvOCMCMCNCMCMCNvOvOvOCMCM HHrHHHHcoCOCOcOCOfOHHHCOCO voinminvovOiHi-HOcMcMcoooHmm inmmininmi^.r^r^f->.r^r-»oicrichOO H H r H i - H H i H H H r H H H r H H H H C M C M O O O O O O O O O O O O O O O O O ooooooooooooooooo • J » H - t H H 0 0 N C 0 r l O * * C 0 i H W O mcNicMoooor^r^vococMHOcor^cMP^r~CMCMCMiHHr-lrOCOCOCOCOCOrOcOCO-<r-tf cocofocococococococococorococococo p^or~>.incMcoova\cOi-imfOvovocMr»-<f O N f f l i f l - T - J M N v O t n - 1 H W N v O M ( 0 CO CTir-)COO\i-ICO<rvOC0<fvO00HCMCOr-.00 <fminmvovominmvovovoooooa\coco CMcMCMCMCMCMCMeMcNCMCMCMCNCMcMCMcM H H H H H r - H r - l H H H i - l r H O O U O U o o o o o u u o a o o o m i n i-» m m mininr^i^r^ininmr^rvr^cococococo cocococOcorOcococOcOfOco SSCS3QQ S S B H i S K P Q i a f l Q O i J t J I ) <y U U U ^ u p j j U ^ U U U p<> ^ X X X ^ i l ^ ^ W W W W W W W ! ^ Wov CM en o> CM C M « a - v O C M ^ r ^ O C M < - v O CMO-vOCMCOCMCNrO oocococoeoooeococowooeoriHriHH 35 depths and equilibrium bond lengths. The striking observation, made in examining the results of Table II-8, is that the potential parameters exhibit a large difference between the H and D isotopes of a given complex. The true intermolecular, on the other hand, should be iso- topically invariant within the Born-Oppenheimer approximation. It must be kept In mind that the potential parameters derived above do not describe the true intermolecular interaction but only an averaged effective potential. They will be expected to vary with D substitution due to strong coupling of the radial and angular potentials and the greatly different angle averaging experienced by the two species. The deuterated complexes, which undergo smaller amplitude bending vibrations, are, on the average, sampling nearer to the potential minimum than H, giving the appearance of a stronger bond. In particular, the estimates of the potential well depths made here cannot be expected to give more than an order of magnitude estimate of the interaction energy. They still find use, however, in comparing the interaction energies of similiar complexes. D. Argon Hydrogen Bromide and Krypton Hydrogen Bromide The studies of KrHCl and XeHCl presented in the previous section, along with the prior investigation of ArHCl, permit the rare gas-hydrogen halide Interaction to be examined as a function of the rare gas atom. To explore the interaction as a function of the halogen, studies of the ArHBr and KrHBr van der Waals molecules were undertaken. The 36 rotational spectra of four isotopic species of ArHBr and eight isotopic species of KrHBr were assigned. In broad outline, the information obtained about ArHBr and KrHBr paralleled that derived from the spectra of KrHCl and XeHCl. However, additional small effects such as a centri- fugal distortion of the Br nuclear quadrupole coupling constant and the Br spin-rotation interaction were first observed here. Their effects demonstrate, once again, that small perturbations in the rotational spectra can be studied using the Fabry-Perot Fourier transform spectrometer. Finally, an analysis of the centrifugal distortion of the Br nuclear quadrupole coupling constant yielded additional information about the intermolecular potential. In particular, it is a measure of the coupling between the radial and angular motions. Microwave transitions of ArHBr and KrHBr were measured m quency band covering 8.0 to 12,4 GHz. the fre- The observed transition frequencies for ArHBr and KrHBr are listed in Tables II-9 and 11-10, respectively. Due to the high J transitions measured here (J=3+J=4 and J=4-*-J=5 for ArHBr and J=5-»J=6 up to J=8^J=9 for KrHBr), no resolvable hyperfine structure attributable to deuterium nuclear quadrupole coupling was observed. However, these high J transitions provided a unique opportunity to study the centrifugal distortion of the Br nuclear quadrupole coupling constant, an effect which was found to increase with J(J+1). Poor signal- to-noise resulting from isotopic dilution caused the second order and centrifugal distortion effects of the Br nuclear quadrupole coupling to be incompletely resolved for 82 86 KrHBr and KrHBr. In these cases, center frequencies of the second order doublets are reported. The estimated Table II-9. Observed and Calculated Transition Frequencies for ArHBr. Isotope J,F -*• J',F' Observed (MHz) Calculated3 (MHz) Residue (kHz) Calculated'5 (MHz) Residue (kHz) 3,5/2 3,7/2 3,9/2 3,7/2 3,5/2 3,3/2 3,9/2 4,7/2 4,9/2 4,11/2 4,9/2 4,5/2 4,7/2 4,11/2 4,5/2 4,7/2 4,11/2 4,9/2 4,7/2 4,5/2 4,9/2 5,7/2 5,9/2 5,13/2 5,11/2 5,7/2 5,9/2 5,11/2 8810.4083 8833.8364 8848.8732 8848.8983 8853.8595 8853.8693 8892.2805 11019.0770 11047.4871 11059.5804 11059.6210 11062.5235 11062.5513 11103.0270 8810.4116 8833.8383 8848.8704 8848.8965 8853.8558 8853.8691 8892.2837 11019.0791 11047.4903 11059.5778 11059.6175 11062.5234 11062.5485 11103.0308 -3.3 -1.9 2.8 1.8 3.7 0.2 -3.2 -2.1 -3.2 2.6 3.5 0.1 2.8 -3.8 8810.4095 8833.8362 8848.8738 8848.8991 8853.8572 8853.8690 8892.2807 11019.0761 11047.4874 11059.5816 11059.6206 11062.5237 11062.5504 11103.0275 -1.2 0.2 -0.6 -0.8 2.3 0.3 -0.2 0.9 -0.3 -1.2 0.4 -0.2 0.9 -0.5 3,5/2 3,7 3,9/2 3,7/2 3,5/2 3,3/2 3,9/2 4,7/2 4,9/2 4,11/2 4,9/2 4,5/2 4,7/2 4,11/2 4,5/2 4,7/2 4,11/2 4,9/2 4,7/2 4,5/2 4,9/2 5,7/2 5,9/2 5,13/2 5,11/2 5,7/2 5,9/2 5,11/2 8744.2856 8763.8321 8776.4224 8776.4452 8780.5969 8780.6036 8812.7075 10935.0812 10958.8068 10968.9300 10968.9656 10971.3934 10971.4186 11005.2501 8744.2856 8763.8344 8776.4203 8776.4432 8780.5968 8780.6017 8812.7098 10935.0827 10958.8084 10968.9300 10968.9626 10971.3939 10971.4172 11005.2520 -1.5 -2.3 2.1 2.0 0.1 1.9 -2.3 -1.5 -1.6 0.0 3.0 -0.5 1.4 -1.9 8744.2843 8763.8331 8776.4224 8776.4448 8780.5976 8780.6016 8812.7079 10935.0808 10958.8066 10968.9313 10968.9645 10971.3941 10971.4184 11005.2500 0.2 -1.0 0.0 0.4 -0.7 2.0 -0.4 0.4 0.2 -1.3 1.1 -0.7 0.2 0.1 Table II-9. Continued Isotope ArD79Br ArD81Br a J,F -y J',F' Observed (MHz) Calculated (MHz) Residue (kHz) 3,5/2 3,7/2 3,7/2,9/2 3,5/2 3,3/2 3,9/2 4,7/2 4,9/2 4,11/2 4,9/2 4,7/2 4,5/2 4,11/2 4,5/2 4,7/2 4,9/2,11/2 4,7/2 4,5/2 4,9/2 5,7/2 5,9/2 5,13/2 5,11/2 5,9/2 5,7/2 5,11/2 8634.4250 8672.0002 8695.8376 8703.6936 8703.7919 8765.0150 10804.8226 10850.2344 10869.4041 10869.4298 10874.0712 10874.0890 10938.6084 8634.4277 8672.0015 8695.8339 8703.6918 8703.7915 8765.0169 10804.8238 10850.2377 10869.4032 10869.4242 10874.0713 10874.0879 10938.6084 -2.7 -1.3 3.7 1.8 0.4 -1.9 -1.2 -3.3 3,5/2 3,7/2 3,7/2,9/2 3,5/2 3,3/2 3,9/2 4,7/2 4,9/2 4,11/2 4,9/2 4,7/2,9/2 4,11/2 4,5/2 4,7/2 4,9/2,11/2 4,7/2 4,5/2 4,9/2 5,7/2 5,9/2 5,13/2 5,11/2 5,9/2,11/2 5,11/2 8573.5899 8604.9064 8624.8826 8631.4705 8631.5348 8682.6943 10726.5936 10764.4880 10780.5517 10780.5727 10784.4712 10838.3841 8573.5927 8604.9072 8624.8791 8631.4684 8631.5352 8682.6960 10726.5960 10764.4900 10780.5498 10780.5690 10784.4681 10838.3884 Calculated15 (MHz) Residue (kHz) 5.6 -0.1 1.1 -3.2 8634.4263 8672.0001 8695.8370 8703.6933 8703.7918 8765.0147 10804.8212 10850.2352 10869.4065 10869.4269 10874.0729 10874.0882 10938.6055 -1.3 0.1 0.6 0.3 0.1 0.3 1.4 -0.8 -2.4 2.9 -1.7 0.8 -0.3 -2.8 -0.8 3.5 2.1 -0.4 -1.7 -2.4 -2.0 1.9 3.7 3.1 -4.3 8573.5914 8604.9060 8624.8823 8631.4701 8631.5353 8682.6934 10726.5929 10764.4874 10780.5529 10780.5719 10784.4708 10838.3854 -1.5 0.4 0.3 0.4 -0.5 0.9 0.7 0.6 -1.2 0.8 0.4 -1.3 0.9 Calculated from Eq. (II-6) without the spin-rotation interaction and the data in Table 11-11. ^Calculated with spin-rotation interaction. U3 00 39 Table 11-10. Observed transition frequencies for KrHBr and those calculated with Eq. (II-6) and the constants in Table 11-12. Isotope 82 84 86 KrH 7 9 Br KrH ? 9 Br KrH 7 9 Br J,F ->• J',F' Observed (MHz) Calculated (MHz) 5,11/2,13/2 5,7/2,9/2 6,13/2,15/2 6,9/2,11/2 7,15/2,17/2 7,11/2,13/2 6,13/2,15/2 6,9/2,11/2 7,15/2,17/2 7,11/2,13/2 8,17/2,19/2 8,13/2,15/2 8304.6126 8307.1662 9688.0939 9689.9217 11071.0618 11072.4406 8304.6127 8307.1666 9688.0931 9689.9218 11071.0629 11072.4398 5,13/2 5,11/2 5,9/2 5,7/2 6,15/2 6,13/2 6,9/2 6,11/2 7,17/2 7,15/2 7,11/2 7,13/2 8,19/2 8,17/2 8,13/2 8,15/2 6,15/2 6,13/2 6,11/2 6,9/2 7,17/2 7,15/2 7,11/2 7,13/2 8,19/2 8,17/2 8,13/2 8,15/2 9,21/2 9,19/2 9,15/2 9,17/2 8207.6450 8207.6570 8210.2011 8210.2126 9574.9845 9575.0017 9576.8177 9576.8259 10941.8204 10941.8467 10943.2014 10943.2204 12308.1250 12308.1578 12309.1972 12309.2278 8207.6449 8207.6571 8210.2021 8210.2085 9574.9834 9575.0029 9576.8173 9576.8270 10941.8223 10941.8477 10943.2021 10943.2219 12308.1232 12308.1537 12309.2004 12309.2274 5,11/2,13/2 5,7/2,9/2 6,13/2,15/2 6,9/2,11/2 7,15/2,17/2 7,11/2,13/2 8,17/2,19/2 8,13/2,15/2 6,13/2,15/2 6,9/2,11/2 7,15/2,17/2 7,11/2,13/2 8,17/2,19/2 8,13/2,15/2 9,19/2,21/2 9,15/2,17/2 8115.1330 8117.6909 9467.0846 9468.9074 10818.5304 10819.9159 12169.4657 12170.5371 8115.1349 8117.6902 9467.0801 9468.9097 10818.5350 10819.9126 12169.4634 12170.5392 Residue (kHz) -0.1 -0.4 0.8 -0.1 -1.1 0.8 0.1 -0.1 -1.0 4.1 1.1 -1.2 0.4 -1.1 -1.9 -1.0 -0.7 -1.5 1.8 4.1 -3.2 0.4 -1.9 0.7 4.5 -2.3 -4.6 3.3 2.3 -2.1 40 Table 11-10. Isotope 82 84 86 KrH 8 1 Br KrH 8 1 Br KrH 8 1 Br Continued J',F' Observed (MHz) Calculated (MHz) Res idue (kHz) 5,11/2,13/2 5,7/2,9/2 6,13/2,15/2 6,9/2,11/2 8,15/2,17/2 8,11/2,13/2 6,13/2,15/2 6,9/2,11/2 7,15/2,17/2 7,11/2,13/2 9,17/2,19/2 9,13/2,15/2 8201.5542 8203.6992 9567.8264 9569.3653 10933.6016 10934.7620 8201.5547 8203.7018 9567.8246 9569.3622 10933.6039 10934.7617 -0.5 -2.6 5,13/2 5,11/2 5,7/2,9/2 6,15/2 6,13/2 6,9/2 6,11/2 7,17/2 7,15/2 7,11/2 7,13/2 8,19/2 8,17/2 8,13/2 8,15/2 6,15/2 6,13/2 6,9/2,11/2 7,17/2 7,15/2 7,11/2 7,13/2 8,19/2 8,17/2 8,13/2 8,15/2 9,21/2 9,19/2 9,15/2 9,17/2 8104.5871 8104.5966 8106.7341 9454.7158 9454.7330 9456.2522 9456.2618 10804.3706 10804.3894 10805.5270 10805.5432 12153.5098 12153.5327 12154.4136 12154.4356 8104.5847 8104.5949 8106.7312 9454.7172 9454.7328 9456.2541 9456.2628 10804.3720 10804.3920 10805.5287 10805.5447 12153.5088 12153.5327 12154.4119 12154.4334 2.4 1.7 2.9 5,11/2,13/2 5,7/2,9/2 6,13/2,15/2 6,9/2,11/2 7,15/2,17/2 7,11/2,13/2 8,17/2,19/2 8,13/2,15/2 6,13/2,15/2 6,9/2,11/2 7,15/2,17/2 7,11/2,13/2 8,17/2,19/2 8,13/2,15/2 9,19/2,21/2 9,15/2,17/2 8012.0718 8014.2148 9346.8145 9348.3458 10681.0831 10682.2376 12014.8492 12015.7498 8012.0736 8014.2146 9346.8123 9348.3455 10681.0836 10682.2381 12014.8487 12015.7504 J,F -»• 1.8 3.1 -2.3 0.3 -1.4 0.2 -1.9 -1.0 -1.4 -2.6 -1.7 -1.5 1.0 0.0 1.7 2.2 -1.8 0.2 2.2 0.3 -0.5 -0.5 0.5 -0.6 41 Table 11-10. Continued Isotope J,F + J',F' Observed (MHz) Calculated (MHz) 5,11/2 5,13/2 5,9/2 5,7/2 6,13/2,15/2 6,11/2 6,9/2 7,17/2 7,15/2 7,11/2,13/2 8,19/2 8,17/2 8,13/2 8,15/2 6,13/2 6,15/2 6,11/2 6,9/2 7,15/2,17/2 7,13/2 7,11/2 8,19/2 8,17/2 8,13/2,15/2 9,21/2 9,19/2 9,15/2 9,17/2 8107.1076 8107.0990 8110.6176 8110.6571 9457.9654 9460.5050 9460.4878 10808.3710 10808.3830 10810.2820 12158.3028 12158.3151 12159.7912 12159.8039 8107.1055 8107.1000 8110.6153 8110.6575 9457.9676 9460.5055 9460.4902 10808.3708 10808.3813 10810.2813 12158.3013 12158.3170 12159.7930 12159.8019 5,11/2,13/2 5,9/2 5,7/2 6,13/2,15/2 6,11/2 6,9/2 7,17/2 7,15/2 7,11/2,3/2 8,19/2 8,17/2 8,13/2 8,15/2 6,13/2,15/2 6,11/2 6,9/2 7,15/2,17/2 7,13/2 7,11/2 8,19/2 8,17/2 8,13/2,15/2 9,21/2 9,19/2 9,15/2 9,17/2 8005.9785 8008.9313 8008.9566 9339.9358 9342.0544 9342.0643 10673.4468 10673.4627 10675.0560 12006.5084 12006.5230 12007.7588 12007.7711 8005.9811 8008.9330 8008.9592 9339.9343 9342.0542 9342.0608 10673.4477 10673.4599 10675.0527 12006.5086 12006.5251 12007.7600 12007.7716 Residue (kHz) 2.1 -1.0 2.3 -0.4 -2.2 -0.5 -2.4 0.2 1.7 0.7 1.5 -1.9 -1.8 2.0 -2.6 -1.7 -2.6 1.5 0.2 3.5 -0.9 2.8 3.3 -0.2 -2.1 -1.2 -0.5 42 no frequency uncertainty is 5 kHz for the Qfl Kr and Kr species. The observed data were fit to the ground vibrational state, K=0 symmetric top energy levels given by Eq. (II-6). In the first analysis, the Br spin-rotation constant was assumed to be zero. It was necessary, however, to make the Br nuclear quadrupole coupling constant, xa> a function of J and Eq. (II-6) was modified by replacing X with X ao +D v J(J+l), where X ao is Che coupling constant extrapolated to the J=0 state and Dv represents the centrifugal distortion of X a . The need for D Y became readily apparent in trying to assign the spectra. Figure II-2 compares the observed AF=+1 hyperfine structure for the J=4->J=5 transition in 79 ArH Br with that calculated with and without D... As can be seen, there is a large difference in the calculated splittings, especially for the F=ll/2->F=13/2 and F=9/2-*F=ll/2 pair. is also reversed for both pairs. The ordering of the transitions In addition to the effects on the AF=+1 transitions, the AF=0 frequencies are shifted by up to 120 kHz. The J(J+1) dependence of the energy levels upon Dy as well as its interpretation m terms of vibrational bend-stretch interaction constant will be explained below. The spectroscopic constants, as determined from the least squares fit of the data to Eq. (II-6) without the Br spin-rotation interaction, are listed in Table 11-11, Column I, for ArHBr and Table 11-12 for KrHBr. The transition frequencies calculated with these constants and Eq. (II-6) are listed in Tables II-9 and 11-10, respectively. The quoted uncertain- ties represent the standard deviations from the least squares fit. Upon comparing the measured and calculated frequencies for ArHBr in Table 11-11, Column I, a clear trend emerges for all isotopes; all AF=0 lines PORTION OF OBSERVED J = 4 - ^ J = 5 SPECTRUM IN ArHBr 11/2-13/2 9/2-11/2 5/2-7/2 7/2-9/2 j , /- Calculated With Dv V / • \ / v \ / \ A I 11059.57 Figure II-2. / A Calculated Without Dv W 11059.63 / / V i 1062.52 11062.56 The effect of the centrifugal distortion of the Br nuclear quadrupole coupling constant on the AF = + 1 hyperfine structure of the J=4-KJ=5 transition in ArHBr. .eu> 44 Table 11-11. Spectroscopic constants for ArHBr. Column I is the fit without and Column II is the fit with the Br spin-rotation interaction. ArH 79 Br I B0(MHz) Dj(kHz) X (MHz) D„(kHz) II 1106.66928(48) 12.397(11) 173.204(17) 18.25(74) 1106.66951(17) 12.397(4) 173.199(6) 18.42(26) 1.09(13) ctkHz) ArD 1087.50854(48) 8.494(11) 276.227(17) 14.58(74) Table 11-12. Isotope 8 84 W 9 7B9 r KrH Br 86KrH79Br 82 KrH 8 1 Br KrH 8 1 Br 86 KrH 8 1 Br 84 KrD 7 9 Br 84KrD81Br 84 a I II 1097.58095(31) 12.226(7) 144.796(11) 15.10(48) 79 Br 1097.58109(15) 12.226(3) 144.793(5) 15.21(23) 0.67(12) ArD8 1 Br II I BQ(MHz) Dj(kHz) Xa (MHz) Dy(kHz) c(kHz) ArH8 1 Br :L087.50889(27) 8.497(6) 276.223(9) 14.73(41) 0.94(20) I 1078.59281(51) 8.350(11) 230.884(20) 12.00(87) II 1078.59315(22) 8.352(5) 230.878(8) 12.26(37) 1.02(16) Spectroscopic constants for KrHBr. B 0 (MHz) 692.31400(14) 684.22952(15) 676.51549(36) 683.71030(41) 675.62442(14) 667.91018(13) 675.84862(14) 667.40217(20) Dj(kHz) 2.6579(14) 2.5973(11) 2.5364(30) 2.6003(40) 2.5266(11) 2.4634(10) 2.1900(10) 2.1384(15) Xa(MHz) 228.18(9) 228.26(14) 228.41(36) 191.26(30) 190.78(14) 190.78(14) 318.18(14) 265.85(19) D x (kHz) a (7.57) 7.57(29) (7.57) (5.88) 5.88(28) (5.88) 5.05(30) 4.76(40) Values of D Y in parentheses were fixed in the least squares fit. 45 are shifted down in frequency relative to the AF=+1 components. This trend is characteristic of the spin-rotation, thus, a second fit of the ArHBr data was made in which the Br spin-rotation constant was included in Eq. (II-6). The results of this fit are shown in Column II of Table 11-11 and the frequencies calculated with these constants and Eq. (II-6) are listed in Table II-9. The measured spin-rotation constants are generally about 1 kHz, implying a shift in the J=4>J=5, AF=0 lines of about 5 kHz. Since this is approximately the uncertainty in the fre- quency measurements, c is imprecisely determined. However, its value is close to that which would be predicted on the basis of the molecular structures as shown below. The KrHBr data were not fit with a spin- rotation constant since the value of the constant is expected to be smaller and its effects on the AF=+1 transitions measured in KrHBr much less pronounced. In this case, the spin-rotation interaction was not expected to add meaningfully to the fit. The vibrationally averaged structures for ArHBr and KrHBr were determined, in the usual way, from the rotational constants, Br nuclear quadrupole coupling constants and known properties of HBr. The appro- priate free molecular properties of HBr are listed in Table 11-13, and the structures of ArHBr and KrHBr are shown in Table 11-14. The struc- tures of ArHBr and KrHBr listed in Table 11-14 indicate that 6 is acute. This was determined by examining the effects of Br isotopic substitution. Table 11-15 compares the structures of ArHBr and KrHBr computed assuming a (and thus 0) is acute in one case and obtuse in the other. Clearly, only for the acute choice does the isotope effect have the proper magnitude 46 Table 11-13. Isotope H 81Br H 7 9 BB rr D «i D 81 Br a Spectroscopic and structural properties of HBr. ,A b B (MHz)* 250358.510 250280.582 127357.639 127279.757 1.42426 1.42426 1.42144 1.42144 X0(MHz) c Br (kHz) v 532,.30590 444,,68066 530,,6315 443.,2799 290.796 313.370 145.82 157.26 See Ref. 43. Calculated from B . See Ref. 46 for HBr constants and Ref. 47 for DBr constants. Table 11-14. Structures of ArHBr and KrHBr.' Isotope a(deg) O(deg) R0(A) ArH 79 Br ArH 81 Br ArD 79 Br ArD ai Br 82 KrH 7 9 Br 84 KrH 7 9 Br 86KrH79Br 42.12 42.11 34.43 34.42 38.11 38.10 38.09 38.05 38.10 38.10 31.11 31.10 42.08 42.07 34.38 34.38 38.04 38.03 38.02 37.98 38.02 38.02 31.00 31.00 4.1464 4.1463 4.1820 4.1819 4.2573 4.2572 4.2571 4.2572 4.2571 4.2570 4.2809 4.2808 82 KrH°J-Br 86 KrH 8 jBr 84 KrD 7 9 Br KrD 8 1 Br 84 1 rR_Br=l.42426 A, rD_Br=l.42144 A. 'See Figure II-l. 47 Table 11-15. Comparison of the Br Isotope Effect in the Structures of ArHBr and KrHBr when a is assumed acute and obtuse. Acute Angle Isotope RQ(A) ArH 79 Br 4.1464 Obtuse Angle R0(79Br)-R0(81Br)(A) R0(X) 4.1198 -0 .0006 0.0001 ArH 8L Br 4.1463 ArD 79 Br 4.1820 82 4.1204 4.1238 -0 .0003 0.0001 Q1 ArD Br KrH 7 9 Br 4.1819 4.1251 4.2573 4.2291 -0 .0005 0.0001 82 KrH 8 1 Br 4.2572 84 KrH 7 9 Br 4.2572 4.2296 4.2290 -0 .0005 0.0001 84 KrH 8 1 Br 4.2571 86 KrH 7 9 Br 4.2571 4.2295 4.2288 -0 .0006 0.0001 86 KrH 8 1 Br 4.2570 84 KrD 7 9 Br 4.2809 4.2294 4.2204 -0 .0013 0 .0001 84 KrD 81 Br 4.2808 R0(79Br)-Ro(8lBr)(A) 4.2217 48 and sign (see Section II-B). The ArHBr and KrHBr structures reported in Table 11-14 are appropriate to the ground vibrational and J=0 rotational states of the respective complexes. Since the bond angles are determined from x a and x a is a function of J, there are small but measurable changes in 9 as J increases. The Br nuclear quadrupole coupling constant for any state J is given by *aJ = X ao + D J(J+1) X ' (11-27) For the J=5 level of ArH 79 Br, xa=173.752 MHz as compared to x a =17 3.199 MHz for J=0. 0.03°. This difference translates into a decrease in 8 of about While this structural change is small, it has a dramatic effect on the transition frequencies and must be accounted for m any potential surface derived for these systems. It is also of interest to compare the measured spin-rotation constants of HBr with those predicted on the basis of the structures derived above. It is again assumed that the electronic properties of HBr are unchanged upon complex formation. The spin-rotation constant is, in 33 general, proportional to the inverse moment of inertia of a molecule, thus, the free HBr spin-rotation constant, c Q , must be reduced by Io/lD=b/B0, where I D and b are the moment of inertia and rotational constant of free HBr and I 0 is the average of the moments of inertia perpendicular to the a-axis of the complex. Also, since only the average of the two perpendicular components of the spin-rotation tensor are measured, the average of the projections of c 0 onto the b- and c-axes of the complex 49 must be taken. c = Following the method of Appendix II, -^-{l+cos 2 a> . (11-28) Table 11-16 lists the measured and predicted values of c for ArHBr. Table 11-16. In Measured values of the spin-rotation constant of ArHBr and those calculated with Eq. (11-28), Isotope c(measured)(kHz) ArH 79 Br ArH°J;Br ArD 79 Br ArD 81 Br 1.09 0.67 0.94 1.02 (13) (12) (20) (16) c(calculated)(kHz) 1.00 1.06 1.05 1.12 Q-l all cases, save ArH Br, the measured and predicted values are the same within the standard deviations from the least squares fit. It must be emphasized, however, that due to the small size of c, the measured values should be considered more uncertain than the standard deviations indicate. The harmonic, bending and stretching, force constants and frequencies for ArHBr and KrHBr as well as the Lennard-Jones estimates of their effective radial potentials are given in Table 11-17. These were calcu- lated according to the prescriptions of Section II-A and the related discussion involving KrHCl and XeHCl applies equally well here. The new effect observed in this study which had not previously been seen in the rare gas-HCl complexes is the centrifugal distortion, D^, of the halogen nuclear quadrupole coupling constant, X a . D\ and the J(J+1) Table 11-17. Isotope Harmonic Force Constants and Frequencies and Lennard-Jones parameters for ArHBr and KrHBr. ys(amu) o , ks(mdyn-A-J-) i vs/c(cm~-L) o^ ub(amu A'1) < o k^mdyn-A) _i v,/ (cm -1) —l e(cm x ) o R e (A) 79 Br 26.64174 0.0076 22.1 2.027210 0.001136 30.8 89 4.0816 ArH 81 Br 26.86010 0.0076 21.9 2.027483 0.001136 30.8 89 4.0817 26.75261 0.0106 26.0 4.001671 0.001291 23.4 126 4.1280 26.96738 0.0106 25.9 4.002752 0.001291 23.4 127 4.1282 KrH 79 Br 40.45381 0.0132 23.6 2.030124 0.001698 37.7 165 4.2191 VKrHU 7 9 Br R 40.93517 0.0132 23.4 2.030190 0.001700 37.7 165 4.2192 ArH 79 ArD Br ArD 81 Br 82 8 86 KrH 79 Br 41.40519 0.0133 23.3 2.030253 0.001702 37.7 165 4.2193 82 KrH 81 Br 40.95941 0.0132 23.4 2.030405 0.001708 37.8 164 4.2192 84 KrH 81 Br 41.45295 0.0133 23.3 2.030472 0.001700 37.7 165 4.2194 86 KrH 81 Br 41.93501 0.0133 23.2 2.030535 0.001700 37.7 166 4.2195 84 KrD 79 Br 41.19752 0.0152 25.0 4.013155 0.001948 28.7 192 4.2458 84 KrD 81 Br 41.70903 0.0152 24.9 4.014269 0.001948 28.7 192 4.2459 51 dependence of the energy levels on D can be explained in terms of coupling between the radial and angular potentials. Consider a simple potential consisting of harmonic terms in the radial and angular coordinates, as assumed before, along with a single bend-stretch interaction term, v(R,e)=|ks(R-Re)2 + ike 2 + lk I e 2 (R-R e ), (11-29) where the measured bending force constant is given by 2 kb - |g|- = k + kx(R-Re) . (11-30) Consider, a l s o , t h e change i n x a with R. From Eq. ( I I - 7 ) , assuming 9^a, &* £ - 3 x 0 c o s 9 s i n 9 ^ | . (11-31) Using t h e chain r u l e and r e a l i z i n g h2 / 0 a and /~2~j 4ir^u b k b dk b IT dR that = k l , (II-32) l we have i n terms of v a r i a t i o n s |-tibe5kIcosesine6R. <5X = (11-33) 4h 6R is easily related to J using the simple diatomic theory of centrifugal distortion by 38 R D o J 6R = ^ - J ( J + l ) ^Bo (11-34) thus, 5x =—p ° ° 8TTB0 J i e5cos9sin9J(J+l) (11-35) 52 and 3u, X R Djk D O o j i e5cos6slne> (11-36) 8h z B Q * To test Eq. (11-36), a rough calculation of the expected value of DY for A 79 ArH 79 Br and KrH Br can be made. To a first approximation, k-r = dkb/dR can be estimated from the changes in k b and R Q upon deuterium substitution. D is predicted to be 13.0 kHz for ArH Br and 6.4 kHz for KrH 79 Br. A These values compare very favorably with the measured values of 18.4 and 7.6 kHz, respectively, considering the crudeness of the calculation. It is also noted that upon °^-Br substitution, Dv is reduced by the rctio of X (H 9 Br)/x(H 81 Br) as is predicted by this model. The absence of signi- ficant effects in KrHCl and XeHCl is consistent with the roughly ten-toone ratio in magnitude of the quadrupole coupling constants in free HBr and HC1, respectively. In conclusion, D.. appears to be a sensitive measure of the coupling between the angular and radial potentials and should be of great help in determining the true potential surfaces for these molecules. E. Argon Deuterium Fluoride In the previous two sections, it has been shown that to determine the structures of complexes containing HC1 or HBr, extensive use must be made of the halogen nuclear quadrupole coupling constant. For complexes containing HF, however, this information is unavailable since H a quadrupolar nucleus. 19 F lacks In this section.the assignment of the rotational spectrum of ArDF is reported. The structure of ArDF is obtained from 53 the rotational constant and analyses of the DF nuclear spin-spin and D nuclear quadrupole coupling constants. These constants are found to yield structural information consistent with one another and equivalent to that obtained from the halogen coupling constants in the corresponding HC1 and HBr complexes. In addition, the rotational spectrum of ArHF has been re-examined and the structure of ArHF determined with help of the HF spin-spin constant. The observed transition frequencies for ArDF and ArHF are listed in Table 11-18. Figure II-3 shows the time domain record for the J=C->-J=l transition in ArDF and Figure II-4 is the corresponding frequency spectrum. The hyperfine structures of the J=0+J=1 transitions indicate the presence of D nuclear quadrupole coupling in ArDF and the H(D)F nuclear spin-spin interaction m both complexes. The spin-spin interaction can be described by the classical interaction between two magnetic dipoles • .u _3(m-&)Qt2-$. H s s = R 3 Ul ^2 Jl where R is the vector from nucleus 1 to nucleus 2. (H-37) The nuclear magnetic moments, U J , are related to the nuclear g-values and angular momenta,1^, by JJ± = "N»l^i (II ' where u^ is the nuclear magneton. d = y N 2 g 1 g 2 /R 3 , ~38) Defining the spin-spin constant, d, as • (11-39) 54 Table 11-18. Isotope Observed and calculated transition frequencies for ArDF and ArHF. J,F, ,F ->• J',Fj_,F' Label Observed (MHz) Calculated (MHz) Residue (kHz) a 6079.3690 6079.3700 -1.0 b 6079.4525 6079.4526 -0.1 c 6079.4628 6079.4611 1.7 d 6079.5110 6079.5112 -0.2 e 6079.5283 6079.5287 -0.4 12157.4101 12157.4095 0.6 12157.5037 12157.5043 -0.6 6131.1112 6131.1111 0.1 6131.1355 6131.1361 -0.6 6131.1420 6131.1415 0.5 12260.5707 12260.5707 0.0 for Figure II-4 ArDF 0,1,3/2 1,0,1/2 0,1,1/2 1,0,1/2 0,1,3/2 1,2,3/2 0,1,1/2 1,2,3/2 0,1,3/2 1,2,5/2 0,1,3/2 1,1,3/2 0,1,1/2 1,1,3/2 0,1,3/2 1,1,1/2 0,1,1/2 1,1,1/2 1,1,3/2 2,1,3/2 1,1,1/2 2,1,1/2 1,2,3/2 2,3,5/2 1,2,5/2 2,3,7/2 0,1/2,1 1,1/2,1 0,1/2,0 1,1/2,1 0,1/2,1 1,3/2,1 0,1/2,0 1,3/2,1 0,1/2,1 1,3/2,2 1,1/2,1 2,3/2,2 1,3/2,1 2,5/2,2 1,3/2,2 2,5/2,3 } } } } } ArHF } } 240 TIME (^s) Figure I I - 3 . Time domain record of the J=0-KJ=1 transition in ArDF. The signal was digitized at 1 us per point and averaged over 10 pulses. Ul Ul 6079.350 6079.525 MHz Figure II-4. Frequency spectrum corresponding to the time domain in Figure II-3. 1.95 kHz per point and +/- label the Doppler pairs. The resolution is Ul 57 Ramsey 34 has shown that (II-40) H + ss - (2J-1K2J+3) K V ^ V ^ 3<l2'J)(Il-J> " 2 ^ 1 ' ^ ) J 2 ] • The total Hamiltonian used to fit the data for ArDF is given by Eqs. (II-l)-(II-3). For ArHF, the same Hamiltonian is used with the exception that no quadrupole term is included. The data were fit using a nonlinear least squares routine involving diagonalization of the Hamiltonian matrix as described in Section II-A. A preliminary fit was made by assuming that the spin-rotation constants, c^, were zero. The structures of ArDF and ArHF were then roughly determined and the spin-rotation constants calculated from Eq. (11-28). For the final fit, the c L were held fixed at these calculated values. The derived spectroscopic constants for ArDF and ArHF are listed in Table 11-19. Table 11-19. B0(MHz) Dj(kHz) Xa(kHz) da(kHz) c„(kHz) b c£(kHz) b Spectroscopic constants for ArDF and ArHF . ArDF ArHF 3039.8529(4) 59.53(6) 194.7(20) 12.4(11) -0.05 1.25 3065.7099(4) 70.90(6) 49.0(19) -0.3 1.2 Uncertainties represent one standard deviation from the least squares fit. b'Held fixed in the least squares fit, 58 As with the halogen nuclear quadrupole coupling constants in the HC1 and HBr complexes, the measured spin-spin and D nuclear quadrupole coupling constants in ArDF are simply those components, along the a-inertial axis, of the respective second rank tensors. Making the usual assumption that all properties of DF are preserved on complex formation, the D nuclear quadrupole coupling constant, X a , is related to that in free DF, X 0 , by X a = i<3cos2a-l> x 0 . (11-41) Similarly, the measured spin-spin constant, d , is given by d a = Y<3COS 2 CX-1> d0. (11-42) The appropriate properties of free DF and HF are listed in Table 11-20. Table 11-20. Spectroscopic constants for HF and DF. HF » X 0 (kHz) d d(kHz) d c H (kHz) d c F (kHz) d DF 616365.6(4)a 0.925595 143.375(25) -71.128(24) 307.637(20) aSee Ref. 48. c Calculated from b d See Ref. 49. See Ref. 43. 325585.0(3)b 0.923243 354.238(78) 22.170(35) -5.755(19) 158.356(45) V Since both X a and d a are measured in ArDF, two values for a are obtained, a=33.2(2)° from X a and ct=32.7(22)° from da. As shown 59 by Eq. (11-39), the free DF value of the spin-spin constant can be calculated from the known properties of DF and depends on (l/r-^), where r is the DF internuclear separation. X 0 , on the other hand, depends on the electric field gradient, q, at the deuterium nucleus. Since the two values of a are the same within experimental uncertainty and it is unlikely that two such diverse properties as (l/r^) and q would change by the same amount on complex formation, it appears that the assumption made above holds for these quantities within the resolution of this experiment. Having determined a from either X a or d a , the structures for ArDF were derived in the usual manner (see Section II-A). The structures of ArDF and ArHF are shown in Table 11-21. Table 11-21. 6(deg) b R0(A)b 9(deg) c R0(A)C a It is noted that unlike in the Structures of ArDF and ArHFa. ArDF ArHF 32.2(21) 3.5352(15) 32.7(2) 3.5349(2) 41.1(5) 3.5445(2) See Figure II-l; r D F = 0.923243A, r H F = 0.925595A. Spin-spin interaction used to determine the angle. C D quadrupole coupling used to determine the angle. HC1 and HBr complexes, the Ar-F distance, R Q , decreases upon D substitution. A possible explanation for this phenomenon is that the lighter HF molecule is undergoing much larger amplitude vibration in 60 the radial mode than is the corresponding HCl or HBr complex. The motion is consequently more complicated and the arguments made for HCl and HBr no longer hold. In general, the heavier of two isotopic species will exhibit the shorter bond. With the greatest fractional mass change upon D substitution occuring in HF, it is likely that the mass effects more than compensate for the repulsion effects seen in the HCl and HBr species and result in the shortening of the bond. The harmonic force constants and frequencies and Lennard-Jones potential parameters for ArHF and ArDF are listed in Table 11-22. Table 11-22. These require Harmonic force constants and frequencies and Lennard-Jones potential parameters for ArDF and ArHF. ks(mdyn/A) v s / c (cm" ) ArDF 0.0170 45.8 ArHF 0.0142 42.5 vjb/,c(cm_1) Rg(A) e(cm -1 ) 0.00392 64.1 3.463 143.0 0.00301 78.1 3.465 119.2 kb(mdyn-A) no comments other than those already made with respect to KrHCl and XeHCl in Section II-B. In summary, the studies of the spectra and structures of ArDF and ArHF presented here show that the nuclear spin-spin interaction, when observable, can be used to gain structural information about van der Waals molecules in complexes which lack quadrupolar nuclei. There is also no evidence that either the bond length of free DF or the electric 61 field gradient at the D nucleus is significantly perturbed on complex formation. F. Summary of Results and Conclusion The studies of rare gas-hydrogen halide van der Waals molecules presented here, along with prior investigations of X-HY, allow an opportunity to examine the X-HY interaction as a function of both the rare gas and halogen atoms. Table 11-23 summarizes the structures, force constants and Lennard-Jones potential parameters for the major isotopes of all studied complexes involving Ar, Kr or Xe and HF, HCl or HBr. These constants were calculated using the methods of Section II-A. The struc- tural constants display expected trends. The rare gas-halogen distance, R 0 , increases with increasing size of the respective atoms. For a given rare gas atom, the structural angles, 9, remain relatively constant in going along the series from HF to HBr. This latter effect results from a decreased anisotropy in the potential, along that series, as measured by kb. The trend is easily explained by realizing that as the halogen becomes bigger, the corresponding hydrogen halide looks more spherical. One question concerning the structure of van der Waals molecules has been whether the van der Waals bond length in a dimer can be explained in terms of the van der Waals radii of the constituent molecules. rare gas hydrogen halides, it appears to be possible. In the If it is assumed that RQ is the sum of the van der Waals radius of the rare gas atom, R ^ and of the hydrogen halide, R^y, and R x * s taken to be 5 1.9, 2.0 and 2.2 A for 62 Table 11-23. Summary of the structures, force constants and LennardJones potential parameters for the rare gas-hydrogen halides as calculated using the method of Section II-A. *o 9 k s k b R e e HY radiusa (A) (deg.) o (mdyn/A) ArHF 3.5445 41.1 0.0142 0.00301 3.465 119 1.64 ArDF 3.5349 32.7 0.0170 0.00392 3.463 143 1.63 KrHF 3.6467 38.9 0.0182 0.00378 3.585 163 1.65 KrDF 3.6399 30.2 0.0210 0.00533 3.584 189 1.64 XeHFC 3.8152 35.6 0.0210 0.00537 3.763 208 1.62 XeDF 3.8111 29.4 0.0237 0.00594 3.763 235 1.61 ArHCld 4.0065 41.3 0.0117 0.00151 3.942 128 2.11 ArDCld 4.0249 33.4 0.0134 0.00180 3.966 148 2.12 KrHCl 4.1106 37.8 0.0155 0.00216 4.064 179 2.11 KrDCl 4.1251 30.5 0.0171 0.00259 4.081 199 2.13 XeHCl 4.2753 34.5 0.0190 0.00311 4.237 239 2.08 XeDCl 4.2875 27.9 0.0205 0.00370 4.251 259 2.09 ArHBr 4.1464 42.1 0.0076 0.00114 4.082 89 2.25 ArDBr 4.1820 34.4 0.0106 0.00129 4.128 126 2.28 KrHBr 4.2572 38.0 0.00132 0.00170 4.219 165 2.26 KrDBr 4.2809 31.0 0.0152 0.00195 4.246 192 2.28 0 0 (mdyn•A) (A) (cm"1) (X) Calculated assuming RQ is the sum of the X and HY van der Waals radii where the radii for X are taken from Ref. 50. b See Ref. 25. c See Ref. 26. d See Ref. 16. 63 Ar, Kr and Xe, respectively, Rjjy can be computed. results of this calculation. Table 11-23 shows the It shows that remarkably constant values are obtained for the van der Waals radii of each of HF, HCl and HBr. In particular, R ^ equals 1.63(1), 2.11(2) and 2.27(2) A for HF, HCl and HBr, respectively. Using tabulated values for the van der Waals radii of the 51 halogens, is obtained. o a constant value for the contribution of H to Ryy* 0.30(2) A, The magnitude of this contribution is approximately that which is expected. Consider KrHCl, where the angle, 8, between HCl and the Kr-Cl mternuclear axis is 37.8 and the van der Waals radius of H 51 ° is taken to be 1.2 A. If Kr is brought up along the Kr-Cl internuclear axis until it touches H, the Kr-H distance is given by the sum of their o o van der Waals radii, 3.2 A. Since the H—Cl distance is known to be 1.28 A, the Kr-Cl distance can be computed from simple trigonometry and is found o to be 4.11 A which is exactly the measured value. While this exact agreement is certainly fortuitous, it shows that a consistent picture of the structures of X-HY can be obtained by considering the van der Waals radii of the monomers. As an added bonus in the above analysis, a con- sistent set of van der Waals radii for the hydrogen halides have been determined. The force constants and potential parameters listed in Table 11-23 show the large variations upon D substitution that were discussed at length in Section II-B. These variations indicate the limits of present analysis since, within the Born-Oppenheimer approximation, the intermolecular potential should be invariant to isotopic substitution. To further the analysis, a clue is taken from the investigation of the centrifugal distortion of 64 the Br nuclear quadrupole coupling constant (see Section II-C), and bendstretch interaction constants are added to the simple harmonic potential. The more complete potential is given by V(R,8)=2k sl (R-R e ) 2 +2k bl 6 2 +k b2 e 2 (R-R e )+k b3 e 2 (R-R e ) 2 . (11-43) The measured stretching and bending force constants are then approximated by k = k ^ = khl+ s si + 2k b3(^ (II " 44) and 21^(R-R e ) + 2k b 3 (R-R e ) 2 . (11-45) It is also noted that RQ is a function of 9 that is approximated here as the first two terms of a Taylor series expansion about 6=0, R 0 = R'0 + c<B>2 . (11-46) R'0 is the rare gas-halogen distance averaged over the stretching vibration but fixed at 6=0, and ks-j_ and k,, are the isotopically invariant stretching and bending force constants for X-HY in its equilibrium configuration. Using the measured stretching force constants and bond angles for the hydrogen and deuterium isotopes of X-HY, Eqs. (11-44) and (11-46) are readily solved to give k , and RQ. These, in turn, are used in the development following Eq. (11-11) to obtain isotopically invariant estimates of the Lennard-Jones well depth, e, and internuclear separation, R e . With R e known, Eq. (11-45) can be solved, using the data from isotopic substitution, to give kbx- The isotopically invariant potential parameters 65 for the X-HY complexes are given in Table 11-24. Table 11-24. ksl(mdyn/A) ArHF KrHF* XeHFb ArHClc KrHCl XeHCl ArHBr KrHBr a The bonding trends exhi- Isotopically invariant force constants and potential parameters for the rare gas-hydrogen halides. The attractive energy, E, calculated from the multipole potential is also given. k ^ (mdyn'A) 0.0218 0.0252 0.0295 0.0166 0.0202 0.0233 0.0166 0.0192 0.0116 0.0194 0.0133 0.0015 0.0022 0.0032 0.0014 0.0021 R£(A) e (era- ) e/eArHF 3.454 3.577 3.758 4.005 4.112 4.277 4.211 4.297 182 226 291 186 239 298 206 247 1.0 1.2 1.6 1.0 1.3 1.6 1.1 1.4 E(ciri 101 123 143 96 119 142 98 121 ) E /EArHF 1.0 1.2 1.4 1.0 1.2 1.4 1.0 1.2 See Ref. 25. b See Ref. 26. c See Ref. 15. bited in Table 11-24 follow, for the most part, simple chemical intuition. It is expected that the binding energy should increase with the polarizability of X for a given HY. the polarizability. The binding energy does not scale, however, as It is also surprising, perhaps, that the binding energy remains relatively constant as the halogen is changed. To put the expected trends in binding energy on a somewhat more quantitative basis, consider the intermolecular potential as expanded in a multipole series. Buckingham has given a multipole expansion for the attractive potential between a linear and spherical molecule. While 66 this potential does not contain repulsive terms and may not be convergent at the small distances considered here, it is thought that the attractive energies should serve as a rough guide to the bonding interaction. The leading term in the multipole expansion is given (for the linear configuration by where a is a polarizability, p is a dipole moment and U is an ionization potential. Table 11-24 lists the multipole energies calculated with Eq. (11-47) and reduced binding and multipole energies where all energies have been normalized to the values of ArHF. It is seen from Table 11-24 that the multipole expansion does an excellent job of explaining the relative binding energies observed in the rare gas-hydrogen halides. The absolute values of the binding energies cannot be determined from the multipole expansion. Comparison is possible, however, with detailed potential surfaces derived from spectroscopic data for ArHCl 52 and KrHCl 41 52 , ArHBr -1 . The binding energies of 183, 186, and 224 cm for ArHCl, ArHBr and KrHCl, respectively agree reasonably well with the values of 186, 206, and 239 cm" listed in Table 11-24. This good agreement indicates that the simple potential in Eq. (11-43) is doing a reasonably good job at reproducing the potential near the potential minimum. As shown above, a multipole expansion of the potential can be used to account for the trends in the binding energy observed in the rare gashydrogen halides. Evidence is presented here that the same potential can also be used to account for the anisotropy of the binding interaction as 67 measured by the angle 9. The full expression, given by Buckingham , for the long range part of the potential between a linear and spherical molecule is (C±+Cd) v( R cm5 ecm') - V(R -' e ^ R*m , / - 1 (3cos e -1) " "'cm - -; I 6c, ^ + l£d Rcm( HY ^ ^ m a u HY HY a «_U I ll- L +c 'i- -^ lC± + Cd 3«HY (11-48) . 4A 3 cose„„ ll ' 3~ cos 9 cm+ cj&d a^y cm A where u u 3 U„Ui: y HY -± = <*xu a 2 UX+UHY XaHY (11-49) Q is the molecular quadrupole moment of HY, A., and Aj_ are components of the tensor describing the dipole-quadrupole polarizability of HY. R c m is the length from X to the center of the mass of HY and 6 c m is the angle HY makes with R,cm. The other quantities have been previously defined. R c m and e c m are easily derived in terms of the structures listed in Table 11-23 by using simple trigonometry. In the limit of harmonic oscillation, the vibrationally averaged angle is given by (see Eq. (11-22)) < 9 cm 2 > = H / t t i r V b ) * (11-50) where the reduced mass, u b , is approximately given by the moment of inertia of free HY and the force constant, k b , is related to the intermolecular potential by 1 S= 32V 3 fl 2 9 cm (11-51) cm = 0 D 68 Using Eqs. (11-48), (11-50) and (11-51), the vibrationally averaged angles, <^9 )= ( G 2 ) > can be calculated. Table 11-25 summarizes the structures of X-HY and includes the angles calculated from the multipole expansion. The agreement between the observed and calculated angles is very good considering the approximations inherent in this model, namely, that the multipole expansion is valid at the small distances considered here, that the bending motion is harmonic, that the repulsive potential is isotropic, and that <9cm> = ( O 1 " 2 " coa-1«co8necm>1/n). (11-52) In addition, the average angle is relatively insensitive to k b since it varies only as the inverse fourth root of the force constant. However, all observed trends in the observed average angle are reproduced by the multipole potential. The calculated angles increase along the series X-HF to X-HBr and decrease along the series Ar-HY to Xe-HY as is found experimentally. The large difference in the average angle between the H and D isotopes of a given complex is also reproduced but is due mainly to the factor of 2 difference in the bending reduced mass. The present results indicate that the anisotropy in the X-HY potential surface is dominated by long range forces and that the multipole expansion may be useful m modeling the attractive part of the X-HY potential. In conclusion, it is believed that the foregoing analysis, while still fairly crude, is giving a more realistic and consistent picture of the rare gas-hydrogen halide van' der Waals interaction than has generally been the case thus far. 69 Table 11-25. Comparison of the observed angles in the rare gas-hydrogen halides with those calculated from the multipole expansion in Eq. (11-48). O o 0 Complex Rcm(A) ArHF 3.5096 41.6 38.3 a ArDF 3.4607 33.5 31.9 a KrHF 3.6105 39.3 36.4 25 KrDF 3.5637 31.0 30.3 25 XeHF 3.7772 35.6 35.0 26 XeDF 3.7339 29.5 29.2 26 NeDCl 3.799 47 42.4 32 ArHCl 3.9795 41.7 38.5 16 ArDCl 3.9668 34.0 32.5 16 KrHCl 4.0822 38.1 36.5 a KrDCl 4.0652 31.0 30.7 a XeHCl 4.2457 34.8 34.9 a XeDCl 4.2260 28.3 29.4 a ArHBr 4.1331 42.2 43.4 a ArDBr 4.1529 34.7 36.9 a KrHBr 4.2431 38.2 41.0 a KrDBr 4.2506 31.2 34.7 a T h i s Chapter. <8 c m >(observed) <8>(calculated) Reference 70 CHAPTER III A HINDERED ROTOR MODEL FOR ESTIMATING ANGULAR EXPECTATION VALUES IN THE RARE GAS-HYDROGEN HALIDES A. Introduction The internal electric fields and electric field gradients in van der Waals molecules which arise from monomer charge distributions have found great utility in explaining polarization effects in these complexes. 53-55 In the rare gas (X)-hydrogen halide (HY) complexes, they have been used to explain the rare gas nuclear quadrupole coupling 16 31 as well as the X-HY dipole moments. ' arising from HY m ' ' The fields and field gradients X-HY are generally expressed by a multipole series involving expectation values of the Legendre polynomials, (P^(cos8)) , where 6 is the angle between HY and the vector R connecting the HY center of mass and X. (P2(cos8)) can be determined from the halogen nuclear quadrupole or the hydrogen-halogen nuclear spin-spin coupling and (P,(cos9)> can be derived from the X-HY dipole moment, u, if it is available. Unfortunately, it is not possible to measure u using the current spectrometer owing to the difficulty of placing Stark electrodes into the Fabry-Perot cavity and (P2(cos9)) is the only angular expectation value available in this case. There are currently no ways to directly obtain (P,j/(cos8)) for l>2 from the microwave results. Defining e^as the angle derived from <Pa(cos9)) , it is not, in gen- 71 eral, true that 6 = 6 for m ^ n. In fact, in the limit that HY undergoes free internal rotation, 9^ decreases monotonically from 90 30.56 for 1= 4. for A = 1 to In this chapter, a simple method is presented to estimate (p2,(cos9)> with Z i 2 if (P9i.(cos9)) is known. The (P-,(cos9)> i computed with this method are compared with experimental values derived from spectroscopic data and with calculations based on detailed potential surfaces where available. Finally, a multipole expansion of the potential between X and HY is used to calculate (P2(cos8)) and some insight into the nature of the true X-HY potential surface is gained. B. Hindered Rotor Model To estimate the angular expectation values, (P^(cos9)), the bending motion of the linear X-HY complex is treated as a hindered internal rotation of the HY subunit. Fixing the HY center of mass to X distance at its vibrationally averaged value,R, HY experiences a potential V(6) and the Hamiltonian for the hindered rotor is given by H = BJ 2 + V(9) , (III-D where B is the ground state rotational constant and J is the angular momentum of HY. V(9) is cylmdrically symmetric about R and is conveni- ently expressed in terms of Legendre polynomials as 00 V (8) = g 1 V £ [1-P£ (cos8)] (III-2) In the limit that all V. go to zero, HY becomes a free rotor and its motion 72 is governed by the rigid rotor eigenfunctions |JM> = ^ ( 9 , 4 0 , where Y (III-3) (9,<|>) is a Spherical Harmonic. Since only the ground state (J = M = 0) expectation values are desired here and V(9) does not couple states with different M, the hindered rotor eigenfunctions are expanded in the basis |J) = |JO) = Y JO (8,4) = ( 2 J±I) 1/2 P J (cos9). (III-4) In the basis of Eq. (III-4), the Kamiltonian matrix elements are given by H J K = [BJ(J+1) + ^ v j 6 J K - JJ1V^<j|PJl(cose)|K), (III-5) where the matrix elements of the Legendre polynomials are given by37 P and 6 JK = <J|P£(cos6)|K> = [(2J+1) (2K+1)] 1/2 ( J Q 2 * M (III-6) is the Kronecker delta. In Eq. 6, L Q Q is a 3-j symbol. The expectation values,(Pjj, (cos8)^, are then obtained by constructing and diagonalizing H and transforming P to the same basis. At most, 15 basis functions were required to attain convergence for the potentials considered here. C. Results and Discussion To a first approximation, the potential experienced by HY is given by the leading term in Eq. (III-2) 73 V(9) = Vj-jVP^cose)] = V1(l-cos8) where V^ is half the barrier height. , (III-7) The one adjustable parameter in V(8) can be fixed by requiring V(9) to reproduce the measured value of (P2(cos6)) within the hindered rotor approximation. all (P^(cos9)} are readily calculated as above. With V^ determined, Since (P^(cos9)>depends only on the ratio V^/B, the calculation need only be done once with (PJ2/(cos9)) tabulated as a function of V^/B. (P£(cos9)) can also be fit to a polynomial in (^(cose)) given by (III-8) <P£(cos6)> = az + b^<P2(cos6)> + c£(P2(cos6)>2 + dJl<P2(cose)>3. The coefficients for Eq.(III-8) with 1= 1, 3 and 4 are listed in Table III-l. Table III-l. Coefficients for the equation <P2,(cos6)> = az + b£<P2(cos9)> + ^ ^ ( c o s g ^ 2 + dJl(P2(cos6)>3. 1 H H H 1 3 4 0.2697 0.0022 -0.0100 1.8959 0.0482 0.1321 -2.3510 0.8511 -0.4722 Use of Eq. (III-8) reproduces the (P£(cos9)> d * 1.3032 0.1112 1.3246 to within 0.5% of their tabulated values over the range of angles appropriate to the X-HY complexes. Figure III-l shows 6^ as a function of V,/B. Clearly, this model gives the correct limiting behavior and it is seen that the algebraic approximation (i.e., 9 m = 92 for all m) becomes increasingly bad as the most weakly bound 0 10 20 30 40 V,/B Figure II-l. 6 £ as a function of the hindered rotor barrier height. 50 75 complexes are approached. In the course of this calculation, the height of the barrier to internal rotation is obtained. X-HY complexes. Table III-2 gives the barrier height for the These barriers are probably too high, in absolute terms, because a P-, (cos 8) potential is relatively broad thus requiring a greater well depth to achieve the correct expectation values. however, all expected trends are observed. In relative terms, For a given HY, the barrier height scales roughly as the polarizability of X and for a given X, the anistropy decreases from HF to HBr. This last trend is expected since HY looks more spherical as the halogen becomes larger relative to H. The hindered rotor model can be tested by comparing <P1(cos6)> as computed above with those derived from dipole moment measurements. a simple electrostatic model, ° /» , ^ » ^ (l (P^cose)) = ~ - - ^ . S?k\ [1 " — — ) _1 R + f 2 "HY\ Using . , [ ^ - + - 2 ^ ) <P2(cos9)> , where uand u Q are the X-HY and HY dipole moments, respectively, Q is the HY quadrupole moment,a is a static polarizability and ct^y is the polarizability of HY along R given by (see Appendix II) ai a HY = 3(a|| ~a0.V < V c o s 6 ^ Table II-2 compares the 9 +a HY - for several complexes. (111-10) In general, the hindered rotor method is much better than the algebraic method for 6 2 > 35°. Below 35°, the algebraic method is comparable to slightly better than the hindered rotor. This agrees with Figure III-l where the 9^ are seen to greatly diverge at^ 35°. While the algebraic method seems to work in some 76 Table III-2. Comparison of measured values of 6, with those estimated from the hindered rotor model and detailed potentials where available. 2xVL 9° (cm~^-) NeDCl ArHCl ArDCl KrHCl KrDCl XeHCl XeDCl ArHF ArDF KrHF KrDF XeHF XeDF ArHBr ArDBr KrHBr KrDBr a 30 112 149 173 221 261 326 222 322 292 450 456 550 84 108 136 170 See Ref. 41. See Ref. 52. c See Chapter II. b 9° Measured 6° 6j Hindered Rotor Experimental Potential 47 65 56 41.7 34.0 38.1 31.0 34.8 28.3 41.6 33.5 39.3 31.0 35.7 29.5 42.2 34.7 38.2 31.2 47.9 34.5 40.4 29.1 34.6 46.3 36.0 41.2 32.5 37.0 29.3 46.2 35.4 42.9 32.5 38.1 30.8 47.2 36.8 41.4 32.7 — 48.4 34.0 — — 37.8 28.2 50.8 34.0 — — 47.5-50,.5a 37.0-37 ,3a 43.0-44,.3b 32.9-33,.0B Reference 32 16 16 27,c 27,c 28, c c 24,c 26,c * 49.8-53, 5 ' h 37.6-38.,0b 25 25 26 26 31,c 31,c c c 77 cases in estimating (P^(cos6)) , it will do worse in estimating (Po(cos9)) and (P^(cos9)). For X-HY, the algebraic method will actually give the wrong sign for (P3(cos9)) in 1/3 of the complexes and for (P^(cos9)) in all but one complex. Table III-2 also compares (p^(cos6)) as calculated above with those obtained from detailed potential surfaces derived from spectroscopic data. 41 52 ' Both methods give comparable results and, in particular, neither method predicts the relatively small values of 9-^ for 9 2 < 35°. It is interesting that this simple hindered rotor model appears to give angular expectation values as well as a much more sophisticated calculation. This is especially surprising in the light that experimental values of 9j_ were used in deriving these surfaces. Finally, the hindered rotor method can be used to calculate the angular expectation values for any barrier given as a sum of Legendre polynomials. Assuming that the anisotropy in the X-HY potential surface arises primarily from the attractive part of the potential, the barrier 5 57 58 can be described by the multipole expansion ' ' „ V(6) = - Zi c(n,i,ind) n,Jl + where again w e assume R is fixed. P (cos9) c(n,S,,dis) _ _ „n , (III-ll) Expressions for the induction and dispersion coefficients, c(n,a,type), are listed in Table III-3. Q c(9,£,ind)/R accounts for about 5% of the attractive energy in X-HY and since the dispersion terms are difficult to calculate and drop off faster than the induction terms, c(9,£,dis) was not included in V ( 8 ) . 9o calculated with the barrier in Eq. (III-ll) is compared, in Table 78 Table III-3. Explicit expressions for the induction and disperson coefficients for the multipole potential in Eq.(III-ll).a on c ( 6 , 0 , i n d ) = y 2 a „x HY = c ( 6 , 0 , d i s ) = 42- =Ux+U - = . 2ax a„„ „2„a c(6,2,ind) = = M x is c(7,l,ind) = i | y Q a x ( a I I ~°±) c ( 6 , 2 , d i s ) = c d " 3a ^ HY c(7,3,ind) = ^ Q a d Cj m c(7,l,dis) = j c d ( A | | + 2 ^ ) / ^ c(8,0,ind) = | y 2 q x + | Q2ax 26 ,n. x j+. 2u^q o„2„ c ( 8 , 2 , m d ) = ^funa 7 c(8,4,ind) = ^ W 7 ^ _ 4 _ ,... 4 4 4 c ( 7 , 3 , d i s ) = j c d (Ai i - -j Aj_)/aHY ^+ i12 2„ £ . nQ/a X ~ + £ Q2a 7 X c ( 9 , l , m d ) = 12yQqx + 36 7 ^ ^ na c(8,0,dis) =-|c d (3d[p+7d| 2) ) b 4 c(8,2,dis) = y cd(13dfp+15d_^;)D c(8,4,dis) =lic d (d| ( p-d| 2 >) b on c ( 9 , 3 , i n d ) = ^ i * a x + 6uQqx + 4 fiax W xx + c(9,5,ind) = ± ±^u$a + i 20 ^ pax 3 7 u , Q, £2 and $ a r e the 2 - , 4 - , 8 - , and 16-pole moments of HY, a and q are 2 - and 4 - pole p o l a r i z a b i l l t i e s , U i s an i o n i z a t i o n p o t e n t i a l and Aii and Aj_ are components of the t e n s o r d e s c r i b i n g the d i s t r i b u t i o n of p o l a r i z a b i l i t y in HY. b See Ref. 58. 79 III-4 , with the measured value Table III-4. The agreement 8~ for several complexes computed using the multipole potential in Eq. (III-ll) and the hindered rotor model. 4 °°2 ArHF KrHF XeHF NeDCl ArHCl KrHCl XeHCl for several complexes. Measured Multipole Potential 41.6 39.3 35.7 47.0 41.7 38.1 34.8 40.5 37.5 35.1 45.9 40.7 37.8 35.4 Difference 1.1 1.8 0.6 1.1 1.0 0.3 -0.6 Reference a a 26 32 16 a a See Chapter II. is remarkable and better than that using the harmonic oscillator model proposed in Section II-E. It is noted that 92 from the multipole potential is generally smaller than the measured value and, while they are certainly the same within the uncertainty of the model and properties of HY, anisotropy in the repulsive part of the potential would be expected to have the opposite sign and, in effect, make the potential look more isotropic. The multipole barrier for ArHCl is shown in Figure given by (all values are in cm ) III-2 and is 80 e *> o 60 - 40 - 20 0 -180 -140 -100 -60 -20 20 60 100 140 180 THETA Figure III-2. The multipole potential barrier for ArHCl from Eq. (111-12). oo o 81 V(9) = 58.6 - 25.3 P1(cos9) - 13.6 P2(cos9) - 14.3 P3(cos9) -4.0 P4(cos9) - 1.4 P 5 (cos9). (111-12) While the experimental X-HY potential surfaces ' ' ' are generally parameterized in terms of Pi(cos9) and P 2 (cos9), it is clear from Eq. (111-12) that the P-(cos8) term plays a large role in the anisotropy of the surface and must be accounted for. It is also evident that the octupole and hexadecapole moments as well as the higher order polarizabillties of HY are needed to get good agreement with experiment. Unfor- tunately, these are unknown for most molecules and are difficult to estimate. It is hoped, however, that the long range potential in Eq. (III-ll) can be matched to an ab initio or empirical repulsive potential to reproduce the microwave data as well as scattering and line broadening results. 82 CHAPTER IV XENON NUCLEAR QUADRUPOLE COUPLING IN XENON HYDROGEN CHLORIDE A. Introduction The first determination of a nuclear quadrupole coupling constant for a rare gas atom in a van der Waals molecule was reported in a recent paper on KrHF. That study was extended by investigating the nuclear quadrupole coupling in XeHCl. Xe The nuclear quadrupole inter- action arises through a coupling of a nuclear quadrupole moment with the 62 electric field gradient at that nucleus. In a free ground state rare gas atom, the electronic distribution about the nucleus is spherically symmetric, thus, the field gradient and coupling constant are identically zero. The measurement of a nonzero nuclear quadrupole coupling 131 constant for Xe in XeHCl indicates that the free Xe electron distri- bution is to some extent perturbed in the complex. field gradient is interpreted m The observed electric terms of Sternheimer antishieldmg and formation of the weak Xe-HCl van der Waals bond. Subsequent to the study presented here, slight modifications to the spectrometer and experimental technique permitted the rotational spectrum of 131 XeHCl to be examined in lower rotational states than had been previously possible. The lower J rotational transitions have a more highly resolved hyperfine structure which allowed better values of the Xe nuclear quadrupole coupling constants to be obtained and a more 83 complete analysis of the effect to be made. study are presented elsewhere. The results of this latter The major conclusions concerning the nature of the Xe quadrupole coupling remain unchanged, however. The main purpose of the present chapter is to complete the reporting of the experimental results and to outline the origin of the rare gas nuclear quadrupole coupling in van der Waals molecules. As an aside, it is indicated how useful information may be obtained from poorly resolved spectra by using the effects of isotopic substitution as a predictive tool. B. Spectra and Results The spectra of 131 XeHCl and 131 XeDCl are greatly complicated by the presence of two quadrupolar nuclei in these species (deuterium coupling is ignored). There are more than 100 hyperfine components in the spectra observed here. This large number of components greatly reduces the intensity of any particular component, thus lowering the signal-to-noise ratio. In addition, severe overlapping of the spectral lines and the Doppler doubling phenomenon lower the effective resolution and make the unambiguous assignment of the spectra difficult. The hyperfine Hamiltonian is given in Eq. (II-3) and is the sum of quadrupole Hamiltomans for Cl and Xe. The Hamiltonian matrix elements were calcu- lated using the method described in the discussion following Eq. (II-3). For 129 131 XeH XeHCl, the Cl coupling constant, x . » was fixed at its value in Cl (-34.76 MHz) and the hyperfine Hamiltonian matrix was con- structed and diagonalized as a function of the Xe coupling constant, x v , AS 84 until plots of the predicted spectra suitably matched the observed transition frequencies. value of x x After the spectral assignment and an approximate were obtained in this manner, the data were fit by least squares involving diagonalization of the Hamiltonian matrix to give Xv = -4.9(2)MHz. are listed in Table The observed transition frequencies for IV-1 . 131 35 Ji XeH Cl Figures IV-1 and IV-2 show the predicted spectra as a sum of Lorentzians as well as the measured frequencies. A second evaluation of x X e can be obtained by considering the centrifugal distortion in XeHCl. In a pseudodiatomic model of the van der Waals stretching vibration, the centrifugal distortion constant, Dj, is proportional to the inverse square of the stretching reduced mass (see Eqs. (II-8), (11-10) and (11-16)). Using the precisely known values of Dj in 129 XeH 3 5 Cl (3.8130(15)kHz) and 132 XeH 35 Cl(3.7762(31)kHz), this scaling law is found to hold within the experimental uncertainly. the scaling law and the value be 3.7882(45)kHz. of Dj in With 129 35 XeH Cl, Dj is calculated to It is now noted that as x v changes, the line positions move with respect to the hypothetical center frequency, v Q . Thus, by using the best determined line from each R branch (J=4,F1=5/2, 7/2, F=8/2, 10/2-*J=5, F 1 =7/2, 9/2, F=10/2, 12/2 and J=6, F-,=9/2, 11/2, F=12/2, 14/2+ J=7, F^ll/2, 13/2, F=14/2, 16/2), the center frequencies, \>0, can be calculated as a function of Xx e obtained from and the rotational constant, B 0 , and Dj 33 v Q = 2B0(J+1) - 4Dj(J+l) 3 . (IV-1) Figure IV-3 shows a plot of D, as a function of x x • Using this graph 85 Table IV-1. Observed Isotope J 131 4 -*• 5 XeH 3 5 Cl J' L31 XeH 3 5 Cl and 131 XeD 3 5 Cl. Fx,F F1',F' 7/2,8/2 9/2,10/2 7,2,10/2 5/2,8/2 9/2,12/2 7/2,10/2 9/2,8/2 11/2,10/2 9906.855 11/2,10/2 9/2,10/2 13/2,12/2 11/2,12/2 } 9906.888 11/2,14/2 13/2,16/2 9906.934 9/2,12/2 11/2,14/2 9906.973 11/2,8/2 9/2,10/2 13/2,10/2 } 11/2,12/2 13866.704 11/2,14/2 9/2,12/2 13/2,16/2 11/2,14/2 } 13866.731 13/2,12/2 15/2,14/2 13866.956 15/2,14/2 17/2,16/2 13866.967 15/2,12/2 13/2,10/2 13/2,14/2 15/2,16/2 17/2,14/2 15/2,12/2 15/2,16/2 17/2,18/2 15/2,18/2 17/2,20/2 13867.004 13/2,16/2 17/2,18/2 13867.016 7/2,10/2 5/2,8/2 9/2,12/2 7/2,10/2 } 9758.986 6 +7 11/2,14/2 9/2,12/2 13/2,16/2 11/2,14/2 } 13660.713 7 + 8 13/2,16/2 11/2,14/2 15/2,18/2 13/2,16/2 } 15610.676 6 -> 7 XeD JJ Cl Frequencies for 4 -»-5 Frequency (MHz) 9906.271 } 9906.387 13866.981 J=4-vJ =5 F F—-F 1 , F' 7/2, 10/2^9/2, !2/2 5/2, 8/2~*7/2, 10/2 J 11/2, 14/2—13/2, 16/2 11/2, 10/2—13/2,12/2 9/2, 10/2—N/2, 12/21 J 9/2,12/2—11/2,14/2 7/2, 8/2—9/2,10/2 9 / 2 , 8/2—11/2,10/2 9906.2 9906.9 MHz '0 Figure IV-1. The J=4+J=5 transition in 131,XeHCl predicted with the constants in Table IV-2. The vertical lines are the measured frequencies. 03 J = 6—J = 7 F' F' F r rF r 11/2, 14/2—13/2, 16/2 9/2, 12/2— H/2, 14/2 15/2, 16/2—17/2, 15/2, 12/2—17/2, 13/2, 10/2—15/2, 13/2, (4/2—15/2, 18/2 14/2 12/2 16/2 Jj 11/2, 8/2—13/2,10/2 9/2, 10/2—11/2,12/2 15/2, 18/2—17/2, 2 0 / 2 15/2, 14/2—17/2,16/2/ 13/2,16/2—15/2,18/2 13/2, 12/2—15/2,14/2J 13866.7 Figure IV-2. 13867.0 MHz The J=6-hJ=7 transltxon in 131, ^""-XeHCl predicted with the constants in Table IV-2. The vertical lines represent the measured frequencies. oo 3.90 X Q 3 3.7882 3.70 Xa( Xe)MHz Figure IV-3. Graph of Dj vs assumed values for Xa( 131, Xe) 00 oo 89 and the calculated value of Dj for 131 35 XeH Cl, x X e is estimated to be -5.0(3) MHz, which is consistent with the value obtained from the spectra. 131 35 XeD Cl, onlyone hyperfine component was obtained in In the case of each R branch and the above method was the only way to obtain x y • The value derived for x_ in 131 35 XeD Cl is -6.0(5) MHz. The spectroscopic AS constants of IV-2. 131 XeH 3 5 Cl and The values of x v 131 XeD 3 5 Cl obtained here are listed in Table derived in the subsequent study 55 of the Xe nuclear quadrupole coupling are -4.641(50) MHz and -5.89(20) MHz for 131 XeH 3 5 Cl and 131 Table IV-2. X e D 3 5 C l , respectively. Spectroscopic constants for XeD 3 5 Cl. 131 35 XeH Cl and 131 B0(MHz) 131 XeH 3 5 Cl 131 X e D 35 c l a C. D J (kHz) 3 X c l (MHz) a X (MHz) 990.86264(40) 3.7844 -34.76 -4.9(2) 976.11581(9) 3.4321 -44.82 -6.0(5) Assumed value. Analysis of the Xenon Nuclear Quadrupole Coupling Constant. The existence of a nonzero nuclear quadrupole coupling constant for Xe in XeHCl indicates the presence of a nonzero electric field gradient at the Xe nucleus. The electric field gradient,qaa,along the a-inertial axis is related to the measured coupling constant, x> °y 90 eQ where h is Planck's constant, e is the proton charge and Q is the nuclear quadrupole moment. For x = -4.9(2) MHz and Q = -0.12x10 the electric field gradient at the q aa = -5.6(2)xl014SC-cm~3 where SC is a statcoulomb. will be considered here. 131 Xe cm , 131 Xe nucleus is found to be , (IV-3) Four contributions to this field gradient Two contributions result from the electrostatic potential experienced at the Xe nucleus due to the neighboring HCl charge distribution. The effects of overlap of the Xe and HCl electronic distributions and possible charge transfer will also be considered. The coordinate system used to describe the interaction between an atom and a cylindrically symmetric molecule is shown in Figure IV-4. The electrostatic potential energy experienced by an elctron at the atomic position (r,8) due to the neighboring molecule can be expressed in terms of a multipole series as -eV = -ERerP,(cos9) - q^er P2(cos9) - ... , (IV-4) where P (cosG) is a Legendre polynomial and E R and q ™ are the electric field and field gradient, along R, measured at the atomic nucleus and arising solely from the molecular charge distribution. E R and q ^ can be expressed in terms of the permanent electric moments of the linear molecule as 2u 3Q_ 4Q En = -o Pi (cos9) + — J f P„(cos8) + __ P.(cos6) + ... R i I ^ 3 Rj R4 (IV-5) 91 R Figure IV-4. Coordinates used to describe the electrostatic interactions between an atom and a linear molecule, 92 and q RR 6„ " ~f p i ( c o s = e) 12Qm 2 on 1 p (cos6) —r ? " ~T p 3 ( c o s e> " • • ' R where u, Q R 5 <IV_6> R and P, are the permanent electric dipole, quadrupole and octupole moments, respectively. E„ and q„„ can be calculated for XeHCl by using the known structure of XeHCl (see Table II-5) and properties of HCl. They are found to be E_ = 4.59xl04SC-cm~2 R (IV-7) and q„„ = 13.87xl02SC-cm~3 RR . (IV-8) The first term on the right hand side of Eq. (IV-4) gives rise to dipole polarization of Xe in XeHCl. When the polarizable Xe atom is placed in an electric field, its initial spherical symmetry is distorted by the perturbation. The interaction energy arises in second order of perturbation theory. To estimate the effect of dipole polarization on q, the mixing of the 5p and 6s states of Xe is considered. accounts for about 20% of the Xe polarizability. This mixing Using the perturbation Hamiltonian Ex = - E R er cos 6, (IV-9) the wave function for the p electron correct to first order is given by i> | = ( l - b 2 ) ^ <j>5p + hj>6s • (IV-10) 93 where 6 5p and A. are zero order Xe atomic wave functions, and assuming & 6s b«l, = _ ^6e\eT cos9 l<l>5p>= ER<uz> _- EEr_5^„ 5 (IV-11) hv '5P6 . 5p 6s where v is the frequency of the corresponding transition in Xe. (u ) can be estimated from the oscillator strength, f, of this transition in X e 6 6 by (u) = (3fhe2/8Tr2mv)'2 where m is the electron mass. , (IV-12) The excitation can now be related to the field gradient at the Xe nucleus. The field gradient operator is given by q - -2(^3") (3cos 2 e 1 -i), (iv-13) where the sum runs over all charges, e., outside of the Xe nucleus. Realizing that s electrons and closed shells do not contribute to q, determination of the field gradient due to polarization, q , reduces to the evaluation of q p = b2(<(>5p[^3(3cos29-l)|t(,5p). Integrating over angles for a p i SP - "*2<5»-< 3>„ i \ r 3/ av where b (IV-14) electron gives .' (IV 15) " represents the fractional p hole character in the 5p electron 94 distribution. Using analytical Hartee-Fock atomic orbitals for Xe <l/r3> is found to be 112xl0 24 cm -3 . av 67 , Thus, q„ = -7.0xl012SC'cnT3 P and accounts for about 1% of the observed field gradient. The second term on the right of Eq. (IV-4) causes quadrupole polarization of Xe and results in the Sternheimer antishielding effect. In this effect, the external field gradient, q ^ , arising from HCl, induces a quadrupole moment in the electronic distribution of Xe which in turns induces a field gradient, qin<j> at the Xe nucleus. The Sternheimer shielding factor, You, is defined in terms of the external and induced field gradients by Y» = - lind q (IV-16) RR and arises in first order of perturbation. the Xe nucleus, q Thus, the field gradient at , is the sum of the external and induced field grad- ients or = 1 R R C 1 ^-) • %s < IV " 17 > The difficulty in this analysis is the determination of the shielding factor. Too for Xe was calculated, in this lab, by using a variational procedure to solve for the Xe atomic wave function perturbed by H l = iRR er2 p 2(cos9>- (IV-18) The details of the calculation have been given before relativistic and electron reorganization effects. ' and neglect Once the perturbed wave function is known,qincl is computed with the aid of Eq. (IV-13) 95 and Ym obtained from its definition in Eq. (IV-16). Y*. for Xe is found to be -138 and the field gradient becomes q sg = -5.4xl0 14 SC-cm"3 . (IV-19) This effect is quite large and accounts for the entire measured field gradient in 131 XeHCl within the experimental uncertainties. subsequent studies of the Xe coupling in XeHCl In the and XeHF, it was found that all field gradients could be explained solely in terms of Sternheimer shielding if YM = -153. This experimentally derived value compares well with the calculated value of Yoo= -138 shown above and a value of Yoo = -177 calculated from relativistic Hartree-Fock-Slater -i, theory. 7 0 For completeness, two additional contributions to the field gradient are considered here. In XeHCl, it might be expected that the electron distributions of Xe and HCl will begin to overlap. This will necessitate electron rearrangements to maintain conformance to the Pauli principle. It appears unlikely that this reorthogonali2ation will have much effect on the field gradient at the Xe nucleus since only the shape of the outer electron distribution should be severely affected and q depends on (l/r3) which Is only large near the nucleus. Thus,it is assumed that q ro &0. Finally, if charge is transferred from Xe to HCl in a Lewis acid- Lewis base type interaction, a field gradient, q ct , will arise from the partial depletion of a p electron. for polarization: The treatment is similar to that 96 %t - - c 2 4 K p > a v > where c <iv-2o> represents the fraction of an electron transferred. In the model considered here, q aa » %s + q + P q ro + q ct • < IV " 21 > Using the previous results, q ct is found to be -1.8xl0^3 SC-cm c 2 = 0.0004. or In other words, on the order of 0.04% of an electron appears to be transferred from Xe to HCl. In conclusion, the electric field gradient at the Xe nucleus in XeHCl is found to arise primarily through the Sternheimer shielding effect, and that charge transfer plays a minor, if any, role in this complex. 97 CHAPTER V ARGON CYANOGEN CHLORIDE A. Introduction The assignment of the rotational spectrum of the weak complex formed between an argon atom and the linear cyanogen chloride molecule is reported here. ArClCN is a planar asymmetric rotor and its rotational constants, centrifugal distortion constants and diagonal elements of the Cl and N nuclear quadrupole coupling tensors are determined. This study represents the first time that complete centrifugal distortion and normal mode analyses have been performed for the weak van der Waals bond between a rare gas atom and a molecule. These data allow a reasonably quantita- tive determination of the bending and stretching frequencies and average molecular structure in the part of the molecule involving the van der 1L. Waals bond. Furthermore, the measurement of the N and 35 Cl nuclear quadrupole coupling in ArClCN allows the first comparison between the average structure as described above with that obtained by projecting the free C1CN molecule's nuclear quadrupole coupling constants onto the principal inertial axes of the ArClCN van der Waals molecule. The agree- ment is quite good thereby justifying the use of this latter method of determining structures of weakly bound rare gas-molecule complexes. The van der Waals binding Interaction between an atom and a linear molecule has been previously investigated in inertially linear complexes 98 such as the rare gas-hydrogen halides asymmetric ArOCS is T-shaped. 72 and ArC0 2 71 (see Chapter II), as well as the complexes. Like ArOCS and ArC02, ArClCN However, ArClCN provides certain features which permit a more intimate look at the van der Waals interaction. Only the structure was determined for ArOCS, but for ArC0 2 , an analysis of the centrifugal distortion was performed. Upon complexation, two weak modes of vibration are introduced into the atom-molecule system. These correspond to a stretch and an in-plane bend of the van der Waals bond. C 2 „ symmetry, only two of the Kivelson-Wilson T'S Since ArC0 2 has depend on the weak modes and the bending and stretching force constants are directly calculated from them. ArClCN, on the other hand,has C s symmetry and all four of the linearly independent T'S depend on the two weak modes. This allows the force constants to be fitted and the coupling between the modes to be obtained from a bend-stretch interaction constant. methods of vibrational analysis can also be used to obtain further information about the complex once the force field is known. tional frequencies, structure of the average configuration estimate of the inertial defect Standard are derived for ArClCN. 76 The vibraand an The fact that the estimated inertial defect differs by only 2% from the experimental value indicates that a normal mode analysis retains a large measure of validity for these weakly bound complexes and, in addition, confirms the 78 planarity of ArClCN. Finally, the spectrum of ArClCN shows the effects of Cl and N nuclear quadrupole coupling. In an inertially linear complex, nuclear quadrupole coupling is used to determine the vibrationally averaged 99 structure of the complex. This procedure depends on the assumption that the measured coupling constant simply represents the averaging of a monomer coupling constant over the van der Waals modes. While there is evidence that this assumption is reasonable in the rare gas-hydrogen 53 halides, a study of the N coupling in NCCN"-HF 54 and (HCN)2 that it certainly is not valid in hydrogen bonded species. shows In planar ArClCN, however, the component of the nuclear quadrupole coupling tensor perpendicular to the plane does not depend on an average over the van der Waals modes. This allows a quantitative measure of how much the coupling constants of free C1CN are perturbed by complexation with Ar. It is found that there is definitely a small change. However, it is reassuring that the angle between C1CN and the b-inertial axis as determined from the rotational constants alone and from N and Cl quadrupole coupling agree quite well. B. Spectra and Results The Hamiltonian for ArClCN is the sum of rotational and nuclear quadrupole parts H = HR + HQ . (V-l) 79 The rotational Hamiltonian, HR, is appropriate to the semi-rigid rotor and involves the A, B and C rotational constants as well as the quartic centrifugal distortion constants. The quadrupole Hamiltonian, HQ, includes both Cl and N nuclear quadrupole coupling and is given by H 0 = Q(C1):V(C1) + Q(N):V(N) , (V-2) where Q(i) and V(i) are the nuclear quadrupole moment and electric field gradient tensors, respectively, for nucleus i. Matrix elements of H are calculated in the symmetric rotor basis and with the angular momentum coupling scheme fcl + fo - I (V 3) " I + J =F 1 <\l <\l Following Flygare and Gwmn 80, matrix elements of the 2 quadrupole Hamiltonian in the basis of Eq. (V-3) are given by (IC1INIJFMF|HQ|IC1INI'J'F'M'F) , ^ . u t f . 1/2/ J 2 J'\-l Cl I' 2) x[(2I+l)(2I'+l)] (_J0 j ] j j, j FJ I I' 2 I 1 ^ W (V-4) fl l'2 2 (-1) (^l5ci%) e Q c l (q J J f ) c l + (-l) (!ININIClJe QN(qjji)N I C1 2 IC1^ "^1° W I / XN 2 % ^N ° h where e is the proton charge, QJ is the nuclear quadrupole moment of nucleus i, and (qTT,) is the electric field gradient at nucleus 1. JJ l The quantities in parentheses and braces are 3-j and 6-j symbols, respectively, and 6., is the Kronecker delta. Using the method of Benz, 81 et.al., ^ J J ' \ is readily calculated in the symmetric rotor basis 101 eq J J t = <JKj|Mr [J'K'J) = <JKJ| V 2Q | J'K' J> , 3z (V-5) 3 V —j ^-s the electric field gradient in the direction defined by 8z the angular momentum and V„_ is the corresponding spherical component where of the field gradient tensor. V' 2q The electric field gradient components, ,are constant in a body-fixed axis system and are related to Vor. by <£U V2o = q=ZV2qD^(-Y-3Hx) , 37 (V-6) (2) where D M , M is the matrix representing the 5-dimensional irreducible representation of the 3-dimensional rotation group and (aBY) are the Euler angles locating the body-fixed axes in the space-fixed frame. Since V 2 Q is real, 2 v2o=v2o= . ZD^WX-DVJ q=-2 M (v-7) 4 In addition, the symmetric rotor wave functions are given by ,(J) J2J+1^ *&'Wq D(J) (aeY) MK ' (v 8) " thus , <JKJ|V20|J'K'J) = [(2J+l)(2J'+l)] ^(-l) J+K (_j I ^ _\ ^ V 2 _ q . (V-9) Defining the spherical components of nuclear quadrupole coupling tensor as X ^ = e2QV'q , " (V-10) 102 Eq. (V-4) becomes I'+I (^I^JKFMJIHQIICX^I'J'K'F'M'P) I' 2 ] N = 6FF,6MpM, i z | i x[(2l+l)(2I'+l)(2J+l)(2J , +l)] ^ , j ' F ] (l +1 +F+K _K _2q K-) (V *n) ("I I'2 c - i ^ l ^ c i V (x2.q)cl + (-D1 }jb&±.{ (X2.q)N I 2 X C1 Cl\ °y -^I X Nz % 1-%°% For a near prolate planar asymmetric rotor, the axes x, y and z corres- pond to the principal inertial axes b, c and a, respectively, and the spherical components of X are related to the usual Cartesian components by x 20 = x X2+1 = aa i ^ X a b *2±2 - & 2 (V"12> (X bb -X cc ) • It is seen from Eq. (V-ll) that H Q is diagonal in F and M F < Thus,to calculate the exact quadrupole hyperfine energies it is simply necessary to set up the matrix of Eq. (V-ll) in blocks of F, add the rotational energies calculated in the same basis and diagonalize the resulting matrix. The first order quadrupole energies can be obtained in the scheme above by considering only the AJ=0 and AK>0,±2 matrix elements. The data for ArClCN were fit by an iterative procedure consisting of the following steps: 1. Estimates were made of the rotational parameters and quadrupole coupling constants, and both the first order and exact energy levels were calculated as shown above. 2. The differences between the calculated first order and exact energies were used to adjust the measured frequencies which allowed a least squares fit based on the first order expression to be performed. The results of this fit gave the diagonal elements of the coupling constant tensors and center frequencies of the transitions. 3. The line centers were fit to the Watson parameters A", B", C", T l> T 2 ' T aaaa' T bbbb and T cccc us inS 82 tne program written by Kirchhoff. 82 To was calculated by invoking the planarity conditions and was held fixed in the fit as were the off-diagonal elements of the quadrupole coupling tensors which were estimated from the molecular structure (see Section V-E). Since the higher order quadrupole corrections are fairly insensitive to small changes in rotational and quadrupole parameters, only 2 iterations were required to attain self-consistency. Table V-l shows the observed transition frequencies as well as those calculated from the fit of the hyperfine structure in 2 above. The fit is very good with the standard deviation of the residuals being about 2 kHz. The line centers are listed in Table V-2 and Table V-3 shows the spectroscopic constants for ArClCN. C. Molecular Structure The vibrationally averaged structure of ArClCN can be derived from 104 Table V-1. JK Observed and c a l c u l a t e d frequencies for ArClCN. The c a l c u l a t e d frequencies are obtained from Eq. (V-ll) and data i n Tables V-2 and V-3. - 1 C J ' K 1 I K «hi [,F -*• I',F' Observed (MHz) Calculated (MHz) Residue (kHz) Ar 35 ClCN 3 13" 4 04 5/2,11/2 3/2 ,9/2 3/2 3/2 5/2 5/2 ^O"1!! 5 05~ 5 14 1 -2 ^ 1 ^12 7/2 5/2 9/2 7/2 5/2,13/2 3/2 ,11/2 3/2 ,9/2 3/2 7/2 5/2 11/2 5/2 ,9/2 7293.5468 7294.2585 7301.1910 7306.5914 7306.7150 7307.2577 7293.5489 7294.2583 7301.1900 7306.5923 7306.7137 7307.2571 1/2 5/2 3/2 3/2 5/2 1/2 3/2,3/2 5/2 5/2 1/2 5/2 3/2 3/2 5/2 1/2 1/2 5/2 3/2 5/2 7/2 3/2 3/2, 1/2 5/2 3/2 7382.3654 7382.9091 7383.5704 7402.3848 7403.1015 7403.3984 7418.8524 7418.9009 7382.3647 7382.9086 7383.5696 7402.3842 7403.0979 7403.3977 7418.8546 7418.9057 5/2 15/2 3/2 13/2 5/2,13/2 5/2 11/2 5/2 3/2 5/2 5/2 15/2 13/2 13/2 11/2 7662.4832 7663.6175 7682.1778 7683.1972 7662.4810 7663.6160 7682.1815 7683.1972 5/2 5/2 5/2 5/2 3/2,3/2 1/2 1/2 5/2 5/2 3/2 3/2 3/2 5/2 5/2, 7/2 1/2 3/2 3/2 5/2 5/2 7/2 3/2 5/2 5/2 7/2 1/2 3/2 5/2 3/2 3/2,1/2 5/2 3/2 3/2 5/2 3/2 3/2 5/2 5/2 3/2 5/2 3/2 3/2 3/2 1/2 5/2 1/2 5/2 1/2 5/2 5/2 7/2 5/2 1/2 5/2 3/2 5/2 7/2 3/2 7/2 7/2 5/2 9/2 5/2 5/2 3/2 1/2 9876.2879 9876.4974 9876.6020 9883.6233 9883.9012 9884.2403 9885.5922 9886.1392 9886.3490 9895.7508 9896.0942 9896.4123 9896.5883 9896.8947 9901.0766 9911.4650 9911.5038 9876.2900 9876.4989 9876.6048 9883.6246 9883.9028 9884.2406 9885.5899 9886.1394 9886.3496 9895.7498 9896.0904 9896.4111 9896.5860 9896.8938 9901.0746 9911.4672 9911.5049 1/2 5/2 3/2 3/2 5/2 1/2 -2.1 0.2 1.0 -0.9 1.3 0.6 0.7 0.5 0.8 0.6 3.6 0.7 -2.2 -4.8 2.2 1.5 -3.7 0.0 -2.1 -1.5 -2.8 -1.3 -1.6 -0.3 2.3 -0.2 -0.6 1.0 3.8 1.2 2.3 0.9 2.0 -2.2 -1.1 105 Table V - 1 . "V+f Continued JK K ' :I +I I,F I',F' Observed (MHz) Calculated (MHz) Residue (kHz) Ar 35 ClCN 4 14" 5 05 2 02""3i3 5 14" 5 23 4 13 -4 22 1/2,9/2 5/2,13/2 3/2,11/2 3/2,5/2 3/2,9/2 5/2,7/2 3/2,7/2 5/2,11/2 5/2,9/2 1/2,11/2 5/2,15/2 3/2,13/2 3/2,7/2 3/2,11/2 5/2,9/2 3/2,9/2 5/2,13/2 5/2,11/2 10478, 4498 10478, 5578 10479, 1345 10485, 3092 10485, 4204 10485. 8510 10488, 7376 10488, 8226 10489. 3186 10478 .4505 10478 .5569 10479 .1357 10485 .3081 10485 .4197 10485 .8518 10488 .7350 10488 .8252 10489 .3186 -0.7 0.9 -1.2 3/2,5/2 5/2,7/2 3/2,3/2 3/2,5/2 5/2,7/2 3/2,7/2 5/2,9/2 5/2,3/2 5/2,5/2 5/2,3/2 3/2,7/2 5/2,9/2 1/2,5/2 5/2,3/2 5/2,5/2 1/2,3/2 5/2,1/2 1/2,3/2 5/2,1/2 5/2,7/2 5/2,9/2 3/2,5/2 5/2,5/2 3/2,7/2 5/2,7/2 5/2,9/2 5/2,5/2 3/2,7/2 3/2,3/2 3/2,9/2 5/2,11/2 1/2,7/2 5/2,3/2 1/2,5/2 5/2,3/2 5/2,3/2 1/2,5/2 5/2,1/2 12213.2438 12213.7278 12213.8580 12217.2176 12217.6096 12221.5084 12222.2299 12223.3418 12223.5600 12223.6625 12229.7545 12230.5968 12230.8318 12231.6507 12231.9218 12240.0669 12240.0831 12240.3551 12240.5969 12213.2444 12213.7274 12213.8583 12217.2191 12217.6110 12221.5124 12222.2341 12223.3394 12223.5573 12223.6610 12229.7544 12230.5973 12230.8359 12231.6480 12231.9185 12240.0692 12240.0811 12240.3534 12240.5949 -0.6 0.4 -0.3 -1.5 -1.4 -4.0 -4.2 2.4 2.7 1.5 3/2,13/2 5/2,15/2 1/2,11/2 3/2,13/2 5/2,15/2 1/2,11/2 12403.1276 12403.7029 12403.7870 12403.1270 12403.6995 12403.7910 3/2,11/2 5/2,13/2 1/2,9/2 5/2,5/2 3/2,11/2 5/2,13/2 1/2,9/2 5/2,5/2 12858.8818 12859.6706 12859.8196 12863.9040 12858.8820 12859.6683 12859.8187 12863.9070 -0.8 2.6 -2.6 0.0 1.7 2.0 -0.2 2.3 0.9 -3.0 106 Table V-1. Continued I,F -1K+1 K I',F' -1 K +1 Observed (MHz) Calculated (MHz) Residue (kHz) Ar 35 ClCN 12' 3 21 5/2,9/2 3/2,9/2 5/2,11/2 1/2,7/2 5/2,9/2 3/2,9/2 5/2,11/2 1/2,7/2 13312.5822 13330.9137 13331.9352 13332.1766 13312.5822 13330.9144 13331.9335 13332.1776 0.0 -0.7 1.7 -1.0 5 15-606 1/2,11/2 5/2,15/2 3/2,13/2 1/2,13/2 5/2,17/2 3/2,15/2 13638.5364 13638.6056 13639.0566 13638.5342 13638.6065 13639.0579 2.2 -0.9 -1.3 2 H~220 3/2,7/2 5/2,9/2 1/2,5/2 3/2,7/2 5/2,9/2 1/2,5/2 13750.1401 13751.4827 13751.9306 13750.1394 13751.4835 13751.9306 0.7 -0.8 0.0 3 03" 4 14 5/2,7/2 5/2,9/2 3/2,5/2 5/2,5/2 3/2,7/2 3/2,9/2 5/2,11/2 1/2,7/2 1/2,5/2 5/2,9/2 5/2,11/2 3/2,7/2 5/2,7/2 3/2,9/2 3/2,11/2 5/2,13/2 1/2,9/2 1/2,7/2 14412. 2356 14412. 8662 14413. 0025 14418. 2724 14418. 7803 14426. 8087 14427. 6076 14427. 7782 14433. 4958 14412 ,2363 14412 .8660 14413 .0008 14418 .2725 14418 ,7816 14426 ,8062 14427 .6083 14427 ,7791 14433 ,4964 -0.7 0.2 1.7 -0.1 -1.3 2.5 -0.7 -0.9 -0.6 2 -2 12 Z 21 1/2,5/2 3/2,7/2 5/2,5/2 5/2,7/2 3/2,5/2 1/2,5/2 3/2,7/2 5/2,5/2 5/2,7/2 3/2,5/2 14716.2840 14716.3485 14716.5308 14717.1908 14717.2134 14716.2839 14716.3482 14716.5288 14717.1917 14717.2149 0.1 0.3 2.0 -0.9 -1.5 3 -3 13 J 22 1/2,7/2 5/2,11/2 3/2,9/2 1/2,7/2 5/2,11/2 3/2,9/2 15217.1295 15217.2348 15217.6790 15217.1288 15217.2355 15217.6790 0.7 -0.7 0.0 4 1/2,9/2 5/2,13/2 3/2,11/2 1/2,9/2 5/2,13/2 3/2,11/2 15892.3797 15892.4990 15893.1368 15892.3775 15892.4997 15893.1383 2.2 -0.7 -1.5 3 14 -4 23 107 Table V - 1 . Continued v+r'^+i I , F I',F' Observed (MHz) Calculated (MHz) Residue (kHz) Ar 35 ClCN 4 04- 5 15 3/2,11/2 5/2,13/2 1/2,9/2 3/2,13/2 5/2,15/2 1/2,11/2 16521.9899 16522.7221 16522.8438 16521.9914 16522.7200 16522.8444 -1.5 2.1 -0.6 5 15" 5 24 1/2,11/2 5/2,15/2 3/2,13/2 1/2,11/2 5/2,15/2 3/2,13/2 16744.2662 16744.3898 16745.1402 16744.2684 16744.3885 16745.1393 -2.2 1.3 0.9 Ar 37 ClCN OOO" 1 !! 5/2,5/2 5/2,5/2 5/2,3/2 7/2,5/2 7269.866 7270.577 1 X -2 01 n.2 5/2,3/2 7/2,5/2 3/2,1/2 7/2,3/2 9/2,5/2 5/2,1/2 9724.302 9725.131 9725.438 02" 3 i3 7/2,3/2 9/2,5/2 5/2,1/2 9/2,3/2 11/2,5/2 7/2,1/2 2 12021.547 12022.384 12022.626 Table V-2. J Line centers for ArClCN . -*•.]•', „, -lK+l K_iK+i Observed (MHz) Calculated (MHz) Residue (kHz) Ar35ClCN !3-404 7298.2691(8) 7298.2706 -1.5 ^O^ll 7398.9402(7) 7398.9404 -0.2 7670.3592(10) 7670.3597 -0.5 or2i2 9891.5025(5) 9891.5036 -1.1 4 14" 5 05 10482.5052(7) 10482.5047 0.5 2 02-313 12225.1817(5) 12225.1781 3.6 5 14" 5 23 12399.7199(12) 12399.7213 -1.4 4 13~ 4 22 12854.3276(10) 12854.3251 2.5 3 12- 3 21 13325.2184(10) 13325.2192 -0.8 5 15" 6 06 13641.7369(12) 13641.7361 0.8 2 H-220 13743.0464(12) 13743.0455 0.9 3 03~ 4 14 14422.2449(7) 14422.2454 -0.5 210-2 12 21 14716.5635(9) 14716.5696 -6.1 3 5 05" 5 14 1 3 13" 3 22 15220.1227(17) 15220.1202 2.5 4 14~ 4 23 15896.8037(12) 15896.7983 5.4 4 04~ 5 15 16517.7069(12) 16517.7086 -1.5 5 15~ 5 24 16749.5614(12) 16749.5644 -3.0 109 Table V-2. K Continued -l K +l"' K-1K+1 Observed (MHz) Calculated (MHz) Residue (kHz) Ar 37 ClCN ^O"1!! 7267.264 'oi-'u 9721.081 2 a 02~ 3 i3 12018.070 The uncertainties represent one standard deviation from the least squares fit of the hyperfine structure. Table V-3. Spectroscopic constants for Ar Constant Value (MHz) a A" 6152.5411(21) B" 1577.0362(8) C" 1246.7514(6) T l -0.5760(2) T 2 -0.09728(8) x3b 2.059(3) T aaaa -0.2348(17) T bbbb -0.05597(10) T cccc -0.02160(7) Xaa Cl bb Cl Xcc Cl b x ab N x x *aa N Xbb N XCc Nb x x a C1CN. 37.9468(23) -79.5239(20) 41.5771(43) 21.9 1.6403(22) -3.4571(20) 1.8168(42) 0.95 ab Values in parentheses represent one standard deviation from the least squares fit. Assumed value held fixed in fit. Ill its rotational constants by treating the CICN subunit as a rigid linear rod having properties identical to those of free CICN. The appropriate properties of CICN are shown in Table V-4. In the limit that CICN remains Table V-4. Molecular constants of CICN. Constant Value B0(MHz)a 5970.820(10) X C1 (MHz) b o -83.27519(40) x*0mz)h r -3.62277(90) ci-c& C 1.627(1) rc_N&c 1.166(1) O—l c kg-, _c(mdyn-A ) 5.284(14) °-l c kc_N(mdyn-A ) 17.982(80) k 0.3502(5) ci-c-N(mdyn^)C See Ref. 83. See Ref. 84. c See Ref. 85. b linear, the structure of ArClCN is necessarily planar and can be described by 2 parameters. These are conveniently taken to be the distance R from the CICN center of mass to Ar and the angle 9 cm between Rcm and the CICN axis (see Figure V-1). In this coordinate system, the non-zero moments of inertia are given by (s ee Appendix I where z is along Rcm and y is perpendicular to the molecular plane) i a-axis Figure V-1. Coordinates used to describe the structure of ArClCN. X x xx = yy = I h + ^ l C N 0 0 3 9 cm D + IC1CN (V-13) 2 X X where zz = ^lCN 3 1 1 1 9cm xz = x zx = "^lCN^^cm^^cm I Q ^ C N *"S t ie m o m e n t * °f inertia of free CICN and I_ is a pseudo- diatomic moment of inertia given by h _ mArmClCN 2 mA + 0 1 * 0 1 1 Ar CICN (V_14> • The 2 structural parameters can be obtained from any pair of A, B and C by solving the moment equations. The structures determined for the 3 possible choices of rotational constants are listed in Table V-5. Table V-5. The structures of ArClCN as derived from the effective moments of inertia. Pair of l°a useda 1° and 1° aa bb 1° and 1° aa cc 1° and 1° bb oc 9. R^tf) 9 cm (deg) 3.629 81.5 3.645 81.5 none b 3.645 O 0 The effective moments of inertia are obtained by I =(h/8ir A) etc. where A, B and C are the T free rotational constants in Table V-7. No angle is consistent with the observed I.. . 114 It is apparent from Table V-5 that the calculated structures vary considerably with the choice of rotational constants. Indeed, no rigid planar structure is consistent with the measured B and C. This is a consequence of the nonrigidity of the van der Waals complex as indicated by its large inertial defect, AI = I c c - I a a ^ b b = 2 - 7 5 The zero superscripts on the 1° amu ^2 (v"15> • indicate that these are effective moments of inertia which are derived directly from the measured rotational cono stants. The structures obtained from the I a a are the usual r 79 structures. The problems posed by the inertial defect in calculating the structure can be largely overcome if the harmonic force field is known. It then becomes possible to compute the moments of inertia, I^a,and the structure of the average configuration. Discussion of this point will be taken up in Section V-D. Consideration of Eq. (V-13) shows that the 2 structures having the same R and with angles related by 8 c m + 9^m = IT will yield the same rotational constants. This ambiguity in the structure determination, namely, whether 6 c m is acute or obtuse can be settled by reference to the results of isotopic substitution in the linear molecule. In this 37 "^S case, Cl was substituted for J J C1 in CICN. Assuming the geometry of the complex remains constant upon isotopic substitution, the rotational 37 constants of Ar CICN were predicted on the basis of the 2 possible structures (T-shaped when 6 c m is acute or L-shaped when 9 c m is obtuse). These predicted rotational constants are compared with the measured ones in Table V-6. Clearly, ArClCN is T-shaped with the C-Ar line nearly Table V-6. Measured rotational constants of Ar 37CICN and those predicted on the basis of the two structures consistent with the Ar3->C1CN rotational constants. Measured A(MHz) B(MHz) C(MHz) 6040.4 1554.0 1226.9 perpendicular to the CICN axis. T-Shaped L-Shaped 6046. 1552. 1226. 6006. 1562. 1231. Additional structural parameters such as the C-Ar distance and the Ar-C-N angle are easily derived from R c m and 9 c m by simple trigonometry. D. Intramolecular Force Field The effects of centrifugal distortion in the rotational spectrum of ArClCN can be used to obtain information about its intramolecular force field. T abab Assuming planarity, four linearly independent T'S ( T a a a a , ^bbbb' and T aabb^ are related to the force field by c.3 J n ag F = i 3N-6 3N-6 (i) 74 , ,.s la^ i where R., is one of the internal coordinates in Figure V-2, 3^ i is the inverse of the force constant matrix and N=4 is the number of atoms. t a a a a and T b b b b are taken directly from the fit of the line Figure V-2. Internal coordinates of ArClCN. a, and cu are the m-plane and out-of-plane angles, respectively. 117 centers whereas T a b a b and T a a b b must be derived from any two of T^, T and TCCCC. 82 2 In analogy with the inertial defect, ArClCN exhibits a 82 large Tplanarity defect (AT = -1.54(10)kHz). This causes the calcu- lated T a b a f a and T a a b b to vary with which pair of to compute them. T;J.} T2 and TCCCC is used Table V-7 shows this variability in T a b a b and T a a b b and reflects the range of uncertainty expected on the basis of model error alone. It is to be noted that the model error greatly exceeds the measurement error. Strictly speaking, Eq. (V-16) is valid only when the laa refer to the equilibrium configuration of the molecule. and Jag These are seldom known and,in practice, vibrationally averaged moments of inertia and structures are used. In the procedures which follow, 2 sets of calculations were performed. The first set employed the effective moments of inertia, I^a, and the r Q structure. ft The second set used the average moments, I^a, and the structure of the average configuration which were derived in the course of the first set of calculations as shown below. Only the results of this second set of calculations are reported here. The J^g in Eq. (V-16) depend solely on the masses and geometry of 74 ArClCN. They can be calculated using the method of Kivelson and Wilson and expressions for the J^g are given in Table V-8. Alternatively, the (i) 86 Jog can be obtained from the matrix expression -1 a^6 Jn.»2fi B I I X •top % fy % ^ ffc (V-17) where J a o i s a 3N-6 dimensional column v e c t o r c o n t a i n i n g the J a g , G is Table V-7. Derived Molecular Constants of ArClCN. Tn and T 2 T 1 and T C C C C x 2 and T C C C C A(MHz)b 6152.369 6152.381 6152.384 B(MHz)b 1576.893 1576.905 1576.907 C(MHz)b 1246.701 1246.683 1246.679 T aaaa( kHz ) -234.8(17) -234.8(17) -234.8(17) T bbbb( kHz ) -55.97(10) -55.97(10) -55.97(10) T aabb(kHz) 53.7(14) 25.3(18) 25.3(18) ^abab^Hz) -310.3(11) -286.5(15) -281.7(17) AI(amu>A ) 2.739 2.747 2.748 50(13) 63.93(9) 62.52(16) 40(2) 35.17(1) 34.60(1) 12(7) 17.55(2) 17.06(4) F ^ (mdyn-A L) F (mdyn*A) F (mdyn) 25 X a Numbers in parentheses represent one standard deviation from the least squares fit. This represents the T-free rotational constant . 119 Table V-8. The non-zero derivatives J in terms of the internal coordinates shown in Figure V-2. R-^, R 2 and RQ are the C-Cl, C-Ar, and C-N lengths, respectively, cc^ and o, are the in-plane and out-of-plane Cl-C-N angles and a 2 is the Ar-C-N angle, m- is the mass of Cl, m 2 of Ar and nig of N. The X£ and z^ are the principal axis coordinates of atom i and 9. is the counterclockwise angle between the positive x-axis and R J . i Internal Coordinate 1 I, R, 2m,z,sm9., 2m-,x-LCOs9.. 2 R2 2m 2 z 2 sin9 2 2m 2 x 2 cos9 2 - — = - I Izz,z cos9 2 +I xx x 2 sin6 1 3 R„ 3 2m 0 z.sin6o 3 3 3 2nux 0 cos9o 3 J 3 - (l) xx (i) zz j(i) xz 1 fl Iyy 2m 3 4 5 2 -2m1R;.z cos9 1 2m..R,x,sin9, 2m2R2Z2cos92 ^n^RjX^inG;, z cos8-+I x-sineJ r I z„cos9»+I x„sin9 "I I yy L zz 3 3 xx 3 3J 2m, Ri1 r •, 2m 1 R =—- I z z ; z.. sin9..-I x x x.. cos9-j| yy 2m9R_ _ —£_£ I Iyy L ZpSinOg-I » Xgcose. J 120 42 the inverse of the well known G matrix it and B is the matrix which trans% forms the Cartesian coordinates,Xi9to the internal coordinates,Rj,by R = B X . (V-18) $ is the 3N dimensional column vector whose components are the atomic coordinates in the principal inertial axis system and I a is the 3N x 3N 87 dimensional auxilliary matrix introduced by Meal and Polo. It is formed by placing N identical 3x3 blocks along the diagonal. These blocks have the form /0 0 0\ i"" /0 0 0 l U N f l \0 - 1 0 / \l 0 -l\ /0 1 0> 0 0 , ^ Z = (-1 0 0 ]. 0 0/ \0 (V-19) 0 0, If J is the matrix whose columns are the Jag then %~%I $ and Eq. (V-16) becomes W ' f i x'x !., ^ 8 * (V 21) " 4TT •LaccLBB-LYY SS The force field of ArClCN was obtained by assuming the force constants in CICN remain unchanged upon complex formation and using Eq. (V-16) to fit the 4 t's to the (F~ ) . of the van der Waals modes. It was found that 3 inverse force constants were required to obtain a good fit of the centrifugal distortion constants. These correspond to the stretching of the Ar-C distance, F~T, the bending of the Ar-C-N angle, F j 5 , and a term 121 -1 coupling the two modes, F2^. from F Table by matrix inversion. V-7 The force constant matrix, F, was obtained The inverse force constants are shown in for the three sets of T ' S . The force constants listed in Table V-9 were obtained by taking the weighted averages of the corresponding (F ) . . in Table V-7 before the matrix inversion. While there is still a fair amount of uncertainty in F due to the model errors, it is reassuring that the interaction constant is negative as would be ex<- A 4 2 pected. The normal coordinate analysis is the starting point from which molecular information can be obtained from the force field. This analy- sis involves solving the secular equation nij r\j ?\j *Xi f\j A.^ = X x = 4ir^c2o)2 where oi^ is a vibrational frequency m cm - 1 , c is the speed of light in vacuum and L transforms the normal coordinates, Q x , to internal coordinates by <v 23) 1 1 1 ' " G is the kinetic energy matrix in internal coordinates given by «- « _ 1 £ where M~ (v 24) - is the inverse of the 3N x 3N dimensional diagonal matrix having the atomic masses along the diagonal. The vibrational frequencies for the van der Waals modes are listed in Table V-9. Information from the normal coordinate analysis can be used to 122 Table V-9. Vibrational force constants and frequencies for the van der Waals modes of ArClCN and properties of ArClCN calculated from the force field. The structure of the average configuration of ArClCN is also given. Constant Value °-l k s = F22(mdyn-A ) kB = F55(mdyn-A) k I ~ F25^mdyn^ 0.0185 0.0333 -0.0092 vs/c(cm-1) 36.0 vB/c(cm"1) 26.3 Taaaa(kHz) -234.6 Tbbbb(kHz) -55.86 Taabb(kHz) 24.99 ^abab(kHz) "284-3 AI(amu-A2) 2.81 I* (amu-A2) 82.626 aa Ibb(amu-A2) * 323.67 02 Icc(amu-A ) 406.03 W*> 3 649 9cm(deg) 82.40 Rc_ Ar (£) ' 3.622 0C1-C-Ar(des) 87 a(deg)a 10.25 a a is the angle between CICN and the b inertial axis. '09 a 87 compute the Coriolis coupling constants ?ij. These, in turn, can be used to obtain the moments of inertia of the average configuration and an estimate of the inertial defect. ? =%($ < v - 25 > • where <^ transforms the normal coordinates to the mass weighted Cartesian coordinates, qx, according to a = M ^ X = %Q. (V-26) -1 ^-1 -1 From Eqs. (18), (23) and (26) and the fact that G = (L )L , it is seen that <»? = M~ ** B G L . (V-27) ft 76 The moments of inertia of the average configuration, Iact, are given by x ft aa = C 3N_6 " T Z d s e s a thar) * s=l , (V-28) where the sum runs over all vibrational modes having degeneracy d and e g (har) depends only on the harmonic force constants. .aap a d n e« (har)\ -= - . Aaa_ 4 y 6h 8-rr (osc L t^s a \^_ 2 c s , (V-29) . where A aa o »2 t_^ 6Qg N = x __£ ^ ^ i-l 2 f (v _ 30) The structure of the average configuration is derived from the L ^ as shown in Section V-C and is listed in Table V-9. The average structure computed in this fashion has the advantage that the inertial defect is 124 largely corrected for. This permits a single structure to reproduce the ft three I o a . In addition, the structure of the average configuration, which depends on (Rx)» has a well defined physical meaning. This con- trasts with the structures derived from the effective moments,Iaa,which are defined only in terms of their method of computation. Finally, the inertial defect, AL. can be calculated and compared with the measured value in Eq. (V-15). AI is the sum of vibrational, centri- fugal and electronic parts, AI = A I v i b + A I c e n t + A I e l e c . (V-31) The vibrational part accounts for 96% of AI in ArClCN and is the part taken into consideration m the computation of the 1 ^ above. Following 77 78 Oka and Morino, ' AI ., f° r a planar molecule in its ground vibra- tional state and having the ArClCN geometry is given by 6 AI vib = ~2 2TT C 8=1 , _ 2. ° S'#S (U)s-0J 3_ t (V-32) ) where u>t refers to the out-of-plane bending mode. The much smaller centrifugal contribution is given by Alcent " ^abab ( £ T + if + |r) (V "33) and the electronic contribution is assumed to be zero. Using Eqs. (V-31), (V-32) and (V-33), AI is calculated to be 2.81 amu-A as compared to the measured value of 2.75 amu-A . This good agreement indicates that useful information about the intramolecular potential can be obtained from an 125 analysis of the centrifugal distortion and that a normal mode analysis can be used to advantage in weakly bound complexes. The agreement also helps to confirm the planarity of ArCLCN since it fits the planar model 11 so well. E. 7 8 Nuclear Quadrupole Coupling In the limit that the electric field gradients at the Cl and N nuclei are not perturbed upon complexation, the observed nuclear quadrupole coupling constants are obtained by rotating the free CICN coupling constant tensors X to the principal axes of the complex and averaging over the ground vibrational state. As shown in Appendix II, — cos a- i 0 0 - i 0 - y cosasincT 0 ) , (V-34) 3 2 1 - T cos a where xlj is the coupling constant of the atom l in free CICN with l = Cl or N and a is the angle between CICN and the b-inertial axis of the complex. From Eq. (V-34), it is seen that the out-of-plane components do not depend on a vibrational average but are simply given by X1 = - - x j cc 2 ° • (V-35) In ArClCN, -2x C 1 = -83.1542(86) and -2x^c = -3.6336(84) as compared to X C 1 = -83.2752(4) and x? = -3.6228(9). r* O * While -2xJL and x? a *e the same 1*1— o Cl Cl within 2 standard deviations, -2x c c and x o clearly differ slightly. indicates that the field gradient along the c-axis is perturbed to the This 126 extent of about 0.15^. Previous work on complexes containing a rare gas atom has not provided any evidence that the field gradients in the binding partner are perturbed on complex formation. It is clear, however, from studies of the rare gas nuclear quadrupole coupling in 131 55 83 25 XeHCl and KrHF and 53 N coupling in hydrogen bonded complexes such as NCCN---HF 54 and (HCN)„ that dipolar species severely perturb the field gradients in their partners. Unfortunately, it is difficult to separate the perturbation of the field gradient from the effects of vibrational averaging in the cases Cl above. The measurement of x in ArClCN represents the first quantitative cc measure of how much the field gradient changes on forming a weak complex Cl Cl with a rare gas atom. x a a and x b b cannot be used in the same fashion, however, since both undergo vibrational averaging. While /a) is known from the rotational constants, the relation (cos 2 a) = cos 2 (a) does not, in general, hold. Cl X (V-36) It is conceivable that the small change in Cl Cl results from large offsetting changes in x a a and x b b . However, the Cl magnitude of the change in x c is about that to be expected from polar- ization of CICN by a dipole moment induced in Ar. In addition, while Eq. (V-36) does not hold exactly, it is found to be approximately correct as shown in the next paragraph. The main use of nuclear quadrupole coupling in van der Waals molecules has been to aid in the determination of the vibrationally averaged structures of inertially linear molecules. This procedure depends on the validity of Eqs. (V-34) and (V-36). Since a can be determined from the rotational constants of ArClCN, it is interesting to compare it with a as obtained from the coupling constants. PI Using Eqs. (V-34) and (V-35) , TVT a is found to be 9.98°(1) from x b b and 10.1(1) from x b b • This compares o with a = 10.25(4) from the rotational constants. As the most pronounced effects of changes in x 0 on the structure will be felt at the small pa angles considered here, it appears that no serious problems should be encountered in using quadrupole coupling data in the structural determination of a rare gas containing complex. 128 APPENDIX I THE ROTATIONAL CONSTANTS FOR AN ATOM-LINEAR MOLECULE COMPLEX The moments of inertia, I, for a molecule in an arbitrary axis system, (r,s,t), are given in terms of the atomic Cartesian coordinates by 3 8 I rr =2> x (s x 2 + tx2) , i ^s - (AI-1) -ZWi i and cyclic permutations, where the sums run over all atoms having mass mx. The purpose of this appendix is to express the moments of inertia in terms of the structural parameters of a molecular complex instead of the Cartesian coordinates. The appropriate parameters for an atom-linear molecule complex are the distance, R, from the linear molecule center of mass to the atom and the angle, 9, between the linear molecule and R. Figure AI-1 shows an atom-linear molecule complex as located in two coordinate systems, (u,v,w) and (x,y,z). Both coordinate systems have their origin at the linear molecule center of mass. The w and z axes coincide with one another and are perpendicular to the plane of the complex. In the (u,v,w) system, the linear molecule lies along v and in the (x,y,z) system, R lies along x. The moments of inertia of Lhe complex are easily derived in the (u,v,w) system by using Eq. (AI-1) and realizing that Y complex cm V near cm \6 R X U Figure AI-1. Coordinate systems used to describe the structure of an atom-linear molecule complex. The complex lies in the uv and xy planes and the w and z axes are perpendicular to this plane. to VO 130 1m H mivi = h <AI-2> * where the sum runs over all atoms in the linear molecule and I. is the moment of inertia of the linear molecule. Using primed I's to denote the moments of inertia with the linear molecule center of mass as origin, the nonzero moments of inertia are I' = I 0 + m„R cos'9 , uu *• a x•' vv == m m R p 2s ei.l» n 2 9 a ^ ^ > (AI-3) *ww = h + m a R 2 and I' uv = -nuR a sin9cos9 , where m a is the masa of the atom. These moments of inertia can be re- lated to those in an axis system parallel to (u,v,w) but having its origin at the center of the mass of the complex by using the Parallel Axis Theorem.89 Using unprimed I's to denote these moments of inertia, I u u becomes 2 I,,,, = I„ + m-R2cos29 - (ma+mj) maRcos9 ma + m^ where m^ is the mass of the linear molecule. > (AI-4) Defining a pseudodiatomic rotational constant, I., by = mam0 2 R h , 1^ » Eq. (AI-4) becomes (AI 5) " 131 X uu = Z Z + I d cos2 9 . (AI-6) In a similar fashion, the other moments of inertia are obtained: V " I d sin2e I ™ = I £ + Id (AI-7) I u v = - I.cos8sin9 . To obtain the principal axis moments of inertia, the inertia tensor, whose components are given by Eqs. (AI-6) and (AI-7) can be diagonalized. The rotational constants are then obtained from A = (h/8ir I aa ) , etc. In practice, it is found that one principal axis nearly coincides with $. To put the moment of inertia tensor into a more diagonal form, it can be transformed from the (u,v,w) axes to the (x,y,z) system by a rotation of TT/2-6 about w. Using the usual two dimensional coordinate transformation and realizing that cos (IT/2-8) = s m 8 and that sin(Tr/2-9) = cos9 I(x,y,z) ={l(u,v,w) jg, (AI-8) where sin8 -cos9 0\ C =1 cose sine 0 * * 0 0 1 . (AI-9) Using Eqs. (AI-6) - (AI-9), the moments of inertia in an axis system parallel to (x,y,z) whose origin is at the center of mass of the complex are found to be x xx = I*sin29 x yy " Td + I £cos e (AI-10) x T zz - d + Z l I x y = Iyj^ = - l£COs9sin9 . The moment of inertia tensor in Eq. (AI-10) can be diagonalized by a rotation of coordinates through an angle Y about the z-axis 38 where Y is given by tan 2Y = 2I x y /(I x x -I y y ) (AI-11) and the principal moments of inertia are I a a = I x x cos 2 Y+ 2IxycosysinY + I yy sin 2 Y 2 Ibb = I U xx sin Y - 21 XX"" I CC = 1 ZZ cosYsmY + I yy cos 2 Y -^y^-W-u, . o.yy (AI-12) • In the case that IJj<<I(3 in Eqs. (AI-10), the principal moments of inertia in Eqs. (AI-12), can be expanded in powers of (^/I^). l(x,y,z) is nearly diagonal, thus, Y is small and tan 2Y can be approximated by tan 2Y % 2Y = 2I x y / (I xx -I yy ) • (AI-13) Using Eqs. (AI-10) and (AI-12), the binomial expansion gives (through second order in I^/I^) Yft (I„7ld)cos9sin9 - (I!!//Id)2(cos29-sin20). (AI-14) Likewise, sinY^Y and cosY%l-Y/2, hence Eqs. (AI-12) become X aa = W 1 " 7 * ) + 21^7(1-^/2) + hyy2 I b b = IxyY2 - 2IxyY(l-Y/2) + I y y ( l - Y 2 ) Ice = (AI-15) •'•zz' S u b s t i t u t i n g Eqs. (AI-10) and (AI-14) i n t o Eqs. (AI-15) and expanding through second o r d e r m (I^/I^), x aa = I d [ ( l £ / I d ) s m 2 e x bb = Id[L Ice - ^ + - (I£/Id)2cos29sin29] (I)i/Id)cos29 + (l£/ld)2cos26sin2e] (AI-16) + xe • The rotational constants measured for the rare gas hydrogen halides are the averages of those about the b and c inertial axes, B 0 = (B+C)/2 = (h/8TT2) (1/I bb + l/lcc)/2 (AI-17) Applying the binomial expansion to Eqs. (AI-16), 1/I bb £ [l-(I£/Id)cos29 + (Iji/Id)2(cosA9-cos2esin29)]/Id 1/I CC % [l-(I£/Id) + (I^/Id) 2 ]/^ , (AI-18) and the rotational constant is given by B Q = Bd[l-(Bd/2b)(l+cos26) + (Bd/b)2(l-cos20sin26+cos49)/2] , (AI-19) 2 where B d = (h/8tr Id) and b is the rotational constant of the linear 134 molecule. For ArHCl, use of Eq. (AI-19) predicts B 0 to within spectro- scopic accuracy. Eqs. (AI-14) and (AI-15) can be used to derive the structures for complexes that fulfill the requirement that l£ < < ^j. In the rare gas- hydrogen halides, the angles, a, obtained from analyses of the hyperfine structure are related to 9 and Y by 9 = a + Y % a + (B~ 0 /b)cosasina , where it has been assumed that B, £ B Q and 9%a. (AI-20) B d is then obtained as the solution to Eq. (AI-19) and R from R = (h/8TT2uBd)J5 • (AI-21) The structures derived in this manner compare well with those obtained using the exact numerical technique of Chapter II. 135 APPENDIX II RELATION OF MONOMER TENSOR PROPERTIES TO ATOM-LINEAR MOLECULE COMPLEXES The spectroscopic properties of atom-linear molecular complexes that relate to tensorial properties of the linear molecule are simply those tensor components referred to the principal inertial axes of the complex. Under the assumption that all monomer properties remain un- changed upon complex formation, the tensors, P 0 , describing linear molecule properties, are diagonal in an axis system where the molecule lies along one of the axes (taken to be the x-axis) and have the form Px 0 0 \ 0 Py 0 P0 - % *00P2/ - / Pn 0 o\ 0 Pj. 0 \o . (AII-1) o ?J Due to the cylindrical symmetry of the linear molecule, one principal plane can be chosen arbitrarily. It is chosen here to coincide with the plane defined by the atom-molecule system and z is taken to be perpendicular to this plane. P Q is transformed to the principal axes of the complex by a rotation of coordinates through a (see Figure II-l) about z: cosa sina 0\ /P|| 0 o\/cosa -sina 0\ F. *I-sina cosa 0 0 B 0 sina cosa 0 , l * o o i/\o Cl i ) \ 0 0 1/ (AII-2) where P_ is the tensor in the principal inertial axis system of the complex. Performing the indicated matrix multiplications, ~ f B| cos2a+Pxsin2a (Pn -P,)sinacosa \ -*- 0 (P||-F, )sinacosa P,|Sin2a+PJ_cos2a 0 0 \ 0 . (AII-3) P:JLI 136 Eq. (AII-3) is general but three specific applications are shown below. 1. The nuclear quadrupole and nuclear spin-spin coupling are des34 cribed by traceless tensors. 1 In these cases, ?]_ = - "T P,i 1 = ~ "T p where P Q is the free linear molecule coupling constant and Eq. (AII-3) becomes /P0(3cos2a-l)/2 P r = -3P 0 sinacosa/2 * \ 0 -3P 0 sinacosa/2 P Q (l-3cos 2 a/2) 0 0 \ 0 .(AII-4) - P 0 /2 / For the rare gas-hydrogen halides, x+a and the nuclear quadrupole coupling constant, Xa> or nuclear spin-spin constant, d a , for the complex is given by P a = P0(3cos2a-l)/2 2. , P = x, d . (AII-5) The spin-rotation tensor, c 0 , for the linear molecule has c,. = 0 and c, = c Q where c Q is the spin-rotation constant. The spin-rotation tensor for the complex, c, is then found from Eq. 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Edwards, Molecular Spectroscopy (The Chemical Society, London, 1973). 86. S.J. Cyvin, B.N. Cyvin, and G. Hagen, Z. Naturforschg. A23, 1649 (1968); G.O. Sorensen, G. Hagen, and S.J. Cyvin, J. Mol. Spec. 35., 489 (1970). 87. J.H. Meal and S.R. Polo, J. Chem. Phys. 24., 1119 (1956). 88. K.C. Janda, J.M. Steed, S.E. Novick, and W. Klemperer, J. Chem. Phys. 67, 5162 (1977). 89. K.R. Symon, Mechanics, 3rd ed. (Addison-Wesley, Reading, Mass. 1971). 143 VITA Michael Robert Keenan was born on September 6, 1953, in Rochester, New York where he attended Cardinal Mooney High School. After obtaining a Diploma in June, 1971, he studied medical laboratory technology at Monroe Community College in Rochester and was awarded an A.A.S. degree, With Distinction, in May, 1974. He enrolled in the State University College at Brockport, New York, majoring m B.S. degree, Summa Cum Laude, in May, 1976. chemistry, and received a In August, 1976, he entered graduate school at the University of Illinois. He served both as a teaching assistant and a research assistant and held a DuPont Industrial Fellowship. His Ph.D. degree in physical chemistry will be obtained in October, 1981. He is a member of the American Physical Society, the American Association for the Advancement of Science and Phi Kappa Phi and is co-author of the following publications: T.J. Balle, E.J. Campbell, M.R. Keenan, and W.H. Flygare, A New Method for Observing the Rotational Spectra of Weak Molecular Complexes: KrHCl, J. Chem. Phys. 71, 2723 (1979), (Communication). T.J. Balle, E.J. Campbell, M.R. Keenan, and W.H. Flygare, A New Method for Observing the Rotational Spectra of Weak Molecular Complexes: KrHCl, J. Chem. Phys. 72, 922 (1980). E.J. Campbell, M.R. Keenan, L.W. Buxton, T.J. Balle, P.D. Soper, A.C. Legon, and W.H. Flygare, 8 ^Kr Nuclear Quadrupole Coupling in KrHF: Evidence for Charge Transfer, Chem. Phys. Lett. 70, 420 (1980). M.R. Keenan, E.J. Campbell, T.J. Balle, L.W. Buxton, T.K. Minton, P.D. Soper, and W.H. Flygare, Rotational Spectra and Molecular Structures of ArHBr and KrHBr, J. Chem. Phys. 72, 3070 (1980). A.C. Legon, P.D. Soper, M.R. Keenan, T.K. Minton, T.J. Balle, and W.H. Flygare, The Rotational Spectra of Weakly Bound Dimers of Carbon Monoxide and the Hydrogen Halides HX (X=F, Cl, and Br), J. Chem. Phys. 73, 583 (1980). 144 M.R. Keenan, L.W. Buxton, E.J. Campbell, T.J. Balle, and W.H. Flygare, !^lxe Nuclear Quadrupole Coupling and the Rotational Spectra of XeHCl, J. Chem. Phys. 73, 3523 (1981). M.R. Keenan, T.K. Minton, A.C. Legon, T.J. Balle, and W.H. Flygare, Microwave Spectrum and Molecular Structure of the Carbon Monoxide-Hydrogen Bromide Molecular Complex, Proc. Natl. Acad. Sci. USA 77^, 5583 (1980). E.J. Campbell, L.W. Buxton, T.J. Balle, M.R. Keenan, and W.H. Flygare, The Gas Dynamics of a Pulsed Supersonic Nozzle Molecular Source as Observed with a Fabry-Perot Cavity Microwave Spectrometer, J. Chem. Phys. 74, 829 (1981). L.T«. Buxton, E.J. Campbell, M.R. Keenan, T.J. Balle, and W.H. Flygare, The Rotational Spectrum, Nuclear Spin-Spin Coupling, Nuclear Quadrupole Coupling, and Molecular Structure of KrHF, Chem. Phys. 54., 173 (1981). M.R. Keenan, L.W. Buxton, E.J. Campbell, A.C. Legon, and W.H. Flygare, Molecular Structure of ArDF: An Analysis of the Bending Mode in the Rare Gas-Hydrogen Halides, J. Chem. Phys. 74, 2133 (1981). 83 E.J. Campbell, L.W. Buxton, M.R. Keenan, and W.H. Flygare, Kr and 132-Xe Nuclear Quadrupole Coupling and Quadrupole Shielding in KrHCl and XeDCl, Phys. Rev. A (to be published). M.R. Keenan, D.B. Wozniak, and W.H. Flygare, Rotational Spectrum, Structure, and Intramolecular Force Field of the ArClCN van der Waals Complex, J. Chem. Phys. (to be published). M.R. Keenan and W.H. Flygare, A Hindered Rotor Method for Estimating Angular Expectation Values m the Rare Gas-Hydrogen Halides, Chem. Phys. Lett, (to be published).

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