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The microwave spectra of cyclopropyl bromide Li, Hao 1989

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THE MICROWAVE SPECTRA OF , CYCLOPROPYL BROMIDE by HAO U B. Sc., The University of Science and Technology of China, 1985 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMISTRY We accept this as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1989 ©HaoLi.1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive c o p y i n g of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is u n d e r s t o o d that c o p y i n g or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British C o l u m b i a Vancouver, Canada DE-6 (2/88) ii A B S T R A C T The microwave spectra of cyclopropyl bromide, C3H5BT, in its ground state, has been observed and analyzed in the frequency range 15-90 GHz. The spectra of two species were measured: C^US^BT and C3H58 1Br. They contain strong a-type tran-sitions and very weak c-type transitions which could not initially be assigned; all show Br quadrupole hyperfine structure with several large perturbations. A procedure specially developed for analysis of such spectra, which uses perturbations in the Br structure, was used to evaluate all the rotational constants accurately, as well as the Br quadrupole tensor, entirely from a-type R branch transitions. This has allowed some c-type transitions to be assigned. The principal values of the Br quadrupole tensor have been evaluated and have provided some information about the type of bonding involved in the C-Br bond.The centrifugal distortion constants have also been obtained. iii TABLE OF CONTENTS Page 1 6 II. Theory 2.1 The Rigid Rotor 7 2.2 The Rigid Rotor Selection Rules 13 2.3 Centrifugal Distortion 15 2.4 Nuclear Quadrupole Coupling 21 2.5 Stark Effect 26 Bibliography 32 HI. Experimental methods 3.1 The Microwave Spectrometer 34 3.2 Sample, Sample Handling and Measurement Techniques 38 Bibliography 41 IV. Microwave Spectrum of Cyclopropyl Bromide 4.1 Introduction 42 4.2 Assignment and Analysis 43 Bibliography 77 Chapter I. Introduction Bibliography iv V. Discussion 5.1 Comparison of Data 78 5.2 Diagonalization of The Brorj^e (^adrupole Tensor 81 5.3 The Bonding of Br in Cyclopropyl Bromide 86 5.4 Structure of Cyclopropyl Bromide 89 Bibliorgraphy , 9 2 VI Conclusion 93 V LIST OF TABLES Table Page 4.1 The near-degenerate energy levels of cyclopropyl bromide C^Br) 47 4.2 Weighting scheme for the transitions of cyclopropyl bromide 51 4.3 Spectroscopic constants of cyclopropyl bromide 52 4.4 Correlation coefficients of the spectroscopic constants of CsHsBr derived from total transitions 53 4.5 Measured rotational transitions (in MHz) of C3Hs79Br 56 4.6 Measured rotational transitions (in MHz) of C3Hs81Br 66 5.1 Spectroscopic constants of cyclopropyl bromide 09Br) in comparison with those of Lam and Dailey " 79 5.2 Spectroscopic constants of cyclopropyl bromide (^ Br) in comparison with those of Lam and Dailey 80 5.3 Bromine quadrupole coupling constants in principal inertial axes in comparison with those of Lam and Dailey 84 5.4 Principal values of the bromine quadrupole coupling in comparison with those of Lam and Dailey 85 5.5 Comparison of of cyclopropyl bromide with those of similar molecules 91 v i LIST OF FIGURES Figure Page 3.1 Block diagram of microwave spectrometer 40 4.1 Rotational energy levels of C3H579Br 54 4.2 The energy levels of the 826 - 725 transitions of CsHs^Br 55 Vll ACKNOWLEDGEMENTS I would like to thank my supervisor, Mike Gerry for all the encouragement and support he has given me throughout my time in UBC. His enthusiasm for microwave spectroscopy and approachability as a teacher will always be remembered. I would also like to thank Dr. W. Lewis-Bevan, David Cramb, Dr. Allan Adam and Chris Chan for the help and useful advice. 1 CHAPTER I INTRODUCTION Each molecule possesses an unique set of energy levels, associated with the motions of the nuclei and electrons in the molecule. These levels depend on such factors as atomic masses, interatomic distances, the electron distribution and interparticle forces. Molecular spectroscopic research examines the absorption and emission of radiation between these molecular energy levels. The frequency of the radiation depends on the type of the molecule and the type of transition. The microwave region of the electromagnetic spectrum covers approximately the frequency range 1 -1000 GHz. In this region, the absorption of radiation usually corresponds to transitions between rotational levels of gaseous molecules having a permanent dipole moment. In most cases, the microwave spectrum shows transitions between rotational states, not only of molecules in the ground vibrational state but also of those in excited vibrational states, depending on their populations. It is possible, as well, to detect transitions due to different isotopic species present in the sample, often in natural abundance; the intensities of the transitions are directly proportional to the percent abundance. By assigning a microwave spectrum for a particular molecule in a particular vibrational state, we can obtain the rotational constants and then use them to determine the principal moments of inertia which depend on the molecular structure. It is possible, by studying enough isotopic species, to determine the structure completely and accurately. 2 The centrifugal distortion constants, which arise because of an inte action of the vibrational and rotational motions of the molecule, can also be obtained, and can be used in the determination of the force field and the frequencies of the normal modes of vibration. More structural information can be obtained when the molecule contains a nucleus with a spin I > 1/2. In such a case, the rotational transitions will show hyperfine structure because of the coupling of the nuclear spin with the rotational motion of the molecule. From measurements of these splittings, the nuclear quadrupole coupling constants can be obtained. These constants can be related to the electronic enviroment in the immediate vicinity of the coupling nucleus and the nature of its chemical bonds. In the presence of an electric field, the rotational transitions will be split and shifted. These rotational transition frequencies are called Stark transitions or Stark components. The accurate measurement of electric dipole moments can be obtained by measuring these shifts as a function of electric field. Computers have been introduced into microwave spectroscopic studies. Small computers are used for control of spectrometers and for data processing. Both small and large computers have allowed increasingly complex formulations to be used to analyse microwave spectra. Not only have stable gaseous molecules been investigated, but also the microwave spectra of many unstable gases have been analysed (1). More recently, gaseous free radicals and molecular ions have been studied spectroscopically (2). 3 Cyclopropyl bromide This thesis is concerned with the microwave spectrum of cyclopropyl bromide, C3HsBr, ^ ~ B r . This substance is a clear, volatile liquid at room temperature. As one of the derivatives of cyclopropane, it gives an excellent opportunity for studying by microwave spectroscopy the structure of the cyclopropane ring and especially the intemuclear bond angles. There have been several studies of cyclopropane series in vibrational and rotational spectra (3 - 8), such as cyclopropyl chloride (4,5 ) and cyclopropyl cyanide (4). Initial microwave work on this molecule was done by Lam and Dailey many years ago (3 ). It yielded values of the rotational constants B and C with good accuracy (± 0.03 MHz) from the a-type spectra of both 7 9Br and 8 1Br species. This work also provided information on the structure of the molecule including bond angles and bond lengths. From this information it was deduced that there are C-C bent bonds in the ring because the C-C bond lengths are significanUy shorter than 1.53 A, the normal value of a C-C single bond (4). There were, however, several flaws in the earlier results. In the first place, only two of the three rotational constants were accurately obtained for each isotopic species. For C3H579Br the uncertainty in the third rotational constant, A, was ± 65 MHz; for C3H581Br it was ± 160 MHz (3). Ideally, the precision of this constant should be similar to those of B and C (± 0.1 MHz). The reason was that only a few transitions were measured, and these were all so-called a-type R branches of a prolate near-symmetric rotor, from which A is notoriously diffcult to obtain (3). The usual way of obtaining an accurate A is to assign so-called b- or c-type transitions, but if they are 4 weak or show large hyperfine structure they are diffcult to identify. Such is the case for this molecule. Secondly, the quadrupole hyperfine structure due to both 7 9Br and 8 1Br was accounted for using a simple first-order theory. As a result one of the possible quadrupole coupling constants, called X^, was not directly obtained. It was then necessary to make assumptions using the derived structure in order to get information about the electron distribution. It was not possible to say, for example, whether the C-Br bond is straight or bent. A knowledge of Xac would remove this difficulty. Very recently a new method has been developed in which the precision of A obtained from a-type transitions is much improved by taking advantage of perturbations in quadrupole structure. The method was applied successfully to the molecules BrNCO (9), IN CO (10) and CH2CHI (11), all of which had spectra dominated by strong a-type R branches like those of cyclopropyl bromide. The procedure involves an exact global least squares fit of all a-type transitions to rotational, centrifugal distortion and all quadrupole coupling constants, including an "off-diagonal" constant such as X^ for cyclopropyl bromide, which was missing in previous work (3). This constant produces perturbations in the quadrupole structure dependent on certain near-degeneracies, which are themselves functions of A. The global fit thus produces accurate values of both A and the "off-diagonal" constant, the two poorly measured parameters of the previous work. The aim of the work described in this thesis was thus to make very extensive measurements of the microwave spectrum of cyclopropyl bromide, in order to locate all perturbations due to X^ and apply the global fitting procedure in an attempt to measure accurately both A and X^. This would permit refinement of the structural parameters. In 5 addition the quadrupole coupling could now be used to obtain bonding information without resorting to the previous assumption. In the process it was also hoped that the centrifugal distortion constants could also be measured 6 BIBLIORAPHY 1. M. C. L. Gerry, W. Lewis-Bevan, and N. P. C. Westwood, J. Chem. Phys., 79, 4655-4663, (1983). 2. G. Winnewisser, E. Churchell, C. M. Walrnsley, Astrophysics of Interstellar  Molecules, pp. 313-503, in Modern Aspectes of Microwave Spectroscopy, Ed. G. W. Chantry, Academic, London, 1979. 3. F. M. K. Lam, B. P. Dailey, J. Chem. Phys., 49, 1588-1593, (1968). 4. J. P. Friend, B. P. Dailey, J. Chem. Phys., 29, 577-582, (1958). 5. R. H. Schwendeman, G. D. Jacobs, T. M. Krigas, J. Chem. Phys., 40, 1022-1028, (1964). 6. Takeshi Hirokawa, Michiro Hayashi, Hiromu Murata, J. Sci. Hiroshima Univ., Ser. A., 3" No.2, 301-324, (1973). 7. Jacques Maillols, Vlado Tabacik, Spectrrcrumica Acta, 35A, 1125-1133, (1979). 8. C. Marsden, L. Hedberg, K. Hedberg, J. Phys. Chem., 92, 1766-1770, (1988). 9. H. M. Jemson, W. Lewis-Bevan, N. P. C. Westwood, M. C. L. Gerry, J. Mol. Spectrosc, 118, 481, (1986). 10. H. M. Jemson, W. Lewis-Bevan, N. P. C. Westwood, M. C. L. Gerry, J. Mol. Spectrosc, 119, 22-37, (1988). 11. D. T. Cramb, M. C. L. Gerry, J. Chem. Phys., 88, 3497-3507, (1988). 7 CHAPTER H THE BASIC THEORY OF MICROWAVE SPECTROSCOPY 2.1 THE RIGID ROTOR If a rotor is made up of a series of mass points mj, m2,..., rigidly connected, at respective perpendicular distances xi, X2,... from a lins, the moment of inertia about that line is X m ^ . In a molecule there are always three mutually perpendicular axes running through the centre of mass, called principal inertial axes, which correspond to principal moments of inertia Ia, lb and Ic, with I a £ lb ^ Ic- la and Ic are the smallest and largest possible moments, respectively. The Hamiltonian for a rigid, rotating molecule is given by: H r = B / x + B yJ 2 y + B2J^ (2.1) where J x , J y and J z represent the operator for the components of the angular momentum about the principal axes, arbitrarily designated x, y and z. Bx, B y and B z are the rotational constants of the molecule, related to the principal moments of inertia by: Here the rotational constants are in frequency units. 8 It is pos: ible to assign the ~, y, z axes to the a, b, c axes mentioned above in six different ways or "representations" (1). The three "right handed" representations are: F: x - » b , y - » c , z - » a ; (Ix, Iy, I*) ^» (lb. Ic, Ia) (Bx, By, Bz) -4 (B, C, A) IF: x->c, y - » a , z - » b ; (Ix, Iy, Iz) -> (Ic, Ia, lb) (Bx, By,Bz)-*(C, A, B) I I rr: x->a, y->b, z->c; (Ix, Iy, Iz) ^  (Ia, Ib, U (Bx, By, Bz) _»(A, B, C) !i Which designation is used in a given case depends on the properties of the molecule concerned (see later); the commonest-are F and UF. In a symmetric top molecule two of the principal moments are equal. Molecules with a three fold or greater symmetry axes fall in this category, with the moment of inertia about the symmetry axis different from those about the two axes perpendicular to it. In a prolate symmetric top, the symmetry axis is the a-axis and Ia < lb = Ic (so that A>B = C);inan oblate symmetric top the symmetry axis is the c-axis, and Ia = lb < Ic (with rotational constants A = B>C). Ina symmetric top it can be shown that the operators J 2 (which designates the square of the total angular momentum), J 2 (which designates the angular momentum about the symmetry axis) and Jz (which designates 9 the angular momentum about a space-fixed Z axis) all commute. There is a function IJ,K,Mj > which obeys the eigenvalue equation: J 2 U K M j > = J(J+l)n2|JKMj> (2.2) J2IJ K Mj > = KM IJ K Mj > (2.3) J Z U K Mj > = Mjh IJ K Mj > (2.4) where J = 0, 1, 2, 3, ... K = 0 , ± l , ± 2 , . . . , ± J Mj = 0, ± 1 , ± 2 , . . . , ±J are rotational quantum numbers. The non-zero matrix elements of the rotational Hamiltonian in the IJ K Mj > basis are: <J K Mj I Hr IJ K Mj > = i (Bx+ By)J(J+l)+[B2 - i (Bx+ B y )]K 2 <J K±2 Mj I Hr IJ K Mj > = j(B x- By){[J(J+l) - K(K±1)] (2.5) x [ J ( J + l ) - ( K ± l ) ( K ± 2 ) ] } i/2 10 In the two representations F end JTF these thus become: F : <J K Mj I H r IJ K Mj > = i (B+C)J(J+1)+[A - j C B + Q j K 2 <J K±2 Mj I H r IJ K Mj > = j (B-C) {[J(J+1) - K(K±1)] (2.6) x[J(J+l)-(K±l)(K±2)]} i/2 B F : <J K Mj I H r IJ K Mj > =i(A+B)J(J+l)+[C - j(A+B)]K 2 <J K±2 Mj I H r IJ K Mj > =i<A-B){[J(J+l) - K(K±1)] (2.7) x [ J ( J + l ) - ( K ± l ) ( K ± 2 ) ] } 1 / 2 For a prolate symmetric top B = C, and the F representation (with z designated as a) is the most convenient because the off-diagonal element is zero. The rotational energy is E r = BJ(J+l) + (A-B)K2 (2.8) Because Er depends on K 2 there are only (J+l) distinct rotational energies; the levels with IKI > 0 are doubly degenerate. There is no dependence of Ej on Mj in the absence of an external field. 11 Many molecules, including cyclopropyl bromide, are slighdy asymmetric prolate rotors, with A » B ~ C. For these molecules the off-diagonal element in the F representation is very small, and Er is given by the diagonal elements plus a small correction. This would not be found if the HF representation were used. It is thus usually more convenient to use F rather than HF for these molecules, though both representations are correct. The remaining discussion of this section considers the F case, except where otherwise noted. For an asymmetric rotor the IKI degeneracy of a symmetric rotor is lifted; the quantum number K is no longer good, and there is no internal component of the angular momentum which is a constant of the motion. There are now, in general, 2J+1 distinct rotational sublevels for each value of J. With an increase in asymmetry the " K splitting" increases until there is no longer any close correspondence between the two levels and the symmetric top levels. Following King, Hainer and Cross (1), K is retained as a label, however. An asymmetric rotor level is designated JKa,Kc (or J T ; x = Ka- Kc), where Ka represents the K value of the limiting prolate top and Kc represents the K value of the limiting oblate top. The degree of asymmetry of a molecule is usually defined by an asymmetry parameter. A very common one is Ray's asymmetry parameter (2): X = (2-9) 1 2 The limiting values are X = -1 and +1 for a prolate and obkte symmetric tops, respectively. Another asymmetry parameter is bp, Wang's asymmetry parameter (3 ), defined by bp=2Xilc- (2.10) with the limits of bp being 0 and -1 for the prolate and oblate symmetric tops, respectively. The rotational energy E r can be written as (4): E; =i(B+C)J(J+l) + (A-5±CT)W(bp) (2.11) W(bp) is the Wang "reduced energy", and is the energy which would be obtained in the F represention if the rotational constants were A = 1, B = -bp, C = bp (4). This is a particularly convenient equation for sh'ghdy asymmetric prolate rotors. If bp is very small W(bp) can be expressed as a power series in bp (4): W(bp) = + cibp + c2b* + ... (2.12) where Ka has been previously defined and the q are coefficients that have been tabulated (5). 2.2 T H E R I G I D R O T O R S E L E C T I O N R U L E S 13 Microwave absorption transitions are induced by interaction of a radiation electric field, defined in terms of space-fixed axes, with electric drpole moment components fixed in the rotating molecule, defined along the principal axes of inertia. We designate the space-fixed axes by F (= X,Y,Z), and the molecule-fixed axes by g (= x,y,z), and the direction cosine of the angle between F and g as $pg. For plane-polarized radiation, such as the microwave radiation used in these experiments, the direction of radiation field will be defined as being along the Z axis. The interaction of the dipole moment and radiation field is given by: H(t) = -|iE(t)=-jizEz(0 (2.13) w i u l U - Z ^ x t e x + ^ y ^ Z y + fe^Zz For a transition to be allowed between the levels IJ x Mj > and IJ' x' Mj'> the matrix element <J x Mj I \iz 0* Mj' > must be non-zero. This integral is separated into factors that depend on the different quantum numbers as follows: <J x Mj I nz IJ' x' Mj' > = X <J x Mj I teg LT t' Mj'> s = I M<J 1 ta* I J , > <J 1 1 « z « ^ f > < J M j I ^ W Mj'>] (2.14) 8 For a symmetric top x is replaced by K in equation 2.14. In this case only Hg= fa* 0, and the following selection rules can be derived: for K = 0 AJ= ±1 A K = 0 AMj = 0 14 for.K*0 AJ = 0 , ± 1 A K = 0 AMj = 0 For an asymmetric rotor the first and third term give similar selection rules: A J = 0 , ± 1 ; AMj = 0 . The second term < J x I tyzg  IJ' x'> gives restrictions on the subscripts K a , Kc in the notation JKS,KC- They are stated in terms of the evenness or oddness of the K a , IQ subscripts as follows: Dipole moment Permitted transitions Type of component between K a Kc levels transitions u a * 0 ee <-> eo oe <-> oo a-type u,b^0 e o e o e e e n o o b-type | i c * 0 eof ) oo o e n e e c-type Any given transition is due to only one component of the molecular dipole. For both symmetric and asymmetric rotors the selection rules involving Mj become important when an external field is applied to the system and lifts the Mj degeneracy of the rotational levels. 1 5 2.3 CENTRIFUGAL DISTORTION The rigid rotor theory treats the nuclear framework as rigid. However, in reality molecules are flexible, and bond distances and angles vary with the rotation, giving rise to a "centrifugal distortion". The moments of inertia can no longer be considered constants, but are functions of rotational state. The changes in the moments of inertia shift the rotational transition frequencies away from those predicted using the rigid rotor Hamiltonian by a significant amount, which in extreme cases can be as much as hundreds of MHz for some asymmetric rotors. Although we must consider the influence of centrifugal distortion in order to account accurately for the positions of rotational transitions, its effects still represent a small fraction of the rotational energy, which is accounted for mainly by the rigid rotor term. It can thus be treated as a perturbation of the rigid rotor Hamiltonian, in which corrections are made involving higher order angular momentum terms. The new Ftamiltonian becomes: From a perturbation treatment of centrifugal distortion, Wilson (6) expressed the Hamiltonian as: H j = HR + Hrj (2.15) (2.16) 16 i i i with a, p\ y, 8 = x, y or z. The rotational constants Bx, By , Bz are slightly different from those of the rigid rotor because they now have small contribution from distortionThe distortion constant Xafrfi is given by : 1 ^ dl „ 81 a a BB YY 58 , J 1 J (2.17) Ri and Rj belong to the set of 3N -6 internal displacement coordinates, while (f *)ij is an element of the inverse force constant matrix. Kivelson and Wilson ( 6 , 7 , 8 ) showed by application of commutation rules and symmetry restrictions that only six distortion constants are needed to account for the rotational energies. The rotational Hamiltonian becomes: H D - T X X a a B B J a j 2 4 &md a a B B a B The corrected rotational constants B , B , B have a further marginal distortion x y z contribution. The x ftR are called quartic centrifugal distortion constants, because the angular momentum terms are raised to the fourth power. The relationship between the 17 old and new coefficients is: x ctaaa aaaa (2.19) aapp + 2T afkxf> ) There is a problem with the application of this Hamiltonian because it contains six quartic distortion constants and they can not all be (ktermined simultaneously from experimental data (9,10). However, Watson (11,12) showed that the maximum number that can be obtained from the analysis of the rotational spectrum of an asymmetric rotor is five. It is necessary to "reduce" the Hamiltonian by taking linear combinations of the x' constants. This reduced Hamiltonian contains only five quartic distortion constants. There are two different reductions which are commonly used. In the asymmetric, or A-, reduction the Hamiltonian becomes: H D = V - AJi - v* - ™Ai+$ - v i<i - JJ>+ii - £ £ H T = H R + H D H„ = B A J 2 + B V + B A J 2 R x x y y z z (220) 1 8 here BA, BA and BA are effective rotational constants. The distortion constants are given by: AT = - 4 (X + X ) J g v xxxx yyyy' i * t 3 • • 1 A _ = — (X + X ) - v (X + X + X ) JK g v xxxx yyyy7 4 v yyzz xxzz xxyy' * • t 1 • • 1 A „ = - — (X +X +X ) + T ( X +X +X ) K 4 v xxxx yyyy zzzz' 4 v yyzz xxzz xxyy' 8T = ~ ( x -x ) (2.21) J 16 xxxx yyyy 1 , B -B 1 . B -B 5 = _ L T (-1 L u i t (-*• -) K g xxxxv A -A' g yyyy A A 7 Bx • By Bx ' By 2 BA- BA- BA + [x -x +x ( — - - )] yyzz xxzz xxyy v A _ A * " y The big advantage of the A-reduction is that its non-zero matrix elements in the basis IJ K Mj > are of the form <JK Mj I H A IJ K Mj> and <J K±2 Mj l Hf^U K Mj >, the same as for the rigid rotor. Thus, a rigid rotor computer program is readily applied to 19 the A-reduction simply by adding small terms. The matrix elements are: < J K M ; I i J K Mj > = I (Bx+ By)(J+l)J + [B 2-1 (Bx+By)]K2 - A ; /(J+l) 2 - A J K JCJ+ljK2 - A K K 4 (2.22) i 2 + K2]} x{[J(J+l) - K(K±1)][J(J+1) - (K±1)(K±2)]} 1/2 Although the A-reduction is very useful, and has found wide application, problems can arise in some cases. The most common case occurs when the F representation is used for a slightly asymmetric prolate rotor. In this case B x - B y = B - C can be very small, and the constant 8K in particular tends to infinity. This is unfortunate because, as was shown earlier, the F representation, so far as the rigid rotor is concerned, is very convenient. To remove this difficulty, one possible method is to use a different representation, such as HF. This is perfecdy legitimate because now B x - B y = A - B, and is large, so that §K is small. The disadvantage is that it is difficult to retain an intuitive physical picture of the constants. A second method is to use a different reduction, the so-called symmetric, or S-, 2 0 reduction. The Hamiltonian in this case is : = H R + H D Hp = B S J 2 + B S J 2 + B S J 2 (2.23) R xx y y z z ^ ' HD = " DJ ^  " DJK j 2 jz - DK ^ + dlJ^J! + $ + ^ + with J± = J x ± Uy The matrices are more complex, with the non-zero elements being: < J K Mj I | J K Mj > = I (Bx+ By)(J+l)J +. [B z -1 (Bx+ B y ) ]K 2 - Dj J2(J+1)2 - D J K J ( J + 1 ) K 2 - D K K 4 < J K±2 Mj I | J K Mj > = [ I (B x- B y ) + dj(J+l)J] {[J(J+1) - K(K±1)] x [ J ( J + l ) - ( K ± l ) ( K ± 2 ) ] } 1 / 2 (2. < J K±4 Mj | H ^ | J K Mj > = <L([J(J+1) - K(K±1)][J(J+1) - (K±1)(K±2)] x [J(J+1) - (K±2)(K±3)][J(J+1) - ( K ± 3 ) ( K ± 4 ) ] } 1 / 2 The advantage of this reduction can be seen from the definition of constants: D T = A. +2d0 = - 4 (x + x ) + 2d, J J 2 g v xxxx yyyy' 3 ' ' 1 • D W = A „ - 12d0 = — (x + X ) - - ( T + X + x ) - 12cL JK JK 2 g v xxxx yyyy' 4 v yyzz xxzz xxyy' '•^2 1 . . . J . D „ = A„ + 10do=-— (x +x +x ) + T ( X +X +X ) + 10d„ K K 2 4 v xxxx yyyy zzzz' 4 v yyzz xxzz xxyy' 2 21 6* 1 • B * - B * j . B ' - B f 2 4 a 32o " x x B S - B S 32a y y y y B S - B S i . . 2B S - B S - B S L-[ T -x +x < — : — i — X ) ] 32a 3 0 , 2 2 x x y y B S - B S * y „ s „ S ' s 2B - B -B z x y G =" S S B - B x y In the I r representation the denominator Bx - By = B - C has been replaced by 2BZ - Bx - By = 2A - B - C. For a slightiy asymmetric prolate rotor this is always large, so that no distortion constant becomes mdeteiroinate, even at the smallest asymmetries, and the S-reduction can be used where the A-reduction fails. The disadvantage of this reduction is that it is much more complex to program in a computer because of the greater number of off-diagonal matrix elements. 2.4 NUCLEAR QUADRUPOLE COUPLING Nuclear hyperfine structure in molecular rotational spectra arises when nuclear spin angular momenta couples with the rotational angular momenta. The most common cause is the interaction of an electric quadrupole moment of a nucleus with I > 1/2 with an electric field gradient Since the electric field gradient is caused mosUy by the electron distribution, inforrration about electronic structure and chemical bonding can be obtained. The nuclear quadrupole coupling results because the nucleus has a non-spherical charge distribution which interacts with an asymmetric distribution of external (mostly electronic) charge. This interaction puts a torque on the nucleus tending to align its spin moment in the field gradient Because the field is rotating the spin and rotational angular moments couple to give a resultant F, around which they process: J +1 = F (2.26) The rotational energy levels split to 21+1 components (2J+1 if J < I) with different F, whose values are given by: F = J+I, J+I-l I J-II J is no longer an entirely good quantum number; its place is taken by F. Selection rules for hyperfine transitions in the rotational absorption spectra are AF = 0, ± 1 , AI = 0. The most intense hyperfine components are these with AF = AJ. Usually quadrupole coupling energies are small relative to rotational energies of an asymmetric rotor and the quadrupole Hamiltonian HQ can be considered as a perturbation of the rotational Hamiltonian HR . The quadrupole Hamiltonian is (13): HQ = ^ Q : V E (2.27) 23 Q is the quadrupole moment tensor and V E is the field gradient tensor due to the extra nuclear charge. The matrix elements are diagonal in I and F but can be off diagonal in J and T (or K ) . In many cases, however, a first-order approximation, diagonal in J, accurately accounts for the hyperfine structure. The non-vanishing matrix elements of HQ, < J x IFI HQ I J' x' I F >, for an asymmetric rotor are of the form (14): < J x I F | H Q | J x ' I F > = C 0 e Q < J x M ^ J | V ^ | J x' M ^ J > <J x IF | HQ | J+l x' I F > = Cj e Q < J x M ;=J | V K | J+l x' M ^ J > (2.28) < J x I F | HQ | J+2 x' I F > = C 2 e Q < J x M ^ J | | J+2 x' Mj=J > where eQ is the quadrupole moment of the nucleus and Vzz is the electric field gradient along a space-fixed Z axis. Co, C i and C 2 are constants, functions of I, J and F (14). Vzz is related to the molecule-fixed axis system by: V z z - X v ^ ^ . (2.29) g . g ' where <|>zg and fag are direction cosines mentioned earlier. Usually one works in 24 terms of "quadrupole coupling constants", %zz = eQVzz, and Xgg- = eQV g g ' so that: g.g Symmetry considerations can be used to decide which terms of the matrix elements are non-zero. If X g g- * 0 the rnatrix elements will be non-zero if <J X Mj=J l^gteg'U' Mj=J> * 0. For an asymmetric rotor the non-zero matrix elements are related to the parities of K aKc as follows: i t Elements K aKc <-» K K a c X a a X b b X c c ee H ee , oo H oo, eo <-> eo , oe <-» oe, X a b eo <-» oo , ee <-> oe, X a c eo <-> oe , ee «-» oo, Xbc eo <-> ee , oe <-> oo, Thus, for example, if X ^ * 0 (as is the case for cyclopropyl bromide) there are non-zero matrix elements between K aKc = eo and oe levels and between KaKc = ee and oo levels. The first order expression is obtained by setting T x' = J x. From the table above it can be seen that the energy in this case depends only on X^, X b b and X^.. In the F 25 representation it is (15,16): E Q = | l | ([3<J>- J(J+1)] X a a + f [<J 2> - WCb^tx^-Xjl (2-31) where f(I J F) is Casirnir's function, defined by: 3C(C+1)-4I(I+1)J(J+1) ^ } 8I(2I-l)(J+l)(2J+3) C = F(F+1) - J(J+1) -1(1+1) 2 2 < J a> is the average of J a in the rigid asymmetric basis. This expression is written in terms of only two constants, and (%bb - Xcc). This is possible because Laplace's equation holds for the coupling constants and X a a + X b b + X c c = 0 (2.32) Since < J T Mj=J I <pzg<PZg'l J 'V Mj=J > are awkward to evaluate in an asymmetric rotor basis, it is easier in practice to start a full calculation in a symmetric rotor basis and diagonalize the rotational and quadrupole Hamiltonians simultaneously. Matrix elements of H Q in such a basis are given by Benz, Bauder and Gunthard (17). The analyses described in this thesis were done using a program written in this basis. 26 2.5 S T A R K E F F E C T The Stark effect is particularly useful in microwave spectroscopy. When a rotating molecule is placed in an electric field E there is an interaction between its dipole moment (i and this field. This interaction perturbs the rotational energy levels, and lifts or partially lifts the degeneracy in Mj. When the external field is along a space fixed Z-axis the Stark effect Hamiltonian can then be expressed as: u g is a component of the permanent dipole moment of a molecule along its molecule-fixed reference system g chosen as the principal inertial axes x, y, z. Since Stark energies are usually much smaller than rotational energies they can be calculated using perturbation theory. The first-order energy is simply the average of Hs over the unperturbed rotational state. For symmetric top molecule, which have | i x = \iy = 0 and | i z = u, the first-order Stark energy can be written as (18): (233) 8 uEKMj (2.34) J(J+D This effect, for which the Mj degeneracy is completely lifted, is observed at low field. There is no first-order Stark effect for a linear molecule nor for the K = 0 levels of a symmetric top molecule since E£ = 0. At high fields, for all K, mcluding K second-order effect (19 ) becomes significant 27 = 0, the The Stark effect for asymmetric top molecules is calculated using perturbation theory in a manner similar to that for linear or symmetric rotors. However, the expre-ssion for the Stark energy is complicated by the possibility of permanent molecular dipole components along each of the three principal axes of inertia. Since the (jirection cosine elements <f>zg are not symmetric in a rigid asymmetric rotor basis IJ x Mj >, the matrix elements < T x' Mj I tyzg I J x M T > are non-zero only if J',T' * J,T. The interacting levels must be different, and second-order perturbation theory can usually be applied. The Stark energy of the level IJ x Mj > is given by: Since the only non-vanishing direction cosine matrix elements are those for which J' = J - l , J or J+l, the energy shift arising from the gth component of the permanent dipole is (20): (2.35) Jx J - l X ' 2 8 M? ^ |< J x | d>Zg 1J x' >|2 4J2(J+1)2 f E° - E ° JT JT' (2.36) (J+1) 2 -MJ V |<JT|(|>Z 8|J+1T'>12 4(J+ir(2J+l)(2J+3) <• E° - E ° J T J+l T' The Stark energies are seen to depend on M j . The (2J+1) fold degeneracy in Mj is partially removed, and a given rotational level is split into (J+l) distinct sublevels. In asymmetric rotors rotational degeneracies frequently occur, both approximate symmetric rotor degeneracy and various types of accidental degeneracies are possible. When < J' x' I <)>zg I J x > * 0 and E j T ~ Ejy equation 2.35 fails, since it will contain terms with vanishing denominators, and this "simple" second-order perturbation treatment is no longer applicable. Instead, for these two levels a secular determination of the following term must be solved: E° (J x) - e E^2 Eh2 E'(J ' x')-e = 0 (2.37) E°(J x) and E°(T x') are the unperturbed energies of the two near-degenerate levels; e is the perturbed energy to be calculated; E is the magnitude of the applied electric field; and l i ^ ^ r t ' M j I ^ I J x M L x 2 9 The two possible solutions are: c± = \ I E'CJ t) + E-(J- <0 ] ± [(E' ( J T ) 2 E ' ( J ' T' }) 2 + E 2 a l 2 ] 1 / 2 (2.38) It can be seen from this solution that if the levels are very close in energy, that is IE* (J x) - E° (J' x') I « I E Jin I, then the equation may be expanded to give an energy that is a linear function of the electric field: e± = ^ [E-(Jt) + E'(J*t ,)]±E Iu 1 2 (2.39) This can be referred to as pseudo first-order Stark effect. If, on the other hand, I E ° ( J T) - E°(J't') I » I E u.121, the expansion will give the conventional second order result, although possibly a large second order effect because of the presence of a small energy denominator. For the case when U x > and IT x'> are actually degenerate, the equation 2.39 reduces to ^ = E°(JT)±EU12 (2.40) giving a linear Stark effect In a near symmetric prolate rotor this often occurs at high Ka, and the Stark effect with analogous is that of a syinmetric top with K * 0 . - E ° f J T i ± * * * * * e ± R ( j * ) 7 ^ T T y (2.41) 30 In general, transitions are between non-degenerate energy levels and will show a s jcond-order Stark effect. This effect is small and the Stark components (the perturbed lines) shift very slowly from the unperturbed lines with an increase of the external electric field The K a = 0 transitions also show a second-order Stark effect Any degenerate or near-degenerate asymmetry pair will exhibit a "first-order" Stark effect and the Stark components will move away very rapidly from the unperturbed lines when the electric field is increased In the work described in this thesis the Stark field is parallel to the microwave electric field inducing transitions. The selection rule is AMj = 0, and the intensities of the Stark components are given by: I ~ M 2 , J <-> J I oc [(J+l)2-M 2] , J+l <-> J In microwave spectroscopy the Stark effect has three major applications: 1) Measurement of dipole moments Stark shifts are measured as a function of a calibrated homogeneous electric field The dipole moment is then evaluated using equations such as 2.34 or 2.36 appropriate to the transition in question. 3 1 2) Stark modulation A high frequency electric field from 0 to E , usually in the form of a square wave, is applied to the molecules, causing the rotational levels and transitions to split and shift as a function of field The lines are detected with an amplifier tuned to the modulation frequency, thus increasing the signal-to-noise ratio by several orders of magnitude. In a relatively slow microwave frequency scan (slow relative to the modulation frequency) both unsplit and split lines are displayed simultaneously, and it is easy to see how the lines split and the fields at which they split. 3) Line assignments In general, transitions between non-degenerate levels, such as those for K a = 0 and 1 of a slightly asymmetric prolate rotor like cyclopropyl bromide, will show a second-order Stark effect. The Stark components (the perturbed lines) shift very slowly with an increase in electric field and high fields are required to modulate such lines. Any degenerate or near-degenerate levels, such as those at high K a will exhibit a first-order Stark effect, and will be modulated at low field This simple method of confirming line assignments was used very frequendy in the work described in this thesis. In favourable cases, the number and pattern of Stark components can also be used to make assignments. 3 2 BIBLIOGRAPHY 1. G. W. King, R. M. Hainer, P. C. Cross, J. Chem. Phys., 11, 27-42, (1943). 2. B. S. Ray, Z. Physik, 78, 74-91, (1983). 3. S. C. Wang, Phys. Rev., 34, 243-252, (1929). 4. W. Gordy, R. L. Cook, Microwave Molecular Spectra. 3rd. ed., in Techniques of Chemistry, Ed., A. Weissberger, Vol. 18, Wiley, New York, 1984, Chapter VU, page 251. 5. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy. McGraw-Hill, Book Company Inc., New York, 1955, Appendix HJ. 6. E. B. Wilson, J. B. Howard, J. Chem. Phys., 4, 260-268, (1936). 7 D. Kivelson, E. B. Wilson, J. Chem. Phys., 20, 1575-1579, (1952). 8. D. Kivelson, E. B. Wilson, J. Chem. Phys., 21, 1229-1236, (1953). 9. H. Dreizler, G. Z. Dendl, Z. Naturforsh, 20a, 30-37, (1965). 10. H. Dreizler, H. D. Rudolph, Z. Naturforsh, 20a, 749-751, (1965). 11. J. K. G. Watson, J. Chem. Phys., 46, 1935-1949, (1967). 12. J. K. G. Watson, J. Chem. Phys., 48, 4517-4524, (1968). 13. W. Gordy, and R. L. Cook, o^ cJL, Chapter 9. 14. J. K. Bragg, Phys. Rev., 74, 533-538, (1948). 15. H. P. Benz, A. Bauder, Hs. H. Gunthard, J. Mol. Spectrosc, 21, 156-164, (1966). 16. C. H. Townes, A. L. Schawlow, Qrj^  ciL, Chapter 10, page 249. 17. C. H. Townes, QJL. cit, page 251. 33 18. C. H. Townes, or^ cjt., page 255. 19. W. Gordy, W. V. Smith, R. F. Trambarulo, Microwave Spectroscopy. Dover Publications, Inc., New York, 1966. 20. H. M . Jemson, Ph. D. Thesis, University of British Columbia, 1986. 21. D. Anderson, Ph. D. Thesis, University of British Columbia, 1986. 34 CHAPTERm EXPERIMENTAL METHODS 3.1 THE MICROWAVE SPECTROMETER The microwave spectra were measured using a conventional 100 kHz Stark modulated microwave spectrometer, in the frequency range of 8 - 90 GHz. The essential elements of such a spectrometer are a tunable source of microwave radiation, a frequency measurement system, an absorption cell, a modulation system, a detector, and a means of storing and displaying data. Each of these will be described in rum. A block diagram is given in Fig 3.1. In the microwave region radiation is electronically generated; with respect to the lines this is essentially monochromatic, but tunable. The WatMns-Johnson 1291A Microwave Synthesizer, which has a step size as small as 10 Hz, was used as a fundamental source in the region 8-18 GHz. Measurements were referenced to a crystal in the synthesizer, of accuracy 1 in 109, which was checked against the atomic standard owned by Dr. W. Hardy, Dept. of Physics. Higher frequencies were obtained by multiplying the output with a Honeywell - Space Kom 14-27 Doubler or T-K al Tripler, or a Millitech FEX-10 sktimes Multiplier. The specified output ranges of the 35 multipliers were: SPECIFIED MULTIPLIER FREQUENCY RANGE (GHz) Space Kom 14 - 27 18.0 - 26.5 Space Kom T - K a l 26.5 - 40.0 Millitech FEX-10 80.0-97.0 Two Hewlett-Packard (HP) X band Stark cells (8425B), approximately one meter each in length, were connected in series. They had an inner cross sectional dimension of 0.9 inch x 0.4 inch. A flat metal strip or septum of gold plated copper was placed halfway between and parallel to the broad faces df the cell, so that when a 0 - 2000 volt zero-based square wave voltage (see below) was applied between the septum and wave guide, the resulting electric field, which could thus be varied from 0 - 4000 volt cnr 1, was effectively uniform. Teflon strips were used to insulate the septum from the rest of the cell. Brass vacuum ports were connected to each end of the cell, and permitted sample to be flowed through it. The cell, which was connected to a vacuum system, was sealed at both ends using O-ring and mica windows. The latter were transparent to microwave radiation. The synthesizer was connected to the cell using a flexible microwave cable and a series of microwave parts which included a multiplier (if necessary), a ferrite isolator, a variable attenuator and an appropriate tapered transition. The isolator was used to prevent the possibility of reflected power reaching the source and interfering with the frequency stablization. The attenuator was used to prevent power saturation of strongly absorbing lines. 36 The microwave spectrum was measure d at low pressures, ~10 - 40 microns, as higher pressures cause collisional broadening of the lines, with consequent reduction in resolution(they also result in discharging problems from the Stark modulation to be described below). At this low pressure the absorption coefficients are very small, with less than 0.1% of the incident power being absorbed. Any absorption signals can be swamped by other fluctuations in power, such as reflections in the waveguide or thermal noise in the detector. Modulation methods are usually used to enhance the signal to noise ratio; Stark modulation is the most common method, and was used in this work. A 0 - 2000 volt zero-based 100 kHz square wave voltage was applied directly to the Stark plates; the field was on for half the cycle and off for the other half. Transitions between unsplit levels were thus modulated at 100 kHz, as were transitions between levels split by the field (Stark components). Different detectors were used within certain frequency region. The table listed below indicates the detector used and the frequency range. DETECTOR T Y P E BAND FREQ-RANGE HP H06 X422A B A C K DTODE X 8-12 GHz HPH06P422A B A C K DIODE P 12-18 GHz HP K422A POINT C O N T A C T K 18-26.5 GHz DIODE HP 11586A POINT C O N T A C T R 26.5-40 GHz DIODE HUGHES 47324H-1100 SCHOTTKY BARRIER M DIODE 40-100 GHz The output signal of the detector was passed through a preamplifier to a phase sensitive (lock-in) amplifier. A reference signal (100 kHz) from the square wave generator also went to the lock-in amplifier. Only those signals oscillating at the reference frequency at the correct phase were detected, and other power fluctuations were not selected. Since the background characteristics are ignored by the detection system, the strength of the signal and the signal-to-noise ratio are enhanced by Stark modulation. The output of the lock-in amplifier was passed to the Y plates of an oscilloscope or to a chart recorder, and was also stored on a hard disc of a Digital Micro PDP 11/23 + computer. The frequency sweep of the oscilloscope was provided by a ramp from the synthesizer applied to the X plates. The lines and Stark components were displayed simultaneously on the oscilloscope or chart, or stored in the computer. They were distinguished from each other because the lock-in amlification caused them to be displayed on opposite sides of the baseline. In addition, the frequencies of the Stark components varied with the applied field. Because the Stark field was parallel to the electric vector of the electromagnetic radiation only transitions with AMj = 0 occurred when the Stark field was on. Frequencies were swept using the microprocessor in the Watkins-Johnson synthesizer, controlled using the Micro PDP 11/23 + computer. The microprocessor was interrupt driven so that signal averaging and oversampling could take place in real 38 time, and hence the signal-to-noise ratio and sensitivity of the signal could be improved significantly. Signal averaging was done by summing the intensity at o le frequency point for several time units, and averaging the sum. Oversampling was accomplished by collecting data at given intervals between the pre-set sampling points averaging these together and storing the result as the datum for a designate sampling point Each line was recorded using sweeps both up and down in frequency, and the average of the two peak frequencies was taken to remove the effects of different time constants in the system. Estimated accuracies for all measured transitions were ± 0.04 MHz. 3.2 S A M P L E , S A M P L E H A N D L I N G A N D M E A S U R E M E N T T E C H N I Q U E S A sample of cyclopropyl bromide was purchased from Aldrich Company, Inc., and was not purified further. At room temperature it is a colourless liquid and its boiling point is 69° C under 1 atm pressure. Since it is a flammable liquid lachrymator, it was stored in the dark room. A fresh sample of cyclopropyl bromide (10 drops approx.) was introduced to a 100 mL round bottomed single-necked flask. The flask was attached to the spectrometer vacuum line via a stopcock, and the sample was cooled to liquid nitrogen temperature; the air was removed by pumping. A small amount of sample, now back to at room temperature, were introduced into the system by letting a small amount into the space between the container outlet and the vacuum line inlet. The desired frequency range was obtained on the spectrometer, and the signal was displayed on the oscilloscope while the pressure and Stark voltages were adjusted for optimum conditions of best resolution and minimum interference by Stark components. At this point the pressure was usuaMy 10 ~ 40 microns (the low presure was for strong lines, and the high pressure was for weak lines, in order to enhance their intensities). At this point lines as close together as 0.27 MHz could be resolved. The sample could last in the cell for more than 1 hour. After detailed study and measurement of a given transition, the sample was pumped from the cell and replaced for measurement of a new transition. Below 18 GHz it was also necessary to adjust the microwave power level, in order to remove saturation broadening. Above 18 GHz, where the multipliers were used, the sources did not produce enough power to saturate the lines. Frequency scanning ranges when measurements were being made were about 10 MHz or less. At this point the step size and dwell time used were 0.003 MHz and 0.03 seconds, respectively. The signal average count was set to 6 for stronger lines, and 20 for weak lines (c-type transitions). The average scanning times were about 10 minutes for the strong lines, and 30 minutes for the weak lines. Attenuator Source of Microwaves Stark Cell To Vacuum Line I Square-wave Generator L Computer Chart Recorder Oscilloscope Detector Preamplifier Lock-in Amplifier Figure 3.1 Block diagram of microwave spectrometer 4 1 BIBLIOGRAPHY 1. W. D. Perkins, J. Chem. Education, 63, A5-10, (1986). 2. H. W. Kroto, Molecular Rotation Spectra, pp. 250-256, Wiley, London, 1975. 3. H. M. Jemson, Ph. D. Thesis, university of British Columbia, 1986. 4 2 C H A P T E R IV M I C R O W A V E S P E C T R U M O F C Y C L O P R O P Y L B R O M I D E 4.1 INTRODUCTION A moderate amount of spectroscopic work has been done on cyclopropyl bromide. The vibrational spectrum of this molecule has been measured several times. In 1973 Hirakawa et al. (1) measured the gaseous and liquid phase infrared spectrum of cyclopropyl halides and calculated the force constants using their data. The irifrared absorbtion and Raman scattering spectrum of cyclopropyl bromide was reported in 1979 (2). The gas electron diffraction spectrum was reported by Marsden et al. in 1988 (3) for detenriining the molecular structure of cyclopropyl bromide. As was mentioned in Chapter I, the microwave spectrum of cyclopropyl bromide, measured by Lam and Dailey (4), consisted of a limited number of a-type transitions of both ?9Br and 8 1Br species. From it were obtained values of rotational constants B and C to good accuracy (± 0.03 MHz), and fairly rough estimates of the Br nuclear quadrupole coupling constants diagonal in the inertial axes system. These data were used to make an estimate of the structure, and it was indicated that the molecule had Cs symmetry, with the symmetry plane the ac plane. By assuming that the C-Br bond is a principal axis of the quadrupole tensor the tensor was diagonalized. However, the rotational constant A of both species was not well determined, since only few a-type R branch transitions were measured. No value for the coupling constant was given, and it could not be obtained from their data because they reported no measurements. Finally, no centrifugal distortion constajits were evaluated. In the work described in this thesis extensive measurements were made of a-type R branch transitions, over a very much wider frequency range than was used previously. These included accurate quadrupole hyperfine splitting measurements. From the analysis of these lines it was possible to detect and assign some weak c-type transitions. The accuracy of all three rotational constants has been improved by several orders of magnitude. The centrifugal distortion constants have been obtained for the first time. All the quadrupole coupling constants, including X^, which was missing in the previous work, have been determined to high accuracy. 4.2 A S S I G N M E N T A N D A N A L Y S I S Since the a-type spectra previously reported for the 7 9 B r and 8 1 B r species are rather insensitive to the value of the rotational constant A the determination of an accurate value for it was the initial aim. It could have been determined by assigning c-type transitions. However, at this stage these were very difficult to find because they were very weak compared to the a-type transitions. Furthermore, predictions of their frequencies were very poor ( ± 500 MHz), because they were very dependent on the value of A. Initially a prediction of the a-type spectrum was made with the published rotational constants and quadrupole constants, and considering only first order contributions to the quadrupole structure. A search was made for new transitions over the frequency range 25 - 45 GHz, which was a considerably wider range than was previously reported. For most of these the Br hyperfine splitting patterns were essentially symmetric quartets as predicted from first order theory. However, some of the transitions had different splitting patterns that could not be explained in this way. In this case higher order theory was necessary to explain the pattern; this had been hoped for, since it would provide information necessary to the value of A. The reason for this was the following. As was mentioned in section 2.4 for molecules containing a single quadrupolar nucleus the field gradient at this nucleus is: In the present case because of the symmetry of the molecule, V ab = Vbc = 0. The quadrupole coupling constant to be used in equation 2.30 is thus Only the first three elements contribute to the first order theory. They determine most of the hyperfine sphttings. The element Xacfefea is the high order term, and often makes a significant contribution to the spectrum in the case of the bromine nucleus (5 ). An initial value of was estimated from the constants and structure in ref. (4) using (4.1) + 2 Vbc< l )Zb < 1 >Zc + 2Vab<t>Za<t>Zb = eQV^ = x M <4 + X b b <& + Xcc 4>zc + 2Xac <t>zAc (4.2) 45 the following equation: tan 2ti = -2% ac Xan Xg£ (4.3) where is the angle between the a-inertial axis and the C-Br bond given by Lam and Dailey as 19° 13'. These values were 254.1 MHz for C3H579Br and 217.3 MHz for C 3 H 5 8 1 Br. When has such large values it can perturb the hyperfine structure severely especially when rotational near degeneracies occur (6). From the symmetry point of view the selection rules for these near degeneracies for CsHsBr are AJ = 0, ± 1, ± 2; AF = 0, and K a Kc = ee <-> oo or oe <-> eo. Since these degeneracies are closely related to the three rotational constants and centrifugal distortion constants (6), we can get extra information on these constants and on X ^ from the difference between first order prediction and the observed spectra. Therefore, it should be possible to get an improved A rotational constant solely from a series of a-type R branch transitions that include several perturbed transitions. Interactions between other levels much further apart also contribute to the accuracy of these constants, and cannot be neglected. In the past, spectra of molecules having quadrupole hyperfine structure were analysed by first accounting for the hyperfine structure with perturbation theory, subtracting it off, and then analysing the frequencies of the "unsplit" lines for rotational and centrifugal distortion constants. Although this is a relatively simple procedure to apply in general, it is rather awkward when high order terms are needed, and the accuracy of the resulting constants is poor. Recently, however, Jemson developed a 46 procedure which greatly improves the accuracy of the A rotational constant measured solely from a-type R branch transitions (7). She used a computer program which does a simultaneous global least squares fit to the rotational, centrifugal distortion and quadrupole coupling constants, using an exact Hamiltonian including bom A and an off-diagonal constant (6,7,8). The program was very large and was run using Cray vector computer of die Atmospheric Envlrouaent Service of Enviroment Canada in Dorval, Quebec. It was proposed to use this method for cyclopropyl bromide. It is much easier to apply the method when the largest potential perturbations are known. This was done by predicting approximately the rotational energy levels of both species, again using the constants of I .am and Dailey. An energy level diagram was constructed, from which it was very easy to see the closest near degeneracies which might cause perturbations. A similar diagram for C^HS^T is given in Fig. 4.1; it was calculs . d from the final constants given later in this chapter. The nearest degeneracies causing perturbations are listed in Table 4.1. The first measurements were for the lines K, = 0, K, = 1 and KK = 2 for J = 5 to J = 9 between 25 - 45 GHz. This region was chosen because of the calculated near degeneracies between the levels 8n - 726 and the levels 9i9 - 826. so that several of the lines were expected to be perturbed. Assignments of transitions were first made on the basis of the first order predictions, in combination with the Stark effect The lines with K 8 = 2 occurred at relatively low Stark field, and their Stark components moved "inwards" in frequency towards each other. Those with K* « 0 required very much higher Stark fields. Use of these Stark properties was very important because the lines having K 4 = 0 and K, > 2 are all very closely spaced; thus a method of distinguishing these lines clearly was essential. Even then, sometimes lines with K» = 2 overlapped with lines with K, > 2, and had to be ignored. 47 Table 4.1 The near-degenerate energy levels of cyclopropyl bromide ( 7 9Br) The near-degenerate 1The energy difference energy levels between energy levels / MHz. 3fJ3 2l2 1477 8] 7 72 6 1027 9l9 826 934 142,13 133,10 1063 142.12 133.11 2521 These are unsplit energies of levels resulting in large effect of X 48 By using the global fitting program, with initial approximations to the constants those already published and calculated above, improved rotational constants and quadrupole coupling constants were obtained from these lines. Since the near degeneracies shifted some lines from the first order predictions (see Figure 4.2) some wrong assignments were initially made. However, these errors were quickly spotted because of their much bigger residuals (obs-calc) than for the other transitions. The least squares refinements were made using Watson's S-reduction in its F representation since problems arose in determining good values for some of the distortion constants, caused by large correlation between some of the constants, using the A-reduction Hamiltonian. A new modified version of the program for the fits, modified for the Micro Vax IT by W. Lewis Bevan was used in this thesis. After correction of the assignments the values of the constants, particularly A and X^, were improved considerably. There were, in addition, no correlations among the constants. The deviation of A, in particular, was much reduced to ± 1.5 MHz, which was much better than the previous ±160 MHz and ± 65 MHz. Because C3H5BT is a fairly light molecule, centrifugal distortion will have a noticeable effect, even for the lower rotational levels (9,10). The distortion effect increases with J, and is seen most clearly in high J transitions. Therefore, to be able to determine accurate values of the distortion constants it was necessary to measure and assign transitions to as high J and K a as possible. As well, to improve the accuracy of the quadrupole coupling constants it seemed that further measurements at both low and high J would be helpful. There were two near degeneracies of about 1000 MHz: 3Q3 -2i2 and 142,13 - 133,io- Values of the constants from the early fits described above were used to predict transitions over a much wider range of frequency (14-90 GHz) and of J (2 -17 ). A large number of these were sought and found, and another fit was carried out. Further improvement of the rotational constants and quadrupole coupling constants was obtained, along with initial values for the centrifugal distortion constants. Many of these further measurements required special effort and care. The sensitivity of the spectrometer was somewhat lower at the low frequencies than was used for the early measurements; in addition the quadrupole sphttings were larger and asymmetry sphttings smaller, so there was always a danger of overlapping lines. Careful attention to the Stark effect and the predictions resolved these problems. At higher frequencies the equipment was used often beyond its specified ranges. Between 40 and 50 GHz, for example, the T-K al Tripler (specified for 26.5 - 40 GHz) and the Hughes detector (specified for 50 - 75 GHz) were used; both were beyond their ranges. Above 54 GHz the T-K al Tripler was used as 6x multiplier, this use was not specified at all, yet the multiplier produced enough power at frequencies six times the fundamental that many useful measurements could be made. As was mentioned above, the c-type transitions were much weaker than the a-type lines and were very sensitive to the A rotational constants, so they could not initially be assigned. The new much more precise constants evaluated from a-type lines were then used to predict c-type transitions, and a search was carried out For both isotopic species several such transitions were found within at most a few MHz of their predicted frequencies. They were identified from their quadrupole patterns, which could now be predicted accurately, even including the Xac contributions. Identification of many of the c-type transitions was still not particularly easy, however, not only because they were very weak because of the small value of Uc, but also because they were overlapped with strong a-type lines. Still, some K a = 0,1, and 2 R and Q branch lines were measured. All the transitions, both a and c-type, were then put in a global least-squares fit to (tetemiine the 5 0 final rotational, centrifugal distortion and bromine quadrupole coupling constants. Once again, Watson's S-reduction in the F representation (10) was used throughout. The measurements were given relatively arbitrary weights reflecting the confidence in the measurement accuracies. In particular, when it was realized that a given line represented two overlapped transitions then it was weighted down by a factor of 100. An exception was made when the (obs-calc) deviations for these lines were less than the measurement accuracies. Accordingly a weighting scheme was developed and applied to the lines account for the fitting problems. This scheme took into account problems caused by the overlapping, interference and an inablity to modulate the lines properly. A weight of 1.00 corresponded to a measurement uncertainty of ± 0.03 MHz. A low weight corresponded, roughly, to a measurement uncertainty of (1/ weight) x 0.03 MHz. The weighting scheme is given in Table 4.2. The values of the constants are in Table 4.3 and Table 4.4 gives the correlation coefficients obtained in the fits. With the possible exception of A and DK , which are at most weakly correlated, the derived constants are uncorrelated. All the measured transitions are in Table 4.5 for C3Hs 7 9 Br and in Table 4.6 for C3H581Br.These tables also give the deviation between the observed frequencies and those calculated from the derived constants, both with XK included and with X ^ omitted. The very small deviations in the former case indicate that the fit is excellent. In the latter case there are several large deviations, indicating that Xac makes several significant contributions to the frequencies in those cases. 5 1 Table 4.2 Weighting scheme for the transitions of cyclopropyl bromide Weight Quality of lines 1.00 Accurate: includes a quartet, well spaced doublets with narrow internal spUtting, a narrow singlet and no interference. 0.50 Very good: includes two doublets with internal spUtting > 0.3 MHz and < 0.6 MHz, and slight interference. 0.10 Good: includes a doublet with internal spUtting > 0.3 MHz and < 0.6MHz, and a singlet with a spread of > 0.3 MHz and < 0.4 MHz. 0.01 Fair: includes same as those with a weight ing of 0.1, but now have a poor shape or their modulation is dubious. Singlets have a spread of > 0.4 MHz and < 0.6 MHz. 0.0001 Poor: includes quartets or doublets with inter-ference so only half the quartets or doublets could be measured, and singlets with a spread of > 0.6 MHz and < 0.8 MHz. 0.000001 Bad: includes a singlet with a spread > 0.8 MHz. Table 4.3 Spectroscopic constants of cyclopropyl bromide Parameter C3H 5 7 9 Br C ^ i B r Rotational constants / MHz Ao 16334.241(H)1 16333.034(23) B 0 2579.92044(40) 2560.54002(67) Co 2457.71605(38) 2440.14986(52) Centrifugal distortion constants / kHz Dj 0.64572(74) 0.63754(84) D J K 0.902(36) 0.899(42) DK 14.4(32) 14.9(38) di 0.01615(79) 0.0139(10) d 2 - 0.00207(43) -0.00238(47) Br quadrupole coupling constants / MHz X „ 463.69(22) 388.14(25) Xb b - X^ - 105.838(71) - 88.156(78) Xac 266.36(34) 221.98(35) 1 Numbers in parentheses are one standard deviation in units of the last significant figures. 53 Table 4.4 Correlation coefficients of the spectroscopic constants of CsHsBr derived from total transitions C 3 H 5 7 9 B r Ao 1.00 B 0 -0.33 1.00 Co 0.65 -0.06 1.00 Dj -0.02 0.67 0.40 1.00 D J K 0.26 0.04 0.28 -0.32 1.00 D K 0.96 -0.36 0.59 0.04 0.03 1.00 dj -0.49 0.64 -0.64 0.31 -0.40 -0.44 1.00 d2 0.40 -0.25 0.22 0.04 -0.13 0.46 0.01 1.00 Xaa 0.00 -0.03 -0.07 -0.08 0.07 -0.03 0.01 -0.02 1.00 Xbb "Zee "0-01 0.05 0.00 0.01 0.05 -0.04 0.04 0.02 -0.13 1.00 Xac -0.10 -0.04 -0.04 0.00 -0.10 -0.08 0.00 0.02 -0.17 0.01 1.00 C 3 H 5 8 1 B r A0 1.00 B 0 -0.76 1.00 C 0 0.75 -0.51 1.00 Dj -0.10 0.47 0.31 1.00 D J K -0.18 0.30 -0.12 -0.28 1.00 DK 0.96 -0.69 0.69 -0.02 -0.30 1.00 di -0.73 0.81 -0.79 0.24 -0.00 -0.63 1.00 d2 0.31 -0.12 0.17 0.13 -0.07 0.39 -0.01 1.00 XM 0.02 -0.06 -0.05 -0.08 0.03 0.02 -0.02 0.02 1.00 Xbb-Xcc-0 09 0.03 -0.05 0.00 -0.03 -0.11 0.02 0.04 -0.25 1.00 Xac 0.02 -0.07 0.05 -0.05 -0.09 -0.02 -0.06 -0.16 -0.16 0.03 1.00 5 4 900- 1 7 3 . U 800 -700" 6 0 0 -5 0 0 -A00-3 0 0 -200 -100-0 -uo,o •17, 0.17 16, '0.16 •15 0.15 o,u 13 0.13 12 0.12 - Ilc.n •IO0.IO ' 9o,9 ' 8c,8 •7c.7 ' 60,6 ' 5o,5 Mi HTTMHz ? z z : A1ToT M.1 1 71.16 • ' ' U 7 •15 1 , 1 4 • 1.15 . . , 1 . 1 3 1 V ' 1 2 1^1.13 . . , 1 . 1 1 • •^1 .12 : . 1.10 ' ' 1 , 1 1 :io!:?o 934 MHz • A y 1027 MH2 1/2.16 'K>2,15 , 1 « ; 2 ' 1 3 2.12 2,13 1063 M H i • 1 J 2 , 1 2 . 1 ? 2 , 1 0 ' ^ 2 , 1 1 •11 2.10 2,11 •wi:! l t ) 3 , B 1&3.13 . 3 . 1 1 W 3 . 1 2 •i3|:!f i2l:?0 •113'8 M 3 , 9 -10? Q 3 . 6 o 3 . 5 ° 3 , 6 .4 • 5 e|i 53:! 71:33: Figure 4.1 Rotational energy levels of CjRs^T- Four important near-degeneracies are indicated. JK*Kc OBS HQ. MB 110-IW IV* ISA IV-•w euct CALC. limankr lOMHl 0») Figure 4.2 The energy levels of the 826 - lis transitions of C3Hs 7 9Br. (a) The energy levels: (i) the hypothetical unsplit rotational energies; (ii) predicted first-order bromine quadrupole energy levels; (iii) bromine quadrupole energy levels derived from the exact Hamiltonian. (b) The observed transitions compared with the calculated fust-order and exact patterns. TABLE 4.5 Measured rotational transitions ( in MHz ) of C 3 H 5 7 9 B r 56 Transition Normalised1 Observed Residuals 2 F - F" Weight Frequency Without X a c with X 3 o 3 3 / 2 -5 / 2 -9 / 2 -7 / 2 -1/2 3 / 2 7 / 2 5 / 2 2 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 5 1 3 7 . 8 0 9 1 5 1 3 4 . 4 2 5 1 5 1 0 4 . 2 4 3 1 5 1 0 6 . 2 6 3 4 . 7 1 4 2 . 0 8 1 0 . 0 1 4 2 . 2 7 9 0 . 0 6 0 - 0 . 0 0 7 - 0 . 0 0 6 0 . 0 5 3 3 1 3 7 / 2 -5 / 2 -9 / 2 -3 / 2 -5 / 2 3 / 2 7 /2 1/2 2 1 2 0 . 0 1 0 0 . 0 1 0 1 . 0 0 0 1 . 0 0 0 1 4 9 4 6 . 7 9 5 1 4 9 4 6 . 2 2 7 1 4 9 1 8 . 6 2 7 1 4 9 1 4 . 2 8 3 0 . 8 4 8 2 . 7 4 0 1 . 2 4 0 0 . 0 6 3 0 . 0 3 4 0 . 0 8 2 0 . 0 2 5 0 . 1 1 7 3 1 2 5 / 2 -7 /2 -3 / 2 -9 / 2 -3 / 2 5 / 2 1 / 2 7 / 2 2 i 1 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 5 3 1 5 . 5 2 7 1 5 3 1 1 . 3 5 8 1 5 2 8 6 . 2 2 0 1 5 2 8 2 . 7 4 0 - 0 . 1 1 1 - 0 . 0 3 5 - 0 . 0 3 9 - 0 . 0 2 0 - 0 . 0 1 0 - 0 . 0 3 3 - 0 . 0 8 3 - 0 . 0 5 5 4 o « 5 / 2 -7 / 2 -11 /2 -9 / 2 -3 / 2 5 / 2 9 /2 7 /2 3 0 31 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 2 0 1 4 9 . 6 1 6 2 0 1 5 1 . 1 2 6 2 0 1 3 8 . 9 5 9 2 0 1 3 7 . 4 5 4 - 2 . 7 3 6 - 0 . 8 2 9 0 . 0 4 6 -1 . 1 8 5 - 0 . 0 4 2 0 . 0 5 6 0 . 0 3 2 0 . 0 0 4 4 , « 7 /2 -9 / 2 -5 / 2 -11 /2 -5 / 2 7 /2 3 / 2 9 /2 3 1 31 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 9 9 1 4 . 1 5 5 1 9 9 0 9 . 4 3 0 1 9 9 0 2 . 2 0 9 1 9 8 9 8 . 2 1 6 0 . 2 8 1 0 . 0 5 6 - 0 . 0 0 1 0 . 2 1 0 0 . 0 3 2 0 . 0 0 0 0 . 0 5 9 0 . 0 4 1 4 1 3 7 / 2 -9 / 2 -5 / 2 -1 1 / 2 -5 / 2 7 /2 3 / 2 9 / 2 3 1 21 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 2 0 4 0 4 . 8 8 9 2 0 3 9 7 . 4 0 2 2 0 3 9 3 . 2 3 4 2 0 3 8 5 . 9 4 0 - 0 . 1 2 7 - 0 . 0 0 3 - 0 . 1 1 2 - 0 . 0 6 7 - 0 . 0 2 0 0 . 0 0 4 - 0 . 0 6 1 - 0 . 0 8 8 5 o 5 7 / 2 -9 / 2 -1 3 / 2 -1 1/2 -5 / 2 7 / 2 1 1/2 9 / 2 4 0 * 0 . 0 1 0 0 . 0 1 0 0 . 2 5 0 0 . 2 5 0 2 5 1 7 6 . 8 9 9 2 5 1 7 6 . 8 9 9 2 5 1 6 9 . 3 6 0 2 5 1 6 8 . 9 4 0 - 0 . 4 2 9 - 0 . 0 6 5 - 0 . 0 2 7 - 0 . 1 2 7 - 0 . 1 1 2 0 . 0 3 0 - 0 . 0 3 9 0 . 0 2 7 1 Measurements were weighted a c c o r d i n g to 1 / a2, where o i s the u n c e r t a i n t y i n the measurements. Unit weight corresponded to an u n c e r t a i n t y of 0 . 0 3 MHz. 2 Observed frequency minus the frequency c a l c u l a t e d using the constants i n Table4.3 57 Table 4.5 (continued) Transition Normalised Observed Residuals p F" Weight Frequency Without X a c with X a c 5 , 5 9 / 2 -1 1 / 2 -7 / 2 -1 3 / 2 -7 / 2 9 / 2 5 / 2 1 1 / 2 4 i " o . o i o 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 2 4 8 8 5 . 1 7 0 2 4 8 8 0 . 5 5 2 2 4 8 7 9 . 0 7 4 2 4 8 7 4 . 9 7 2 0 . 4 2 6 0 . 0 1 6 0 . 1 0 0 0 . 0 7 8 0 . 0 5 6 0 . 0 1 7 - 0 . 0 3 0 0 . 0 0 8 5 j « 9 / 2 -7 / 2 -1 1 / 2 -1 3 / 2 -7 / 2 5 / 2 9 / 2 1 1 / 2 4 i 3 1 . 0 0 0 0 . 0 1 0 o . o r o 1 . 0 0 0 2 5 4 9 7 . 2 0 6 2 5 4 9 1 . 2 4 2 2 5 4 9 0 . 9 4 9 2 5 4 8 5 . 3 0 9 0 . 2 2 7 0 . 0 3 1 0 . 0 1 3 0 . 0 2 7 0 . 0 5 0 - 0 . 0 5 6 0 . 0 3 3 0 . 0 1 3 6 1 e 1 1 / 2 -9 / 2 -13 / 2 -1 5 / 2 -9 / 2 7 / 2 1 1 / 2 1 3 / 2 5 1 5 1 . 0 0 0 0 . 5 1 0 0 . 5 1 0 1 . 0 0 0 2 9 8 5 6 . 0 9 2 2 9 8 5 2 . 7 7 4 2 9 8 5 2 . 7 7 4 2 9 8 4 9 . 6 6 9 - 0 . 0 5 5 - 0 . 1 3 0 - 0 . 0 0 9 0 . 0 6 7 0 . 0 3 0 0 . 0 0 6 0 . 0 1 7 0 . 0 2 8 ° 1 5 1 1 / 2 -9 / 2 -13 / 2 -1 5 / 2 -9 / 2 7 / 2 1 1 / 2 1 3 / 2 5 1 * 1 .000 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 3 0 5 8 9 . 8 6 8 3 0 5 8 6 . 5 1 4 3 0 5 8 5 . 3 7 1 3 0 5 8 2 . 2 1 8 - 0 . 1 1 3 - 0 . 2 2 8 - 0 . 0 2 4 0 . 0 0 7 - 0 . 0 1 9 - 0 . 1 0 5 0 . 0 2 9 - 0 . 0 0 3 7 O 7 1 1 / 2 -1 3 / 2 -1 7 / 2 -1 5 / 2 -9 / 2 1 1 / 2 1 5 / 2 1 3 / 2 6 0 6 1 . 0 0 0 1 . 0 0 0 0 . 0 1 0 0 . 0 1 0 3 5 2 2 0 . 7 8 1 3 5 2 1 9 . 2 9 6 3 5 2 1 6 . 1 0 4 3 5 2 1 5 . 7 1 4 - 0 . 0 0 4 - 0 . 0 2 7 - 0 . 0 2 7 0 . 0 1 0 0 . 0 6 8 - 0 . 0 0 9 - 0 . 0 3 5 0 . 0 4 1 7 1 7 1 3 / 2 -1 1 / 2 -1 5 / 2 -1 7 / 2 -1 1 / 2 9 / 2 1 3 / 2 1 5 / 2 6 1 6 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 3 4 8 2 7 . 1 8 5 3 4 8 2 5 . 0 7 3 3 4 8 2 4 . 4 6 1 3 4 8 2 2 . 5 6 8 0 . 0 4 8 - 0 . 0 8 7 - 0 . 0 2 0 0 . 0 3 0 0 . 0 3 7 0 . 0 1 3 0 . 0 4 6 0 . 0 0 5 7 1 6 1 3 / 2 -1 1 / 2 -1 5 / 2 -1 7 / 2 -1 1 / 2 9 / 2 1 3 / 2 1 5 / 2 6 1 5 1 . 0 0 0 1 .000 1 . 0 0 0 1 .000 3 5 6 8 2 . 6 6 1 3 5 6 8 0 . 4 2 1 3 5 6 7 8 . 8 3 7 3 5 6 7 7 . 1 3 6 - 0 . 0 0 6 - 0 . 2 7 7 - 0 . 2 9 8 - 0 . 0 6 0 0 . 0 3 1 - 0 . 0 1 5 - 0 . 0 9 6 - 0 . 0 6 8 7 2 5 1 3 / 2 -1 5 / 2 -1 1 / 2 -1 7 / 2 -1 1/2 1 3 / 2 9 / 2 1 5 / 2 6 2 * 1 . 0 0 0 1 . 0 0 0 0 . 0 1 0 0 . 0 1 0 3 5 3 0 6 . 6 2 7 3 5 3 0 5 . 4 8 6 3 5 2 9 7 . 6 6 2 3 5 2 9 7 . 1 8 8 0 . 0 4 6 - 0 . 0 6 6 - 0 . 2 1 5 0 . 2 0 3 - 0 . 1 1 4 - 0 . 0 7 2 - 0 . 1 2 0 - 0 . 0 5 1 Table 4.5 (continued) 58 Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c with X a c 7 2 6 13 /2 -15 /2 -11 /2 -17 /2 -1 1/2 1 3 / 2 9 / 2 1 5 /2 6 2 5 0 . 0 0 1 1 . 000 0.001 1 . 000 3 5 2 5 8 . 4 6 6 3 5 2 5 9 . 9 9 0 35252 .961 3 5 2 5 0 . 0 9 3 - 2 . 6 5 9 - 0 . 0 5 7 0 . 124 - 1 . 8 0 8 0 . 2 2 3 0 . 0 6 2 0 . 2 1 6 0 . 0 0 2 8 • 8 15/2 -13/2 -17/2 -19/2 -1 3 / 2 1 1/2 1 5/2 1 7 / 2 7 1 7 1 . 0 0 0 1 . 0 0 0 1 .000 1 . 0 0 0 3 9 7 9 7 . 1 5 9 3 9 7 9 5 . 5 9 0 3 9 7 9 4 . 8 2 5 3 9 7 9 3 . 8 2 9 0 . 0 1 9 - 0 . 2 8 1 - 0 . 1 9 3 0 . 0 6 0 0 . 0 1 5 0 . 0 1 4 0 . 0 5 3 0 . 0 4 3 8 - 7 15/2 -13/2 -17/2 -1 9/2 -1 3/2 1 1/2 1 5/2 1 7/2 7 1 6 1 . 0 0 0 1 . 000 1 .000 1 . 000 4 0 7 7 4 . 3 6 7 4 0 7 7 5 . 7 4 4 4 0 7 7 3 . 3 0 2 4 0 7 7 0 . 2 4 8 0 . 0 9 2 2 . 7 2 7 1 .810 - 0 . 0 0 4 - 0 . 0 1 0 - 0 . 0 2 7 - 0 . 0 1 3 - 0 . 0 1 1 8 2 6 17/2 -15/2 -1 3/2 -19/2 -15/2 1 3/2 1 1/2 1 7 / 2 7 2 5 1 . 0 0 0 1 .000 1 .000 1 .000 4 0 3 6 0 . 0 2 9 4 0 3 5 8 . 9 3 4 4 0 3 5 5 . 5 7 7 4 0 3 5 2 . 4 2 4 - 0 . 1 1 5 - 2 . 4 5 1 0 . 1 50 -1 .844 - 0 . 0 1 6 - 0 . 0 0 1 - 0 . 0 1 5 - 0 . 0 0 4 8 2 7 1 5/2 -17/2 -13/2 -19/2 -1 3/2 1 5/2 1 1/2 1 7/2 7 2 6 1 . 0 0 0 1 .000 0 . 0 1 0 0 . 0 1 0 4 0 2 9 5 . 6 2 6 4 0 2 9 2 . 1 7 3 ' 4 0 2 8 8 . 0 0 3 4 0 2 8 8 . 0 0 3 2 . 2 3 2 0 . 0 7 3 0 . 0 9 9 1 .298 0 . 0 2 3 - 0 . 0 0 1 - 0 . 0 6 6 0 . 0 9 4 9 o 9 15/2 -17 /2 -2 1 / 2 -19 /2 -1 3 /2 1 5 / 2 1 9 / 2 1 7 / 2 8 0 8 1 . 0 0 0 1 . 000 1 .000 1 . 000 4 5 2 4 1 . 8 4 3 45241 .381 4 5 2 3 9 . 6 4 9 4 5 2 3 9 . 1 0 6 - 0 . 0 4 2 0 . 0 2 9 0 . 0 1 5 0 . 0 1 2 - 0 . 0 1 5 0 . 0 3 4 0 . 0 1 0 0 . 0 2 4 9 1 9 17 /2 -1 5 / 2 -19 /2 -2 1 / 2 -1 5 / 2 1 3 / 2 1 7 / 2 1 9 / 2 8 1 8 0 . 5 1 0 1 . 000 0 . 5 1 0 1 . 000 4 4 7 6 5 . 8 9 8 4 4 7 6 7 . 5 9 6 4 4 7 6 5 . 8 9 8 4 4 7 6 3 . 2 5 2 0 . 107 2 . 6 4 6 1.831 0 . 0 1 3 0 . 0 0 4 - 0 . 0 3 5 - 0 . 0 1 3 0 . 0 0 0 9 , . 8 17 /2 -1 5 / 2 -19 /2 -2 1 / 2 -1 5 / 2 1 3 / 2 1 7 / 2 1 9 / 2 8 1 7 1 . 0 0 0 0 . 0 1 0 1 . 000 0 . 0 1 0 4 5 8 6 4 . 2 0 8 4 5 8 6 1 . 4 9 2 4 5 8 6 0 . 8 7 5 4 5 8 6 1 . 2 2 6 - 0 . 0 8 9 - 1 . 9 8 1 - 1 . 1 8 1 - 0 . 0 1 7 0 . 0 0 0 - 0 . 0 0 2 0 . 0 1 4 - 0 . 0 2 3 Table 4.5 (continued) 59 Transition F - F" Normalised Observed Residuals Weight Frequency Without X a c with X a c 9 2 8 - 8 2 7 1 7 / 2 - 1 5 / 2 1 . 000 1 9 / 2 - 1 7 /2 1 . 000 1 5 / 2 - 13 /2 1 . 000 2 1 / 2 - 19 /2 1 . 0 0 0 9 2 7 . - 8 2 6 1 7 / 2 - 15 /2 1 . 000 1 9 / 2 - 17 /2 1 . 000 1 5 / 2 - 1 3 / 2 0 . 5 1 0 2 1 / 2 - 1 9 / 2 0 . 5 1 0 10 , 1 0 - 9 1 9 1 9 / 2 - 1 7 / 2 1 . 000 1 7 / 2 - 15 /2 0 . 0 1 0 2 1 / 2 - 1 9 / 2 0 . 0 1 0 2 3 / 2 - 2 1 / 2 1 . 0 0 0 10 , 9 - 9 1 8 1 9 / 2 - 1 7 / 2 0 . 2 5 0 1 7 / 2 - 1 5 /2 0 . 2 5 0 2 1 / 2 - 1 9 / 2 0 . 2 5 0 2 3 / 2 - 2 1 / 2 0 . 2 5 0 10 2 8 - 9 2 7 1 9 / 2 - 1 7 / 2 1 .000 2 1 / 2 - 19 /2 1 . 000 1 7 /2 - 1 5 /2 1 . 000 2 3 / 2 - 2 1 / 2 1 . 0 0 0 10 2 9 - 9 2 B 1 9 / 2 - 17 /2 1 . 000 2 1 / 2 - 19 /2 1 . 000 1 7 / 2 - 15 /2 1 . 0 0 0 2 3 / 2 - 2 1 / 2 1 . 000 11 O ,1 1 0 O 1 0 1 9 / 2 - 1 7 / 2 0 . 0 1 0 2 1 / 2 - 19 /2 0 . 0 1 0 2 5 / 2 - 2 3 / 2 1 . 0 0 0 2 3 / 2 - 2 1 / 2 1 . 0 0 0 11 1 10 - 10 1 9 2 1 / 2 - 1 9 / 2 0 . 2 5 0 1 9 / 2 - 1 7 / 2 0 . 2 5 0 2 3 / 2 - 2 1 / 2 0 . 0 1 0 2 5 / 2 - 2 3 / 2 0 . 0 1 0 4 5 3 2 5 . 0 3 2 4 5 3 2 3 . 6 4 3 4 5 3 2 1 . 0 3 0 4 5 3 2 0 . 0 1 3 4 5 4 2 3 . 4 3 0 4 5 4 2 0 . 5 3 3 45417.33" , 45417 .331 4 9 7 3 2 . 7 4 5 4 9 7 3 0 . 3 2 4 4 9 7 3 0 . 3 2 4 4 9 7 3 0 . 8 3 4 5 0 9 5 2 . 3 0 9 5 0 9 5 1 . 6 2 8 5 0 9 5 0 . 3 8 7 5 0 9 4 9 . 9 7 5 5 0 4 8 7 . 9 9 2 . 5 0 4 8 6 . 7 1 7 5 0 4 8 4 . 4 8 8 5 0 4 8 3 . 5 7 9 50355 .761 5 0 3 5 4 . 1 9 2 5 0 3 5 2 . 5 6 7 5 0 3 5 2 . 1 6 2 5 5 2 3 5 . 7 1 0 5 5 2 3 5 . 5 2 4 55234 .621 5 5 2 3 3 . 3 9 2 5 6 0 3 7 . 9 5 8 5 6 0 3 7 . 5 8 9 5 6 0 3 6 . 4 0 5 5 6 0 3 6 . 1 3 8 0 . 1 2 9 0 . 0 1 9 • 0 . 0 5 5 0 . 122 1 . 728 0 . 0 4 8 • 0 . 0 4 0 1 . 125 • 0 . 0 8 6 •1 . 9 4 2 •1 . 0 8 2 • 0 . 0 1 7 • 0 . 0 2 0 • 0 . 160 • 0 . 102 0 . 0 1 9 0 .121 0 . 0 3 6 0 . 0 7 3 0 .101 0 . 0 8 4 0 . 0 3 9 • 0 . 0 1 0 0 .121 • 0 . 0 9 4 • 0 . 0 0 6 • 0 . 0 5 3 • 0 . 0 5 6 • 0 . 0 3 5 • 0 . 0 6 5 • 0 . 0 6 6 0 . 0 0 6 0 . 0 3 4 0 . 0 2 2 0 . 0 0 0 0 . 0 0 7 •0 .040 0 . 0 0 0 0 . 0 1 5 •0 .020 •0 .035 0 . 1 1 5 0 . 0 5 8 •0 .027 •0 .007 •0 .006 0 . 0 0 6 0 . 0 1 5 •0 .023 •0 .023 •0 .039 0 . 0 2 2 0 . 0 4 5 0 . 0 5 3 0 . 0 2 2 0 . 0 7 6 • 0 . 0 8 0 • 0 . 0 0 3 • 0 . 0 5 7 • 0 . 0 5 0 • 0 . 0 2 7 •0 .006 •0 .027 0 . 0 0 3 Table 4.5 (continued) 60 Transition P - F" Normalised Observed Residuals Weight Frequency Without X a c with X a c 11 2 10 -21/2 - 19/2 23/2 - 21/2 19/2 - 17/2 25/2 - 23/2 11 2 9 -21/2 - 19/2 23/2 - 21/2 19/2 - 1 7/2 25/2 - 23/2 12 o 12 -21/2 - 1 9/2 23/2 - 21/2 27/2 - 25/2 25/2 - 23/2 12 , 1 2 -23/2 - 21/2 21/2 - 1 9/2 25/2 - 23/2 27/2 - 25/2 12 2 , , -23/2 - 21/2 25/2 - 23/2 21/2 - 19/2 27/2 - 25/2 12 2 10 -23/2 - 21/2 25/2 - 23/2 21/2 - 1 9/2 27/2 - 25/2 1 3 o 13 -23/2 - 21/2 25/2 - 23/2 29/2 - 27/2 27/2 - 25/2 13 , 1 3 -25/2 - 23/2 23/2 - 21/2 27/2 - 25/2 29/2 - 27/2 10 10 11 11 11 11 1 2 1 2 1 .000 1 .000 1 .000 1 .000 000 000 000 000 1 1 1, 1, 0, 0, 1 1 0, 0, 0, 0, 000 000 010 010 010 010 010 010 1 0 1 .000 1 .000 1 .000 1 .000 1 .000 1 .000 1 .000 1 .000 1 ?.000 1 .000 1 .000 1 .000 10.001 0.001 0.010 0.010 55383.964 55383.298 55382.344 55380.720 55560.281 55559.199 55557.536 55556.687 60221.887 60220.789 60220.189 60220. 189 59661.330 59661.041 59660.240 59660.066 60411.989 60410.960 60410.387 60409.589 60639.006 60638.025 60636.702 60635.979 65198.152 65197.468 65197.098 65196.360 64622.353 64622.353 64621.491 64621.491 0.020 •0.041 •0.044 0.011 0.063 •0.016 0.041 0.088 0.014 0.037 0.033 0.053 •0.009 •0.069 •0.053 0.014 0.008 •0.067 •0.062 0.037 0.005 •0.085 •0.114 0.037 •0.002 0.009 •0.003 0.006 •0.065 0.100 •0.084 0.061 0.001 0.012 •0.003 •0.014 0.012 0.016 0.005 0.039 0.026 0.039 0.030 0.058 •0.014 0.005 •0.007 0.007 •0.003 0.0 0.017 0.021 •0.020 •0.004 •0.018 0.010 •0.009 •0.002 •0.006 •0.002 •0.069 0. 133 •0.059 0.056 61 Table 4.5 (continued) Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c with X a c 13 2 1 1 - 12 2 1 0 25/2 - 23/2 1 .000 65724.374 0.003 -0.006 27/2 - 25/2 1 .000 65723.178 -0.402 0.005 23/2 - 21/2 0.250 65722.065 -0.442 -0.012 29/2 — 27/2 0.250 65721.753 0.024 0.007 13 3 i 1 - 12 3 1 0 25/2 - 23/2 1 .000 65517.685 -1 .431 0.031 27/2 - 25/2 1 .000 65518.555 -0.012 0.017 23/2 - 21/2 1 .000 65516.152 -0.011 0.006 29/2 — 27/2 1 .000 65514.406 -1 .236 0.004 13 3 1 0 - 12 3 s 25/2 - 23/2 1 .000 65526.673 -2.508 -0.030 27/2 - 25/2 1 .000 65528.595 -0.041 0.000 23/2 - 21/2 1 .000 65526.132 -0.043 -0.025 29/2 — 27/2 1 .000 65523.817 -1 .841 0.006 u , 1 3 - 13 1 1 2 2 7/2 - 25/2 0.010 71277.338 -0.041 -0.037 25/2 - 23/2 0.010 71277.338 -0.031 -0.017 29/2 - 27/2 0.510 71276.375 -0.033 -0.026 31/2 — 29/2 0.510 71276.375 -0.026 -0.029 14 2 1 3 - 13 2 1 2 27/2 - 25/2 1 .000 70461.791 0.025 -0.015 29/2 - 27/2 1 .000 70462.874 1 .852 0.001 25/2 - 23/2 1 .000 70463.282 2.403 -0.061 31/2 — 29/2 1 .000 70460.164 0.013 0.005 14 2 1 2 - 1 3 2 1 1 27/2 - 25/2 0.010 70816.266 -0.076 -0.109 29/2 - 27/2 1 .000 70816.851 1.212 -0.031 25/2 - 23/2 0.010 70816.266 1 .554 0.105 31/2 — 29/2 1 .000 70814.027 0.009 -0.003 14 3 1 2 - 13 3 1 1 27/2 - 25/2 1 .000 70562.746 0.561 0.023 29/2 - 27/2 1 .000 70561.652 -0.002 -0.001 25/2 - 23/2 0.010 70559.768 -0.039 -0.023 31/2 — 29/2 0.010 70559.768 0.469 0.047 14 3 1 1 - 13 3 1 0 27/2 - 25/2 1 .000 70578.430. 1 .635 -0.027 29/2 - 27/2 1 .000 70576.230 -0.041 -0.054 25/2 - 23/2 1 .000 70574.355 0.003 0.020 31/2 - 29/2 1.000 70574.970 1 .122 -0.025 62 Table 4.5 (continued) Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c with X a c 15 o 1 5 - 14 0 1 « 27/2 - 25/2 1 .000 75125.510 -0.019 -0.012 29/2 27/2 0.510 75124.754 -0.003 -0.002 33/2 - 31/2 0.510 75124.754 0.032 0.030 31/2 — 29/2 1 .000 75123.965 0.020 0.023 15 , 1 5 - 14 1 1 4 29/2. - 27/2 0.510 74537.954 -0.016 -0.019 27/2 - 25/2 0.510 74537.954 0.004 0.019 31/2 - 29/2 0.510 74537.295 -0.012 -0.001 33/2 — 31/2 0.510 74537.295 0.006 0.002 15 , 1 U - 14 i 1 3 27/2 - 25/2 0.510 76350.067 -0.043 -0.033 29/2 - 27/2 0.510 76350.067 0.018 0.021 33/2. - 31/2 0.510 76349.236 -0.026 -0.028 31/2 — 29/2 0.510 76349.236 0.038 0.043 15 2 1 « - 14 2 1 3 29/2 - 27/2 1 .000 75483.655 -0.024 -0.005 31/2 - 29/2 1 .000 75481.886 -1 . 129 0.015 27/2 - 25/2 1 . 000 75481.313 -1.683 -0.013 33/2 — 31/2 1 .000 75482.347 0.005 -0.002 15 2 1 3 - 14 2 1 2 29/2 - 27/2 1 .000 75914.770 0.005 0.006 31/2. - 29/2 1.000 75913.728 -0.410 0.010 27/2 - 25/2 •0.51C 75912.738 -0.574 -0.028 33/2 — 31/2 0.510 75912.738 0.046 0.037 15 3 1 2 - 14 3 1 2 29/2 - 27/2 1 .000 75605.924 0.115 0.019 31/2 - 29/2 1 .000 75605.358 0.051 0.064 27/2 - 25/2 0.250 75603.856 -0.007 0.013 33/2 — •31/2 0.250 75603.536 0.157 0.072 15 3 1 2 - 14 3 1 1 29/2 - 27/2 1 .000 75626.572 0.113 -0.030 31/2 - 29/2 1 .000 75625.957 -0.007 0.005 27/2 - 25/2 0.010 75624.370 -0.062 -0.042 33/2 — 31/2 0.010 75624.089 0. 135 0.013 16 o i e - 15 0 1 5 29/2 - 27/2 1 .000 80076.216 0.010 0.016 35/2 - 33/2 0.010 80075.487 -0.008 -0.010 31/2 - 29/2 0.010 80075.487 0.071 0.072 33/2 - 3 1 / 2 1 .000 80074.738 0.036 0.038 63 Table 4.5 (continued) Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c with X a c 16 , 1 6 - 15 i 1 5 29/2 - 27/2 0.510 79492.205 -0.012 -0.001 31/2 - 29/2 0.510 79492.205 0.010 0.008 35/2 - 33/2 0.510 79491.612 -0.021 -0.025 33/2 — 31/2 0.510 79491.612 0.003 0.011 16 , 1 5 - 15 1 1 4 29/2 - 27/2 0.010 81418.588 -0.036 -0.028 31/2 - 29/2 0.010 81418.588 0.085 0.088 35/2 - 33/2 0.010 81417.864 -0.011 -0.013 33/2 — 31/2 0.010 81417.864 0.113 0.117 16 2 1 <i - 15 2 1 3 31/2 - 29/2 1.000 81019.307 -0.009 -0.013 33/2 - 31/2 1.000 81018.657 -0.097 -0.014 29/2 - 27/2 0.250 81017.829 -0.174 -0.072 35/2 — 33/2 0.250 81017.458 0.01 1 0.004 16 3 1 U - 15 3 1 3 31/2 - 29/2 0.250 80649.943 0.041 0.001 33/2 - 31/2 0.250 80649.443 0.011 0.035 29/2 - 27/2 0.010 80648.276 -0.011 0.019 35/2 — 33/2 0.010 80647.931 0.098 0.061 16 3 1 3 - 15 3 1 2 31/2 - 29/2 0.250 80678.473 0.056 0.004 33/2 - 31/2 0.250 80677.951 -0.005 0.019 29/2 - 27/2 0.010 80676.677 -0.028 0.003 35/2 — 33/2 0.010 80676.367 0.111 0.064 17 o 1 7 - 16 0 1 6 31/2 - 29/2 1 .000 85018.584 0.017 0.022 37/2 - 35/2 0.010 85017.879 -0.058 -0.060 33/2 - 31/2 0.010 85017.879 0.112 0.112 35/2 — 33/2 1.000 85017.120 -0.016 -0.014 17 , 1 7 - 16 1 1 6 31/2 - 29/2 0.010 84444.024 -0.020 -0.011 33/2 - 31/2 0.010 84444.024 0.038 0.035 37/2 - 35/2 0.010 84443.501 -0.023 -0.026 35/2 — 33/2 0.010 84443.501 0.036 0.042 17 2 1 6 - 16 2 1 5 33/2 - 31/2 1.000 85520.616 -0.036 -0.037 31/2 - 29/2 0.010 85520.190 -0.143 -0.087 35/2 - 33/2 0.010 85520.190 -0.044 0.002 37/2 - 35/2 1.000 85519.847 0.009 0.005 64 Table 4.5 (continued) Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c withX, 17 3 1 5 - 16 3 1 « 33/2 - 31/2 0.250 35/2 - 33/2 0.250 31/2 - 29/2 0.010 37/2 - 35/2 0.010 17 3 1. - 16 3 1 3 33/2 - 31/2 0.250 35/2 - 33/2 0.250 31/2 - 29/2 0.010 37/2 - 35/2 0.010 10 o ,0 9 1 8 17/2 - 15/2 1.000 23/2 - 21/2 1.000 19/2 - 17/2 1 .000 21/2 - 19/2 1 .000 1 2 o 12 - 1 1 1 1 0 21/2 - 19/2 1 .000 27/2 - 25/2 1 .000 23/2 - 21/2 1 .000 25/2 - 23/2 1 .000 9 0 9 8 1 7 15/2 - 13/2 1 .000 21/2 - 19/2 1 .000 17/2 - 15/2 1 .000 85694.408 0.039 0.018 85693.896 -0.034 0.015 85692.950 -0.060 -0.005 85692.706 0.118 0.097 85732.961 0.018 -0.008 85732.511 -0.004 0.046 85731.383 -0.087 -0.030 85731.104 0.052 0.027 33482.037 -0.267 -0.027 33478.719 0.020 0.030 33470.392 -0.033 0.039 33466.579 -0.138 0.012 41950.478 -0.061 -0.008 41947.442 0.000 0.011 41936.329 -0.056 0.000 41933.317 -0.023 -0.011 29100.790 -2.176 0.024 29098.903 -0.031 -0.022 29092.292 -0.190 -0.032 7 2 5 " 7 , 7 11/2 - 1 1/2 0.010 43270.887 -0.109 -0.125 17/2 - 17/2 0.010 43270.664 0.454 -0.072 13/2 - 13/2 1.000 43267.776 0.524 0.001 15/2 - 15/2 1.000 43266.828 0.162 0.048 12 2 10 - 12 , 1 2 25/2 - 25/2 1 .000 47083.370 -0.356 -0.001 23/2 - 23/2 1.000 47082.290 0.019 0.019 27/2 - 27/2 1.000 47072.272 -0.028 -0.002 21/2 - 21/2 1.000 47070.448 -0.424 -0.045 65 Table 4.5 (continued) Transition Normalised Observed Residuals F - Weight Frequency Without X a c with X a c 13 2 .1 1 - 13 , 1 3 27/2 - 27/2 1.000 48184.972 -0.759 -0.022 25/2 - 25/2 1.000 48184.237 0.013 0.007 29/2 - 29/2 1.000 48172.612 0.012 0.027 23/2 - 23/2 1.000 48170.389 -0.737 0.038 10 2 9 - 10 , 9 17/2 - 17/2 1.000 36136.418 -0.315 -0.048 23/2 - 23/2 1.000 38136.033 -0.048 0.027 19/2 - 19/2 1 .000 38116.605 -0.120 0.000 21/2 - 21/2 1 .000 38112.886 -0.247 -0.042 66 TABLE 4.6 Measured rotational transitions ( in MHz ) of C 3 H 5 8 i B r Transition Normalised1 Observed Residuals2 F - F" Weight Frequency Without X a c with X a c 3 o 3 3 / 2 5 / 2 9 / 2 7 / 2 - 1/2 3 / 2 7 / 2 5/2 2 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 5 0 2 1 . 9 9 3 1 5 0 1 9 . 4 1 8 1 4 9 9 4 . 4 0 1 1 4 9 9 5 . 8 3 6 3 . 4 9 4 1 . 4 7 0 0 . 0 4 9 1 . 6 7 8 0 . 0 5 5 - 0 . 0 2 5 0 . 0 3 6 0 . 0 2 7 3 1 3 7 / 2 5 / 2 9 / 2 3 / 2 5/2 3/2 7/2 1/2 2 i 2 0 . 0 1 0 0 . 0 1 0 1 . 0 0 0 1 . 0 0 0 1 4 8 3 5 . 7 6 1 1 4 8 3 5 . 2 0 3 1 4 8 1 2 . 1 2 8 1 4 8 0 8 . 6 5 0 0 . 6 4 3 2 . 1 2 8 0 . 9 6 8 0 . 0 3 1 0 . 0 1 9 0 . 0 7 5 0 . 0 1 7 0 . 0 6 7 3 • j 5 / 2 7 / 2 3 / 2 9 / 2 - 3/2 5/2 1/2 7/2 2 i 1 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 5 1 9 8 . 8 1 1 1 5 1 9 5 . 2 7 6 1 5 1 7 4 . 2 3 0 1 5 1 7 1 . 3 0 1 - 0 . 0 1 3 - 0 . 0 2 6 - 0 . 0 6 0 0 . 0 0 8 0 . 0 5 7 - 0 . 0 2 5 - 0 . 0 9 2 - 0 . 0 1 7 4 o • 5/2 7 / 2 1 1 / 2 9 / 2 3/2 5/2 9/2 7/2 3 0 3i.ooo 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 2 0 0 0 1 . 0 7 1 2 0 0 0 2 . 2 0 9 1 9 9 9 1 . 9 2 8 1 9 9 9 0 . 7 6 8 - 2 . 1 0 1 - 0 . 6 5 6 0 . 0 1 2 - 0 . 9 2 8 - 0 . 0 2 3 0 . 0 1 9 0 . 0 0 2 0 . 0 0 5 4 1 , 7/2 9 / 2 1 1 / 2 5 / 2 5/2 7/2 9/2 3/2 3 1 3 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 9 7 6 8 . 4 1 2 1 9 7 6 4 . 5 3 9 1 9 7 5 5 . 0 9 9 1 9 7 5 8 . 5 5 4 0 . 1 7 9 0 . 0 7 8 0 . 1 7 7 0 . 0 6 8 - 0 . 0 0 3 0 . 0 3 8 0 . 0 5 5 0 . 108 4 1 3 7 / 2 9 / 2 5 / 2 1 1 / 2 - 5/2 7/2 3/2 9/2 3 1 2 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 2 0 2 5 1 . 6 5 0 2 0 2 4 5 . 3 2 9 2 0 2 4 1 . 9 0 7 2 0 2 3 5 . 8 2 7 - 0 . 0 6 6 - 0 . 0 2 8 - 0 . 0 5 8 0 . 0 2 9 0 . 0 0 5 - 0 . 0 2 4 - 0 . 0 2 5 0 . 0 1 4 5 o 5 7 / 2 9 / 2 1 3 / 2 1 1 / 2 5 / 2 7 / 2 1 1/2 9 / 2 4 0 Voio 0 . 0 1 0 0 . 0 1 0 0 . 0 1 0 2 4 9 9 1 . 8 4 5 2 4 9 9 1 . 8 4 5 2 4 9 8 5 . 5 0 5 2 4 9 8 5 . 1 6 6 - 0 . 3 2 6 - 0 . 0 3 5 - 0 . 0 1 3 - 0 . 0 9 2 - 0 . 0 9 9 0 . 0 3 2 - 0 . 0 2 1 0 . 0 2 0 1 Measurements were weighted according to 1 / a 2 , where a is the uncertainty in the measurements. Unit weight corresponded to an uncertainty of 0 . 0 3 MHz. 2 Observed frequency minus the frequency calculated using the constants inTable4.3 67 Table 4.6 (continued) Transition Normalised Observed Residuals p F" Weight Frequency Without X a c with X a c 1 s 9 / 2 -7 / 2 -1/2 -3 / 2 -7 / 2 5 / 2 9 / 2 1 1 / 2 4 i * 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 2 4 7 0 3 . 9 8 1 2 4 6 9 8 . 9 5 4 2 4 7 0 0 . 1 7 4 2 4 6 9 5 . 4 7 9 0 . 2 7 6 0 . 0 7 4 - 0 . 0 0 2 0 . 0 3 8 - 0 . 0 0 1 - 0 . 0 2 8 - 0 . 0 0 2 - 0 . 0 1 2 1 4 9 / 2 -7 / 2 -1/2 -3 / 2 -7 / 2 5 / 2 9 / 2 1 1 / 2 4 i 3 1 . 0 0 0 0 . 0 1 0 0 . 0 1 0 1 . 0 0 0 2 5 3 0 6 . 7 9 3 2 5 3 0 1 . 7 1 9 2 5 3 0 1 . 7 1 9 2 5 2 9 6 . 8 6 8 0 . 1 4 2 - 0 . 1 1 0 0 . 1 2 2 0 . 0 1 4 0 . 0 1 3 - 0 . 1 7 4 0 . 1 3 5 0 . 0 0 4 1 6 1/2 -9 / 2 -3 / 2 -5 / 2 -9 / 2 7 / 2 1 1 / 2 1 3 / 2 5 i 5 1 . 0 0 0 0 . 5 1 0 0 . 5 1 0 1 . 0 0 0 2 9 6 3 9 . 4 2 6 2 9 6 3 6 . 6 6 5 2 9 6 3 6 . 6 6 5 2 9 6 3 4 . 0 5 0 - 0 . 0 5 8 - 0 . 1 0 5 0 . 0 0 3 0 . 0 5 8 0 . 0 1 9 0 . 0 0 . 0 2 0 0 . 0 3 0 1 5 1/2 -9/2 -3 / 2 -5 / 2 -9 2 7 / 2 1 1 / 2 1 3 / 2 5 i * 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 0 . 2 5 0 3 0 3 6 2 . 2 3 9 3 0 3 5 9 . 4 4 9 3 0 3 5 8 . 4 0 4 3 0 3 5 5 . 7 6 3 - 0 . 0 6 5 - 0 . 1 4 5 - 0 . 0 6 2 - 0 . 0 3 1 0 . 0 0 7 - 0 . 0 5 8 - 0 . 0 2 8 - 0 . 0 3 9 2 4 1/2 -3 / 2 -9 / 2 -5 / 2 -9 / 2 1 1 / 2 7 / 2 1 3 / 2 5 2 3 0 . 0 1 0 0 . 0 1 0 0 . 0 1 0 0 . 0 1 0 3 0 0 3 3 . 3 7 5 3 0 0 3 3 . 3 7 5 3 0 0 2 2 . 1 3 1 3 0 0 2 2 . 1 3 1 - 0 . 0 2 7 0 . 0 4 2 0 . 1 6 3 0 . 0 6 1 - 0 . 0 6 4 0 . 0 4 1 0 . 1 7 8 0 . 0 0 9 2 5 1/2 -3 / 2 -9 / 2 -5 / 2 -9 / 2 1 1 / 2 7 / 2 1 3 / 2 5 2 * 0 . 0 1 0 0 . 0 1 0 0 . 0 1 0 0 . 0 1 0 3 0 0 0 5 . 8 2 7 3 0 0 0 5 . 8 2 7 2 9 9 9 4 . 8 8 7 2 9 9 9 4 . 8 8 7 0 . 0 1 4 0 . 1 2 2 0 . 2 1 3 0 . 1 5 0 - 0 . 1 2 6 0 . 1 1 5 0 . 2 2 8 0 . 0 1 3 O 7 1/2 -3 / 2 -7 / 2 -5 / 2 -9 / 2 1 1 / 2 1 5 / 2 1 3 / 2 6 O 6 0 . 0 0 . 0 0 . 0 0 . 0 3 4 9 6 1 . 9 2 5 3 4 9 6 1 . 9 2 5 3 4 9 5 9 . 0 3 7 3 4 9 5 8 . 6 6 5 - 0 . 3 7 4 - 0 . 0 4 8 - 0 . 0 4 7 - 0 . 0 7 0 - 0 . 3 2 9 - 0 . 0 3 8 - 0 . 0 5 3 - 0 . 0 4 8 1 7 3 / 2 -1/2 " 5 / 2 " 7 / 2 -1 1 / 2 9 / 2 1 3 / 2 1 5 / 2 6 1 * 1 . 0 0 0 0 . 2 5 0 0 . 2 5 0 1 . 0 0 0 3 4 5 7 4 . 7 9 6 3 4 5 7 3 . 0 0 6 3 4 5 7 2 . 4 7 9 3 4 5 7 0 . 9 0 0 0 . 0 4 5 - 0 . 0 8 9 - 0 . 0 4 3 0 . 0 0 9 0 . 0 3 8 - 0 . 0 2 2 - 0 . 0 0 1 - 0 . 0 0 9 68 Table 4.6 (continued) Transition Normalised Observed Residuals F - P Weight Frequency Without X a c with X 7 1 6 1 3 / 2 -1 1 / 2 -1 5 / 2 -1 7 / 2 -1 1 / 2 9 / 2 1 3 / 2 1 5 / 2 6 i 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 3 5 4 1 7 . 4 6 9 3 5 4 1 5 . 6 9 5 3 5 4 1 4 . 4 5 5 3 5 4 1 2 . 9 9 2 - 0 . 0 4 1 - 0 . 1 6 6 - 0 . 0 9 8 0 . 0 6 7 - 0 . 0 1 6 - 0 . 0 0 1 0 . 0 2 6 0 . 0 6 1 7 2 5 1 3 / 2 -1 5 / 2 -1 1 / 2 -1 7 / 2 -11/2 1 3 / 2 9 / 2 1 5 / 2 6 2 * 1 . 0 . 0 0 1 . 0 0 0 0 . 2 5 0 0 . 2 5 0 3 5 0 4 6 . 0 3 3 3 5 0 4 5 . 0 5 8 3 5 0 3 8 . 5 4 5 3 5 0 3 8 . 1 4 0 0 . 0 9 3 - 0 . 0 2 5 - 0 . 1 2 5 0 . 2 3 0 - 0 . 0 1 0 - 0 . 0 2 8 - 0 . 0 6 8 0 . 0 7 5 8 o 8 1 3 / 2 -1 5 / 2 -1 9 / 2 -1 7 / 2 -11/2 1 3 / 2 1 7 / 2 1 5 / 2 7 0 7o . 0 1 0 0 . 0 1 0 0 . 2 5 0 0 . 2 5 0 3 9 9 4 0 . 0 8 1 3 9 9 3 9 . 7 5 1 3 9 9 3 7 . 7 6 0 3 9 9 3 7 . 3 7 8 - 0 . 1 0 1 - 0 . 0 3 5 - 0 . 0 1 4 0 . 0 - 0 . 0 7 3 - 0 . 0 2 9 - 0 . 0 1 8 0 . 0 1 3 8 y e 1 5 / 2 -1 3 / 2 -1 7 / 2 -1 9 / 2 -1 3 / 2 11/2 1 5 / 2 1 7 / 2 7 1 7 1 . 0 0 0 0 . 2 5 0 0 . 2 5 0 1 . 0 0 0 3 9 5 0 8 . 9 9 5 3 9 5 0 7 . 7 3 6 3 9 5 0 7 . 1 1 2 3 9 5 0 6 . 2 6 5 - 0 . 0 0 4 - 0 . 1 9 9 - 0 . 1 0 6 0 . 0 9 6 - 0 . 0 0 8 - 0 . 0 1 6 0 . 0 4 3 0 . 0 8 4 8 1 7 1 5 / 2 -1 3 / 2 -1 7 / 2 -1 9 / 2 -1 3 / 2 11/2 1 5 / 2 1 7 / 2 7 1 6 0 . 2 5 0 1 . 0 0 0 0 . 2 5 0 1 . 0 0 0 4 0 4 7 1 . 7 3 3 4 0 4 7 3 . 4 8 4 4 0 4 7 1 . 1 3 6 4 0 4 6 8 . 1 9 3 0 . 1 3 5 2 . 9 4 0 1 . 8 6 8 - 0 . 0 3 3 0 . 0 2 9 0 . 0 1 8 0 . 0 8 1 - 0 . 0 3 8 8 2 7 1 5 / 2 -1 7 / 2 -1 3 / 2 -1 9 / 2 -1 3 / 2 1 5 / 2 1 1 / 2 1 7 / 2 7 2 6i . 0 0 0 0 . 2 5 0 0 . 0 1 0 0 . 0 1 0 4 0 0 0 0 . 0 8 6 3 9 9 9 6 . 5 5 5 3 9 9 9 3 . 1 6 9 3 9 9 9 3 . 1 6 9 2 . 5 2 9 0 . 0 7 7 0 . 2 0 2 1 , 2 1 5 - 0 . 0 1 5 - 0 . 0 1 0 0 . 0 9 0 - 0 . 1 4 9 8 2 6 1 5 / 2 -1 7 / 2 -1 3 / 2 -1 9 / 2 -1 3 / 2 1 5 / 2 1 1 / 2 1 7 / 2 7 2 5i . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 - 4 0 0 6 0 . 8 4 1 4 0 0 6 2 . 2 9 6 4 0 0 5 8 . 5 6 8 4 0 0 5 5 . 6 7 0 - 2 . 6 0 0 - 0 . 1 0 7 0 . 1 0 3 - 1 . 8 1 6 0 . 0 1 6 - 0 . 0 1 4 - 0 . 0 0 9 - 0 . 0 5 6 9 o 9 1 5 / 2 -1 7 / 2 -2 1 / 2 -1 9 / 2 -1 3 / 2 1 5 / 2 1 9 / 2 1 7 / 2 8 0 * 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 1 . 0 0 0 4 4 9 1 2 . 0 2 2 4 4 9 1 1 . 6 3 0 4 4 9 1 0 . 1 5 7 4 4 9 0 9 . 7 1 9 - 0 . 0 4 0 0 . 0 0 5 - 0 . 0 1 8 - 0 . 0 1 4 - 0 . 0 2 1 0 . 0 0 8 - 0 . 0 2 1 - 0 . 0 0 6 69 Table 4.6 (continued) Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c with X a c 9 1 9 ~ 8 1 8 17/2 -15/2 -19/2 -21/2 -15/2 13/2 17/2 19/2 0.010 1.000 0.010 1 .000 44442.143 44443.949 44442.143 44439.797 0.244 2.755 1 .690 0.041 0.146 -0.013 -0.074 0.032 9 1 8 17/2 -15/2 -19/2 -21/2 -15/2 13/2 17/2 19/2 8 i 71.000 0.010 0.010 1 .000 45524.021 45520.973 45520.973 45521.559 -0.088 -2.444 -1 .259 0.01 1 0.009 -0.054 0.098 0.007 9 2 8 17/2 -19/2 -15/2 -21/2 -15/2 17/2 13/2 19/2 8 2 71.000 1 .000 1 .000 1 .000 44992.514 44991.371 44989.229 44988.374 0.102 0.025 0.007 0.161 0.032 0.026 0.050 0.072 9 2 7 17/2 -1 9 / 2 -15/2 -21/2 -15/2 17/2 13/2 19/2 8 2 61.000 1 .000 0.250 0.250 45088.298 45085.241 45082.604 45082.887 2.067 0.029 -0.013 1 .252 -0.073 -0.032 0.029 -0.039 10 o i o 17/2 -19/2 -23/2 -21/2 -15/2 17/2 21/2 19/2 9 0 91.000 1.000 1 .000 1.000 49877.093 49876.676 49875.602 49875.137 -0.078 -0.023 -0.053 -0.028 -0.064 -0.020 -0.056 -0.022 10 , 1 0 19/2 -17/2 -21/2 -23/2 -17/2 15/2 19/2 21/2 9 1 91.000 0.010 0.010 1.000 49373.120 49370.666 49370.666 49371.545 -0.092 -2.071 -1.351 -0.003 -0.030 0.116 -0.063 -0.010 10 2 9 19/2 -21/2 -17/2 -23/2 -17/2 19/2 15/2 21/2 9 2 "l.000 1.000 1.000 1 .000 49986.295 49985.203 49983.838 49983.106 0.008 0.008 -0.044 0.040 -0.021 0.017 -0.021 0.006 10 2 8 19/2 -21/2 -17/2 -23/2 -17/2 19/2 15/2 21/2 9 2 71.000 1 .000 1 .000 1 .000 50114.766 50113.737 50111.865 50111.106 0.100 0.007 -0.041 0.112 -0.012 0.015 -0.018 0.016 70 Table 4.6 (continued) Transition Normalised Observed Residuals P - F" Weight Frequency Without X a c with X 11 0 11 - 10 0 1 O 19/2 - 17/2 1 .000 21/2 - 19/2 1 .000 25/2 - 23/2 1 .000 23/2 — 21/2 1 .000 11 t 1 0 - 10 i 9 21/2 - 19/2 0.010 19/2 - 17/2 0.010 23/2 - 21/2 0.010 25/2 — 23/2 0.010 11 a 1 0 - 10 2 9 21/2 - 19/2 1 .000 23/2 - 21/2 1 .000 19/2 - 17/2 1 .000 25/2 — 23/2 1 .000 11 2 9 - 10 2 8 21/2 - 19/2 1 .000 23/2 - 21/2 1 .000 19/2 - 17/2 1 .000 25/2 — 23/2 1 .000 12 o 1 2 1 1 O 1 1 21/2 - 19/2 1 .000 23/2 - 21/2 1 .000 27/2 - 25/2 1 .000 25/2 — 23/2 1 .000 12 , 1 2 - 1 1 1 1 1 23/2 - 21/2 0.001 21/2 - 19/2 0.001 25/2 - 23/2 0.001 27/2 — 25/2 0.001 12 , 1 1 - 1 1 1 1 0 23/2 - 21/2 0.010 21/2 - 19/2 0.010 25/2 - 23/2 0.010 27/2 — 25/2 0.010 12 2 1 1 - 1 1 2 1 0 23/2 - 21/2 1 .000 25/2 - 23/2 0.250 21/2 - 19/2 0.250 27/2 - 25/2 1 .000 54834.759 -0.026 -0.016 54834.390 0.006 0.008 54833.631 -0.018 -0.021 54832.990 -0.006 -0.002 55622.897 -0.032 -0.027 55622.594 -0.043 -0.001 55621.598 -0.048 -0.019 55621.358 0.001 -0.002 54978.371 -0.063 -0.078 54977.738 0.002 0.020 54976.784 -0.150 -0.122 54975.854 -0.041 -0.059 55149.055 -0.027 -0.065 55148.176 -0.066 -0.045 55146.780 -0.101 -0.072 55146.070 0.011 -0.025 59784.937 0.015 0.023 59784.211 0.023 0.025 59783.713 0.030 0.028 59783.329 0.023 0.026 59230.332 -0.002 -0.004 59230.071 -0.061 -0.014 59229.426 -0.072 -0.038 59229.426 0.133 0.128 60668.556 -0.020 -0.016 60668.556 0.156 0.178 60667.389 -0.089 -0.075 60667.389 0.082 0.080 59969.646 0.064 0.055 59968.688 -0.095 -0.054 59968.245 -0.054 -0.004 59967.507 -0.039 -0.050 7 1 Table 4.6 (continued) Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c withX a c 12 2 1 0 - 1 1 2 9 23/2 • - 21/2 1 .000 25/2 - 23/2 1 .000 21/2 - 19/2 1 .000 27/2 — 25/2 1 .000 12 3 1 0 - 1 f 3 9 23/2 - 21/2 1 .000 25/2 - 23/2 1 .000 21/2 - 19/2 0.010 27/2 — 25/2 0.010 12 3 9 - 1 1 3 e 23/2 - 21/2 1 .000 25/2 - 23/2 1 .000 21/2 - 19/2 0.010 27/2 — 25/2 0.010 13 o 1 3 - 12 0 1 2 23/2 - 21/2 1 .000 25/2 - 23/2 0.250 29/2 - 27/2 0.250 27/2 — 25/2 1 .000 13 , 1 3 - 12 1 1 2 25/2 - 23/2 0.010 23/2 - 21/2 0.010 27/2 - 25/2 0.010 29/2 — 27/2 0.010 13 , 1 2 - 12 1 1 1 25/2 - 23/2 0.010 23/2 - 21/2 0.010 27/2 - 25/2 0.010 29/2 — 27/2 0.010 13 2 1 1 - 12 2 1 0 25/2 - 23/2 1 .000 27/2 - 25/2 1 .000 23/2 - 21/2 1 .000 29/2 — 27/2 1.000 13 3 1 1 - 12 3 1 0 25/2 - 23/2 1 .000 27/2 - 25/2 1 .000 23/2 - 21/2 1 .000 29/2 - 27/2 1 .000 60189.765 0.020 0.001 60188.958 -0.040 0.009 60187.866 -0.060 0.000 60187.221 0.030 0.010 60032.518 0.292 0.089 60031.802 0.033 0.032 60029.055 -0.046 -0.020 60029.055 0.385 0.169 60038.800 0.166 0.042 60038.206 0.026 0.032 60035.347 -0.128 -0.102 60035.347 0.300 0.167 64726.255 -0.029 -0.018 64725.652 -0.061 -0.053 64725.412 0.012 0.009 64724.773 -0.023 -0.021 64155.857 -0.051 -0.053 64155.857 0.083 0.108 64155.086 -0.072 -0.054 64155.086 0.053 0.049 65711.148 -0.087 -0.085 65711.148 -0.003 0.011 65710.215 -0.084 -0.076 65710.215 -0.002 -0.005 65236.838 0.017 0.009 65235.968 -0.190 0.023 65235.026 -0.244 -0.012 65234.648 0.032 0.020 65036.515 -1.109 0.019 65037.145 -0.020 0.001 65035.144 -0.011 0.001 65033.812 -0.904 0.022 72 Table 4.6 (continued) Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c with X 13 3 1 0 - 12 3 9 25/2 - • 23/2 1 .000 27/2 - 25/2 1 .000 23/2 - 21/2 1 .000 29/2 — 27/2 1 .000 14 , 1 3 - 13 1 1 2 27/2 - 25/2 0.010 25/2 - 23/2 0.010 29/2 - 27/2 0.510 31/2 — 29/2 0.510 14 2 1 2 - 13 2 1 1 27/2 - 25/2 0.010 29/2 - 27/2 0.010 25/2 - 23/2 0.010 31/2 — 29/2 1 .000 15 o 1 S - 14 0 1 4 27/2 - 25/2 1 .000 29/2 27/2 0.510 33/2 - 31/2 0.510 31/2 — 29/2 1 .000 15 , 1 5 - 14 1 1 4 29/2 - 27/2 0.010 27/2 - 25/2 0.010 31/2 - 29/2 0.510 33/2 — 31/2 0.510 15 , 1 4 - 14 1 1 3 27/2 - 25/2 0.510 29/2 - 27/2 0.510 33/2 - 31/2 0.510 31/2 — 29/2 0.510 15 2 1 4 - 14 2 1 3 29/2 — 27/2 1 .000 31/2 - 29/2 1 .000 27/2 - 25/2 1.000 33/2 — 31/2 1 .000 15 2 1 3 - 14 2 1 2 29/2 - 27/2 1 .000 31/2 - 29/2 1 .000 27/2 25/2 0.510 33/2 - 31/2 0.510 65043.429 -3.791 -0.016 65046.660 -0.105 -0.054 65044.656 -0.052 -0.040 65041.920 -2.352 -0.025 70750.550 -0.030 -0.028 70750.550 -0.017 -0.008 70749.749 -0.018 -0.014 70749.749 -0.007 -0.009 70290.364 0.024 -0.001 70290.651 0.901 -0.027 70290.364 1.380 0.262 70288.428 0.026 0.018 74583.861 0.001 0.006 74583.219 -0.005 -0.005 74583.219 0.036 0.034 74582.550 0.007 0.008 74000.380 0.010 0.008 74000.380 0.031 0.041 73999.828 0.015 0.023 73999.828 0.034 0.031 75786.233 -0.046 -0.039 75786.233 0.0 0.001 75785.534 -0.035 -0.036 75785.534 0.013 0.016 74931.655 -0.050 -0.013 74929.282 -1.866 -0.009 74927.893 -3.237 -0.010 74930.590 0.008 0.003 75350.141 -0.038 -0.036 75349.233 -0.420 0.001 75348.425 -0.546 0.005 75348.425 -0.025 -0.032 Table 4.6 (continued) 7 3 Transition Normalised Observed Residuals F F" Weight Frequency Without X a c with X a ( 15 3 1 3 - 14 3 1 2 29/2 - 27/2 1.000 75050.250 0.084 0.007 31/2 - 29/2 1 .000 75049.773 0.027 0.035 27/2 - 25/2 0.010 75048.519 -0.020 -0.007 33/2 — 31/2 0.010 75048.243 0.111 0.043 15 3 1 2 - 14 3 1 1 29/2 - 27/2 1 .000 75069.950 0.090 -0.027 31/2 - 29/2 1 .000 75069.428 -0.018 -0.010 27/2 - 25/2 0.010 75067.991 -0.177 -0.164 33/2 — 31/2 0.010 75067.991 0.226 0.126 16 o 1 6 - 15 0 1 s 29/2 - 27/2 1 .000 79499.905 0.007 0.011 31/2 - 29/2 0.010 79499.296 0.050 0.050 35/2 - 33/2 0.010 79499.296 -0.006 -0.008 33/2 — 31/2 1.000 79498.660 0.012 0.013 16 1 6 - 15 1 1 5 31/2 - 29/2 0.510 78919.158 0.006 0.004 29/2 - 27/2 0.510 78919.158 -0.010 -0.003 33/2 - 31/2 0.510 78918.678 0.017 0.023 35/2 — 33/2 0.510 78918.678 0.0 -0.003 16 2 1 5 - 15 2 1 4 31/2 - 29/2 1 .000 79914.993 -0.011 -0.012 29/2 - 27/2 0.010 79914.387 -0.174 -0.053 33/2 31/2 0.010 79914.387 -0.118 -0.020 35/2 — 33/2 0.250 79914.063 -0.005 -0.009 16 2 1 4 - 15 2 1 3 31/2 - 29/2 1 .000 80416.039 -0.007 -0.011 33/2 - 31/2 1.000 80415.511 -0.064 0.002 29/2 - 27/2 0.250 80414.849 -0.106 -0.025 35/2 — 33/2 0.250 80414.542 0.054 0.049 16 3 1 4 - 15 3 1 3 31/2 — 29/2 1 .000 80057.201 0.040 0.010 33/2 - 31/2 1.000 80056.787 0.019 0.034 29/2 27/2 0.250 80055.758 -0.053 -0.034 35/2 — 33/2 0.250 80055.473 0.044 0.016 16 3 1 3 - 15 3 1 2 31/2 - 29/2 0.250 80084.376 0.017 -0.023 33/2 - 31/2 0.250 80083.974 0.001 0.016 29/2 - 27/2 0.010 80082.846 -0.084 -0.065 35/2 - 33/2 0.010 80082.594 0.041 0.005 7 4 Table 4.6 (continued) Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c with X a c 17 o 1 7 - 16 0 1 6 3 1 / 2 - 2 9 / 2 1 . 0 0 0 3 7 / 2 - 3 5 / 2 0 . 0 1 0 3 3 / 2 - 3 1 / 2 0 . 0 1 0 3 5 / 2 — 3 3 / 2 1 . 0 0 0 17 , 1 7 - 16 1 1 6 3 1 / 2 - 2 9 / 2 0 . 0 1 0 3 3 / 2 - 3 1 / 2 0 . 0 1 0 3 7 / 2 - 3 5 / 2 0 . 0 1 0 3 5 / 2 — 3 3 / 2 0 . 0 1 0 17 , 1 6 - 16 1 1 5 3 1 / 2 - 2 9 / 2 0 . 0 1 0 3 3 / 2 - 3 1 / 2 0 . 0 i 0 3 7 / 2 3 5 / 2 0 . 0 1 0 3 5 / 2 — 3 3 / 2 0 . 0 1 0 17 2 1 6 - 16 2 1 5 3 3 / 2 - 3 1 / 2 0 . 2 5 0 3 1 / 2 - 2 9 / 2 0 . 0 1 0 3 5 / 2 - 3 3 / 2 0 . 0 1 0 3 7 / 2 — 3 5 / 2 0 . 2 5 0 17 2 1 5 - 16 2 1 « 3 3 / 2 - 3 1 / 2 0 . 2 5 0 3 5 / 2 - 3 3 / 2 0 . 2 5 0 3 1 / 2 - 2 9 / 2 0 . 2 5 0 3 7 / 2 — 3 5 / 2 0 . 2 5 0 17 3 1 5 - 16 3 i a 3 3 / 2 - 3 1 / 2 0 . 0 1 0 3 5 / 2 - 3 3 / 2 0 . 0 1 0 3 1 / 2 - 2 9 / 2 0 . 0 1 0 3 7 / 2 — 3 5 / 2 0 . 0 1 0 17 , 1 » - 16 3 1 3 3 3 / 2 - 3 1 / 2 0 . 0 1 0 3 5 / 2 - 3 3 / 2 0 . 0 1 0 3 1 / 2 - 2 9 / 2 0 . 0 1 0 3 7 / 2 — 3 5 / 2 0 . 0 1 0 9 o 9 8 1 7 1 5 / 2 - 1 3 / 2 1 . 0 0 0 2 1 / 2 - 1 9 / 2 1 . 0 0 0 1 7 / 2 - 1 5 / 2 1 . 0 0 0 1 9 / 2 - 1 7 / 2 1 . 0 0 0 8 4 4 0 7 . 7 5 8 0 . 0 0 4 0 . 0 0 7 8 4 4 0 7 . 1 7 6 - 0 . 0 5 0 - 0 . 0 5 1 8 4 4 0 7 . 1 7 6 0 . 0 8 2 0 . 0 8 2 8 4 4 0 6 . 5 6 7 0 . 0 0 3 0 . 0 0 3 8 3 8 3 5 . 5 9 4 - 0 . 0 0 7 - 0 . 0 0 1 8 3 8 3 5 . 5 9 4 0 . 0 3 8 0 . 0 3 6 8 3 8 3 5 . 1 7 3 0 . 0 0 9 0 . 0 0 6 8 3 8 3 5 . 1 7 3 0 . 0 5 5 0 . 0 5 9 8 5 8 4 5 . 0 0 6 - 0 . 0 2 9 - 0 . 0 2 5 8 5 8 4 5 . 0 0 6 0 . 1 1 1 0 . 1 1 2 8 5 8 4 4 . 4 6 7 - 0 . 0 1 0 - 0 . 0 1 1 8 5 8 4 4 . 4 6 7 0 . 1 3 3 0 . 1 3 4 8 4 8 9 6 . 0 0 1 - 0 . 0 1 8 - 0 . 0 1 9 8 4 8 9 5 . 5 6 9 - 0 . 1 1 2 - 0 . 0 6 9 8 4 8 9 5 . 5 6 9 - 0 . 0 0 5 0 . 0 2 9 8 4 8 9 5 . 2 6 9 0 . 0 3 0 0 . 0 2 6 8 5 4 8 7 . 5 2 0 0 . 0 3 4 0 . 0 3 1 8 5 4 8 7 . 1 0 3 0 . 0 4 9 0 . 0 7 6 ' 8 5 4 8 6 . 4 8 8 - 0 . 0 0 3 0 . 0 3 0 8 5 4 8 6 . 1 4 3 0 . 0 8 1 0 . 0 7 7 8 5 0 6 4 . 4 9 9 - 0 . 0 2 3 - 0 . 0 3 9 8 5 0 6 4 . 1 3 9 - 0 . 0 1 7 0 . 0 1 3 8 5 0 6 3 . 3 4 1 - 0 . 0 4 6 - 0 . 0 1 2 8 5 0 6 3 . 1 1 9 0 . 0 8 7 0 . 0 7 1 - 8 5 1 0 1 . 3 0 0 - 0 . 0 2 1 - 0 . 0 4 1 8 5 1 0 0 . 9 4 3 - 0 . 0 1 9 0 . 0 1 1 8 5 1 0 0 . 0 1 0 - 0 . 0 8 2 - 0 . 0 4 7 8 5 0 9 9 . 8 0 6 0 . 0 6 5 0 . 0 4 6 2 8 7 8 9 . 0 7 1 - 2 . 3 8 5 0 . 1 7 2 2 8 7 8 8 . 0 9 7 0 . 0 0 5 0 . 0 1 2 2 8 7 8 2 . 6 0 3 - 0 . 1 6 3 - 0 . 0 1 7 2 8 7 7 7 . 8 3 9 - 1 . 5 0 4 - 0 . 0 3 7 75 Table 4.6 (continued) Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c with X a c 10 o 1 0 " 9 , 8 17/2 - 15/2 1.000 23/2 - 21/2 1.000 19/2 - 17/2 1.000 21/2 — 19/2 1.000 12 o 1 2 - 11 i 1 0 21/2 - 19/2 1.000 27/2 25/2 0.010 23/2 - 21/2 1 .000 25/2 — 23/2 1 .000 12 2 1 0 - 12 , 1 2 25/2 - 25/2 1 .000 23/2 - 23/2 1.000 27/2 - 27/2 1.000 21/2 — 21/2 1 .000 10 2 B - 10 , 1 0 21/2 - 21/2 1 .000 19/2 - 19/2 1 .000 23/2 - 23/2 0.000 17/2 — 17/2 0.000 7 1 6 " 6 o 6 13/2 - 1 1/2 1 .000 15/2 - 13/2 1.000 1 1/2 - 9/2 1 .000 17/2 — 1 5/2 1 .000 7 2 s - 6 , 6 1 1/2 - 9/2 1 .000 17/2 - 15/2 1 .000 13/2 - 1 1/2 1 .000 15/2 — 13/2 1.000 9 , e - 8 o 8 17/2 - 15/2 0.010 19/2 - 17/2 0.010 15/2 - 13/2 1 .000 21/2 — 19/2 1.000 14 2 1 2 - 14 , 1 4 29/2 - 29/2 1.000 27/2 - 27/2 1.000 33144.977 - 0 . 2 3 3 - 0 . 0 5 2 33142.160 - 0 . 0 3 8 -0 .031 33135.406 0.050 0.102 33132.281 0.005 0.120 41568.054 - 0 . 0 1 3 0.026 41565.442 - 0 . 0 5 4 - 0 . 0 4 6 41556.236 - 0 . 0 9 4 - 0 . 0 5 5 41553.709 - 0 . 0 9 7 - 0 . 0 8 6 47037.402 - 0 . 1 7 4 0.064 47036.427 0.004 0.003 47028.128 - 0 . 0 2 3 - 0 . 0 0 5 47026.732 -0 .291 - 0 . 0 3 5 45231.384 - 0 . 1 9 5 0.101 45230.546 -0 .130 - 0 . 0 7 7 45225.378 - 0 . 2 4 8 - 0 . 1 8 5 45224.774 -0 .001 0.337 50559.006 - 0 . 0 1 4 - 0 . 0 3 6 50556.914 -0 .321 0.042 50552.085 - 0 . 4 5 8 -0 .001 50550.783 0.068 0.053 77768.659 - 0 . 0 7 2 - 0 . 0 2 3 77764.187 - 1 . 4 0 5 0.060 77763.165 - 2 . 6 9 8 -0 .091 77762.783 - 0 . 0 4 5 0.037 61652.951 - 0 . 0 1 7 - 0 . 0 6 3 61652.713 0.091 -0 .011 61644.204 0.181 0.032 61643.693 0.062 0.048 49330.180 0.477 - 0 . 0 2 9 49328.441 0.023 - 0 . 0 0 5 76 Table 4.6 (continued) Transition Normalised Observed Residuals F - F" Weight Frequency Without X a c withX, 25/2 - 23/2 1.000 49574.134 - 0 . 0 2 6 - 0 . 0 1 2 31/2 - 29/2 1.000 49571.903 - 0 . 0 5 6 - 0 . 0 4 7 27/2 - 25/2 1.000 49560.932 - 0 . 0 4 9 - 0 . 0 1 4 29/2 - 27/2 1.000 49558.849 0.059 0.052 BIBLIOGRAPHY 77 1. T. Hirokawa, M. Hayashi, H Murata, J. Sci. HirosMma Univ., Ser. A, 37, 301-324, (1973). 2. J. Maillols, V. Tabacik, Sr^trochimica Acta, 3SA, 1125-1133, (1979). 3. C. Marsden, L. Hedberg, K. Hedberg, J. Phys. Chem., 92, 1766-1770, (1980). 4. F. M. K. Lam, B. P. Dailey, J. Chem. Phys., 49, 1588-1593, (1968). 5. See for example, C. Flanagan, L. Pierce, J. Chem. Phys., 38, 2963-2969, (1963). W. Gordy, J. W. Simmons, A. G. Smith, Phys. Rev. 74, 4655-4663, (1983). 6. H. P. Benz, A. Bauder, Hs. H Gunthard, J. Mol. Spectrosc, 21, 156-164, (1966). 7. H. M. Jemson, Ph. D. Thesis, University of British Columbia, 1986. 8. M. C. L. Gerry, W. Lewis-Bevan, N. P. C. Westwood, J. Chem. Phys. 79, 4655-4663,(1983). 9. W. Gordy, R. L. Cook, Microwave Molecular Spectra. 3rd. ed., in Techniques of Chemistry, Ed., A. Weissberger, Vol. 18, Chapter 8, Wiley, New york, 1984. 10. J. K. G. Watson, in Vibration of Spectra and Structure, in A Series of Advances, Ed., J. R. Durig, Vol. 6, pp. 1-89, Elsevier, New York, 1977. 7 8 C H A P T E R V D I S C U S S I O N 5.1 C O M P A R I S O N O F D A T A In this thesis extensive measurements of the microwave spectrum of cyclopropyl bromide are presented. Since the dipole moment component is much larger than u«, the spectrum contains strong a-type and very weak c-type transitions. It has provided an excellent application of the global fitting program, and an excellent example of how perturbations in nuclear quadrupole hyperfine structure can be used to evalute several spectroscopic constants which might otherwise be unavailable. Comparison of the present results with those of Lam and Dailey indicates that we have obtained a great improvement in the accuracy of the constants. This comparison is listed in Table 5.1 and Table 5.2. In this microwave study, the rotational constants of cyclopropyl bromide have been measured very precisely. For C3Hs 7 9Br and C3H5 8 1Br, the A constants were evaluated to within ± 0.017 MHz and ± 0.023 MHz respectively, and have been improved by several orders of magnitude. The B and C constants have also been measured more precisely. Since the A constant for CjHs^Br is slightly lower than that for C ^ M B r , both of the new A constants are reasonable physically. The difference between these constants is small because the Br atom is very close to the a-principal inertial axis. 79 Table 5.1 Spectroscopic constants of cyclopropyl bromide ( 7 9 B r ) in comparison with that of Lam and Dailey Parameter Present work Ref.3 Rotational constants / MHz A 0 1633424K17)1 16504(65) B 0 2579.92044(40) 2579.87(3) Co 2457.71605(38) 2457.69(3) Centrifugal distortion constants / kHz D j 0.64572(74) D J K 0.902(36) D K 14.4(32) d i 0.01615(79) d 2 - 0.00207(43) Numbers in parentheses are one standard deviation in units of the last significant figures. 80 Table 52 Spectroscopic constants of cyclopropyl bromide ( 8 1 B r ) in comparison with those of Lam and Dailey Parameter Present work Ref .3 Rotational constants / MHz Ao 16333.034(23)* 16239(160) Bo 2560.54002(67) 2560.49(3) Co 2440.14986(52) 2440.12(3) Centrifugal distortion constants / kHz Dj 0.63754(84) D J K 0.899(42) D K 14.9(38) di 0.0139(10) d 2 - 0.00238(47) 1 Numbers in parentheses are one standard deviation in units of the last significant figures. 81 It has also been possible to evaluate centrifugal distortion constants for the first time; four quartic centrifugal distortion constants Dj, DJK, di and 62, for both isotonic species were well determined. The uncertainty in D K is rather large compared with that of the other distortion constants because more information is needed (probably more c-type branches). We did not try adding sextic distortion constants to the fits because the fits are more than adequate with quartic constants. The distortion constants show the right trends with isotopic species, because Dj and di for C3Hs8 1Br are less than those for C3H579Br. This is consistent with the inverse dependence of the distortion constants with principal moment of inertia, given in Eq. 2.17. 5.2 D I A G O N A L I Z A T I O N O F T H E B R O M I N E Q U A D R U P O L E T E N S O R Nuclear hyperfine structure in molecular rotational spectra arises from either magnetic or electric interactions of the molecular fields with the nuclear moments. The quadrupole coupling constants of a nucleus in a molecule give a measure of the electric field gradient at the nucleus due to external charges, particularly the electrons. Thus they provide information about die electronic structure and chemical bonds in neighborhood of the nucleus. Although the diagonal elements X^, X ,^ and X^ of the quadrupole coupling tensor, measured with reference to the principal inertial axes, are die coupling constants directly observable from the quadrupole hyperfine structure (see Table 5.3), they are not necessarily convenient for interpretation of the quadrupole coupling in terms of the properties of chemical bonds. It is better to transform to the principal axes of the 82 quadrupole coupling tensor, if possible. For cyclopropyl bromide the measurement of has enabled this to be done. The unitary transformation is: X?T 0 0 x = 0 Xyy 0 0 0 m Xaj 0 Xac x= 0 Xbb 0 Xac 0 Xpp (5.1) U = cosfl 0 sinfl za za 0 .1 0 -surd za 0 cosd za where $za is the angle between the a-inertial axis and the z-principal quadrupole axis, and x' is the quadrupole tensor in the principal axis system of the quadrupole (and therefore will have no off-diagonal elements ) and XQ, Xyy and are the principal values of the bromine quadrupole coupling tensor. The y-axis is taken to be the out-of-plane axis. The resulting equations are: 83 X22 = XaaCos2^+XccSin 2^+2X^08^8^ (5.2) In order to calculate XQ, Xyy and X „ it was necessary to determine Since Xbb and Xcc are well-determined, d u can be calculated from the relationship (1) The principal values of the bromine quadrupole coupling tensor were calculated for both isotopic species. The results are given in Table 5.4 along with the previously published values. For cyclopropyl bromide a good measure of the accuracy of the constants can be tetemined by how well the ratio %a (T^Br) / %& (^Br) agrees with the value obtained when this ratio is taken for the quadrupole moments of the nuclei The experimental value 1.1964 ± 0.0012 is in excellent agreement with this value, 1.19707 ± 0.00003 ( 2 ). 2X, 'ac tan 2i3 = (5.3) 'CC 8 4 Table 5.3 Bromine quadrupole coupling constants in principal inertia! axes in comparison with those of Lam and Dailey C3H5WBr C3H 5 8 1Br Parameter Present work Ref.3 Present work Ref.3 8H *bb X 463.69(22)1 284.76(16) 178.93(7) 266.36(34) 462.3(35) • 284.3(70) -178.0(53) 388.14(25) 238.15(16) 149.99(9) 221.99(35) 391.2(10) - 234.7(33) - 156.5(29) 1 Numbers in parentheses are one standard deviation in units of the last significant figures. 85 Table 5.4 Principal values of the Bromine quadrupole coupling in comparison with those of I .am and Dailey Parameter C3H 5 7 9 Br C3H 58 1Br Present work Ref.3 Present work Ref.3 X *"XX -284.76C16))1 -284.3(70) -238.15(16) -234.7(33) Xyy -274.96(31) -266.3(90) -229.76(29) -231.8(80) 559.72(35) 550.6(90) 467.90(36) 446.5(80) flM/deg 19.829(1) 19'13' 19.762(1) 19* 13' 1 Numbers in parentheses are one standard deviation in units of the last significant figures. 8 6 5.3 T H E BONDING O F Br IN C Y C L O P R O P Y L B R O M I D E The bromine quadrupole coupling constants give several pieces of information about the bonding of the bromine nucleus to the rest of the molecule. In the first place the angle $ z a between the a-inertial axis and the z-principal axis is within 1* of the angle between the a-axis and the C-Br bond (3). Within the limits of accuracy of the structure these are equal The conclusion from mis is that the maximum electron density of the C-Br bond is directly between the two nuclei; ie. that the bond is not bent This work has thus been able to show mat the assumption of Lam and Dailey ( 3 ) was correct. The field gradient which gives rise to the nuclear quadrupole coupling of halogens is due primarily to an unequal filling of the p-orbitals of the valence shell of the coupling atoms (4). The field gradient can therefore be related to the number or fractional number of electrons in the valence p-orbitals; in the case of Br these are the 4p orbitals. Townes and Dailey ( 5 ) have related the quadrupole constants and orbital populations with the following equations: = eQqx = - (Up)xeQqnio X y y = eQq y = -(Up)yeQqn lo (5.4) Xzz - eQqz - - (UpJzeQqnio where (Up)x = 1/2 ( n y + n 2 ) - n x 87 (U p ) y =l/2(n z + n x ) - n y (U p ) z =l/2(n x + n y ) - n z n x, n y and n z represent the number of electrons in the valence p x, Py and p z orbitals, eQqnio is the quadrupole coupling constant of an atom with a singly occupied valence np orbital. For Br n = 4. The Br atom itself has a nearly full set of 4p orbitals (one electron missing), and can be considered as an atom with a single positive hole. Thus for Br eQq 4 1 0 = -eQq a U ) m . When the bromine atom is bound to another atom, the electric field gradients about the nucleus are modified, so that comparisons of experimental quadrupole coupling constants with those of a free bromine atom will give some measure of the type of bond involved, i.e. the degree of TC bonding or the ionic or covalent character. There are three structures having different forms of bonding that can contribute to the Br quadrupole constants of cyclopropyl bromide: I n m 88 The first (I) is normal covalent bonding, the second (II) is an ionic form, and the third (HI) is a double bonded structure. The population difference between the 4px and 4py orbitals indicates how far from cyclindrical symmetry is the electron population of the orbitals round the o bond. This is given by (6,7): This came to 0.85% for C3H 5 7 9Br and 0.87% for C3H5 8 1Br. Thus the C-Br bond is essentially cyclindrically symmetric, and the contribution of form in can be taken as negligible. Xzz for cyclopropyl bromide was considerably smaller than the value for the Br atom (769.76 MHz for 7 9 Br and 643.03 MHz for 8 1 Br (8)). This results chiefly because of contributions of structure TL the ionic form, for which all 4p orbitals are full, Up z = 0, and X^ , = 0. If form II is given weight i, and form HI is given weight it, then the ionic character i of the o bond is ( 9 ): X B = [ CU/, (1-i - K)(-l + s) + ( U ^ i + it (Up™ ] (5.6) 89 where I 2+2 V' 2 1 = 1 2 = 0 s is the percentage of s character in the hybrid We assume s = 8.5% as in alkyl bromides (10). Substituting in for all the known variables gives a value of 19.2% for i, the ionic character. This made the amount of covalent character in the Br-C bond around 80.0%. In Table 5.5 we have presented the quadrupole coupling data for a series of compounds for comparison. 5.4 S T R U C T U R E O F C Y C L O P R O P Y L B R O M I D E Cyclopropyl bromide has C s symmetry, i.e., it has a plane of symmetry which contains the bromine-carbon-hydrogen angle and bisects die C1-C2 bond; also in this plane he the principal axes a and c. A total of 12 parameters are necessary to fully determine the structure of this molecule. Since we have only two new A constants there is nothing definitive we can say about the structure. Much more isotopic information is needed. 9 0 Because the two A constants of cyclopropyl bromide are nearly equal the Br atom is very close to the a-axis. It is well known (11) that in this case getting an accurate distance from the a-axis is virtually impossible. However, Lam and Dailey have calculated two structures. Structure I was calculated by assuming the ring to. be the same as in cyclopropyl chloride, and structure II was derived with the assumption that all d C-H = 1.0800 A and the ring bisecting the Z H-C-H's. Both structures, I and IL are rather similar but slighdy different The rotational constants presented in this thesis agree well with structure I, but not so well with structure II (see below). Calculated (MHz) Observed (MHz) Structure I Structure n C3H579Br A 16327.5 16467.7 16334.24 B 2579.87 2579.88 2579.92 C 2457.68 2457.68 2457.72 C 3 H 5 8 1 B r A 16326.4 16466.5 16333.03 B 2560.49 2560.49 2560.54 C 2440.11 2440.11 2440.15 It is thus an excellent approximation to say mat cyclopropyl bromide is essentially the cyclopropyl ring in cyclopropyl chloride attached to Br. A C-Br bond distance of 1.905 A, also given by I .am and Dailey is thus very reasonable. 91 Table 5.5 Comparison of XJZ of cyclopropyl bromide with those of similar molecules Xzz/MHz ™Br " B r « B r / " B r Methyl bromide1 577.1 481.2 1.1993 Ethyl bromide2 541.0 450.0 1.2022 Vinyl bromide3 551.3 461.4 1.1948 Cyclopropyl bromide4 559.72 467.92 1.1962 Cyclobutyl bromide5 512.2 419.6 1.2208 Bromobenzene6 567.0 480.0 1.1812 1 Y. Morino and C. Hirose, J. Mol. Spe., 24, 204-224, (1967). 2 C. Flanagan and L. Pierce, J. Chem. Phys., 38, 2963-2969, (1963). 3 J. A. Howe and J. H. Goldstein, J. Chem. Phys., 22, 1477, (1954). 4 Present work. 5 W. G. Rothschild and B. P. Dailey, J. Chem. Phys., 36, 2931-2940, (1962). 6 E. Rosenthal and B. P. Dailey, J. Chem. Phys., 43, 2093, (1965). 9 2 BIBLIOGRAPHY 1. E M , Jemson, Ph. D. Thesis, University of British Columbia, 1986. 2. W. Gordy, R. L. Cook, Microwave Molecular Spectra. 3rd. ed., in Techniques of Chemistry, Ed., A. Weissberger, Vol. 18, Appendix E, Wiley, New York, 1984. 3. F. M. K. Lam, B. P. Dailey, J. Chem. Phys., 49, 1588-1593, (1968). 4. W. Gordy, R. L. Cook, OJL £LL, Chapter XTV. 5. a) C. H. Townes, B. P. Dailey, J. Chem. Phys., 17, 782-796, (1949). b) C. H. Townes, B. P. Dailey, J. Chem. Phys., 20, 35-40, (1952). c) C. H. Townes, B. P. Dailey, J. Chem. Phys., 23, 118-123, (1955). 6. R. Bersohn, J. Chem. Phys., 22, 2078-2083, (1954). 7. J. H. Goldstein, J. Chem. Phys., 24, 106-109, (1956). 8. Gordv. Cook, op. cit.. Page 737. 9. Gordy, Cook, 2C cJL, Chapter 14. 10. B. P. Dailey, J. Chem. Phys., 33, 1641, (1960). 11. C. C. Costain, J. Chem. Phys., 29, 864, (1958). CHAPTER VI CONCLUSIONS The microwave spectra of the two bromine isotopes of cyclopropyl bromide were measured in the range 15-90 GHz, The extensive measurements of a-type R branch transitions, as well as c-type transitions from different branches which were not available before, were used to determine the previously measured rotational constants and quadrupole coupling constants more accurately, and to determine five quartic centrifugal distortion constants which were not reported before. In addition, the off-diagonal quadrupole coupling constant, X*., was also obtained. The C-Br bond has been demonstrated! to be essentially covalent with small amounts of ionic and K bonded character, and not bent. In general, the method of using perturbations in die quadrupole hyperfine structure in the microwave spectrum of a molecule to evaluate otherwise undetenninable rotational and centrifugal distortion constants, will be useful when analyzing the spectra of molecules containing bromine and iodine, particularly when the spectrum has predominandy a-type R branch transitions. If there are only a few near degeneracies, then the accuracy of the constants obtained by this method will be limited. 

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