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High resolution infrared study of vibration-torsion-rotation interactions in CH3CF3 and CH3SiF3 Wang, Shixin 2000

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High Resolution Infrared Study of Vibration-Torsion-Rotation Interactions in C H 3 C F 3 and C H 3 S i F 3 By Shixin Wang B. Sc. Peking University, 1990 M . Sc. The University of British Columbia, 1993 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E STUDIES PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A September, 2000 @ Shixin Wang, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract Vibration-torsion-rotation interactions in CH3CF3 and CH3SiF3 have been investigated using Fourier transform spectroscopy at a resolution of 0.00125 c m - 1 . Although CH3CF3 has a high barrier at ~ 1100 c m - 1 and CHsSiF3 has an intermediate barrier at ~ 415 c m - 1 , internal rotation plays an important role in determining the form of the spectrum in both molecules. The internal degree of freedom in both molecules provides excellent opportunities to study the Hamiltonian from small amplitude oscillation to large ampli-tude internal rotation. The general theory of the vibration-torsion-rotation interactions for A B 3 X Y 3 type molecule has been reviewed. Sequential contact transformations have been applied to the vibration-torsion-rotation Hamiltonian. The formalism for the effective Hamiltonian has been developed in terms of the fundamental molecular parameters. Earlier works have used the principal axis method and the internal axis method at the same time. A new approach has been introduced to simplify the effective Hamiltonian in the principal axis method so that it has the advantages of both methods. For CH3CF3, four bands have been investigated, namely the vibrational fundamentals (v\2 = 1 <— 0), (vn = 1 <— 0), (t>5 = 1 <— 0), as well as the torsional overtone (ug = 2 <— 0). For each of the three vi-brational fundamentals, both the spectrum originating in the ground torsional state (t>6 = 0) and the first torsional hot band originating in the torsional state (v6 = 1) were studied. A total of about 12,000 frequencies were included in the global analysis. Effective parameters which characterize the effective vibration-torsion-rotation Hamiltonian in the three excited vibrational states were determined. For each of the (v\2 = 1) and (vn = 1) vibrational states, the magnitude of the centrifugal distortion constant e that characterizes the (A£ = 0, AA; = ±3) matrix elements has been determined. The Fermi-like interaction parameters which couple the torsional states in the ground vibrational state with those in the (i^ = 1) v i -brational state were obtained. The theory of vibrational contact transformations was tested by comparing the results of different models. For CHsSiF3, the second and third lowest lying vibrational fundamentals, namely (vn = 1 <— 0) and (i>5 = 1 <— 0), have been studied. A total of about 3,000 frequencies were identified. The («n = 1 <— 0, VQ — 0 <— 0) band was strongly affected by the resonance of the (vn = 0, ve = 3) state with the (vu = l,t>6 = 0) state through the Coriolis-like interactions. The Coriolis-like parameters were deter-mined along with the parameters which describe the effective vibration-torsion-rotation Hamiltonian for the (vn = 1) vibrational state. The parallel band (v5 = 1 <— 0, v§ = 0 <— 0) was resonantly perturbed by the Fermi-like interactions between the (^ 5 = 0,t>6 = 4) state and the (1*5 = \,v§ = 0) state. The Fermi-like interaction parameters were obtained together with the effective parameters for the (v5 = 1) vibrational state. ii Table of Contents Abstract ii Table of Contents iii List of Tables vii List of Figures ix Acknowledgments xi 1 Introduction 1 1.1 Studies of the Torsion-Rotation Hamiltonian in Symmetric Tops 2 1.2 Studies of Vibration-Torsion-Rotation Interactions in Symmetric Tops 4 1.3 Theoretical Background 7 1.4 Thesis Organization 8 2 General Theoretical Considerations 12 2.1 Classical Kinetic Energy of AB3XY3 Molecule 12 2.1.1 Definitions of the Molecular Fixed Coordinate Systems 17 2.1.2 Hamiltonian Form of the Kinetic Energy 19 2.2 Potential Energy 23 2.3 Harmonic Oscillator, Rigid Rotor Approximation 24 2.3.1 Harmonic Oscillator 26 2.3.2 Rigid Rotor 29 2.3.3 Separation of Torsion and Overall Rotation 30 2.3.4 The Zeroth-Order Torsion-Rotation Hamiltonian in the Principal Axis Method 32 2.3.5 The Zeroth-Order Torsion-Rotation Hamiltonian in the Internal Axis Method 36 2.3.6 Comparison of the Torsion-Rotation Hamiltonian in the PAM and the IAM 38 2.4 Symmetry Classification of the Eigenstates of AB3XY3 39 2.4.1 Symmetry Classification of the Vibrational States 39 2.4.2 Symmetry Classification of Free Rotor Torsion-Rotation Wavefunctions 42 2.4.3 Symmetry Classification of the Vibration-Torsion-Rotation States 44 2.5 Expansion of the Vibration-Torsion-Rotation Hamiltonian 47 2.6 General Symmetry Requirements for the Vibration-Torsion-Rotation Hamiltonian 52 2.7 Expanded Hamiltonian for AB3XY3 Molecule 52 2.7.1 Transformation Properties of Vibrational, Torsional, and Rotational Operators in the Molecular Symmetry Group G\& 52 2.7.2 Centrifugal Distortion 54 2.7.3 Coriolis Coupling 57 2.7.4 Anharmonic Coupling 59 2.8 Theory of Contact Transformation 59 iii 2.9 Vibrational Contact Transformation for Molecule AB3XY3 62 2.10Effective Hamiltonians for AB3XY3 65 2.10.1 Vibrational Hamiltonian H": Effective Band Origins 65 2.10.2 Vibrational Dependence of Torsion-Rotation Hamiltonian: H'{2 66 2.10.3 Reduction of the Torsion-Rotation Hamiltonian 70 2.10.4 Effective Torsion-Rotation Parameters 75 2.10.5 Effective Torsion-Rotation Hamiltonian: 3a —> Ja Transformation 77 2.10.6 Effective Hamiltonian for a vt = 1 Doubly Degenerate Vibrational State 81 2.10.6.1 The z-type Coriolis Interaction 82 2.10.6.2 Quartic Centrifugal Interaction 83 2.10.6.3 Effective Hamiltonian 84 2.10.7 Reduction of the Hamiltonian Terms Off-Diagonal in A; 84 2.10.7.1 Reductions of H$ for a Nondegenerate Vibrational State 85 2.10.7.2 Reductions of H*s and Hl0 for vt = 1 State 86 2.11 Electric Dipole Selection Rules 88 2.11.1 Dipole Moment Operators 88 2.11.2 Selections Rules for Electric Dipole Transitions 91 2.12Comparison with the "High Barrier Model" 94 3 Numerical Procedures 98 3.1 Single Band Models 98 3.1.1 Single Band Model: One-Step Procedure 98 3.1.2 Single Band Sextic Model: Two-Step Procedure 105 3.1.2.1 Nondegenerate Vibrational States 106 3.1.2.2 Doubly Degenerate Fundamental Vibrational States I l l 3.2 Three Band Model: One Step Procedure 115 3.3 Two Band Sextic Model: Two Step Procedure 118 4 Experimental 119 5 Analysis of CH3CF3 123 5.1 Spectrum Simulation 124 5.2 The Torsional Overtone (ve = 2 <- 0) in the Ground Vibrational State 127 5.2.1 Spectra and Data Set 127 5.2.2 Frequency Analysis and Results 131 5.2.3 Measurement of the Torsional Distortion Dipole Moment 137 5.3 The Torsional Fundamental and First Hot Band in the (v5 = 1 <- 0) Band 140 5.3.1 Characteristics of the Spectra and Data Set 140 5.3.2 Analysis and Results 144 5.3.3 Discussion 146 iv 5.4 The vn and ^ 12 Bands for (vq = 0) and for (vq = 1) 149 5.4.1 Characteristics of the Spectra and Identifications 149 5.4.2 Analysis and Results 155 5.4.3 Discussion 160 5.5 Summary 165 6 Analysis of CH 3SiF 3 167 6.1 The Ground Vibrational State 168 6.2 The Torsional Fundamental in the (t>5 = 1 <— 0) Band 171 6.2.1 Characteristics of the Spectra and Data Set 171 6.2.2 Analysis and Results 178 6.2.3 Discussion 180 6.3 The Torsional Fundamental in the (vn = 1 <— 0) Band 183 6.3.1 Characteristics of the Spectra and Data Set 183 6.3.2 Analysis and Results 187 6.3.3 Discussion 189 6.4 Three-Band Analysis 190 6.5 Summary 194 7 Summary and Discussion 196 Bibliography 202 Appendix A Theoretical Derivations 206 A.l Derivations of Vibration-Torsion-Rotation Kinetic Energy 206 A.2 Commutation Relations 209 A.3 Symmetry Considerations 210 A.4 Derivation of H'2'2 212 A.5 Derivations of Centrifugal Distortion Constants 214 A. 5.1 Properties of Rotational Operators 214 A.5.2 Derivation for Nondegenerate Vibrational State 216 A. 5.3 Derivations for a Doubly Degenerate State 216 A. 6 Derivations for M 0 3 218 Appendix B Observed Frequencies for CH3CF3 221 B. l Observed Frequencies for the Ground Vibrational State 221 B. l . l Observed Frequencies for the Torsional Overtone (vq = 2 <— 0) in CH3CF3 221 B.l.2 Observed Mm-wave Frequencies for CH3CF3 in the Torsional States (ve = 0,1,2,3) 224 B.2 Observed Frequencies for the (v^ = 1) Vibrational State 227 B.2.1 Mm-wave Frequencies in the (^ 5 — 1,vq — 0) State 227 B.2.2 Frequencies in the (v5 = 1 <- 0, ve = 0 <- 0) Band 228 v B.2.3 Frequencies in the (v5 = 1 <— 0,f6 = 1 <— 1) Band 243 B.3 Observed Frequencies for the (vu = 1) Vibrational State 247 B.3.1 Mm-wave Frequencies 247 B.3.2 Frequencies in the (yu = 1 <- 0, v6 = 0 *- 0) Band 251 B.3.3 Frequencies in the (v\2 = 1 <— 0, VQ = 1 <— 1) Band 263 B. 4 Observed Frequencies for the (vu = 1) Vibrational State 267 B.4.1 Mw and Mm-wave Frequencies 267 B.4.2 Frequencies in the (vn — 1 <— 0, v% = 0 <— 0) Band 268 B. 4.3 Frequencies in the (vn = 1 <— 0,^ 6 = 1 <— 1) Band 288 Appendix C Observed Frequencies for CH3S1F3 294 C l Frequencies for the (v$ = 1) Vibrational State 294 C. 2 Frequencies for the (vu = 1) Vibrational State 303 C. 2.1 Mw Frequencies 303 C.2.2 Frequencies in the (vn = 1 <- 0, v6 = 0 <- 0) Band 303 vi List o f Tables 2.1 The molecular symmetry group of AB3XY3: G\s a 41 2.2 Symmetry coordinates of AB3XY3 a 42 2.3 Symmetry properties of vibration-torsion-rotation eigenstates. a < b < c < d > e 46 2.4 Lower order terms of the vibration-torsion-rotation Hamiltonian. a'b 49 2.5 Torsion-rotation operators. a'b 51 2.6 The transformation properties of the vibrational, torsional, and rotational operators of AB3XY3 in the G\s group 53 2.7 Contact transformation functions a'b 62 2.8 Nonvanishing parameters in H^9 67 2.9 Centrifugal distorsion constants for AB3XY3 72 2.10 Effective parameters affected by the torsion-rotation contact transformations. 0 74 2.11 Summary of contributions to the effective torsion-rotation parameters in the ground vibrational state."'6 76 2.12 Effective parameters affected by the J Q —• J a transformation 80 2.13 Parameters characterizing interactions off-diagonal in k and/or 1 85 3.1 Summary of numerical models used for the analysis of CH3CF3 and CH3S1F3 99 3.2 Matrix elements of torsional operators in the torsion-rotation basis 100 3.3 Matrix elements of torsional operators for CH3CF3 (k = 1, cr = 0) 101 3.4 Torsion-rotation eigenvectors for the ground vibrational state of CH3CF3 as a function of (ve,a).a 110 4.1 Experimental details on the fundamentals of CH3CF3 and CHsSiF3 120 5.1 Nuclear spin statistical weight for CH3CF3 and CH 3 SiF 3 . a - 6 ' c 125 5.2 Data set in the ground state of CH3CF3: components and their characteristics 128 5.3 Spectroscopic constants for the ground vibrational state of CH3CF3 136 5.4 Data set in the 1/5 band of CH3CF3: components and their characteristics 144 5.5 Spectroscopic parameters from the two-band analysis of CH3CF3 in comparison with the single band analysis of the ground vibrational state 147 5.6 Data set in the 1/12 band of CH3CF3: components and their characteristics 156 5.7 Spectroscopic parameters from the GS/1/5/1/12 analysis of CH3CF3 157 5.8 Data set in the v\\ band of CH3CF3: components and their characteristics 158 5.9 Spectroscopic parameters from the GS/v^/vw analysis of CH3CF3 159 5.10 Effective parameters in the ground vibrational state, the (v\2 = 1) state, and the (vu = 1) state 163 6.1 Data set in the ground state of CH3SiF3: components and their characteristics 168 6.2 Spectroscopic constants for the ground vibrational state of CHsSiF3 169 6.3 Data set in the 1/5 band of CH3SiF3: components and their characteristics. a 177 6.4 Spectroscopic parameters from the GS/1/5 analysis of CH3S1F3 in comparison with the single band analysis of the ground vibrational state 179 vii 6.5 Data set in the v\\ band of CH3S1F3: components and their characteristics. a 186 6.6 Spectroscopic parameters from the GS/vn two band analysis of CH3SiF3 in comparison with the single band analysis of the ground vibrational state 188 6.7 Spectroscopic parameters from the GS/vs/vn analysis of CHsSiF3 in comparison with the results of the GS/V5 analysis and the GS/Vn analysis 192 6.8 Comparison of the x 2 in different analyses 193 viii List of Figures 1.1 The configuration of molecule AB3XY3 2 1.2 Zeroth-order torsional potential and its associated torsional energy levels 3 1.3 Vibration-torsion energies below 800 c m - 1 for CH3CF3 4 2.1 The space-fixed axis system XYZ, the molecule-fixed axis system xyz, and the top-fixed axis system x'y'z 13 2.2 The hindered internal rotational enery levels as a function of the barrier height 36 2.3 The reference configuration of AB3XY3 40 3.1 -F(ja) (•) and V 3 ((1-cos 3^/2)^ (V) as functions of 102 4.1 Schematic diagram for Bruker IFS 120 HR Fourier transform infrared spectrometer. 119 5.1 Vibration-torsion energy levels for CH3CF3 below 1100 c m - 1 124 5.2 Part of the spectrum of CH3CF3 in the region of the Q-branch of the torsional overtone (ye = 2 <- 0) 130 5.3 A portion of the spectrum of CH3CF3 containing i?(22) of the torsional overtone («6 = 2<-0) 131 5.4 Part of the rotational spectrum of CH3CF3 containing the i?(12) envelope of lines measured in the state (v6 = 2) 132 5.5 Part of the rotational spectrum of CH3CF3 containing the R(12) envelope of lines measured in the state (v6 = 3) 133 5.6 General torisonal energies for (K = 1) in the ground vibrational state of CH3CF3 135 5.7 A portion of the spectrum of CH3CF3 containing the Q-branch of the (v\<z = 1, ve = 2 <— 0) band 139 5.8 A portion of the spectrum containing i?(19) of the (v5 = 1 <— 0, v 6 = 0 <— 0) band 142 5.9 A portion of the spectrum containing i?(14) of the (V5 = 1 <— 0, VQ = 1 <— 1) band 143 5.10 Energy level diagram illustrating the effect of the off-diagonal quartic distortion interaction involving e for (v\2 = 1,VQ = 0; J = 46, cr = 0) 151 5.11 A portion of the rR branch of the bands (v\2 = 1 <— 0, Ave = 0) with VQ = 0 and 1 152 5.12 Energy-level diagram illustrating the effect of the off-diagonal quartic distortion interaction involving e for (vn = l,£n = —1; J = 40,i>6,c) with ve = 0 and ve = 1 153 5.13 A portion of the spectrum containing the PQ\5 sub-branch and the perturbation-allowed mQi5 sub-branch in the (vn = 1 <— 0, ^ 6 = 0 •»— 0) band 154 6.1 Vibration-torsion energy levels for CH3SiF3 below 450 c m - 1 167 6.2 General torisonal energies for the ground vibrational state of CR3SiF3 170 6.3 Energy level diagram to illustrate the effect of Fermi-like interactions on the energy levels for (J = 30) 172 6.4 Energy level diagram to illustrate the effects of the nonresonant Fermi-like interactions and the resonance Fermi-like interactions for (J = 30) 173 6.5 A portion of the spectrum containing the strongly perturbed CfK subbranches of the (vs = 1 «- 0, v6 = 0 «- 0) band 175 6.6 A portion of the spectrum containing the R^1 (J = 24) and Rj} (24) transitions of the (V 5 = 1 <_ 0) band 176 6.7 Probability densities P (a) for (J = 30, K = 13, a = 0) level in the ground vibrational state and the (v$ = 1) state for various values of v§ 181 6.8 Probability density P (a) as a function of K for (v5 = 1, v6 = 0; J = 30, a = 0) 182 6.9 Energy level diagram and P-branch transitions showing near-resonance Coriolis-like shifts for K" = 35 183 6.10 Portion of the PP branch showing the cr-splittings in the spectrum of the u\\ band due to the Coriolis-like interactions 185 7.1 Energy levels for (J = 30, a = 0) in the (v$ = 0,ve = 4) state and the (vs = 1, VQ = 0) state of CH3SiF3, showing the level crossing at K = 13 197 Acknowledgments I would like to thank my supervisor, Dr. I. Ozier, for his support and guidance during the course of the project, for his patience in reading this thesis, for his encouragement during the moments of frustration, and for his help to my family. I would like to thank Dr. F. W. Dalby for his valuable instructions, discussions, and encouragements. I would like to thank J. Schroderus, both as a colleague and a friend, for his cheerful assistance with the experiments and valuable discussions on the theory. Special thanks also goes to his wife, A. Tolvanen, and his family, for their hospitality during the period the experiment was done. I would like to thank Dr. J. Booth, Dr. A. Chanda, and Dr. W. Ho for their initial instructions with the laser system. I would also like to thank Dr. C. Boone for making the working environment enjoyable. I would like to thank Dr. N. Moazzen-Ahmadi for valuable discussions. Special thanks also goes to Dr. V. -M. Horneman and Dr. R. Anttila for their assistance on the experi-ments. Finally, I would especially like to thank my wife, Yunfei Lu, for her constant support, for her faith in my abilities, and most of all, for making my life a better place. xi C h a p t e r 1 I n t r o d u c t i o n The large amplitude interactions in molecules have been of interest in molecular spectroscopy for many years. Molecules with a single internal degree of freedom provide excellent opportunities to study these interactions. Symmetric rotors such as CH3CF3 and CH3S1F3 form one of the simplest classes of molecules that can undergo large amplitude motion. The axial symmetry remarkably simplifies the treatment of the vibration-torsion-rotation Hamiltonian. The coupling between the torsional mode and other vibrational modes in isolated molecules is impor-tant in the dynamics of the processes which involve vibrationally excited molecules. For example, it has been suggested that the torsional motion strongly enhances the re-distribution of the vibrational energy [1] and that this motion is responsible for a mechanism which drives infrared induced conformer changes in molecules trapped in rare-gas matrices [2]. To assess the importance of hindered internal rotation in processes such as those mentioned above, it is necessary to understand in detail the mechanisms that couple the torsional coordinate and other vibrational modes. This requires the formulation of the vibration-torsion-rotation Hamiltonian and subsequently the determination of the leading coupling parameters. The symmetric rotors considered in this thesis have the general formula AB3XY3. This consists of two C^-type rotors connected by a single A-X bond (Figure 1.1). In the lowest energy state, at equilibrium, the molecule is in the staggered conformation. In addition to the vibrational and overall rotational motion, the AB3 top can undergo torsional motion relative to the XY3 frame. The angle that measures the deviation from the staggered conformation is called a. As a varies, the top and frame move within a potential barrier. In lowest order, this can be written just as V (a) = V3 (1 — cos 3a) /2, where V3 is the height of the barrier. The torsional states are labelled by the principal quantum number ^6- This potential is plotted in Figure 1.2. In the lowest torsional state (VQ = 0), the torsional motion generally can be approximated by a harmonic oscillator. For excited torsional states close to the top of the barrier, a can vary almost freely over all 2n radians. Within each ve state below the top of the barrier, the top can tunnel through the barrier and move from one equivalent configuration to another. This is one of the simplest cases of quantum mechanical tunneling known and is one of the major reasons for the interest in this problem. The three torsional sublevels are labelled by the index a which takes the values 0, +1 and —1. For molecules like CH3CF3 considered here, the torsional energy is in the same order of the vibrational energy. The vibration-torsion energies for lower vibrational states of CH3CF3 are shown in Figure 1.3. The non-zero vibrational quantum number vk are specified. Both vibrational and torsional energies are orders of magnitude larger than the rotational energies. 1 Chapter 1 Introduction B B B Y Y Figure 1.1 The configuration of molecule AB3XY3. 1.1 Studies of the Torsion-Rotation Hamiltonian in Symmetric Tops One of the major interests in studying internal rotation is the torsion-rotation Hamiltonian. For symmet-ric tops, in the zeroth-order, the torsion-rotation Hamiltonian can be written as a sum of the rotational and torsional Hamiltonians. The torsional energy depends on the barrier height as well as the moments of iner-tia of both the top and the frame about the symmetric axis. In terms of quantum numbers, the zeroth-order torsional energy depends on only VQ, a, and k. Here k is the signed eigenvalue of J 2 , the component of the rotational angular momentum J along the symmetry axis. The effect of internal rotation in symmetric tops has been known for over half a century since this motion in ethane was first discussed by Kemp and Pitzer in 1936 [3]. However, relatively little progress was made until the late 1970's. The reason lies in the electric dipole selection rules. To first order, these require the conservation of the three quantum numbers VQ, k, and a. One result of the selection rules is that the conventional microwave rotational spectrum is insensitive to the leading terms in the torsional Hamiltonian. Earlier measurements of these terms had to rely on indirect techniques. The most widely used of these established methods is the torsional satellite technique intro-duced by Kivelson [4-6]. Centrifugal distortion effects in the leading torsional terms enter the rotational spectrum and produce splittings that, at least in higher torsional states, can be resolved with conventional microwave spectrometers. This method has been used for several symmetric rotors including CHsSiF3 [7,8], CF 3 SiF 3 [9], and CH 3 SnH 3 [10]. The main weakness in this approach is that the height and shape of the potential enter only through their effect on the torsional wavefunctions used in calculating the distortional perturbations. Furthermore, the moments of inertia about the symmetry axis of both the top and the entire molecule are required and generally cannot be determined directly. As a result, the same data can often be interpreted in different ways, as has been done with methyl silane [11,12]. In order to measure directly both the leading torsional terms and the required moments of inertia, the selection rules have to be broken. One way of doing this is to apply the avoided-crossing molecular-beam method, which was first developed by Ozier and Meerts [13,14] in 1978. In this technique, two appropriate 2 Chapter 1 Introduction v r m <c L A r A -180 -120 -60 0 60 120 180 a (degree) Figure 1.2 Zeroth-order torsional potential and its associated torsional energy levels. The values of the principal torsional quantum numbers VQ are shown on the left. The sublevels are denoted by their symmetry T under the C3 group. The free rotor quantum numbers m are shown on the right. The labels are discussed in Section 2.3.3. sublevels of different k and/or different a, which are coupled by either centrifugal distortion dipole moment or by the hyperfine effects, are brought to a near-degeneracy by applying an electric field. If the mixing between the two sublevels is strong enough, transitions between them can be observed and the difference in the corresponding zero field energies can be determined. The selection rules possible include AA; = 0, ± 1 , ± 2 , ± 3 and ACT = 0, ± 1 , ± 2 [15]. In favorable cases, the change in the zeroth-order torsional energy between the two levels can be determined to 20 ppm. The technique has been applied to the ground torsional states of CH3CF3 [15], CH 3 SiF 3 [16], CH3SH3 [17], CH3CD3 [18], and CH 3 SiD 3 [19]. One disadvantage of this method, however, is that the measurements are confined to the lowest torsional level because the states with VQ > 0 are depopulated in the nozzle used to form the molecular beam. A second method that can be adapted for direct measurements of the required moments of inertia is distortion moment spectroscopy. An effective transition moment perpendicular to the symmetry axis is generated by centrifugal distortion [20]. The resulting dipole moment, here denoted u.D [21], has been determined for CH3CF3 [15]. The selection rules are A J = 0, ± 1 , AA; = ± 3 , Aa — 0. Recently, Ozier et al. [22] used this method to study CH3CF3 in the ground and the first excited torsional states of the ground vibrational state. In comparison with the beam method, distortion moment spectroscopy has the advantage that these excited torsional levels can, in favorable cases, be investigated. One limitation with the distortion moment spectroscopy is that it requires a relatively large u.D. The third method for measuring the leading torsional terms is to study the pure torsional spectrum in the infrared. Torsional bands with (AVQ ^ 0, AA: = 0, ACT = 0) are weakly allowed because of the second 3 Chapter 1 Introduction Figure 1.3 Vibration-torsion energies below 800 c m - 1 for CH3CF3. Only the non-zero vibrational quantum numbers are shown. The torsional quantum number v§ is given beside each level. The rotational and torsional fine structure is neglected. The vibration-torsion fundamentals (solid line) and first torsional hot bands (dashed line) are indicated, as is the (v$ = 2 <— 0) torsional band (solid vertical line). derivative of the dipole moment and intensity borrowing from allowed vibrational bands. Among these, tor-sional bands with (Ave = 2n), where n is an integer, tend to be the strongest. However, even these torsional bands are generally very weak compared to vibrational fundamentals. Fourier transform spectroscopy [23] and diode laser spectroscopy [24] have been used to study the torsional band with (AVQ = 2n) in CH3SiH3. The torsional band (ve = 1 <— 0) for CHsSiH3 was also reported in Ref. [23]. In the current work, this method is used to study the weak torsional overtone (ve = 2 <— 0) in CH3CF3. By combining these three methods with conventional microwave spectroscopy, the torsion-rotation Hamiltonian can be studied in a more complete manner. By fully characterizing the torsion-rotation Hamil-tonian in the ground vibrational state, one can subsequently work up the vibrational energy ladder to obtain a more complete description of the vibration-torsion-rotation interactions. 1.2 Studies of Vibration-Torsion-Rotation Interactions in Symmetric Tops The current work is focused on the interactions between the vibrational fundamentals and the torsional mode in symmetric tops. The major vibration-torsion interactions will couple the torsional stacks in the ground vibrational state (GS) with those in the first excited vibrational states. These can be separated into two classes: the Coriolis-like interactions which couple the GS with the first excited degenerate vibrational states, and the Fermi-like interactions which couple the GS with the first excited nondegenerate vibrational states. As is well known, if the motion of a particle is referred to a uniformly rotating coordinate system, there is a Coriolis acceleration, which may be thought of as due to an apparent force, i.e. the Coriolis force. 4 Chapter 1 Introduction This leads to an additional coupling between rotation/torsion and vibration (Coriolis coupling). The Fermi-like interactions are due to the anharmonicity of the potential energy, which depends on both the torsional angle a and the normal vibrational coordinates. This leads to a direct coupling between the vibrational modes and the torsional mode. As will be seen in this thesis, both interactions can be classified as off-diagonal centrifugal distortion terms which involve torsional operators. Once large amplitude internal motion is involved, the interactions can have large matrix elements with (Avk = 1, Ave ^ 0), where vk refers to the vibrational quantum number for kth vibration. The names Coriolis-like and Fermi-like come from the nature of these coupling matrix elements between different torsional states. These matrix elements have forms very similar to those of Coriolis and Fermi interactions in molecules with no internal rotation. For polar symmetric tops with general formula A B 3 X Y 3 , the Coriolis-like interactions were first derived by Moazzen-Ahmadi et al. [25] The theory was based on the assumption that the torsional motion can be treated as a normal vibrational coordinate. Then the normal coordinate is extended so that it can allow for large amplitude motion. By doing so, the well established vibration-rotation theory for a C3v molecule can be directly applied to derive the Coriolis-like interactions. This interaction is referred as Coriolis interaction in Ref. [25]. For discussion purposes, this model is called the "High Barrier Model" in this thesis. Measurements of the effective barrier height in the excited vibrational states with conventional mi-crowave spectroscopy have difficulties similar to those for the ground vibrational states. As mentioned before, the molecular-beam method only works for the ground torsional state in the ground vibrational state. Distortion moment spectroscopy is limited to molecules with large distortion moments. Furthermore, the Boltzmann factors for the excited vibrational state are smaller than that for the ground vibrational state. This makes distortion moment spectroscopy even more difficult. A more effective method is to study the cr-splittings in vibrational fundamentals. The selection rules on vibrational and torsional quantum numbers are (Avk = 1, Avq = 0, ACT = 0). High resolution infrared Fourier spectroscopy has provided an excellent tool to study vibrational fundamentals. However, the vibra-tional dependence of the barrier height, which arises from the second derivative of the potential with respect to the corresponding vibrational coordinate, is generally small compared to the zeroth-order barrier height. Without strong intervibrational interactions, the cr-splittings normally cannot be resolved for the vibrational fundamentals (Avk = 1 <— 0, Avq — 0 <— 0) in a high barrier molecule like CH3CF3. An alternative method is to study the torsional hot bands of the vibrational fundamentals which originate from (vq > 0) states at the same time. The band origins of the torsional hot bands will provide information on the torsional energies for excited vibrational states. The difficulty with this method, however, is that these transitions are generally weaker in intensity compared to the vibrational fundamentals. Furthermore, most of these transi-tions appear in the same frequency region as the vibrational fundamentals; the spectrum is very congested and thus difficult to identify. To our knowledge, no high resolution study on the torsional hot bands in a 5 Chapter 1 Introduction vibrational fundamental of a symmetric top has been previously reported. In the current work, this method has been used to study three vibrational fundamentals in CH3CF3, namely (v\2 = 1 <— 0), (v\\ = 1 •*— 0), as well as (^ 5 = 1 <— 0). For each of the two perpendicular fundamentals (v 12 = 1 <— 0) and (i>n = 1 <— 0), the cr-splittings were not resolved. However, the torsion distortion parameters, along with the effective barrier height V3 in the excited vibrational states, were obtained from the simultaneous analyses of the vibrational fundamentals and their corresponding torsional hot bands. Quite often the (vk = 1, f6) state is strongly perturbed by the nearby ve states in the ground vibrational state. If the nearby v§ states are close to the top of the torsional barrier, they involve large amplitude torsional motion and hence have large cr-splittings. In these cases, the a structure in the (vk = 1,VQ) state can be strongly perturbed and the cr-splittings can be resolved in the vibrational spectrum. One example is the (us = 1,VQ = 1) state and the (v$ = 0,ve = 4) state of CH3CF3 in Figure 1.3. These two states are coupled by the Fermi-like interactions. To date, only four cases have been reported on global analysis of high resolution, resonantly-perturbed vibrational bands where the perturbing torsional states in the ground vibrational state involving large ampli-tude motion. The first three cases were for the V12 band of CH 3 SiH3 [25-27], the U12 band of CH3CD3P8], and the v\2 band of CHsSiD3 [29,30]. The fourth case was for the vg band of CH3CH3 [31], which has a higher symmetry compared to AB3XY3 type molecules. Al l four bands were measured using Fourier trans-form interferometer. For each of the four cases, the lowest degenerate vibration-torsion fundamental was studied and the Coriolis-like constants were determined. The High Barrier Model has been successfully used to analyze the four different bands. However, some difficulties exists with the theoretical basis of the model. In the current analysis of the v\\ band of CH3SiF3, although a tentative fit can be obtained with the High Barrier Model, the model introduced in this thesis clearly gives better results. The Fermi-like interactions have only been previously observed in CH3SiH 3 for the lowest nondegen-erate 1/5 band [32], which was resonantly perturbed by the excited torsional state (ye — 5) in the ground vibrational state. One motivation in the current work is to study the vibration-torsion interactions in a high barrier molecule CH3CF3 and an intermediate barrier molecule CH3SiF3. The strong Fermi-like interactions have been in-vestigated in the global analyses of the high resolution vibrational fundamentals (v5 = 1 <— 0) for both CH3CF3 and CH3SiF3. The Coriolis-like interactions have been studied in the resonantly-perturbed per-pendicular fundamental (vn = 1 <— 0) of CH3SiF3. Because of the difficulty with the symmetric tops due to the selection rules, most of the studies on internal rotation were focused on asymmetric rotors. Methanol, for example, has been of both practical and theoretical interest to spectroscopists for several decades. The most recent work can be found from Lees and Xu [33] on the identification of the v\\ out-of-plane CH3-rocking mode. Although it is not 6 Chapter 1 Introduction the intention here to review the works on methanol, it is worth mentioning that due to the asymmetry of the molecule and the low potential barrier (373.594 (7) c m - 1 [34]), the global analysis of the methanol vibrational fundamentals is still far from being able to fit the data to within the experimental error. The analyses of methanol are still limited to the torsional states in the ground vibrational state [34,35]. On the other hand, the axial symmetry for AB3XY3 molecules allows one to fit vibrational fundamentals to well within the experimental errors. For example, the resonantly perturbed 2^ 12 and V5 bands of CH3S1H3 [25,32] were successfully analysed by using a three-band model. 1.3 Theoretical Background Most of the earlier studies of the theory on molecules with internal rotation were focused on the effective torsion-rotation Hamiltonians. Perturbation theory has been used to treat the coupling between torsion and rotation; see reports as by Nielsen [36], by Kirtman [7,37,38], and by Lin and Swalen [39]. Generally, the torsion-rotation Hamiltonian for a molecule with internal rotation is treated in two differ-ent molecule fixed axis systems (MFS): the principal axis method (PAM) uses a MFS fixed to the frame, and the internal axis method (IAM) uses a MFS in which the zeroth-order torsion-rotation coupling vanishes [39]. A significant advantage of the IAM is that at the infinite high barrier limit, the rotational and distor-tion constants have the same meanings as those for a C$v molecule. On the other hand, the PAM has the advantages that the symmetry of the molecule is well defined in the molecular symmetry (MS) group. In the PAM, the MS group of A B 3 X Y 3 is the group G i g [40,41]. If the MFS is defined in the IAM, the symmetry of some of the rotational and vibrational operators are not properly defined in the Gis group [42]. Recently, Hougen et al. [42] have introduced an extended permutation inversion group so that the symmetries of the molecule can be treated properly in the IAM. However, the complicated character tables make it difficult to use. Consequently, one would like to use the much simpler G\% group for symmetry considerations. Earlier studies of Coriolis-like interactions as mentioned before have been done using both the PAM and IAM at the same time, i.e. the vibrational operators are treated in the PAM and the torsional/rotational operators are treated in the IAM. This introduces difficulties in the physical interpretations of the Coriolis-like interactions, which involve products of torsional, rotational as well as vibrational operators, since the operators are not defined in the same molecular fixed frame. For AB3XY3 type molecules, the reduction of the torsion-rotation Hamiltonians using contact transfor-mation theory has been discussed by Moazzen-Ahmadi and Ozier [43]. Their results were given in both the PAM and IAM. Various theoretical works can be found on interactions which involve large amplitude internal rotation in other molecules. For example, the perturbations in the vibration-torsion-rotation energy in ethane has been discussed by Hougen [44]. A Hamiltonian is also developed by Hougen [45] for the treatment of the small 7 Chapter 1 Introduction amplitude vibrations in the presence of the large amplitude motion of a methyl top internal rotor molecules like CH3CHO. Recently, a series papers has been published by Duan et al. [46—48] on the theories of vibration-torsion-rotation interactions for a molecule containing a three-fold symmetric internal rotor. A formal theory has been developed for centrifugal distortion [46], and sequential contact transformations for the vibration-torsion-rotation Hamiltonian [47], as well as the vibration-torsion-rotation dipole moment operator [48]. Unlike the "High Barrier Model", the torsional degree of freedom was treated as a large amplitude motion. The general formalism was developed for asymmetric molecules, however, it can be easily modified so as to be applied to the symmetric rotors. Although some of the basic ideas have been discussed elsewhere, Refs. [46-48] provide a consistent systematic development of the theory that acts as a useful starting point. Despite a considerable amount of previous works on symmetric tops with internal rotation, a similar systematic approach on the reduction of the vibration-torsion-rotation Hamiltonian has not been reported previously. One of the challenges of the current work is not only to analyze numerically the observed spectrum using a highly correlated model, but also to present a vibration-torsion-rotation theory, so that the observed molecular parameters can be used directly to calculate molecular structure. 1.4 Thesis Organization The current work is divided into two parts. In the theory part, the general formalism for vibration-torsion-rotation interactions for AB3XY3 type molecules is presented. The Coriolis-like and Fermi-like interactions are derived from the classical Hamilto-nian. Following the contact transformations presented by Duan et al. [47], the vibrationally diagonal Hamil-tonian is derived by applying vibrational, torsional, and rotational contact transformations. The Fermi-like and Coriolis-like interactions are derived as well as their effects on the torsion-rotation Hamiltonian through the vibrational contact transformation. Some of the effective parameters, for example, the centrifugal dis-tortion constants which describe the torsion-rotation interactions, are derived in terms of the fundamental molecular parameters. The theory presented in the current work differs from the previous works [25,43] in two aspects. First, the torsional motion is treated from the beginning as part of the rotational motion. This allows the torsional motion to be a large amplitude motion. Only vibrational motions are assumed as harmonic oscillation. Second, all the effective Hamiltonians are derived in the molecular fixed axis sys-tem in the PAM; hence the difficulty with the symmetry considerations in the IAM is avoided. By taking advantage of the axial symmetry, a new approach has been introduced so that in the high barrier limit, the effective Hamiltonian reduces to that of a normal C$v molecule. Thus the effective Hamiltonian has the ad-vantages of both the PAM and the IAM. Since the theory is presented in such a way that it can be applied 8 Chapter 1 Introduction to AB3XY3 molecules in general, some of the results are not used in the analyses of the current work. For readers interested in experimental results, it is suggested the theory chapter be omitted on first reading. In this case, the reader can start with Chapter 3, where the effective Hamiltonians used in the analyses are given. In the experimental part, the vibrational fundamentals of CH3CF3 and CH3SiF3 are studied using Fourier transform spectroscopy. The vibration-torsion-rotation interactions are investigated for a high bar-rier (~ 1100 cm - 1) molecule CH3CF3 and an intermediate barrier (~ 415 cm - 1) molecule CH3SIF3. The vibration-torsion-rotation Hamiltonian derived in the theory are used to successfully analyze the measured fundamentals in both molecules. Both C H 3 C F 3 and CH 3SiF 3 are near-spherical symmetric tops, (A - B) / \ (A + B) « 6% for CH3CF3 and « 9% for CH 3SiF 3. Furthermore, the rotational B constants (« 0.173 cm - 1 for CH3CF3 and 0.124 cm - 1 for CHsSiFs) are small compared to the more extensively studied C H 3 S U 4 3 for which B « 0.37 cm - 1 . This allows one to study transitions with high J and A; values for both CH3CF3 and CR3SiF3 so that the higher order terms in the Hamiltonian can be determined. In fact, for the degenerate vibrational fundamentals studied in both molecules, the upper limits on J and k for on the identified transitions are about 60 and 50, respectively. Of course, one disadvantage of these two molecules is that the spectrum is very congested and transitions are generally more difficult to identify. Besides the torsional mode v§, an A B 3 X Y 3 molecule has 11 normal vibrational modes. For CH3CF3, the most recent studies of the vibrational spectra was by Nivellini et al. [49] using Fourier transform and Raman spectroscopy. The band origins of vibrational bands were reported. For CH 3 SiF3, the vibrational bands were studied by Clark and Drake using Fourier interferometry and Raman spectroscopy [50]. No torsion-rotation analysis was reported in either work. Most of the previous high resolution studies of CH3CF3 were carried out for the ground vibrational state. Recently, the torsional states (ve = 0,1, 2) of CH3CF3 have been studied by Ozier et al. [22] with molecular beam, microwave and mm-wave methods, and various torsion-rotation constants in the ground torsional state have been determined. (See also references cited in this work for earlier studies.) However, because of the high potential barrier, some of the correlations among torsional distortion constants were so high that only combinations of these constants could be determined. The infrared spectrum of CH3CF3 near 970 cm - 1 has been measured at a resolution of ~ 3 MHz by Fraser et al. [51] using an electric resonance optothermal spectrometer. A triad of states was observed: the combination levels (i>n = 1, 6^ = 2) and (v\2 = 1, v$ = 1), as well as the fundamental (t>io = 1). The work [51] demonstrates that the determination of the torsional splittings in excited vibrational states is very useful in investigating vibrational couplings. In the current work, the torsional overtone (i^ = 2 <— 0), the vibrational fundamentals (v\2 = 1 <— 0), (vn = 1 «— 0), and (v$ = 1 <— 0) are studied for CH3CF3 with high resolution Fourier transform spec-9 Chapter 1 Introduction troscopy. The torsional overtone (VQ = 2 <— 0) was measured by Dr. A. R. W. McKellar with a Bomem Fourier Transform Spectrometer at a resolution of 0.005 c m - 1 . All the analysis of the spectrum has been done at the University of British Columbia. The three vibrational bands were measured in the present thesis with a Bruker IFS 120 HR Fourier transform infrared spectrometer at a resolution of 0.00125 c m - 1 . The four bands studied are indicated in Figure 1.3. From the analysis of the torsional overtone (VQ = 2 <— 0), improved effective values for the ground vibrational state are determined for the height V3 of the hindering barrier and the first order correction Vis. For each of the three vibrational fundamentals, both the torsional fundamental and the first torsional hot band are investigated. For each of the (v\2 = 1) and (v\\ = 1) vibrational states, the effective barrier height V 3 has been obtained along with the ^-doubling constant. Here £ represents the vibrational angular momentum quantum number for a degenerate vibrational state. The torsional-dependence of the ^-doubling constant and that of the zeroth-order Coriolis constant, as pre-dicted in the current theory, are determined. This is the first time that these effects have been observed for molecules of this type. The magnitude of the centrifugal distortion constant e that characterizes the (Ak = ±3 , A£ — 0) matrix elements has been determined for both vibrational states. For the (v^ — 1) state, the Fermi-like interaction parameters for the 1/5 mode were determined along with the effective bar-rier height V3. A good fit to well within the experimental error is obtained for the global data set, which includes about 11,000 transitions from the infrared measurements. The vibrational contact transformation theory is tested by the changes in V3 in different models. About 650 mm-wave frequencies measured by Ilyushyn et al [52] are included in the analysis of CH3CF3. The collaboration started when the current analyses for the infrared spectra were finished. As a different type of measurement, the mm-wave data provide a good test for the current infrared work. The identifications and analyses for the mm-wave data were all based on the current infrared work. For most mm-wave transitions, the infrared results predict the frequencies to within 50 kHz. For transitions within the (vn — 1) state and the (vs — 1) state, the infrared analyses predict the frequencies to within 10 kHz, which is well within the experimental error. An exception occurs with the ground vibrational state. The mm-wave measurements extended the data to the (VQ = 3) torsional state. By adding the mm-wave data, some higher order parameters are determined for the ground vibrational state. For CH3SiF3, earlier high resolution studies were all carried out for the ground vibrational state. Various techniques, e.g. avoided-crossing molecular beam method [16] and mw/mm-wave Fourier interferometry [8,53,54], were used. Al l these measurements have been used in this study to provide the best results for the ground vibrational state. In the current work, the second and third lowest vibration-torsion fundamentals v\\ and 1/5 are studied using a Bruker IFS 120 HR Fourier transform infrared spectrometer at a resolution of 0.00125 c m - 1 . Both bands are resonantly-perturbed by the torsional states in the ground vibrational state. The Coriolis-like constants for the v\\ mode and Fermi-like constants for the 1/5 mode are determined. The effective barrier 10 Chapter 1 Introduction height in both excited vibrational states are determined along with the vibrational dependence of various torsion-rotation constants. The remainder of this thesis is divided into six chapters. Chapter 2 give a general theory for vibration-torsion-rotation interactions for AB3XY3 molecule. Chapter 3 discuss the various different numerical pn> cedures used in this thesis. Chapter 4 describes the Fourier transform interferometry experiments. Chapter 5 gives the analysis results of CH3CF3. Chapter 6 discusses the analysis of CH3SiF3. Chapter 7 compares briefly the results of the two molecules. Some potential future work involving the two molecules is outlined. 11 C h a p t e r 2 G e n e r a l T h e o r e t i c a l C o n s i d e r a t i o n s In this chapter, the general formalism of the vibration-torsion-rotation interactions is presented for a A B 3 X Y 3 type molecule. The theory discussed is concentrated on the electronic ground state. Therefore, the electronic energy will be neglected. The nuclear hyperfine energy is also neglected as it makes no significant contributions to the energies studied. 2.1 Classical Kinetic Energy of A B 3 X Y 3 Molecule Three types of motions of molecule A B 3 X Y 3 are of interest here, namely vibrational, torsional, and rotational motions. To understand where and how the various types of vibration-torsion-rotation interactions arise, we begin with a derivation of the kinetic energy. The model used here consists of a nonrigid C$v symmetric top (i.e. AB3) attached to a nonrigid Czv symmetric framework (i.e. X Y 3 ) , which is normally taken as the part of the two rotors with the larger moment of inertia about the symmetry axis. All internal degrees of freedom must be taken into account. The classical kinetic energy for an ethane like molecule (the top and the frame are identical) has been derived by Hougen [55], and by Papousek and Aliev [56]. Here the approach by Papousek and Aliev will be followed with the modification necessary so that the top and the frame can be different. The results for ethane can be easily obtained from the current results by making the top identical to the frame. The kinetic energy can be most simply derived by the vectorial method. It is helpful in considering this problem to visualize four coordinate axes systems. These four systems are [55]: (1) the space-fixed coordinate system (SFS) XYZ (fixed in the laboratory); (2) the molecule-fixed axis system (MFS) xyz; (3) the top-fixed axis system (TFS) x'y'z (fixed to the top); (4) the frame-fixed axis system (FFS) x"y"z (fixed to the frame). The origins of the MFS, the TFS, and the FFS are all fixed to the instantaneous center of mass of the entire molecule, with a common z-axis which coincides with the A-X bond. The TFS and the FFS can be brought to the MFS through a rotation about the z-axis by an angle of (1 — A) a and —Aa, respectively, a is the torsional angle which can be defined as the angle between x'- and a/'-axis. For the moment, A can be considered as a parameter which defines the ratio of the angles through which the TFS and the FFS should be rotated about the 2-axis to coincidence with the MFS. Other variables used here are defined as shown in Figure 2.1. ro is the Cartesian position vector in the 12 Chapter 2 General Theoretical Considerations Figure 2.1 The space-fixed axis system XYZ, the molecule-fixed axis system xyz, and the top-fixed axis system x'y'z. ro and rj are defined in XYZ. Here it is assumed that nucleus i is in the top. aj and dj are defined in x'y'z. ti is defined in xyz. SFS of the instantaneous center of mass of the whole molecule. 0, (fi, x (which are not shown in Figure 2.1) are the Euler angles through which the MFS xyz must be rotated to the SFS XYZ. aj represent the reference position of the nuclei so that all bond lengths and angles, except the torsional angle a, have their equilibrium values, dj represent the infinitesimal vibrational displacements. The subscript i labels atomic nuclei, with i = 1,2, • • •, 3iV. N is the total number of nuclei for the molecule. For a nucleus in the top, ai and dj are defined in the TFS. Similarly a; and d; are defined in the FFS for nuclei in the frame. These variables are related to the spaced-fixed Cartesian coordinates rj of the ith nuclei (rj is a column matrix of the three coordinates of the i-th atom nucleus) by the equation [56]: rj = r 0 + S-1{6,(j),x)Sr\0,0,-f)(aii+di). (2.1.1) S_1((9, (fi, x) is a 3 x 3 matrix with the form: / cos 6 cos <fi cos x — sin (fi sin x ~ c o s # c o s ^ sin x — sin <p cos x sin 6 cos (fi \ s-1(^,0,x) = cos 6 sin (fi cos x + cos (fi sin x — cos 9 sin (fi sin x + cos (fi cos x sin 6 sin (fi \ — sin 0 cos x sin 6 sin x cos 8 j (2.1.2) 7 is the angle through which the top or the frame axis system is rotated in order to be brought to the MFS, according to the definition: 7 = (1 — A)a, i C top; (2.1.3) 7 = —Aa, i C frame. The first rotation ST/1 (0,0,7) brings (aj+dj) (which is defined in the TFS or the FFS) to the MFS. The 13 Chapter 2 General Theoretical Considerations second rotation S 1 (0,4>, x) converts the MFS vector to the SFS. The following conditions are used to constrain the quantities ro, 0, </>, x, and a [56]: 0 = J ^ m i S r ^ O . O ^ d i , (2.1.4) i 0 = ^mIS-1(0,0,7)(ai xdi), i = [Sr^O.O.T)*] x [ S - ^ C C T H ] , (2.1.5) i 0 = k • ^ ra^a* x dj. (2.1.6) T-F Y1t-f m ^9- 1 -6) represents a sum that: ^ D i = ( l -A) ^ D , - A D i ( 2- 1 J) T-F iCTop iCFrame in Eqs. (2.1.4) and (2.1.5) represents a sum over all nuclei. The symbol k represents the unit vector along the z-axis. From the definition of a*, we find that the above three equations reduces the number of independent r^  from 3iV to 3N — 7, since Eqs. (2.1.4), (2.1.5) and (2.1.6) represent 3, 3 and 1 constraints, respectively, for a total of 7. Eq. (2.1.4) is just the ordinary center-of-mass or first Eckart condition [57]. Eq. (2.1.5) is similarly called the second Eckart condition. Loosely speaking, we can say that Eq. (2.1.4) requires the quantities S^"1(0,0,7)d, to be orthogonal to the three translations of the molecule as a whole, and that Eq. (2.1.5) requires the quantities S " 1 (0,0, j)d{ to be orthogonal to the three infinitesimal rotations of the molecule computed when the molecule has the shape given by S~1(0,0,7)ai. Eq. (2.1.6) is just the modified Sayvetz condition [58], requiring the quantities S^"1(0,0,7)dj to be orthogonal to an infinitesimal torsion of the molecule about the 2-axis, when the molecule has the shape given by S~1(0,0,7)aj. According to Eq. (2.1.1), the velocity f j of the nucleus with respect to the space-fixed axes X, Y, Z is: v{ = v0 + S - 1 ^ , <t>, X)ti + S - 1 ^ , <f>, X)ii, (2.1.8) where S - 1 = diS-^/dt ( not (dS/dt)~l). In addition: U = S~l(0, O^Xarr-di); (2.1.9) U = S-1(0,0,7)(ai+dl) + S- 1(0,0 ,7)d i . (2.1.10) Here the expression of t* has been derived under the assumption that the reference positions of the atomic nuclei do not depend on the torsional angle a, i.e. den/'dt = 0 [56]. In general, the total kinetic energy of a molecule system consists of two parts: the total kinetic energy of iV atomic nuclei and the total kinetic energy of -/Ve electrons. Since the mass of an electron is much smaller than that of a nucleus, the kinetic energy of electrons will be neglected for the rest of the chapter. 14 Chapter 2 General Theoretical Considerations Using Eqs. (2.1.8), (2.1.9) and (2.1.10), the total kinetic energy can be written in the form: 2T = ^ miiii i = "wSSf-j i i = ?ofo5^mi + 2Sro ( S S " 1 J ^ m ^ + ^rmU ) + ( s S - 1 ^ ( s S - 1 t ; ) i \ i i / i ^ ' +2j2mi ii + ^rmiiii. (2.1.11) Here ~ means matrix transposition. The term (YJj rrii/2) tqTq represents the kinetic energy of the trans-lation of a molecule with total mass V \ m,i concentrated in the center of mass. The second term on the right-hand side of Eq. (2.1.11) represents the energy of the interaction of translation with other molecular motions. Because the origin of the MFS is by definition at the center of mass, it holds that: ^^mjtj = 0, ^ r a j t j = 0. (2.1.12) i i Therefore, the second term on the right side of Eq. (2.1.11) vanishes and the kinetic energy of the trans-lational motion can be exactly separated from other types of molecular motion. The kinetic energy of translation is of no spectroscopic interest in the absence of external fields and will not be considered in further discussions. The expression for the kinetic energy then assumes a simpler form: 2T = ^ m * ^ S S _ 1 t ^ ( s S - 1 t i ) + 2 ^ ™ * ^ S S _ 1 t ^ U + ^ m j t j t ; (2.1.13) Because S is an orthogonal matrix, S _ 1 = S, S _ 1 = S. From SS= E , it follows that SS + SS = 0, thus SS = —SS and SS is an antisymmetric matrix. By using the expression for S _ 1 given by Eq. (2.1.2) we have: / 0 ~U)Z UJy \ SS = s s 1 = UJZ 0 -LOX \ -UJy CVX 0 / (2.1.14) where UJX = sin x • 6 — sin 6 cos x • <t>] (2.1.15) ujy = cos x " ® + sin 6 sin x • & (2.1.16) UJZ = cos6-(f) + x- (2.1.17) Here 9 — d6/dt, <$ = dcfi/dt and x — dx/dt. It can be shown that u> is the total angular velocity vector of the moving MFS xyz and u> = \UJX + jujy + kujz, (2.1.18) where i, j and k are the unit vectors along the x, y and z axis directions, respectively. Because the elements of the column matrix SS ^ are the components of the vector product uj x ti, 15 Chapter 2 General Theoretical Considerations the expression for the kinetic energy given by Eq. (2.1.13) can be written in the following vector form: 2T = ] T m i (o> x tj) • (OJ x ti) + 2 Y m i (u x *0 ' + Ymi^i ' **• (2.1.19) i i i The expression for T in Eq. (2.1.19) does not contain the torsional angle a. explicitly because both ti and ti are functions of a. Now we can substitute expressions for ti and ti in Eqs. (2.1.9) and (2.1.10) into Eq. (2.1.19). The resulting expression of T can be further simplified by using the Eckart and Sayvetz conditions in Eqs. (2.1.4), (2.1.5) and (2.1.6). (See Appendix A.l.) For discussion purposes, we write the total kinetic energy as T = Tv + Ttr + T c . (2.1.20) Here Tv represents the vibrational kinetic energy; Ttr represents the torsional and rotational kinetic energy; and T c represents the coupling between the vibration and torsion or overall rotation. These terms have the following forms (see Appendix A.l): Tv = l / 2 ^ m i d i - d i ; (2.1.21) i Tu- = l/2ZIu + 1/26? [(1 - Xfll + A 2 / £ ] +d Y, "/J [(1 - A)/J. - A/&] ; (2-1-22) P=x,y,z Tc = w J ^ m i S r ^ C C T ) d, x d, +ak- midixd T-F +d(k x w) • Y mi [ St _ 1(0,0,7)(ai x di)] . (2.1.23) T-F Here IT, IF and I are the moment of inertia tensors of the top, the frame and the whole molecule, respec-tively. Their elements have the following form: i iCTop iCFrame ^ j 24) i iCTop tCFrame where a, 8 and 7 assume the values x, y and z, with a ^ 3 7^  7 . It can be easily shown that at equilibrium, i.e. d, = 0, the moment of inertia tensors for molecule AB3XY3 are all diagonal. The diagonal elements of these moment of inertia tensors are 4 = = E mi(a% + al), I°zz = £ m i ( c £ + a%), &° = Iyf = E rm{a% + <&), j £ ° = * E ™i(<4 + n 1 25) iCTop " tCTop " 1.^.0) I«° = lff= £ mi{al + al), J5°= E rnicl+ a*,). iCFrame icFrame All the components are independent of the torsional angle a and A (which simply represents a rotation of 16 Chapter 2 General Theoretical Considerations the MFS about the symmetry z axis by an angle). In comparison with the results for an ethane like molecule given by Papousek and Aliev [56], the third term on right hand side of Eq. (2.1.23) has been neglected in Ref. [56]. This term can also be written as u- mjS^ "1(0,0,7)[a* x dj] (see Eq. (A.16)), which is clearly a Coriolis type kinetic energy. As will be seen later, this is the classical counterpart of the Coriolis-like interaction. (See Section 2.12.) Except for the difference in this term, the results of Ref. [56] can be easily obtained from the results given in this section by setting the top to be identical as the frame and A = 1/2. 2.1.1 Definitions of the Molecular Fixed Coordinate Systems The MFS is generally chosen so that at the vibrational equilibrium, the torsion-rotation part of the kinetic energy has a simple form. In this section, we only consider the zeroth-order torsion-rotation kinetic energy. The general forms of the kinetic energies given by Eqs. (2.1.21), (2.1.22) and (2.1.23) are derived in a MFS, which is defined in such a way that the TFS and the FFS can be brought to coincidence with the MFS by rotating by an angle of (1 — A) a and —Aa about the z-axis, respectively. The three axis systems share a common z axis. The orientation of the z axis in the SFS are defined by Euler angles 6 and cf). If we let x T and xF to be the Euler angles of the top and the frame (so that xT ~ XF = A)> respectively, then from Eq. (2.1.1) and (2.1.3) we can see that they are related to the Euler angle x of the MFS by: XT = x + ( l - A ) a ; XF = X - A a . (2.1.26) Each A value defines an Euler angle x through the above equations. In principle, many MFS can be defined by selecting different values of A. In order to see the effect of A on the kinetic energy, let us first look at the torsion-rotation kinetic energy Ttr given by Eq. (2.1.22). Here we only consider this term at the equilibrium of vibration, by using the results given by Eq. (2.1.25), we have: # = \ E ^ + ^ 2 [ ( 1 - A ) 2 / S ° + A 2 ^ 0 ] P=x,y,z +auz [(1 - A) J£° - A/£°] . (2.1.27) The first term on the right hand side of Eq. (2.1.27) is just the rigid rotor kinetic energy. The second term is the torsional kinetic energy. The last term represents the coupling between the torsion and overall rotation through the term auz. Two different kinds of MFS can be introduced according to different forms of lfr. In the principal axis method, A = 0, so that from Eq. (2.1.26) we have XP = XF- (2-1.28) This means the MFS is chosen to be the FFS. In this MFS, the coordinates will be denoted as xyz. The 17 Chapter 2 General Theoretical Considerations torsion-rotation kinetic energy from Eq. (2.1.27) is T?r = l E Q w + l # £ f i + ™.%°. (2-1.29) P=x,y,z Note that the components of the moment of inertia tensor in Eq. (2.1.25) are independent of A . In this MFS, the frame is stationary and the top is moving with an angular velocity ak; therefore, the angular momentum T 0 of internal rotation is lzz a. In the internal axis method, the MFS is chosen so that the term in Eq. (2.1.27) which represents the coupling of the internal motion and overall rotation vanishes. To make auz [(1 - A)l£° - A/£°] = 0, (2.1.30) we have X = p = I^/I°zz. (2.1.31) In this MFS, the coordinates will be denoted as xyz in order to distinguish from xyz defined by the PAM. Note that the z axis in the PAM is the same as the z in the IAM. From Eq. (2.1.26) we have: Xi = Xp + pa. (2.1.32) The torsion-rotation kinetic energy then has the form: ^r = \ E ^ / 3 + ^ 2 [ ^ ° ( l - P ) ] • (2-1-33) /3=x,y,z A bar on top of u> means that it is defined in the MFS xyz. To an observer located in this frame of reference, both the framework and the top appear to be moving. For the top, the angular velocity is simply xT ~ Xi = (1 - p) a, and for the frame it is xF ~ Xi — —pet. This can be seen directly from Eq. (2.1.26). The angular momentum (z component) arising from the internal rotation is: I z r i ° ( l - p ) a - I ^ p a = 0. (2.1.34) The angular velocity u> in the PAM is defined in the MFS with the Euler angles 8, <f> and Xp through Eqs. (2.1.15), (2.1.16) and (2.1.17). Similarly, u> (in the IAM) is defined by the Euler angles 8, <j> and X l = xP + pa. By substituting Eq. (2.1.32) into Eqs. (2.1.15), (2.1.16) and (2.1.17), it can be shown that the components of OJ in xyz (PAM) are related to those of in xyz (IAM) by the following equations: u>x = eos paujx + sin pauy , u>y = — sin paujx + cos paujy , uz = uz-\-pa. (2.1.35) 18 Chapter 2 General Theoretical Considerations 2.1.2 Hamiltonian Form of the Kinetic Energy Having discussed the effects of the MFS on the torsion-rotation kinetic energy, we now derive the Hamil-tonian form of the kinetic energy. For simplicity, we assume that the MFS is in the PAM in this section, i.e. A = 0, although the general results in this section should be similar when the MFS is in the IAM. However, the IAM form could be more complicated. The convention used for the labelling throughout the thesis need to be summarized. The index i refers to the label of nuclei, k, I, m, n etc. refer to the vibrational normal coordinates. 3,7, 6 etc. refer to the x, y, and z directions of the MFS. When these are used to label the components of the angular momentum, they could assume the value a which refers to the angular momentum of the torsional motion. The total kinetic energy has been expressed as a sum of three terms, i.e. Tv, and Tc, which are given by Eqs. (2.1.21), (2.1.22) and (2.1.23), respectively. In order to express the kinetic energy in terms of normal coordinates of vibration, the definition of the normal coordinates in terms of the Cartesian displacements should be introduced. The mass-weighted Cartesian displacement coordinates QX, Q 2 , • • •, Q^N a r e defined as Qi/3 = mk/2dip-> i = 1 ' 2 ' - " -,Nand/3 = x,y,z. (2.1.36) Here rrii are the masses for the corresponding atomic nuclei and are the Cartesian components of the displacement vectors. Let g be the column matrix of the coordinates rnj2dip and Q be the column matrix of the normal coordinates of vibration Qk. Here Q has a dimension of (3iV — 7) since the internal rotation is excluded. Then: ff = L » Q , (2.1.37) where lv(a) is a 3N x (3/V — 7) matrix of the coefficients hpyk, which are in general functions of a. Therefore g=aH0(a)Q + \v(a)Q, (2.1.38) where the elements of the matrix 1^  are l'i/3 k = dlip^/da. The matrix LJ^ is an antisymmetric matrix since LJ,; = E3;v_7. (Here E3JV-7 is a unit matrix with dimension of (3N — 7) x (3iV — 7).) Furthermore, d (T„l„) /da = Yvlv + \VVV = 0; therefore, Yvlv = -lvl'v. By using the properties of lv(a) and 1^  we can express the vibrational kinetic energy in terms of normal vibrational coordinates: Tv = l / 2 ^ m i d i - d i = l/2ff-e i = 1/2 [6.Q+a (Sl^Q+Qi&Q) + "2QKKO\ • (2-1-39) 19 Chapter 2 General Theoretical Considerations It can be shown that: where Q U Q = E ( C S ) ' Q f c Q i , k,l (Chi)' — - (Czfc)' — E'w^.fc • (2.1.40) (2.1.41) Similarly QLJLJQ can be found to have the same expression. Thus the second term on the right hand side of Eq. (2.1.39) can be written h ( Q U Q + Q I ^ Q ) = & E (Cki)' QkQi k,l (2.1.42) This term is referred to as a Coriolis coupling term which represents the coupling between the vibrational motion and the internal rotation. The index a on (CM)' indicates that it is a Coriolis constant associated with the internal rotation. The last term on the right hand side of Eq. (2.1.39) is where d 2 Q l ^ Q = d 2 ^ ( ^ ) ' Q f c Q z , k,l (hi)' = (kk)' = ^I'ip/w (2.1.43) (2.1.44) a Ql^l^Q contributes to the effective potential energy. Finally, Tv can be expressed in terms of Qk as Tv = 1/2 { k,l k,l (2.1.45) The kinetic energy Tc in Eq. (2.1.23) involves three terms which all depend on the Cartesian vibrational displacements di. In order to evaluate terms like dj x dj, we can introduce permutation symbol or the unit antisymmetric [56] ep1s, defined as: &xyz — &yzx — &zxy — 1> ^•xzy — e.yzx — &zyX — 1 • (2.1.46) (2.1.47) epyg — 0 if two or more of the indices are equal. It can be shown that component (5 of vector product (A x B) can be expressed as: (A x B)p = ep^AyBs (2.1.48) Therefore, the first term on right hand side of Eq. (2.1.23) can be expressed as w - E m i S : - 1 (0,0,7) dj x d, ^ w J ^ & Q i + d ^ Q , ] (2.1.49) P=x,y,z \ k,l k,l 20 Chapter 2 General Theoretical Considerations where 1,6 7,(5 i Here = E [sr 1 (0,0,7)]^^,*; 6=x,y,z ($*) ' = E [ s r H o . o . T ) ] ^ ^ , * ) ' . 7=x,i/,z The second term on right hand side of Eq. (2.1.23) can be written: dk-LT-F m,'d,- x d = a I E &iQkQi + « E VkiQkQi , k,l k,l where Ckl — ^ ] eZ7t5 ^ ] lvy,kh6,li 1,6 T-F = ^ ^ 6Z76 ^ ^ hi,k(.h6,i) • 7,6 T - F The last term in T c can be expressed in the following form: d(k x w) • ] P m i [S~ 1(0,0,7)(ai x di)]. = A ^ ^ E ^ - * ^ ' /3=x,?/ fc T—F where Therefore Atf,*= E [S . -^O.O.T)] 07 7=x,y,z E ^ E mi / 2°ii5^e,fc (5,e T-F (2.1.50) (2.1.51) (2.1.52) (2.1.53) (2.1.54) (2.1.55) (2.1.56) (2.1.57) (2.1.58) Tc = E UP\ E + A E ^QfcQ* P=x,y,z y fe,Z fc,i +a I E (kiQkQi + VklQkQi \+os E ^ E A^Qk (2.1.59) \ k,i k,i J P=x,y k Finally, from Eqs. (2.1.22), (2.1.45) and (2.1.59), the total kinetic energy can be expressed as (remember in the PAM A = 0 in Eq. (2.1.22)): 3N-7 2T = £ If^upw^ + a2Ijz + 2a ] T / J 2 ^ + £ Q 2 P,1=x,y,z P=x,y,z k=l 21 Chapter 2 General Theoretical Considerations p V fc,« fc,i +2d k,l k,l + 2a Y vp^2Ai0ikQk P=x,y k (2.1.60) where Ckl — (Ckl)' + Ckh nh = vh + (hi)'/2. (2.1.61) (2.1.62) Cki (with 0 = x,y, z) is given by Eq. (2.1.50). Eq. (2.1.60) can be written in the momentum form, 3N-7 2T= J2 Hi (J/3 - Pp) ( J7 -Pi)+/Z Pk • ( 2- 1 -63) P,-y=x,y,z,a k=l In Eq. (2.1.63), Jp (0 = x, y, z) are the molecule-fixed components of the total angular momentum exclu-sive of nuclear spin, = dT/dujp E fp^-y + E IkiQkQi + a 7 k,l /24iQkQi + ipz + (i - <W E A^Qk k,l k ,(2.1.64) where Spz is the Kronecker 6 function. The momentum conjugate to the torsional angle a is J a = dT/da = a I L + ^2CkiQkQi+ E IPzUJP + z^UJiiz~lr^iQkQi k,l P=x,y,z P k,l (2.1.65) KI P=x,y The momentum conjugate to Qi is Pt = dT/dQt = Ql + Y,upJ2 ~&iQk + *ECliQk P k k (2.1.66) The vibrational angular momentum pp is defined as 7/5 Pp = E (kiQkPi ;0 = x,y,z,a k,l (2.1.67) Here the parameters (,kl (0 = x,y, z, and a) are called the Coriolis coefficients. From definitions of ( k t given by Eqs. (2.1.50) and (2.1.61), one can easily find that the Coriolis coeffi-cients have the following properties Cu = -Cifc, 0 = x,y,z,a. (2.1.68) 22 Chapter 2 General Theoretical Considerations These are functions of the torsional angle a. The Coriolis coefficient Qkl corresponding to a z direction Coriolis coupling involves the torsional angular momentum. The quantities p,p1 are complicated functions of a and Q. The form of fip1 can be worked out by substituting Jp, pp (3 = x, y, z, a) and Pi into Eq. (2.1.63) and comparing the results with Eq. (2.1.60). 2.2 Potential Energy The potential energy is more easily discussed in terms of the symmetry coordinates Sk of the molecule. Here the approach of Kirtman [7] will be followed. In order to determine explicitly the transformation from 5fc's to normal coordinates, it is necessary to know the complete potential energy V as a function of the Sfc's. Expanding V about Sk = 0 in Taylor series for all k, we have V = V?r{a) + Vo + 5 > ( a ) S f c + \ Ekkk>(a)SkSk' + W'(a)SfcSfc<Sfc» + • • •, (2.2.1) where the force constant kk, kkk>,... are functions of a, and V^r is the potential energy of internal rotation for the nonvibrating molecules. V®r (a) is the pure torsional potential and Vq + V^. (0) gives the zero point energy. For an operation which rotates the top about the z-axis by ±27r/3 rad (at the same time, keeping the vibrational displacements tied to their positions in space), V must be invariant. It follows then the transformation a —> a ± 27r/3, Sk —> Sk leaves kk, kkk> • ••, and V? (a) unchanged. Therefore, all the force constants and V?. can be represented by Fourier series expansions in a with a period of 27r/3, i.e. oo oo kk(a) = fc^ cos 3na + A:^ sin 3no;; n=0 n=l oo oo hk'((~) = Y^k,cos3na + Yklkk'sin^nai (2.2.2) n=0 n = l ^ oo oo V® = ~Y v~ln cos 3na + ^ vf1 sin 3na. n=l n=l Here a has been denned so that a = 0 at equilibrium. The index c and s indicate that the corresponding force constant is the expansion term corresponding to a cosine and sine function of a, respectively. The total potential energy V must have a minimum at the equilibrium conformation. Thus, for all k (a) (dV/da)0 = 0, and (b) (dV/dSk)0 = 0. (2.2.3) Hence oo oo (a) Y 3nv*n = 0, and (b) £ k% = 0. (2.2.4) n = l n=0 It is reasonable to assume that the sine series expansion in will be rapidly convergent and therefore only the leading term will be important. However, then the equilibrium condition in Eq. (2.2.4) implies that this leading term must vanish. For A B 3 X Y 3 type molecule, since it has a plane of symmetry, all the sine series 23 Chapter 2 General Theoretical Considerations in V vanish. Then, by making use of Eq. (2.2.4). it can be shown that V = -^Y^Kkk'SkSk' + ; ^ Kkk'k"SkSk'Sk" + kk' 4 E n = l 3n V3n + J2KckSk + YK!kk'SkSk' + kk' (1 — cos3na) where Kkk' — E^cfefc'' Kkk'k" — y^^cfcfc'fc"' n=0 nck — V3n = „-;3n n=0 -"•cfcfc' — ~Kckk'> (2.2.5) (2.2.6) (2.2.7) (2.2.8) and the arbitrary additive constant Vo + \ Y^=i ^c" n a s been set equal to zero. The terms in the first line of Eq. (2.2.5) constitute the pure vibrational potential energy while the terms in the second line constitute the potential energy for the internal rotation. The terms in the second line of Eq. (2.2.5) that depend on the vibrational displacements should be small because they describe how the barrier to internal rotation changes as the molecule vibrates. Therefore, they can be considered as a perturbation. In terms of normal coordinates Qk, the potential energy is simply V(Q, a) = ]- Y XkQl + ^ E Xkk'k"QkQk'Qk" + ••• n = l (l-cos3no). (2.2.9) 6 VZn + Y V&Qk + V3n'QkQk' + Here Xk, Xkk'k", - are the force constants. It has been shown by Papousek and Aliev [56] that the total quantum mechanical vibration-torsion-rotation Hamiltonian with a classical kinetic energy given by Eq. (2.1.63) and potential energy given by Eq. (2.2.9) has the following form l l m ~ 7 h2 H"tr =2 E Hi ( J / 3 - P/3) ( J 7 - P 7 ) + 2 E ^ + V ( Q > a ) - j J 2 HP > (2-2-1 °) P,-y=x,y,z,a k=l p where pp (8 = x, y, z and a) is given by Eq. (2.1.67). The last term in Eq. (2.2.10) can be considered as a mass-dependent contribution to the potential energy, which will be neglected in later discussion. 2.3 Harmonic Oscillator, Rigid Rotor Approximation So far we have derived the complete Hamiltonian for molecule AB3XY3. However, Eq. (2.2.10) is still too complicated to be solved exactly. Therefore it is necessary to introduce more approximations. The approximate vibration-torsion-rotation Hamiltonian is chosen so that its known eigenfunctions form the basis vectors for higher order calculations. Later on, the high order terms in the Hamiltonian, including 24 Chapter 2 General Theoretical Considerations terms which involve coupling between large amplitude internal rotation and the vibrational motion, will be set up in these known basis vectors. These approximations are based on the assumption that the vibrational and torsional displacements of the atomic nuclei in rigid molecules are limited to a small region around the equilibrium configuration. In the first approximation, it is then possible to neglect the vibrational dependence of because fip are functions of the small amplitude vibration coordinates Qk- Similarly, the torsional dependence of u-p1 can also be neglected. The coefficients / i ^ 7 in Eq. (2.2.10) then become /zjg7. (u-^ can be easily worked out; see Section 2.5.) In this approximations Hvtr can be written: 1 1 3 N ~ 7 Hvtr = 5 J M j , (J/3 - Pp) ( J 7 - Pi) + 2 E P * + V(Q> «)• (2-3-1) P,1=x,y,z,a fc=l In the second approximation, the vibrational angular momenta pp, which are small with respect to the angular momentum of the overall and internal rotation, and the anharmonicity of molecular vibrations are both neglected. Then the harmonic oscillator, rigid rotor approximation is obtained Hvtr = Htr + Hv (2.3.2) where Htr = \ E Pp^p^ + Vf, (2.3.3) P,l=x,y,z,a 37V-7 1 3AT-7 H* = \ Y.PZ + \ E X*Ql> <2-3.4) fc=i fc=i Vt = E ^ 3 n ( l - c o s 3 n a ) / 2 . (2.3.5) n=l H® is just the harmonic oscillator Hamiltonian. H®r is the zeroth-order torsion-rotation Hamiltonian. The solutions of the Schrodinger equation H^0V = E°*V (2.3.6) are the harmonic oscillator energy levels and wavefunctions (See Section 2.3.1.) Hp,, can be separated into a sum of rigid rotor Hamiltonian and zeroth-order torsional Hamiltonian (Section 2.3.3). The solutions of the Schrodinger equation can be expressed as « = E t ° r < (2.3.7) where E^r and ^ r are the zeroth-order torsion-rotation energy levels and eigenfunctions, respectively. These will be discussed in Section 2.3.3. According to Eqs. (2.3.2), (2.3.6) and (2.3.7), the overall vibration-torsion-rotation wavefunction ^ v t r 25 Chapter 2 General Theoretical Considerations of the molecule can be written to a first approximation as the following product function * « t r = * 2 ( Q i , Q 2 , - - - ) * f r ( ^ ^ X , a ) - (2-3.8) The vibration-torsion-rotation energy, Evtr can be written as a sum of the vibrational energy E® and the torsion-rotation energy E^r Evtr = El + E°tr. (2.3.9) Of course, all these energies are in joules. In molecular spectroscopy, these energies are normally used in the units of cm - 1 , which are simply obtained by multiplying these energies by a factor of 1 /he. 2.3.1 Harmonic Oscillator The Schrodinger equation (2.3.6) can be separated into 3N — 7 independent equations one for each Qk. Each equation has the form - f dHQQ*k) + \^Ql*vk{Qk) = Evk^Vk(Qk) . (2.3.10) This is the Schrodinger equation for a one dimensional harmonic oscillator with potential energy XkQl/2, and mass equal to 1. Then the solutions of Eq. (2.3.6) are just 3 7 V - 7 EQV = YI Evk); (2-3.11) fc=i 3 J V - 7 < = n • ( 2 - 3 - 1 2 ) fc=i It is convenient in the theory of the vibrational-rotational spectra to introduce dimensionless normal coordinates qk and operators of linear momenta pk which are defined by the equations / \ . \ 1/4 qk = [4\ Qk, (2.3.13) Pk = -i£-k=\-k1/Ah-^Pk, (2.3.14) where Xk = 47r2z/| and Pk = —ihd/dQk. Here vk is the frequency of the fc-th normal vibration. The commutation relations of these operators are: [Qk,Pi] = ifal 5 [qk,qi] = 0 ; \pk,Pi] = 0 . (2.3.15) where [A, B] = AB — BA is the normal commutator. The Schrodinger equation (2.3.6) in terms of the dimensionless coordinates qk and momenta pk assumes the following form 1 3N-7 - Y (*"*) (Pl + <&) ®S(?*) = $*°v (Qk) • (2.3.16) fc=i 26 Chapter 2 General Theoretical Considerations It is well known [56] that the solutions of the linear harmonic oscillator Schrddinger equation (2.3.10) are the vibrational energy levels Ev ' given by = hvk [vk + 1/2), vk = 0,1,2,- •• (2.3.17) where vk is the vibrational quantum number. Therefore, the expression for the vibrational energy levels of a molecule can be written in the following form 3AT-7 Gv = E°Jhc = z~2UJk K + 1/2) • (2-3-18) k=l where u>k = uk/c is in cm - 1 . The eigenfunctions of the linear harmonic oscillator equations (2.3.10) are: *Vk=NVke-W2MHVk(qk), (2.3.19) where HVk(qk) is the Hermite polynomial in qk; NVk is a normalization constant. For any one-dimensional harmonic oscillator, the coordinates q and momenta p have the following prop-erties in the basis \v) = \tyv(q)): l\v) = \l—\v + l) + ^2\v-Vi ; (2-3.20) = * y ^ l « + l > - » y | l « - l > • (2-3.21) For a doubly degenerate normal vibration, the same frequencies va = v\, = v correspond to the two independent normal vibrations described by coordinates qa and The Schrddinger equation for a two-dimensional isotropic harmonic oscillator can be written in the form \ ihu) [{pI+pI) + {ql + ql)] * v = EVV!V. (2.3.22) It is convenient to introduce polar coordinates g and <p by the equations qa = gcosip; (2.3.23) qb = Q sin ip . (2.3.24) where 0 < g < oo and 0 < ip < 2n. Eq. (2.3.22) then has the form 0. (2.3.25) f d2 18_ 1 d \ d2g g dg g2 dtp2 2EV o Q hu The eigenfunctions of above equation depends on g and <p as variables and on the vibrational quantum numbers v and £ as parameters. It can be shown that this function can be written as a product of the radial (g) and angular (</?) wavefunctions * V (ft <P) = Nv,ee-^2gW W J . M w 2 (q)] • (2.3.26) 27 Chapter 2 General Theoretical Considerations Vibrational quantum number v assumes the following values: v = 0,1,2,-••; (2.3.27) and quantum number £ assumes v + 1 values: £ = v,v-2,---,-v + 2,-v. (2.3.28) In Eq. (2.3.26), £ ^ + | £ | ) / 2 (&) 1S m e associated Laguerre polynomial; Nv>e is the normalization factor. The expression for the energy levels of the two-dimensional isotropic harmonic oscillator can be written in the form Gv = {Ev/hc) (v + 1) = uiv (v + 1) (2.3.29) i.e. Ev is independent of £. Therefore, each energy level of this oscillator for a given v has (v + l)-fold degeneracy. The fact that exp (i£<p) appears in the expression for the wavefunction (2.3.26) indicates the presence of a vibrational angular momentum; £ is therefore called the vibrational angular momentum quantum number. Classically, the vibrational angular momentum can be interpreted as arising through superposition of the normal vibrations qa and qt, with a 90° phase shift. If qa and qt, do not represent motions on the same line, their superposition results in a circular or elliptical vibrational motion of the atomic nuclei. By defining q± = qa±iqb, (2.3.30) P± = Pa±ipb; (2.3.31) the Schrodinger equation for a two-dimensional isotropic harmonic oscillator (2.3.22) can be written in the form \ fal'2) ]p+P- + Q+Q-) *v,e = EVVV (2.3.32) Following Papousek and Aliev [56], the phase factor of the wavefunctions is chosen so that the matrix elements (v,£\q±\v,£) are real and the matrix elements (v,£\p±\v,£) are pure imaginary. In the basis \tyv e) = \v,£), the operators q± and p± have the following properties: q±\v,£) = ^ V ± e 2 + 2\v + l,£±l) + ^ ^ \ v - \ , £ ± l ) ; (2.3.33) P±\vJ) = i^V±l2 + 2\v + l,£±l)-isJ^^\v-l,£±l) . (2.3.34) 28 Chapter 2 General Theoretical Considerations 2.3.2 Rigid Rotor A rigid rotor Hamiltonian has the following form " \ xx ±yy xzz J where I%x, and P]z are the components of moment of inertia in equilibrium. For a symmetric top, 1° =1° •'•xx yy The components of a angular momentum vector operator J satisfy the following commutation relations in the SFS (X, Y, Z) and the MFS (x, y, z) [56]: [Jx, Jy] = ifciz, [Jy, 3Z\ = iH3x, [Jz, 3X] = ihJy ; (2.3.36) [3x,3y] = -ihJz, [Jy,3Z] = -ihJx, [3Z,3X] =-ih3y . (2.3.37) The square of the angular momentum operator, 32=3l + 32y + 3l = 3\+32y + 32z (2.3.38) commutes with 3X, 3y, 3Z as well as 3x, Jy, 3z'-[J ,J 2 ]=0 (2.3.39) As J 2 commutes with 3Z as well as with 3z, operators J 2 , J 2 and 3Z have a common complete set of eigenfunctions which are denoted as | J, k, m). The eigenvalues of these three operators are 32\J,k,m) = h2J(J+ l)\J,k,m) , (2.3.40) 3z\J,k,m) = hk\J,k,m) , (2.3.41) 3z\J,k,m) = hm\J,k,m) . (2.3.42) It can be shown that, if J is the angular momentum operator for the rotation of the reference configuration of the atomic nuclei, J can assume only integral values and J, k and m can assume the following values: J = 0,1,2,3 - •• , (2.3.43) k = -J,-J+l,---,J-l,J, (2.3.44) m = -J,-J + 1,- • -,J -1,J . (2.3.45) In the basis of | J, k, m), it is more convenient to define two kinds of ladder operators J i m ) = 3x±i3y, (2.3.46) J i s ) = 3x±i3Y; (2.3.47) where m and s stand, respectively, for the MFS and the SFS. The commutation relations for the ladder 29 Chapter 2 General Theoretical Considerations operators are T T H = T ^ J ± m ) , (2.3.48) = ±hJ{±) . (2.3.49) J ± ^ are ladder operators for the eigenfunctions of Jz. ladders the eigenvalues of Jz down by h~ and ladders them up by h~. Similarly are ladder operators for the eigenfunctions of Jz- J^+ ladders the eigenvalues of Jz up by h and ladders them down by h. For a symmetric top, the only nonvanishing matrix elements of in the J, k, m representation are: E°jkm = (J, Km\HQR\ J, k,m) = B0J{J + 1) + (Aq - B0)k2. (2.3.50) This is diagonal in J , k, and m. Here B ° - 2 ^ k ' <"- 5 1> >^ = 2 ^ k ; <2-3-52> are the rotational constants in cm - 1 . The quantum mechanical operators for Jx, Jy, and Jz are [56]: Jx = cot(9cosx(—iftd/dx) + smx(—ifrd/d6) — csc#cosx(—ih~d/d<fi); = -cot6smx(-ifid/dx)+ cosx(-ihd/d9)+csc9sinx(-ihd/d(f>); (2.3.53) J z = -ihd/dx-For a symmetric top, the rotational wavefunctions in terms of the Euler angles 0, <p, x can be written as: | J, k, m) = SJkm (6,4>) eik* = eJkm (9) eim*eik*. (2.3.54) 2.3.3 Separation of Torsion and Overall Rotation In Section 2.1.1, we have shown that Tfr has different forms depending on whether the MFS is in the PAM (Eq. (2.1.29)) or in the IAM (Eq. (2.1.33)). Similarly, flg. will have different forms in the PAM and IAM. This has been discussed in detail by Lin and Swalen [39]. In the principal axis method, the angular momenta for the Hamiltonian have been given by Eqs. (2.1.64) and (2.1.65). For the zeroth-order torsion-rotation Hamiltonian, the angular momenta can be obtained directly from these equations by neglecting the terms containing vibrational operators. These have the following forms: Jx = (2-3.55) J „ = J£ ,w y ; (2.3.56) 30 Chapter 2 General Theoretical Considerations J z = P]zcoz + Lji°a ; (2.3.57) 3a = i £ ° ( w z + a) . (2.3.58) From the above definitions of 3X, and J z , we see that they are simply the components of the total angular momentum J (including the internal rotation) about the x, y and z axis. J Q is the total angular momentum of the top including both the internal and external rotation. Then the torsion-rotation Hamiltonian can be written: Hi =B0 ( J 2 + J j ) + A Q . F J 2 . + F032 + Vt + ( - 2 ^ ) f J z J q ) (2.3.59) where h2 h2 2hc ( i ? , - 2 f c c j £ ° ft2/0 2hdTJ (l°zz -In the internal axis method, we can use Eq. (2.1.33) to introduce the angular momenta: 3S = &T%./du>x = , (2.3.61) 3y = dT^/dujy = I^yuy, (2.3.62) J 2- = trLfr/dcuz = r]zuj-z , (2.3.63) J Q = tnfr/da = I^{l- p)a. (2.3.64) Again, 3X, 3y and J z are the components of the total angular momentum J (including the internal rotation) about the x, y and z axis. J a is the angular momentum of the top when measured in the MFS xyz. This occurs because the top is moving at an angular velocity (1 — p) d k in this MFS. Then the Hamiltonian in the IAM can be written in the following form: Hi =B0(32X + J?) + AoJz- + F 0 J 2 + VT . (2.3.65) * J- ' v ' Vt(a) is the torsional potential at the equilibrium. From Eq. (2.2.9), we have T t / \ _ _ 1 — cos 3a Vt(a) = Y V3N ~0 • (2.3.66) n=l Z The zeroth-order torsional potential is just V3 (1 — cos 3a) /2 and V3 is generally called the barrier height. It can be seen that the barrier is actually given by the sum E vok+3 = Vz + V9 + Vis + • • • • (2.3.67) fc=0 In general, Vq is much smaller than V3 term, the ratio being the order of one-hundredth or less. So the threefold term provides a good approximation for the potential. This potential is plotted in Figure 1.2. 31 Chapter 2 General Theoretical Considerations In both the PAM and the IAM, H^T includes the sum of a rigid rotor Hamiltonian (H® or H®) and a torsional Hamiltonian (H® or H®). However, H^r in the PAM has an extra coupling term between the torsion and overall rotation, namely — 2 A O , FJQJZ . This is due to the difference in the definitions of the MFS in the two methods. From the relations of the components of u> given by Eq. (2.1.35) as well as the definitions of J ^ and J ^ given in this section, it can easily be seen that the components of the total angular moment J defined in the PAM and the IAM are simply related by the following transformation: Jy v j . y / cos pa sin pa 0 \ — sin pa cos pa 0 0 - 0 1 / Jy V J 2 / (2.3.68) Ja is related to Ja by J a = J a - pJz (2.3.69) At the limit of a very high potential barrier, H^r in the PAM in Eq. (2.3.59) does not reduce readily to the case of a rigid rotor plus a torsional harmonic oscillator. The reason is that the operator Ja defined in Eq. (2.3.58) contains not only the angular momentum of the internal rotation of the top but also a contribution from the overall rotation. When the internal rotation is completely frozen, the classical angular velocity d becomes zero and thus Ja becomes Izz uz = pjz. When this expression for Ja is substituted into Eq. (2.3.59) , the usual energy equation for a rigid symmetric top is recovered. On the other hand, Ja in the IAM given by Eq. (2.3.64) depends solely on d; it vanishes in the high barrier limit since a is a constant. 3%. in the IAM reduces to a sum of a rigid rotor plus a torsional harmonic oscillator in the high barrier limit. The different forms of H^T in the PAM and IAM correspond to different eigenfunctions. However, since they describe the same H^T, they both give the same eigenvalues. This will be discussed in the next two sections. 2.3.4 The Zeroth-Order Torsion-Rotation Hamiltonian in the Principal Axis Method In this section, we will use the zeroth-order torsion-rotation Hamiltonian as an example to discuss the general behavior of the torsion-rotation energies in the PAM. In the PAM, Jx, Jy and Jz are the components of the total angular momentum (including the internal rotation) about the x, y and z axis. Therefore they satisfy the usual commutation relations. Since the MFS in the PAM is chosen so that x, y and z axis are fixed on the frame, the quantum mechanical operators for Jx, Jy and J z in the coordinate representation depend only on the Euler angles 0, (fi and Xp> but not on a. The quantum mechanical forms of J x , Jy and Jz are given by Eq. (2.3.53). As a quantum mechanical 32 Chapter 2 General Theoretical Considerations differential operator, 3a may be expressed as: 3a = -ih (•?-) . (2.3.70) Therefore 3a commutes with 3X, 3y and 3Z. The Schrodinger equation for the Hfr in Eq. (2.3.59) in terms of 9, <j>, Xp, and a can be obtained by substituting operators 3X, 3y, 3Z, and J Q by their corresponding quantum mechanical forms. Because of the term Hfr = — 2Aq<f3z3a, the variable a cannot be separated from the three Euler angles. However, one can determine the complete basis set for H®T and solve the Schrodinger equation by diagonalizing the matrix for Hfr. In the limit the barrier height goes to zero, the Hamiltonian iff becomes: Hfr = Bo (J 2 + 3y) + AQ,F32 + F03l - 2A0<F3z3a. (2.3.71) The first two terms form the rigid rotor Hamiltonian of a symmetric top, which has the eigenfunctions Sjk (9,4>) etkxr as given by Eq. (2.3.54). So the eigenfunctions of H^T are simply ^SJk(9,<p)eik^eima, (2.3.72) 2TT where m must be an integer since this eigenfunction must have a period of 2TT. 1/\/2n is the normalization factor for the torsional part of the wavefunction. This is simply a product of a symmetric top rotational wavefunction and the free rotor wavefunction. In the case V3 is not zero, from the periodicity of Vt (a), we can see that eirna must have a 27r/3 period. Therefore, one can define a basis function in terms of the free rotor torsion-rotation wavefunction as \J,k;kf,a) = -^=Sjk(9,<f>)e^e^k^, (2.3.73) where a = —1, 0, and +1. kf is an integer with —00 < kf < 00. The torsional operators 3a and (1 — cos 3a) /2 have the following properties in this basis: 3a\J,k;kf,a) = (3kf + a) \J,k;kf,a); (2.3.74) cos 3a I J, k;kf,a) — ^ \J, k; kf — I, a) + ^ \J, k; kf + l,er). (2.3.75) 3X, 3y and 3Z have the same properties as those for a rigid rotor since they depend only on 9, <f>, and xP-The symmetry of | J , k; kf,a) will be discussed in Section 2.4.2. The zeroth-order torsion-rotation Hamiltonian H^r is diagonal in J, k, and a. We can set up a matrix for each J , k, and a. The matrix elements Mkfk>f (J, k, a) = (J, k; kf,a\ Htr \ J, k; kf,a) have the following form: Mkfk-f = (Ao,F-Bo)k2 + BoJ(J + l)-2A0,F(3kf + a)k + F0(3kf + o-)2 + ^j, for kf = k'f; (2.3.76) 33 Chapter 2 General Theoretical Considerations Mkfk> = ~ , forkf = k'f±l; (2.3.77) (2.3.78) Mkfk>f = 0, in other cases. It is clear that Mkfk'f (J, k, a) has the following properties M f c / , f c > (J, k, a) = M _ f c / , _ f c > (J, -fc, -cr) . (2.3.79) The diagonal matrix elements contain a term ( A Q , F - #o) k2 + Z?o«7 (J + 1). This is not the total rotational energy of the whole molecule since A Q , F A ; 2 is just the rotational energy (in the z axis direction) of the frame. However, by using p = A O , F / F 0 and A0 = A ) , F / (1 - p), the diagonal matrix elements Mkjkf can be rearranged as follows: Mk/k> = (Ao,F-Bo)k2+B0J{J + l) + Y +F0 \(Skf + cr)2 - 2pk (3kf + a) + p2k2} - F0p2k2 = (A0-Bo)k2 + BoJ(J + l)+F0(3k + o--pk)2 + ^-. (2.3.80) The term (A0 — Bq) k2 + B0J (J + 1), denoted as Er (J, k), is just the rotational energy for the whole molecule. For each matrix M (J, A;, cr), this term only appears in the diagonal matrix elements as a constant. Thus we can write: M(J,A:,cr) = £ R ( J , A ; ) E + M '(A;,cr), (2.3.81) where E is a unit matrix with the same dimension as M . The new matrix M ' does not depend on J, and can be labelled by just k and a. Clearly, matrix M and M ' have the same eigenvectors. M ' (k, a) has the following form: J , k, u kf -1 0 +1 -1 F 0 (-3 + a - pkf + V3/2 -V3/4 0 0 -V3/4 F 0 (Cr - pk)2 -V3/4 +1 0 -V3/4 F 0 (3 H - Cr - pkf + V3/2 (2.3.82) M ' (k, a) is a real and symmetric matrix. It is written in a 3 x 3 matrix without losing any generality except the kf range. The M ' (A:, a) and M ' ( - A ; , -a) matrix are related by a unitary similarity transformation: M ' (A;, cr) = U+M' (-jfe, -cr) U (2.3.83) where 1 U = ± (2.3.84) The eigenvalues and eigenvectors can be found by diagonalizing M ' (A;, cr) numerically. For each A; and cr, the eigenvalues are labelled by the index v with v = 0 for the lowest eigenvalue. From Eq. (2.3.83), we can 34 Chapter 2 General Theoretical Considerations see that the eigenvalues Et (k; v, a) have the following property: Et(k;v,a) = Et(-k;v,-a). (2.3.85) Et (k; v, a) cannot be classified as the eigenvalues of H^+Hfr, see Eq. (2.3.59). Instead, it is the eigenvalue of the operator if t° + Hfr + Fp232z. Because the rotational energy of the whole molecule is contained in Er, Et (k; v, a) is just the pure torsional energy. It follows from Eq. (2.3.85) that the torsion-rotation energy Etr = Er + Et has the same property: Etr (J, k; v, a) = EtT (J, -k; v, -a). (2.3.86) The eigenfunction | J, k; vq,<t) corresponds to the eigenvalue Etr (J, k; v, a) has the following form: \J,k;v6,a) = Sjk{9,4>)eik^UKv,a(a), (2.3.87) where UkiVtff (a) = -±=Y Ak^'aei^+^a. (2.3.88) Akk^'a are the coefficients of the eigenvector corresponding to the eigenvalue Etr (J, k; v§,a). All A^J^ are real numbers. From Eq. (2.3.83), it can be seen that they have the following property: = ^ - M a , - ^ ( 2 3 G 9 ) where g is the phase factor which can be determined numerically only to within a factor of ±1 . It follows that UkiV}Cr (a) has the following properties Uk,v,*(«) - ^ Y e A - k T ' ~ a e i { 3 k f + a ) kf = g x _ L A-k;ve,-<7e-i(3kf-a) kf = QU*-k,v,-A*) • (2.3.90) The fact that the rigid symmetric top rotational energy Er can be separated from the eigenvalues does not mean that the variables have been separated. In fact, the basis functions of matrix M ' are still \ J, k; kf, a), which is a function of 9, <f), \p, and a. The symmetry of \ J,k;ve,a) depends on J , k, vq as well as a. (See Section 2.4.2.) As a simpler example, we can consider A; = 0. In this case, UktVt(T (a) is an eigenfunction of just iff and can be classified in the representations of the C3 group [41] as follows Uk=0,v,cr=0 (oi) A; Uk=0,v,*=+i (a) Ea; (2.3.91) Uk=0,v,a--\ (a) Eb. 35 Chapter 2 General Theoretical Considerations Potential Barrier • Figure 2.2 The hindered internal rotational enery levels as a function of the barrier height. The free rotation quantum numbers m are at the left, i.e. a zero barrier, and the torsional quantum number v are on the right, i.e. a relatively high barrier. The diagonal line V 3 shows the actual height of the barrier in relation to the energy levels. Those energy levels below the top of the barrier are clearly discernible. Uk=o,v,a=o (a) forms the nondegenerate eigenfunction; Uk=otV,+i (a) and Uk=otV,-i (a) form a doubly degenerate pair. In Figure 2.2 the energy levels for Et (k = 0; v, a) are plotted as a function of the barrier height V3 from free internal rotation to a relatively high barrier. Well below the top of the barrier, the spacings between the nondegenerate and degenerate energy levels associated with a given v are much smaller than those between levels with different v. (See Figure 1.2.) For this reason, the index v is called the principal torsional quantum number and the different energy levels associated with a given v are thought of as the torsional sublevels belonging to the same torsional state. The sublevels are distinguished by the index a with cr = 0 for the nondegenerate level (A) and cr = ± 1 for the levels (E) in the figure. 2.3.5 The Zeroth-Order Torsion-Rotation Hamiltonian in the Internal Axis Method From Eqs. (2.3.68) and (2.3.69), Jx, Jy, Jz and J a can all be put in terms of the angular momenta defined in the PAM. From the commutation relations of the PAM angular momenta, we can easily show that in the IAM, Jx, Jy and J 2 all commute with Ja, and Jx, and J 2 satisfy the usual commutation relations for a rigid rotor. The variables for H^r are 9, <f>, Xi, and a'. Xi's related to Xp by Eq. (2.1.32) and a' = a = xT — XF • Since J x , Jy and J 2 are only functions of the Euler angles 6, <j>, and Xi, w e have 36 Chapter 2 General Theoretical Considerations [Hr, H?] = 0. The quantum mechanical differential operator Ja has the following form [39]: 'd_ da' 3 « = ~ i h ( 7£7 ) • (2.3.92) Therefore, the differential equation for H®r can be separated into a sum of two differential equations: the one with variables (9, <fi,Xi) a r , d the one with a'. The first one is just the Schrodinger equations for the rigid symmetric top. The wavefunction of H^r can then be written as a product of the symmetric rotor wave function and the torsional wavefunction: i> = SJk (9, <t>) eik*-Mkv<7 (a') . (2.3.93) Mkva (a1) represent the solutions of the differential equation H?M (a') = [-Fd2/d2a' + Vt{a')] M (a') = EtM (a') , (2.3.94) with the appropriate boundary conditions. As in the case for the PAM, the differential equation (2.3.94) can be set up in the basis of the free rotor functions eima. However, the boundary conditions are different since in this case xi and a' are the variables. Since the wavefunction ip must by invariant under the transformation: XT - XT + 27 rn i ; X F -» + 27rn 2 ; (2.3.95) it can be shown [39] that m' must satisfy the following condition rri = I - pk, (2.3.96) where I is an integer. Therefore, the free rotor functions have the following form in the IAM \k,kf,a) = - ^ ( 3 * / + * - ^ (2.3.97) V27T where kf is an integer and a = — 1, 0, and +1. The 3kf + a runs over all integers and thus is equivalent to I. The bar symbol \k, kf, a) indicates that this free rotor function is defined in the IAM. It can be seen that the torsional operators have the following properties: 3a\k,kf,a) = (3kf + a - pk)\k,kf,a); (2.3.98) 1 1 cos3a\k,kf,a) = -\k,kf + l,a) --\k,kf - l,a). (2.3.99) Clearly, when is set up in the basis \k, kf, a), the matrix obtained will be identical to the matrix M ' in the PAM; refer to Eq. (2.3.82). Therefore, the eigenvalues of H® must have the same properties as that for Et (k; ve, cr) in the PAM. The coefficients of the eigenvectors of this matrix give the eigenfunctions for H®: Mkva (a') = -^=Y Ak-,v,aei{3kf+a-pk)a'_ (2.3.100) V27T V"' 1 kf 37 Chapter 2 General Theoretical Considerations 2.3.6 Comparison of the Torsion-Rotation Hamiltonian in the PAM and the IAM From mathematical point of view, the PAM and IAM methods differ in the fact that the variables of the wavefunctions are different. The differential equation of Hfr in the PAM is set up in terms of 6, <j>, Xp and a; while in the IAM 3$. is set up in terms of 6, c/>, Xi a n d a ' - As has been shown the previous two sections, once the torsion-rotation Hamiltonian is set up in the corresponding free rotor torsion-rotation basis, the matrix obtained by the two methods are identical. In fact, the eigenfunctions in" the two methods can be easily converted to each other by using xi = Xp + Pa a n d a' = a. Although the IAM has the advantage that the torsion-rotation interaction term vanishes in the zeroth-order, it has a serious difficulty in that the rotational operators (J x ,3 y) as well as vibrational operators (qta,qtb) do not transform by the irreducible representations in the molecular symmetry group of G\& [42]. This will make the derivation of the vibration-torsion-rotation interactions difficult. For this reason, all the theoretical work done later on is based on the MFS denned in the PAM. For both methods, in a matrix form, the rigid rotational energy can be separated from the torsional energy. This is a direct result of the symmetry of the symmetric top. The MFS in the IAM is related to the MFS in the PAM by a rotation of —pa about the z axis. For a symmetric top, this rotation does not alter the components of moment inertia in the x and y directions in either MFS. However, this is not the case for asymmetric tops. The IAM has significant advantages for asymmetric tops where k is no longer a good quantum number in the zeroth-order. In that case, the separation of the torsion-rotation interaction in the zeroth-order is important since one wants to solve the torsional and rotation problems separately to reduce the computing time. Since the two methods are identical in numerical results, one can use either method to discuss the be-havior of the torsional energies. In the IAM, at the very high barrier limit, the eigenfunctions of the torsional Hamiltonian can be ap-proximated by the harmonic oscillator functions. In addition, the eigenvalues can be simply obtained by the following approximate method: when pk — a is replaced by pk — a + 3, the form the eigenfunction is not altered, and therefore the eigenvalues remain unchanged. One may then regard the eigenvalues as periodic functions of (27r/3) (pk — a) and expand them in a Fourier series as [39]: Et(k;v,a) = FYan)\^s(27rn/3)(pk-a)] . (2.3.101) n In principle, this equation will not be limited to the high barrier case. However, the Fourier series converges (v) faster in the high barrier limit. In the lowest order, a n ; are functions of the reduced barrier height [39]: sm = W3/9F. (2.3.102) Eq. (2.3.101) is normally used to estimate the qualitative behavior of the a levels. The cr-splittings between the torsional sub-levels increase as sm decreases. Eq. (2.3.101) also means that the matrix elements of J 2 38 Chapter 2 General Theoretical Considerations and (1 — cos 3a) /2 in the basis of \k; v, a) = Mv^>a (a) can also be expanded in a Fourier series: (k;v,a\ 32a\k;v,a) = ^ bn] c o s (27rn/3) {pk - a) , (2.3.103) n (k;v,a\ 1 ~ c o s 3 a \k-v,a) = cos(27rn/3) (pfc - cr) . (2.3.104) n In the PAM, with a very high barrier, the torsional motion reduces to a harmonic oscillation, and the potential energy V 3 (1 — cos 3a) /2 may be expanded about the equilibrium point yielding (9/4) Vsa2. The torsional wavefunction in the vicinity of the equilibrium configuration can be approximated by the harmonic oscillator functions. The corresponding eigenvalues are Ev = 3(V3F)l/2 (v + ^j . (2.3.105) This equation can be used to estimate the torsional frequencies. 2.4 Symmetry Classification of the Eigenstates of A B 3 X Y 3 2.4.1 Symmetry Classification of the Vibrational States The reference configuration of molecule AB3XY3 is shown in Figure 2.3. All bond lengths and angles have their equilibrium values (except the torsional angle a); the top-fixed, frame-fixed and molecule-fixed axis system (x'y'z), (x"y"z) and (xyz), respectively, are shown in the same figure. The orientations of the TFS, FFS and MFS in the SFS are defined by the Euler angles (M , X i ) » {6A,X2) m A {Q,<t>,x), respectively. All the symmetry discussions are based on the PAM, so the MFS (xyz) is the same as FFS (x"y"z) in Figure 2.3 and x = X2- As before, the torsional angle a is taken as the angle between the x' and x" axis projected on the xy plane; in terms of Euler angles, a is just %i — Xi- The symmetry properties of a AB3XY3 type molecule have been discussed in detail by Hougen [55] and Bunker [40] based upon the formalism of the permutation-inversion (PI) group [41,59]. The present treatment will follow that of Bunker [40]. The molecular symmetry (MS) group is formed by a set of operations which leaves the molecular Hamil-tonian unchanged. The MS group of AB3XY3 is the group G\& [40,41]. The character table of Gi8 is shown in Table 2.1, where the labeling of the atoms in AB3XY3 is shown in Figure 2.3. The transformation prop-erties of the Euler angles, 9, <f> and x2 a r e a ' s o g i y e n m the same table. In the limit of infinite high barrier, the molecule AB3XY3 has a MS group of Csv- The species of Gi& are correlated with those of C^v by [11]: Ai -» A i , A2 A2, Ei -> E (i = 1,2,3) and £ 4 Ai + A2 . (2.4.1) 39 Chapter 2 General Theoretical Considerations The reverse correlation (C$v —> Gis) is -» Ai + A2-^A2 + EA, and E^E1+E2 + E3. (2.4.2) In other words, each C3v species in the high barrier limit splits into two or three Gis species as the barrier height is lowered. For AB3XY3 with IB = ly = 1/2 and IA — IX — 0, where I A and Ix are the nuclear spins of nucleus A and X, respectively, there are total of 64 nuclear spin functions. These are reduced as follows 8A1 + SA2 + 8 £ i + 8E2 + 4E 3 + 4 £ 4 • (2.4.3) The spin statistical weights are of great use in assigning the spectra. To determine to species of the normal coordinates in G\%, we first define the symmetry coordinates which are independent of the torsional angle. There are total of 3AT — 7 = 17 symmetry coordinates. The species of these symmetry coordinates are easily determined to be [40]: hAi + 2>EX + 3E2 (2.4.4) The internal coordinates of the molecule are defined in Figure 2.3. The symmetry coordinates [40] are listed in Table 2.2. As can be seen from the table, the three doubly degenerate symmetry coordinates S\o, Su and S12 are all localized in the frame group and have a symmetry of E\, while the other three S-j, S$ and Sg which localized in the top have a symmetry of E2. In the case when we can neglect the end-to-end coupling these symmetry coordinates can be approximated as the normal coordinates. However, for most molecules the end-to-end coupling cannot be neglected. Since all the doubly degenerate symmetry coordinates localized in the top are defined in the TFS, a new set of symmetry coordinates should be defined 40 Chapter 2 General Theoretical Considerations Table 2.1 The molecular symmetry group of AB3XY3: C?i8 a E (123) (456) (123)(456) (123) (465) (12) (45)* (132) (465) (132) (465) (132) (456) (12)(56)* (12) (64)* (23)(45)* (23)(56)* (23) (64)* (31) (45)* (31) (56)* (31)(64)* Equ. rot. R° R° R2,/3 R-2n/3 K/2 ^1 1 1 1 1 1 1 T A2 1 1 1 1 1 - 1 •Izj J Q £1 2 2 - 1 - 1 - 1 0 (TX, Ty) , ( J X , Jy) E2 2 - 1 2 - 1 - 1 0 Ez 2 - 1 - 1 2 - 1 0 Ei 2 - 1 - 1 - 1 2 0 e e e e 0 6> -e + ir 4> 4> 4> 0 + Xi Xi Xi - (2TT/3) Xi Xi - (27T/3) Xi - (27T/3) - X l +7T X2 X2 X2 X2 + (27T/3) X 2 + (27T/3) X 2 - (27T/3) - X 2 + 7T a This table is quoted from Bunker [41]. by projecting these coordinates onto the molecule fixed frame, i.e. the FFS. These are St,a W c o s a - s i n a \ / St \ . . . < = 6 7 8 ( 2 4 5 ) St'b J \ sin a cos a J \ Stb J v ' It can be shown (See Appendix A.3) that all these three symmetry coordinates have E\ symmetry in the d s group. Therefore, the symmetry coordinates representation is now: 5.4.1+6£i. (2.4.6) Symmetry coordinates with the species given by Eq. (2.4.6) are the best to use in interpreting the vibration-41 Chapter 2 General Theoretical Considerations Table 2.2 Symmetry coordinates of A B 3 X Y 3 a . 5Ai Si = 3- J/2 (Adi + Ad2 + Ad3) S2 = 3 - 1 / 2 (Ad 4 + Ad5 + Ade) s3 = AR S4 = 6 - 1 / 2 ( A / J 1 2 + A/3 2 3 + A / 3 3 1 - A 7 l - A 7 2 - A 7 3 ) s5 = 6- 1 /2 (A/? 4 5 + A/3 5 6 + A/3 6 4 - A 7 4 - A 7 5 - A 7 6 ) 3E2 5Va = = 6- 1 / 2 (2Adi - Ad2 - Ad3) S7b = 2~ ^ 2 (Ad 2 - Ad 3 ) -58a = = 3 - 1 / 2 ( A / 3 2 3 - A / 3 3 1 - A / 3 1 2 ) £>8b = 2- 1 / 2 (A/3 3 1 -- A/3 1 2) S9a -= 3 - 1 / 2 ( A 7 l - A 7 2 - A 7 3 ) Sgb = 2~ 1 / 2 ( A 7 2 - A 7 3 ) SlOa = 6- x/2 (2Ad 4 - Ads - Ad6) SlOb = 2 -Ade) •Silo = 3 - 1 / 2 ( A / 3 5 6 - A / 3 6 4 - A / 3 4 5 ) Sub = 2 - 1 / 2 (A/? 6 4 - A / 3 4 5 ) <5l2a = 3-^2 ( A 7 4 - A 7 5 - A 7 6 ) Sl2b = 2" - 1 / 2 ( A 7 s - - A 7 6 ) a The symmetry coordinates are quoted from Ref. [40]. torsion-rotation spectrum quantitatively since with them the G matrix [60] is independent of the torsional angle a and the F matrix can be added by perturbation theory just as the torsional barrier can. Since normal coordinates are obtained as the linear combinations of symmetry coordinates which belong to the same irreducible representation of the symmetry group G\%, each normal coordinate must have the same symmetry as the corresponding symmetry coordinate [60]. Eq. (2.4.6) also defines the symmetry species of the normal coordinates. Following the conventions for A B 3 X Y 3 type molecule, the 5 nondegenerate vibrational normal coor-dinates are labelled from 1 to 5 with their vibrational frequencies in descending order; similarly, the six doubly degenerate vibrational coordinates are labelled from 7 to 12. The torsional motion, although it is not considered here as a vibrational coordinate, is labelled as v§ following the earlier conventions in the literature. The torsional motion actually accounts for one degree of freedom in vibration (with symmetry A2). Clearly, in the limit of a infinite height barrier, the torsional motion has the character of the harmonic oscillation. 2.4.2 Symmetry Classification of Free Rotor Torsion-Rotation Wavefunctions The free rotor torsion-rotation wavefunction (in the PAM) is a product of a symmetric top rotational 42 Chapter 2 General Theoretical Considerations wavefunction and a free rotor basis function with the following form: \J, k,m; kf,a) = \J,k,m) x —-jL= exp [i(3fc/ + a)a]. (2.4.7) V 27T The barred symbol here (in the PAM) is not to be confused with the I A M free rotor basis functions. It is used in order to indicate explicitly the free rotor nature of the basis function. To determine the effect of a Gi$ element on the rotational wavefunction \J, k, m), we can replace each element by its equivalent rotation which describes the effect of the Gi& element on the Euler angles of the molecular fixed axis system. As can be seen from Table 2.1, the operations in Gis involve rotations like R2 and Hp. Rj is a rotation of the molecule fixed (x,y,z) axes through 7 radians about the z axis. Rp is a rotation of the molecule fixed (x,y,z) axes through tr radians about an axis in the xy plane making an angle /? with the x axis ((5 is measured in the right handed sense about the z axis). A l l these rotation operations are in the active picture, i.e. the operations are interpreted as a rotation of the whole molecule in the space fixed frame. According to Bunker [41], the general effect of the rotation operations Rp and R2 on the symmetric top function \ J, k, m) (where J is an integer) are: Rl\J,k,m) = eilc"<\J,k,m); (2.4.8) R}\J,k,m) = (-1)Je-2ikP\J,-k,m). (2.4.9) Since these operations do not depend on the magnetic quantum number m, m is suppressed henceforth. The effect of a Gis element on exp [i(3kf + a)a] can be easily worked out using the transformation properties of Xi and X2 from Table 2.1 and a — Xi — X2- T n u s by using Eqs. (2.4.8) and (2.4.9) the effect of a G18 element on \J, k; kf, a) can be determined. It can be seen that the pair of functions \ J, k; kf, a) and I J , —k; —kf, —a) generates a 2 x 2 matrix representation with the following character in the Gis group: 2,2 cos (27rcr/3), 2 cos [2TT/3 (k -a)},2 cos [2TT/3 (k - 2a)}, 2 cos (2TTA;/3) , 0. (2.4.10) Since cosine is an even function, the (k, a) chosen can be either partner. In the case that k ^  3n or a ^ 0, this representation can be directly written in terms of the irreducible representations of Gis- These belong to Ei, Ei, Ez, and E± species. When k = 2>n and a = 0, the representation gives A\ © A2. Let us define \kf, A±) as \kf, A±) = ^ {\J, kfi 0) ± (-l) f c \J, -k; -kf,0)} . (2.4.11) \kf\A+) belongs to A\ for J even and A2 for J odd; \kf\AJ) belongs to A2 for J even and Ai for J odd. This can be seen from the fact that the basis functions | J , k; kf,a) transform under the Gis element (12) (45)* as follows: (12) (45)*\J,k;kf,a) = (-l)J~k \J,-k;-kf,-a). (2.4.12) \kf, A±) is defined as having (12) (45)* character ± (—1)J. 43 Chapter 2 General Theoretical Considerations 2.4.3 Symmetry Classification of the Vibration-Torsion-Rotation States In the PAM, the eigenfunction of a zeroth-order torsion-rotation Hamiltonian given in Eq. (2.3.87) has the following form: \J,k;v6,<r) = Y s A k 7 ' ° \ J ^ k f i a ) > <2A13) k, where A ^ 6 ' * are real numbers. The representation of \J, k; ve,cr) is denoted as TTR-In the case that k ^  3n and/or a ^  0, \J, k; kf, a) has an Ei symmetry. \ J, k; VQ, a) must have the same symmetry as | J, k; kf, a) since it is a linear combination of the latter. These symmetry species, as a function of k and cr, are collected in Table 2.3 as TTR [25]. However, when | J, k; kf,cr) has an A\ or A2 symmetry, the properties of the coefficients A^6'c must be considered in order to work out the symmetries of | J, k; VQ,<T). A S can be seen from Eq. (2.4.11), the symmetry of \kf, A±) depends on the relative phase between |J, k; kf, 0) and |J, —k; —kf, 0). For each | J, k; VQ,a), the set of Ak'^6,cr coefficients can be determined numerically to within a phase factor, which is either +1 or —1. Although, for a given eigenstate, the phase factor does not the alter A\ or A2 symmetry, it is difficult in practice to keep track of the relative phase. Therefore, a proper control of the phase is necessary in order to simplify the symmetry identification. For any given k and cr, the phase convention used here is the following (see Section 3.1.2): Avk6;k'c = (-l)v°Av*'-k'~a- (2.4.14) For k = 0 and cr = 0, by using Eq. (2.4.14), the eigenfunction can be written as: |J,k = 0;v6,a = 0) = AvoB'k'°\J,0;0,0) + £ Avkf'° [\J,0;kf,0) + (-l)Ve |J,0; -fc/,0)] , (2.4.15) kf>0 where A Q 6 ^ ' 0 = 0 for odd VQ. Note the kf range is different. This will generate the representation with the following character in the MS group Gis'. l,l,l,l,l,(-ir+J. (2.4.16) This has a symmetry of A\ for (VQ + J) even and A2 for (^6 + J) odd. For A; = 3n ^  0 and a = 0, the \A±) states for a given torsional VQ state can be written: | J, A; = 3n; v6, a = 0, A±) = -j= {| J, k; v6,0) ± (-l)fe \ J, -k; v6,0)} • • • -for k ^ 0. (2.4.17) Using the properties of A^k'a given by Eq. (2.4.14), this can be written as in terms of the free rotor torsion-rotation wavefunctions as: | J, k = 3n; ve, cr = 0, A ± ) = i = £ Avkffi [\J, k; kf, 0) ± (-l)v°+k\J,-k;-kf,0)} . (2.4.18) kf Thus the A± symmetry for a ve state is defined as having (12) (45)* character ± (—1)V6+J. This differs 44 Chapter 2 General Theoretical Considerations from the free rotor torsion-rotation wavefunctions \kf,A±), (Eq. (2.4.11)) which has a (12) (45)* character ± (—l)J. The correspondence between A+/A- state and the A\/A2 is determined by the factor: V(J,V6,A±) = ±(-iy°+j = | + J ; ; ; . (2.4.19) For J even, A+ belongs to Ai for ve = 0 and A2 for U6 = 1. This is consistent with the fact that the torsional motion has an A2 symmetry. Since \J,k = 0;ve,(T = 0) state always has the same symmetry as a A+ state with k ^  0, it can be considered as a A+ state. The A+1 A- states are introduced for convenience, as can be seen in later sections. It should be noted that the correspondence of A+/A- <-> A\/A2 depends on the choice of the phase factor for Avkjk'(J coefficients. This can be seen from the fact that the factor (—l)Ve, which according to Eq. (2.4.14) defines the phase convention of AvkJk'c coefficients, appears in the character in Eq. (2.4.19). In Table 2.3, only the A+/A^ representations are given. The A+/A- levels can split in the presence of sextic interactions. The vibration-torsion-rotation wavefunction for a vibrational state can be written as f ®VTR = \vk,ek;J,k;v6,a) . (2.4.20) Here we only consider vibrational fundamentals, so that it = 0, ± 1 . Since the torsion-rotation Hamiltonian for a certain vibrational state vk is different than that of the ground vibrational state, \J,k;ve,a) here should be written as | J , k; v6, a) = Y Ak^K°\J,k-kf,a), (2.4.21) kt where the coefficients Ai£Ve,k'cr have an extra dependence on it for a doubly degenerate vibrational state. The representation TVTR of ^ VTR equals Ty <8> TTR, where Ty is the representation of vibrational wave-function |ffc,4>-For a nondegenerate vibrational state (including the ground vibrational state), the representation Ty = Ai so that TVTR = TTR. For a doubly degenerate vibrational state of vt = 1, Ty = E\. The phase convention for the A e ^ V e ^ k ^ coefficients are (see Section 3.1.2): Ae^v6,k,a _ f_i)VeA-ttyoe-k-<f_ (2.4.22) The transformation properties of Qta ± iQtb (E\) under the G\% group elements can be found in Section 2.7.1. It can be proved that the vibrational wavefunctions \vt,± \£t\) of the two dimensionally isotropic oscillator transform like Qta ± iQtb [56]. For example (12) (45)* \vt,± \£t\) = \vt,T |£t|> • (2.4.23) For given it, k and a, the sum of the TVTR for the different possible combinations of ± \it \ and ± | / t | equals Ei <8> TTR. 45 Chapter 2 General Theoretical Considerations Table 2.3 Symmetry properties of vibration-torsion-rotation eigenstates. aAc>d>e vs = 0, vt = 0 vt = i , r v = E1' k a lt FTR TVTR 0 0 A+ +1 0 +1 Ex A+ -1 0 -1 Ei A-0 +1 E4 +1 +1 +1 E2 E4 -1 +1 -1 E3 E4 0 -1 E4 +1 -1 +1 E3 E4 -1 -1 -1 E2 E4 +1 0 E* 0 0 -1 A+ Ei 2 0 +1 E2 Ei +1 +1 E2 0 +1 -1 E4 E2 2 +1 +1 E3 E2 +1 -1 E3 0 -1 -1 E4 E3 2 -1 +1 E2 E3 +2 0 Ei +1 0 -1 Ei Ei +3 0 +1 A+ + A_ Ei +2 +1 E3 +1 +1 -1 E2 E3 +3 +1 +1 E4 E3 +2 -1 E2 +1 -i -1 E3 E2 +3 -1 +1 E4 E2 +3 0 A+ + A_ +2 0 -1 Ei A+ + A-+4 0 +1 Ei A+ + A-+3 +1 E4 +2 +1 -1 E3 E4 +4 +1 +1 E2 E4 +3 -1 E4 +2 -1 -1 E2 E4 +4 -1 +1 E3 E4 a Except for introduction of the symmetry labels A+/A-, this table is identical as the one given by Ref. [25]. b This table applies to any (vg, J). 0 For G > 3, the Table applies modulo 3. d For G = 3n, a = 0 and vt = 1, k and £ t a r e artificial since they are no longer good quantum numbers. e The correspondence between A+/A- and Ai/A2 is discussed in the text. It is convenient for a doubly degenerate state to define a new quantum number G as G = k-£t. (2.4.24) From the transformation properties of \vt, ± \£t\) and |J, A;; ^6, cr), it can be shown that TVTR f ° r specific G, it, k and cr equals TTR for k = G, the same cr [25]. The symmetry properties of ^VTR for vt = 1 are also summarized in Table 2.3. For G — 3n (n is an integer) and a = 0, there are two different representations present, namely A\ 46 Chapter 2 General Theoretical Considerations and Ai- As was the case for k = 3n and cr = 0 for \J, k; ve, cr)(Eq. (2.4.18)), an A± wavefunction can be constructed using a Wang transformation: \vt; J,G; ve,0, A±) = {\vt,£t = +1; fe; «6,0) ± (-1)G \vt,£t = -1; J, -fc; «6,0)} , (2.4.25) where G = 0, ± 3 , ± 6 - and A; = G-t-1. For G = 3, we can have either (A; = 2,£t = -1) or (A; = A,£T = -bl). However, in Eq. (2.4.25), G is defined as the G quantum number for £ t = +1 state. Therefore, the A± state for (A; = 2,£t = -l) should be labelled by G = -3 and for (k = 4,£ t = +1) by G = +3. From the properties of A^j.'*'*' given by Eq. (2.4.22), we have: K J,G;VQ,(J = 0, A±) = -j=Yl 4 ^ {\vt,+l;J,k;kf,0) ± ( - l ) ^ " 1 \vt, -1; J, -k; -kf,0)^ . (2.4.26) The A± representations are defined as having (12) (45)* character ± (—l)Ve+J+1. The correspondence between A+/A- and AyjAi is determined by the factor: r,(vt = l,va; J,G;vE,A±) = ± ( - i ) « e + J + « . = | + j ; ; ; ^  . (2.4.27) Clearly, this correspondence between and Ai/A% also depends on the phase convention defined in Eq. (2.4.22). As will be seen later, the A+/A- can be split in the presence of couplings with A G = 0 (/-type doubling for G = 0) or A G ^ 0 (sextic interactions). By letting £t — 0 and hence G = k, Eq. (2.4.25) simply reduces to a product of a nondegenerate vibrational wavefunction with A± states given by Eq. (2.4.17). Thus Eq. (2.4.25) can be considered as a general definition for A± state for vibrational state vt = 0 or 1. 2.5 Expansion of the Vibration-Torsion-Rotation Hamiltonian The complete vibration-torsion-rotation Hamiltonian of molecule A B 3 X Y 3 has been given by Eq. (2.2.10). In practice, it is more convenient to use the dimensionless vibrational coordinates qi and their conjugates Pi introduced by Eqs. (2.3.13) and (2.3.14). The expanded form of the vibration-torsion-rotation Hamilto-nian is obtained by expanding p,^, the Coriolis coefficients cf; (refer to Eqs. (2.1.50) and (2.1.61)) and the potential energy V (refer to Eq. (2.2.9)) in terms of the dimensionless vibrational coordinates c/, and the torsional angle a [46]. In all the expanded forms of the Hamiltonian, the MFS is considered as in the PAM form. In terms of c/, and pi, the vibration-torsion-rotation Hamiltonian in Eq. (2.2.10) can be written as 1 1 3 N ~ 7 P,-y=x,y,z,a fc=l where the eigenvalues of the Hamiltonian are in joules. Here pp is given by Eq. (2.1.67). In terms of pi and 47 Chapter 2 General Theoretical Considerations qi, it has the form: P/3 = /~ZCki\ —QkPi , 0 = x,y,z,a. k,l V Wfc (2.5.2) The potential energy given by Eq. (2.2.9) can be written in terms of dimensionless vibrational coordi-nates qi as v(q, a)/he = i YUkqk + \z~Z ®umqkqiqm + ••• fc klm +\ E V^n + zZ V3n9k + \zZ V£«W + • • • (1 - cos3na) , (2.5.3) n=l |_ fc kl where uk, $kim>--- a ° d V£n, V£n,... are all in cm - 1 . Here ojk is the vibrational frequency introduced by Eq. (2.3.18) when the higher order terms in the expression for V (q, a) are neglected. p.p^ is the (/?7) component of a four dimensional tensor with 0,7 = x, y, z and a. This can be expanded in a Fourier series in a and a Taylor series in qi as: = j ^ + E^on U -cos3na) + ^ s i n 3 n a | C 0 0 + E * \ & + E - c o s 3 n Q ) + " £ s i n 3 n a n=l + j E j ^ 0 + E A (1 - cos 3na) + ] T ^ sin 3na kl I n=l n=l + • • •, (2.5.4) where subscript k and n are used in the expansion coefficients as p^ and v®kn. The nonvanishing expansion coefficients can be obtained from the symmetry considerations once the expanded operators are obtained. The PAM expression for ffl can be easily determined from Eq. (2.3.59): \ I B° ° 0 1 -1 1° (h2/2hc) piQ = (h2/2hc) 0 0 0 0 jo- 0 0 0 r F . O J-zz V 0 0 J-zz 1° r F , 0 r T , O r F , 0 / • * z z J z z J z z / 0 0 \ 0 730 0 0 0 0 A 0 , F - A Q , F \ 0 0 - A 0 , F F0 J (2.5.5) where i^'s are given by Eq. (2.1.25). For further applications, it is more convenient to introduced the "rotational derivatives" defined by ' B £ = (&/2hc)& = {hi/2hc)(d^/dqk)Q; B* = {h2/2hc)p{]Q=(h2/2he){d2pp1/dqkdql\ 48 (2.5.6) (2.5.7) (2.5.8) (2.5.9) Chapter 2 General Theoretical Considerations Table 2.4 Lower order terms of the vibration-torsion-rotation Hamiltonian. a , b H20 = ^ W f c ^ + P ? ) A k klm H\l} = 7^ E kklmnqkqiqmqn + E Bk'm<ikPnqmPl klmn knml kl ^3 { i 2 } = E ^ ^ klm fc ^2{22> =YRkl(*kqi kl — E Rklmqkqiqn klm #2{3} = E ^ ' kl #i{43} = E T ^ * fe kiqkqi kl H02 = V ^ J , + ^ (1 - cos3a) = ^ | J B ^ . C (1 _ C O S 3a) + B^y s in3a} JpJ7 + - y (1 - cos 6a) 07 H ™ = E {^0,6 C (1 - cos 6a) + B$y s in6a} + y (1 - cos 9a) 07 0 This table is quoted from Ref. [47] after conversion to the current notation and units. 6 For simplicity, the operators are not symmetrized. The symmetrization can be done in a straightforward manner. Here all the B's are in c m - 1 while /zf7 and / X Q 7 are given in (a.m.u) - 1A - 2 units. In introducing B^Q and # f ]jn, we have assumed that angular momentum operators 3p are dimensionless. In fact, h has been absorbed into the rotational derivatives. For molecules with no internal rotation, /x is a three dimensional rotational tensor. The (3N — 6) internal degrees of freedom are all classified as vibrational motions, /J, is expanded in Taylor series with regards to the (3N — 6) normal coordinates. Watson [61] has shown that all terms in the expansion of ppy depend only on the first derivatives of Ipy with respect to qk evaluated in the reference configuration, a ^ 7 = (dlp^i/dqklQ, and on the moments of inertia Ipy. A direct result of this is the sum rules [56] for the Coriolis 49 Chapter 2 General Theoretical Considerations C constants. However, for molecules with internal rotation, the (37V — 6) internal degrees of freedom are separated as (37V — 7) vibrational motions plus the internal rotation. As a result, fi is a four dimensional rotational tensor. Unlike the case for molecules with no internal rotation, the expansion of \x involves a Fourier series in a. It is not clear that the expansion of p^ has relations to af 7 similar to those for molecules with no internal rotation. Similar difficulty exists for the sum rules of the £ constants. Therefore, all the rotational derivatives introduced here are used directly in the expansion terms of the Hamiltonian. The Coriolis coefficients £f j can be expanded in a Fourier series in a as: oo ~di = + £ [C£ , 3 n (1 - cos 3na) + ( g 5 ' 3 " sin 3a] , (3 = x, y, z, a, (2.5.10) n=l where the Fourier expansion coefficients can be derived from the expression for (fkl given by Eqs. (2.1.50) and (2.1.61). When these expansions are substituted into Eq. (2.5.1), the vibration-torsion-rotation Hamiltonian be-comes H = H20 + #30 + #40 + #50 + (vibrational terms) +H21 + #23 + #31 + #41 + (Coriolis terms) +#02 + #12 + #22 + #04 + #14 H (torsion-rotation terms). (2.5.11) Here # m n is the set of terms of degree m in the vibrational operators (qk or pk), and of degree n in the rotational operators ( J x , Jy and J z ) and the torsional operators ( J a , sin 3a, 1 — cos3na). The operators (1 — cos 3na) and sin 3na appearing in Hmn are considered of order 2n. Following the approach by Duan et al. [47], the torsional and rotational operators are considered as of the same order since the zeroth-order torsional and rotational Hamiltonian cannot be separated in the PAM. As it turns out in later discussions, the vibrationally off-diagonal terms involving both torsional and rotational operators can be removed by the same vibrational contact transformation. Therefore, the torsional and rotational operators are not distin-guished in Hmn. In the perturbation procedure, the Hamiltonian in Eq. (2.5.11) is separated into terms of different orders of magnitude. Following the ordering presented by Aliev and Watson [62], # can be expressed as follows # = ]T An#<n> , n = 0,1,2,3, • • • . (2.5.12) n Here A is a bookkeeping parameter. Then one has up to fourth order of approximation [47]: # { 0 } = # 2 0 , (2.5.13) HW - # 1 { 2 1 } + # 2 { 1 1 } + # 3 { 0 1 } + # 0 2 , (2.5.14) #<2> = # 2 { ? + # 3 { i 2 } - l - H i o } + < 2 } > (2-5.15) 50 Chapter 2 General Theoretical Considerations Table 2.5 Torsion-rotation operators. a ' b / \ 1/2 * B = - ( ^ ) E (»l/2B%CkmJ? + ^B%Qm3p) Rk = E Bfco J/3 J 7 + y (1 - cos 3a) = 5 j E ^ 0 J 0 J 7 + -cos 3a) I 1 I a y f c l m I Rkim = jg < E B ^ o J 0 J 7 + (1 - cos 3a) \ T k = E { 5 M C ^ ~ c o s 3 a ) + Bk!i' s i n 3 a } J 0 J 7 + - ^ ( 1 - cos6a) 07 Tfe< = ^ E {Bki,3 ( ! - c o s 3 a ) + s i n 3 a } J 0 J 7 + ^ (1 - cos6a) E J ? + ^ 7 C £ f 3 ) (1 - cos 3a) + + < C f f 3 ) sin 3a} 07 / \ i / 2 R n i _ / \ ST~ R 0 7 A 0 , - 0 7 \UkUJmJ ^ 07 07 -1/2 ) 07 ° This table is quoted from Ref. [47] after conversion to the current notation and units. 6 For simplicity, the operators are not symmetrized. The symmetrization can be done in a straightforward manner. ffW = ffg>+JF/g>+flg} + flif}+^3}. (2-5-16) l f W = H^ + H^ + H^ + H ^ + H ^ + H^. (2-5.17) Here the superscript n in denotes the order of the terms in the expanded Hamiltonian. The detailed expressions with their orders in the expansion are list in Table 2.4. The various torsion-rotation operators R and T introduced in Table 2.4 are given in Table 2.5. The operators in Tables 2.4 and 2.5 are not written explicitly in the Hermitian form. The operators can be easily symmetrized into the Hermitian form. For example, a product of any two torsion-rotation operators RnRn' can be written as RnRn' = g [Rn, Rn'}+ , (2.5.18) where [A, B]+ = AB + BA is the anticommutator. A product of vibrational operators can be symmetrized in a similar way. All the operators are quoted from Duan et al. [47], with the modifications for the A B 3 X Y 3 type molecules and using the rotational derivatives iff 7 instead of / i g r A simple physical meaning can be assigned to the individual Hmn. if20 is the harmonic oscillator Hamil-tonian expressed in dimensionless normal coordinates qk and pk. H30 and i/40 describe the anharmonicity of molecular vibrations. iio2 is the zeroth-order torsion-rotation Hamiltonian whose form in the PAM is 51 Chapter 2 General Theoretical Considerations given by Eq. (2.3.59). Hyi and H.22 are centrifugal distortion operators. H2i describes Coriolis interaction between torsion-rotation and vibration. 2.6 General Symmetry Requirements for the Vibration-Torsion-Rotation Hamiltonian There are two general symmetry properties of the vibration-torsion-rotation Hamiltonian valid for any molecular model (with or without internal rotation). The Hamiltonian is invariant to the operations of Hermitian conjugation +, and to the operator of time reversal T [41]: H = H+ = THT~l . (2.6.1) Vibrational, torsional and rotational operators transform under Hermitian conjugation and time reversal as follows: J + = J , T J T " 1 = - J ; (2.6.2) p+ = p, TpT~l = -p; (2.6.3) q+ = q, TqT-l=q. (2.6.4) Here the torsional operator J a is taken as a component of J , so that J should be taken as a four dimensional operator. Since time reversal consists of a change of sign of the time followed by complex conjugation, for any coefficient B (independent of time) in the expansion of the Hamiltonian we have: B+ = TBT-i = B* (2.6.5) By taking into account the reversal rule for the Hermitian conjugate of a product of operators, it can be easily shown that all coefficients in the expansion of the Hamiltonian we have given in Table 2.4 must be real. 2.7 Expanded Hamiltonian for A B 3 X Y 3 Molecule In Table 2.4, the general forms of the lower order terms of the Hamiltonian are given. In this section, we utilize the symmetry considerations to give the exact forms of the Hamiltonian for A B 3 X Y 3 type molecules. 2.7.1 Transformation Properties of Vibrational, Torsional, and Rotational Opera-tors in the Molecular Symmetry Group Gig A molecular symmetry (MS) group consists of the feasible operations of the complete permutation inversion group of the molecule, which leaves the complete molecular Hamiltonian unchanged [41]. The 52 Chapter 2 General Theoretical Considerations Table 2.6 The transformation properties of the vibrational, torsional, and rotational operators of A B 3 X Y 3 in the G\& group. E R° (123) R° ' (456) (123)(456) (123) (465) (12) (45)* 7~>7T KTT/2 r a a - 2TT/3 a - 2?r/3 a - 4TT/3 a —a Qs Qs qs qs qs Ita qt-1ta Qtb 1t+ qt-qtac + qtbs qtbc-qtas w2qt+ uqt-qt„c-qtas uJ2qt+ uqt-Qtac-qtbs qtbc + qtas v<lt+ u2qt-1ta 1 -9tb J <h+ J Ei Ei Sta Stb Stac- Stbs Stbc + Stas Sta St„ Stac — Stbs Stbc + Stas Stac - Stbs Stbc + Stas Sta 1 J E2 St> 5 * St- Sfac + St'bs St'bc-St>as St>ac + St,s St>bc-St>as St>ac-St>bs St,c+St,as ^ \ Ei J x Jy Jx Jy •Ja;^ -f- JyS JyC Jx$ JxC J y ^ JyC Jx& J X C JyS J y C -j- JXS - J x 1 J y J Ei J + J _ J + J _ w 2 J + wJ_ LJ2J+ U)J+ CJ2J_ Ei Jz Jz Jz J , J , - J , A2 'Ifv J a Ja J a J a J a A2 cos 3a cos 3a cos 3a cos 3a cos 3a cos 3a Ai sin 3a sin 3a sin 3a sin 3a sin 3a — sin 3a A2 sin a cosa c sin a — s cos a s sin a + c cos a c sin a — s cos a s sin a + c cos a c sin a + s cos a -ssina + ccosa sin a cosa — sin a 1 cosa J E4 T: symmetry species of the corresponding operator. qs : nondegenerate vibrational modes; qt : doubly-degenerate vibrational modes, c = cos (2-7r/3); s = sin(27r/3); ui = exp (27ri/3) = c + is; cu2 = c — is . Qt± = Qta ± iqtb] J ± = J ± " ) = Jx±iJy. St- doubly degenerate symmetry coordinates localized in the top in the TFS. St'b'- projections of St on the FFS. St>a = Sta cosa - Stb sin a; St'b = Sta sin a + Stb cosa. 53 Chapter 2 General Theoretical Considerations MS group of A B 3 X Y 3 is Gi$. In order to derive various terms in the Hamiltonian for molecule A B 3 X Y 3 , the transformation properties of vibrational, torsional and rotational operators must be considered. For example, there are three different kinds of terms off-diagonal either in vibrational or in torsional quantum numbers, the leading term of each are H12, H21 and H 3 0 . Each of these terms is the product of a coupling coefficient, such as B^, and a coupling operator. Since each of these terms must be totally symmetric in the MS group Gis, i.e. it must have A\ symmetry, the coupling coefficients must vanish if the coupling operator is not totally symmetric. Following the approach by Bunker [41] for a C$v molecule, the transformation properties of the normal coordinates, angular momentum components, as well as the torsional operators of A B 3 X Y 3 in the G\s group can be worked out. The results are summarized in Table 2.6. The convention for a doubly degenerate vibrational mode used here is the same as that used by Bunker [41], i.e. the degenerate coordinates (qta,qtb) transform like (Tx,Ty). Here Tx, Ty, and Tz represent the translational coordinates. It can be seen from Table 2.6, in the PAM, the transformation properties of the vibrational (T (qs) = A i , T(qta,qtb) = Ei) and pure rotational operators (T(JZ) = A2, T(Jx,Jy) = E{) in the Gi$ group are identical as those in the C^v group, provided one makes the correspondence E\ (Cis) to E (C^v)- As a result, terms in the Hamiltonian for a A B 3 X Y 3 molecule involving only vibrational and rotational operators have the same forms as those for a C^v molecule. However, the definitions of the coupling coefficients are different. These terms are considered here together with terms involving torsional operators. In the lower order terms listed in Table 2.4, the operators diagonal in vibrational quantum numbers are H2Q, HQ2, HQ4, HQQ etc. Unlike molecules with no internal rotation, for which the highest order in the rotation is two in the expanded Hamiltonian, the A B 3 X Y 3 has terms like HQ4 and HQQ. Of course, strictly speaking, all these terms should be classified as torsional terms which all arise from the expansion of the potential and ^ 7 with regards to the torsional angle a. The exact forms of these terms for AB3XY3 molecule will be given in Section 2.10.3 when the torsion-rotation Hamiltonian is discussed. The other lower order vibrationally off-diagonal terms will be discussed in the following sections. 2.7.2 Centrifugal Distortion The general form of H12 is given by Table 2.4 (where it is written as H$) as: This operator has matrix elements off-diagonal in both the vibrational quantum numbers and torsional quantum number VQ. Since H12 only contains one power of qk, it has a coupling selection rule Avk = ±1. Terms in H12 involving either 3a or (1 — cos 3a) operators will couple the torsional states in the first vibrational states with those in the ground vibrational state. The sum over the vibrational states can be separated into a sum over 5 nondegenerate Ai vibrational (2.7.1) 54 Chapter 2 General Theoretical Considerations states and a sum of 6 doubly degenerate E\ vibrational states. Since cos 3a has a symmetry of A\, the only nonvanishing Vk are those when qk has a symmetry of A\, i.e. the nondegenerate vibrational states. The coefficients B^Q can be nonvanishing only if r(qk)®T(Jp)®T(J1)DA1. (2.7.2) For a nondegenerate A\ vibrational state with normal coordinate qs, from Table 2.6 we see that 32zqs, (J2, + J2) qs, J 2Ja9s, and J 2c/ S are totally symmetric; no other A\ type terms can be constituted. Conse-quently, the only nonvanishing coefficients are: BT = Bf5 = B%; (2.7.3) BT = B&, (2.7.4) B™ = B% = B%; (2.7.5) >s0 — ^ sO ' JCHCX BT = B%. (2.7.6) For a doubly degenerate vibrational state, from a detailed consideration of the transformation properties of qta and qtb (or of qt+ and qt-) in Table 2.6, the following operators have been shown to be totally symmetric (see Appendix A.3): Qta ( J * J Z + J * J x ) + qtb ( J j / J z + J 2 J j / ) = ^ [(ft+J- + 9 t - J + ) , J z ] + ; (2.7.7) qta ( J a J x + J X J Q ) + qt„ (3aJy + J ^ J a ) = ^ [ (gt+J-+gt -J+), J a ] + ; (2.7.8) 2 ? t o J x - f t . J j - f t t ( J x J » + V * ) = \(qt+Jl+qt-32-). (2.7.9) No other Ai type terms can be constituted. Although J Q commutes with J + and J _ , \ [(qt+J- + qt- J + ) , J a is written in an anticommutator form for later discussion purposes. It follows the above equations that the following coefficients can be nonzero: — T3XZ TJZX - Bta0 ~ C o = C o ; (2.7.10) r>xa = Bta0 - Bta0 ~ C o = C o ; (2.7.11) J^XX = Bta0 - -Ryy r>xy _ T>yx - ~ D t b Q - n t b 0 -(2.7.12) Therefore, i f 12 has the following form: where # 1 2 = E ^ i 2 + E ^ i 2 > < 2 - 7 - 1 3 ) s=A1 t=Ei H{2 = | / 3 f x J 2 + ( B f - Bxsx) J 2 . + Bfa3l + 2Bzsa3z3a + ^ (1 - cos3a) j qs; (2.7.14) # 1 2 = \BT[(3+qt-+3-qt+),3z}+ + \Br(3lqt++32-qt-) 55 Chapter 2 General Theoretical Considerations +-Bfa [(3+qt-+3-qt+),Ja}+. Similarly, the #14 term can be shown to have the following form: # 1 4 = Y H U + Y " ^ 1 4 ' s=A! t=E1 (2.7.15) (2.7.16) where H14 = 2 { # - 3 J 2 + (Bt% - B»*3) 32z} h i - cos 3a) qs + B% J Q > 2 ( l - C O s 3 a ) +2BZS^3Z J a , 2 C1 ~~ cos3a) + 9s + - y (1 - cos6a)gs; « 1 4 = BZlc[(3+qt-+3-qt+),3z}+-(l-cos3a) + B ^ ( J + f t _ + J _ f t + ) 1 J a , 2 ( 1 ~ cos 3a) + 2i : ( J _ g t + - J + f t - ) , J 2 sin 3a. J + The parameters in above equations have the following properties: (2.7.17) (2.7.18) TDXX Bs,3 - Bvy- (2.7.19) Bs,3 r>az. - ^ , 3 . (2.7.20) JDXZ - ° t , 3 c _ gXZ,C t a ,3 r>zx,c _ r>yz,c _ R z t / , c . nta,Z — ntb,3 — utb,3 ) (2.7.21) Bt,3c _ JjX(X,C — ta ,3 gctx,c _ g y a , c _ g a y , c . ta,3 tb,3 tb,3 > (2.7.22) JDXZ Bt,3s _ __gz3/.s — _ gZX,S i<i)3 ^b)3 i&)3 (2.7.23) All these constants arise from the expansion of the original Hamiltonian. The coupling between the torsional mode and the first excited vibrational states are dominated by terms in #12 and #14 which are associated with torsional operators. The (v6,va) coupling is dominated by torsional operators (1 — cos 3a) /2, J Q , and J 2 . It has a selection rule on the vibrational and rotational quantum numbers Avs = 1, AJ = 0, and AA; = 0. Since all the operators involved are diagonal in cr, it has a selection rule Acr = 0. The selection rule on the principal torsional quantum number ve depends on the barrier height. In principle, Ave can be arbitrary. This is particularly important when the large amplitude internal rotation is involved in some of the torsion states coupled by this term. In a high barrier limit, the torsional operators (1 — cos 3a) /2 and J 2 have large matrix elements with even Av$. The (VQ, V%) coupling is dominated by odd torsional operator J Q in Eq. (2.7.15). Thus the selection rules are Avt = 1, A J = 0, Aft; = A£t = ± 1 and Acr = 0, with Ave arbitrary. In the high barrier limit, the selection rule on vQ can be approximated by Ave = 2n + 1, where n is an integer. Each vibrational state has torsional states which form a so-called torsional stack. #12 and #14 couple 56 Chapter 2 General Theoretical Considerations the ground state torsional stacks with those of the first excited vibrational states. In a high barrier approxi-mation, the torsional motion can be treated as a harmonic oscillation. Furthermore, the matrix elements of H12 and Hf4 are similar to that of a Fermi interaction between two nondegenerate vibrational states for a molecule with no internal rotation. For this reason, H[2 and Hf4 are called "Fermi-like" interactions. Sim-ilarly, H{2 and H{A are called "Coriolis-like" interactions. However, the detail forms of these operators shown in this section are not exactly the same as those for molecules with no internal rotation. For exam-ple, in the high barrier approximation, a normal coordinate qe can be defined for the torsional motion by the fact that (1 — cos 3a) /2 becomes proportional to q\. Thus the last term on right hand side of Eq. (2.7.14) can be written as being proportional to qsq\. This has the same form as a normal Fermi interaction term for two nondegenerate vibrational states for a C3v molecule. However, the third term in Eq. (2.7.14) with the operator qs J 2 has no counterpart in the Hamiltonian of a normal C3v molecule. 2.7.3 Coriolis Coupling The H21 term in Table 2.4 has the following general form kl ^ p-y The only totally symmetric Coriolis coupling operators are: > QkPl- (2.7.24) -qsptb Jx + QsPtJy = (3-Pt+ ~ 3+Pt-) qs; (2.7.25) {qt'aPtb-qt'bPta)3z = ji(qv-Pt+-qv+Pt-)3z; (2.7.26) {Qt'aPtb ~ Qt'bPta) J« = 2I (qt'~Pt+ ~ Qt'+Pt-) (2.7.27) (QtaPt>a ~ QtbPt'„) J y + (qtaPt'b + qtbPt'a) Jx = ^ (qt+Pt'+3+ - qt-Pv-J-) • (2.7.28) The nonvanishing Coriolis coupling coefficients are: Ct = = (2-7.29) (tv = (t't — (tat'b = ~(tbt'a'i (2.7.30) Cw = Ct't = Ctat'b = ~ Ctbt'a '•> (2.7.31) Cw = Ctat'a = ~(tbt'b = Ctat'b — Ctbva- (2.7.32) The index tf can equal to t in Eqs. (2.7.30) and (2.7.31). However, t' ^ tin Eq. (2.7.32) because of the antisymmetry of the £ constants (see Eq. (2.1.68)). From the nonvanishing £ constants, the Coriolis coupling can be classified into three different types. The first type couples two components of the same degenerate mode, vta and vtb, through rotation about the symmetry z axis. The second type couples the nondegenerate mode vs with the degenerate mode ut through 57 Chapter 2 General Theoretical Considerations rotation about the x or y axis. The third type couples two different doubly degenerate vibrational modes vt and Uf through rotations about the x, y axis and/or z axis. The second and third types of interactions can be removed by the contact transformation provided there is no accidental resonance between the coupled states. The exact forms of the three different types of interaction can be easily obtained from the nonvanishing £ constants. From Eq. (2.7.29), the xy Coriolis interaction term between a nondegenerate and a doubly degenerate vibrational states has the following form: st 1/2 (J-p«+ - 3+Pt-)qa UJ e 1/2 ( J - f t + - J+Qt-)Ps (2.7.33) This operator will couple two states with \Avs\ = 1, |At>t| = 1 and AA; = A£t — ± 1 . This coupling has no torsional dependence. Clearly, the selection rules on the torsional quantum numbers are Aft;/ = 0, ACT = 0. From Eq. (2.7.32), the xy Coriolis interaction term which couples two different doubly degenerate states has the form: HlfV = -2B0Cyttl { ( ^ _u>t uv\~2 1 2» : (qt+pt>+J+ - qt-Pt'-J-) U)f 2i (qt'+Pt+3+ - qv-Pt-J-) (2.7.34) where t / t' and Bxx = B%y = Bo. This operator will couple states with, for example, (vt — l,vt> = 0) and (0,1), provided Aft; = A£ = ±1 and A J = ±1 . This operator also has no torsional dependence. The z Coriolis interaction term which couples two different doubly degenerate states has the form: H, z;tt'. _ 21 / , \ l y / 2 1/2 (qt'-Pt+ - qt'+Pt-) - Yi(qt-Pt'+-qt+Pt.-)+{-) ^ (2.7.35) for t ^ t'. H^x will couple two degenerate states, for example, (vt = l,£t = l\Vf = 0,£f = 0) and (vt = 0, £ t = 0; Vf = 1, £f = 1), provided AA; = 0 and A J = 0. This operator has a torsional dependence in J a and thus can couple torsional stacks in the two doubly degenerate vibrational states. In the case that t = t', H^1 has the following form = Hzi-r = - 2 { A Q , F (Cu ~ Qt) J , + ( -A ) ,FC« + Fo(?t) J * } — (qt-pt+- qt+Pt-) . (2.7.36) For Vt = 1, this term will lift the degeneracy for £t — +1 and £% = —1 levels. 58 Chapter 2 General Theoretical Considerations 2.7.4 Anharmonic Coupling The HSQ operator has the general form (refer to Table 2.4): # 3 0 = ^J~]himqkqiqm- (2.7.37) klm In ds, the only totally symmetric operators involved in H3Q are qsqS'qS"; QstftaQt* + qsqt'bqtb Qt-qt'aqta - Qt-qvaqtb - qt-Qt'bqta - qv-qt'bqtb As a result, the nonvanishing k^im are (2.7.38) = Qs (Ot'+qt- + qv-qt+) /2; (2.7.39) = (Qt"+qt'+qt+ + qr-qv-Qt-) /2. (2.7.40) ksss\ (2.7.41) ksVata = ksrbtb; (2.7.42) kW>ata = -h'b>tatb = -h'b't'bta = -h<>t>btb- (2.7.43) The exact form of H30 between different vibrational states can be easily worked out from above equations. It should be noted that the coupling due to # 3 0 has no torsion-rotation dependence. This has exactly the same forms as that for a C3v molecule. 2.8 Theory of Contact Transformation The idea of the contact transformation approach is to transform the original Hamiltonian operator H into an equivalent operator H, which in the order of interest is diagonal in the basis of the zeroth-order Hamiltonian. Normally, the zeroth-order Hamiltonian is taken as the harmonic oscillator Hamiltonian. The transformation must be unitary so that the eigenvalues and the normalization of the eigenfunctions are preserved. This is ensured by writing the unitary transformation operator as U = exp {iS), where S is a Hermitian operator. The transformation for the Hamiltonian and the eigenfunction are [56]: H - UHU~l = exp(iS)H exp(-zS); (2.8.1) $ = Uij) = exp (iS) ip. (2.8.2) Here ip and ip are the original and the zeroth-order wavefunctions, respectively. The transformed Hamiltonian H must have the same symmetry properties as H; that is, it must be invariant to Hermitian conjugation, time reversal, and to the operations of the appropriate MS group. From these requirements as well as Eq. (2.8.1), it is clear that S must be invariant to the operations in the MS group, but must change sign under time reversal. Otherwise, it can be chosen arbitrarily within broad order 59 Chapter 2 General Theoretical Considerations of magnitude constraints. By properly choosing S, H can be obtained in the desired form. In the perturbation process, the Hamil-tonian i f is expanded in a convergent series of terms of different magnitudes: H = H0 + \H1 + X2H2 + (2.8.3) The parameter A is introduced for mathematical convenience in defining the various orders of perturbation; it can be considered to have a value between 0 and 1. The terms belonging to a given order of magnitude j are denoted by if,-. In the representation of the wavefunctions of HQ, higher order terms Hi, H2 etc. in the expansion (2.8.3) have diagonal as well as off-diagonal matrix elements. We can carry out a contact transformation to obtain the first reduced form if W of the Hamiltonian: Ui can be expanded as Then we obtain for i f [56]: H(1) = UiHU^1 = eiXSlHe~iXSl Ui = exp (iXSi) = 1 + iXSi - xA 2 5i + HW = H + Y (* nA n/«!) [Si, [Si,-- •[ SUH]] • ••]. (2.8.4) (2.8.5) (2.8.6) n=l Here [5, H} = SH - HS is the normal commutator. By substituting Eq. (2.8.3) into Eq. (2.8.6) and equating terms of the same order of A, one has: 2tf> = HQ ; (2.8.7) if?) = Hi+i[Si,HQ] ; (2.8.8) H? = H2 + i[Si,Hi}-±[Si,[Si,H0}} ; (2.8.9) = H3+i[Si,H2}-±[Si,[Si,Hi}]-1. 6Z [Si [Si,[Si,H0]}} ; (2.8.10) = Hi + i[Si,H3}-±[Si,[Si,H2)}-1. r [Si [Si,[Si,Hi]}} +j-4[Si,[Si,[Si,[Si,H0}}]} . (2.8.11) In carrying out the transformation in Eq. (2.8.4), Si is chosen so that, in Eq. (2.8.6), not only HQ1^ but also H[^ is diagonal in the representation of the wavefunctions of Ho. This procedure can be repeated by using a second contact transformation on i f W such that H^ + Aff{2) + A 2 i f f ) is now diagonal. Similarly, higher order contact transformations can be applied the way. The successive contact transformations can be achieved if we replace U by the product same U = UnUn-i • • • U2Ui 60 (2.8.12) Chapter 2 General Theoretical Considerations where Un = eiX"s" . (2.8.13) Transformed Hamiltonians will be denoted by the superscript n: H ( n ) = e i \ " S n H ( n - l ) e - i \ » S n (2.8.14) The evaluation of a transformation function S at any stage of the perturbation calculation is reduced to the solution of a commutator equation of the form H = H + i[S,H0] (2.8.15) where HQ is a zeroth-order Hamiltonian, H is an ordinary perturbation term, and H is a transformed form of H. As we want S to be found such that H is diagonal in the representation which diagonalizes H 0 , H commutes with HQ and we have: H,HQ = [H, HQ] + i [[S, H0] ,H0}=0. (2.8.16) As an alternative method, S can also be found by solving the above equation. In the case that HQ is the harmonic oscillator Hamiltonian H2o, the function S can be represented as a linear combination of the commutators [56]: [H0,H], [H0, [H0,H]}, [Ho, [H0, [H0,H]}}, etc. (2.8.17) such that S = r i „ [HQ, [Ho, [••-,[ Ho,H}} • ••] (2.8.18) n  > * ' n where Xn are parameters to be determined by the use of Eqs. (2.8.15) or (2.8.16). The commutators in Eq. (2.8.18) must be linearly independent. It can be easily seen that in the basis of HQ, the diagonal matrix elements of [S, Ho] are zero and hence the diagonal matrix elements of H are equal to those of H. Thus from Eq. (2.8.15) we can see that S should be chosen so that (a \H\ b) = -i (a[[S,H0][b) = -i (Eb - Ea) [a\S\ b) . (2.8.19) In the case that Eb ~ Ea, the method of contact transformations as well as the usual methods of perturba-tion method fail. Thus the Hamiltonian cannot be transformed through contact transformation to be block diagonalize and should be set up in the basis which includes both states \a) and |6). For molecules with no internal rotation, the steps for the calculation of the vibration-rotation contact transformations are well documented in the literature [56,62]. The expressions for various 5 m n and Hmn operators can be found in Tables IV and V of Aliev and Watson [62], for example. 61 Chapter 2 General Theoretical Considerations Table 2.7 Contact transformation functions a ' b J2 B0JQiJ/3 (wfcwj) 1 / 2 (ul - uf) X[(u\+ Jf) qkqi + 2uku>lPkPl Y (wfc"i) 1 / 2 RiR?qk - i £u~k 2 [Jfc, #02] <7fc _ 6 £ fcfcl»*nfcim [2wfcW(Wmp f cp ipm + 3wj (u>\ - u 2 + u>2m) qkqiqm W Pi k klm a Operators in this table are quoted from Ref. [47] with the conversion the current notation and units. b Operator in this table are not symmetrized. The symmetrization can be done in a straightforward manner. 2.9 Vibrational Contact Transformation for Molecule A B 3 X Y 3 In the case that the molecule A B 3 X Y 3 can be assumed as a semirigid molecule, i.e. the molecule is strongly bound and has no low potential energy barriers for large amplitude internal vibrations (except the internal rotation), the higher order terms in vibration-torsion-rotation Hamiltonian can be treated by perturbation, provided there is no resonance between the vibrational states coupled by those terms. The most convenient way is to perform the higher order perturbation treatment is based on the method of contact transformations. The lower order interaction terms can be treated by the conventional Rayleigh-Schrddinger perturbation method. The general formalism of vibration-torsion-rotation contact transformations for molecules containing a threefold symmetric internal rotor has been discussed in detail by Duan et al. [47] Although the theory was presented for asymmetric tops, it can be used for the current AB3XY3 type molecule after allowance is made for the molecular symmetry. The Hermitian operator S which defines the unitary transformation operator U is best represented by the two dimensional array [47]: where each line is suitably truncated for any particular problem. As before, the subscripts m and n of various terms Smn denote, respectively, their combined total power in the vibrational operators and their total power in the torsion-rotation operators. Using a scheme similar to that used for the operator S Sl2, S13, • • • S30,S3\, • • • (2.9.1) 540, Sn , • • • 62 Chapter 2 General Theoretical Considerations can be separated into terms of different orders of magnitude. U is written as the product of seven contact transformation functions in order to obtain the first and second order transformed Hamiltonian: U = exp (*A2SJo}) exp (iA2sg}) exp ( iA 2 S 2 { 2 } ) exp ( i A 2 ^ ) x exp ( i A S § } ) exp (iA5J}}) exp (i\S[^ . (2.9.2) The lower order transformation functions Smn are listed in Table 2.7, along with S^2}. The contact transformations are applied sequentially. In the first sequential contact transformation, trans-formation exp ^ i A S i ^ is applied first, followed by exp ^ A S l i ^ - The transformation exp ^AS|o^ is applied at last. The transformation functions S^, S r ^ , and are chosen so that the following equations hold: S$,H2o £ 3 0 * ' # 2 0 = iH 12 J %n2\ 1 = in in 30 (2.9.3) (2.9.4) (2.9.5) Operators H$ and H$ have only nonvanishing matrix elements off-diagonal in the vibrational quantum numbers. On the other hand, the operator H^ (refer to Section 2.7.3) has nonvanishing matrix elements both diagonal and off-diagonal in vibrational quantum numbers. Here the superscript od denotes the vi-brationally off-diagonal part of the operator. As a result, after the first contact transformation, the terms in H^ (due to Hy^ , H^, and H^) which have matrix elements off-diagonal in vibrational quantum numbers are removed. The first order Hamiltonian becomes: H>W = H02 + HW. (2.9.6) Here H$A, which represents part of the operator H2} that is diagonal in vibrational quantum numbers, only has nonvanishing matrix elements for degenerate vibrational states. After the first sequential contact transformation, the second-order transformed Hamiltonian H'^2\ which consists of terms from H^ as well as contributions from the first sequential contact transformation, has matrix elements both diagonal and off-diagonal in vibrational quantum numbers. We can denote these vibrationally off-diagonal terms as H'{2}°d. In the second sequential contact transformation, the transfor-mation exp (iXS^2f) is applied to the once transformed Hamiltonian. After this transformation, H'{2}°d i s removed. The second-order transformed Hamiltonian becomes diagonal in vibrational quantum numbers which has the following form [47]: H"{2} _ H»m , r r " { 2 } , ff»{2} where o-"{2} _ ^ 4 0 — H{2}d + -i °30 ' - " 3 0 i d (2.9.7) (2.9.8) 63 Chapter 2 General Theoretical Considerations vi d 1. r „ m „ m " i d H'2P = Hg» + i Hn{2} _ r r { 2 } , l . "04 — -"04 + 2 D12 '-"30 J + Q [ 2 1 ' 2 1 o(l} r r W ° 1 2 '-"12 (2.9.9) (2.9.10) where the commutators can be either vibrational or torsion-rotation commutators. Only those contributions whose vibration-torsion-rotation indexes ran match those of the left-hand side are retained. The vibra-tionally diagonal part of an operator is denoted with superscript d. As a result of the vibrational contact transformations, the Hamiltonian diagonal in vibrational quantum numbers can be written as follows: H" = H'l + H'lr + #™ + H"Cor (2.9.11) where H'l = Hh + Hb + Hh, (2.9.12) K = Hn + Hh + Hfa, (2.9.13) H«cent = H^ + H'^ + Hh, (2.9.14) H"Cor = H$ + H$ + H!g. (2.9.15) H'l collects all the pure vibrational terms. H"r is the torsion-rotation Hamiltonian which collects all the terms that are not associated with vibrational operators. H'{rcent and H"CoT are terms which describe cen-trifugal distortion and the Coriolis interaction, respectively. Each term in Eq. (2.9.11) is a linear combination of commutators [Sm>n',Hmn]. In evaluating the trans-formed forms of various Hamiltonian terms, the general forms of [Sm>n<, Hmn] should be considered. Both Hmn and Sm'n' can be written as sums of product operators Hmn = £ (VmRnJi \ Sm'n> = £ {Vm'Rn')j , (2.9.16) » 3 where V and R represent the vibrational and torsion-rotation operators, respectively. Then the evaluation of the commutator [Sm'n', Hmn] reduces to evaluations of the following commutators [Vm> Rn', VmRn] = [Vm., Vm] Rn/Rn + Vm> Vm [Rn', Rn] • (2.9.17) Here the first term on the right hand side of the equation is referred as the vibrational part of the commutator; and the second term is referred as the torsion-rotation part. For a vibrational commutator involving qk and Pk (refer to Eq. (2.3.15)), it reduces the vibrational order by 2. For a normal rotational commutator involving Jp (ft = x, y, z) (refer to Eq. (2.3.37)), it reduces the rotational order by 1. Torsional operators like 3a and cos 3a commute with both vibrational and rotational operators. The torsional operators have the following commutation relations: [JQ,cos3na] = 3msin3na; (2.9.18) 64 Chapter 2 General Theoretical Considerations [Ja,sin3no;] = —3mcos3na. (2.9.19) Therefore, since both cos 3na and sin 3na are denoted as an order of In in torsion-rotation, a torsional commutator reduces the torsional order by 1. Note is denoted as order of n. This is the same as the ordering for a normal rotational commutator. The vibrational part of the commutator [Smini, Hmn] has the order of (m + m! — 2, n + n') and the torsion-rotation part has the order of (m + m', n + n' — 1). For in Eq. (2.9.9) has an order of 22. example, the vibrational part of the commutator sffl, Using the conventions for the ordering of torsional operators defined in this thesis, the results from Aliev and Watson[62] can be used directly. Of course, the modifications must be included so that Hmn and Smn have the current forms appropriate for molecules with internal rotation. Except for JJ"Cor, detailed expressions for terms in if" can be found from Duan et al. [47] General expressions for terms in H"Cor can be found from the corresponding expressions given by Aliev and Watson [62]. 2.10 Effective Hamiltonians for A B 3 X Y 3 The general formalism of vibrational contact transformations has been discussed in the previous section. In practical calculations, a frequently asked question is what are the terms in the original Hamiltonian that contribute to a given term in a transformed Hamiltonian. This can only be answered by evaluating the transformed Hamiltonian, and by expressing the results in a form which is determinable in practice (i.e. without linear relations among the parameters). In this section, by using the symmetry properties for molecule A B 3 X Y 3 , the exact forms of different order terms in Eqs. (2.9.12), (2.9.13), (2.9.14), and (2.9.15) will be discussed. The term H"r in Eq. (2.9.13) can be further subjected to torsion-rotation transformations. This reduction will also be discussed in this section. 2.10.1 Vibrational Hamiltonian H": Effective Band Origins The matrix element of if" diagonal in vibrational quantum numbers (Eq. (2.9.12)) has the following form [56]: G (v,e) = + + (Vt + \ ) + E *«' [Vs + \) (Vs' + \) + J2dtM', (2.10.1) t>t" where the subscripts s and t represent, respectively, the nondegenerate and doubly degenerate vibrational 65 Chapter 2 General Theoretical Considerations modes. The expressions for Xkk' a n d 9tt> are given in Table 18.1 of reference [56]. Here the fourth order term has been neglected. For the ground vibrational state Go = l + \ £ w * + jJ2xSf (2.10.2) s t s,t Here we are only interested in the case where only one vibrational quantum has been excited. The effective band origins are defined as follows ufa = G ( « , = l ) - G 0 ; u% = G(vt = l)-G0. (2.10.3) (2.10.4) Clearly, the effective band origins are equal to the vibrational frequencies when H'$^ is neglected; 2.10.2 Vibrational Dependence of Torsion-Rotation Hamiltonian: E'l 22 The quartic vibration-torsion-rotation interaction is described by the term H'2^2\ This has the following form [47,62]: o-"{2} _ rr{2} . [9{1> „{1}1 1 . [^ {1} „ { 1 } ] 1 . [ q{l} „{l}d , 9 „ n22 — n22 + l a\2 '-"30 +2 2 1 ' 2 1 o 2 1 ' 2 1 + 2 ""0 2  + 2* (2.10.5) The last term on the right hand side is purely off-diagonal in the vibrational quantum numbers. It will therefore not be considered in the following explicit expression for if2'2. The operators in Eq. (2.10.5) are given in Tables 2.4, 2.5 and 2.7. has both diagonal and off-diagonal matrix elements in vibrational quantum numbers. Here we are only interested in the diagonal part. After evaluating the commutators (see Appendix A.4), the diagonal part of this operator has the following general form: H22 kl 1 — cos 3a kl QkQl + £ J / 3 J 7 £ BuoQkqi, (2.10.6) kl where kl nkl0 ~ (2.10.7) 6e . 3ul + u>: (ClmCkm CkmClrn^J (2.10.8) Here the asterisk denotes that tok ^ tom when {UJ\ — u>m) appears in the denominator. Once the symmetry of A B 3 X Y 3 molecule is considered, it can be seen that this operator has different forms for nondegenerate ( A i ) vibrational states and degenerate (E\) vibrational states. 66 Chapter 2 General Theoretical Considerations Table 2.8 Nonvanishing parameters in H22ag Non Degenerate Vibrational State V3SS - a r = B » 0 = B™0 -«r = B*A*M - a r = m% -a™ = £»<*o Doubly Denerate Vibrational State 6 0 7 _ p 7 0 nxx 1 ( fin 1 S u \ 1 / S y y 1 R j / y \ _ a t — 2 ^ t o t o O + "^tbtbOj — 2 ^ t a t a O ^ tb tbOJ ~,zz p z z Tizz a t — n t a t a O — a t b t b O a t — n t a t a o — n t b t b O a t ~ ntataO — n t b t b 0 „ r>xx T>xx 5w r>VV Q22 — &tata0 - n t b t b o — & t b t b o ~ n t a t a o „ 1 5 i2 1 p x z 1 f>yz 912 — 2 n t a t a 0 — ~ 2 n t b t b 0 — ~ 2 D t a t b 0 _ 1 p x a 1 D I Q 1 f>Va 9l2m — 2 n t a t a 0 — ~ 2 a t b t b 0 — 2 n t a t b 0 For a nondegenerate vibrational state, H22 has the following form: (#22)s = \vrl~c2os2,a - «r J 2 - « - <x) 2«r - £ • (2.10.9) The detailed expressions and the properties of the parameters in the above equation are given in Table 2.8. Note the operators in the bracket of Eq. (2.10.9) have exactly the same form as #02 as given in Eq. (2.3.59). For a doubly degenerate vibrational state, of the various operators in Eq. (2.10.6), the following are the only ones totally symmetric in Gig: 1 - cos 3a 2 T2 t I T2 2 Qt+qt-, J qt+qt-, 3zqt+qt-, 3z3aqt+qt-, 3aqt+qt-, 32+q2_ + 32_q2+, [J2, J + ] + q 2 + + [J2, J_]+<?2_, 3a3+q2+ + 3a3zq2_. (2.10.10) Thus H'22 has the following form for a doubly degenerate vibrational state ut (H^t=(H'22)dt + (H^0td, (2.10.11) where the superscript d denotes the diagonal in £ part and od denotes the off-diagonal in £ part of H"2-67 Chapter 2 General Theoretical Considerations These can be written: (H'^ = { v 3 « ( l - c o s 3 a ) / 2 - a r j 2 - ( a r - a n ^ -2azta3z3a - a?a32a} qt+qt^ (2.10.12) {H^)°td = \q22{3lql+32_q2+) + [(qU3z+q12m3a),{3+q2++3.ql)}+. (2.10.13) Although J a commutes with J±, in the expression of (H22)°d it is still written in a symmetrized form for the purpose of later discussions. The expressions and the properties of the parameters in above equations are listed in Table 2.8. The diagonal matrix elements of q2 and q+q~ can be written as: (vs\q2s\vs) = + (2.10.14) {vtJt\qt+qt-\vt,tt) = vt + l. (2.10.15) It then follows that Eqs. (2.10.9) and (2.10.12) simply give the vibrational dependence of the torsional barrier height, and of the leading rotational and torsional constants. Effective parameters for each vibrational state can be introduced as follows: Vs = Vf + ^ Vr^ + ^ +Y^Vfivt + l)-, (2.10.16) B* = ^ - £ ^ ( ^ + 0 - £ a f 7 ^ + l ) . (2.10.17) It should be noted that for a certain vibrational state, the effective barrier height, rotational constants and torsional constants also contain contributions from higher order terms like H'l2 ^ , etc. However, these contributions are small in magnitude compared to those from #22 and hence will not be discussed here. (H22)°d has matrix elements off-diagonal in the quantum number £ and k which can, in some cases, split and shift the zeroth-order levels. The associated effects are called £-type resonance. The selection rules for the matrix elements are AG = Ak - A£ = 0, ±3 . (2.10.18) This relation leads to off-diagonal matrix elements of two types for the operator (H'22)t. Here we are only interested in the first excited vibrational state vt — 1. These matrix elements are better written in terms of the zeroth-order vibration-torsion-rotation basisfunctions (Eq. (2.4.20)) as follows: , od h vt,£ = =f l;J,k^fl,v6,a (vt,£ = ±l;J,k±l,v6,a\(HZ2) = \q22\]{vt + l ) 2 - l2F (J, k — l)F (J, k) I (+1, k + 1; - 1 , k - 1; VQ, a); (2.10.19) vt,£ + 1; J,k,v&,a od vt,£- l;J,k- l,ve,cr 68 Chapter 2 General Theoretical Considerations = quy/(vt + l ) 2 - l 2 (2k - 1) F (J, k - 1) 7 (+1, k;-l,k - 1; a) +912™^(«t + I) 2 - l 2 ^ (J, A; - 1) (J a ); (2.10.20) where F(J,k) = {J,k + l\3_\J,k) = ^ / J (J + 1) - fc (fe + 1); (2.10.21) I(e",k";£',k';v6,a) = £ ^ W ^ ' ; f c > e , a . ( 2 1 0 2 2 ) */ (J a ) = ^ ( a f c z - r - c r ) ^ 1 * " 6 ^ ; 1 : * - 1 : ^ ^ . (2.10.23) */ 7 (£", A;"; £',k';ve, cr) plays the role of an overlap integral as indicated by the notation, 7 is a function of both VQ and cr. (J a) is the matrix element of J a between the two states coupled by (H^yf. The effect associated with the matrix element in Eq. (2.10.19) is denoted as (2,2) (A£ — ±2, AA; = ±2) interaction. <j»22 is called -^doubling constant. Similarly, the matrix element in Eq. (2.10.20) is denoted as (2,-1) (A£ = ±2, Ak = T l ) interaction. In the (vt = 1) vibrational levels, the matrix element of (2,2) interaction gives rise to a splitting of the (k = £t = ±1,(7 = 0) pair of levels into A\ and A2 components. It is more convenient to label these as A+/A- using the correspondence in Eq. (2.4.25). The splitting of the two components can be written: E (A+) - E (A-) = q22J (J + 1)1 (1,1; - 1 , - 1 ; v6,0) (2.10.24) The expressions of 922 and q\2 given by Table 2.8 follow the phase convention defined by Eq. (2.4.14). With the current phase convention, 1(1,1; — 1, — 1; VQ, 0) is a positive number for all VQ. Thus the order of A+ and A_ components is determined by the sign of q22. For a torsional state where the high barrier approximation can be assumed, e.g. ve = 0, 7(1,1; - 1 , - 1 ; u6,0) is a number very close to 1. For higher torsional states, it can be much smaller than 1. The effect of (2,2) interaction on the (k = £t = ± 1 , a = 1) pair of levels is more complicated due to the cr splitting between the two levels. The same is true for the (A; = £t = ± 1 , cr = — 1) pair. These effects will be discussed in Chapter 3. The exact form of H'2^ can be worked out from the formula given by Duan et al. [47] Like H^2^, H2l2^ has both diagonal and off-diagonal matrix elements in vibrational quantum numbers. From symmetry considerations, the part of H^2^ diagonal in vibrational quantum numbers has the following form: Py6e t a,b \ /3-rSe J has matrix elements which simply give the vibrational dependence of the quartic centrifugal distor-tion constants and Ve- The sextic centrifugal distortion constants also depend on the vibrational quantum 69 Chapter 2 General Theoretical Considerations numbers due to the term #2'6; the exact form will not discussed. 2.10.3 Reduction of the Torsion-Rotation Hamiltonian After the sequential vibrational contact transformations, the torsion-rotation Hamiltonian has the follow-ing form: K = #02 + #04 + #06 • (2-10.26) where # 0 4 has the form [47]: -"04 — -"04 + 2 ° 1 2 '-"12 (2.10.27) 12) The terms #02 and # 0 l 4 / arise from the expansion of the untransformed Hamiltonian. It follows from the symmetry considerations that the torsion-rotation terms in the expanded Hamiltonian have the following forms: #02 = B0(3l+32y)+Ao!F32z + -2A0}F3z3a + F32a + V3e #0 { 4 2 } = 2 {#0^J 2 + (#0"3 - #-) 1 — cos 3a 1 — cos 3a (1 — cos 6a) (2.10.28) (2.10.29) The term in sin 3a vanishes because of the symmetry; furthermore: ^0,3 — D< /3-y,c 0,3 ' 5ox,3 = The vibrational part of the commutator -i .. F 2 L c(l} TTW ° 1 2 '-"12 (2.10.30) (2.10.31) in Eq. (2.10.27) leads to a contribution in # 0 4 of the following form: 1. 2* ° 1 2 '-"12 k ~ \ E E ^BSBfeJ / jJ^J , - E E ^ l j B f e X J / 3 J 7 | (1 - cos 3a) fc /37(5e fc /3 7 - E \VWVz\ (1 - cos3a) + £ ^ * " k ^ (1 - cos6a) (2.10.32) fc=Ai fc=A1 It can be seen from the above equation that \i s\2; ,#12 contains a term — 3^ ^  — L Jv fc 2 2 cos 3a), which has a vibration-torsion-rotation indexes 02 hence should be included in #02. The order of magnitude of this contribution is still two orders smaller than the leading term V3e in #02- The potential 70 Chapter 2 General Theoretical Considerations constant in H02 is modified to become the effective potential constant V^' where v3" = vi- \VJU?VJ• (2.10.33) s=Ai Hence H02 after this stage with be denoted as H 0 2 . The sum over the vibrational states only runs for the nondegenerate A \ vibrational states because only the nondegenerate states have nonvanishing VK. (Refer to section 2.7.2). By using the results of Eqs. (2.10.29) and (2.10.32), the general form of HQ\ can be written as ^04 = \ E r ^ £ J ^ J 7 J F I J E + J2 3P3i®p*\ (1 - cos3a) + V&"\ (1 - cos6a), (2.10.34) PjSe Pi where rp^ = ~ 2 E Bko 1 Bio > ®Vi = 2Bg - J2 B%»7lVi , (2.10.35) (2.10.36) (2.10.37) s=Ai Now the potential constant VQ in the Hamiltonian H 1 ^ is modified to become the effective potential constant VQ . It can be easily seen from the definitions of 7 that r^se has the following properties: TpiSe — Tple6 = TfPSe = T^PeS = T Sep-y = T fieyP = T esyp — T€spy. (2.10.38) The general form of HQ4 applies to all molecules with a threefold rotor. For A B 3 X Y 3 , some distortion constants will vanish because of the symmetry. To derive H'Q\ for AB3XY3, the form of H12 restricted to Gis symmetry given by Eq. (2.7.13) should be used. The evaluation of \i for A B 3 X Y 3 is straightforward but tedious. However, the contributions from this commutator to various centrifugal distortion constants is clearer from the result of the evaluation. The direct derivation is given in Appendix A.5. The nonvanishing distortion coefficients and their properties are listed in Table 2.9. By using the results in Appendix A.5 and Eq. (2.10.29), H'^ for A B 3 X Y 3 can be written as: ZT" — r V ' T 4 r>" I 2 T 2 n" T 4 n" T 2 T 2 n" T 2 " r 2 n" i 4 - ^ J 2 J Z J Q - d"Krzla - <&J ZJ| + VI'- (1 - cos 6a) „ t 2 1 - cos 3a „ 2 1 - cos 3a + - ^ 3 J J ~ rfzK-J, + 0 2 ; 2 J 3K°z~ 1 — cos 3a + e 2:2 + 1 — cos 3a 2 +e" [3Z, Jl + J3_] + + e'^ [JA, J3+ + J3_]_ 71 (2.10.39) Chapter 2 General Theoretical Considerations Table 2.9 Centrifugal distorsion constants for AB3XY3. Nonvanishing r's and Their Properties Nondegenerate Vibrationa Ai Doubly Degenerate Vibration E\ 7 xxxx — Tyyyy 7zzzz 7~cx.cta.ct T ZCCZOL Txxzz ~ Tyyzz T xxza — Tyyz<x T xxcxct Tyyaa ~~~ ZZZa T zzctct 7~ zctctct 7xxxx — Tyyyy 7xxxy 7xzxz — Tyzyz Txotxa. — 7 xzxa TyzXQc = Tyzya ~~~xxxz Txxyz — ~1~yyya Txxxct — ~Tyyya 1 yyxy Distortion Coefficients = _ i ( T M + T E 1 \ 4 V xxxx 1 'xxxx) = 1 (TAi + T B i ) _ 2 \ xxxx 1 xxxxJ 2 xxzz _ T B i ' X 2 X Z 70" = - I f r A l + T B l ) 4 \ 'xxxx * 'xxxx} 1 2 X X Z J Z 4 ' 1 ' xzxz n" = 2 xx act ' xaxa D'km = _lTAi _ TAi 1 2 ZZOLOL ZOLZOt 1 lTAi 2 x x a a 1 ' r a x a </ = ' xxza ' xzxa d"K = ' zzzoi ' ' xxza 1 xzxa d" aaaz D'm = _ I T A i 4 ' a a a a = Q 2 ; 2 TP" *3K — @2;2 _ @2;2 ^ z z ^ x x e" = 4'xxxz e" = lTEi 4 ' x x x a Although J a commutes with all J + and J _ , [JQ, J + + Ji]+ is used in Eq. (2.10.39) to compare the operators corresponding to the parameters e" and e'm. All the parameters in Eq. (2.10.39) are written in terms of r or ©. The definitions of these parameters are listed in Table 2.9. As can be seen from Table 2.9, all the constants in H$, which originate from the expansion of the original Hamiltonian, are modified by the contributions from the H{2. These include VEE and all the BQ 3 o.2 constants (which are modified as 0^) in Eq. (2.10.29). V3E in #02 is also modified. The commutator \i , #12^  also produces operators in HQ4 which do not exist in the expansion of the original Hamiltonian. The corresponding r constants have meanings similar to those of the centrifugal distortion constants for a normal molecule with no internal rotation. These constants can be classified as following three different types according to their origins as listed in Table 2.9. The first consists of constants which arise from contributions from both H[2 and H\2. These can be further arranged into two 72 Chapter 2 General Theoretical Considerations groups: (1) pure rotation-type constants, namely D", D"JK, D"K\ (2) all the torsion-rotation constants which are associated with rotational operators and with only torsional operators 3a or J 2 , namely d"j, d"K, D"Jm, D'^m. The second type consists of constants which arise only from contributions of Hf2, namely D'^ and d'^. The last type consists of constants which arise from only contributions of H\2, namely e" and e^ . The corresponding operators in this last type have only nonvanishing matrix elements off-diagonal in k with selection rule AA; = ± 3 . Most of the signs of the r constants can only be determined experimentally. The only exception is Tpppp, which it is always negative since it is proportional to — [B^Q^ • Therefore, both D'^ and D'j are positive. In the reduction of HQ\, some of the constants in HQ2 are also modified. (Refer to Eq. (A.96).) The modified HQ2 has the form: = B" (J 2 + J2,) + AF32Z - 2 (AF + AA'p) 3z3a + F"32a + ^  (1 - cos 3a), (2.10.40) where V^' is given by Eq. (2.10.33) and the rest of the constants are listed as follows: B" = B 0 - ^ X X X + ^ Z X Z ; (2.10.41) AnF = 4 , , F + \T^XXX - rfzxz; (2.10.42) 2 xzxa ^' F" = F0. (2.10.44) A.A'F — n^ lxa. A Txxxx + Txzxzi (2.10.43) If the properties of T^ST are considered, the modifications of the rotational constants B" and A"F here are the same as that for C3v molecules, which were discussed in detail by Gordy and Cook (Table 8.4 of reference [63]). It should be noted that once the correction of the quartic centrifugal distortion is made, the coefficient of the operator J 2 is no longer equal to that of —2JZ J Q . However, the difference AAF is on the order of a quartic centrifugal distortion constant. A torsion-rotation contact transformation can be performed to removed some of the parameters in HQ4 [43]. In this section we will discuss only the transformation which remove the parameters associated with diagonal matrix elements in A;, namely J 2 [(1 — cos 3a) /2, J a ] + and [(1 — cos 3a) /2, J 2 ] . The transfor-mation function SQ3 is given by Ref. [43]: 5Q3 = Ci Jz sin 3a + C2 [3a sin 3a + sin 3aJ a ] + , (2.10.45) where Ci - -(e% + le%AF/F^/6F", (2.10.46) C2 = -OW/24F" . (2.10.47) Clearly, this transformation only modifies the torsion-rotation Hamiltonian. The transformed Hamiltonian 73 Chapter 2 General Theoretical Considerations Table 2.10 Effective parameters affected by the torsion-rotation contact transformations. a AF = AF-AAF — AA'F F = F0 + © a a _ Ao,F @ a a 2 ~ p 2 v3 = vi' + F3K = TP" *3K ' t- 2Pe% + p 2 0 2 ^ = O 2L 2 - e 2 £ + 2 P e 2 i 2 + p 2e 2^ 2 V6 = V" — \&llvi' = vi + \ Es v?^v? - Ivi'Qlil A0,F TO,T Lzz P = F0 1° * r.r. up to the fourth order in rotation has the form: #02 + #04 = #02 + # 0 4 2 } + ' cd TT" '-'03' -"02 where #02 = B"32 + (AF - #") J 2 - 2 (AF + A A F ) Jz J c #04 = + - ^ J a + y (1-cos 3a); — £ ) j j 4 — DjKJ23l — -Dj^ J4; — Z ) ' J m J 2 J 2 — D ^ m J 2 J 2 -d'jJ 2J 2JQ - d ^ j 3 j a - d'^mJ zJ a - £CJ Q + Ve J (1 - cos 6a) cos 3a (2.10.48) (2.10.49) 2 + F 3 K J 2 2 1 2 ° S 3 Q + [e"3z + e'mJa,J3+ + J 3 _ ] + . (2.10.50) A tilde has been used for the symbols #02 and #06 to denote the torsion-rotation contact transformation has been applied. It should be noted that i [ 5 Q 3 , #o2] also produced terms which have the order 02; hence #02 has also been modified. A modified parameter is represented by a symbol with a tilde in above equations. They are summarized in Table 2.10. After the vibrational contact transformation, #064^ has the general form [62]: o-"{4} _ rr"{4} • -"06 — -"06 ^ 1 qd rr"{2} ^03» -"04 qd ^03» od TT" ^03I -"02 [505, #02] • (2.10.51) The sextic centrifugal distortion terms can be easily obtained: #06 { 4 } = £ * / J 7 & e C J 0 J 7 J * J « J e J C + E J / 3 J 7 J « J e 0 f t e "COS 3a) + J/3J-yQp* (1 - cos 6a) + e°0'6 (1 - cos 9a) . 07 (2.10.52) 74 Chapter 2 General Theoretical Considerations The coefficients in Eq. (2.10.52) can be obtained with a simple algebraic processing according to Eq. (2.10.51). This term can be reduced by the torsion-rotation contact transformation function S05 [47]: 5 0 5 = J2 J / 3 J 7 J 7 J « J , + E [e/37<5 ( ! - C O S 3 « ) + S i n 3 a ] J / 3 J 7 J « • (2-10.53) Instead of going through tedious evaluations of this transformation, the reduced Hoe is presented here by using only symmetry considerations: Hoe = H'JJ6 + HjK3A32 + HKJJ2JZ + H'x3z + 77 " J m J 4 J 2 + 7 7 j ^ m J 2 J 2 J 2 + i f j m m J 2 J ^ , + ^ m m J 2 J a + + hjj3A3z3a + h'^K^z^a + h"JK323z3a + hjKm323z3^ +F^jj3i\ (1 - cos3a) + F'iJK3232z\(1 - cos3a) + F'{KK3Z\ (1 - cos3a) +F^j32\ (1 - cos6a) + FgK32z\ (1 - cos6a) + ^ (1 - cos9a) + h'i {3% + 3%_) + je ' j J 2 + e"K32z + e ^ J 2 + (1 - cos3a)| 3Z , (J 3 + + J3_) (2.10.54) + 2.10.4 Effective Torsion-Rotation Parameters The effective Hamiltonians have been subjected to sequential vibrational contact transformations and torsion-rotation contact transformations. In general, an effective parameter contains contributions from each stage of the contact transformations. Various notations for the parameters at different stage have been used. Consequently, it is necessary to summarize the contributions from all the contact transformations applied so far. These contributions to the effective torsion-rotation parameters in the ground vibrational state are summarized in Table 2.11. All the parameters at this stage will be written with a barred symbo