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J. . ofln‘ 11x . 11 . v. 1 .1 1 . 1 A I 119111H41111A. ..|.-\.1\1t v I 1.11.. -1} .. ...§mm.fi§ 1.11. n - -- LIBRARY Michigan State University ,Hggtfl This is to certify that the dissertation entitled STUDIES IN INFRARED LASER-AND MICROWAVE SPECTROSCOPY: I. TWO-PHOTON SPECTROSCOPY OF THE \) BANDS 0F 14 15 12 2 13 NH3 AND NH3 and v3 BANDS of CH3F and CH3F; II. MICROWAVE SPECTBQGQQRYy 0F 3-ME'I‘HYL-1-BUTENE. Parvaneh Shoja-Chaghervand has been accepted towards fulfillment of the requirements for Ph . D . degree in Chemis try Q/I/JVZW/WR Major professor DMe August 12, 1982 MS U is an Affirmative Action/Equal Opportunity Institution 042771 RETURNING MATERIALS: 1V1531_1 Place in book drop to LJBRARJES remove this checkout from m your VQCOY‘d. FINES W111 be charged if book is returned after the date stamped beiow. STUDIES IN INFRARED LASER AND MICROWAVE SPECTROSCOPY: I. TWO-PHOTON SPECTROSCOPY OF THE v2 BANDS OF l4NH3 AND 15NH3 AND v3 BANDS of 12CH3F and 13CH3 II. MICROWAVE SPECTROSCOPY OF 3-METHYL-l-BUTENE F; BY Parvaneh Shoja-Chaghervand A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1982 ABSTRACT STUDIES IN INFRARED LASER AND MICROWAVE SPECTROSCOPY: I. TWO-PHOTON SPECTROSCOPY OF THE v2 BANDS OF 14NH3 AND l5NH3 AND v3 BANDS of 12CH3F and 13CH3 II. MICROWAVE SPECTROSCOPY OF 3-METHYL-l-BUTENE. F; BY Parvaneh Shoja-Chaghervand The recently developed technique of infrared-microwave two-photon spectroscopy has been employed to study the v2 bands of 14NH3 and 15 12 l3CH NH3 and the v bands of CH3F and 3 3F. A CO2 or N20 laser was used as the infrared source and a backward wave oscillator served as the microwave source. The molecular parameters for the inversion-free ground and v2=l excited states were obtained for both species of ammonia by least-squares fits of the frequencies of the assigned two-photon transitions combined with previously reported frequencies measured relative to CO2 or N20 lasers. The two-photon frequencies in this work have been compared to previous two-photon measurements and infrared frequencies obtained by other methods. For 12CH3F, the assigned two-photon frequencies were Parvaneh Shoja-Chaghervand combined with previous two-photon measurements, with laser stark measurements, with far—infrared rotational frequencies in the ground and v3=l states, and with microwave frequencies of the rotational transitions in both states. Least-squares analyses of the data were carried out and the molecular con— stants in both states were determined. Molecular parameters of 13 CH3F were derived by a least- squares fitting of the frequencies of the two-photon transi- tions assigned in this work, the two-photon frequencies previously reported, the infrared frequencies obtained by analysis of the laser Stark spectrum, and the frequencies of the J=l+0 rotational transitions in the ground and v3=l states measured by microwave spectroscopy. From relative intensity measurements on several transi- tions in the ground and first excited torsional states of two conformers of 3-methyl-l-butene, the torsional excita- tion energies have been estimated to be 90:10 and 104i10 cm-1 for the species with the double bond trans to the ring and rotated leG’from the trans position "gauche", respectively. The gauche-trans energy difference has been estimated to be 130:20 cm—l. The torsional excitation energies and the gauche-trans energy difference have been used to estimate the torsional potential constants. By analysis of Stark effects, the dipole moments have been determined to be “a =0.312i0.003D, =0 (assumed), uc=0.07li0.042D, and “T: “b 0.320i0.010D for the trans conformer. For the gauche con- formen they are ua=0.367i0.004D, =0,uc=0.15410.006D, and “b uT=0.398i0.004D. To My Parents For Their Everlasting Love ii ACKNOWLEDGMENT I wish to express my sincere appreciation to my advisor, Professor R. H. Schwendeman, for his enlightening guidance, whole-hearted support and friendship throughout the course of this study. Thanks are also extended to Professors Katharine L. C. Hunt, Andrew Timnick, William W. Reusch, and George Leroi for serving on my Committee. Gratitude is also extended to the Department of Chem- istry, Michigan State University, The National Science Foundation, and the people of Iran for their partial finan- cial support. The author would like to thank Peri-Anne Warstler for her invaluable typing skill. Special thanks to all my friends for their support and all my colleagues for their assistance in the laboratory. Above all, my deepest appreciation goes to my parents for their love, my brother for his moral and financial sup- port, my sisters and my uncle for their constant encourage- ment during these years. To my parents, I dedicate this Dissertation. iii TABLE OF CONTENTS Chapter Page LIST OF TABLES. . . . . . . . . . . . . . . . . . . Vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . x CHAPTER I. INTRODUCTION. . . . . . . . . . . . . . 1 CHAPTER II. THEORY . . . . . . . . . . . . . . . . 3 2.1. Introduction. . . . . . . . . . . . . . . 3 2.2. Energy Levels . . . . . . . . . . . . . . 5 2.2.a. Rotational Energy Levels. . . . . 5 i) Symmetric Top Energy Levels . . . ii) Asymmetric Top Energy Levels. . . 10 2.2.b. Vibrational Energy. . . . . . . . l3 2.2.c. Torsional Levels. . . . . . . . . 15 2.3. Stark Effect. . . . . . . . . . . . . . . 20 2.4. Molecular Spectra . . . . . . . . . . . . 23 2.4.a. One-Photon. . . . . . . . . . . . 23 i) Symmetric Top Selection Rules . . 23 ii) Asymmetric Top Selection Rules. . 24 2.4.b. Two-Photon. . . . . . . . . . . . 25 i) Single Parity Levels. . . . . . . 25 ii) Double Parity Levels. . . . . . . 27 iii) Density Matrix Formulation. . . . 30 CHAPTER III. EXPERIMENTAL PROCEDURES . . . . . . . 45 3.1. Infrared-Microwave Two-Photon Spectroscopy. . . . . . . . . . . . . . . 45 iv Chapter 3.2. CHAPTER IV. 3.1.b. Radiation Sources . . . . . . . . 3.1.c. Sample Cell . . . . . . . . . . 3.1.d. Samples . . . . . . . . Microwave Spectroscopy. . . . . . . . . . 3.2.a. Description of the Spectrometer and Experimental Procedure. . 3.2.b. Sample. . . . . . . . . . . . . . TWO-PHOTON SPECTROSCOPY OF AMMONIA AND METHYL FLUORIDE. . . . . . Introduction. . . . . . . . . . . . . The v2 4.2.a. 4.2.b. 4.2.c. i) 4.2.f. 4.2.g. The v3 4.3.a. Bands of 14NH3 and 15NH3 . Introduction. . . . . . . . 14NH 3 . . . . . . . . . . Some Checks of the Accuracy of the Data. . . . . . . . . . Inversion Frequencies in the v2=l State of 14NH3 . . . . . . Difference Between Frequencies of the Laser Lines. . . . . . . . Calculation of Ground State B" l4NH3 and lsNH . Constants for 3 Comparison of the Frequency of the (J=1,0+0,0) Rotation- Inversion Transition of NH3 with the Value from Micro- wave Spectroscopy Discussion. . . . . . . . 12 13 Bands of CH3F and CH3F . . . . Introduction. . . . . . . Page 51 53 55 55 55 57 58 58 61 61 68 84 84 85 89 100 103 103 112 112 Chapter Page 4.3.b. 12CH3F. . . . . . . . . . . . . . 115 4.3.c. 13CH3F. . . . . . . . . . . . . . 121 4.3.d. Discussion. . . . . . . . . . . . 124 CHAPTER V. MICROWAVE SPECTRUM OF 3-Methyl-1- BUTENE.................130 5.1. Introduction. . . . . . . . . . . . . . . 130 5.2. 3—Methy1-l-Butene . . . . . . . . . . . . 132 5.2.a. Dipole Moment . . . . . . . . . . 132 5.2.b. Internal Rotation . . . . . . . . 137 5.3. Discussion. . . . . . . . . . . . . . . . 143 APPENDIX A. Inversion Frequencies of Ammonia in the Ground (v2=0) and Excited State (v2=1) in MHz. . . . . . . . . . . . . 148 APPENDIX B. Input Data for the Least-Squares Fitting Program in 14NH3 . . . . . . . 150 APPENDIX C. Input Data for the Least-Squares Fitting Program in l5NH3 . . . . . . . 153 APPENDIX D. Input Data for the Least-Squares Fitting Program in l5NH3 . . . . . . . APPENDIX E. Input Data for the Least-Squares Fitting Program in 12CH3F. . . . . . . 156 155 APPENDIX F. Input Data for the Least-Squares Fitting Program in l3CH3F. . . . . . . 159 APPENDIX G. Comparison of Observed and Ca1— culated Frequencies in the v2 Band of 15NH3 . . . . . . . . . . . . . . . 160 APPENDIX H. Comparison of Observed and Cal- culated Frequencies in the v2 Band of 14NH3 . . . . . . . . . . . . . . . 164 REFERENCES 0 O O O O O O O O O O O O O O O O O O O I 17]- vi Table LIST OF TABLES Polynomial representation of vINV’ the Inversion Frequency in the v2=l State of 14NH3. The 21 Parameters were Obtained by Fitting the Data From Ref- erence 63 . . . . . . . . . . . . . . . . Ground State and v2 Excited State Molecular Parameters of 14NH3 (cm-l). . . . Comparison of Observed and Calculated Frequencies of s-s Two-Photon Transi- l4 tions in the v2 Band of NH 3 I O O O O O 0 Comparison of Observed and Calculated Frequencies of a-a Two-Photon Transi- tions in the v2 Band of 14NH3 . . . . . . . Inversion Frequencies of 14NH3 in the v2=1 State (MHz). . . . . . . . . . . . . . Comparison of Differences in Laser Fre- quencies Obtained from Duplicate Measure- 14 ments of NH3 Two-Photon Frequencies with Differences Calculated from Vibration- 72 73 Rotation Constants of CO2 or N20 . . . . vii Page 62 74 78 79 86 88 Table Page Comparison of Observed and Calculated Frequencies of s-s Two-Photon Transi- 15 tions in the v Band of NH 90 2 3 O O O 0 O 0 0 Comparison of Observed and Calculated Frequencies of a-a Two-Photon Transi- 15 tions in the v2 Band of NH 92 3 O 0 O O o o Vibration—Rotation Parameters of 15NH 3 in the Ground State m" and the v2=1 1) o o o o o o o o o 94 15 Symmetric (s) State (cm- Vibration-Rotation Parameters of NH3 in the v2=1 Asymmetric (a) State cm-1 . . . 97 Inversion Frequencies of 15 NH3 in the v2=1 State Calculated From Differences in Two-Photon Frequencies . . . . . . . . . 98 Comparison of Differences in Laser Frequencies Obtained from Duplicate Measurements of Two-Photon Frequencies with Differences Calculated from Vibra- tion-Rotation Constants of CO2 or N20 . . . 99 Comparison of the Calculated Frequency of the J,K=1,0+0,0 Rotation-Inversion Transition in l4NH3 and 15NH3 with the Experimental Value. . . . . . . . . . . . . 104 Frequencies of the IR Transitions in 14NH3 . . . . . . . . . . . . . . . . . . . 108 viii .1 Table 4:. l 15 uh I 16 Frequencies of the IR Transitions in 15 NH3 . . . . . . . . . . . . . . . . Comparison of Observed and Calculated Frequencies of Two-Photon Transitions in the v3 Band of 12CH3F. . . . . . . 12CH3F in the Ground State and v3=1 Excited State . Molecular Parameters of Comparison of Observed and Calculated Frequencies of Two-Photon Transitions Band Of 13CH F . . . . . . . . 3 3 Molecular Parameters of 13 the v CH3F in the Ground State and v3=l Excited State . Comparison of Two-Photon Frequencies )a and Zero-Field Frequencies De- b (VTP rived from Laser Stark Spectra (vLS) Stark Coefficients and Dipole Moments for Trans 3-Methy1-l-Butene . . . . . Stark Coefficients and Dipole Moments for Gauche 3-Methy1-l-Butene. . . . . Parameters Related to the Internal in Torsion of the Vinyl Group in 3-Methy1- l-Butene. . . . . . . . . . . . . . . ix Page 110 117 122 123 125 129 135 136 142 LIST OF FIGURES Figure Page 2-1 Energy-level schemes for IR—MW two- photon process. Three level system (NH3) . . . . . . . . . . . . . . . . . . . 26 2-2 Energy-level schemes for IR-MW two- photon processes. Two level system (CH3F). The microwave radiation Vm is added (c) or subtracted (d) from the infrared v2 radiation. The (+) and (-) signs indicate the parity of levels. Each level has double parity (1) . . . . . . . . . . . . . . . . . . . . 28 2-3 Perturbation diagram used to calculate the density matrix elements . . . . . . . . 33 3-1 IR-Microwave two-photon spectrometer. . . . 46 3-2 Mirror configuration for two-photon experiment. . . . . . . . . . . . . . . . . 48 3-3 Diagram of the absorption cell used in infrared-microwave two-photon ex- periment. . . . . . . . . . . . . . . . . . 54 4—1 Energy level diagrams involved in two- photon transitions of NH3 . . . . . . . . . 69 Figure Page IR-MW two-photon absorption signal. The sQ(5,5) two-photon transition of 14 NH observed by using the CO2 P(36) 3 laser line (v£=929.01743 cm’l) and micro- wave frequency of vm=16651 MHz. The offset frequency Av=7.882 GHz and the sample pressure was 0.6 torr. The width of the sweep is 400 MHz . . . . . . . . . . 76 IR-MW two-photon absorption signal. The aQ(l,1), two-photon transition of 14NH3 observed by using the CO2 R(8) line (vl=967.707233 cm-l) and microwave frequency of vm=l4987 MHz. The sample pressure was 1.25 torr, and the offset frequency Av=8.707 GHz. The width of the sweep is 400 MHz. . . . . . . . . . . . 77 Energy-level diagrams for the J=0 and 14 15 J=1, v=0 states of NH3 and NH 3 showing the observed transitions and the hypothetical levels corresponding to no inversion . . . . . . . . . . . . . . . . . 101 Trace of the QQ(6,K), (K=6,5,4,3) two- photon transitions in 12CH3F. The P(l8) CO2 laser line in 9.5 um region was used with a microwave sweep from 13900-15600 MHz xi Figure Page to record this Spectrum. The K value is given at the top of each transition. . . 118 Trace of the QQ(6,K), (K=6,5,4,3,2) two-photon transitions in 13CH3F. The P(40) CO2 laser line in the 9.5 pm region was used with a microwave sweep from 8200-10100 MHz to record this spec- trum. The K value is given at the top of each transition. . . . . . . . . . . . . 126 Projection of trans 3-methy1-l-butene in its ac plane of symmetry. The angle a measures internal rotation about the indicated cc bond . . . . . . . . . . . . . 133 A plot of the variation of calculated values of A—AO, B-BO and C-CO with tor- sional angle a for 3-methyl-1-butene. Here, A B and Co are constants for o’ o the trans configuration calculated for an assumed structure. The vertical scale is in MHz. The crosses mark the experi- mental values of A -At, Bg-B and 9 t' Cg-Ct o o o o o o o o o o o o o o o o o o o 140 Plot of the torsional potential function for 3-methyl-l-butene calculated from the potential constants given in Table 5-3. xii Figure Page 5-3 The calculated energies of the first few torsional levels in each well are indicated . . . . . . . . . . . . . . . . . 144 xiii CHAPTER I INTRODUCTION This thesis is concerned with studies of the struc- tural parameters of molecules by two different molecular spectroscopy techniques. In one study, the infrared-micro- wave two-photon spectra of the NH3 and CHBF molecules have been examined by using an N20 or CD2 infrared gas laser com— bined with a tunable microwave source. In the second study, the dipole moment and torsional potential function of 3-methy1-l—butene have been determined from an investi- gation of the microwave spectrum of this molecule. The theoretical and experimental aspects involved in these two studies are similar. Therefore, Chapters II and III are devoted to discussions of the theory and des- criptions of the experiments, respectively. In Chapter II the energy levels, the effect of the application of an electric field to a system of molecules (Stark effect), the selection rules governing one—photon and two-photon transitions, and finally the response of a molecular system irradiated simultaneously from two sources are described. Chapter III includes a brief description of the infra- red-microwave spectrometer and the experimental procedure employed to observe the vibration-rotation spectrum of am- monia and methyl fluoride. A description of the micro- wave spectrometer and the experimental procedure used to determine the dipole moments and torsional potential func- tion for 3-methyl-1-butene are also included in this chapter. The eXperimental results of the two-photon studies of ammonia and methyl fluoride are presented in Chapter IV, whereas the results of the microwave study of the 3-methyl- l-butene are covered in Chapter V. Each of these chapters includes a review of previous work and a discussion sum- marizing the results of the present experiments. CHAPTER II THEORY 2.1. Introduction Molecular spectrosc0py is essentially a method for the determination of the energy levels of molecules. The energy of the electromagnetic radiation absorbed by a molecular system is always approximately equal to an energy differ- ence between two allowed states of the molecule. The funda- mental equation of spectroscopy is the quantum condition: hv = E — E , (2-1) where: h is Planck's constant; v is the frequency of the radiation; and and E E are the initial and final energy 1 2 states or levels of the system absorbing the radiation. The molecule is considered as a collection of atoms held together by a set of interatomic bonds. The molecule is not a rigid body; electrons move, atoms vibrate relative to each other, and the molecule rotates and translates as a whole. The energy of the motion of the electrons and of the rotational and vibrational motions of the nuclei in a mole- cule can only assume certain discrete values, and so leads to discrete frequencies of transitions. An understanding of molecular motions starts with the total energy expres- sion ET' To a high level of accuracy the total energy of a molecule can be expressed as the sum of its electronic E vibrational Evib' rotational Er and translational elec' Ot' . 1 energies Etrans’ ET = Eelec + Evib + Erot + Etrans ' (2-2) In some cases it is necessary to add terms to Equation (2-2) which allow for the interaction between the electronic and vibrational motions or between the vibration and rotational motions. To the same degree of approximation that Equation (2—2) is valid the molecular Hamiltonian may be written as the sum of corresponding operators, and the molecular wave func- tion as a product of separate wave functions, as follows: + H + H (2-3) elect vib rot trans w = welect X Ipvib X 1prot x wtrans (2'4) The terms in Equation (2-3) are related to the cor- responding factors in Equation (2—4) by the time independent Schr6dinger equation: Hw = Em . (2-5) 2.2. Energy Levels 2.2.a. Rotational Energy Levels. The rotational energy, Erot’ of a molecule is the solu- tion of Equation (2-5) in which H is the rotational Hamil- tonian operator for the system and w is the wave function describing the system. The value of Erot is a constant which defines the stationary state energy levels of the system. The rotational Hamiltonian can be expressed in terms of the angular momentum components Pa' Pb, and PC about the three principal axes,2 as follows: P: Pf; Pi rot 21a 21b 2IC Equation (2-6) may also be written as H - 213 (AP2 + BP2 + CPZ) (2-7) rot _ h a b c ' where A, B, and C are rotational constants and are defined as _ h . _ h h A ’ ' B " 2 2 8n2I 8n I 8n I a c ('3 ll (2-8) b In Equations (2—6)and (2-8) Ia' Ib' and Ic are the prin- cipal moments of inertia where by convention, A Z B 1 C or I i I i I . (2-9) i) Symmetric Top Energy Levels - When two moments of inertia are equal, the molecule is a symmetric tOp. A molecule is considered to be a prolate symmetric tOp if Ia < Ib = IC and an oblate symmetric top if Ia = Ib < IC. In terms of the rotational constants the energy for a pro- late rotor is 2 Erot = h[BJ(J+1) - (A-B)K ] . (2-10) For an oblate top the energy is _ _ _ 2 _ Erot — h[BJ(J+1) (c B)K ] . (2 11) It is obvious from Equations (2-10) and (2-11) that each rotational level is specified by two quantum numbers, J and K. J represents the total rotational angular momentum of the molecule. The quantum number K represents the com- ponent of angular momentum about the symmetry axis of the molecule, the a axis for a prolate top or the c axis for an oblate top. The J quantum number takes positive integer values including zero, while the K quantum number takes all the negative and positive values between -J and +J including zero. The energy for a symmetric rotor is independent of the sign of K; therefore, levels with K > O are doubly de- generate. In a prolate rotor the value of (A-B) is posi— tive; this causes an increase in energy level with increase in K for a given J. For an oblate rotor the opposite holds because (C-B) is a negative quantity. Up to this point, a molecule has been considered as a rigid body. However, in reality a molecule is not rigid. The effects of vibration-rotation interaction including centrifugal distortion should be considered. For a polyatomic molecule, there are several normal modes of vibration, each with its own vibrational quantum number Vi' The strongest effect of vibration-rotation interaction can be taken into account by defining effective rotational constants,3 B =13 -Zo113(v.+—£) . (2-12) i Here Be is the hypothetical B rotational constant for the rigid molecule at equilibrium, BV is the effective rota- tional constant for a given vibrational state, and the sum runs over all of the vibrational modes of the molecule. Also in Equation (2-12) di is equal to one for non-degen- erate and two for doubly degenerate modes, respectively, and a? is called a vibration-rotation coupling constant. Similar expressions exist for the A and C rotational con- stants. The centrifugal forces resulting from the rotational motion tend to alter the effective moments of inertia. For rotation about any axis in the molecule, this effect forces the atoms away from the axis of rotation and in- creases the moment of inertia about that axis. The effect of centrifugal distortion on rotational energies in many cases is small, so it can be treated as a perturbation of the rigid rotor Hamiltonian. The rotational energy including centrifugal distortion constants for a prolate symmetric tOp is:4 E(J,K) = BJ(J+1) + (A-B)K2-DJJ2(J+1)2-DJKJ(J+1)K2 4 3 3 2 2 2 -DKK + HJJJJ (J+1) + HJJKJ (J+1) K + H J(J+l)K4 + H K6 + (2 13) JKK KKK .'. ° Here D D and D are the first order distortion constants, J’ JK K HJJJ, HJJK' HJKK and HKKK are the second order distortion constants, etc. For oblate symmetric tops A is replaced by the C rotational constant. Experimental values of the centrifugal distortion constants can be combined with values of vibrational wave numbers to determine the force field of a molecule or can be used to check the validity of results obtained from vibra— tional data alone.5'6 General formulas relating DJ, D JK and DK to the vibrational force constants have been given by Wilson7 for the case of a molecule with symmetry point group C3V; 31 31 DJ = (34/321134) [ 2 53-1- (F'1)i. 33—1— + i,j(Al) i 3 j l . (E ) ::2 (F-l)i - ::2] I _ - 4 .2 .2 DJK — zoJ + (6 /81B IA ) X[ Z gi‘F-l)i';:73+ i,j(Al> i 3 j 31 31 . .3 3‘7“? l)ij3“s’4‘] ' 1,](Ey) 1 j 31 31 __ _ 4 .4 ___:_a_ -1 __3_ DK ' DJ DJK + (h /81A ) 2 35. (F )ij 3 . i,j(A1) 1 j 10 Here, I1 = IB + IC = i m (x + y + 22a) , (Al) _ _ 2 _ 2 12 — IB - IC - : ma(xa Ya) , (E ) I = E m (x + y2) 3 o o ' o I4 = 2 Z mayaza . (2—14) In these formulas,‘fi is Planck's constant divided by Zn; 1; is the moment of inertia about the threefold axis of sym- metry (figure axis 2): I; is the moment of inertia about an axis (y) perpendicular to the z axis; Si is an internal symmetry coordinate; (F-l) is an element of the matrix ij inverse to the force constant matrix; I I and I are A' B’ C the instantaneous moments of inertia; x, y, and z are th Cartesian coordinates and m is the mass of the o atom. a The derivatives of II and 13 are non-zero only for internal symmetry coordinates which have the symmetry Al' Similarly, I and I4 derivatives vanish unless the Si have symmetry E. 2 ii) Asymmetric Top Energy Levels - A molecule is considered to be an asymmetric top when it possesses three unequal moments of inertia, Ia # Ib # Ic' In contrast to symmetric tops, the rotational energy for an asymmetric 11 tOp cannot be given by a simple formula. For an asymmetric top molecule no component of the total rotational angular momentum along a molecule-fixed axis is a constant of the motion. Therefore, while J and M (M represents the projection of the total rotational angular momentum along a space fixed axis) are still good quantum numbers, K is not. However, K is kept to label the energy levels: K is the value of K for the limiting case 1 of an oblate symmetric tOp, K for the limiting prolate -1 tOp. Each level may be labeled by JK . -1K1 The asymmetry of a molecule is conveniently described in terms of an asymmetry parameter K prOposed by Ray,8 _ 2B-A-C _ K — —IA:ET_ (2 15) The value of K is between +1 for an oblate tOp and -l for a prolate symmetric top. The rotational Hamiltonian in standard form is given P q r r q P _ hpqr (JXJyJz + JszJx) (2 16) where p + q + r is even; Jx' J , Jz in units of‘h are Y the components of the total angular momentum vector J projected on molecule-fixed axes. These compon- ents satisfy the reversed commutation relations 12 [JX ,Jy ] = JXJy-JyJX = -iJz , etc. (2-17) The coefficients hpqr are the rotational and distortion constants of the vibrational state under consideration. The Hamiltonian of Equation (2—16), after a suitable unitary transformation is given as a reduced Hamilton- ian.9 For terms up to sextic in the angular momentum, this reduced Hamiltonian is: H = H + H + H (2-18) or H = [%(X+y)J2 + {z-% (X+y)}Jz _ AJ(J2)2 2 2 4 2 3 2 2 2 - AJKJ Jz — AKJZ + HJ(J ) + HJK(J ) Jz 2 2 2 2 2 + [%(x-y)(Jx-Jy) - 26JJ (Jx-Jy) 2 2 - 5 Kz{J (J i-Jy) + (Ji- %)Jz} 2 2 2 2 2 2 2 2 + 2hy(J ) (Jx-Jy) + hJKJ {Jz(Jx-Jy) 2 2 2 4 2 2 2 2 4 + (JX-Jy)Jz} + hK{Jz(JX-Jy) + (JX—JY)JZ}] . (2-19) 13 In this equation: x, y, z are the effective principal rotational constants; A A A 6 6 are the quar- J' K H J' JK' K’ tic distortion coefficients; H H H J' JK' KJ' K' hK are the sextic distortion coefficients. The matrix elements of the first bracket are diagonal in K and those of the second bracket have AK = :2. The term H2 represents the effective rigid rotor Hamil- tonian,Equation (2-6). The first order contributions to the rigid rotor energy levels are obtained by taking the diagon- al matrix elements of H4 and H6 in the rigid rotor basis. The rotational energy levels correSponding to the Hamil- tonian in Equation (2-19) are computed by evaluating the matrix elements for a given J and diagonalizing the matrix. 2.2.b. Vibrational Energy For a molecule consisting of N atoms, the number of vi- brational degrees of freedom is 3N—6 for a non-linear system and 3N-5 for a linear system. The kinetic and potential energy of vibration may be written in terms of 3N-6 or 3N-5 internal coordinates. The kinetic energy T and po- tential energy V of vibration, in terms of the normal co- ordinates Qi and their conjugate momenta Pi,are given by the expressions10 T = % 22 P. (2-20) 14 v = g 2: A. Q. . (2'21) The vibrational Hamiltonian of a molecular system can be written as 3N-6 2 3N-6 H = %[ 2: Pi + i=1 i 110%] (2-22) 1 IIM and the Schrédinger wave equation becomes 2 3N-6 2 3N-6 hz 2 L41 + 12 z 1.9% = Eu) (2-23) 871 i=1 3Q. i= PM The advantage of using normal coordinates is obvious from the fact that Equation (2-23) is separable into 3N-6 equations, one for each vibrational mode 1113.1 w. = E.w. (2-24) 8n2 BQE L $1 I 2 A10 Equation (2-24) is the well known harmonic oscillator wave equation. Therefore, the vibrational wave function is a product of harmonic oscillator wave functions, and the vibrational energy is a sum of harmonic oscillator energies. = 11 mi; E . = 2: E. (2-25) 15 Here Ei is given by E. = (v. + %)h\)i v. = 0, l, 2, ... (2-26) In Equation (2-26) vi is the frequency associated with normal coordinate Qi and vi is the vibrational quantum number, a positive integer or zero. By substitution of Bi from Equation (2-26) into Evib' Equation (2-25), the vibrational energy of a molecule with several modes of vibration can be written as Evib = (vl + 15)h\)l + (v2 + %)h02 + ... + (V3N-6 + %)hv3N-6 (2-27) A vibrational transition is induced by radiation only if the derivative of the dipole moment with respect to a normal co- ordinate is non-zero at equilibrium. The selection rule for the ith normal mode turns out to be Avi = :1, Av. = 0, J 195i. 2.2.c. Torsional Levels Internal rotation occurs when one part of molecule (top) rotates relative to the rest (frame) about a bond connect- ing the two parts. A simple example is ethane in which one methyl group can rotate with respect to the other *- ‘A-‘n—AmW-a. ~-.. 16 about the C-C bond. This torsional motion is not a free rotation; in most molecules a potential energy barrier has to be overcome in turning from one configuration to another. Microwave spectroscopy provides a powerful technique for the determination of potential barriers. The rotational spectrum will be affected by coupling between internal and overall rotations of the molecule, and the complexity of the Spectrum depends on the height of the potential barrier hindering internal rotation and the moment of inertia of the rotating group. 11 in which We follow the procedure of Quade and Lin the torsional angle a is assumed to be the only internal degree of freedom. The resulting kinetic energy expres- sion may be written 2T (2-28) II IE 2H 88 or T = E E 2 (2-29) In these equations the transpose of the column matrix w is defined as (i) = (w w (3 0L) (2-30) and the transpose of the corresponding momentum matrix as 17 + P = (PXP Pzp) . (2-31) Y The mg and P9 are components of the molecular velocity and angular momentum about the 9 axis, respectively, and p is the momentum conjugate to o. The matrix F is the inverse of the matrix 1/2. After substitution into the usual ex- pression for the Hamiltonian,lo'12 4 + V ; (2-32) in which I is the determinant of I. After simplification, it is found that13 H = Z 22 F . P P . + Z (P F +F P )p + pF p + V' + V g g' 99 9 9 g 9 99 99 9 99 (2-33) In this expre551on Fgg" Fgo’ and Foo are appropriately de- fined components of the 4 x 4 matrix F, and V' is a mul- tiplicative operator which remains from the evaluation of the quantum-mechanical kinetic energy operator after separat- ing out the P9 and p dependent terms. It is found that 1 d1 3 d1 2 -—- F -) (2-34) on do do 16I2 do (do Fourier expansion of V' leads to terms which simply add to the corresponding terms of V and lead to slightly altered 18 values of the Vn' where V represents the potential energy associated with the internal rotation and the Vn are the potential constants, V = Z —— (l - cosno) . (2-35) n The first three terms of the Hamiltonian (2-33) lead to an effective Hamiltonian for a rigid rotor with cen- trifugal distortion, from which the usual rotational energy levels are obtained. The last three terms of the Hamil- tonian make up the torsional energy Operator, as follows: H = pFaap + v (2-36) in which the Fourier components of V are the sums of the Vn and the corresponding components of V'. The torsional energies can be obtained by direct di— agonalization of a matrix for H Linear combinations of T' the free-rotor functions, exp (ima) with m = O, :1, :2, etc., can be used as basis functions for the calculation. The potential function for an internal rotor with three fold symmetry (-CH3) has three identical minima. If the potential barriers betweentfluaminima are high, all energy 1 However, levels within the minima are triply degenerate. as the barrier is reduced, the possibility of quantum mechanical tunnelling increases and the degeneracy is 19 partially removed, giving a non-degenerate A level and a doubly degenerate E level. This causes many microwave transitions to appear as doublets, each component cor- responding to a transition between torsional levels of the same symmetry; i.e., A +*A or E++ E. The extent of the splitting is a sensitive function of the potential bar- rier to internal rotation. The mode of vibration known as inversion is another hindered motion. This vibration exists, in principle, for all non-planar molecules and usually involves the inter— change of two equivalent configurations via a planar inter- mediate. The potential function has two minima separated by a potential hump. The energy levels for a particle moving with a harmonic potential are equally spaced. If the potential is dis- torted by a barrier rising in its center, then pairs of energy levels approach each other. In the limit of a very high potential barrier, the particle again has equally spaced energy levels, but in two sets corresponding to vibration on either side of the barrier. Therefore, the energy levels are doubly degenerate. Classically, the motion of the particle from one side of the barrier to the other side is forbidden, unless the particle has enough energy to overcome the barrier. But quantum mechanically, the tunnelling effect allows penetra- tion of the particle from one minimum to the other. In NHB' the potential barrier is moderately high, and a 20 transition between the two lowest vibrational levels falls in the microwave region, AE m 0.8 cm-l. In CH3F, the poten- tial barrier is so high that the two lowest vibrational levels coincide. 2.3 Stark Effect When a molecule is subjected to a static electric field, the total rotational angular momentum J is constrained to 2J+l possible orientations with respect to the electric field direction. These orientations may be identified with a quantum number M which measures the projection of J in the direction of applied field a. The 2J+l values of M are integer values from +J to -J. In the absence of an electric field, the 2J+l possible directions of J have the same energy. The existence of a permanent dipole moment u causes an interaction between the molecule and an electric field. The perturbation of the rotational energy levels of the mole- cule by this interaction is called the Stark effect, and the 2J+l degeneracy of the rotational levels can be lifted. The interaction energy of a molecule subjected to an electric field may be written as H = -u-€ (2-37) Since the Stark energies are usually small in comparison 21 with rotational energies, perturbation theory can be used to calculate the Stark splittings. For symmetric top molecules a first order perturbation treatment leads to E<1) = kM S '“6 JTJIIT (2‘38) where k, the signed value of K, is the projection of the angular momentum along the figure axis of the molecule. The second order Stark energy in a symmetric top is (l) d s an usually much smaller than the first order correction E is ignored unless k or M = 0, in which case the first order contribution vanishes. A linear molecule may be thought of as a special case of a symmetric top molecule with K = 0. Then, the first order correction to the rotational energy is zero for this calss of molecules and the second order effect is E(2) = “282 [ J(J+l)-3M2 S 2hB .J(J+l)(2J-1)(2J+3) 1 (2-39) where B is the rotational constant of the molecule. Stark effects in asymmetric rotors are usually second order or proportional to 82 (2) 2 2 2 E = Z A + B M 2— O J,M E: [( g 9 )ug] ( 4) 9 22 In this equation, g = a,b,c; the Ag's and Bg's are Stark coefficients, which depend on the rotational constants and the quantum numbers of the specific level. At Michigan State University, these coefficients can be calculated by a computer program called "EIGVALS". The second order energy depends only on M2 and so does not remove the (2J+l) degeneracy completely. It splits the levels into (J+1) components. Equation (2-40) provides a means by which the dipole moment of a molecule can be obtained from experimental results. The selection rules are AM = 0 or :1 depending on whether the electric field of the radiation and Stark field are parallel or perpendicular to each other, respec- tively. For AM = 0 selection rules, the frequency of a transition v can be written as M v = v + v (2-41) where V0 is zero field frequency and Vs is the Stark ef- fect contribution to the frequency. From Equation (2-40), vs turns out to be ’ 2 VS " 8 92': (AAg + ABgM )Ug] (2 4 ) where AAg and ABg are the difference in Ag and B9 values for the two levels involved in the transition. The Stark shift in frequency, Vs' can be measured for 23 different values of the Stark field a for a particular M component. A plot of the Stark contribution vs vs. 82 2 2 )U I will be a straight line whose slope is Z(AAg + ABgM g from which the u: can be calculated. 2.4. Molecular Spectra 2.4.a. One-Photon i) Symmetric Top Selection Rules - A molecule under certain conditions can interact with electromagnetic radia- tion and undergo a transition from one state to another. Relations between the quantum numbers of the two states are called selection rules. Rotational transitions can be induced by radiation provided that the molecule possesses a permanent dipole moment. The selection rules governing rotational transi- tions in a symmetric top molecule are: AJ = 0, :1; AK = 0; AM = 0, :1; + + - . (2.43) The last selection rule applies to symmetric top molecules, such as NH in which the inversion degeneracy is removed. 3! In this case transitions are allowed between two inversion levels with opposite symmetry. For molecules without inversion doubling, the last selection rule is unimportant, because symmetric and 24 asymmetric inversion levels coincide in pairs. The mole- cule CH3F is an example of this group of molecules for which each level has double parity. ii) Asymmetric Top Selection Rules - The dipole moment in an asymmetric top molecule is not restricted to lie along a figure axis as in the symmetric top, but may have components pa, along any or all of the principal Lib: UC axes. Therefore, the spectrum of an asymmetric rotor may be complicated by as many as three different types of rotational transitions. These transitions are called a- type, b-type, and c-type transitions depending upon which component of the dipole moment is responsible for the tran- sition. If the dipole moment has three significant com- ponents along three principal axes, all three types of transitions will be present in the spectrum. The selection rules for the J quantum number are: AJ = 0 for Q branch transitions and AJ = -l, and +1 are for P and R branch transitions, respectively. The selection rules on the quantum number K_l and K+1 are the following: #0 ee++eo a- transi ions: type t ua oe++ oo 25 . . ee +>oo b type tran51tions. “b # 0 oe +*eo O I ee H De c type tranSitions. “c # 0 eo ++oo (2 44) Here, e and 0 represent the quantum numbers K_1 and K+l as even or odd. 2.4.b. Two-Photon i) Single Parity Levels - The energy levels involved in a microwave-infrared two-photon process can have single parity, like NH or double parity, as in the case of CH3F. 3! Figure (2-1) is a diagram of some energy levels of NH3 showing possible two-photon transitions in which a micro- wave quantum is added (a) or subtracted (b) from an infra- red quantum. For this case, the transition moment is given by M _ <1IupEmI2><2lqufll3> 2 — 2hAv ' (2-45) In this equation up and “v are the permanent dipole moment and the vibrational transition moment, respectively; Em and B2 are the electric fields of the microwave and the laser radiation, respectively; and Av is the offset fre- quency, which is the difference between the laser fre- quency and the frequency of the allowed transition 3 + 2. 26 3————1 _, _ 3 K + “2 V2 rJ5 w / ffl‘d 2 Av l _ V 4‘ m Vm l 2 ‘ Jk___.+ (a) -..-.J_!_AV__ (b) I (The microwave radiation vm is added (a) or subtracted (b) from the infrared V2 radiation. The (+) and (-) signs in- dicate the parity of levels.) Figure 2-1. Energy-level schemes for IR-MW two-photon pro- cesses. Three level system (NH3). 27 It is apparent from Equation (2-45) that in order for the two-photon transition 3 ++ l to be allowed, it is necessary that there be at least one third level, 2, such that the one—photon transitions 2 + 1 and 3 + 2 are al- lowed. It is also apparent from Equation (2-45) that two- photon absorption is favored when up and “v are large, microwave and laser powers are high, and Av is small. If we compare the transition moment for a two-photon process with that for a single photon process (2—46), M1 = <2|UvEfll3> , (2-46) we find that a two—photon transition is weaker than a nor- mal vibrational transition by a factor of 2 2hAv (2-47) However, if microwave radiation of sufficient power is used, this factor is not very small, even for relatively large Av. ii) Double Parity Levels - In case II Figure (2-2), where the molecular levels involved in the process are of double parity, a third level is not necessary and we can add or subtract the microwave radiation directly. Whether the microwave frequency has been added or subtracted \\ \\ Y1, \\ §\ \\ "'1’- (c) (3) - i J ..vm Figure 2-2. Energy-level schemes for IR-MW two-photon processes. Two level system (CH3F). The microwave radiation vm is added (c) or subtracted (d) from the infrared 0% radiation. The (+) and (-) signs indicate the parity of levels. Each level has double parity (:). H- i 29 can be easily checked experimentally by manually increas- ing the frequency of the laser and seeing whether the micro- wave frequency satisfying the two-photon condition de- creases or increases. For each of the cases for double parity levels there are two processes shown in Figure (2-2), corresponding to whether the required third state is part of the lower level or part of the upper level. Since these two processes are not distinguishable by the experiment, the overall process is a superposition of the two. Unfortunately, this is a destructive interference; the transition dipole is, M2++1 = [<1luEmIl>-<2lu'Em|2>]/2hvm , (2-48) where u and u' are permanent dipole moments corresponding to states 1 and 2, respectively, and v is the frequency m of the microwave radiation required to observe the two- photon transition. (In this case Vm is the same as the offset frequency.) In Equation (2-48), the microwave matrix element ap- pears in the transition moment as a difference. For a Q branch transition, the magnitude of the microwave matrix element for AM = 0 transitions depends only on the dif- ference between u' and n, which is normally very small. However, with a correct experimental arrangement, AM = :1 transitions are allowed and we were able to observe Q 30 branch CH3F transitions. iii) Density Matrix Formulation - The response of the molecules to simultaneously applied infrared and microwave radiation is calculated by the density matrix method.14 5 = - %[H.o] - P(p-o(o)) (2-49) in which H is the matrix representation of the complete Hamiltonian and p represents the density matrix. The term P(p-p(°)) accounts for the random perturbations that cause relaxation of the system to thermal equilibrium. We consider a three level system, as in Figure (2—1). The molecules are irradiated with two beams; therefore, the applied field is cos w t (2—50) E = Em cos wmt + 82 2 where am and 5% are the amplitudes of the microwave and infrared laser beams, respectively, and mm and (32 are the corresponding angular frequencies (w = va if v is the circular frequency of the radiation). The Hamiltonian (2—49) for this system can be written as 0) + H(l) (2-51) 31 (0) In this equationfi is the Hamiltonian for a free molecule and H‘l) is the interaction Hamiltonian due to the applied fields; = -u°E (2-52) where u is the molecular dipole moment and E is defined by Equation (2-50). It is assumed that mm % w21 and mg m _ (0) (o) (o) . w32 where wij —(Hii - Hjj)/h and Hii is the energy of the ith level. The matrix elements of H(l) are H(l) = -u (6 cos w t + e cosw t) (2-53) JK JK m m 1 2 For the system in Figure (2-1), only p12 and u23 # 0; _ , (1) _ (1) “11' p22' and u33 ’ 0' and HJK ' HKJ We also assume that -iwmt 021 = d21 e ; (2‘543) -iw£t 032 = d32 e ; and (2—54b) -i(w +w )t 031 = d3l e m k . (2‘54C) By using the rotating wave approximation, in which all terms with fast oscillation in time are eliminated, the 32 density matrix elements can be given by . ix2 (d d _ ) _ ( _ (o 32 23 Y1 01 01 ix ' _ 1 . (o) _ . . ix — ' _ 1 __Z _ - d21 ‘ 1(wm ”21)d21 + 2 x1‘31 + 2 d31 Y21d21 (2 550’ . 1x2 1x1 d32 = l““53"”32’8132 + ‘2‘ p2 ‘ ‘2‘ d31 ' Y32d32 (2‘55d) . 1X2 1X1 d31 = l(‘*’rn+‘*’sz"“31)C331 + ‘2“ d21 ' 2 d32 ' Y31d31 (2-55e) where 01 = p11 ' 022' p2 = 022 ’ 933' x1 = “alsm/fi ' and 2 = “2351/f1 Equation (2-55) may be solved exactly for the steady- state approximation in which all of the derivatives with respect to the time, pl, p2, d21, d32, d3l,are zerols, or they may be solved approximately. One approximate solution is based on the perturbation diagram given in Figure (2-3).16 The equations are solved by iteration. In the zeroth order of approximation all off-diagonal matrix 33 .pxmu map on umwmn mEHmu osu mo coflumcmaoxw Uwaflmuwo mom .mucmewam xfluume muflmcmp on“ mumHsoamo ou poms EmH06flp cofiumnusunmm N N Q 133 \3 NM NM . mum 980; 126 SEA am am 1 Et 53 - HN Hm on oA 53 3 HQ :1 5 Amy H 133 — .—~ —_- -9 —k v .b‘u—g a-.. 34 (o) and p2 are non- (0) elements of p are zero and only 01 zero. In order to derive the relation between the absorption coefficient and the density matrix elements, we begin with the expression for the absorption coefficient a given by Flygare,17 Pm - _ _§_ _ o — (4an/C) e . (2 56) 2) In this equation, P; is the coefficient of the sine com- ponent of the polarization and e is the amplitude of the R detected radiation (here the infrared radiation). The induced polarization resulting from two radiation fields in the sample can be written as (m) _ (2) _ (2) - (m) P — PC coswgt P Slnwgt + P s cosw t - P s c m sin . wmt (2-57) The polarization is also related to the density matrix elements by the expression P = NTr(uo) (2-58) where N is the number of molecules per cubic centimeter. The symbol Tr(up) represents the trace or sum of the di- agonal elements of the matrix that is the product of the 35 dipole moment and density matrices. For our case, Equa- tion (2-58) can be expanded as P = N[“12(021+012) + “23(932+923)] ° (2’59) If we substitute 921, p32, and their complex conjugates from Equation (2-54) and compare Equation (2-59) and (2-57), we obtain (Q) _ _- _ _ PS — iNu23 (d23 d32) . (2 60) If we write d32 and d23 in terms of their real (déz) and imaginary (dgz) parts, then (2) _ _ n - PS — 2Nu23d32 . (2 61) (R) After substitution for PS in Equation (2-56), 8HNw£ O. = __C—€£—_ U23d32 . (2-62) Therefore, the absorption coefficient depends on the imag- inary part of d32. Two-photon absorption turns out to be a third order effect and dég) can be evaluated according to the perturbation scheme Figure (2-3). 36 o 2 2 2 2 2 . (3) (3) _ X292 Y32x2 X1 26253Y32+Y31(‘52 Y32) i(d -d ) - - —— 23 32 (32+ 2 )2 yz 4 52 + 2 2 Y32 3 Y31 2 o _ x2x191 [Y32Y31 2 2 2 2 y 2(51+Y21)(52+Y32) 2 _ 5152Y31+6153Y32+5253Y21+Y31721Y32] (2_63) 2 2 2(53“)(31) In this equation, Only the second and the fourth terms in Equation (2-63) contribute to the two-photon absorption. Then,since for our experiments, 5i>> Ygl' 6§>>y§2, and 61a,62, 2 o 0 5X X y(p -p ) . (3) (3) g 2 1 2 l l 2 2 2 461 (63+y ) To obtain this expression y21, y32, and Y31 have all been 2 taken to be equal (i.e , y), and y has been ignored relative 37 2 . to 61 wherever it occurs. After substituting in Equation (2‘60), 2 o o 5X2X1Y (Dz-pl) 23 2 461 (Q) _ _ Ps — Nu (2-65) 2 2 (63+Y ) The absorption of the radiant power will be proportional to the incident power, where P is the power, x is the coordinate along the axis of the cell, and o is the absorption coefficient. From this equation the signal at the detector, the difference in the power with and without sample, turns out to be Signal = AP = P - P 5 aQPO . (2-67) In this expression Po is the incident power of the radia- tion, Pf is the output power after absorption occurred, and R is the length of the cell. If we substitute P0 = P2 = é% Si, and use a from Equation (2-56), the signal due to the detection of the infrared will be m 1 - —£— P(£)€ Signal = 2 s 2 , (2—68) 38 and finally from Equation (2-65), the signal is o o 2 2 Signal — ——————— . (2-69) 2 432 (62+y2) l 3 Since 2 2 2 2 2 2 x1X2 “21“32Em52 2 4 2 ’ 461 4h 61 the two-photon signal is proportional to the square of the product of the electric fields of the two radiation fields, is inversely proportional to the square of the offset fre- quency 61, and depends on the product of the square of the microwave and infrared transition moments u21u§2. For the exact solution, used by Dr. R. H. Schwendeman,ls Equations(2-55) are solved by setting up an eight by eight matrix. In this treatment d21, d32 and d31 are separated into real and imaginary parts, d = d' + id" 21 21 21 d32 = d32 + id32 d31 = 51 + id31 (2-70) Then, 39 . __ _ n n _ _ 0 p1 ‘ 2X1d21 + X2d32 Y1 (91 91) . _ u __ H _ - 0 p2 ‘ X1d21 2X2d32 Y2‘92 92) . x2 I _ _ n _ __ n _ | d21 ‘ 51d21 2 d31 Y21d21 n = l _ __ n _ n d21 ‘31‘321‘+ 2 D1 + :2 d31 Y21521 . x1 I _ _ n __ n _ 1 d32 ‘ 52‘132 + 2 d31 Y32‘332 . x2 x1 I; _ I __ _ _ ' .. " d32 " 52d32 I 2 D2 2 8‘31 Y32‘332 . x2 x1 d3i = ’ 53d31 ‘ if d21 + if d32 ‘ Y31d3i . x2 x1 d31 = 63531 I If d21 ’ if d32 ’ Y31d3i ° (2'71) The equations of motion for the density matrix elements in Equation (2—71) may be written in matrix notation, L = -AL + C , (2—72) in which the transposes of the column matrices C and L are I " ' II " (pl d21 d21 d31 d3i 02 d' d 32 32) (2’73) L = O l [(ylpi) o o o o (yng) o 0] (2-74) 40 and Y1 0 2x1 0 o o 0 -x2 0 y21 31 o x2/2 o o o -Xl/2 '51 Y21 ’x2/2 O 0 0 O o o x2/2 y3l (51+52) o o —x = 0 ’X2/2 0 ’(51+52) Y31 0 X1/2 0 0 0 -X1 0 0 Y2 0 2X 0 o o o -xl/2 o y32 62 0 0 0 x1/2 0 X2/2 ‘52 Y32 (2-75) L is the time derivative of the column matrix L. The steady state solution of Equation (2-72) can be obtained by setting L = 0, therefore, L = A c . (2-76) dgz can be evaluated from Equation (2—76) either by approxi— mate methods or by exact solution. To study the response of a two-level system of double parity levels irradiated by two simultaneous photons, as in 41 Figure (2-2), the method used by Dr. R. H. Schwendeman15C will be described. In this treatment the basis functions chosen are the functions of mixed parity, which can be shown to have no dipole matrix elements connecting them. The effect of the microwave radiation is neglected at first, and the density matrix elements are calculated for a two- level one-photon (here, infrared) case. With this assump- (1) tion H in Equation (2-51) can be written as Hm _ JK — -“JK £2 cosw t . (2-77) 2 where only “ab = “ba # 0. The equations of motion can be obtained as follows, A = -2XV - yl (A-AO) u = -6v - Y2u 3 = 3n + 5 A - v (2-78) 2 Y2 ' In these equations, 6 = wR-wba' X = “abEI/fi' and A = paa - pbb . (2-79) Also, u and v are the real and imaginary parts of dba' respectively, where 42 -iw t and d = u + iv (2-80) 0ba ba e ' ba ' The effect of the microwave radiation is introduced as a high frequency Stark effect. In this case of levels of mixed parity the Stark effect can be treated by first-order perturbation theory. Then, even for strong microwave fields, iw t -iw t 6 — 6a + 6b (e + e ) (2 81) in which 6a = wQ u)ba and 5 _ b — (uaa-ubb)em/2n Also, although the energies are changed by the perturbation, to a good approximation the eigenfunctions are not. We then let ikw t A=ZAke m k ikmmt u = 2 uk e k ikwmt V = Z vk e (2-82) 43 After substituting the expressions (2-81) and (2—82) into Equation (2-78) and solving for the steady state, v u = A = 0, the third order expression for v0 is 2 b _ ___ __ (2-83) 2 [v§+<3m-3a)21 Only vé3) is needed,as the oscillating components of v(3) will average to zero as a result of the filtering of the absorption signal. The imaginary part of the radiation-induced polarization Pég) due to the laser radiation is S = 2N11ab VO . (2-84) Therefore, the signal at the detector, . 2 E Signalauab (uaa-ubb)2 2 . (2-85) As for the three level system, the signal is proportional to the product of the squares of the electric fields of the two radiant beams and inversely proportional to the square of the offset frequency 6a (Figure 2-2). The signal is also prOportional to the square of the infrared transition moment. It is, however, proportional to the difference in the 44 diagonal matrix elements of the dipole moment for the upper and lower levels. Thus, as mentioned above, some care must be exercised in the experimental arrangement to avoid can— cellation of these two terms. CHAPTER III EXPERIMENTAL PROCEDURES This chapter describes the spectrometers and experi- mental procedures used for the investigations of the in- frared-microwave two-photon spectroscopy of 14NH3, 15 12CH3F, and l3CH3F and for the study of the microwave NH3 , Spectrum of 3-methyl-l-butene. 3.1. Infrared-Microwave Two-Photon Spectroscopy_ 3.1.a. Description of the Spectrometer and Experim- tal Procedure A block diagram of the spectrometer used in this in- vestigation is shown in Figure (3-1). Microwave radiation produced by a backward wave oscil- lator (BWO) is passed through an isolator to reduce micro- wave reflections and through an attenuator to control the microwave power in the system. The output from the at- tenuator enters a 3 db directional coupler, where it is divided into two parts. One part is sent to a diode mixer- multiplier for comparison with a harmonic of a precisely known reference frequency. The second part is amplitude 45 46 [mjm i! [mm] 57.. 93.3.3.3] fps. WOMLERJ—[m | IR " MICROWAVE TWO " PHOTON SPECTROMETER Figure 3-1. 47 modulated by means of a PIN diode controlled by a 10 KHz square wave voltage and amplified in a traveling wave tube amplifier (TWTA). The maximum output power of the TWTA was measured to be about 20 watts. The amplified radia- tion is sent to the sample cell. A 20 db directional coupler between the TWTA and the cell samples the microwave radia- tion for power measurement. The radiation that passes through the absorption cell is absorbed by a microwave termination at the end of the cell. A CO2 or N20 laser creates infrared radiation which leaves the laser cavity through a partially-transmitting mirror. The output power of the laser is about 1 watt. A correctly aligned configuration of 5 mirrors allows the beam to pass through the absorption cell. Mirror 1 in Figure (3-2) can be changed to a combination of two mir— rors in order to rotate the plane of the polarization of the radiation by 90°. This causes the electric field of the infrared radiation to oscillate in a plane perpendicular to that of the microwave electric field. A particular laser transition is selected by rotating a plane grating at one end of the laser cavity and in order to identify the transi- tion selected, the laser beam is deflected by inserting a mirror between mirrors 1 and 2 in the diagram. The re- flected beam enters an Optical Engineering CO2 Model l6-A laser spectrum Analyzer where its wavelength can be esti- mated well enough to identify the particular laser transition. 48 Laser Beam Beam Splitter Power Meter Absorption Cell '*< Figure 3-2. Mirror configuration for two-photon experi- ment. 49 The beam splitter shown between mirrors 2 and 3 has two uses. One application is to use the reflected component of infrared radiation for power measurements. But the primary use of the beam splitter is to allow insertion of a beam from a He-Ne laser into the infrared optical path for assist- ance in alignment. For this purpose, a pyroelectric de- tector is placed between mirror 5 and the absorption cell. The infrared beam is chopped, monitored by this detector, and the detector output is displayed on an oscillosc0pe. The position and size of the two irises between mirrors 3 and 4 and the position of the detector are changed until the amplitude of the oscilloscope pattern is maximized when the Openings of the irises are as small as possible. This shows that the infrared beam is centered at the openings of the two irises. Next the position of the He-Ne laser is ad- justed until its beam after reflection from the splitter is exactly aligned through the irises; at this point the He-Ne beam is superimposed on the infrared beam. Next the detector is removed and by adjustment of mir- rors 4 and 5 the beam is sent through the absorption cell. A low-noise, liquid-nitrogen-cooled Barnes Engineering Pb- Sn-Te photovoltaic detector, placed at the exit of the sample cell, is used to monitor the infrared beam. The detected signal is amplified and coherently detected by a Keithley Model 840 Autoloc Amplifier. The reference signal for the phase sensitive detector is obtained from the square wave that drives the PIN switch in the microwave circuit. 50 The output of the Autoloc Amplifier can be displayed on an oscilloscope, a chart recorder, or can be sent to a Digital Equipment Corporation PDP8/E computer, where the absorption as a function of frequency of the microwave radiation can be recorded on a flexible disk. The computer program BWOHP, which was written by Dr. Erik Bjarnov, was used to scan the frequency of the micro- wave radiation in any region of interest. For each single laser line the microwave frequency is scanned in the two ranges of 8-12.4 GHz and 12.4-18 GHz to identify the posi- tion of the transitions. Survey spectra were usually re- corded at 2 MHz intervals in 800 MHz sections. In order to record a transition, a shorter sweep - 400 MHz in most cases - is used with the transition in the center of the sweep. It is possible to scan over a particular region several times and average the spectra. In this investi— gation l to 3 sweeps were used depending on the strength of the observed transition. Normally, 5 readings at each frequency were averaged during each sweep. The output of the phase sensitive detector was filtered with time constants ranging from 3-30 ms and the time between read- ings was typically 2-5 time constants. The data were taken in 2 MHz steps; therefore, a 400 MHz sweep included 201 data points. A second computer program LFITZ, was used for least squares fitting of the line shape of the observed transitions 51 to a Lorentzian function in order to determine the fre- quency of the transitions. At the sample pressures used (ml torr) the lineshapes were some combination of Lorentz and Gauss functions, but the Lorentz fitting provided a rapid and accurate method of obtaining the frequencies of the transitions. 3.1.b. Radiation Sources It is obvious that in order to be able to carry out an infrared-microwave two-photon experiment, a microwave source and an infrared source are necessary. In the MSU two-photon spectrometer,microwave radiation is produced by one of two Varian Backward Wave Oscillators (BWO). The frequency ranges of the BWO's are 8-12.4 GHz and 12.4-18.0 GHz. The frequency-controlling helix voltage for the BWO is generated by a Kepco Model 2000 Operational Power Supply (OPS) which in turn is driven by a voltage from a D/A converter controlled by the computer. The microwave fre- quency is brought into the lock range of a phase-sensitive synchronizer by the OPS output. The synchronizer locks the frequency to a synthesized frequency also controlled by the computer. A flowing gas N20 or CO2 laser is used as the source of infrared radiation for the MSU spectrometer. The N20 or CO2 laser oscillates at single frequencies separated l by 1 or 2 cm' throughout the 900-1100 cm"1 range. The 52 laser used in this experiment consists of a 2.8 m long water cooled gain cell with an outer diameter of 2.8 cm. The laser tube is sealed with NaCl windows at the Brewster angle at each end; therefore, the output radiation is plane polarized with the electric field of the radiation parallel to the floor. The laser cavity is about 4.4 m long with a 150 lines/mm plane grating blazed at 10 pm at one end and a concave di- electric-coated, partially-transmitting germanium mirror of 10 m radius at the other end. Mirrors with reflectivity of 80% and 95% are available; which mirror is used depends on the strength of the chosen laser line. This mirror is mounted on a piezoelectric translator. To stabilize the laser the position of the cavity mirror is sinusoidally modulated (520 Hz), causing modulation of the laser cavity length and therefore modulation of laser frequency. The modulated laser output is detected by a Santa Barbara Re- search Center Hg-Cd-Te detector. The detected signal is preamplified and coherently detected by a Model 80-214 Lansing Lock-in Stabilizer. The output from the stabi- lizer is a discriminator signal which is zero when the laser is oscillating at the top of its gain profile 00 and non-zero otherwise. This discriminator signal is applied as a DC bias to the piezoelectric translator to stabilize the cavity length, and therefore maintains oscil— lation of the laser at V0“ 53 By manual adjustment of the angle of the grating a particular CO cn: N20 vibration-rotation line can be 2 chosen for laser operation. The active medium of the laser is a mixture of helium, nitrogen, and CO2 or N20 with a total pressure of less than 10 torr. The composition of the gas mixture is controlled by five needle valves. 3.1.c. Sample cell Two brass absorption cells were used for the infrared- microwave two-photon experiments. One cell, a P-band wave guide (0.790 cm x 1.580 cm, i.d.) one meter long was used for microwave frequencies between 12.4 and 18 GHz. The second cell is a 1.5 meter long, X band (1.016 cm x 2.286 cm, i.d.) cell for microwave frequencies in the 8-12.4 GHz range. Figure (3-3) is a tOp view of the absorption cell. Two holes, 6 mm diameter in X-band cell and 5 mm diameter in the P-band cell, are drilled in positions 1 and 2 in Figure (3-3). The holes are sealed by NaCl windows of about 4 mm thickness to allow transmission of the infrared radiation. At posi- tions 3 and 4 the sample cell is sealed by vacuum tight mica windows and "0" rings. The cell is attached to the vacuum line by a brass or glass tubing through the sample inlet. In all measurements the absorption cell was placed outside the laser cavity. 54 .ucmeflummxw couonm :03» m>mzonofl51pmumumcfl ca pom: Hamo COHDQHOmbm may mo Emummfla .35 Boson A .— .mum gnomes 55 3.1.d. Samples The four samples studied by infrared-microwave two- photon spectroscopy for this thesis include l4NH3, 15NH3, 12CH3F and 13CH3F. The sample of 14 NH3, which was purchased from Matheson and had a stated purity of 99.9%, was kindly provided by Professor J. L. Dye's research group at Michigan State University. The 15NH3 sample, obtained from Prochem was enriched in 15NH3 to 99.8%. A sample of 12CH3F was obtained from PCR Research Chemicals, Inc. and 13CH F with 90% enrichment 3 was purchased from Merck & Co., Inc. All of the samples were used as they were received without any further puri- fications. 3.2. Microwave Spectroscopy 3.2.a. Description of the Spectrometer and Experimental Procedure A Hewlett-Packard 8460A Molecular Rotational Resonance (MRR) Spectrometer was used to study the rotational spec- tra of 3-methyl-l-butene. This MRR spectrometer uses Stark modulation techniques with a modulation frequency Of 33.333 kHz. The microwave radiation is produced by a backward wave oscillator (BWO), and sent into the Stark cell by means of 56 the proper wave guide. The frequency of the BWO is con- trolled by a frequency synthesizer whose frequency can be stepped at rates and step sizes under operator control. The Stark cell is a 2.5 m long, X band cell. A metal plate inserted in the middle of the cell along its length serves as an electrode or septum. It is parallel to the broad dimension of the cell and insulated from the cell by Teflon strips. When square-wave 33.333 KHz modulation is applied to the septum, the absorption frequencies of the molecule are modulated by the Stark effect. A modulated absorption sig- nal is detected by a Si crystal diode. The detected signal is preamplified and sent to a phase—sensitive detector, where it is compared against the 33.333 KHz reference sig- nal. The output of the phase sensitive detector is filtered, in order to reduce the noise, and finally displayed on a strip chart recorder. As a result of phase-sensitive detection the zero field transition and its Stark compon- ents are recorded in opposite directions with respect to each other. The 8456A sweep control provides a continu- ous display of the microwave frequency to 1 KHz. The MRR spectrometer can provide markers on the chart paper at selected intervals between 1 KHz to 10 MHz. To measure the frequency of an MRR absorption line, the microwave frequency is swept up and down over the line at the same rate. The frequency from both traces is 57 extracted from the markers and an average is taken. To determine the Stark effect, the frequencies of the Stark components as a function of the electric field are measured. These frequencies are plotted against the square of the electric field. The observed slopes are used to determine the dipole moment components. Relative intensity measurements were made by recording pairs of transitions at constant crystal current under conditions of low microwave power. For many of the transi- tions the peak intensities were measured and compared at several pressures and Stark fields. All of the intensity measurements were made with the sample cell surrounded by dry ice. The temperature of the cell at several points was measured by taping a Pt resistance thermometer to the cell walls. The temperature was taken to be the mean of the measured values (204 K). 3.2.b. Sample The sample studied by microwave spectrosc0py for this thesis is 3-methyl-l-butene. This sample was obtained from Chemical Samples Company, Columbus, OH, and used without further purification. CHAPTER IV TWO-PHOTON SPECTROSCOPY OF AMMONIA AND METHYL FLUORIDE 4.1. Introduction The invention of the laser18 provided spectrosc0pists with a light source of extremely high intensity and extremely narrow bandwidth and has led to the development of the ex— tensive field of laser spectroscopy for characterizing the interaction of radiation with matter. Under low light level conditions, the absorption cross section is linear in the laser intensity. As the laser intensity increases, new phenomena appear that no longer vary linearly with intensity. The two-photon process, which is involved with the excitation of rotational, vibrational, and electronic states of mole- cules by the simultaneous interaction with two photons, is an example of a non-linear phenomenon. Two-photon spectrosc0py, using both absorption and fluorescence techniques, is now well established in the study of molecular electronic transitions where, because of the operation of different selection rules, information complementary to that from one—photon processes may be obtained.19”21 The theory of two-photon absorption was first 58 59 22 investigated by Goppert-Mayer. She was able to extend Dirac's theory of dispersion to cover the case where two photons were absorbed simultaneously in a single process. In 1970, Oka and Shimizu reported the observation of microwave double-photon absorption in the rotational spectra of CD3CN and PF3. The double-photon transitions involving quantum numbers J = 3 ++ 2 and J = 2 ++ 0 in CD3CN and J = 2 ++ 0 in PF3 were monitored by using a second microwave field as a probe.23 The first infrared-microwave two-photon transition 15NH3. In this technique a tunable microwave frequency is added to, or subtracted was observed in the v2 band of from, the fixed laser frequency by using the nonlinearity of the molecular transition process. The P(15), N20 laser line was employed to detect the ss()(4,4) two-photon tran- 15 24 NH3. of infrared spectroscopy being carried out with a resolu- sition in This paper demonstrated the possibility tion and accuracy of frequency measurement approaching that of microwave spectroscopy. After observation of infrared—microwave two-photon 26 saturation dips (Lamb dipszs) by Freund and Oka in 1972, a theoretical explanation of two-photon Lamb-dips was developed by Shimizu.27 The first example of systematic spectroscopy carried out by the infrared—microwave two-photon technique was reported by Freund and Oka in their study of the 0 band 2 60 of ammonia with N20 and normal CO2 lasers.28 An extension 13 CO2 and 2 lasers were used with the cell placed within the 29,30 Of this work was reported by H. Jones in which C180 laser cavity. In addition, a number of two-photon transitions have been observed in fundamental and hot- bands of the linear molecules fluoroacetylene (HCCF) and cyanogen fluoride (FCN) by H. Jones.31 Recently, Doppler—free infrared-infrared two-photon band of NH absorptions have been observed in the v by 2 3 using a fixed-frequency CO2 laser and a diode laser. Beams from the two lasers were passed in Opposite directions through the sample cell and the transmitted intensity of the diode laser beam was monitored as a function of its frequency.32 In 1979 a computer—controlled infrared-microwave two- photon spectrometer was assembled at Michigan State Uni- versity. Ammonia was chosen as the first molecule to be studied with this spectrometer. It was chosen because of its large transition dipole moment, and because a number of transitions in the v2 vibration-rotation band are known to be near CO2 laser frequencies.33-35 Furthermore, since this molecule is of considerable astrophysical sig- 36’37 it is important to have as precise informa- nificance, tion as possible for its energy level structure. Methyl fluoride was selected as the second molecule to be studied in order to compare the spectra of symmetric 61 top molecules with (NH3) and without (CH3F) inversion. The theory of two-photon spectroscopy was presented in Chapter II and the two-photon spectrometer used in this study was described in Chapter III. 15 14 NH3 and NH3 4.2. The 0 Bands of 2 4.2.a. Introduction Numerous investigations of the microwave spectrum of ammonia, a pyramidal molecule with C3V symmetry and the rotational structure of a symmetric tOp, have been pub- lished since the early stages of microwave spectroscopy.38 The microwave spectrum of ammonia in the centimeter region is the result of transitions between the members of inver- 39 sion doublets. These doublets result from the fact that the potential barrier to the motion in which the NH3 pyra- rmniis inverted is relatively low. At the present time, the most accurate ground state inversion frequencies have been obtained in beam maser 40-42 studies by Kukolich and co-workers, but these are limited to a small number Of ammonia lines. Poynter and Kakar have measured the frequencies of 119 inversion transi- 14 tions of NH3 in the ground state to high accuracy by conventional microwave spectroscopy. The accuracy of 43 their measurements is reported to be 10.005 MHz. The frequencies of 15 high J inversion transitions have been 62 44 measured by Sinha and Smith with an accuracy of :0.03 MHz by thermally populating higher rotational levels. Ac- curate measurements (:1 MHz) of the frequencies of the in- f 14 version transitions O NH3 in the 02 excited state have 45 been carried out by Belov et 31. The inversion spectrum of NH3 in the ‘Q = 1 state lies in the far-infrared or sub— millimeter region (W35 cm-l) 15 . The most recent published inversion frequencies of NH3 in the ground vibrational 46 state are measurements made by Sasada. In this work frequencies Of 115 microwave lines were measured to an ac- curacy of £0.01 MHz. The analysis of the inversion spectrum of ammonia is based on numerical fits of the frequencies to selected functional forms. In one approach a power series in J(J+l), 2 the square Of the total angular momentum, and K , the square of the projection of the angular momentum on the symmetry axis of the molecule, has been used.47 48 In a second approach an exponential model due to Costain has been used. More recently, Young and Young used a Padé approximation to fit Poynter and Kakar's measurements with a standard deviation which was close to the accuracy of their experimental measure- ments.49 The vibration-rotation spectrum of NH3 has also been studied since the early days of infrared spectrOSCOpy.50 Ammonia is predicted to have 4 fundamental frequencies, two totally symmetric (al symmetry in the C3V group) and two doubly degenerate modes (e symmetry). all of 63 them infrared active. The strong V1 and 02 bands of al symmetry are near 3 and 10 pm, respectively. The very weak v3 band is close to V1 in frequency and the very strong 04 band is near 6 um. In 1941, Sheng gt 31. published the results of a study of the 10 um.band, in which a grating spectrometer was 51 used. In 1958, Benedict 33,31. studied the (v2 + 03) perpendicular and (01 + 02) parallel combination bands of ammonia which lie between 2.15 and 2.5 pm. They also gave a general analysis of the ammonia vibration-rotation spec- tra.52 Later, in a high resolution study of NH 1 and ND 3 3 with an effective spectral resolution of 0.1 to 0.2 cm-l, they determined the over the range of 1750 to 7100 cm- rotational constants and the molecular dimensions Of am- . ° 53 monia to be r = 1.0124 A and h = 0.3816 A (r and h e e e e are the equilibrium NH bond distance and the distance Of the nitrogen atom from the plane Of the three hydrogens, respectively). Measurements and analysis of the v band Of ammonia 1 2 under a resolution of 0.1 cm- were reported by Mould and his co-workers in 1959. They derived the molecular constants 54 of ammonia in the ground and v2 states. In the same year an independent investigation of ammonia in the regions 1 and 1440-1840 cm-1 55 510-1280 cm- with a vacuum grating spectrometer was carried out. The accuracy of the fre- quencies of the transitions is reported to be :0.03 cm-1 64 1 for the 10-16 pm region and :0.05 cm- in the 6 pm region. Moreover, a theoretical discussion which involves the Cor- iolis interaction between the 02 and v4 bands and Q—type doubling in the v4 band was presented. Infrared laser Stark spectra of the v2 bands<3fl4NH3 15 34,35 and NH3 were published by Shimizu in 1970. In these experiments CO2 and N20 lasers were employed as infrared sources, and quite a few coincidences between the ammonia lines and CD2 or N20 laser lines were revealed. Since then, the ammonia spectrum has proved to be a favorite example of infrared transitions for the study of novel methods of laser spectroscopy. 15 Analysis of the 02 band of NH3 was carried out by Shimizu and Shimizu.56 In this study the spectrum of 15NH3 was obtained by using an infrared vacuum spectrom- eter. About 170 absorption lines in the frequency range front 1 842-1154 cm- were Observed. They believe that the error in the frequency determination Should be less than 0.02 cm-l. Rotational constants and the v band origin were 2 Obtained from the least squares analysis of the observed 15 NH3. In 1976 Freund and Oka published a paper frequencies of 28 in which they presented the result of a high resolution study of the v2 band of ammonia by two-photon spectroscopy, where the sources of radiation were a tunable microwave source and a fixed frequency CO2 or N20 laser. Both straightforward and Lamb-dip two-photon techniques were used. Thirty-nine 65 14 NH 15 and 11 NH3 transitions were assigned by straight- 3 forward two-photon technique and the accuracy of their fre- quency measurements was estimated to be i30 MHz. For Lamb- dip measurements to be carried out, the infrared transition has to be saturated. Therefore, a high power infrared beam is required and in order to satisfy this condition, Freund and Oka placed the sample cell inside the laser cavity. A total of 25 l4NH3 transitions and 6 transitions in 15NH3 were Observed with this method and the accuracy of these data is reported to be :3 MHz and :6 MHz for measurements done with CO2 or N20 lasers, respectively. Additional measurements of the v2 band of ammonia by the infrared-microwave two-photon method were made by H. Jones. In these experiments the frequencies of 36 transitions of 14NH329 and 11 transitions Of 15 330 13 18 isotOpic CO2 and C 02 laser lines. By combining these results with those of previous two-photon measurements and NH were determined with laser Stark data, the spectroscopic constants of this band 15 were calculated for NH 3. By employing an infrared heterodyne technique, precis— ion measurements Of NH3 spectral lines near 11 um have been made.57’58 Although this technique is very accurate, it is limited by the frequency-tuning range of the laser local Oscillator and the intermediate—frequency (IF) bandwidth of the optical mixer. Diode laser measurements of NH3 absorption lines in 66 the 9.6 and 10.6 um region have been performed by different investigators.59—62 In 1980 there appeared two papers concerned with the high resolution infrared study of ammonia. One paper is concerned with the laser Stark spectroscopy of the 02 hand 33 of ammonia. In this study the molecular constants for the ground and v vibrational states are Obtained by apply- 2 ing a simple Hamiltonian. The dipole moments in the two states were determined for both l4NH3 and 15NH3. The second paper compiles the results of experiments carried out by a number of investigators. This paper63 is unique in that it reports for the first time the measured fre- quencies of the pure inversion and rotation-inversion transitions in the 02 state of 14 NH3 in 700—1100 GHz region. A submillimeter wave spectrometer was used for these measure- ments. A vibration-inversion—rotation Hamiltonian, which 64 was used for parameterization of was developed previously the energy levels of ammonia. In this approach, the Coriolis interaction between the v2 and 04 states and 2- type doubling in the v4 band were considered. Rotational and centrifugal distortion constants of the molecule in the 202, v4, 302, and (02 + 04) states as well as in the ground and v2 states were derived in this study. In an extension Of this work experimental data, including Fourier transform infrared Spectra of the inversion—rotation 1 transitions between 40-300 cm_ , submillimeter wave spectra of the inversion and inversion-rotation transitions in the 67 v2 state Of 14NH3, and a few AK = i3 "perturbation-allowed" transition frequencies are combined with the microwave ground-state transition frequencies and 02 infrared-micro- wave two-photon frequencies and are simultaneously analyzed. The ground state and 02 state molecular parameters are determined in this analysis.65 The IR spectrum of 15NH3 between 510-3040 cm-1 with a Fourier transform infrared spectrometer has been re- recorded ported.66 The 02, 202, 302, v4 and v2 + 04 bands were measured and analyzed on the basis of the vibration-rota- tion Hamiltonian of Ref. 64-d. The molecular parameters for 66 The rotation inversion spectrum in the ground state of 15NH 1 the v2 = 1,2,3 states were derived in this study. 3 has been Obtained between 38 and 280 cm- by Fourier trans- form spectroscopy.67 68 Recently, Sattler gt gt. have made diode laser heterodyne measurements of the frequencies of a number of transitions in the 9 and 10 um region of NH3 with an ac- curacy of :2 x 10-4 cm-l. Molecular absorption frequencies were determined by heterodyning the emission of the lead- salt diode laser and a reference CO2 laser and using a spectrum analyzer to measure the beat frequency. We have reinvestigated the two-photon spectrum of 14 15 NH3 and NH3 by using normal CO2 and N20 lasers and microwave oscillators in the 8-18 GHz region. A least- squares fit of all of the two-photon frequencies measured in this work, all of the previous two-photon frequencies,28-30 1’46 68 and all Of the diode laser heterodyne measurements68 has been carried out. The microwave, millimeter wave, and sub- millimeter wave values Of the inversion frequencies were as— sumed in this analysis. For this fitting the simple Hamil- tonian that was used by Shimoda gt g1.33 was used. A number of simple tests of the accuracy and consistency of the two- photon results have been performed. Rotational and centrifu- gal distortion constants Of the ground states and v2 states have been determined and are compared with previous results. 4.2.b. 14NH 3 The study of the 02 band of NH3 by two-photon Spectros— copy was begun by calculating approximate frequencies of the possible two-photon transitions. The ground and 02 vibrational levels of 14NH3 are shown in Figure (4-1). Inversion splitting causes a separation between rotational levels (J,K) of approximately 0.7934 cm.1 in the ground 70 1 71 state and 35.6881 cm- in the v2=l state. The rota- tional energy of the (J,K) state of NH3 in a particular vibrational state is expressed in the form, BJ(J+l)-(B-C)K2-D J2(J+1)2 (J,K)/h J Erot 2_ 4 3 3 DJKJ(J+1)K DKK + HJJ (J+1) 2 2 2 4 6 + HJJKJ (J+1) K + HJKKJ(J+1)K + HKK (4-1) A\ \V Figure 4-1. 69 <'. ll H AL \S I AV 0 T"..-I-..-.._Jr_ ..- *7 A\ i_\ M _J_-_-- (b) Energy level diagrams involved in two-photon transitions of NH3. 70 where B and C are rotational constants, and the D's and H's are centrifugal distortion constants. Equation (4-1) is an expression for the rotational energy of a symmetric tOp molecule. In this equation, the B, C, D's and H's are mean values of the constants for the symmetric (s) and antisymmetric (a) inversion levels. The two-photon selection rules are the same as vibra- tion—rotation selection rules (i.e., for a parallel band, AJ = 0,:1, AK=O) with one exception: two-photon transitions are allowed only between s++s or a+>a levels while vibra- tional transitions happen for s4+a levels. In order to calculate the frequencies of the a+a and s+s two-photon transitions, the effect of the inversion Splitting must be added to Equation (4-1). Therefore, E (v,J,K) = E (J,K) : l/2AE (v,J,K) . (4-2) INV—rot rot INV Here + and - signs refer to the a and 5 states, respectively, and AE (v,J,K) is the energy difference between the mem- INV bers Of the (J,K) inversion doublet in the vibrational state v. By using the AK=0 selection rule and assuming (J",K) and (J',K) as the lower and upper state quantum numbers, respectively, the following expression is obtained for the frequency of an s+s or a+a two-photon transition. 71 0TP = 00 + B'J'(J'+l) - [(B'—C') - (B"-c")]K2-D3J'2(J'+1)2 U I3 l 3 y '2 | 2 2 JJ (J +1) +HJJKJ (J +1) K D'JKJ'(J'+1)K2—(Dfi-D§)K4+H + ' l l 4 I_ n 6 HJKKJ (J +1)K + (HK HK)K H II II II II 2 ll 2 H I! II 2 B J (J +l)+DJJ (J +1) +DJKJ (J +1)K - H3Jn3(Jn+l)3_Hu J"2(J"+l)2K2-H" JH(JH+1)K4 JJK JKK 1/2[AE (1,J',K) - AE (0,J",K)] (4—3) INV H INV Again the + or - sign is used for an a+a (n: an s+s transi— tion, respectively. In this equation 00 is the band origin, which is the energy difference between the J" = K" = O and J' = K' = 0 states in the absence of inversion splitting. In Equation (4-3) all Of the rotational constants with a prime belong to the upper state and those with a double prime belong to the lower state. To calculate inversion frequencies in the ground state, AEINV(0,J",K"), the 21- parameter equation and parameters given by Schnable gt gl.70 were used. In order to reproduce the required inversion frequencies in the 02 state, AEINV(1,J',K'), a 21-term polynomial in the quantum numbers J and K was used (Table 4-1). These parameters were Obtained here by a least- squares fit of the inversion frequencies in the 02 state 72 , the In- INV 14 version Frequency in the v2 = 1 State of NH3. The 21 parameters were obtained by fitting the Table 4-1. Polynomial Representation of v data from Reference 63. . _ 2 2 2 2 VINV _ v0 + A1J(J+l) + A2 K + A3J (J+1) + A4J(J+1)K + A K4 + A J3(J+l)3 + A J 2(J+1)2K2 + A J(J+1)K4 + A K6 5 6 7 8 9 4 4 3 3 2K 4 6 + AlOJ (J+1) + AllJ (J+1) K2 + A12 J2 (J+1) + A13J(J+1JI< 8 5 4 2 3K 4 + A14K + AlSJ (J+1)5 + A16J4 (J+1) K + A17J3 (J+1) 2K 6 8 10 + A18 32 (J+1) + A19J(J+1)K + AZOK Parameter Value Parameter Value V0 = 0.106990428-107 A1 = -o.540014712-1o4 All = 0.109025839-10'2 A2 = 0.754257405-104 A12 = -o.27152418o-1o'2 A3 = 0.128582772-102 A13 = 0.284822774-10"2 A4 = -o.35018524o-1o2 Al4 = -o.106401210-1o‘2 A5 = 0.236303331-102 A15 = 0.105158889-10’5 A6 = -o.997400177-1o'2 A16 = -o.873633.923-1o'5 A7 = 0.24922715240’l Al7 = 0.277375788-10'4 A8 = -o.104487519-1o‘l A18 = «0.424652581-10’4 A9 = -o.698633839-1o'2 A19 = 0.315335845-10'4 Alo= -o.153245109-1o"3 A20 = -o.914279839-1o’5 73 of 14NH3 obtained by Beloxrgtht. spectroscopy. The K = 3 transitions were excluded from 63 by submillimeter wave the fit since they are strongly affected by AK = :3 per- turbations arising from centrifugal distortion effects.71 The rotational parameters obtained by ShimOda gt gt.33 (Table (4-2) Column II) for the upper (m') and lower (m") states* and the calculated inversion frequencies in the ground and v2 excited states were used as input data for a computer program that uses Equation (4-3) to cal- culate the two-photon frequencies and compares them to CO2 and N20 laser frequencies. The output of this program is a list of the differences between the frequencies of the two-photon transitions and the nearest laser frequencies. v = VTP - The MSU spectrometer can detect those m VLaser' vm frequencies which are in the microwave range of 8 to 18 GHz, provided that the transition has enough intensity. The frequencies of CO2 laser lines were computed by using the rotational parameters of Freed gt gt.72. The N20 laser frequencies were derived with the parameters pub- lished by Whitford gt £1.73 Figure (4-1) shows examples of the energy levels of ammonia and the possible two-photon transitions. The * The symbols m" and m' are used to designate hypothetical levels halfway between the members of an inversion doublet in the ground and first excited states, respectively. 74 Table 4-2. Ground State and v2 Excited State Molecular Parameters of 14NH3 (cm-l).a This Workb Ref. 33C Parameter I II v0 949.88112(8) 949.88093(18) B' 9.9801426(432) 9.980194(68) AC-AB -o.1404435(122) -o.140441(24) D5 9.1479(126)x10'4 9.1757(280)x10"4 05K -1.82866(450)x10’3 -1.83398(620)x10‘3 Dk-Dg 2.2168(30)x10"4 2.2026(88)x10’4 H& 2.7528(1230)x10‘7 3.1131(360)x10'7 H&JK -1.1205(456)x10-6 -1.2145(1100)x10‘6 H5KK 1.50067(11152)x10"6 1.5759(1240)x10"6 HR-Hfi --2.o328(254)x10'7 -1.785(102)x10'7 B" 9.9441342(420) 9.944190(66) D3 8.4042(122)x10"4 8.4409(260)x10'4 03K -1.55129(448)x10‘3 -1.55886(600)x10'3 83 2.3829(1210)x10‘7 2.8665(346)x10'7 H3JK -8.5268(4518)x10‘7 -9.925<1080)x10‘7 HSKK 1.06253(11096)x10'6 1.1953(1220)x10"6 S.D. (MHz) 5.85 11.1 aThe quoted uncertainties for the constants are two times the standard deviations of the least squares fit, given in units of the last digit. b varied. From fit of frequencies in Appendix B. cReference 33. All 16 parameters 75 l4NH3 sample was used at a pressure ranging from 0.2 to 1.3 torr. By using 18 CO2 and 11 N20 laser lines a total of 63 spectral lines were observed; 44 transitions were assigned and the assignments were confirmed by comparison 28,29 with the results of other two-photon experiments and the known infrared frequencies of 14NH3.63 As a result of generally good line shapes, the MSU spectrometer can measure the microwave frequency Vm with an uncertainty of about i1 MHz. However, because of the uncertainty of the laser frequency arising from our method of stabilization, we estimate the accuracy of our measure- ments to be 35 MHz. Figures(4-2) and (4-3) are photographs of oscillosc0pe displays of the stored data files for two typical two-photon transitions. In both cases the width of the sweep is 400 MHz, corresponding to 200 data points. The assigned transitions are listed in Tables (4-3), and (4-4) for 5+3 and a+a transitions, respectively. The second column of these tables gives the laser line involved, where 10 and 9 indicate a C0 laser line in the 10 um and 2 9 um region, respectively, and N indicates an N 0 laser 2 line. The first column of the tables shows the assignment in the form V(J",K) with V = P,Q and R referring to AJ = -l,0, and +1 transitions, respectively. The frequency of the microwave radiation at which the transition occurs is given in the third column with a sign + or - which shows whether the two photons are added together or whether Figure 4-2. 76 IRrMW two-photon absorption signal. The SQ(5,5) two-photon transition of 14NH3 ob- served by using the C02 P(36) laser line (vg = 929.017435 cm'l) and microwave frequency of Vm = 16651 MHz. The offset frequency Av = 7.882 GHz and the sample pressure was 0.6 torr. The width of the sweep is 400 MHz. Figure 4-3. 77 IR-MW two-photon absorption signal. The aQ(l,1) two-photon transition of 14NH3 ob- served by using the C02 R(8) line (Vg = 967.707233 cm‘ ) and microwave frequency of vm = 14987 MHz. The sample pressure was 1.25 torr, and the offset frequency Av = 8.707 GHz. The width of the sweep is 400 MHz. 78 Table 4-3. Comparison of observed and calculated frequencies of s-s two-photon transitions in the \5 band of14 N83. . . . a . b c d Tran51tion Laser 11ne Microwave Two-Photon Av (MHZ) (cm-l) (GHZ) Q( 1,1) 10P(32) -16251. 932.41833(25) 39.95 Q( 1,1) NP( 8) 10089. 932.41926(118) 13.61 Q( 3,3) NP( 8) -15881. 931.55298(91) 39.75 Q( 3,3) NP( 9) 9956. 931.55333(127) 13.91 Q( 3,3) 10P(34) 16548. 931.55341(135) 7.32 Q( 4,3) 10P(32) -12892. 932.53038(-2) 35.58 Q( 4,3) NP( 8) 13424. 932.53049(l0) 9.26 Q( 4,4) NP( 9) -15517. 930.70362(29) 39.66 Q( 4,4) 10P(34) -8935. 930.70341(8) 33.07 Q( 4,4) NP(10) 10402. 930.70335(2) 13.74 Q( 5,5) 10P(36) 16651. 929.57286(4) 7.88 0( 6,4) NP( 7) 11848. 933.33605(2) 9.15 Q( 6,6) NP(12) -13718. 928.15903(22) 38.77 Q( 7,3) 10P(28) -8956. 936.50501(-200) 26.97 Q( 7,6) NP(11) 13147. 929.92670(16) 9.78 Q( 7,6) NP(10) -12883. 929.92663(9) 35.81 Q( 7,7) 10P(38) -16512. 926.45753(20) 42.23 Q( 7,7) NP(15) 14263. 926.45765(32) 11.45 Q( 8,5) 10P(30) -13222. 934.45346(20) 32.03 Q( 8,6) NP( 9) 17753. 931.81338(25) 2.97 Q( 8,7) 10P(36) -15005. 928.51693(—11) 38.24 Q( 8,8) 10P(40) -15290. 924.46395(51) 41.89 Q( 9,4) 10P(26) -14l45. 938.21643(-27) 29.67 Q( 9,5) 10P(28) -16067. 936.26781(22) 32.87 Q( 9,7) NP(10) 9174. 930.66240(32) 11.56 Q(1l,9) 10P(38) 17360. 927.58740(1) 3.71 R( 4,2) 9P(34) 16543. 1034.03982(3) 5.16 R( 6,1) 9R(16) 13452. 1076.43654(-2) 4.94 R( 6,5) 9R(10) -16019. 1071.34942(8) 38.75 a N, 9, l0 refer to N20, 9 um band of C02, or 10 um band of b cThe numbers in parentheses are observed frequencies in multiples of 0.00001 cm' . C02 laser, respectively. Microwave frequency in MHz. A minus sign indicates that the microwave frequency is subtracted from the laser frequency to obtain the two-photon frequency. 1 minus calculated The parameters for the calculation are in the first column of Table 4-2. dAbsolute value of the difference between the laser frequency and the frequency of the corresponding one-photon allowed transition. 79 Table 4-4. Comparison of observed and calculated frequencies of . . . 4 a-a two-photon tran51tions 1n the 'wzband of l NH3. Transition Laser line? Microwaveb Two--PhotonC Avd (MHZ) (cm-1) (GHZ) P( 2,1) 10P(38) 13581. 927.46135(-7) 36.68 Q( 1,1) 10R( 8) -14987. 967.20731(-5) 8.71 Q( 3,2) 10R( 6) 11834. 966.64509(-11) 34.67 Q( 3,3) 10R( 6) 8987. 966.55013(1l7) 32.86 Q( 5,2) 10R( 6) -13677. 965.79414(28) 6.69 Q( 5,3) 10R( 6) -l740l. 965.66992(-101) 3.88 Q( 5,5) 10R( 4) 16900. 965.33270(-l6) 41.43 Q( 7,4) 10R( 4) -12905. 964.33851(9) 6.31 Q( 9,5) 10R( 2) -16640. 962.70808(-59) 0.16 R( 3,3) 9P(22) 16688. 1045.57832(-23) 40.56 R( 5,1) 9R(28) 14481. 1083.96181(-13) 34.32 R( 5,2) 9R(28) 13537. 1083.93033(19) 33.91 R( 5,3) 9R(28) 12122. 1083.88311(15) 33.41 R( 5,4) 9R(28) 10473. l083.82812(0) 33.13 R( 5,5) 9R(28) 9060. 1083.78098(7) 33.59 aN, 9, 10 refer to N O, 9 Um band of C02, or 10 pm band of C02 laser, respectively. Microwave frequency in MHz. A minus sign indicates that the microwave frequency is subtracted from the laser frequency to obtain the two-photon frequency. c . . The numbers in parentheses are observed minus calculated frequencies in multiples of 0.00001 cm'l . The parameters for the calculation are in the first column of Table 4-2. Absolute value of the difference between the laser frequency and the frequency of the corresponding one-photon allowed transition. 80 the microwave frequency is subtracted from the laser fre- quency to obtain the two-photon frequency. The fourth column gives the frequency of the two-photon transition (v = vgtvm), whereas the last column lists the offset TP frequency, which is the difference betweentflmalaser fre- quency and the frequency of the nearest one-photon transi- tion. The intensity of a two-photon transition is pro- portional to the inverse of the square of Av, so the smaller Av, the stronger the transition. We have been able to ob- serve transitions with a Av as large as W43 GHz. In order to analyze the data and derive a set of mol- ecular constants, Equation (4-3) has been rearranged as follows: _ I _ N - vRV — vO + E(J ,K) E(J ,K) (4 4) where vRV is a hypothetical inversion-free rotation- vibration frequency defined as: (l,J',K) - AE (0,J",K)] (4—5) v = v + %[AE INV RV TP INV with - and + for a + a and s + s two-photon transitions, respectively. The E(J',K) and E(J",K) in Equation (4-4) represent the rotational terms in the upper v2 and ground state, respectively, as defined in Equation (4-1). For this purpose, the AEIN 's were taken from the experimental V 81 71 inversion frequencies of Belov gt gt. for the v2 state, and those of Poynter and Kakar43 for the ground state (Ap- pendix A). The vTP values were obtained from the laser frequencies and the measured microwave frequencies. In this way values of v were determined experimentally. RV A simultaneous least squares fit of the v V derived R from our two-photon data, from those reported by H. Jones 28 68 29 and Freund and Oka, and from diode laser and hetero- dyne measurements58 was carried out. Since the transi- tions reported in Reference 68 and 58, are a + s or s + a one-photon IR transitions, the v corresponding to RV Equation (4-5) was v (1,J',K) + AEIN RV = VSat i ;5[AEINV v (0,J",K)]. For these transitions the + and - signs are for transitions from a+s or s+a levels, respectively. All of the frequencies in the fitting were given weights that are inversely prOportional to the square of their estimated accuracy. The set of linear equations used in the least squares fitting are 16 8v 5\) = E (5')?) 5X1 . (4-6) 1—1 1 ‘where (3V = (Vexp ’ vcalc) I 6X1 = (Xi)eXp " (xi)calc’ and vexp and vcalc are the experimental and calculated frequencies of ro—vibrational transitions. The values of v were obtained by using the parameters of Shimoda calc 2e: 21-33 82 The parameters fitted are: 1 v0 Band origin X = H' 9 JKK x2 = B' x10 = (Hk-Hfi) x3 = [(c'-C")-(B'-B")] xll = B" x4 = D3 X12 = D3 x5 = DJK X13 = DJK x6 = (bk-Di) xl4 = H3 X7 = HJ X15 = HJJK H8 = HJJK x16 = HJKK (4’7) Each (ggL) is the partial derivative of the transition frequency th with respect to the i parameter. These derivatives are: 3v __ 8v ._ . . 4 (3X ) ‘ l (§§_ — J (J +1)K 1 9 8v 3v 6 (———) = J'(J'+l) ( ) = K 8X2 ax10 8v 2 3v (___) = K ( ) = -J"(J"+1) ax3 3x11 (ggL) = -J'2(J'+1)2 (5%X—) = J"2(J"+l)2 4 12 (53:) = -J'(J'+1)K2 (33V ) = J"(J"+1)K2 5 13 (ggL) = —K4 (aiv ) = -J"3(J"+1>3 6 14 83 8v 3 3 8v 2 2 (———) = J' (J'+l) ( ) = -J"(J"+l) K 3X7 3X15 3\’ _ .2 . 2 2 av _ u n 4 (§§—) — J (J +1) K (ax ) — -J (J +1)K 8 16 (4-8) No inversion transitions involving levels with K = 0 are observed, because for K = 0, levels with even values of J are missing in the lower component of inversion doublets (s) and levels with odd values ofJ are missing in the upper component of inversion doublets (a). The missing levels are a result of the effect of nuclear spin statistics. There- fore, the inversion frequencies of the (J,K) = (5,0) and (6,0) levels in the ground state and the (6,0) and (7,0) levels in v2 = 1 state were calculated from the parameters 70 and Belov gt gt.,7l respectively. given by Schnabel gt gt. These values appear in Appendix A, where all of the in- version frequencies used in our fit are given. In order to obtain a set of parameters which are able to reproduce the experimental two-photon transitions, a number of fittings were performed. In the first fitting all of the transitions mentioned above, including transi- tions involving levels with K = 3, were included in the fit; 16 rotational parameters in the ground and v2 = 1 states were varied. With this fitting, rather large residuals for transitions with K = 3 were obtained and the deviation for 84 these transitions increased with J. The fit was repeated after eliminating the K = 3 transitions; in this case a better standard deviation 26.03 MHz was obtained. In the final fit, in addition to the K = 3 transitions, 10 transitions that showed deviations larger than 34 MHz were eliminated. For this fitting, the standard devia- tion dropped to 5.85 MHz, Table 4-2, Column I. The final parameters obtained here are compared to those from the infrared laser Stark study of Shimoda gt gt. in Table 4-2. A large deviation appears in the value of the band origin, v0, which is N6 MHz higher than the Shimoda result when both ground state and v2 excited state parameters are allowed to vary.. The constants H' and H" can not be determined inde- ci' C", I D" K K' K' K pendently because of the AK = 0 selection rule. A list of all of the frequencies used in the fitting is given in Appendix B. 4.2.c. Some Checks of the Accuracy of the Datg i) Inversion Frequencies in the v2 = 1 State of £3§§3_:_It is possible to derive the frequency between inversion doublets in the v2 state of ammonia from the frequencies of two-photon transitions by using the ex- pression, 85 VINV(JIK) = Vaa(JIK) "' vSS(J'K) + V0 (JIK) 0 (4'9) Here, v is the inversion frequency in v2 state, v INV and vSS are two-photon transitions and v0 is the inversion aa frequency in the ground state. The vaa and vSS values in Tables (4-4) and (4-3) may be combined with the ground state inversion frequencies given by Poynter and Kakar (Appendix A) to determine "experimental" v2 inversion frequencies. The results are given in Table (4-5) and are compared to the submillimeter wave measurements of Belov gt gt.71. ii) Difference Between Frquencies of the Laser ttggg - In the cases where the same transition has been observed with two different laser lines, the difference in frequency of the two laser lines can be obtained. As an example, Table (4-3) shows that sQ(7,7) has been id- entified with two different laser lines, P(lS) of the N O 2 laser and P(38) of the CO2 laser in the loimxregion. The measured microwave frequencies are 14263 MHz and -16512 MHz, respectively. Therefore, vENP(15)] + 14263 = v[10P(38)] - 16512 (4-10) 86 .xbos mflsb cg bmcgmuno mmoau can as mocmbmbmm no Ax.bv>zm N > cwm3uwn mwocwummmwp msu mum mammcucmumm :fl nudges: mcu “oumum an > map ca wocmsomum :onuo>:fl vmumaouamo o .mv mucmummwm Eouw madam ocsonm 039 new hocmsmmum coflmuw>cHo .M can b mEmm may nufiz mao>ma HOm m+m can m+m mcofluwwcmuu couonmlosu may mo mmflocoscmum Um>umm£o on» mum mm» paw mm>o .uwansoo GOHmuo>cw on» no madness Educmsv Hmcofiumuou map mum M can on lab.mvmo.bmmbmoa www.mmmqm om.mmmsbmhm m~.bemmmmmm Am.mv mm.svm.mmomsoa = bs.mmm>~mnm = Am.mv Am.mvm.thMhoa .. om.ommhmmhm .. Am.mv Am.mve.m¢omnoa ma.onmmm mb.mbmsmmn~ mm.mevbsmmm Am.mv xem.mvm.avbbboa .. m.mmammm>m .. AH.HV Amm.bmvo.eabbboa bae.vmbmm ma.bmmmmmnm ab.mvabmmmm AH.HC 631.222? 631.222? b8» b6...» 6:12 .Anmzv wumum HHN> map CH mmz mo mowocmskum cowmum>cH .mlw manna vH 87 or v[10P(38)] — v[NP(15)] = 16512 + 14263 = 30775 MHz. The calculated difference from References72 and73 is 30771.88 MHz. The deviation between the two methods of obtaining the frequency difference is 3.09 MHz. Table (4-6) contains a list of the transitions that have been observed with more than one laser line. The third column of this table shows the frequency difference Av of the two laser lines identified in the second column and the last column shows observed minus calculated values. All of the differences are within the estimated error with the exception of [10P(32)-NP(8)] obtained from sQ(l,1) transitions. This is probably due to a less accurate value of the sQ(1,1) frequency determined with the N20 laser, as the same discrepancy is seen in the residual columns of Table (4-3) and Table (4-5). 88 Table 4-6. Comparison of Differences in Laser Frequencies Obtained from Duplicate Measurements of 14NH3 Two-Photon Frequencies With Differences Calcu- lated from Vibration—Rotation Constants of 72 73 CO2 or N20. c d b Avobs. Avobs. Avcal. Transitiona Avl-v2 (MHz) (MHz) sQ(1,1) 10P(32)-NP(8) 26340 27.12 sQ(3,3) NP(9)-10P(34) 6592. 2.9 NP(8)-NP(9) 25837. 10.02 NP(8)-10P(34) 32429. 12.92 sQ(4,3) 10P(32)-NP(8) 26316. 3.12 sQ(4,4) 10P(34)-NP(10) 19337. —1.51 NP(9)-10P(34) 6582. —7.1 NP(9)-NP(10) 25919. -8.61 sQ(7,6) NP(10)-NP(11) 26030 1.88 sQ(7,7) 10P(38)-NP(15) 30775 3.09 aThe transition observed with more than one laser line. bv1 and v2 are the two laser lines whose frequency difference is determined by two-photon spectroscopy. c . AvobS-vzl-vgz-vmz-vml. vm represent the microwave frequency satisfying the two-photon condition. dAvcalc is the calculated frequency difference from Ref. 72 and 73 for CO2 and N20 laser lines, respectively. 89 4.2.d. 15NH3 The initial calculation of the frequencies of the two- 15 photon transitions in the v band of NH3 was carried out 2 by combination of the vibration-rotation constants re- ported in Reference 33, and those in Reference 30. 15NH3, 28 co and 14 N20 laser lines were employed. A total of 101 In the present study of the v2 band of 2 transitions were detected and 60 of them were assigned to two-photon transitions of 15NH Most of the transitions 3. were recorded with a sample pressure of approximately 1 torr. The assignment of the transitions was based both on the results of the initial calculation and on previous two- 28,30 photon results. Tables (4-7) and (4-8) list the 15 assigned NH3 transitions measured in this work. No direct measurements of the inversion frequencies of 15NH3 in the v = l vibrational state have been reported so far. 2 Therefore, it is not possible to obtain the mean values of the vibration-rotation parameters in the v2 state directly from our data, as was done for 14NH3. The present two- photon frequencies were combined with the previous two- 28'30, with 7 transitions measured photon measurements in a laser Stark experiment, transitions measured by a diode laser heterodyne technique,68 and with the pre- viously measured microwave inversion frequencies in the ground state.46 These data were used to obtain experi- mental frequencies from the hypothetical level m" (midway 90 Table 4-7. Comparison of observed and calculated frequencies of s-s two-photon transitions in the\b band of ISNHB. Transition Laser linea Microwaveb Two-PhgtonC AJi (MHZ) (cm ) (GHZ) Q( 2, 1) NP(12) 9866. 928.94570(-9) 12.18 Q( 2, 1) NP(11) -16257. 928.94589(8) 38.30 0( 2, 2) NP(12) -l37l9. 928.15899(2) 36.37 Q( 3, 2) NP(11) -17388. 928.90816(7) 39.17 Q( 3, 2) NP(12) 8739. 928.90811(1) 13.05 Q( 3, 3) 10P(38) 17669. 927.59770(—22) 5.12 Q( 4, 2) NP(ll) 11605. 929.87527(-8) 9.08 Q( 4, 2) NP(10) -14429. 929.87507(-27) 35.11 Q( 4, 3) 10P(36) -12547. 928.59891(9) 34.19 Q( 5, 3) NP(11) 9321. 929.79908(-6) 10.95 Q( 5, 3) NP(10) -16699. 929.79935(21) 36.97 0( 5, 4) NP(12) -17864. 928.02073(40) 39.46 Q( 5, 5) NP(16) 16486. 925.64688(8) 6.94 Q( 6, 4) 10P(36) 13177. 929.45697(-7) 6.81 Q( 7, 3) 10P(32) -8728. 932.66929(63) 25.83 Q( 7, 7) 10P(42) —l0239. 922.57276(-6) 34.79 Q( 8, 2) 10P(30) 13504. 935.34494(19) 1.30 Q( 8, 5) 10P(34) -10759. 930.64255(12) 28.61 Q( 8, 5) NP(10) 8567. 930.64213(-30) 9.29 Q( 8, 7) NP(16) —12523. 924.67924(27) 34.66 Q( 8, 7) 10P(40) -8838. 924.67918(21) 30.97 Q( 9, 4) 10P(30) -13210. 934.45386(1l) 27.89 0( 9, 5) 10P(32) -l3831. 932.49907(1) 29.74 Q( 9, 6) NP(11) 15258. 929.99712(-22) 2.29 Q( 9, 9) NP(24) 13275. 918.34064(-35) 12.97 Q( 9, 9) NP(23) -14050. 918.34069(—30) 40.29 Q(10, 7) NP(12) 14614. 929.10408(—21) 2.72 Q(10, 8) 10P(40) 15763. 925.49978(-12) 4.05 Q(10, 8) NP(16) 12084. 925.50004(14) 7.73 Q(10, 9) NP(Zl) 14476. 921.10529(26) 8.58 Q(10, 9) NP(20) -12553. 921.10526(23) 35.61 Q(11, 7) 10P(34) 9723. 931.32576(30) 5.33 Q(11, 9) NP(18) 16662. 923.87291(-5) 3.35 Q(ll, 9) NP(17) -10046. 923.87361(65) 30.06 Q(12, 7) 10P(32) 15914. 933.49125(—35) 2.99 Q(12,10) NP(20) 14006. 921.99117(-23) 6.30 Q(13, 9) NP(12) 17537. 929.20158(34) 3.00 Q(14,12) NP(ZS) 14251. 917.45837(-10) 6.96 Table 4—7 (cont.) 91 Transition Laser linea Microwave Two—PhotonC A9 (MHZ) (cm-1) (GHZ) R( 1, 1) 10R(10) -15751. 968.61415(-11) 38.38 R( 2, 1) 10R(44) -15439. 989.13151(-3) 37.48 R( 4, 2) 9P(38) 13696. 1029.89894(—8) 6.99 R( 4, 4) 9P(40) -9298. 1027.07202(15) 32.34 R( 5, 2) 9P(16) 9961. 1050.77355(5) 9.43 R( 5, 4) 9P(18) -16925. 1048.09625(-1) 38.52 R( 6, 1) 9R(10) 11422. 1072.26476(17) 6.04 R( 6, 5) 9R( 4) -1l478. 1067.15625(4) 33.15 R( 7, 3) 9R(42) 15279. l09l.53985(-99) 1.82 R( 7, 5) 9R(36) 10655. 1088.30372(-l) 9.14 R( 7, 7) 9R(28) -16250. 1082.93674(27) 40.80 a Microwave frequency in MHz. N, 9, or 10 refer to N20, 911m band of C02, or 10 Um band of C02 laser, respectively. A minus sign indicates that the microwave frequency is subtracted from the laser frequency to obtain the two-photon frequency. CThe numbers in parentheses are observed minus calculated frequencies in multiples of 0.00001 cm'"1 . The parameters for the calculation are in the first column of Table 4—9. dAbsolute value of the difference between the laser frequency and the frequency of the corresponding one-photon allowed transition. Table 4-8. 92 a-a two-photon transitions in the v2 band of 15NH Comparison of observed and calculated frequencies of 3. Transition Laser Linea Microwave}D Two-PhotonC And (MHZ) (cm *1) (GHZ) P( 2, 1) 10P(42) -16285. 922.37108(24) 5.76 Q( 6, 4) 10P( 2) 14966. 959.89096(54) 34.95 Q( 7, 5) 10P( 2) -10023. 959.05741(-31) 9.77 Q( 7, 7) 10P( 4) 17778. 958.39355(-19) 42.33 Q( 8, 6) 10P( 4) 8645. 958.08890(14) 28.59 Q( 8, 8) 10P( 4) -14848. 957.30526(3) 10.48 0( 9, 6) 10P( 4) -11091. 957.43058(251) 6.46 Q(10, 5) 10P( 4) -l7400. 957.220l4(—8) 3.40 Q(10, 8) 10P( 6) -13284. 955.74188(8) 6.53 Q(ll, 8) 10P( 8) 12852. 954.97378(—5) 30.06 Q(13,13) 10P(14) 11358. 949.85818(0) 43.00 aN, 9, 10 refer to N20, 9 um band of C02, or 10 um band of C0 2 laser, respectively. b Microwave frequency in MHZ. A minus sign indicates that the microwave frequency is subtracted from the laser frequency to obtain the two-photon frequency. CThe numbers in parentheses are observed minus calculated frequencies in multiples of 0.00001 cm'l. The parameters for the calculation are in the first column of Table 4-10. d Absolute value of the difference between the laser frequency and the frequency of the corresponding one-photon allowed transition. 93 between the s and a inversion levels in the ground state) to the s and a inversion levels in the v2 = 1 state. Eighty-five v2 = l, s + v2 = 0, m" and 27 v2 = l, a + v2 = 0, m" frequencies were used. . A least squares analysis of s+m" transitions was car- ried out to obtain a set of vibration-rotation parameters for the m" ground and v2 = l, s excited states; the final results are given in Table (4-9). The standard deviation of the fit is 6.99 MHz, if all of the transitions with K = 3, two transitions with J,K = 7,7, and one transition with J,K = 5,4, which show rather large residuals, are eliminated from the fit. As for l4NH3, K = 3 transitions were omitted because they are perturbed by a centrifugal distortion effect. For the least squares calculation, the parameters and derivatives used were the same as those in the previous section (Equation (4-7) and (4-8)) except for the following: x3 = CI_CH 8v . . 2 (3)—('— = J (J +1)-K 2 (3%) = -[J"(J"+l)-K2] . (4-11) 11 The inversion frequencies used for the present calculation are listed in Appendix A. Appendices C and D list all the transitions used in the fitting for 15NH Vibration- 3. Mb. m m-OHxAObMHVHOHHm.Hu b muoaxaebvmmbm.an =o m auoaxfiovbvaObe.m o anoaxxemcmme.m mo 6 immacmmaaaa.a b . Amsmvmosmaam.a =m m buoaxasmm.on buoaxlbanvammmm.ou bnoaxlosbcmmaem.on mmuwm m buoaxlmmcoo.m b-oaxxommaquqqm.m bnoaxlbeemcbbbmm.m gems m eucaxlmavmm.mu bsoaxfimeavmomv.~: bnOHonmmvmamm.mu same b buchxbvoa.o bnoaxlmmmvommmm.o bnoaxieommceoavm.o . mm m encaxmwmom.b asoaxxmscvmam.b quoaxlmbvbbm~.b mon«o o 6-0Hxlvavqs.vmu vuoaxxmavamm.em- eucaxlmbvvee.emn Mme M u 6-0Hxlbvam.afl anonxxbcflme.afi quoaxlbmcbam.fia mm m mmabmma.ou Abmavasaamma.on Aomavmcmsmma.ou =o-.o m Aqavummeo.oa Ammmvomaamvo.oa Abmbcommmmeo.oa .m m mmbbao.mmm Aemvbesso.mmm lemcsmsso.mmm o> .umm oHHH bHH 6H mumumembma mumbbo x663 mane m .Aaueuv mumum Ame oauumasxm H n m> wsu pan :8 oumum ocsouw mnu ca m2 m0 mmobmfimumm coaumpomlcoflumunw> .mnv manna ma 95 mummemumm m #mmH cw mHmQEdc one .mm mocouomom ca mumumsmnmm Scum cougasuamum .om mucoummmmm .mm mocmnmwwmm .HHH :ESHOU :w mmsHm> on» ou pmcflmuumcouo .cEsHoo umma msu ca mmucmummmm .mwflooum Hmnuo EOHM muaommmo .HHH :EdHoo ca mosam> um ooxflm “omfium> muouoEmumm oa umufim .0 xflocwmmm ca mmflocmovoum no new Eoumn .muouum oumocmum N mum mononucwumm .Umwum> mnoumfimumm ma Haa .U xflocommm CH moflucmnvmum mo paw Eonmm Nmz ea.m was mm.b .o.m m bnoaxxoesmcbgem.a b bnoaxxommmcmomvc.a meme 0 mloaxxovmmvoaho.al o wnoaxfimmmvvwvm.ou Mbmm m snoaxiomscmom.m b H-0Hxibmsmvmnflm.m mm .mmm oHHH QHH MH muwumEmumm mbwnbo xuoz mane .Uwscwucoo .mlq magma 96 rotation parameters by other workers are included in Table 4-9 for comparison. Again, the major difference is ob- served for the value of the band origin, a difference of m27 MHz (0.00090 cm'l) In a second fitting the ground state parameters were constrained to those obtained from the laser Stark data,33 and upper state parameters were calculated. A comparable difference of W23 MHz (0.00077 cm-l) in the band origin was obtained. The number of experimental frequencies available for transitions from the ground state to the asymmetric excited state is insufficient to allow determination of 16 param- eters. Therefore, a fit of these lines was carried out in which the ground state parameters were constrained to those obtained in the fit of s+m" transitions. The result~ ing upper state parameters are shown in Table (4-10). In Table (4-11) the inversion frequencies in the v2 = 1 state that can be calculated from our data are shown. In Table (4-12) the difference in frequency for pairs of laser lines obtained from our experimental data are com- pared with values calculated from the vibration-rotation parameters of C0272 or N20.73 97 15 Table 4-10. Vibration-Rotation Parameters of NH3 in the v2 = 1 Asymmetric (a) State cm-l. Parameter This WorkC Others Ref. 00 962.514804(412) 962.5140 b B' 9.8713084(448) 9.87121(8) a k c'-C" -0.0690825(368) -0.0690695 b 5 .. _ ( Dd 7.126(14)x10 4 7.09(6)x10 4 a 5 0&K -12.766(36)x1o'4 -12.5(2)x10'4 a ' Dk-Dfi - 1.6116(254)xlo‘4 - 1.74(8)xio"4 a H5 0.06058(1182)xlo'6 0.0692x10'6 a I _ '6 -6 HJJK 0.3069(254)x10 0.139x10 a g -6 -6 HJKK 0.45084(6076)x10 0.0927x10 a Hk-HR 0.24056(2846)xio'6 0.453(10‘6 a S.D. 8.39 MHz aFrom Reference 30. b Calculated from parameters of Reference 33. CThe upper state parameters obtained by constraining the ground state parameters to those values in Table 4-9. column I. 98 Table 4-11. Inversion Frequencies of 15NH3 in the v2=1 State Calculated From Differences in Two— Photon Frequencies.a Transition v v v (0)b v (v )C aa 35 INV INV 2 Q(6,4) 28776807.03 27864418.96 19984.32 932372.4 Q(7,7) 28731915.81 27658035.54 24553.43 1098433.7 Q(9,6) 28703046.69 27880612.25 17548.34 839982.8 Q(10,8) 28652420.74 27745785.39 19810.62 926446.0 Q(10,8) 28652420.74 27745793.19 19810.62 926438.2 aAll frequencies in MHz. bGround state inversion frequency from Reference 46. C vINV(V2) =\) - v + aa SS vINV(0). 99 Table 4-12. Comparison of Differences in Laser Frequencies Obtained from Duplicate Measurements of Two- Photon Frequencies with Differences Calculated From Vibration-Rotation Constants of CO2 or N20. . _ a v -v b Avobs.C Avobs.-Avcalcc.i Tran51tion 21 £2 (MHz) (MHz) SQ(2,l) NP(11)-NP(12) 26123. -5.54 SQ(3,2) NP(11)-NP(12) 26127. -1.54 SQ(4,2) NP(10)-NP(11) 26034. 5.88 SQ(5,3) NP(10)-NP(ll) 26020. -8.13 SQ(8,5) 10P(34)-NP(10) 19326. -12.51 SQ(8,7) NP(16)-10P(40) 3685. -1.85 SQ(9,9) NP(23)-NP(24) 27325. -1.39 SQ(10,8) NP(16)-10P(40) 3679. -7.85 SQ(10,9) NP(20)-NP(21) 27029. 0.95 SQ(11,9) NP(17)-NP(18) 26708. -21.01 aTwo-photon transitionsof 15NH3 observed with more than one laser line. bThe two laser lines used to determine the two-photon fre- quency. C — AvObs - sz quencies of the transitions for the laser lines v22 and v2 , 1 - vml where vml and vm2 are the microwave fre— respectively. dAv = Vi - v calculated from vibration-rotation calc 0 2 constants in Reference 72 and 73. 100 4.2.e. Calculation of Ground State B" Constantsfor 14NH3 and 15NH 3. The ground state rotational constantsEW for 14NH and 3 15NH3 were calculated by Helminger gt gt.74 from the mil- limeter wave frequency of the J = l + 0 rotation-inversion transitions. For this calculation, they used D" values J obtained from conventional infrared spectroscopy.54’56 We can follow their procedure, but use our values of D" and J 14 15 HSJJgiven in Tables (4—2) and (4-9) for NH3 and NH3, res- pectively. Figure (4-4) shows energy-level diagrams for the (J=0, K=0) and (J=1, K=0) levels of NH As was mentioned 3. previously, for K = 0 one of the inversion components is 39 missing because of nuclear spin statistics. Therefore, to eliminate the contributions from inversion, hypothetical inversion frequencies must be calculated for these levels. These were calculated from the empirical formulas of 70 14 Schnable gt gt. for NH3 and of Sasada46 for 15NH 3. The values obtained are shown in Figure (4-4). The rotational frequencies of the (J,K) = (1,0) + (0,0) transitions are 572112.78 MHz for 15 3 14 74 . for NH3. To calculate the rotational constants we NH and 572498.15 MHz need to obtain the frequency at which these transitions would occur if there were no inversion splitting. This can be obtained by adding half the sum of the apprOpriate inversion frequencies, as shown in Figure (4-4). The hypothetical frequencies obtained in this way are 101 14 15 NH3 NH3 J,K—... o- o a. ooooo q’ J,K-I- o — o— o- ...q 0 In 00 m 1,0 “t 1,0 -- -— —-—-—-‘— F; --‘---- --- . G) N '- ... . 6 I m N : N 1 N I ' I l 'U 00 'r-l U U) I m m h m (D :—I <1: p . . > o ' o H N '0 :4 w «a 0 r4 'w a o ox lrn m r) o g (.0 V H ..Q N V : Q C“ I W O r~ la 4 O l‘ ON m In ' I 1 l 00 O l 00 a m o ' . 0’0 .-—---—J--—rg 0'0- --_———-—fi-: F b m N -.-._O—O—O-OJ N 0-0-0—.- 0 - 0‘ N Figure 4-4. Energy-level diagrams for the J = 0 and J = 1, v = 0 states of 4NH3 and 15NH3 showing the ob- served transitions and the hypothetical levels corresponding to no inverstion (observed fre- quencies from Reference 74). 102 596133.49 MHz and 594680.91 MHz for 14NH3 and 15 NH3, respectively. Now by using the usual symmetric tOp rotational energy terms Equation (4-1), the frequency of the (J,K) = (J+l,0) + (J,0) rotational transition is, v = 2B(J+l) - 4D (J+1)3 + H (J+l)3[(J+2)2-J3] J JJJ (4-12) Here, J refers to the quantum number of the lower state, which in this case is J = 0. Therefore, (4-13) C II 2B - 4DJ + 8HJJJ or v + 4D - 8H B = J JJJ (4_14) 2 By substitution of the values of DJ, HJJJ, and v, we cal- culate the ground state B" constants of NH3 to be 298117.107 14 15 MHz for NH3 and 297390.985 MHz for NH3. Comparison of these values with those obtained from our fit of the two-photon data (14NH3: 298117.64i0.63 MHz and 15NH : 3 297390.24:0.87 MHz) shows good agreement. 103 4.2.f. Comparison of the Frequencytof the (J=1,0+0,0) Rotation-Inversion Transition of NH3 with the Value From Microwave Spectroscopy The frequency of the (J,K=l,0+0,0) rotation-inversion transitions can be calculated using our values of the B", II I! DJ, and HJJJ (4-2) and (4-9) for constants in the ground state given in Table 14 15 NH3 and NH3, respectively. For this purpose, B", D3 and HJJJ have been substituted in Equation (4-13), in order to calculate the frequency of the hypothetical rotational transition (J=l,0+0,0). Next the effect of the inversion splitting is eliminated by sub— tracting half of the sum of the frequencies of the hypo- thetical inversion frequencies of the J,K = 1,0 and 0,0 transitions. The calculation of the inversion frequencies J,K = 1,0 and 0,0 are described in the last section. The same procedure was used to calculate the rotation-inversion frequencies from the parameters of Reference 33. The cal- culated frequencies and the observed submillimeter frequen- cies of the J = l + 0 transitions for l4NH3 and 15NH3 are compared in Table 4-13. 4.2.g. Discussion . . . 14 15 The v2 Vibration-rotation bands of NH3 and NH3 have been reinvestigated in this study. Forty-four two- photon transitions in l4NH3 and 60 two-photon transitions 104 Table 4-13. Comparison of the Calculated Frequency of the J,K = l,0+0,0 Rotation-Inversion Transition in l4NH3 and 15NH3 with the Experimental Value.a Calculated Value Molecule Experimentb This Work Ref. 33 14NH3 572498.15:0.15 572499.22 572502.07 l5NH3 572112.78:0.l 572111.30 572115.95 aAll frequencies in MHz. bReference 74. ' 99.3w ‘3 105 in 15NH3 were assigned. The frequencies of the observed two-photon transitions in each case were combined with all ‘ of the previous infrared data that had been measured rela- tive to CO2 or N20 laser lines. By means of least-squares analyses of these data molecular parameters in the ground and v2 = l excited states were derived. The parameters are listed in Table (4-2) for 14NH3 and in Tables (4-9) lSN and (4-10) for H 3. By combining a two-photon frequency with an appropriate ground state or upper state inversion frequency, the fre- quency corresponding to a one-photon allowed a + s or s + a transition can be obtained. The one-photon frequen- cies derived in this way for 15NH3 from the results of the present study and inversion frequencies in the ground state (from Reference 46) are compared in Appendix G to similarly derived frequencies from previous two photon work and to one-photon frequencies obtained by C02 laser-based, diode- 1aser measurements. Also,compared in Appendix G are the results of the theoretical calculation by DiLonardo gt gt. A similar comparison has been made for l4NH3; the re- sults appear in Appendix H. For 14NH3, because experimental inversion frequencies in both the ground and the upper state have been reported (43,71) the frequencies of both the a + s and s + a one-photon transitions can be obtained from the frequency of one, two-photon transition. This has been done for the entries in Appendix H. The results 106 of the theoretical calculations for l4NH3 by Urban gt gt. are also included in this appendix. | Comparison of the same transition measured by different methods provides further verification of the ability to measure the frequencies of Doppler-broadened transitions 1 to a precision of 10.0002 cm- by methods based on the frequencies of CO2 or N20 lasers. 1 Tables (4-3), (4-4), (4-7), and (4-8) and Appendices ? .1 .11. ‘ G and H provide comparisons of observed frequencies with frequencies calculated according to two—different approaches. )- We have chosen to express the energies by a double power series expansion in J(J+l) and K2 (Equation 4-1). In this approach the various perturbations and interactions with other states are assumed to lead to corrections to the energy levels that follow this form. This procedure has the advantage of being easily programmed for the rapid calculations of the frequencies by a computer. It is only 'useful for those levels (J,K) which are not strongly per- turbed. For the range of J and K values shown here, the s + a and a + s one-photon frequencies can be calculated to an accuracy of N20 MHz by means of the constants in Table (4-2) and experimental inversion frequencies in the ground 14 state and v2 = 1 state for NH3, and by constants given in Tables (4-9) and (4-10) and inversion frequencies in the ground state for 15NH 3. The calculated frequencies compared with experimental 107 values in Appendices G and H were obtained by considering the interaction between the v2, 2v2, 3v2, v4, and v2 + v4 levels explicitly. This approach allows a reasonable calculation of the energies of the strongly perturbed states. The advantage of this approach is that the derived rota- tional constants and centrifugal distortion constants are more easily related to structural properties. The dis- advantage of such a calculation is its complexity. Tables (4-14) and (4-15) show the frequencies of the infrared transitions measured in Michigan State University for 14NH3 and 15 NH3, respectively. Whenever there has been an infrared measurement by other investigators for the same transition, it is included in these tables. These two tables show that the lineshape method of detenmining the frequency of Doppler broadened infrared-microwave two- photon transitions used in this investigation gives results that are comparable in accuracy to the measurements of saturation dips when the laser has been stabilized by maximizing the power output. This is probably a result of the fact that in either case the error in determination of the microwave frequency is considerably smaller than the fluctuation in the laser frequency. A real test of the two methods will occur when measurements are made with the laser stabilized to a Doppler-free dip in the fluorescence from a C02 or N20 sample. 108 Table 4-14. Frequencies of the IR transitions in l‘4NH3. Transitiona Ref;b IR Transitiona Ref}D IR FrequencyC FrequencyC sP(2,1) SC 891.88171 sQ(5,5) SC 928.75434 sQ(1,l) SC 931.62767 sQ(5,5) F 928.75393 sQ(1,1) F 931.62746 sQ(5,5) FL 928.75432 sQ(1,l) FL 931.62773 sQ(7,4) SC 934.23571 sQ(3,2) SC 932.09398 sQ(9,5) SC 935.70667 sQ(3,2) F 932.09452 sR(3,3) SC 1011.20350 sQ(3,3) SC 930.75701 sR(5,l) SC 1054.91248 sQ(3,3) F 930.75741 sR(5,2) SC 1054.25276 sQ(3,3) F 930.75654 sR(5,2) F 1054.25109 sQ(3,3) FL 930.7569? sR(5,3) SC 1053.13049 sQ(3,3) SA 930.75708 sR(5,3) FL 1053.13044 sQ(5,2) SC 934.25216 sR(5,4) SC 1051.51210 sQ(5,3) SC 932.99231 sR(5,4) FL 1051.51206 sQ(5,3) F 932.99279 sR(5,5) SC 1049.34639 sQ(5,3) SA 932.99237 sR(5,5) FL 1049.34631 sQ(5,3) HK 932.99228 aQ(l,1) SC 967.99798 aQ(7,6) F 964.59561 aQ(l,1) SC 967.99890 aQ(7,6) SA 964.59570 aQ(l,1) J 967.99785 aQ(7,7) SC 964.42430 aQ(3,3) SC 967.34610 aQ(7,7) SC 964.42442 aQ(3,3) SC 967.34645 aQ(7,7) J 964.42215 aQ(3,3) SC 967.34653 aQ(8,5) SC 964.04123 aQ(3,3) F 967.34502 aQ(8,5) J 964.04152 aQ(4,3) SC 966.90520 aQ(8,5) FL 964.04115 aQ(4,3) SC 966.90531 aQ(8,6) SC 963.79644 aQ(4,4) SC 966.81505 aQ(8,6) F 963.79658 aQ(4,4) SC 966.81484 aQ(8,6) J 963.79678 aQ(4,4) SC 966.81478 aQ(8,6) FL 963.79608 aQ(4,4) F 966.81507 aQ(8,7) SC 963.55855 aQ(4,4) J 966.81477 aQ(8,7) F 963.55820 aQ(5,5) SC 966.15122 aQ(8,7) FL 963.55853 aQ(5,5) F 966.15053 aQ(8,8) SC 963.36329 aQ(5,5) J 966.15083 aQ(9,4) SC 963.53399 aQ(5,5) SA 966.15118 aQ(9,4) J 963.53426 aQ(6,4) SC 965.65207 aQ(9,5) SC 963.26922 aQ(6,4) J 965.65208 aQ(9,7) SC 962.67017 aQ(6,4) JL 965.65211 aQ(9,7) J 962.67042 aQ(6,6) SC 965.35420 aQ(11,9) SC 960.01989 aQ(6,6) F 965.35433 aQ(11,9) F 960.02063 109 Table 4-14 (cont.) Transitior? Ref? IR Transitiona Ref]? IR FrequencyC FrequencyC aQ(6,6) J 965.35422 aQ(11,9) FL 960.01978 aQ(6,6) JL 965.35415 aR(4,2) SC 1065.58180 aQ(7,3) SC 965.13796 aR(6,1) SC 1103.48587 aQ(7,6) SC 964.59604 aR(6,5) SC 1103.43445 aQ(7,6) SC 964.59597 aR(6,5) J 1103.43421 aV(J",K) where v = P, Q, or R for AJ = -1, 0, or +1, res- pectively. s and a represent transitions from ground state s to upper state a and from ground state a to upper state s ,respectively. b.J, Ref. 29 by Jones; F, Ref. 28 by Freund et al.; SA, diode laser measurements by Sattler, Ref. 68; SC, this work. c:All frequencies are in cm'du T" 110 Table 4-15. Frequencies of the IR transitions in 15NH3. Transitiona Ref.b IR Transitiona Ref.b IR Frequency - FrequencyC: sP(2,1) SC 923.10640 sQ(8,8) D 958.15001 sP(2,1) J 923.10657 sQ(9,6) SC 958.01593 sQ(6,4) SC 960.55756 sQ(10,5) SC 957.68709 sQ(7,5) SC 959.71764 sQ(l0,8) SC 956.40269 sQ(7,7) SC 959.21256 sQ(l0,8) SA 956.39927 sQ(8,6) SC 958.74609 sQ(11,8) SC 955.54763 sQ(8,8) SC 958.14996 sQ(13,13) SC 950.91534 aQ(2,1) SC 928.21039 aQ(9,5) SC 931.96844 aQ(2,1) SC 928.21057 aQ(9,6) SC 929.41177 aQ(2,2) SC 927.40348 aQ(9,9) SC 917.46528 aQ(3,2) SC 928.18153 aQ(9,9) SC 917.46532 aQ(3,2) SC 928.18148 aQ(l0,7) SC 928.52593 aQ(3,3) SC 926.83752 aQ(l0,8) SC 924.83897 aQ(3,3) F 926.83793 aQ(l0,8) SC 924.83923 aQ(3,3) SA 926.83771 aQ(l0,9) SC 920.33626 aQ(4,2) SC 929.18536 aQ(l0,9) SC 920.33622 aQ(4,2) SC 929.18516 aQ(l0,9) J 920.33613 aQ(4,2) SA 929.18536 aQ(11,7) SC 930.82348 aQ(4,3) SC 927.87716 aQ(11,7) SA 930.82286 aQ(5,3) SC 929.12288 aQ(11,9) SC 923.20546 aQ(5,3) SC 929.12315 aQ(11,9) SC 923.20616 aQ(5,3) F 929.12241 aQ(11,9) D 923.20502 aQ(5,3) FL 929.12287 aQ(12,7) SC 933.06019 aQ(5,3) SA 929.12274 aQ(12,7) SA 933.06062 aQ(5,4) SC 927.30030 aQ(12,10) SC 921.31381 aQ(5,4) J 927.29996 aQ(13,9) SC 928.71667 aQ(5,4) F 927.29820 aQ(14,12) SC 916.75101 aQ(5,5) SC 924.86560 aR(1,1) SC 967.85946 aQ(6,4) SC 928.79037 aR(1,1) F 967.86024 aQ(6,4) J 928.79067 aR(1,1) SA 967.85943 aQ(6,4) SA 928.79018 aR(2,l) SC 988.39620 aQ(7,3) SC 932.09898 aR(4,2) SC 1029.20904 aQ(7,7) SC 921.75374 aR(4,4) SC 1026.30329 aQ(7,7) J 921.75469 aR(5,2) SC 1050.12685 aQ(7,7) F 921.75462 aR(5,4) SC 1047.37583 aQ(8,2) SC 934.85127 aR(6,1) SC 1071.68213 aQ(8,2) SA 934.85121 aR(6,1) SA 1071.68178 aQ(8,5) SC 930.04696 aR(6,5) SC 1066.43348 aQ(8,5) SC 930.04654 aR(7,3) SC 1090.96955 111 Table 4-15 (cont.) Transitiona Ref.b IR Transitiona Ref.b IR FrequencyC FrequencyC: aQ(8,5) J 930.04671 aR(7,3) J 1090.96971 aQ(8,7) SC 923.94090 aR(7,5) SC 1087.64349 aQ(8,7) SC 923.94084 aR(7,7) SC 1082.11772 aQ(9,4) SC 933.96415 aR(7,7) SA 1082.11719 aV(J",K) where V = P, Q, or R for AJ = -1, 0, or +1 ,res- pectively. s and a represent transitions from ground state s to upper state a and from ground state a to upper state s ,respectively. bJ, Ref. 30 by H. Jones; F, Ref. 28 by Freund et al.;FL, Lamb-dip measurements, Ref. 28; SA, diode laser measurenmnits by Sattler, Ref. 68; SC, this work. CAll frequencies are in cm'l. 9E___ . ra-u 112 4.3. The v3 Bands of 12CH3F and 13CH3F_ 4.3.a. Introduction Methyl fluoride is a prolate symmetric tOp with C3V symmetry. This molecule possesses six fundamental vibra- tional modes, three totally symmetric and three doubly de- generate, all infrared active. The frequencies of rotational transitions in 12CH3F have been measured by Gilliam gt gt.,75 69 76 77 Johnson gt gt. Sullivan and Fren- 80 Orville gt gt., Winton and Gordy, 78 kel, Tanaka and Hirota,79 and Hirota gt gt. The most precise frequencies were obtained by Lamb dip procedures 77 Sullivan and Frenkel measured sev- 79 by Winton and Gordy; eral high J transitions;78 and Tanaka and Hirota and Hirota 80 measured the J = 1 + 0 transition in excited vibra- tional states. The frequency of the J = 1 + 0 transition in 13CH3F was originally measured by Gilliam gt gt.75 but their value has been corrected by Tanaka and Hirota,79 who also measured the frequencies of this transition in the v and V excited states. 3 6 A detailed study of the infrared spectrum of CH3F was 81 carried out by Yates and Nielsen. Andersen and coworkers reported and analyzed all of the perpendicular fundamentals 82 83 and discussed the structure of CH3F. Smith and Mills re-analyzed the rotational structure of some of the bands, particularly the two fundamentals at 1049 cm-1 (parallel band) and 1182 cm-1 (perpendicular v6 band). Similarly, V3 113 l3CH3F.84 All of this work was performed with conventional grating Duncan gt gt. studied the infrared spectrum of spectrometers. The v3 band of CH3F, corresponding to the C-F stretch- ing vibration, lies close to the 9.55 pm region of CO2 laser lines. Therefore, this molecule, like NH3, has been used for various laser Spectroscopic experiments. g In an optical double-resonance experiment, Brewer de- E termined accurate values for dipole moments in the ground = 85 and v3 states. In this experiment two CO2 lasers were used as sources of radiation. Chang and Bridges reported the observation of laser action on six rotational transitions in a sample of CH3F that was optically pumped by a C02 86 laser. This was the first Optically pumped far infrared or submillimeter laser. Several additional optically pum- ped laser lines were observed later 137 Chang gt gt.87'88 Laser Stark spectroscopy of the v3 band of CH3F was carried out by Freund gt gt. by using CO2 laser lines near 9.4 pm. In this investigation, the dipole moment and vibration-rotation parameters of the v3 state were de- 89,90 termined. Doppler-free, two-photon absorption in the v3 band of CH3F has been observed, in which two fixed-fre- quency infrared radiation sources (CO2 lasers), in combina- 91 tion with molecular Stark tuning were used. The ground state rotational constant A0 and the centrifugal distortion constant D: of CH3F were determined by Graner from the fre- quencies of perturbation-allowed infrared transitions92 114 measured with a tunable diode laser. Infrared-radio fre- quency two-photon and multiphoton Lamb dips have been ob— served for methylfluoride in the cavity of a C02 laser oscillating in the 9.4 pm band.93 Sattler and Simonis studied the v3 band of 12CH3F by 94 diode laser Spectroscopy. They measured frequencies by means of a Ge etalon and therefore only their frequency dif- ferences are very precise. Magerl gt gt.95 measured the frequencies of numerous high J transitions in the v3 band 12 of CH3F by a tunable laser sideband technique. Her- 96 lemont gt gt. measured the frequencies of more high J transitions in this band by using heterodyne methods with a 97 diode laser and a waveguide laser. Blumburg gt gt. used a tunable submillimeter wave source to measure the fre- quencies of rotational transitions in the v3 = 1 state of 12 CH3F, but reported only the resulting D and D values J JK and not the frequencies themselves. Freund gt gt.89’90 Arimondo and Inguscio,98 Magerl gt gt.95 96 and Herlemont gt .gt. have all reported the results of fitting the v3 band transitions to rotational and centrifugal distortion con— stants. The v3 band of methyl fluoride was studied at Michigan State University with the same infrared-microwave two- photon spectrometer that was used for NH3. Two-photon transitions were assigned for both species, 12CH3F and 13 CH3F. From a least squares analysis of the two-photon 115 data obtained in this study and from the earlier measure- ments combined with the data available from microwave, FIR, laser-Stark, and laser infrared investigations of 12CH3F, rotational and centrifugal distortion constants in the ground and v3 = 1 excited states were determined. The molecular constants of the ground and the v3 fundamental mode were obtained for 13 CH3F as well. For this fit the two-photon data, the zero-field frequencies obtained from a laser-Stark experiment, and the frequencies of the rota- tional transitions (J = l + 0) in the ground and v3 states were used. lzCH F ____3_ 4.3.b. Methyl fluoride is an example of a symmetric top mole— cule in which the inversion splitting of the levels is not resolvable. Therefore, all of the levels with K # 0 have double parity and there is no need for a third level to ob- serve a two-photon transition. A pair of ground and v3 vibration-rotation levels of CH3F are shown in Figure (2-2). In this figure vm and v1 are the frequencies of the microwave and laser satisfying the two—photon condition, respectively. The offset fre- quency - the frequency difference between the laser and the molecular transition - is equal to vm in this case. The study of the v3 vibrational band of 12CH3F by two- photon spectroscopy was begun by calculating the frequencies .'. 'v. 91 LY- fi . i 116 of the possible two-photon transitions. For this calcula- tion, the ground state rotational parameters, B", D3, and DJK’ obtained by analysis of the frequencies of mil- 77 limeterwave Doppler-free spectra, and the vibration—ro— tation parameters of the v3 = 1 state, reported in a laser Stark study of CH3F,89 were used. Since the v3 band of CH3F is a parallel vibrational band, the selection rule for K is AK = O. The frequency of a transition from the ground to the v3 excited state can be obtained by applying the selection rules to Equation (4-4). It should be noted that for CH3F, vRV in Equation (4-4) corresponds to the frequency of the one-photon as well as to the two-photon transition, and v in the equa- 0 tion is the energy difference between the J' = K' = 0 and J" = K" = 0 states. The primes and double primes mark the levels for the upper and lower states, respectively. By using eight CO2 laser lines in the 9.4 pm region, 12CH3F in the available microwave range of 8 - 18 GHz; Table (4-16) 24 two-photon transitions were observed for lists the two-photon transitions and their assignments. The 12 CH3F sample was used at a pressure of N1 torr. Figure (4-5) shows a trace of the Q branch two-photon transitions (J = 6, K = 3,4,5,6). These transitions were observed by using the 9P(18) laser line. Since microwave transition moments appear as a difference in Equation (2-48), we were able to see Q branch transitions of CH3F 117 Table 4-16. Comparison of observed and calculated frequencies of two-photon transitions in the v3 band of 12CH3I?. Transitiona Laser Lineb Microwave(MHz)C Two-Photon(cm"l)‘i P(5,2) 9P(28) 15145. 1039.87450(24) P(5,3) 9P(28) 15385. 1039.88250(20) P(5,4) 9P(28) 15727. 1039.8939l(7) P(4,1) 9P(26) 11508. 1041.66294(1) R(4,2) 9P(26) 11651. 1041.66771(6) P(4,3) 9P(26) 11891. l04l.67571(6) P(3,l) 9P(24) 8111. 1043.43379(-l7) P(3,2) 9P(24) 8261. l043.43880(l6) Q(4,2) 9P(18) -8103. 1048.39052(-9) Q(5,5) 9P(18) ~10551. 1048.30887(-2) Q(5,4) 9P(18) -10965. 1048.29506(-6) Q(5,3) 9P(18) -11276. 1048.28468(-ll) Q(6,6) 9P(18) —14116. 1048.18995(12) Q(6,5) 9P(18) -14630. 1048.17281(18) Q(6,4) 9P(18) -15036. 1048.15926(10) Q(6,3) 9P(18) -15333. 1048.14936(30) R(2,2) 9P(12) -9962. l053.59121(-1) R(2,1) 9P(12) -10080. l053.58727(16) R(3,2) 9R(10) —12613. 1055.20435(37) R(4,4) 9P( 8) -14701. l056.80979(8) R(4,3) 9P( 8) -14981. 1056.80045(4) R(4,2) 9P( 8) -15177. 1056.79391(-5) R(5,3) 9P( 6) -17430. 1058.36731(4) R(5,4) 9P( 6) -17161. 1058.37629(14) aV(J",K) where V = P, Q, or R for respectively. AJ = -1, 0' or +1, bAll the laser lines are in the 911m region. cMicrowave frequency in MHz. A minus sign indicates that the microwave frequency is subtracted from the laser frequency to obtain the two-photon frequency. dThe numbers in parentheses are observed minus calculated frequencies in multiples of 0.00001 cm-l. The parameters for the calculation are given in Table 4-17. Figure 4-5. 118 Trace of the QQ(6,K), (K = 6,5,4,3) two- photon transitions in 12CH F. The P(18) CO2 laser line in the 9.5 3m region was used with a microwave sweep from 13900 - 15600 MHz to record this Spectrum. The K values are given at the top of each transition. 119 £511....le . . .. r '1' .mue musmflm E oeoma smz\ > _ m . ooova NOIIdHOSEIV 120 only when the electric field of the laser and microwave radiation were perpendicular to each other, in which case AM = i1. Figure (4-5) shows that for the same J quantum number, transitions with larger K quantum numbers have higher in— tensity. Another feature of this spectrum is that the transitions involving rotational levels with K = 3n Show higher intensity. This higher intensity is associated with the fact that the statistical weight of levels with K = 3n is twice that of levels for which K is not a multiple of three.39 To obtain a set of vibration-rotation parameters for 12CH3F, the two-photon frequencies observed in this study 89 the zero 89,90 (Table 4-16), the previous two-photon data, field frequencies determined in a laser Stark study 12 of the v band of CH3F, the frequencies of the rotational 77,78-80 3 transitions in the ground and v3 = 1 states, 88 the rotation-vibration transition P(32,3), the FIR transitions 98 in the ground and v3 = 1 states, the diode laser and 96 and the transitions ob- wave guide laser measurements, served with a tunable sideband laser,95 were combined. The transitions were given weights proportional to the inverse of the square of their reported uncertainties and a simultaneous least squares fitting of the weighted data was carried out. A total of 172 frequencies were used as input data for the least squares fitting; they are given 121 in Appendix E. The resulting vibration-rotation parameters are given in Table (4-17). 13 4.3.c. CH3§ Equation (4-4) and the vibration-rotation parameters 13 of the ground state and v3 = 1 state of CH3F given in Reference 89 provided the means by which the frequencies of the two-photon transitions of the v3 band of l3CH3F were first calculated. In this investigation three CO2 laser lines were used to detect fourteen two-photon transitions in the 9 pm region. The frequencies and assignments of the observed two-photon transitions are given in Table (4-18). The 13 pressure of the CH3F sample was approximately 1 torr. The molecular parameters for the ground state and v3 13 = l excited state of CH3F were obtained from a least squares analysis of a combination of two-photon data ob- 89 tained in this work and by a previous study with zero- field frequencies from a laser Stark study of the v3 89,90 band, and with the microwave frequencies of the 79 The total number of transitions J = l + 0 transitions. was 44. Since this number was not sufficient to determine all of the 16 parameters in the ground and excited states, the values of the sextic centrifugal distortion constants (H's) were constrained to zero. As a result, 9 parameters— D' (D'-D"), B", D" and D" _ u |_ u _ l_ n I J! 122 Table 4-17. Molecular Parameters of 12CH3F in the Ground State and v3 = 1 Excited State. Parametera Ground Stateb v3 = 1b v0 0 31436558.6(18) MHz 25536.1500(12) 25197.563(18) MHz AA-AB 0 44.109(218) MHz DJ 60.26(6) 57.43(20) KHz é DJK 439.47(20) 518.22(222) KHz v_ n - DK DK 0 101.35(255) KHz E HJ 1.2(7) 1.4(10) Hz HJJK —9.6(90) 15.6(112) Hz HJKK 46.3(330) -ll9.0(332) Hz I_ n HK HK 0 106.3(145) Hz Standard DeviationC 5.71 MHz aPrime and double primes refer to the upper and lower state, respectively. bFrom fit of frequencies in Appendix E. The numbers in parentheses are 2 standard errors in multiples of the last digit in the parameter. cStandard deviation is for observations of unit weight. 123 Comparison of observed and calculated frequencies 13 of two-photon transitions in the V3 band of CH F. 3 Transitiona Laser Line b Microwave(MHz)C Two-Photon(cm") 1d P(4,3) 9P(46) ~9461. 1020.74133(9) P(4,2) 9P(46) -9638. 1020.73542(3) Q(6,6) 9P(40) -8993. 1027.08220(-14) Q(6,5) 9P(40) -9358. 1027.07002(-9) Q(6,4) 9P(40) -9646. 1027.06042(-10) Q(6,3) 9P(40) -9862. 1027.05321(-11) Q(7,7) 9P(40) -l3053. 1026.94677(2) Q(7,6) 9P(40) -13500. 1026.93186(-4) Q(7,5) 9P(40) -13855. 1026.92002(7) Q(7,4) 9P(40) -l4139. 1026.91055(-4) Q(7,3) 9P(40) -14349. 1026.90354(-2) Q(8,8) 9P(40) -l7685. l026.79226(8) R(3,2) 9P(34) 12870. 1033.91730(5) R(3,3) 9P(34) 13014. 1033.92210(22) a ll V(J ,K) where V = P, b C d Q, or R for AJ 0, or +1, respec- ~mm . ... r. tively. All the laser lines are in the 9 pm region. Microwave frequency in MHz. A minus Sign indicates that the microwave frequency is subtracted from the laser frequency to obtain the two-photon frequency. The numbers in parentheses are observed minus calculated frequencies in multiples of 0.00001 cm-l. The parameters; for the calculation are given in Table 4-19. )F 1 124 were determined. The results are shown in Table (4-19). Figure (4—6) shows two-photon spectra of the Q type transitions J = 6, K = 6,5,4,3 and possibly K = 2. This spectrum was recorded in the X band microwave region (8700 - 10100 MHz) with the P(40) CO2 laser line in the 9 um range. In order to observe this spectrum, the electric: field of the laser and the microwave radiation must be per- pendicular to each other. Higher intensity of the transi- tions with larger K (for the same J) and with K = 3n is confirmed in Figure (4-6). 4.3.d. Discussion The v3 vibration-rotation bands of 12CH3F and l3CH3F have been studied in this experiment. Twenty-four two- photon transitions in 12 13 CH3F and fourteen two-photon transi- tions in CH3F were assigned. Even though the intensity of the spectra for the double parity levels in symmetric top molecules without inversion are inversely proportional to the square of the microwave frequency, transitions have been observed with microwave frequencies up to 17 GHz. Considering the fact that the lasers used in this investigation were not particularly high power (ml watt at the sample cell), we can expect that vibrational bands which are considerably weaker than the v band of CH3F will be suitable for study, provided 3 that they have close coincidences with the laser lines. 12S 13 Table 4-19. Molecular Parameters of CH3F in the Ground State and v3 = l Excited State. Parametera Ground Stateb v3 = 1b v0 0 30803472.8(12) MHz B 24862.7901(l96) 24542.2677(176) MHz AA-AB 0 31.356(242) MHz DJ 59.42(626) 56.27(378) KHz DJK 425.18(2550) 478.91(l908) KHz |_ n _ DK DK 0 69.46(1570) KHz Standard DeviationC 4.46 MHz aPrime and double primes represent upper and lower state parameters, respectively. bFrom fit of frequencies in Appendix F. The numbers in parentheses are 2 standard errors. cStandard deviation is for observations of unit weight. Figure 4-6. 126 Trace of the QQ(6,K), (K = 6,5,4,3,2) two- photon transitions in l3CH3F. The P(40) CO2 laser line in the 9.5 um region was used with a microwave sweep from 8700 - 10100 MHz to record this spectrum. The K value is given at the top of each transition. 127 .e-e enemas E $15 > comm NOIldHOSSV 128 Unlike symmetric tops with inversion or asymmetric tops, the selection rules for two—photon transitions in symmetric tops without inversion are the same as the infrared selec- tion rules. Therefore, two-photon spectra of symmetric top molecules without inversion are directly comparable to infrared spectra. A comparison of two-photon frequencies and zero-field frequencies derived from laser Stark spectra is shown in Table (4—20). The agreement is seen to be excellent; in only three cases are the differences greater than the ex- pected uncertainty of :6 MHz. Comparison of the observed and calculated frequencies, given in Tables 4-16 and 4-18, also shows very good agree- ment. Table 4-20. Stark Spectra (vLs) . 129 b Comparison of Two-Photon Frequencies (v a TP) and Zero-Field Frequencies Derived from Laser Transition TP LS P(4,2) P(4,3) R(3,2) R(3,3) P(3,1) P(3,2) P(4,2) P(4,3) P(5,3) P(5,4) R(2,1) R(2,2) R(3,2) R(4,3) R(4,4) 30600878.19 30601055.19 30996060.75 30996204.75 31281358.15 31281508.15 31228412.30 31228652.30 31174893.16 31175235.16 31585751.80 31585869.80 31634230.44 31682080.48 31682360.48 13 12 CH CH 30600876.87 30601054.22 30996058.98 30996199.53 31281365.56 31281506.76 31228410.22 31228650.06 31174887.67 31175234.23 31585745.44 31585871.36 31634218.59 31682068.46 31682349.37 aThis work b From References 89 and 90. CAll transitions are in MHz. CHAPTER V MICROWAVE SPECTRUM OF 3-METHYL-1-BUTENE 5.1. Introduction During the past several years there has been consider- able interest in the characterization and comparison of the potential functions for the internal rotation of groups attached to the cyclopropane ring, to the ethylene oxide ring, and to the isopropyl group. Among the molecules in this class are those in which the attached atom is a double-bonded carbon atom such as the carbon atom in an aldehyde or vinyl group. The internal rotation in these molecules is of interest because of the possibility of conjugation between the attached group and the cyclopropyl or ethylene oxide ring. The corresponding molecules in which the same group is attached to an iSOpropyl group have been studied for comparison. The earliest comparison of this type followed the determination by electron diffrac- tion of the structures of cyclOpropanecarboxaldehyde99 100 It was shown that iso- and iSOpropylcarboxaldehyde. propylcarboxaldehyde occurs in conformations with the CO bond eclipsing either a CC bond or a CH bond in the iso- prOpyl group. These conformations are consistent with 130 131 those found in CH 101 102 COX compounds and most RCHZCOX mole- 3 By contrast, cyclOprOpylcarboxaldehyde was found99 cules. to occur as an almost 50-50 mixture of conformers with the oxygen atom cis and trans to the ring, and this result was later confirmed by microwave spectroscopy.103 Since that time cyclopropylcarboxylic acid chloride}04’105cyclopropyl- methylketone,104'106 and cyclopropylcarboxylic acid fluoride 107' have all been shown to occur with cis and trans conformers, and therefore have a torsional potential which is dominated by a cosZa term. On the other hand, an electron diffrac- 108 was interpreted tion investigation of vinylcyclopropane in terms of species with the vinyl group trans to the ring and rotated approximately 120° from the trans configura- 109 confirmed only the trans con- tion. A microwave study former; no transitions from a second conformer were found. In the present work the microwave spectrum of 3-methyl- l-butene was studied and transitions assigned to species with the vinyl group eclipsing the CH bond of the isopropyl group (referred to here as the "trans" configuration) and with the vinyl group rotated approximately 120° from the trans configuration (the "gauche" configuration). The v==0+l torsional excitation energies and the ground state energy difference of the two species have been evaluated by relative intensity measurements and used to estimate the first four coefficients in the torsional potential func- tion. 132 5.2. 3-Methyl-l-Butene The study of the microwave spectrum of 3-methyl-l-butene in the 18-40 GHz region was begun by R.A. Creswell and M. 110 The a—type transitions were assigned to species Pagitsas. with the vinyl group eclipsing the CH bond of the iSOpropyl group (referred to here as the "trans" configuration) and with the vinyl group rotated approximately 120° from the trans configuration (the "gauche" configuration). No b or c type transitions could be assigned for either species, presumably due to the small “b or “c components of the di- pole moment. Figure (5-1) is a projection of trans 3- methyl-l-butene in its ac plane of symmetry. The transitions for each of the species were accompanied by a single intense series of satellite lines, which have been assigned to rotational transitions in the excited states of the torsional motion of the vinyl group relative to the isoprOpyl group. 5.2.a. Dipole Moment The Stark effects of several transitions for trans-3- methyl-l-butene were studied and combined with the results obtained by Creswell and Pagitsas for gauche and trans con- formers in order to determine dipole moments for the two spe— cies. In each case the slopes of plots of the observed frequen- cies vs.the square of the field 52 were fit by least squares to 133 Figure 5-1. Projection of trans 3-methyl-l-butene in its ac plane of symmetry. The angle a measures internal rotation about the indicated CC bond. 134 the expression ——-2- = 2: (AAg + AB M2)u§ (5‘1) d5 g g which contains the squares of the components of the dipole moment, pg, as adjustable parameters, and in each case the result was that u; was found to be a small negative value. For the trans species this is presumably because “b = 0 by symmetry and the negative value of u: was a result of experimental error. In the gauche species, how- ever, it is expected that “b # 0, and therefore the negative value of Hg is taken to mean that ub is very small. Thus, for each species the fitting of the Stark slopes was re- peated with the assumption that “b = 0. The results of the fittings, including comparisons of observed and cal- culated Stark slopes, are shown in Tables(5-l) and (5-2). The dipole moments for the trans conformer are ua = 0.312 $0.003!» ub = 0 (assumed): uc = 0.07li0.042 D, and “T 0.320i0.010 D; for the gauche conformer, they are “a = 0.367i0.004 D, “b 2 0, uc = 0.15410.006 D and “T = 0.398 2 2 2 2 10.004 D. Here, “T = pa + “b + “c and is the total “T dipole moment. The total dipole moments of several compounds that 109 If the contain a vinyl group were recently compared. present species is included in that comparison, it is found that the total dipole moments of the two conformers 135 Table 5-1. Stark Coefficients and Dipole Moments for Trans 3-methyl-l-butene. . . 2 a 2 a Tran51tion M (dv/ds )obs (dv/de )calc 423 + 322 1 1.09 1.10 2 4.16 4.16b 422 + 321 1 -1.04 -1.05 2 -4.08 -4.08b 625 + 524 2 0.083 0.086 3 0.196 0.197 4 0.353 0.353 5 0.556 0.553 624 + 523 2 -0.080 -0.083 3 -0.182 -0.183 4 -0.318 -0.322 5 -0.509 -0.500 pa = 0.312i0.003 D “b = 0 (Assumed) “C = 0.07110.042 D UT = 0.32010.010 D -!m--. aUnits are MHz/(V/cm)2. bThese slopes were calculated from frequencies obtained by direct diagonalization of the energy matrix. The other slopes were obtained by second-order perturbation. 136 Table 5-2. Stark Coefficients and Dipole Moments for Gauche 3-methy1-l-butene. . . 2 a 2 a,b Tran51tion M ((3)/95 )obs (dv/de )calc 413 + 313 2 0.305 0.314 3 0.746 0.742 524 + 423 1 1.509 1.515 2 0.878 0.852 3 0.383 0.378 4 0.088 0.094 523 + 422 2 -0.336 -0.350 3 -0.773 -0.783 “a = 0.367:0.004 D ub 0 “c = 0.154:0.006 D “T = 0.398i0.004 D aUnits are MHz/(V/cm)2. b Calculated with “b assumed to be zero. 137 of 3-methyl-l-butene are closer to that of prOpylene (0.364 0)111 and methylallene (0.401 0)112 than to trans- vinylcycloprOpane (0.498 D).109 5.2.b. Internal Rotation Relative intensity measurements were made on several transitions in the ground and first excited torsional states of both species of 3-methyl-l-butene with the sample cooled in dry ice. The present measurements were combined with the measurements of Creswell and Pagit- sis. Analysis of the relative intensities led to the fol- lowing estimates of the energy differences: E(t.v=l) - E(t,v=0) = wt = 90510 cm'1 E(g,v=l) - E(g,v=0) = mg = 104i10 cm-l E(g,v=0) - E(t,v=0) = AEgt = 130:20 cm’l (5-2) The excitation energies for the trans (wt) and gauche (mg) species were derived from relative intensity measurements on 9 and 8 transitions, respectively, whereas 9 pairs of transitions were studied to obtain AEgt (energy difference between gauche and trans conformers). The Boltzmann expres- sion was used to derive the three energy differences.1 Thus, Hm?! 138 .1 m m2 = 9 expE-AE/kT] . (5-3) where AB is the energy difference, k is the Boltzmann con- stant, ml/m2 is the equilibrium population ratio of the two rotational isomers at any temperature, T, and g is the statistical weight ratio of the rotamers. In 3-methyl-l- butene, g=l and 2 for the trans and gauche conformer, respectively. The three energy differences together with an estimate of the equilibrium torsional angle for the gauche con- figuration may be used to estimate the first four potential constants in a Fourier expansion of the torsional potential energy, V, Vn V = 7r (l-cosna) . (5-4) 2 n In this equation a is the torsional angle, which is assumed to be zero in the trans configuration, and the Vn are the potential constants. Unfortunately, the determination of the equilibrium angle for the gauche configuration offered even more uncertainty than usual. If reasonable values of bond distances and angles are transfered from the known 113 114 and adjusted structuresof propane and prOpylene slightly, it is possible to compute moments of inertia and rotational constants which agree with those for the trans 139 species. If then the rotational constants are calculated as functions of a, the plot shown in Figure (5-2) is ob- tained. The experimental rotational constants for the gauche conformer are indicated in the plot. It is evident that all one can say is that the equilibrium value of a is near 120°. The fact that the three rotational constants for the gauche species do not indicate the same equilibrium angle is evidence for either some poor assumptions about the structure for the trans species or a correlation between a and some other internal coordinate. The two)C-C-C angles andthe C-C=C angle have been varied to improve the consistency of the predicted a for the gauche species. By adjusting a and any two of the mentioned angles, the con- clusion is always the same: the angles increase and the torsional angle increases to greater than 120° . This result is most simply interpreted by assuming that as the vinyl group rotates from the trans to the gauche configura- tion, a steric hindrance causes the molecule to Open up. Each of therxssible CC distances has been changed by 0.01 A and the above calculation has been repeated without changing the conclusion. The problem then is that if only the torsional angle is changed, the predicted torsional angle for the gauche conformer is less than 120° (112° is the best compromise). On the other hand if the CCC angles are allowed to vary, the predicted gauche torsional angle 140 m m . .uUI U paw .uml m .u (mm mo mosam> Hmucmsflummxm on» xuoE mommouo one .Nmz :a ma mamom HMOfluHm> one .ousuosuum omfismmm so How oouoasoamo coaumusmflwcoo mcmnu ecu How mucmumcou who co one om .od .oumm .mcmuonaalawnuwalm new a wagon Hmnoflmnou sufi3 oUIU pom omlm .ofilfl mo mmoam> omumHsoamu mo :oflumauo> map mo poam fi .Afiiu wfiéfim Nu om. ow. oo o d _ _ A a I. IJOONI o and 1. o oouo I I. 08 l om ..m I 006 _ _ _ _ p 141 is 122-125°. As a result of this uncertainty, the tor- sional angle is kept at 120 and a i5° uncertainty in this angle is included. To obtain the torsional energy levels from HT (2-36) 11 and by we have used expressions given by Quade and Lin Knopp and Quade115 for the moments of inertia as functions of torsional angle for molecules with a planar top (here the -CH=CH2 group) attached to a frame with a plane of sym- metry (here the (CH3)2CH- group). The moment of inertia matrix t was inverted. at 100 intervals in a from 0 - 1800 and the resulting values of Fad were fit to a Fourier series of the form ___ Fm) + Fm (2) G. F cosa + F cosza . (5-5) ad do a do (n) 00. With the F given in Table (5-3) the computed Fad agree with those obtained by inversion of t to within 0.5% (over the entire range of a. The torsional energies were obtained by direct diagon- alization of a truncated matrix for HT’ Equation (2-36). The basis functions used were normalized even and odd linear combinations of the free-rotor functions, exp (ima) . Twenty basis functions were found to give the lowest few levels to sufficient accuracy. The potential constants wereami- justed in this calculation to match the four items of in- formation in Table (5-3). The effects of varying the 142 Table 5-3. Parameters Related to the Internal Torsion of the Vinyl Group in 3—Methyl-l-Butene.a w = 90+10 cm’l t _ wg = 104i10 cm’l _ —1 AEgt — 130:20 cm FO = 1.678 cm-1 Fl = 0.022 cm"1 F2 = -0.102 cm‘1 v1 = 300:100 cm"l (0.86:0.30 kcal/mole) v2 = -50:100 cm”l (-0.14:0.30 kcal/mole) v3 = 750175 cm"1 (2.14:0.21 kcal/mole) v4 = -100:50 cm‘l (-0.29:0.15 kcal/mole) aSee text for the definitions of the parameters. 143 input parameters on the calculated energies were analyzed to obtain rough estimates of the uncertainties in the de- rived potential constants. The results of this calculation show that the potential function is predominantly three-fold with con- tributions from the other terms being considerably smaller than from the V3 term. The Vn obtained are given in Table (5-3) and a plot of the potential function is shown in Figure (5-3). 5.3. Discussion The most important conclusion from the present study is the predominantly 3-fold character of the potential function for 3-methyl-l-butene. This conclusion follows directly from the existence of trans and gauche conformers in approximately equal concentrations. From the relative intensity measurements two additional conclusions can be drawn. From the trans-gauche intensity ratio it follows that the trans conformer is approximately 370 cal/mole more stable than the gauche species. If this fact is coup- led with the fact that the estimated torsional excitation energy for the gauche Species (104 cm-1) is higher than for the trans species (90 cm-l), it is concluded that the gauche-gauche potential barrier is higher than the trans- gauche barrier. All of these conclusions are confirmed by the numerical analysis of the torsional potential with 144 .omymoyocy mum HHm3 comm SH mam>ma HmCOHMHOy 3mm ymyym may mo mmymumcm UmymHso (Hmo mas .mnm magma cw cm>ym mycmymcou Hwyycmyom may Bony omymasu (you mcmyonlalahsymEIM How GOyyocsm anyycmyom Hmcowmuoy mzy mo yon .mum assays 06. ON. C Ow O _ _ 1 _ a _ e \/ _ \ / N \ .. / . \ .._ II N HOE\HMUX\> 145 the resulting constants given in Table 5-3. The coefficient of the cosBa term in the potential func— tion for rotation of the vinyl group in 3-methyl-l-butene (2140 cal/mole) is comparable to that in propylene (1978 111 In addition, the gauche-trans energy dif- cal/mole). ference (AB = 370 cal/mole) is only slightly greater than the closely related cis-skew energy difference in l-butene (AB = 150 cal/mole).116 As already indicated, the dipole moments of the two conformers are similar to that of pro- pylene. Thus, from a number of points of view the elec- tronic structure of 3-methyl-l-butene is very similar to that of propylene and l-butene. One unusual feature of the structure of 3-methyl-l-butene is the evidence presented above for significant steric interaction in the gauche conformation. The molecule appears to become more extended as it moves into the gauche configuration. Also, the gauche- gauche energy barrier predicted by the potential function in Table 5-3 is m2.7 kcal/mole whereas the trans-gauche barrier as seen from the trans configuration is m2.l kcal/ mole. Kondo gt gt. found a higher barrier for internal rotation of the methyl group in the cis form of l-butene (analogous to the gauche form of the present molecule) than in the skew form (analogous to the trans form).116 This difference was attributed to steric hindrance. In addition, they found larger CCC angles in the cis form. Comparison of the internal rotation of the vinyl group 146 in 3-methyl-l-butene with that in corresponding compounds containing the cyclopropane or the ethylene oxide ring is difficult. The corresponding 1,2-epoxy-3-butene does not appear to have been studied by either electron diffraction or microwave spectroscopy. An electron diffraction study of vinylcycloprOpane108 was interpreted to favor a mixture of 3 parts trans to one part gauche at 293 K, which would correspond to a 6.5:1 ratio at 204 K. Here trans refers to the conformer with the vinyl group trans to the ring and the gauche conformer is rotated ~120° from trans. A microwave investigation109 failed to uncover transitions belonging to another species. Also, as indicated in the introduction, a number of related cyclopropyl derivatives have shown trans- cis conformers rather than trans-gauche. However, if ac- cepted, the electron diffraction results predict a gauche- trans energy difference of approximately 1 kcal/mole, which is larger than the 0.37 kcal/mole difference determined for 3-methyl-1-butene. The corresponding aldehyde compounds can be intercom— pared because all three compounds have been studied. An electron diffraction study of 2-methylpropana1 (isopropyl- carboxaldehydegoo Showed that m10% of the molecules have the oxygen atom trans to the isopropyl group (aldehyde hydrogen trans to isoprOpyl hydrogen) and m90% of the molecules are gauche conformers (at 266 K). Electron dif- fraction99 and microwavelOBStudies of cyclopropanecarbox- aldehyde predict an approximately 50-50 mixture (458-558) 147 of species containing the oxygen atom cis and trans to the ring. Thus, the predominantly 3-fold potential in the isoprOpyl compound becomes a predominantly 2-fold poten- tial when the methyl carbons are joined to form a cyclo- prOpane ring. Finally, in a microwave study of glycid- aldehyde,117 only species with the aldehyde oxygen ap- proximately trans to the ring were found. It was con- cluded that any other species, if present, occur with considerably lower concentration. Thus, in the aldehyde compounds there is an increase in the relative stability of the trans species in going from isopropyl to cyclopropyl to ethylene oxide group. A corresponding increase in the relative stability of the trans species occurs for the vinyl compounds in going from the isoprOpyl to the cycloprOpyl group. Comparison of the two groups of molecules shows that the relative stability of the trans species is greater in the vinyl compounds than in the aldehydes. We do not at present have a consistent interpretation of these inter- esting results. APPENDICES .148 APPENDIX A Inversion frequencies of ammonia in the ground (v2 = 0) and excited state (v2 = 1) in MHZ. (Pint-«91.11 Transition 14NH3 15NH3 J K v2=0a v2=1b v2=0c 1 1 23694.496 1066650.82 22624.94 2 1 23098.815 l045318.70 22044.24 2 2 23722.631 1067676.77 22649.84 3 0 - - 21011.889 3 1 22234.504 1014084.00 21202.28 3 2 22834.182 l0358l6.20 21784.01 3 3 23870.130 1073050.70 22789.45 4 1 21134.311 973826.50 20131.40 4 2 21703.358 994747.80 20682.88 4 3 22688.312 1030531.20 21637.73 4 4 24139.417 1082593.49d 23046.02 5 0 19663.821e - - 5 1 19838.346 925657.40 - 5 2 20371.450 945604.80 19387.37 5 3 21285.275 979650.00 20272.04 5 4 22653.022 1029374.50 21597.88 5 5 24532.982 1096591.63d 23421.97 6 0 18230.132e 864695.41 - 6 1 18391.562 870877.055 17466.91 6 2 18884.695 889710.90 - 6 3 - - 18788.04 6 4 20994.617 968810.00 19984.32 6 5 22732.429 1032323.00 21667.92 6 6 25056.025 1115083.22d 23922.38 7 0 - 805159.23f — 7 2 17291.493 828521.90 - 7 3 18017.337 858394.20 17097.18 7 4 19218.465 902459.20 - 7 5 20804.830 961885.20 19793.18 7 6 22924.940 1039360.80 21846.42 7 7 25715.182 1138215.24d 24553.43 8 2 15639.761 763583.33 14799.99 8 5 18808.507 887018.80 17855.33 8 6 20719.221 958827.90 19701.85 8 7 23232.238 1050521.50 22134.88 149 Append ix A (cont. ) Transition 14NH 15 NH 3 3 _ a _ b _ c J K v2 — 0 v2 - 1 v2 — 0 8 8 26518.981 1166172.70d - 9 4 15523.900 759001.20 14681.19 9 5 16798.134 809481.89 15907.90 9 6 18499.390 875368.80 17548.34 9 7 20735.452 959569.00 - 9 8 23657.471 1065868.20 - 9 9 - - 26242.76 10 1 12017.172 615055.022 - 10 5 14822.527 730985.03 13998.96 10 6 16319.324 790818.23 - 10 7 - — 17332.55 10 8 20852.527 964060.30 19810.62 10 9 - - d 23054.96 10 10 28604.737 1237497.36 27323.41 11 3 10536.183 580479.16 - 11 7 - - 15057.77 11 8 18162.253 862810.70 - 11 9 21070.739 972301.60 20009.80 11 10 24881.922 1109551073 23696.82 12 3 10836.127 515386.76 - 12 7 - - 12923.03 12 10 21391.625 984314.106' 20306.75 13 7 11673.171 600861.05d - 13 9 - - 14537.32 13 12 26654.847 1171597.15d - 13 13 33156.849 1387550.49d - 14 11 - - 17312.98 14 12 22353.912 l019895.50 21206.06 14 13 27772.294 1210022.zsd' - 14 14 35134.303 1450547.30d - aRef. 43; bRef. 71; CRef. 46. dCalculated values from Ref. 71. 6Calculated from pparameters given in Ref. 70. fCalculated from parameters given in Ref. 71. 9Calculated from parameters given in Ref. 46. Input data for the least—squares fitting program in 14NH3. 150 APPENDIX B Trans. Frequency“:1 Wt?> Ref? Tyd Trans. Frequencya th.D Ref? Tyd Q( 1, 1) 28474676.59 1.00 SC 1 R( 5, 2) 32060743.95 1.00 SC 2 Q( l, l) 28474704.38 1.00 SC 1 Q( 5, 3) 28470887.76 0.10 F 2 Q( 1, 1) 28474672.80 0.25 J 1 Q( 5, 3) 28470873.55 1.00 SC 2 R( 1, 1) 29670838.l9 0.10 F l R( 5, 3) 32043670.75 1.00 SC 2 Q( 2, l) 2847897l.81 0.10 F 1 Q( 5, 4) 28441987.32 0.10 F 2 Q( 2, 2) 28466383.96 0.10 F l R( 5, 4) 32019271.24 1.00 SC 2 Q( 2, 2) 2846637l.86 0.10 F 1 Q( 5, 5) 28403916.93 1.00 SC 2 Q( 2, 2) 28466358.7l 0.25 J 1 Q( 5, 5) 28403904.74 0.10 F 2 Q( 2, 2) 28466371.7l 4.00 JL 1 R( 5, 5) 3198704l.32 1.00 SC 2 R( 2, 2) 30259929.55 0.10 F 1 Q( 6, 4) 28454617.07 0.10 F 2 Q( 3, 2) 28472808.88 0.10 F 1 Q( 7, 3) 28495881.48 0.10 F 2 Q( 3, 3) 28451846.08 1.00 SC 1 Q( 7, 6) 28386700.50 0.10 F 2 Q( 3, 3) 28451856.65 1.00 SC 1 Q( 7, 7) 28330749.74 0.10 F 2 Q( 3, 3) 28451858.87 1.00 SC 1 Q( 8, 7) 28349869.93 0.10 F 2 Q( 3, 3) 28451813.57 0.10 F 1 Q( 8, 8) 28284545.58 0.10 F 2 Q( 4, 1) 28493353.05 0.25 J 1 Q( 9, 5) 28464920.29 1.00 SC 2 Q( 4, 2) 28481144.98 0.10 F 1 Q( 9, 6) 28422254.73 0.10 F 2 R( 4, 2) 31461684.59 1.00 SC 1 Q( 9, 7) 28370075.71 0.10 F 2 Q( 4, 3) 28460479.02 1.00 SC 1 Q( 9, 8) 28306897.39 0.10 F 2 Q( 4, 3) 28460482.32 1.00 SC 1 Q(10,l0) 28172565.83 0.10 F 2 R( 4, 3) 31443769.38 0.25 J 1 Q(ll, 9) 28283986.22 0.10 F 2 Q( 4, 4) 28431019.50 1.00 SC 1 Q(14,12) 28110720.59 0.25 J 2 Q( 4, 4) 28431013.31 1.00 SC 1 Q(14,12) 28110724.39 4.00 JL 2 Q( 4, 4) 28431011.53 1.00 SC 1 Q( 7, 4) 28468520.88 1.00 SC 2 Q( 4, 4) 28431020.20 0.10 F 1 Q( 1, 1) 28474667.60 1.00 SC 2 Q( 4, 4) 28431011.16 0.25 J 1 Q( 1, 1) 28474661.29 0.10 F 2 Q( 5, 2) 28491167.63 0.25 J l P( 2, 1) 27282815.81 1.00 SC 2 Q( 5, 3) 28470880.12 0.10 F l R( 2, l) 30271494.27 0.25 J 2 Q( 5, 3) 28470863.49 0.10 F 1 Q( 2, 2) 28466358.93 0.10 F 2 Q( 5, 4) 28442037.53 0.10 F 1 R( 2, 2) 30259919.08 0.25.3 2 Q( 5, 4) 28442012.38 0.25 J 1 Q( 3, 2) 28472799.79 1.00 SC 2 Q( 5, 5) 28403922.71 1.00 SC 1 Q( 3, 2) 28472815.99 0.10 F 2 Q( 5, 5) 28403901.94 0.10 F 1 Q( 3, 3) 28451853.54 1.00 SC 2 Q( 5, 5) 28403911.50 0.25 J 1 Q( 3, 3) 28451865.7l 0.10 F 2 R( 6, 1) 32667019.09 1.00 SC 1 Q( 3, 3) 28451839.47 0.10 F 2 R( 6, 2) 32657460.82 0.25 J 1 R( 3, 3) 30842318.81 1.00 SC 2 Q( 6, 4) 28454618.46 1.00 SC 1 Q( 1, 1) 28474669.29 4.00 FL 2 Q( 6, 4) 28454618.85 0.25 J 1 Q( 2, 2) 28466373.53 4.00 FL 2 Q( 6, 4) 28454619.65 4.00 JL 1 Q( 3, 3) 28451852.47 4.00 FL 2 Appendix 8 (cont.) 151 Trans. Frequencya Wt.b Ref? 'Iyd Trans. Frequencya Wtk.J RefC 'Iyd Q( 6, 5) 28417429.57 0.10 F 1 R( 5, 5) 31987039.07 4.00 FL 2 R( 6, 5) 32587823.93 1.00 SC 1 Q( 5, 5) 28403916.24 4.00 FL 2 R( 6, 5) 32587816.46 0.25 J 1 R( 5, 4) 32019269.89 4.00 FL 2 Q( 6, 6) 28370521.34 1.00 SC 1 Q( 5, 4) 28442011.32 4.00 FL 2 Q( 6, 6) 28370525.0l 0.10 F 1 R( 5, 3) 32043669.07 4.00 FL 2 Q( 6, 6) 28370521.77 0.25 J 1 Q( 7, 7) 28330741.94 4.00 FL 2 Q( 6, 6) 28370519.67 4.00 JL 1 Q( 7, 5) 28432333.84 4.00 FL 2 R( 7, 2) 33251080.44 0.25 J 1 Q( 9, 6) 28422290.18 4.00 FL 2 Q( 7, 3) 28495902.37 1.00 SC 1 Q(10,l0) 28172576.73 4.00 FL 2 Q( 7, 4) 28468521.32 0.25 J l Q(11, 9) 28283986.22 4.00 FL 2 R( 7, 4) 33214248.46 0.25 J 1 Q( 8, 8) 28284538.98 4.00 FL 2 Q( 7, 6) 28386705.99 0.10 F 1 Q( 3, 1) 28485234.95 1.00 SA 4 Q( 7, 6) 28386718.95 1.00 SC 1 Q( 5, 1) 28503164.02 1.00 SA 4 Q( 7, 6) 28386716.91 1.00 SC 1 Q( 8, 2) 28528656.29 1.00 SA 4 Q( 7, 7) 28330747.96 1.00 SC 1 Q(13, 7) 28449907.67 1.00 SA 4 Q( 7, 7) 28330751.52 1.00 SC 1 Q(12, 3) 28566954.50 1.00 SA 4 Q( 7, 7) 28330683.39 0.25 J 1 Q( 7, 6) 28386708.72 1.00 SA 3 Q( 8, 5) 28448315.26 1.00 SC 1 Q( 7, 5) 28432330.04 1.00 SA 3 Q( 8, 5) 28448324.l0 0.25 J 1 Q( 7, 4) 28468520.89 1.00 SA 3 Q( 8, 6) 28404116.80 1.00 SC 1 Q( 5, 5) 28403921.40 1.00 SA 3 Q( 8, 6) 28404121.13 0.10 F 1 Q( 5, 4) 28442009.59 1.00 SA 3 Q( 8, 6) 28404126.98 0.25 J 1 Q( 5, 3) 28470872.62 1.00 SA 3 Q( 8, 7) 2834988l.84 1.00 SC 1 Q( 2, 2) 28466367.96 1.00 SA 3 Q( 8, 7) 2834987l.40 0.10 F 1 Q( 2, l) 28478946.85 1.00 SA 3 Q( 8, 8) 28284558.90 1.00 SC 1 R( 1, 1) 29670796.26 1.00 SA 4 Q( 9, 4) 28498759.68 1.00 SC 1 R( 6, 5) 32587822.85 1.00 SA 4 Q( 9, 4) 28498767.93 0.25 J 1 R( 6, 1) 32667015.46 1.00 SA 4 Q( 9, 5) 28464944.73 1.00 SC 1 R( 6, 0) 32670352.78 1.00 SA 4 Q( 9, 6) 28422273.46 0.10 F 1 R( 5, 4) 32019267.72 1.00 SA 3 Q( 9, 6) 28422287.83 0.10 F 1 R( 5, 3) 32043671.5 1.00 SA 3 Q( 9, 7) 28369973.51 1.00 SC 1 R( 5, 5) 31987039.52 1.00 SA 3 Q( 9, 7) 28369980.90 0.25 J 1 R( 5, 2) 32060743.82 1.00 SA 3 Q( 9, 8) 28306920.86 0.10 F 1 R( 5, 1) 32070849.70 1.00 SA 3 Q(10, 5) 28481798.53 0.25 J l R( 5, 0) 32074180.08 1.00 SA 3 Q(10, 6) 28440801.60 0.25 J 1 Q(12,10) 28232155.07 1.00 SA 4 Q(10, 8) 28329924.06 0.25 J 1 Q(11, 9) 28283985.62 1.00 SA 4 Q(10,l0) 28172614.44 0.25 J 1 Q( 7, 6) 28386711.36 1.00 SA 4 Q(ll, 9) 28283986.20 1.00 SC 1 Q( 3, 3) 28451855.70 1.00 SA 4 Q(11, 9) 28284008.12 0.10 F 1 Q(11, 8) 28353025.43 1.00 SA 4 Q(1l,l0) 28202444.52 0.25 J 1 Q( 8, 6) 28404109.77 1.00 SA 4 Q(14,12) 28110744.83 0.25 J 1 Q( 5, 4) 28442007.82 1.00 SA 4 R( 1, 1) 29670797.19 4.00 FL 1 Q( 5, 3) 28470875.23 1.00 SA 4 Q( 2, 1) 28478945.81 4.00 FL 1 Q( 4, 2) 28481135.24 1.00 SA 4 152 Appendix B (cont.) - --..“ ..- c Trans. Frequencya Wt.b Ref. Tyd Trans. Frequencya w? Ref? Tyd Q( 3, 2) 28472800.88 4.00 FL 1 Q( 9, 6) 28422287.66 1.00 SA 4 Q( 5, 4) 28442008.03 4.00 FL 1 Q(10, 1) 28566562.53 1.00 SA 4 Q( 5, 3) 28470877.12 4.00 FL 1 Q(ll, 3) 28552915.58 1.00 SA 4 Q( 5, 3) 2847087l.89 4.00 FL 1 Q(14,14) 27867759.49 1.00 SA 3 Q( 6, 5) 28417415.57 4.00 FL 1 Q(14,13) 27998634.52 1.00 SA 3 Q( 8, 7) 28349881.00 4.00 FL 1 Q(13,13) 27954330.68 1.00 SA 3 Q( 8, 6) 28404105.93 4.00 FL 1 Q(13,12) 28072926.65 1.00 SA 3 Q( 8, 5) 28448312.88 4.00 FL 1 Q( 8, 2) 28528657.65 1.00 SA 3 Q( 9, 8) 28306918.66 4.00 FL 1 Q( 5, 2) 28491162.69 1.00 SA 3 Q(ll, 9) 28283982.72 4.00 FL 1 Q( 2, 2) 28466366.46 1.00 SA 3 P( 4, 1) 26106297.18 0.25 J 2 R( 4, 2) 31461681.24 1.00 SA 4 R( 4, 1) 31472318.32 0.25 J 2 R( 1, 1) 29670802.86 1.00 RX 4 P( 4, 2) 26092774.60 0.25 J 2 Q( 2, 2) 28466368.26 1.00 HR 3 R( 4, 2) 31461703.15 0.25 J 2 Q( 2, 2) 28466369.32 1.00 HK 4 P( 4, 3) 26070012.83 0.25 J 2 P( 4, 3) 26070009.42 1.00 HK 3 R( 4, 3) 31443680.03 0.25 J 2 Q( 5, 4) 28442012.29 1.00 HK 3 R( 4, 4) 31418036.33 0.25 J 2 Q( 5, 3) 28470872.53 1.00 HK 4 R( 5, 1) 32070838.32 1.00 SC 2 Q( 6, 4) 28454617.50 1.00 HK 4 Q( 5, 2) 28491163.27 1.00 SC 2 Q( 6, 6) 28370517.97 1.00 HK 4 R( 5, 2) 32060693.86 0.10 F 2 Q(14,13) 27998643.52 1.00 HK 3 a . . . . The frequency of the tran51tion from the 1nvers1on free ground state (v2 = 0) to the inversion free level in the'v2 = l excited state in MHZ. 1:)Weight for each transition in the least squares fit. C References as follows: SC, this work; F, Ref. 28; FL, Ref. 28, Lamb dip; J, Ref. 29; JL, Ref. 29, Lamb dip; SA, Ref. 68; Ex, Ref. 58. d This column shows the type of transition. 1 = two-photon (s + s) 2 = two-photon (a + a), 3 = IR (a + s) and 4 = IR transition (s + a). 153 APPENDIX C Input data for the least-squares fitting program in 1511113. c Trans. Frequencya Wtb. Recf 'Iyd Trans. Frequencya Wt}? Ref 'Iyd P( 2, l) 26633848.878 8.25 J 1 Q( 8, 7) 277l8118.784 1.88 SC 1 Q( 2, 1) 27838869.685 1.88 SC 1 Q( 8, 7) 27718116.985 1.88 SC 1 Q( 2, 1) 27838875.882 1.88 SC 1 Q( 9, 4) 28886881.248 1.88 SC 1 Q( 2, 2) 2781418l.843 1.88 SC 1 Q( 9, 5) 27947664.?34 1.88 SC 1 Q( 3, l) 27868491.683 1.88 SA 4 Q( 9, 6) 27871837.928 1.88 SC 1 Q( 3, 2) 27837874.294 1.88 SC 1 Q( 9, 9) 27518838.458 1.88 SC 1 Q( 3, 2) 27837872.496 1.88 SC 1 Q( 9, 9) 27518839.957 1.88 SC 1 Q( 3, 3) 27797284.648 1.88 SC 1 Q(l8, 7) 27845173.488 1.88 8C 1 Q( 3, 3) 27797296.932 8.18 F l Q(l8, 8) 27735888.258 1.88 SC 1 Q( 3, 3) 27797298.336 1.88 SA 4 Q(l8, 8) 27735888.844 1.88 SC 1 Q( 4, l) 27889426.452 8.25 J l Q(18, 9) 27682514.277 1.88 SC 1 Q( 4, 2) 27866617.642 1.88 SC 1 Q(18, 9) 27682513.378 1.88 SC 1 Q( 4, 2) 27866611.946 1.88 SC 1 Q(l8, 9) 27682518.679 8.25 J 1 Q( 4, 2) 27866617.642 1.88 SA 4 Q(ll, 7) 27912914.89l 1.88 SC 1 Q( 4, 3) 27827876.862 1.88 SC 1 Q(11, 7) 27912896.384 1.88 SA 4 Q( 4, 4) 27772883.487 8.25 J 1 Q(ll, 9) 27687888.883 1.88 SC 1 Q( 4, 4) 27772867.598 8.18 F l Q(ll, 9) 27687829.869 1.88 SC 1 Q( 4, 4) 27772876.591 4.88 FL 1 Q(11, 9) 27686995.l92 1.88 D 4 Q( 5, 3) 27864539.181 1.88 SC 1 Q(12, 7) 27978982.289 1.88 SC 1 Q( 5, 3) 27864547.275 1.88 SC 1 Q(12, 7) 27978915.l88 1.88 SA 4 Q( 5, 3) 27864525.891 8.18 F 1 Q(12,18) 27638446.548 1.88 SC 1 Q( 5, 3) 27864539.181 4.88 FL 1 Q(12,12) 27257328.858 1.88 D 4 Q( 5, 3) 27864534.984 1.88 SA 4 Q(13, 9) 27849494.896 1.88 SC 1 Q( 5, 4) 27818562.?48 1.88 SC 1 Q(14,12) 27494187.226 1.88 SC 1 Q( 5, 4) 27818552.255 8.25 J l R( l, 1) 29827889.328 1.88 SC 1 Q( S, 4) 27818499.492 8.18 F 1 R( l, 1) 29827832.484 8.18 F 1 Q( 5, 5) 27738484.247 1.88 SC 1 R( l, l) 29827888.128 1.88 SA 4 Q( 5, 5) 27738588.?36 8.25 J l R( 2, l) 29642394.?97 1.88 SC 1 Q( 6, 4) 27854426.882 1.88 SC 1 R( 2, 2) 29619812.831 4.88 FL 1 Q( 6, 4) 27854435.875 8.25 J 1 R( 4, 2) 38865252.131 1.88 SC 1 Q( 6, 4) 27854421.185 1.88 SA 4 R( 4, 4) 38779321.719 1.88 SC 1 Q( 6, 6) 2769648l.788 8.18 F l R( 5, 2) 31491784.746 1.88 SC 1 Q( 7, 3) 27952173.313 1.88 SC 1 R( 5, 4) 314l8336.277 1.88 SC 1 Q( 7, 6) 27751665.222 8.18 F 1 R( 5, S) 31346858.28l 1.88 SA 4 Q( 7, 6) 27751691.884 8.18 F l R( 6, 1) 32136955.549 1.88 SC 1 Q( 7, 7) 27645758.739 1.88 SC 1 R( 6, 1) 32136945.856 1.88 SA 4 Q( 7, 7) 27645787.228 8.25 J l R( 6, 5) 31981785.626 1.88 SC 1 Q( 7, 7) 27645785.121 8.18 F l R( 7, 3) 327l4992.882 1.88 SC 1 Q( 8, 2) 28833535.?87 1.88 SC 1 R( 7, 3) 327l4997.678 8.25 J l 154 Appendix C (cont.) Trans. Frequencya Wt. Refc'Iyd Trans. Frequencya Wtb Refc’Iyd Q( 8, 2) 28033534.288 1.00 SA 4 R( 7, 5) 32616627.978 1.00 SC 1 Q( 8, 5) 27891034.239 1.00 SC 1 R( 7, 7) 32453349.913 1.00 SC 1 Q( 8, 5) 27891021.647 1.00 SC 1 R( 7, 7) 32453334.024 1.00 SA 4 Q( 8, S) 27891026.444 0.25 J 1 a . . . . The frequency of the trans1t1on from the 1nver51on free ground state to the v2= 1 ,s excited state in MHZ. b. . weight for each trans1tion 1n the least squares f1t. C:References as follows; sc, this work; F, Ref. 28; FL, Ref. 28, Lamb dip; J, Ref. 30; SA, Ref. 68; D, Ref. 66. d This column shows the type of transition. Type 1 = two-photon(s + s) , type 4 = IR transition (5 + a). 155 APPENDIX D Input data for the least-squares fitting program in 15NH3. Trans. Frequencya Wtb RefE2 'Iyd Trans. Frequencya WP. Refs3 'Iyd P( 2, l) 27663811.496 1.88 SC 2 Q( 8, 8) 28711952.929 1.88 D 3 P( 2, l) 27663816.592 .25 J 2 Q( 9, 6) 28711821.828 1.88 SC 2 Q( 3, 1) 28837837.588 4.88 FL 2 Q(18, 5) 28783737.116 1.88 SC 2 Q( 3, 2) 28835761.518 4.88 FL 2 Q(18, 8) 28662325.885 1.88 SC 2 Q( 3, 3) 28832485.386 .18 F 2 Q(ll, 6) 28678558.99l 1.88 D 3 Q( 4, 4) 28816572.782 4.88 FL 2 Q(1l, 8) 28637995.628 1.88 SC 2 Q( 6, 3) 28792724.212 .18 F 2 Q(ll, 9) 28621119.711 1.88 D 3 Q( 6, 4) 28786799.114 1.88 SC 2 Q(13,13) 28491877.983 1.88 SC 2 Q( 7, 5) 28761714.579 1.88 SC 2 R( 3, l) 31288173.969 1.88 SA 3 Q( 7, 7) 28744177.628 .18 F 2 R( 3, 2) 31199888.881 1.88 SA 3 Q( 7, 7) 28744212.896 .18 F 2 R( 3, 3) 31197242.898 1.88 SA 3 Q( 7, 7) 28744192.3l8 1.88 SC 2 R( 6, 5) 32987882.456 1.88 D 3 Q( 8, 6) 28732633.512 1.88 SC 2 R( 6, 6) 32985393.178 1.88 D 3 Q( 8, 8) 28711951.438 1.88 SC 2 aThe frequency of the transition from the inversion free ground state bweight for each transition in the least squares fit. to the v 2 = l,a excited state in MHZ. CReferences as follows; SC, this work; F, Ref. 28; FL, Ref. 28, Lamb d dip; J, Ref. 38; SA, Ref. 68; D, Ref. 66. This column shows the type of transition. Type 2 = two-photon(a + a), type 3 =IR transition (a + s). 156 APPENDIX E Input data for the least-squares fitting program in CH3F. Trans. Frequencya WtE> 'ch Trans. Frequency Wtb '13! P( 5, 2) 31174653.158 1.88 2 Q(18, 6) 31321982.588 8.36 2 P( 5, 3) 31174893.158 1.88 2 Q(18, 7) 31322281.588 8.36 2 P( 5, 4) 31175235.158 1.88 2 Q(18, 8) 3l322681.588 8.36 2 P( 4, l) 31228269.382 1.88 2 Q(18, 9) 31323883.588 8.36 2 P( 4, 2) 31228412.382 1.88 2 Q(18,l8) 31323667.588 8.36 2 P( 4, 3) 31228652.382 1.88 2 Q(18,12) 3l325235.588 8.36 2 P( 3, l) 31281358.153 1.88 2 Q(18,13) 31326299.588 8.36 2 P( 3, 2) 31281588.153 1.88 2 Q(18,14) 31327588.588 8.36 2 Q( 4, 2) 31429957.281 1.88 2 P( 5, 2) 31174636.888 8.36 2 Q( 5, 5) 31427589.28l 1.88 2 P( 5, 3) 3ll74878.888 8.36 2 Q( 5, 4) 31427895.28l 1.88 2 P( 5, 4) 3ll75223.888 8.36 2 Q( 5, 3) 31426784.281 1.88 2 Q( 6, 2) 31422581.888 8.18 2 Q( 6, 6) 31423944.281 1.88 2 Q( 6, 2) 31422581.888 8.18 2 Q( 6, 5) 31423438.281 1.88 2 Q( 6, 3) 3l422713.888 8.36 2 Q( 6, 4) 3l423824.281 1.88 2 Q( 6, 4) 31423818.888 8.36 2 Q( 6, 3) 3l422727.281 1.88 2 Q( 6, 5) 3l423415.888 8.36 2 R( 2, 2) 31585869.798 1.88 2 Q( 6, 6) 31423934.888 8.36 2 R( 2, l) 31585751.798 1.88 2 Q(18, 2) 3l399484.888 8.36 2 R( 3, 2) 31634238.435 1.88 2 Q(18, 3) 31399658.888 8.36 2 R( 4, 4) 31682368.478 1.88 2 Q(18, 4) 31399928.888 8.36 2 R( 4, 3) 31682888.478 1.88 2 Q(18, 5) 31488281.888 8.36 2 R( 4, 2) 31681884.478 1.88 2 Q(18, 6) 3l488729.888 8.36 2 R( 5, 3) 31729853.845 1.88 2 Q(18, 7) 31481299.888 8.36 2 R( 5, 4) 31729322.845 1.88 2 Q(l4, 4) 31366874.888 8.36 2 P( 6, 5) 31121264.997 1.88 2 Q(l4, 5) 3l366352.888 8.36 2 P( 5, 3) 31174887.674 1.88 2 Q(l4, 6) 31366716.888 8.36 2 P( 5, 4) 31175234.234 1.88 2 Q(l4, 7) 31367174.888 8.36 2 P( 4, 2) 31228418.221 1.88 2 Q(l4, 8) 31367729.888 8.36 2 P( 4, 3) 31228658.855 1.88 2 Q(l4, 9) 3l368488.888 8.36 2 P( 3, 1) 31281365.561 1.88 2 Q(14,18) 31369221.888 8.36 2 P( 3, 2) 31281586.?68 1.88 2 Q(14,11) 3l378288.888 8.36 2 P( 2, 8) 31333746.499 1.88 2 Q(14,12) 31371396.888 8.36 2 P( 2, 1) 31333787.878 1.88 2 Q(16, 2) 31344776.888 8.36 2 Q(12, 2) 31383944.647 9.88 2 Q(16, 3) 31344982.888 8.36 2 P( 1, 8) 31385488.588 1.88 2 Q(16, 4) 3l345884.888 8.36 2 Q( 4, 4) 31438495.221 1.88 2 Q(16, 5) 31345323.888 8.36 2 Q( 3, 3) 3l432893.868 1.88 2 Q(16, 6) 31345629.888 8.36 2 Q( 2, 2) 3l434786.486 9.88 2 Q(16, 7) 31346823.888 8.36 2 Q( l, 1) 31435926.26l 9.88 2 Q(16, 8) 31346583.888 8.36 2 R( 8, 8) 31486945.438 1.88 2 Q(l9,1l) 31311228.888 8.36 2 R( l, 8) 31536674.515 1.88 2 Q(l9,12) 31311996.888 8.36 2 157 ”<2 Appendix E (cont.) Trans. Frequencya Wt}? 'I‘yC Trans . Frequency a Wt F R( l, 1) 31536714.887 1.88 2 Q(19,13) 31312934.888 8.36 2 R( 2, l) 31585745.443 1.88 2 Q(l9,14) 31314893.888 8.36 2 R( 2, 2) 31585871.356 1.88 2 Q(19,15) 31315477.888 8.36 2 R( 3, 2) 31634218.598 1.88 2 R(l4, 3) 32128863.888 8.36 2 R( 3, 3) 31634428.646 1.88 2 R(l4, 4) 32128963.888 8.36 2 R( 4, 3) 31682868.468 1.88 2 R(l4, 5) 32121111.888 8.36 2 R( 4, 4) 31682349.366 1.88 2 R(l4, 6) 32121387.888 8.36 2 R( 5, 5) 31729664.418 1.88 2 R(l4, 7) 32121558.888 8.36 2 R(12, l) 663488.888 1.88 8 R(l4, 8) 32121868.888 8.36 2 R(12, 2) 663378.888 1.88 8 R(15, 3) 32168948.888 8.36 2 R(ll, 1) 684333.888 1.88 1 R(15, 4) 32161835.888 8.36 2 R(ll, 2) 684297.388 225.88 1 R(15, 5) 32161158.888 8.36 2 R(18, 1) 554829.888 1.88 1 R(15, 6) 32161384.888 8.36 2 R(10, 2) 553995.000 1.00 1 R(15, 7) 32161506.800 0.36 2 Q(12, 3) 31384118.120 9.00 2 R(15, 8) 32161779.000 0.36 2 Q(18,5) 31321653.500 0.36 2 R(15, 9) 32162125.888 8.36 2 P(32, 3) 29477168.8l8 8.84 2 R(l6, S) 32288496.888 8.36 2 R( 8, 8) 58394.922 98888.88 1 R(l6, 6) 32288612.888 8.36 2 R( 1, 8) 188788.348 144.88 1 R(l6, 7) 32288766.888 8.36 2 Q( 1, l) 31435918.288 8.36 2 R(l6, 8) 32288988.888 8.36 2 Q( 2, l) 3l434562.280 0.36 2 R(l6, 9) 32281268.008 0.36 2 Q( 2, 2) 3l434696.288 8.36 2 R(l6,18) 32281629.888 8.36 2 Q( 3, 1) 31432539.288 8.36 2 R(l6,11) 32282897.888 8.36 2 P( l, 0) 31385488.408 0.36 2 R(l6,12) 32202716.000 0.36 2 Q(12, 1) 3138384l.488 144.88 2 R(17, 7) 32239338.888 8.36 2 Q(12, 2) 31383938.588 144.88 2 R(17, 8) 32239487.888 8.36 2 Q(12, 4) 31384345.488 8.36 2 R(17, 9) 32239696.888 8.36 2 Q(12, 5) 31384675.488 8.36 2 R(l7,18) 32239992.888 8.36 2 Q(12, 6) 3l385898.488 8.36 2 R(l7,11) 32248372.888 8.36 2 Q(12, 7) 31385688.488 8.36 2 R(l7,12) 32248887.888 8.36 2 Q(12, 8) 31386230.400 0.36 2 R(l7,13) 32241553.000 0.36 2 Q(12, 9) 31386993.488 8.36 2 R( 1, 8) 182142.666 2.25E+6 8 Q(12,18) 3l387927.488 8.36 2 R( 1, 1) 182148.911 2.25E+6 8 Q(12,12) 31398362.488 8.36 2 R( 2, 8) 153218.399 2.258+6 8 Q(13, 3) 31375286.488 8.36 2 R( 2, 1) 153287.758 2.25E+6 8 Q(13, 4) 31375531.488 8.36 2 R( 2, 2) 153199.843 2.25E+6 8 Q(13, 5) 31375829.488 8.36 2 R( 3, 8) 284273.779 2.ZSE+6 8 Q(13, 6) 31376226.488 8.36 2 R( 3, l) 284278.263 2.ZSE+6 8 Q(13, 7) 31376718.488 8.36 2 R( 3, 2) 284259.716 2.258+6 8 Q(13, 8) 31377328.488 8.36 2 R( 3, 3) 284242.148 2.ZSE+6 8 Q(13, 9) 31378843.488 8.36 2 R( 4, 8) 255331.358 98888.8 8 Q(13,11) 31379968.488 8.36 2 R( 4, 1) 255326.986 98888.8 8 Q(13,12) 31381237.488 8.36 2 R( 4, 2) 255313.887 98888.8 8 Q(18, 3) 31321314.588 8.36 2 R( 4, 3) 255291.845 98888.8 8 Q(18, 4) 31321448.S88 8.36 2 R( 4, 4) 255261.898 98888.8 8 158 Appendix E, Footnotes. aAll frequencies are in MHZ. bMEight for each transition in the least squares fit. CType 2=transitions from the ground state to the V3=l excited state ; Type 1=rotational transitions in the V3=1 excited state ; Type 8=rotationa1 ransition in the ground state. 159 APPENDIX F Input data for the least-squares fitting program in l3CH3F. Trans. Frequencya Wt Frequencya Wt? :2 3 8 P( 4, 3) 30601055.l94 1.0 2 P( 3, 2) 30652526.180 1.0 P( 4, 2) 30600878.194 1.0 2 Q( 5, 4) 30794346.060 1.0 Q( 6, 6) 30791149.681 1.0 2 Q( 5, 5) 30794638.050 1.0 Q( 6, 5) 30790784.681 1.0 2 Q( 3, 2) 30799751.950 25.0 Q( 6, 4) 30790496.681 1.0 2 Q( 3, 3) 30799909.210 25.0 Q( 6, 3) 30790280.681 1.0 2 Q( 2, 1) 30801580.160 25.0 Q( 7, 7) 30787089.681 1.0 2 Q( 2, 2) 30801675.340 25.0 Q( 7, 6) 30786642.681 1.0 2 Q( 1, l) 30802860.880 25.0 Q( 7, 5) 30786287.681 1.0 2 R( 0, 0) 30852558.480 1.0 Q( 7, 4) 30786003.681 1.0 2 R( 1, l) 30901034.690 1.0 Q( 7, 3) 30785793.681 1.0 2 R( 3, 2) 30996058.980 1.0 Q( 8, 8) 30782457.681 1.0 2 R( 3, 3) 30996199.530 1.0 R( 3, 2) 30996060.753 1.0 2 R( 4, l) 31042483.4l0 25.0 R( 3, 3) 30996204.753 1.0 2 R( 4, 2) 31042561.510 25.0 P( 6, 4) 30496133.0l0 1.0 2 R( 4, 3) 31042693.680 25.0 P( 6, 5) 30496473.l60 1.0 2 R( 4, 4) 3l042883.330 25.0 P( 5, 1) 30548502.120 1.0 2 R( 5, 4) 3l088722.770 1.0 P( 5, 2) 30548607.240 1.0 2 R( 5, 5) 31088965.690 1.0 P( 5, 3) 30548784.760 1.0 2 Q( l, 0) 30802835.910 1.0 P( 5, 4) 30549038.900 1.0 2 Q( 2, 0) 30801554.990 1.0 P( 4, 2) 30600876.870 1.0 2 R( 0, 0) 49725.344 90000.0 P( 4, 3) 30601054.220 1.0 2 R( 6, 8) 49084.309 90000.0 afidl frequencies are in MHZ. weight for each transition in the least squares fit. CType 2 Type 1 Type 8 transitions from the ground state to v = 1 excited state; rotational transitio in the v = 1 exc1ted state; rotational transition in the ground state. HGNNNNNNNNNNNNNNNNNNNN .160 APPENDIX G Comparison of observed and calculated frequencies in the V2 band of 1511113. a J ' J" K Tbeef.C IR(obs)d IR(calc)d'e x+ m(obs)f x'<--m(ca1c)f Wt. g l 2 1 2 SC 923.18648 923.18653 922.73874 922.73887 1.88 1 2 1 2 J 923.18657 923.18653 922.73891 922.73887 8.25 3 3 1 2 FL 962.28834 962.28845 961.92672 961.92683 4.88 3 3 2 2 FL 962.22879 962.22892 961.85747 961.85768 4.88 3 3 3 2 F 962.12828 962.12888 961.74819 961.74871 8.18 4 4 4 2 FL 961.68176 961.68197 961.21748 961.21768 4.88 6 6 3 2 F 968.73525 968.73366 968.42198 968.42831 8.18 6 6 4 2 SC 968.55756 968.55699 968.22426 968.22369 1.88 7 7 5 2 SC 959.71764 959.71792 959.38753 959.38781 1.88 7 7 7 2 F 959.21287 959.21286 958.88256 958.88335 8.18 7 7 7 2 F 959.21321 959.21286 958.88371 958.88335 8.18 7 7 7 2 SC 959.21256 959.21286 958.88385 958.88335 1.88 8 8 6 2 SC 958.74689 958.74587 958.41749 958.41728 1.88 8 8 8 2 SC 958.14996 958.15883 957.72761 957.72768 1.88 8 8 8 3 D 958.15881 958.15883 957.72766 957.72768 1.88 9 9 6 2 SC 958.81593 958.81452 957.72326 957.72185 1.88 18 18 5 2 SC 957.68789 957.68831 957.45361 957.45483 1.88 18 18 8 2 SC 956.48269 956.48249 956.87228 956.87288 1.88 11 11 6 3 D 956.57848 956.57843 956.34664 956.34667 1.88 11 11 8 2 SC 955.54763 955.54923 955.26871 955.26231 1.88 11 11 9 3 D 955.83152 955.83145 954.69779 954.69772 1.88 13 13 13 2 SC 958.91534 958.91527 958.38675 958.38669 1.88 4 3 1 3 SA 1841.87948 1841.87962 1848.72578 1848.72688 1.88 4 3 2 3 SA 1841.84997 1841.85819 1848.68665 1848.68687 1.88 4 3 3 3 SA 1841.88818 1841.88831 1848.62881 1848.62822 1.88 7 6 5 3 D 1898.82883 1898.82183 1897.65945 1897.65965 1.88 7 6 6 3 D 1898.88475 1898.88451 1897.68577 1897.68553 1.88 l 2 1 1 J 888.84191 888.84119 888.48957 888.48885 8.25 2 2 l 1 SC 928.21839 928.21844 928.57885 928.57818 1.88 2 2 1 1 SC 928.21857 928.21844 928.57823 928.57818 1.88 2 2 2 1 SC 927.48348 937.48348 927.78124 937.78116 1.88 3 3 l 4 SA. 928.97235 928.97252 929.32597 929.32614 1.88 3 3 2 1 SC 928.18153 928.18152 928.54485 928.54484 1.88 3 3 2 1 SC 928.18148 928.18152 928.54479 928.54484 1.88 3 3 3 1 SC 926.83752 926.83776 927.21761 927.21785 1.88 3 3 3 1 F 926.83793 926.83776 927.21882 927.21785 8.18 3 3 3 4 SA 926.83771 926.83776 927.21788 927.21785 1.88 Appendix G (cont.) 161 K Tbeef .C IR (obs)d IR (calc)d’ e x+m(obs)f 14+m(ca1c)f Wt . g 4 4 1 l J 929.95537 929.95571 938.29113 938.29147 8.25 4 4 2 1 SC 929.18536 929.18555 929.53831 929.53858 1.88 4 4 2 1 SC 929.18516 929.18555 929.53812 929.53858 1.88 4 4 2 4 SA 929.18536 929.18555 929.53831 929.53858 1.88 4 4 3 1 SC 927.87716 927.87724 928.23883 928.23812 1.88 4 4 4 1 J 925.99262 925.99248 926.37699 926.37685 8.25 4 4 4 1 F 925.99289 925.99248 926.37646 926.37685 8.18 4 4 4 1 E13 925.99248 925.99248 926.37676 926.37685 4.88 4 4 4 4 D 925.99241 925.99248 926.37678 926.37685 1.88 5 5 3 1 SC 929.12288 929.12296 929.46898 929.46186 1.88 5 5 3 1 SC 929.12315 929.12296 929.46125 929.46186 1.88 5 5 3 1 F 929.12241 929.12296 929.46851 929.46186 8.18 5 5 3 1 FL 929.12287 929.12296 929.46898 929.46186 4.88 5 5 3 4 SA 929.12274 929.12296 929.46884 929.46186 1.88 5 5 4 1 SC 927.38838 927.38889 927.66852 927.66838 1.88 5 5 4 1 J 927.29996 927.38889 927.66817 927.66838 8.25 5 5 4 1 F 927.29828 927.38889 927.65841 927.66838 8.18 5 5 5 1 SC 924.86568 924.86553 925.25624 925.25617 1.88 5 5 ‘5 1 J 924.86616 924.86553 925.25679 925.25617 8.25 6 6 4 1 SC 928.79837 928.79854 929.12367 929.12384 1.88 6 6 4 1 J 928.7986? 928.79854 929.1239? 929.12384 8.25 6 6 4 4 SA 928.79818 928.79854 929.12348 929.12384 1.88 6 6 6 1 F 923.45354 923.45384 923.85252 923.85282 8.18 7 7 3 1 SC 932.89898 932.89883 932.38414 932.38318 1.88 7 7 6 1 F 925.33155 925.33184 925.69591 925.69628 8.18 7 7 6 1 F 925.33241 925.33184 925.6967? 925.69628 8.18 7 7 7 1 SC 921.75374 921.75338 922.16325 922.16281 1.88 7 7 7 1 J 921.75469 921.75338 922.16428 922.16281 8.25 7 7 7 1 F'h 921.75462 921.75338 922.16413 922.16281 8.18 7 7 7 4 D 921.75343 921.75338 922.16294 922.16281 1.88 8 8 2 1 SC 934.85127 934.85387 935.89818 935.89991 1.88 8 8 2 4 SA 934.85121 934.85387 935.89885 935.89991 1.88 8 8 5 1 SC 938.84696 938.84698 938.34476 938.34469 1.88 8 8 5 1 SC 938.84654 938.84698 938.34434 938.34469 1.88 8 8 5 1 J 938.84671 938.84698 938.34458 938.34469 8.25 8 8 7 1 SC 923.94898 923.94857 924.31887 924.38974 1.88 8 8 7 1 SC 923.94884 923.94857 924.31881 924.38974 1.88 9 9 4 1 SC 933.96415 933.96618 934.28988 934.21184 1.88 9 9 5 1 SC 931.96844 931.96948 932.23375 932.23472 1.88 9 9 6 1 SC 929.41177 929.41186 929.78444 929.78453 1.88 9 9 9 1 SC 917.46528 917.46428 917.98296 917.98196 1.88 9 9 9 1 SC 917.46532 917.46428 917.98381 917.98196 1.88 18 18 7 1 SC 928.52593 928.52572 928.81581 928.81488 1.88 Appendix G (cont .) 162 J ' J" K TykRef.C IR(obs)d IR(ca1c)d'e x—m(obs) x-m(ca1c)f Wt.g 18 18 8 1 SC 924.83897 924.83834 925.16938 925.16875 1.88 18 18 8 1 SC 924.83923 924.83834 925.16964 925.16875 1.88 18 18 9 1 SC 928.33626 928.33621 928.72877 928.72873 1.88 18 18 9 1 SC 928.33622 928.33621 928.72874 928.72873 1.88 18 18 9 1 J h 928.33613 928.33621 928.72865 928.72873 8.25 18 18 9 4 D 928.33583 928.33621 928.72835 928.72873 1.88 11 11 7 1 SC 938.82348 938.82461 931.87462 931.87575 1.88 11 11 7 4 SA 938.82286 938.82461 931.87488 931.87575 1.88 11 11 9 1 SC 923.28546 923.28458 923.53918 923.53831 1.88 11 11 9 1 SC 923.28616 923.28458 923.53988 923.53831 1.88 11 11 9 4 D 923.28582 923.28458 923.53875 923.53831 1.88 12 12 7 1 SC 933.86819 933.86577 933.27572 933.28138 1.88 12 12 7 4 SA 933.86862 933.86577 933.27615 933.28138 1.88 12 12 18 1 SC 921.31381 921.31314 921.65249 921.65182 1.88 12 12 12 4 D 988.78549 988.78514 989.28633 989.28598 1.88 13 13 9 1 SC 928.71667 928.71725 928.95913 928.95971 1.88 14 14 12 1 SC 916.75181 916.75144 917.18478 917.18512 1.88 2 1 1 1 SC 967.85956 967.85961 968.23681 968.23695 1.18 2 1 1 1 F 967.86124 967.85961 968.23758 968.23695 8.18 2 1 1 4 SA 967.85943 967.85961 968.23677 968.23695 1.88 3 2 1 1 SC 988.39628 988.39647 988.76386 988.76413 1.88 3 2 2 1 FL 987.63283 987.63289 988.81858 988.81865 4.88 5 4 2 1 SC 1829.28984 1829.28974 1829.55399 1829.55469 1.88 5 4 4 1 SC 1826.38329 1826.38386 1826.68766 1826.68743 1.88 6 5 2 1 SC 1858.12685 1858.12788 1858.45828 1858.45123 1.88 6 5 4 1 SC 1847.37583 1847.37681 1847.73684 1847.73622 1.88 6 5 5 4 S 1845.28185 1845.28868 1845.59169 1845.59132 1.88 6 5 5 4 D 1845.28887 1845.28868 1845.59151 1845.59132 1.88 7 6 1 1 SC 1871.68213 1871.68411 1871.97345 1871.97543 1.88 7 6 1 4 SA 1871.68178 1871.68411 1871.97318 1871.97543 1.88 7 6 5 1 SC 1866.43348 1866.43323 1866.79487 1866.79461 1.88 8 7 3 1 SC 1898.96955 1898.97282 1891.25478 1891.25717 1.88 8 7 3 1 J 1898.96971 1898.97282 1891.25486 1891.25717 8.25 8 7 5 1 SC 1887.64349 1887.64388 1887.97368 1887.97399 1.88 8 7 7 1 SC 1882.11772 1882.11528 1882.52723 1882.52471 1.88 8 7 7 4 SA 1882.11719 1882.11528 1882.52678 1882.52471 1.88 1 aAll values in cm' . bType: 1 = 5+5 two-photon; 2 = afa two-photon; 3 = afs infrared; 4 = Sta infrared. CReference: so = this work; J = Ref. 30; F = Ref. 28; FL = Ref. 28, Lamb dip; D = Ref. 66; SA = Ref. 68. 163 Appendix G (cont.) dIR frequency is a<~s for first group of frequencies, s+a for second group. ‘ eCalculated values copied from Ref. 66. f x m a for first grOLp of frequencies; x = s for second group. mean of s and a in ground state. 9Weight for least squares fitting. hThese transitions were not included in the fitting. 164 APPENDI X H Comparison of observed and calculated frequencies in the v2 band of 1 a %30 J' J" K TYbRef? a+s(obs) a+s(ca1c)d s+a(obs) s<-a(ca1cfi Wt.e 1 2 l 2 SC 928.23184 928.23285 891.88171 891.88192 1.8 3 4 1 2 J 888.87792 888.87948 853.54676 853.54827 1.8 3 4 2 2 J 887.99888 887.99987 852.72375 852.72481 1.8 3 4 3 2 J 887.87698 887.87685 851.32786 851.32699 1.8 3 4 3 3 HK 887.87687 887.87685 851.32695 851.32699 1.8 1 1 1 1 SC 967.99798 967.99778 931.62797 931.62778 1.8 1 1 1 1 SC 967.99898 967.99778 931.62898 931.62778 8.8 1 1 1 1 J 967.99785 967.99778 931.62784 931.62778 1.8 1 1 1 2 SC 967.99768 967.99778 931.62767 931.62778 1.8 1 1 l 2 F 967.99747 967.99778 931.62746 931.62778 1.8 1 l 1 2 FL 967.99773 967.99778 931.62773 931.62778 4.8 2 2 1 1 F 967.77553 967.77479 932.13696 932.13619 8.8 2 2 l 1 FL 967.77467 967.77479 932.13689 932.13619 4.8 2 2 l 3 SA 967.77478 967.77479 932.13613 932.13619 1.8 2 2 2 1 F 967.73894 967.73844 931.33378 931.33328 1.8 2 2 2 1 F 967.73854 967.73844 931.33337 931.33328 1.8 2 2 2 1 J 967.73818 967.73844 931.33294 931.33328 8.8 2 2 2 1 JL 967.73853 967.73844 931.3333? 931.33328 4.8 2 2 2 2 F 967.73811 967.73844 931.33294 931.33328 8.8 2 2 2 2 FL 967.73868 967.73844 931.33343 931.33328 4.8 2 2 2 3 SA 967.73841 967.73844 931.33324 931.33328 1.8 2 2 2 3 SA 967.73836 967.73844 931.33319 931.33328 1.8 2 2 2 3 HK 967.73842 967.73844 931.33325 931.33328 1.8 2 2 2 4 HK 967.73846 967.73844 931.33329 931.33328 1.8 3 3 l 4 SA 967.44989 967.44916 932.88123 932.88125 1.8 3 3 2 1 F 967.48786 967.48686 932.89429 932.89488 8.8 3 3 2 1 FL 967.48679 967.48686 932.89482 932.89488 4.8 3 3 2 2 SC 967.48676 967.48686 932.89398 932.89488 1.8 3 3 2 2 F 967.48738 967.48686 932.89452 932.89488 8.8 1.. v .....y.- a I‘ h-- _— Appendix H (cont.) 165 b J' J" K TY Ref? a+s(obs) a+s(ca1c) d s+a(obs) 3 3 3 1 SC 967.34618 967.34634 938.75676 3 3 3 1 SC 967.34645 967.34634 938.75711 3 3 3 1 SC 967.34653 967.34634 938.75719 3 3 3 1 F 967.34582 967.34634 938.7556? 3 3 3 2 SC 967.34635 967.34634 938.75781 3 3 3 2 F 967.34675 967.34634 938.75741 3 3 3 2 F 967.34588 967.34634 938.75654 3 3 3 2 FL 967.34631 967.34634 938.75697 3 3 3 4 SA 967.34642 967.34634 938.75788 4 4 1 1 J 967.83811 967.83891 933.84179 4 4 2 1 F 966.98132 966.98184 933.87615 4 4 2 4 SA 966.98899 966.98184 933.87583 4 4 3 1 SC 966.98528 966.98528 931.77358 4 4 3 1 SC 966.98531 966.98528 931.77369 4 4 4 1 SC 966.81585 966.81475 929.89841 4 4 4 1 SC 966.81484 966.81475 929.89828 4 4 4 1 SC 966.81478 966.81475 929.89815 4 4 4 1 F 966.81587 966.81475 929.89843 4 4 4 1 J 966.81477 966.81475 929.89813 5 5 1 4 SA 966.53238 966.53244 934.99484 5 5 2 l J 966.47388 966.47378 934.25231 5 5 2 2 SC 966.47366 966.47378 934.25216 5 5 2 3 SA 966.47364 966.47378 934.25214 5 5 3 1 F 966.38814 966.38881 932.99253 5 5 3 1 F 966.37959 966.38881 932.99198 5 5 3 1 FL 966.38884 966.38881 932.99243 5 5 3 1 FL 966.37987 966.38881 932.99226 5 5 3 2 F 966.38839 966.38881 932.99279 5 5 3 2 SC 966.37992 966.38881 932.99231 5 5 3 3 SA 966.37989 966.38881 932.99228 5 5 3 4 SA 966.37998 966.38881 932.99237 5 5 3 4 HK 966.37989 966.38881 932.99228 5 5 4 1 F 966.27818 966.26928 931.17832 5 5 4 1 J 966.26934 966.26928 931.17748 5 5 4 1 FL 966.26928 966.26928 931.17734 5 5 4 2 F 966.26851 966.26928 931.17665 5 5 4 2 FL 966.26931 966.26928 931.17745 s<—a(calc)d Wt. 938.75786 1.8 938.75786 1.8 938.75786 1.8 938.75786 8.8 938.75786 1.8 938.75786 1.8 938.75786 8.8 938.75786 4.8 938.75786 1.8 933.84256 1.8 933.87587 1.8 933.87587 1.8 931.77359 1.8 931.77359 1.8 929.89813 1.8 929.89813 1.8 929.89813 1.8 929.89813 1.8 929.89813 1.8 934.99488 1.8 934.25228 1.8 934.25228 1.8 934.25228 1.8 932.99235 1.8 932.99235 8.8 932.99235 4.8 932.99235 4.8 932.99235 8.8 932.99235 1.8 932.99235 1.8 932.99235 1.8 932.99235 1.8 931.17737 8.8 931.17737 1.8 931.17737 4.8 931.17737 8.8 931.17737 4.8 .......... UUUUU Appendix H (cont.) 166 . ,. b c d d J J K TY Ref. a+s(obs) a+s(ca1c) s+a(obs) sea(calc) 5 5 4 3 SA 966.26925 966.26928 931.17739 931.1773? 5 5 4 4 SA 966.26919 966.26928 931.17733 931.1773? 5 5 4 3 HK 966.26934 966.26928 931.17748 931.1773? 5 5 5 1 SC 966.15122 966.15116 928.75453 928.75449 5 5 5 1 F 966.15853 966.15116 928.75384 928.75449 5 5 5 1 J 966.15885 966.15116 928.75416 928.75449 5 5 5 2 SC 966.15183 966.15116 928.75434 928.75449 5 5 5 2 F 966.15862 966.15116 928.75393 928.75449 5 5 5 2 FL 966.15181 966.15116 928.75432 928.75449 5 5 5 3 SA 966.15118 966.15116 928.75449 928.75449 6 6 4 1 SC 965.65287 965.65286 932.63574 932.63573 6 6 4 l J 965.65288 965.65286 932.63575 932.63573 6 6 4 1 JL 965.65211 965.65286 932.63578 932.63573 6 6 4 2 F 965.65282 965.65286 932.63578 932.63573 6 6 4 4 HR 965.65284 965.65286 932.63571 932.63573 6 6 5 1 F 965.49985 965.49948 938.38699 938.38663 6 6 5 1 FL 965.49938 965.49948 938.38652 938.38663 6 6 6 1 SC 965.35428 965.35393 927.32325 927.32383 6 6 6 1 F 965.35433 965.35393 927.32337 927.32383 6 6 6 l J 965.35422 965.35393 927.3232? 927.32383 6 6 6 1 JL 965.35415 965.35393 927.32328 927.32383 6 6 6 4 HK 965.35489 965.35393 927.32314 927.32383 7 7 3 1 SC 965.13796 965.13798 935.98482 935.98391 7 7 3 2 F 965.13726 965.13798 935.98332 935.98391 7 7 4 l J 964.97958 964.97969 934.23573 934.23583 7 7 4 2 SC 964.9795? 964.97969 934.23571 934.23583 7 7 4 3 SA 964.9795? 964.97969 934.23571 934.23583 7 7 5 2 FL 964.79888 964.79825 932.8118? 932.81123 7 7 5 3 SA 964.78995 964.79825 932.81894 932.81123 7 7 6 1 F 964.59561 964.59579 929.1615? 929.16174 7 7 6 1 SC 964.59684 964.59579 929.16288 929.16174 7 7 6 1 SC 964.5959? 964.59579 929.16193 929.16174 ? 7 6 2 F 964.59543 964.59579 929.16139 929.16174 7 7 6 3 SA 964.59578 964.59579 929.16166 929.16174 7 7 6 4 SA 964.59579 964.59579 929.16175 929.16174 e Wt. O—‘U—‘t—J .. EGG paper-wasp ....... 8838888 l—‘l—‘bi—‘l—J ..... &&&a& .58 .. GS HAHI—‘H .. ... assent: var-a . as HHH . .. Gas Hub 0 0 SS 0 o o 0 883888 l-‘Hl—‘Hl—‘H Appendix H (cont.) 167 -“A- “"..‘ J' J" K TY Ref.C a+s(obs) a+s(ca1c)d s<-a(obs) s<-a(ca1c)d 7 7 7 1 SC 964.42438 964.42485 925.59976 925.59961 7 7 7 1 SC 964.42442 964.42485 925.59988 925.59961 7 7 7 1 J 964.42215 964.42485 925.59761 925.59961 7 7 7 2 F 964.42436 964.42485 925.59982 925.59961 7 7 7 2 FL 964.42418 964.42485 925.59956 925.59961 8 8 2 4 SA 964.68958 964.61889 938.61758 938.61884 8 8 2 3 SA 964.68963 964.61889 938.61755 938.61884 8 8 5 1 SC 964.84123 964.84128 933.82688 933.82614 8 8 5 1 J 964.84152 964.84128 933.82638 933.82614 8 8 5 1 FL 964.84115 964.84128 933.82688 933.82614 8 8 6 1 SC 963.79644 963.79623 931.12226 931.12283 8 8 6 1 F 963.79658 963.79623 931.12241 931.12283 8 8 6 1 J 963.79678 963.79623 931.12268 931.12283 8 8 6 1 FL 963.79688 963.79623 931.12198 931.12283 8 8 6 4 SA 963.79628 963.79623 931.12283 931.12283 8 8 7 1 SC 963.55855 963.55858 927.74198 927.?4281 8 8 7 1 F 963.55828 963.55858 927.74164 92?.74281 8 8 7 1 FL 963.55853 963.55858 927.74196 927.74281 8 8 7 2 F 963.55816 963.55858 927.74159 927.74281 8 8 8 1 SC 963.36329 963.36273 923.5793? 923.57982 8 8 8 2 F 963.36284 963.36273 923.57893 923.57982 8 8 8 2 FL 963.36262 963.36273 923.57871 923.57982 9 9 4 1 SC 963.53399 963.53473 93?.69861 937.69933 9 9 4 1 J 963.53426 963.53473 937.69889 937.69933 9 9 5 1 SC 963.26922 963.26939 935.78749 935.78766 9 9 5 2 SC 963.26841 963.26939 935.7866? 935.78766 9 9 6 1 F 962.97311 962.97378 933.15688 933.15748 9 9 6 1 F 962.9?359 962.97378 933.15736 933.15748 9 9 6 2 F 962.97249 962.97378 933.15625 933.15748 9 9 6 2 FL 962.9736? 962.97378 933.15743 933.15748 9 9 6 4 SA 962.9?358 962.97378 933.15735 933.15748 9 9 7 1 SC 962.6781? 962.66984 929.97874 929.97848 9 9 7 l J 962.67842 962.66984 929.97898 929.97848 9 9 ? 2 F 962.67358 962.66984 929.97415 929.97848 Wt .9 bi—‘QI-‘H ..... sssss .bsI—o l-‘lv-4 .. .0 ss EGG HbGD—‘H .... sasss ...,_. i-‘H .bb-‘H I—‘bt—‘H .. .. ... .... as as ass sass Hth—‘E .. ... asaaa o‘.‘ a u o o o I O a i o O‘..-‘-. ‘ Appendix H (cont.) 168 s+‘a(ca1c)d Wt.e b J' J" K TY Ref}: a+s(obs) a+s(calc)(i s+a(obs) 9 9 8 1 F 962.38858 962.38842 926.84591 9 9 8 1 FL 962.38858 962.38842 926.84584 9 9 8 2 F 962.38779 962.38842 926.84513 18 18 1 4 SA 963.33648 963.33775 942.41952 18 18 5 1 J 962.4892? 962.48956 937.61188 18 18 6 1 J 962.14463 962.14466 935.22142 18 18 8 1 J 961.41113 961.41184 928.5579? 18 18 18 1 J 968.8535? 968.85238 918.62895 18 18 18 2 F 968.85195 968.85238 918.61933 18 l8 l8 2 FL 968.85232 968.85238 918.61978 11 11 3 4 SA 962.2?982 963.28881 942.5656? 11 11 8 4 SA 968.4481? 968.44829 931.86288 11 ll 9 1 SC 968.81989 968.81993 926.88456 11 11 9 1 F 968.82863 968.81993 926.88529 11 11 9 1 FL 968.81978 968.81993 926.88444 11 11 9 2 F 968.81989 968.81993 926.88456 11 11 9 2 FL 968.81989 968.81993 926.88456 11 11 9 4 SA 968.8198? 968.81993 926.88454 11 11 18 l J 959.65259 959.65253 921.81198 12 12 3 4 SA 961.66749 961.67816 944.11458 12 12 18 4 SA 958.49669 958.49673 924.94996 13 13 7 4 SA 959.28274 959.28281 938.77888 13 13 12 3 SA 956.39673 956.39789 916.42735 13 13 13 3 SA 956.15895 956.15139 938.?6125 14 14 12 1 J 955.8563? 928.2986? 14 14 12 2 J 955.85556 928.28986 14 14 12 2 JL 955.85568 928.28999 926.04576 926.04576 926.04576 942.4211? 937.61208 935.22146 928.55790 918.62058 918.62058 918.62058 942.56672 831.06220 926.88461 926.88461 926.88461 926.88461 926.88461 926.88461 921.8121? 944.11773 924.95002 938.77072 916.42882 908.7673? Appendix H (cont.) 1b J' J" K TY Ref.C a*'s(obs) 169 a+s(ca1c)d 14 14 13 14 14 13 14 14 14 u: n>n1n1n1 m1 P‘F‘P‘F‘ h‘ h‘h‘h‘h‘ WU) N N O\O\ U1U1 F‘Ffl mm mmm mmm UlU'IUl UlU'lU1 WWW NNN U1U1U1 bubob 3 3 N baht-4H WNN UNN MN NH DNH NH 9'38 88% 1’38"” WNN SA HK SA LIL: 98“ 954.57811 954.57841 954.34693 1007.54185 1007.54048 1007.54045 1007.54067 1027.04671 1027.03381 1027.03346 1046.37454 1065.59432 1065.58180 1065.58242 1065.58169 1065.56845 1065.56547 1065.56361 1084.62355 1084.62393 1084.60817 1084.60984 1084.60984 1084.59311 1084.59306 1084.59314 1084.58375 1084.58370 1084.58363 1007.54052 1007.54052 1007.54052 1007.54052 1027.04703 1027.03295 1027.03295 1046.37468 1065.59432 1065.58172 1065.58172 1065.58172 1065.56552 1065.56552 1065.56385 1084.62370 1084.62370 1084.60981 1084.60981 1084.60981 1084.59313 1084.59313 1084.59313 1084.58363 1084.58363 1084.58363 —¢.A OOOOOOOOOOO e s+a(obs) s+a(ca1c)d Wt. 913.28973 1.8 913.29883 1.8 984.78998 1.8 971.88341 971.88286 8.8 971.88284 971.88286 4.8 971.88281 971.88286 1.8 971.88223 971.88286 1.8 992.45881 992.45838 1.8 991.69148 991.69853 1.8 991.69185 991.69853 1.8 1811.28358 1811.28366 1.8 1834.81275 1834.81273 1.8 1833.31587 1833.31579 1.8 1833.31649 1833.31579 1.8 1833.31576 1833.31579 1.8 1832.13484 1832.13186 1.8 1832.13186 1832.13186 1.8 1838.4221? 1838.4223? 1.8 1854.91248 1854.91262 1.8 1854.91286 1854.91262 1.8 1854.25189 1854.25275 8.8 1854.25276 1854.25275 1.8 1854.25276 1854.25275 1.8 1853.13849 1853.13845 1.8 1853.13844 1853.13845 4.8 1853.13852 1853.13845 1.8 1851.51218 1851.51198 1.8 1851.51286 1851.51198 4.8 1851.51198 1851.51198 1.8 Appendix H (cont.) A b 170 J' J" K TY Ref? a+s(obs) a+s(calc) d s+a(obs) s+a(ca1c)d Wt.e 6 5 5 2 SC 1884.59931 1884.59927 1849.34639 1849.34637 1.8 6 5 5 2 FL 1884.59924 1884.59927 1849.34631 1849.34637 4.8 6 5 5 3 SA 1884.59925 1884.59927 1849.34633 1849.34637 1.8 7 6 l 1 SC 1183.48587 1183.48583 1875.82386 1875.82384 1.8 7 6 1 4 SA 1183.48575 1183.48583 1875.82294 1875.82384 1.8 7 6 2 1 J 1183.46886 1183.46982 1875.28242 1875.28329 1.8 7 6 5 1 SC 1183.43445 1183.43435 1878.59115 1878.59184 1.8 7 6 5 1 J 1183.43421 1183.43435 1878.59898 1878.59184 1.8 7 6 5 4 SA 1183.43442 1183.43435 1878.59111 1878.59184 1.8 8 7 2 1 J 1122.16825 1122.16821 1896.11387 1896.11386 1.8 8 7 4 1 J 1122.18436 1122.18486 1893.71179 1893.71158 1.8 a . -1. All values 1n cm . bType: 1 = s+s two-photon; 2 = aea two-photon; 3 = a+s infrared; 4 = s+a infrared. c:Reference: SC = this work; J = Ref. 29 ; F = Ref. 28; FL = Ref.28 , Lamb dip; HK = Ref. dCalculated values copied from Ref. 57,58 SA = Ref. 68. eWeight for least squares fitting. 63,65. REFERENCES 11. 12. 13. 14. REFERENCES D. G. Lister, J. N. Mcdonald, and N. L. Owen, "In- ternal Rotation and Inversion." Academic Press, London, New York, San Francisco, 1978. James E. Wollrab, "Rotational Spectra and Molecular Structure." Academic Press, New York and London, 1967. Walter Gordy, William V. Smith, and Ralph F. Tram- barulo, "Microwave Spectroscopy." New York. John Wiley & Sons, Inc., London. Chapman & Hall, Ltd. (1953). C. C. Costain, J. Chem. Phys. 23, 2037 (1955). R. W. Davis, A. G. Robiette, and M. C. L. Gerry, J. Mol. Spectrosc. 83, 185 (1980). A. G. Robiette, C. R. Parent, and M. C. L. Gerry, J. Mol. Spectrosc. 86, 455-464 (1981). E. B. Wilson, Jr., J. Chem. Phys. 7, 986-987 (1957). B. S. Ray, Z. Physik, 18, 74 (1932). James K. G. Watson, J. Chem. Phys. 48, 4517 (1968). E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, "Molecular Vibrations." McGraw-Hill Book Company, Inc., New York, 1955. C. R. Quade and C. C. Lin, J. Chem. Phys. 38, 540 (1963). E. B. Wilson, Jr. and J. B. Howard, J. Chem. Phys 4, 260 (1936). J. V. Knopp and C. R. Quade, J. Chem. Phys. 53, 1 (1970). (a) M. Takami, Japan. J. Appl. Phys. 15, 1063-1071 (1976). (b) M. Takami, Japan J. Appl. Phys. 15, 1889-189? (1976). 171 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 172 (a) T. Amano and R. H. Schwendeman, J. Mol. Spectrosc. 28, 437-451 (1979). (b) C. Feuillade, J. G. Baker and C. Bottcher, Chem. Phys. Lett. 49, 121-125 (1976). (c) R. H. Schwendeman, Private communication. K. Shimoda, Japan. J. Appl. Phys. ll: 564-571 (1972). W. H. Flygare, "Molecular Structure and Dynamics". Pren- tice-Hall, Inc., Englewood Cliffs, NJ 07632 (1978). (a) T. H. Maiman, Nature, 187, 493-494 (1960). (b) A. Javan, W. B. Benneth, Jr., and D. R. Herriott, Phys. Rev. Lett. 6, 106-110 (1961). W. L. Peticolas, Ann. Rev. Phys. Chem. 18, 233 (1967). W. M. McClain, Accounts, Chem. Res. 1, 129 (1974). D. M. Friedrick and W. M. McClain, Ann. Rev. Phys. Chem. 11, 559 (1980). M. G6ppert-Mayer, Ann. Phys. 9, 273 (1931). T. Oka and T. Shimizu, Phys. Rev. 11, 587 (1970). T. Oka and T. Shimizu, Appl. Phys. Lett. 19, 88 (1971). K. Narahari Rao and C. Weldon .Mathews "Molecular Spectroscopy: Modern Research" Academic Press, New York and London, 1972, Chapter I. S. M. Freund and T. Oka, Appl. Phys. Lett. 31, 60 (1972). F. Shimizu, Phys. Rev. A, 19, 950 (1974). S. M. Freund and T. Oka, Phys. Rev. A, 11, 2178 (1976). H. Jones, Appl. Phys. 11, 261-264 (1978). H. Jones, J. Mol. Spectrosc. lg, 279-287 (1978). H. Jones, J. Mol. Spectrosc. 78, 452-468 (1979). R. Guccione-Gush, H. P. Gush, R. Schieder, K. Yamada, and G. Winnewisser, Phys. Rev. A, 11, 2740-2743 (1981). K. Shimoda, Y. Ueda, and J. Iwahori, Appl. Phys. 11, 181-189 (1980). F. Shimizu, J. Chem. Phys. 53, 3572 (1970). 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 173 F. Shimizu, J. Chem. Phys. 3, 1149 (1970). T. L. Wilson, J. Bieging, and D. Downes, Astron. Astrophys. 88, 1-6 (1978). T. L. Wilson, D. Downes, and J. Bieging, Astron. AstrOphys. 11, 275-282 (1979). (a) B. Bleany and R. P. Penrose, Nature, 157, 339 (1946). (b) W. B. Good, Phys. Rev. 18, 213-218 (1946). (c) C. H. Townes, Phys. Rev. 70, 665 (1946). (d) D. K. Coles and W. B. Good, Phys. Rev. 18, 979 (1946). (e) Dailey, Kyhl, Strandberg, VanVleck, and Wilson, Phys. Rev. 18, 984 (1946). (f) W. B. Good, and D. K. Coles, Phys. Rev. 11, 383 (1947). C. H. Townes and A. L. Schawlow, "Microwave Spectros- copy", Dover Publications, Inc., New York, 1975. S. G. Kukolich, Phys. Rev. 156, 83-92 (1967). S. G. Kukolich and S. C. Wofsy, J. Chem. Phys. 88, 5477-5488 (1970). D. J. Ruben and S. G. Kukolich, J. Chem. Phys. 81, 3780-3784 (1974). R. L. Poynter and R. K. Kakar, Astrophys. J. Suppl. 88, 87-96 (1975). B. V. Sinha and P. D. P. Smith, J. Mol. Spectrosc. 88, 231-232 (1980). S. P. Belov, L. I. Gershstein, A. F. Krupnov, A. V. Maslovskij, 8. Urban, V. Spirko, and D. Papousek, J. Mol. Spectrosc. 88, 288-304 (1980). H. Sasada, J. Mol. Spectrosc. 88, 15-20 (1980). J. W. Simmon and W. Gordy, Phys. Rev. 18, 713 (1948). C. C. Costain, Phys. Rev. 88, 108 (1951). II...- '-“'. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 174 L. D. G. Young and A. T. Young, J. Quant. Spectrosc. Radiat. Transfer, 88, 533-537 (1978). (a) W. F. Colly and E. F. Barker, Phys. Rev. 88, 923 (1927). (b) G. A. Stincomb and E. F. Barker, Phys. Rev. 88, 305 (1929). (c) E. F. Barker, Phys. Rev. 88, 684 (1929). H. Sheng, E. F. Barker, and D. M. Dennison, Phys. Rev. 88, 786 (1941). W. S. Benedict, E. K. Plyler, and E. D. Tidwell, J. Research, Nath. Bur. Standards, 81, 123 (1958). W. S. Benedict and E. K. Plyler, C. J. Phys. 88, 1235 (1957). H. M. Mould, W. C. Price, and G. R. Wilkinson, Spectro- chim. Acta, 18, 313-330 (1950). J. S. Garing, H. H. Nielson, and K. Narahari Rao,.J. Mol. Spectrosc. 8, 496-527 (1959). F. O. Shimizu and T. Shimizu, J. Mol. Spectrosc. 88, 94-109 (1970). J. J. Hillman, T. Kostiuk, D. Buhl, J. L. Faris, J. C. Novaco, and M. J. Mumma, Opt. Lett. 1, 81 (197?). T. Kostiuk, M. J. Mumma, J. J. Hillman, D. Buhl, L. W. Brown, and J. L. Faris, Infrared, Physics, 18, 431 (1977). J. P. Sattler and K. J. Ratter, J. Mol. Spectrosc. 88, 486-489 (1978). N. Nereson, J. Mol. Spectrosc. _8, 489-493 (1978). F. Cappellani and G. Restelli, J. Mol. Spectrosc. 88, 36-41 (1979). G. Baldacchini, S. Marchetti, and V. Montelatici, J. Mol. Spectrosc. 88, 115-121 (1981). S. Urban, V. Spirko, D. Papousek, R. S. McDowell, N. G. Nereson, S. P. Belov, L. I. Gershstein, A. V. Maslovskij, A. F. Krupnov, J. Curtis, and K. Narahari Rao, J. Mol. Spectrosc. 88, 455-495 (1980). ' .n .4. to. urn-.rxu-wnn-A1 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 175 (a) D. Papousek, J. M. R. Stone, and V. Spirko, J. Mol. Spectrosc. 88, 17-38 (1973). (b) V. Daniels, D. Papousek, V. Spirko, and M. Horak, J. Mol. Spectrosc. 88, 339-349 (1975). (c) V. Spirko, J. M. R. Stone, and D. Papousek, J. Mol. Spectrosc. 88, 38-46 (1973). (d) V. Spirko, J. M. R. Stone, and D. Papousek, J. Mol. Spectrosc. 88, 159-178 (1976). S. Urban, V. Spirko, D. Papousek, J. Kauppinen, S. P. Belov, L. I. Gershstein, and A. F. Krupnov, J. Mol. Spectrosc. 88, 274-292 (1981). G. DiLonardo, L. Fusina, A. Trombetti, and Ian M. Mills, J. Mol. Spectrosc. 88, 298-325 (1982). M. Carlotti, A. Trombetti, B. Velino, and J. Vrbancich, J. Mol. Spectrosc. 88, 401-407 (1980). (a) J. P. Sattler, L. S. Miller, and T. L. Worchesky, J. Mol. Spectrosc. 88, 347-351 (1981). (b) J. P. Sattler and T. L. Worchesky, J. Mol. Spec- trosc. 88, 297-301 (1981). (c) T. L. Worchesky and J. P. Sattler, Private com- munication. C. M. Johnson, R. Trambarulo, and W. Gordy, Phys. Rev. 18, 1178 (1951). E. Schnabel, T. T6rring and W. Wilke, Z. Physik, 88, 167 (1965). S. P. Belov, L. I. Gershstein, A. F. Krupnov, A. V. Maslovskij, S. Urban, V. Spirko, and D. Papousek, J. Mol. Spectrosc. 88, 288-304 (1980). C. Freed, L. C. Bradley, and R. G. O'Donnell, IEEE J. Quantum Electron, QE-l6, 1195-1206 (1980). B. G. Whitford, K. J. Siemsen, H. D. Riccius and G. R. Hanes, Opt. Commun. 14, 70 (1975). P. Helminger, F. C. DeLucia, and W. Gordy, J. Mol. Spectrosc. 88, 94-97 (1971). O. R. Gilliam, H. D. Edwards, and W. Gordy, Phys. Rev. 88, 1014 (1949). 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 176 W. Orville, J. T. Cox, and W. Gordy, J. Chem. Phys. 88, 1718 (1954). R. S. Winton and W. Gordy, Phys. Lett. 32A, 219 (1970). T. E. Sullivan and L. Frenkel, J. Mol. Spectrosc. 88, 185-201 (1971). T. Tanaka and E. Hirota, J. Mol. Spectrosc. 88, 437- 446 (1975). E. Hirota, T. Tanaka, and S. Saito, J. Mol. Spectrosc. 88, 478-484 (1976). K. P. Yates and H. H. Nielsen, Phys. Rev. 81, 349 (1947). F. A. Andersen, B. Bak, and S. Brodersen, J. Chem. Phys. 88, 993 (1956). W. L. Smith and I. M. Mills, J. Mol. Spectrosc. 11, 11 (1963). J. L. Duncan, D. C. McKean and G. K. Speirs, Mol. Phys. .81, 553 (1972). R. G. Brewer, Phys. Rev. Lett. 25, 1639 (1970). T. Y. Chang and T. J. Bridges, Opt. Commun. 1, 423 (1970). T. Y. Chang, T. J. Bridges, and E. G. Burkhardt, Appl. Phys. Lett. 18, 249 (1970). T. Y. Chang and J. D. McGee, Appl. Phys. Lett. 18, 103 (1971). S. M. Freund, G. Duxbury, M. R6mheld, J. T. Tiedje, and T. Oka, J. Mol. Spectrosc. 88, 38-5? (1974). M. RBmheld, Ph.D. Thesis, University of Ulm, 1979. W. K. Bischel, P. J. Kelly, and C. K. Rhodes, Phys. Rev. Lett. 88, 300 (1975). G. Graner, Mol. Phys. 81, 1833-1843 (1976). S. M. Freund, M. Ramheld, and T. Oka, Phys. Rev. Lett. 35, 1497 (1975). J. P. Sattler and G. J. Simonis, IEEE J. Quantum Electron, QE-13, 461 (1977). 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 17? G. Magerl, W. Schupita, E. Bonek, and W. A. Kreiner, J. Mol. Spectrosc. 88, 431-439 (1980). Herlemont, M. Lyszyk, J. Lemaire, and J. Demaison, . Naturforsch 36a, 944-947 (1981). N'fl W. A. M. Blumberg, H. R. Fetterman, D. D. Peck, and P. F. Goldsmith, Appl. Phys. Lett. 35(8), 582 (1979). E. Arimondo and M. Inguscio, J. Mol. Spectrosc. 18, 81-86 (1979). L. S. Bartell and J. P. Guillory, J. Chem. Phys. 88, 647 (1965). J. P. Guillory and L. S. Bartell, J. Chem. Phys. 88, 654 (1965). (a) R. W. Kilb, C. C. Lin, and E. B. Wilson, Jr., J. Chem. Phys. 88, 1695 (1957); (b) L. Pierce and L. C. Krisher, Ibid. 81, 875 (1959); (c) K. M. Sinnott, Ibid. 88, 851 (1961); (d) L. C. Krisher and E. B. Wilson, Jr., Ibid. 81, 882 (1959). S. S. Butcher and E. B. Wilson, Jr., J. Chem. Phys. 88, 1671 (1964). H. N. Volltrauer and R. H. Schwendeman, J. Chem. Phys. 54, 260 (1971). L. S. Bartell, J. P. Guillory, and A. T. Parks, J. Phys. Chem. 88, 3043 (1965). K. P. R. Nair and James E. Boggs, J. Mol. Struct. 88, 45 (1976). P. L. Lee and R. H. Schwendeman, J. Mol. Spectrosc. 81, 84 (1972). H. N. Volltrauer and R. H. Schwendeman, J. Chem. Phys. g3. 268 (1971). A. de Meijere and W. Lfittke, Tetrahedron 88, 204? (1969). E. G. Codding and R. H. Schwendeman, J. Mol. Spectrosc. 49, 226 (1974). R. A. Creswell, M. Pagitsas, P. Shoja-Chaghervand, and R. H. Schwendeman, J. Phys. Chem. 88, 1427 (1979). D. R. Lide, Jr. and D. E. Mann, J. Chem. Phys. 88, 868 (1957). 112. 113. 114. 115. 116. 117. 178 D. R. Lide, Jr. and D. E. Mann, J. Chem. Phys. 88, 274 (1957). D. R. Lide, Jr., J. Chem. Phys. 88, 1514 (1960). D. R. Lide, Jr. and D. Christensen, J. Chem. Phys. 88, 1374 (1961). J. V. KnOpp and C. R. Quade, J. Chem. Phys. 88, 3317 (1968). S. Kondo, E. Hirota, and Y. Morino, J. Mol. Spectrosc. 88, 471 (1968). R. A. Creswell, P. J. Manor, R. A. Assink, and R. H. Schwendeman, J. Mol. Spectrosc. 64, 365 (1977).