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Fine and hyperfine structure in the 2II ground electronic state of HBr+ and HI+ Chanda, Alak 1995

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FINE AND HYPERFINE STRUCTURE IN THE 2 fl GROUND ELECTRONIC STATE OF HBr AND HP By  Alak Chanda B. Sc., M. Sc. (Physics) Agra University, 1983 M. Sc. (Physics) University of New Brunswick, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  September 1994  © Alak Chanda, 1994  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.  (Signature)  Department of  Pc.  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  2iLb&cfrls€R,99L-  Abstract  The vibration-rotation spectrum of HBr+ in the  2111,2  and  2113,2  spin substates of  the ground electronic state has been investigated between 1975 cm’ and 2360 cm’ using a tunable diode laser spectrometer coupled to an a.c. glow discharge cell. Both fine and hyperfine studies have been carried out. In the former, about 300 vibration-rotation tran Br+ 7 9 and H Br+. 8 1 These belonged sitions were measured for each of the isotopomers H to the five bands (v’—v”)=(l—O) to (5—4). The observed linewidth was ‘--‘0.006 cm’. In this inverted We  2  state, the difference  (2440 cm’). Here Ae and  We  ((Ael  —  ) compared to 1 We) is small (‘—‘200 cm  are the equilibrium values of the spin-orbit constant  and the harmonic vibrational frequency, respectively. As a result, the energy levels oc cur in neighbouring, but non-resonant, pairs with (v, The one exception is the  2113,2  2113/2)  coupled to  (  —  1,  2111,2).  state with v=0, which is isolated. Centrifugal distor  tion matrix elements between partner states have been shown to effect significantly the A-doubling. A model has been developed in which these distortion matrix elements are treated by a vibrational Van Vleck transformation carried to third order. A good fit has been obtained without introducing any new fitting parameters to characterize the (z.v  0; LS  =  +1) effects. Equilibrium values were determined for the principal pa  rameters which characterize the individual vibrational levels. In the hyperfine study, a combined total of 57 hyperfine splittings were observed in the two spin substates of Br, 79 the transitions in the H  2111,2  spin substate being ob  Br+. These transitions 1 served for the first time. An equal number were measured for HS were distributed over the P,  Q, and R branches of the four lowest vibrational bands. The 11  matrix elements for the magnetic dipole and electric quadrupole interactions have been written in the e/f symmetrized scheme more commonly used in vibration-rotation prob lems. Values have been obtained for the Frosch and Foley magnetic hyperfine constants a, c, and d by using the value of b determined by Lubic et al., J. Mol. Spectrosc. 131, 21-31 (1989). These results have been used to investigate the electronic properties of the ion. The analysis supports a model in which the electron distribution is close to that of a bromine atom perturbed by a proton. A similar study of the HI+ molecular ions has also been carried out. Prior to the current investigation, the spectroscopic information on 111+ was limited to that obtained from low resolution. A total of more than 100 vibration-rotation transitions belonging to the (v’  —  v”)=(l—O), (2—1) and the (3—2) vibrational bands of the  and to the (1—0) vibrational band of the  2111,2  211,  spin substate  spin substate have been recorded in the  frequency range from 1995 cm to 2245 cm. The observed linewidth was -.0.004 cm* Equilibrium values were determined for all the principal parameters characterizing the individual vibrational levels. The precision of the vibrational constants, were improved by a factor of  ‘  We  and  WeXe,  106 over the values determined recently by Böwering et  aL, Chem. Phys. Lett. 1, 467 (1992) and by Zietkiewicz et al., J. Chem. Phys. 101, 86 (1994), using photoelectron techniques. Large hyperfine splittings arising from magnetic dipole and electric quadrupole in teractions were observed for low J transitions belonging to both of the f substates of 111+. The Frosch and Foley magnetic hyperfine constants a, b+c, and d, and the electric quadrupole constant eQqo were determined for the first time in this molecular ion.  In  Table of Contents  Abstract List of Tables  vii  List of Figures  x  Acknowledgments  xiii  1  Introduction  2  General Theoretical Considerations  3  1 12  2.1  Molecular Energy Levels and Transitions  12  2.2  Labelling of Electronic States  17  2.3  Coupling of Angular Momenta: Hund’s Coupling Cases  20  2.3.1  Hund’s Case (a)  21  2.3.2  Hund’s Case (b)  22  2.4  Symmetry Properties of Basis Functions  25  2.5  Allowed Transitions and Selection Rules  28  2.6  Fine and Hyperfine Structures  30  2.6.1  Fine Structure Interactions  30  2.6.2  Hyperfine Interactions  32  General Experimental Considerations 3.1  Experimental Method  46 46  iv  3.2  3.3 4  Diode Laser Spectrometer  3.1.2  The Discharge Cell  3.1.3  General Experimental Set-up  3.1.4  Production of Ions  •  .  •  47  Detection Technique 3.2.1  Modulation Techniques in Molecular Spectroscopy  3.2.2  Velocity Modulation  Spectral Recording and Calibration  Hyperfine Structure in HBr and Hft  64  4.1  Hyperfine Energies  64  4.2  General Features of the Spectrum  73  4.2.1  Hyperfine Structure in HBr  73  4.2.2  Hyperfine Structure in Hft  75  4.3  4.4  5  3.1.1  Data and Analysis  77  4.3.1  HBr  77  4.3.2  HI  89  Results and Interpretation  93  4.4.1  HBr  93  4.4.2  HI  99  Rotational Energies and Fine Structure in HBT+ and HI  101  5.1  General Features of the Spectrum  102  5.2  Data and Analysis of HBr with the Merged Model  108  5.2.1  The Effective Hamiltonian  108  5.2.2  Merged Model Fit  118  5.3  Analysis of HBr with the Split Model V  137  6  5.4  Molecular Constants for HBr  158  5.5  Data and Analysis of Hft with the Merged Model  167  5.6  Molecular Constants for Hft  175  5.7  Vibrational and Rotational Temperatures of the Ions  183  Discussion and Conclusions  190  Bibliography  201  Appendices  207  A Rotational Energy and Fine Structure in  vi  211  Electronic State  207  List of Tables  2.1  Selection rules for electric dipole allowed transitions and for perturbations  4.1  Wavenumbers of the Observed Hyperfine Splittings in the X 11 Electronic 2 79 H State of Br  4.2  80  Wavenumbers of the Observed Hyperfine Splittings in the X 211 Electronic 81 H State of Br  4.3  83  Fitted and Derived Hyperfine Interaction Constants for HBr in the X 211 Electronic State  4.4  86  Wavenumbers of the Observed Hyperfine Splittings in the X 2 f] Electronic State of HI  4.5  29  90  Fitted Hyperfine Interaction Constants and Derived Averages over Func tions of the Electron Space Coordinates for HI in the X 211 Electronic State  4.6  94  Averages over Functions of the Electronic Space Coordinates for HBr+ in the X 211 Electronic State  98  5.1  The Effective Hamiltonian Matrix Elements for the Merged Model  5.2  Wavenumbers of the Observed Vibration-Rotation Transitions in the X 211  .  .  .  Electronic State of HBr 5.3  Molecular Parameters in the  119 Merged Model for H Br 7 9 in the X 211  Electronic State 5.4  Molecular Parameters in the  113  133 Merged  Electronic State  Model for H Br 8 1 in the X 211 134  vii  5.5  The Effective Hamiltonian Matrix Elements for the Split Model  5.6  Molecular Parameters in the Split Model for H Br 7 9 in the X  141 2  Elec  tronic State 5.7  149  Molecular Parameters in the Split Model for H Br 8 1 in the X 211 Elec tronic State  5.8  150  Equilibrium Molecular Constants of the X 211 Electronic State of H Br 7 9 in the Split Model  5.9  159  Equilibrium Molecular Constants of the X  2  Electronic State of H Br 8 1  in the Split Model  160  5.10 Harmonic and Anharmonic Expansion Coefficients for the Effective Potentiall64 5.11 Harmonic and Anharmonic Force Constants for the X 211 Electronic State of HBr  166  5.12 Wavenumbers of the Observed Vibration-Rotation Transitions in the X 11 2 Electronic State of Hft  168  5.13 Molecular Parameters in the Merged Model for Hft in the X 11 Electronic 2 State  176  5.14 Equilibrium Molecular Constants of the X 11 Electronic State of HI 2  .  .  178  5.15 Comparison of Vibrational Constants of the X 11 Electronic State of Hft 2 with Previous Determinations from Photoelectron Spectroscopy  180  5.16 Harmonic and Anharmonic Expansion Coefficients and Force Constants for the X 11 Effective Potential of HI 2 6.1  184  Contributions of the Interstack Interaction to the A-doubling for H Br 8 1 in the 211, State at J=18.5  193  viii  6.2  Comparison of Equilibrium Molecular Constants of the X’E Electronic State of Hydrogen Halides and the X Counterpart  211  Electronic State of their Ionic 196  ix  List of Figures  1.1  Potential energy curves for the ground and the low lying ionic states of HBr  2  1.2  Potential energy curves for the ground and the low lying ionic states of HI  3  1.3  Vibronic energy level diagram for the  211  ground electronic state of HBr  6  1.4  Vibronic energy level diagram for the  2fl  ground electronic state of Hft.  8  2.1  A schematic representation of a bound electronic state  2.2  Vector diagram of Hund’s coupling case (a)  22  2.3  Vector diagram of Hund’s coupling case (b)  23  2.4  Vector diagram of Hund’s coupling case 1 (a 3 )  37  3.1  Schematics of the diode laser monochromator  49  3.2  Design of the discharge cell  51  3.3  A general layout of the experimental set-up  53  3.4  Basic principle of the velocity modulation spectroscopy  60  4.1  Calculated hyperfine splittings of the  2111/2  .  .  13  .  R(O.5)eeff energy levels of  HBr  72  4.2  The hyperfine structure of the  2113,2  (2  4.3  The hyperfine structure of the  2113,2  (1  4.4  Hyperfine structure of the R(2.5) and the P(3.5) transitions in the (1-0) 2  4.5  113/2  —  —  1) Q(1.5) transition of H Br 7 9  74  0) Q(1.5) transition of HI  76  .  .  state of Hft  78  Simulated and observed hyperfine spectra of (1—0) Br 8 H 1 and (21)  211,  79 H R(1.5)ee of Br x  2113,2  ()eeff of 88  4.6  Simulated and observed hyperfine spectrum in (1—0)  2113/2  .S)eeff of 2 R(  HI 5.1  95  A portion of the HBr spectrum showing transitions belonging to each of the five vibrational bands investigated  5.2  Portions of the spectrum of  2113/2  103  P(9.5) vibration-rotation transition of  Br+ belonging to the first three vibrational bands 9 HT  105  5.3  Observation of A-doubling in the  107  5.4  Observed A-doubling in the P-branch of the (1’ tional bands in the  5.5  (1—0) vibrational band of Hft  211,  —  0”) and (2’  —  1”) vibra  spin substate of H Br 7 9  Observed A-doubling in the P-branch of the (1’ tional bands in the  5.6  211,  2113,2  —  130 0”) and (2’  —  1”) vibra  spin substate of H Br 8 1  131  The variation of the effective spin-orbit distortion parameter AD with v in Br 8 H 1 obtained from the fit using the merged model  5.7  135  The variation of the effective A-doubling parameter q with v in Br 81 H obtained from the fit using the merged model  5.8  The merged model fit of the observed A-doublet splittings in branch of the (1’  —  0”) and (2’  —  136 2113,2  P  1”) vibrational bands of H Br 8 1 with the  e and the f assignments interchanged in the fundamental band relative to the assignments in Table 5.2 5.9  138  The variation of the effective spin-orbit distortion parameter AD with v in Br 8 H 1 obtained from fits using the merged model and the split model.  152  5.10 The variation of the effective A-doubling parameter q with v in Br 81 H obtained from fits using the merged model and the split model 5.11 Split model fit of the observed A-doubling in the P-branch of the (1’ and (2’  —  1”) vibrational bands in the xi  211,  153 —  spin substate of H Br 7 9  0”) .  154  5.12 Split model fit of the observed A-doubling in the P-branch of the (1’ and (2’  —  1”) vibrational bands in the  2ll  spin substate of H Br 8 1  5.13 The split model fit of the observed A-doublet splittings in of the (1’  —  0”) and (2’  —  —  2113/2  0”) .  P-branch  1”) vibrational bands of H Br 8 1 with the e and  the f assignments interchanged in the fundamental band 6.1  155  156  A plot of the variation in the value of the harmonic force constant k with the equilibrium bond length Re for the neutrals and the ions of the hydro  6.2  gen halide series  198  ‘In search of HI ion signal’  200  xii  Acknowledgments  Clearly, I did not complete this thesis in a vacuum, and therefore I take pleasure in ac knowledging the contributions of others. Although moments of frustration certainly occu red, and at times the task at hand seemed overwhelming, an excellent cast of faculty and graduate students provided a stimulating and supportive environment. I am very indebted to Dr. F. W. Dalby and Dr. I. Ozier who acted as my supervisors, my teachers and my mentors. I deeply appreciate their accessibility, which is probably the most important requirement in a graduate student’s life. What I learned from them reaches far beyond what is written in the pages of this thesis, and will continue to influence me long after the memory of this experience has blurred. If any further support was needed, it could most certainly be found in the office of Dr. W. C. Ho. He, in fact, taught me the nuts and bolts of the discharge and the diode laser system. His constant help throughout my thesis work is much appreciated. I would like to thank Dr. J. L. Booth with whom I have shared the lab for several years. In many occassions we have worked closely, shared the success as well as the frustrations of tuning an “untunable” laser. My appreciation also goes to Dr. W. Jger and Dr. M. C. L. Gerry for their assistance with the early phases of the experiment, to Dr. A. J. Merer and Dr. A. R. W. McKelIar for fruitful discussions, to Dr. N. Moazzen-Abmadi for the loan of a laser diode, and to X. Zhou, S. -X. Wang, and C. Boone for making the working environment lively and enjoyable. A sentence or two of thanks are inadequate to acknowledge the contribution my parents, my sister and brothers, and my wife have made to this study and my academic career. Their absolute faith in my abilities and pride in my accompishments gave me the confidence to tackle many challenges. This project is only the latest.  xlii  Chapter 1 Introduction  The infrared spectrum of HBr+ and 111+ provides an excellent opportunity to study the electronic, vibrational and rotational degrees of freedom of a simple open shell ion in considerable detail. Two properties of the molecule make it particularly appropriate for such a study. First, the hyperfine splittings are large enough that they can be resolved in the vibrational spectrum with Doppler limited resolution. Second, the vibrational temperature of the ions generated in an a.c. discharge can be —‘4OOO K, so that relatively high vibrational levels can be studied. The singly charged hydrogen halide positive ions form an unusually simple series. The ground state electronic configuration of these ions give rise to an inverted 211 ground elec tronic state, with the  2113/2  spin substate lower in energy than the  2111/2  spin substate.  (See Sec. 2.2 of Chapter 2 for the labelling of the electronic states.) The splitting be tween the two spin substates arises as a result of the magnetic interaction between the magnetic moments associated with the spin and orbital motion of the electrons (called the spin-orbit interaction). The low lying electronic states of HBr and Hft, along with the ground electronic state of their neutrals, are shown in Figs. 1.1 and 1.2, respectively. The first few vibrational levels of the  211  spin substate are also shown. As can be seen  from these Figures, the spin-orbit splitting is considerably larger in HI (‘-.‘5360 cm) than in HBr (‘-‘265O cm ). 1 In HBr the first electronic state, lying approximately 29,000 cm 1 ground electronic state, is a  2+  [1]  above the  2 + state interacts directly with the state. The E 1  2111/2  Chapter 1. Introduction  2  20  H( S 1 ) + Br ( P) 2  10  H( S 2 )+ Br( P) 2  0  1  3  2  Bond length  (A)  Figure 1.1: Potential energy curves for the ground and the low lying ionic states of HBr. The energy in units of cm is related to the energy in units of eV by 1 eV = 8065.54 cm . The first few vibrational levels of 2113/2 spin substate of the ground 1 electronic state of HBr+ ion are shown by horizontal lines. The states on the right are for the dissociated atomic products. The 2111/2 and 2113/2 states correlate to the S)+Br( states of the separated atomic system. The spin-orbit splittings of the H( ) 2 P 3 atomic states are not shown explicitly. There has been a controversy in correlating the 2 state to the separated atomic states. See Refs. [, ]. The atomic states of the A dissociated products shown here are those of Ref. [1.  Chapter 1. Introduction  3  n 4  15  ‘I  E  o  10  C  5  1 z +  0  1  H( P) 2 S)+ I( 2  2  Bond length  (A)  3  Figure 1.2: Potential energy curves for the ground and the low lying ionic states of HI. The energy in units of cm is related to the energy in units of eV by 1 eV = 8065.54 cm. The first few vibrational levels of 2113/2 spin substate of the ground electronic state of 111+ ion are shown by horizontal lines. The states on the right are for the dissociated atomic products. The 2111,2 and 2113,2 states correlate to the 3 S)+ft( states of H( ) 2 P the separated atomic system. The spin-orbit splittings of the atomic states are not shown E+ state to the separated 2 explicitly. There has been a controversy in correlating the A states. atomic See Refs. [2 , ]. The atomic states of the dissociated products shown here are assumed to be similar to those obtained for flBr+ j Ref. [2].  Chapter 1. Introduction  4  spin substate and indirectly (via the  2111/2  state) with the  2113/2  spin substate. The  interaction splits the degenerate e and the f rotational energy levels (see Sec. 5.2.1 of Chapter 5). The e and the f labellings of the rotational energy levels are described in Sec. 2.4 of Chapter 2. The splitting of the rotational energy levels of a 11 state due to admixing with nearby E states of the molecule is referred to as the A-doubling, and is characterized by the parameters p and q. The A-doubling parameters p and q essentially describes the magnitude of the splitting in the rotational energy levels of the the  doubling in the 211  and  spin substates, respectively. Usually, for low rotational energy levels, the A-  2113,2  the  2111,2  2111,2  spin substate is much larger in magnitude than the A-doubling in  spin substate.  + 2 The optical emission spectrum of the A by Norling  []  —  X 1 2 [ system of HBr+ was first observed  in 1935 using a hollow cathode discharge in HBr gas. He identified the  spectrum as arising from the (0—0) and (1—0) vibrational bands, where (v’  —  v”) refers  A E + and the 2 to a transition between vibrational levels v’ and v” of the 2 X 1 1 electronic states, respectively. No direct information about the ground state vibrational constants was obtained from his analysis. In 1953, Barrow and Caunt  [1] measured the (0—0),  (1—0), and (1—1) vibrational bands of the same system using a radio-frequency discharge and a grating spectrometer. They were able to determine the rotational constant B (which is inversely proportional to the moment of inertia of the molecule), the quartic centrifugal distortion constant D (which gives a measure of the centrifugal stretching of the molecule as it rotates), the spin-orbit constant A, the A-doubling constant p, the harmonic vibrational frequency WeXe  e  and the first order anharmonic vibrational constant  for the 2 X 1 1 ground state. In 1968, Marsigny and co-workers [S], and in 1973,  Lebreton [9] extended the emission studies to include the (0—2) and (0—3) vibrational bands of HBr+. They were able to determine the second order anharmonic vibrational constant  WeYe  and obtained improved values of  We, WeXe,  B, D, A and p.  Chapter 1. Introduction  5  Much of the high precision work in the 2 X 1 1 ground state of HBr started in the late 1970’s with the availability of lasers. In 1979, Saykally and Evenson [1Q] measured the pure rotational spectrum of HBr+ using the far-infrared laser magnetic resonance (LMR) technique. They observed the rotational transitions 4 J=5/2 — 3/2 and J=7/2E—5/2 in the v=0 vibrational level of the X 312 1 2 1 electronic state. It was possible to resolve the Adoublet splittings and the hyperfine splittings arising due to the bromine nuclear spin. Later, in 1989, high precision values for the rotational, spin-orbit, A-doublillg, and the Br+ 7 9 and H lBr+ 8 hyperfine constants as well as the g-factors were determined for H by Lubic et al.  [11].  A simultaneous weighted least-squares fit was carried out of the  observed LMR transitions and the optical combination differences from Refs. [7,  ,  9]  with the effective Hamiltonian of Veseth [12]. In its 211 ground electronic state, the hydrogen halide ion HX (X  =  F, Cl, Br, I)  acts much like a halogen atom perturbed by a neighbouring proton [fl]. The hyperfine structure measured for HF [ii, i., i.], HCft [ia,  i7J  and HBr[i, 11] has provided  considerable insight into the distribution of the unpaired electron. Since the HCft and HBr+ studies were based on rotational spectra obtained with laser magnetic resonance, the hyperfine investigation was confined to the ground vibrational state of the param agnetic  2113,2  spin substate. One goal of the current diode laser study of HBr+ is to  extend the hyperfine investigation to the nearby diamagnetic  2111,2  substate and to high  vibrational levels of both substates. A more complete characterization of the magnetic hyperfine structure can then be obtained to probe further the electronic structure. A second goal of the current work on HBr+ is to investigate the coupling off-diagonal in both v and  ,  where v is the vibrational quantum number and  is the eigenvalue of  the component of J on the internuclear axis. The vibronic energy level scheme for J=1.5 is illustrated in Fig. 1.3. The level spacings shown were calculated using the present results, but the general form was known earlier [—9,iS,i9]. The 211 system is inverted  Chapter 1. Introduction  6  HBr  15 12  + V  ;199 1cm1 2649 cm  5 4  0 0  >6 0)  2  C  LU  2  3 2346 cm 1  3  1  12669 cm1 cm  0  0 2  2  Figure 1.3: Vibronic energy level diagram for the 2 ground electronic state of HBr+. The energy spacings are shown for J=1.5, as calculated using the constants determined here. The A-doubling and the hyperfine splittings are to small to be seen on the scale used. The energy in units of cm’ is related to the energy in units of eV by 1 eV 8065.54 cm . 1  Chapter 1. Introduction  7  with the spin-orbit splitting being about 10% larger in magnitude than the vibrational frequency. With one exception, the levels occur in pairs, (v, 1 =  )  and (v + 1, l =  with an energy separation considerably smaller than the vibrational spacing. The one exception is the vibronic ground state (v=0, = ), which is isolated. Any perturbation between partner levels is non-resonant and might be expected to be absorbed into effective constants. However, since the ground state has no partner, these effective constants might be expected to show some anomalous behaviour. The questions to be investigated lie in the nature of this behaviour and in the methods of removing the effects of the non-resonant perturbations. Three  2fl  examples of coupling off-diagonal in v and Q have been investigated previ  ously (to our knowledge). In the A state of figH, the rotational levels of the (2,  )  )  and  80 resonant pertur N 5 ‘ []. In the X state of Se, bations were observed for (3, ) with (2, ) and for (4, ) with (3, ) [i]. For Se, 80 N 4 ‘ similar effects appeared for (6, ) with (5, ) [i}. In both HgH and NSe, the interaction (0,  substates cross at about J=10.5  was treated by diagonalizing the 2x2 Hamiltonian matrix for the interacting partners. In the X state of SeD, non-resonant perturbation was detected for (0,  )  with (1,  )  [2].  In this case, the coupling was taken into account by modifying the form of the leading quartic distortion term and then diagonalizing the usual (v,  )  4—*  (v,  )  matrices for v  = 0 and 1. In each of these cases, interest centered on a pair of interacting levels in isolation. In contrast, here the interest will be on the entire network of interacting levels and the simultaneous analysis of many vibrational bands. As was pointed out by Brown and Fackerell  [n], careful consideration of the matrix elements with (zv  0, LJ  0)  is required whenever the spin-orbit coupling constant A is a significant fraction of the harmonic vibrational frequency Unlike in HBr where Ae  .  ‘  the spin-orbit splitting in the 211 ground electronic  state of Hft is ‘—‘2.5 times the harmonic vibrational frequency. See Fig. 1.4. Prior to the  Chapter 1. Introduction  8  TTT+  ni  V 3  12  v  2  1 5357 cm  2  113/2  2  111/2  Figure 1.4: Vibronic energy level diagram for the 2 ground electronic state of Hft. The energy spacings are shown for J=1.5, as calculated using the constants determined here. The A-doubling and the hyperfine splittings are to small to be seen on the scale used. The energy in units of cm 1 is related to the energy in units of eV by 1 eV = 8065.54 cm . 1  Chapter 1. Introduction  9  current investigation, the spectroscopic information on HI+ molecular ion was limited to that obtained from low resolution work.  Lempka et at.  [2] concluded from their  photoelectron work on 111+ that, unlike HC1+ and HBr+, all vibrational levels of  2+  are  predissociated. It is believed that the 11 state predissociating the A 2 in Hft crosses the A 2 potential curve at a point below the v=0 level of the X  2113,2  spin substate (see  Fig. 1.2). Predissociation of this state is also most likely the reason why the A E2 2 —X 1 1 emission bands were not observed in 111+. From the extrapolation of HF+, HC1+ and HBr photoelectron data, Lempka et at. [2] estimated the frequency of the fundamental band of the  2113,2  spin substate to be --‘2170 cm. In 1977, Eland and Berkowitz  [J,  from their work with photoionization mass spectroscopy, determined the frequencies of the fundamental vibrational bands belonging to the  2113,2  and the  spin substates  2111/2  to be (2181±25) cmand (2163±35) cm, respectively. In 1990, Baltzer et at.  [4)  recorded the proton emission spectrum from dissociation of ionized HI and determined the vibrational frequency of the fundamental band to be 0.20 eV (‘.‘1600 cm). They have also claimed to observe the v=2 and v=3 vibrational levels, though with comparatively low precision. In the past two years, there have been several reports on the determination of the vibrational frequencies of the 2 X 1 1 state of Hft using photoelectron techniques. In 1992, Böwering et at. the X  211  []  determined the effective average of  We  and  WeXe  values for  state to be (2252.7±2.2) cm and (54.7±5.2) cm , respectively. They also 1  determined the equilibrium spin-orbit constant Ae to be —(5358.1±4.3) cm . Values 1 were obtained for the anharmonic constants oA and  A, 3 ,1  but with less precision. In  July 1994, Zietkiewicz et at. [24] reported separate measurements for the the  2111,2  spin substates. For  2113,2,  2113,2  and  it was found that w(2356.3±54.4) cm’ and  wx(97.9+8.7) cm* The corresponding values obtained for the  2111,2  spin substate  were (2457.8±138.7) cm and (103.3±37.4) cm. Also in 1994, Yencha et at. [25] used Penning ionization electron spectrometry and determined the vibrational frequencies of  Chapter 1. Introduction  the fundamental bands of  10  2ll  and  spin substates to be (2170 ± 16) cm’ and  211,  , respectively, along with the frequencies of the first few hot bands. As 1 (2178 + 32) cm is clear, the determination of vibrational frequencies of the ground electronic state of 111+ have shown a large variation in its value over the past few years. The first goal for the current work on 111+ is to observe the vibration-rotation spectra of the fundamental as well as the hot bands of the 2 X 1 1 ground electronic state in order to determine precisely the vibrational and the rotational constants. The second goal is to observe the hyperfine structure in the transitions belonging to both the spin substates. The hyperfine constants can then be determined for the first time for this molecular ion. The remainder of this thesis is divided into five chapters. In Chapter 2, the general theoretical background is summarized. In Chapter 3, experimental methods are discussed along with various modulation and detection techniques for studying ions. Chapter 4 deals with the hyperfine problem of HBr and Hft, including expressions for hyperfine matrix elements in the e/f labelled basis  [],  and the interpretation of the results in  terms of the electronic structure. Chapter 5 deals with the fine structure problem. Here the measurements of HBr+ are analyzed in terms of two different models. In the first model (here called ‘merged’), the usual formulation of the effective Hamiltonian [Z7] for a single vibronic state in the limit  I’4e1  <<  We  is applied. A second model, here called  ‘split’ is introduced. The matrix elements off-diagonal in v are separated into two groups, one with (z1  =  0) and the other with (LS  =  +1). The latter required particular at  tention; they are treated with the vibrational Van Vleck transformation  [2S, 29] carried  to third order. The results for the two models are compared with special attention paid to the A-doubling parameters. In order to calculate the parameters that characterize the (v  0; /M  0) coupling, detailed consideration had to be given to the vibra  tional dependence of the molecular parameters. The equilibrium values for the principal  Chapter 1. Introduction  11  molecular parameters were determined. In order to analyze the vibration-rotation mea surements of 111+, the merged model was found to be adequate. Using this model, the equilibrium values for the molecular parameters were determined. The effective vibra tional and rotational temperatures of the ions in the discharge were estimated in the case of HBr. Chapter 6 concludes with a brief discussion of the implications of the current fine structure study.  Chapter 2 General Theoretical Considerations  2.1  Molecular Energy Levels and Transitions The energy eigenvalues E of the molecular systems to be studied here are solutions  of the time-independent Schrödinger equation, ?t  Ii’)  = E  Iv’)  (2.1)  ,  where 7-is is the Hamiltonian describing the molecule, and ib) represents the eigenvector corresponding to E. According to the Born-Oppenheimer approximation  [n],  which  suggests that since the electrons are much lighter than the nuclei, they move faster and adjust instantaneously to the configuration of the nuclei, the total wavefunction is factored into an electronic part, a vibrational part and a rotational part: b) =  ?iL’e)  ) 7 R1’) b  .  (2.2)  Since Eq. (2.2) implies dividing up the Hamiltonian: (2.3) it follows that the solution of the time-independent Schrödinger equation gives the total energy E of the molecule (excluding nuclear spin interactions and terms that couple different electronic states), represented as EEe+Ev+Er. 12  (2.4)  Chapter 2.  General Theoretical Considerations  13  E  r Figure 2.1: A schematic representation of a bound electronic state. The separation of the electronic states, LEe is much larger than the separation of the vibrational levels, which is, in turn, larger than the separation of the rotational states, /.Er. Typically, LEe >>  >> Er, i.e., the separation between different electronic states  is much greater than the separation of the vibrational states which is, in turn, larger than the separation of neighbouring rotational states. This is shown schematically in Fig. 2.1 where the ordinate and the abscissa represent the potential energy and the internuclear separation, respectively, of the molecule. Eq. (2.4) in conventional spectroscopic notation  [LI  is written as E = Te + G(v) + F(J)  ,  (2.5)  where Te is the electronic term energy, G(v) is the vibrational energy of the state labelled by the quantum number v, and F(J) is the rotational energy of the state labelled by the quantum number J. The subscript v in the rotational energy term takes into account the coupling between the vibrational and rotational motions of the nuclei. In general,  Chapter 2. General Theoretical Considerations  14  the vibrational energy of the molecule can be expressed as [1] 12  1  G(v)  +weYe(V+)  +weze(+)  .  (2.6)  The rotational energy can be written as F(J)  =  BVJ(J  + 1)  —  DVJ ( 2 J  + 1)2 +  HVJ ( 3 J  + i) +•  ,  (2.7)  with (2.8) and (2.9) The signs of the coefficients in the expansions given by Eqs. (2.8) and (2.9) follow the usual convention [.1], although the symbols used in some cases do not. Time-dependent perturbation theory  [i1  is used to describe the interaction of radia  tion with matter producing transitions between the above-mentioned energy levels. Let us consider a transition from a lower state rn) to an upper state  I)•  The transition is  induced via the interaction operator fli, given by =  where  _..,i,.  E(t)  (2.10)  ,  is the molecular electric dipole moment and E(t) is the electric vector of radiation  of frequency v. For simplicity, consideration is confined to plane polarized light with field along the Z-axis. There are two basic criteria which must be met in order for this interaction to produce a transition between the two states  Irn)  and n). First, assuming  that the width of the energy levels of the molecule are infinitely narrow, the energy of the  Chapter 2.  General Theoretical Considerations  15  radiation hv, must equal the energy separation of the two levels involved in the transition [LI, i.e., LE  EnEm  =  hi),  =  (2.11)  where h is the Planck’s constant. Second, the transition moment Rnm, must be non-zero [1], i.e., Rnm  (n  =  t  Im)  0  (2.12)  .  Since the electric field E(t) is defined in a laboratory or space-fixed axis system and it is more useful to define project  L  j.t  in terms of a molecule-fixed axis system, it is necessary to  into the molecule fixed system. We consider only one component of  j.t  with  respect to the space-fixed axis system (since the radiation is chosen to be plane polarized). This component can thus be written as UZ 1  =  (2.13)  Zgttg, g=x,y,z  where ,uz is the projection of the dipole moment along the space-fixed Z axis, zg is the direction cosine which relates the space-fixed Z axis to the x, y and z principal inertial axes of the molecule, and 1 9 is the projection of the molecular dipole moment on to the u g principal inertial axis. Eq. (2.13) can be substituted in Eq. (2.12) to give Rnm  (I  zgItgm)  Under the Born-Oppenheimer approximation Rnm = [Lg  (flel (nI (T1r  g=x,y,z  [Qi,  (2.14)  .  Rnm becomes  IZgI-Lg  me) Imv)  TTir)  .  depends on the bond lengths, and hence on the vibrational normal coordinates  is convenient to expanded =  in a Taylor series in +  ()Q  +  Q  (2.15)  Q.  It  [1]: +...,  (2.16)  Chapter 2. General Theoretical Considerations  where  16  represents the component of the dipole moment operator at equilibrium along  )  the molecule fixed g axis, and  is the  nh  derivative of the g component of the  dipole moment operator with respect to the normal coordinate rium. Retaining terms to first order in  Q  = (e( (nI (rI  g=T,y,z  Zg  +  () Qj  In general, p operates on the electronic variables, nate (vibrations), and  ‘IZg  evaluated at equilib  and substituting Eq. 2.16 into Eq. 2.15, we  obtain Rnm  Q  (me) m) Imr). (2.17) o Q operates on the normal coordi  operates on the rotational variables. The evaluation of Rnm  depends on which type of transition one considers. 1. Rotational transitions (n) = m) and n) = m)  Rnm  (flel (nI (flrl  =  (nIt I’e) (71 nv) (n IIzg(mr)  (4(  (Tir  Ifle>  (nt)  (2.18)  (mr)  (2.19)  Ie) (flr IzgI mr)  (2.20)  =  tie)  for the electronic state of interest and have non-zero  ’Zg( Tflr). 1  2. Vibrational transitions  Rnm  (n( i  4’zgpg°  g—_x,y,z  Imr)).  0 the molecule must possess a permanent electric dipole moment  represented by (e matrix elements  Ir)  =  = Thus for Rnm  but  ,  ((fle)  =  me),  (fle (flv (r  (e  but (ny)  >  g=x,y,z  ()  zg  (flr  ()  Ie) (n Q  The vibrational transition will occur if both non-zero, and the matrix elements  Imv)).  (flel  Zg mr)  0  m)  Q  n) m)  (flr  ()  Imr)  IZgI mr) (ne) and (n  are non-zero.  (2.21) (2.22)  (Q( m)  are  Chapter 2. General Theoretical Considerations  3. Electronic transitions Rnm  (In€) =  (flel (r’ (flr (fle! t  The term (flel  17  g=x,y,z  4)ZgLLgO  me) mu)  (2.23)  mr)  lme) (n Im) (flr IZgI mr)  (2.24)  me) is called the electronic transition moment, nm• The vibra  tional overlap integral (n m) term is called the Franck-Condon factor. Because the two vibrational states n) and m) belong to different electronic states, the Franck-Condon factor can be non-zero even if n,, 2.2  m.  Labelling of Electronic States  An infinite number of electronic potentials are possible depending upon the config urations of the valence electrons. These potentials can be bound or unbound; the latter are called dissociative and the molecules excited into these potentials break apart into the constituent atoms. When several molecular electronic states are known, the letters X, A, B, b, c,  ...  are conventionally used to label them  [1].  ...  and a,  The electronic state lowest in energy  (also called the ground electronic state) is traditionally called X. In itself, this labelling is insufficient to distinguish all the states and determine the allowed transitions from one electronic state to another. One must consider the symmetry properties of the electronic energy levels. A diatomic molecule, visualized as a dumb-bell, possesses cylindrical symmetry; i.e., there is a symmetry axis along the line joining the two nuclei. Therefore, the electric fields in the molecule are symmetric about the internuclear axis. This leads to a precession of the total orbital angular momentum L of the electrons about the internuclear axis. Only the projection L (with eigenvalue ML) of the orbital angular momentum on the axis  Chapter 2. General Theoretical Considerations  18  is here conserved, and we classify the electron terms of the molecules according to the values of this projection. The absolute value of the projected orbital angular momentum along the axis of the molecule is costumarily denoted by the letter A [i 2.]; it takes the By analogy with the atomic spectroscopy, the electronic states are  values 0, 1, 2  labelled according to:  LorA  0  1  2  3  4  Atomic Label  S  P  D  F  G  Molecular Label  E  U  CI’  F  where L=0, 1, 2,  —p  ...  and A=0, 1, 2,  ...  refer to the atomic and molecular labels, respec  tively. Similarly, the total electronic angular momentum S, is the vector sum of the spins of the individual electrons. The corresponding quantum number S is integral or half-integral depending upon whether the total number of electrons in the molecule is even or odd. S is coupled to the internuclear axis by the magnetic field generated from the orbital motion of the electrons. S precesses about the internuclear axis and its projection along the internuclear axis S (with eigenvalues Ms) is conserved. By convention, E  =  Ms,  which can take 2S—}-1 different values. The quantity 2 in symmetrized basis (see Sec. 2.4) is defined as =  A-f-E  .  (In signed (unsymmetrized) basis, the quantum number  (2.25) is defined as  =  A +  .)  Chapter 2. General Theoretical Considerations  19  Each electronic state is labelled with its electron multiplicity, 2S+1. Thus, in general, an electronic state is denoted by the symbolic representation  [1]:  S+lA 2  (2.26)  This labelling is sufficient for most commonly encountered singlet, doublet and triplet electronic states. S > A, the  However, for quartet or higher multiplicity electronic states where  label may be insufficient to distinguish all of the different components.  For example, in a ll state, A + >  =  5/2, 3/2, 1/2 and —1/2 labels the four distinct  components. Using Eq. (2.26) to denote the states leads to the same label for the =1/2 and the 1=—1/2 components, even though they are distinct. Therefore, in such cases it is preferable to represent the levels by (2.27)  S+lA 2  Strictly speaking, the labellings represented by Eqs. (2.26) and (2.27) apply to electronic states for which Hund’s case (a) coupling (see later in this section) is a good approxima tion. For states where other coupling cases dominate, modifications are introduced for their labelling depending on the quantum numbers used to represent such states. Beside rotations through any angle about the internuclear axis, the symmetry of the molecule allows also a reflection in any plane passing through the axis. consider cases where A  Let us first  0. By performing such a reflection, the energy of the molecule  is unchanged. The state obtained from the reflection may not be completely identical with the initial state. On reflection in a plane passing through the axis of the molecule, the sign of the orbital angular momentum about this axis is changed  [n].  Thus, all  electron terms with non-zero values of A are doubly degenerate: to each value of the energy, there corresponds two states which differ in the direction of the projection of the orbital angular momentum on the axis of the molecule. Now, let us consider the  Chapter 2. General Theoretical Considerations  20  case where A=0. Except probably for a phase factor, the state of the molecule is not changed at all on reflection, so that each  electronic state must be non-degenerate.  The wavefunction of the E state can only be multiplied by a constant as a result of the reflection. Since a double reflection in the same plane is an identity transformation, this constant is +1. Thus, we must distinguish the E electronic states whose wavefunctions are unaltered on reflection from those whose wavefunctions change sign. The former are denoted by E, and the latter by E. Other symmetry operations should be considered to complete the labelling of the electronic states if the molecule consists of two similar atoms (isotopomers). See Ref.  [LI, for example.  Since in this thesis the molecules studied  are all heteronuclear diatoms, we will not consider symmetry operations of this type. 2.3  Coupling of Angular Momenta: Hund’s Coupling Cases Other than the electron orbital L and the spin S angular momenta, diatomic mol  ecules possess the nuclear end-over-end rotation R and nuclear spin I angular momenta. There are various schemes for coupling L, S and R together to form the total angular momentum J exclusive of nuclear spin. In general, J=R+L+S.  (2.28)  In zeroth-order approximation, when relativistic effects are entirely neglected, the en ergy of the molecule is independent of the direction of the spin, resulting in a (2S+ 1)-fold degeneracy of the levels [4j. When relativistic effects are taken into account, however, the degenerate levels are split, and the energy consequently becomes a function of the projection of the spin on the axis of the molecule. The chief part in this is played by the interaction of the spins with the orbital motion of the electrons. The nature and classification of molecular levels depend, to a large extent, on the relative parts played by the interaction of the spin with the orbital motion, on the one  Chapter 2. General Theoretical Considerations  21  hand, and by the rotation of the molecule, on the other. The part played by the rotation of the molecule is characterized by the energy differences between adjacent rotational levels. As such, two limiting cases can be considered. First, the energy splitting due to the spin-orbit interaction is large compared to the energy differences between the rotational levels, and second, when it is small. The first case is usually called Hund’s case (a) type coupling and the second is called Huild’s case (b) type coupling, after F. Hund who first classified different coupling schemes for angular momenta [1, 32, 34]. Apart from case (a) and case (b) type of couplings, various other coupling cases exist, depending upon the manner in which the various momenta are coupled together. In most of this thesis work, the electronic states studied are treated with the case (a) coupling and that case (b) coupling is only treated briefly. Therefore, only coupling cases (a) and (b) will be presented here. The nuclear spin will be neglected at this time. The coupling of the nuclear spin with other angular momenta relevant to this work will be included later when the hyperfine interactions are considered. 2.3.1  Hund’s Case (a)  Hund’s coupling case (a) is characterized by both L and S being strongly coupled to the internuclear axis. Thus, the operators L and S (the projections of the electron orbital and the spin angular momenta on the internuclear axis, respectively) have well defined quantum numbers A and E. A vector by A + E  (= )  and whose direction lies along the internuclear axis with origin at the  center of mass of the molecule. (See Fig. 2.2.) angular momenta The quantity  is defined whose magnitude is given  adds vectorially to R to form J. The  and R precess around J, which is itself fixed in space. is integral if the molecule contains an even number of electrons, but it  is half integral if the number of electrons is odd. As is clear, the total angular momentum quantum number J cannot be less than its projection f on the internuclear axis so that  Chapter 2. General Theoretical Considerations  22  3 R  z  A  :  Figure 2.2: Vector diagram of Hund’s coupling case (a). The electron orbital angular momentum L and the S precess rapidly around the molecular axis, making their projec tions A and good quantum numbers. (see text) precesses more slowly around the total angular momentum J. The unshaded circles indicate the precessional motions of L and S.  J has values f, 12 + 1, 12+2,  ....  If the electronic state is a  split components (called the spin substates) are the lowest possible rotational level in the whereas the  2113/2  2111,2  2111,2  211  state, the two spin-orbit  state and the  2113,2  state. The  spin substate will then be the Jz=1/2 level,  spin substate will have J= 3/2 as the lowest rotational level. The  energy levels in this coupling case are labelled by the quantum numbers J, A,  and 12,  and the basis functions are written as ,;A S ;12 J)  ,  (2.29)  where  is used as a label for the rest of the quantum numbers that have been omitted.  2.3.2  Hund’s Case (b)  In Hund’s case (b) the electron spin is coupled more strongly to the vector resultant N of the component of the orbital angular momentum L and the angular momentum  Chapter 2. General Theoretical Considerations  23  S  C I-’  Figure 2.3: Vector diagram of Hund’s coupling case (b). The electron orbital angular momentum L precesses rapidly around the molecular axis. The component of L on molecular axis adds to R to form the intermediate angular momentum N. N and S add to form J. The unshaded circles indicate their precessional motions. arising from the end-over-end rotation of the molecule R, than to the molecular axis. L, however, is still strongly coupled to the molecular axis. The appropriate vector diagram for this case is shown in Fig. 2.3. The electron orbital angular momentum L precesses rapidly around the internuclear axis. L, the component of L along the internuclear axis (with quantum number A), and R are coupled to form the intermediate angular momentum N. N and S add to form the total angular momentum (exclusive of nuclear spin) J. In general, the electron spin is usually coupled to the axis by spin-orbit coupling, i.e., the spin is coupled to L rather than to the axis itself. Hence, for molecules with A=O, coupling between the spin and the molecular axis is small. Such molecules with A=O typically fall in Hund’s case (b). When A=O, N=R, and is normal to the molecular axis. Sometimes, particularly for light molecules, even if A  0  0, S may be only weakly  Chapter 2. General Theoretical Considerations  24  coupled to the internuclear axis. Such molecules usually fall approximately into the case (b) coupling scheme. For the case when A  0, the intermediate angular momentum quantum number N  can take the values (with R=zO, 1, 2, 3,...)  N  =  A,A+1,A+2,•••  (Note that N and R here are referred to as K and N, respectively, in Eq. (V,14) of Ref.  [i])  When the spin-rotation interaction is included, there is a splitting of each term  into 2S+1 terms in general (or 2N+1 terms if N < S), which differ in the value of the total angular momentum quantum number J. (Remember, J excludes the nuclear spin.) According to the general rule for the addition of angular momenta, the quantum number  J takes (for a given N) values from N + S to IN  N—SI  J  —  SI:  N+S  The energy levels in this coupling case are labelled by the quantum numbers N, J and A, and the basis functions are written as  lv’; A S N; J) where, similar to case (a),  i’  ,  (2.30)  is used as a label for the rest of the quantum numbers that  have been omitted. Cases intermediate between (a) and (b) are also possible. It must also be borne in mind that the same electron state can pass continuously from case (a) to case (b) as the rotational quantum number J increases. This is due to the fact that the separation between adjacent rotational levels increase with the rotational quantum number, and hence, for higher J, the separation can become large compared to the energy of the separation of multiplets due to the spin-orbit interaction. As a result of this, the spin  Chapter 2. General Theoretical Considerations  25  gets uncoupled from the molecular axis, and its coupling with the molecular rotation R becomes more important. 2.4  Symmetry Properties of Basis Functions In order to classify basis functions as belonging to either even or odd parity, one  needs to consider various symmetry operations. A detailed explanation of such symmetry operations is given in Ref. [j,  J.  The two fundamental types of transformation are the rotation of the molecule through a definite angle about some axis and the reflection of the molecule in some plane. If the molecule is unaltered on rotation through an angle 2K/n about some axis, then that axis is said to be an axis of symmetry of the  order. The rotation is symbolically denoted  by C. Similarly, if the molecule is left unaltered by a reflection in some plane, this plane is said to be a plane of symmetry. The reflection is denoted by the symbol a-. We consider the case (a) basis function given by Eq. (2.29). Such a basis function is usually called the signed basis function. Here, the electronic basis functions for given values of (A, >, 1) are degenerate with their counterparts obtained by simultaneously reversing the signs of A, E and  .  In order to construct functions with a well-defined  value of , it is necessary to combine functions with specific signed values A and example, a  2113,2  .  For  electronic state is doubly degenerate; the A=+1 and —1 components  lead to =+l and —1 signed basis functions, respectively. These two a-components, degenerate in energy for the non-rotating molecule, are not properly symmetrized eigen functions of the total Hamiltonian fl. Only linear combinations of these + functions have well-defined symmetry (see later in this section). This symmetry is called parity, and o, (which corresponds to a reflection through the molecule-fixed xz plane) is the operator used to classify basis functions belonging to either even or odd parity. The  Chapter 2. General Theoretical Considerations  26  eigenvalues of u, are +1 and —1 and correspond, respectively, to even and odd total parity. A brief but comprehensive description of the use of u, operation to obtain sym metrized basis functions is outlined in Ref.  [n].  Following the phase convention used in  [n], the o, operation leads to the following transformation [33]:  Ref.  IA>  =  (a)  Orbital  (b)  Spin  o., IS, ) =  (_i)5_E  IS, —E)  (2.32)  (c)  Rotation :  o  IJ, 1)  (_i)J_  J, —11)  (2.33)  where s=1 for a E  :  o  (1) —A)  (2.31)  state and s=0 for all other states. This power of s for a E state  appears in the orbital part to classify the A=0 states according to their intrinsic E or symmetry. Thus, a c operation on a case (a) basis function (see Sec. 2.3.1) gives  E  u  IA S ; f J)  where we have dropped  i  =  (_ ) 1 5_J_s  —A S —E;  —  J)  ,  (2.34)  from the wavefunction for convenience. Eq. (2.34) can be  slightly simplified to read:  o IA S E; Q J) For a ‘E state (A  =  0, 5  =  oj0  =  (_i)J_  I—A S —E;  J)  .  (2.35)  0), Eq. (2.35) gives: 00; 0 J)  =  (—i)  000; 0 J)  Therefore, the eigenvalue of the parity operator o, for a D  —  .  (2.36) state is (1) and for a  state is (_1).1. Thus, the sign change of the eigenfunction under the operation o  E+ state, levels with even J have a will alternate with the quantum number J. For a 1  parity and levels with odd J have a a  ‘—‘  ‘—‘  parity. For a ‘E  parity and levels with odd J have a  ‘+‘  parity.  state, levels with even J have  Chapter 2. General Theoretical Considerations  27  For states other than a ‘E state (where the electronic basis functions associated with positive and negative signs of (A, E, IZ) are degenerate), a o operation on the basis function (see Eq. (2.35)) changes the basis function itself. For example, for a  211+3/2  state, +i, +, + +, Here, the basis function +1, +, —1,  —;  —,  )  =  -f-i;  4 —1, +, (i)  +,  _;  _,  j)  (2.37)  .  ) on the left hand side and the basis function  J) on the right hand side are not identical. Hence, (_i)J_ is not  an eigenvalue and +1, +, +; +, j) is not an eigenvector of the parity operator a. Therefore, for states other than a ‘E, it is necessary to use linear combinations of basis functions to obtain well-defined parities of the energy levels. Two methods are in common use for labelling the energy levels. These are the  +1—  total parity scheme and the e/f symmetry scheme. The symmetrized basis functions in the total parity labelling scheme are given by 2 s +lA, J  =  [IA S E; Q J) ±  (_l)J_S+s  I—A S  —  E; —Q J)]  The wavefunctions with the upper and lower signs correspond to the  ‘+‘  and  (2.38)  .  ‘—‘  parity,  respectively, under the operation c, i.e., s+lA J cr 2 2S+1,  j _)  =  24 J + Aç,  =  —  +),  and  2s+lA, J  The above expressions can easily be verified by operating the right hand side of Eq. (2.38) with  Q\.  In Eq. (2.38), the basis functions  2 S +lA,  J  ±)  are the eigenfunctions of the  operator o-, with eigenvalues of +1. The symmetrized basis functions in terms of e/f labels  []  depend on whether the  electronic state in question involves an odd number of electrons or an even number of  Chapter 2.  General Theoretical Considerations  28  electrons. For states in systems with an odd number of electrons, the e/f symmetrized basis functions are given by s+lA J 2 , 0  [] [A S E; Q J) ± (_l)_s+s+4 —A S  =  —  E;  —  J)]  .  (2.39)  Here e and f correspond to upper and lower signs, respectively. The e/f symmetrized basis functions for electronic states in systems with an even number of electrons are given by  [] 2s+lA, J  =  [IA  S >; Q J) ±  (_i)_S+8  —A S  —  E; —1 J)]  .  (2.40)  A careful look at Eqs. (2.38), (2.39) and (2.40) shows that there is a one-to-one cor  respondence between the  +/— total parity of the rotational energy levels and the e/f  symmetry labels. These, for odd and even number of electrons, are given as follows: • For molecules with an odd number of electrons (half integral J values); levels with parity +(—i)—4 are e levels, and levels with parity _(_i)J_ are f levels. • For molecules with an even number of electrons (integral J values); levels with parity +(—l) are e levels, and  levels with parity _(_i)J are f levels. The e and f labels are more convenient for many applications as they remove the awkward (_i) factor from the basis functions [26]. 2.5  Allowed Transitions and Selection Rules The absorption lines studied here follow the electric dipole selection rules. All the  states we have studied and are presented in this thesis follow Hund’s case (a) coupling  Chapter 2.  General Theoretical Considerations  29  Table 2.1: Selection rules for electric dipole allowed transitions and for perturbations. (Nuclear spin effects are neglected.) Transitions Electronic  LA  =  Perturbations  0, +1  LA  =  0, ±1  zf=0,±1 LE=0 +  LSE=0 —*  Rotational  =  0, +1  =  ++-*—  =  0  +-*+;-4-—  LJ=0=e-4f LSJ=+1  4—*  S,)  LJ=0  =  e4-*e;f4-*f  e4-e;f-*f  to a very good approximation. The selection rules for heteronuclear diatomic molecule following coupling case (a), as well as the perturbation selection rules, are listed in Table 2.1. For details, the reader is referred to the monograph by Hougen  {].  In addition to the transition selection rules mentioned in Table 2.1, additional selec tion rules arise which govern the transitions from one hyperfine energy level to another. These are given by LF  =  0, ±1, where F=I+J (see Fig. 2.4 and Sec. 2.6.2) with I as the  nuclear spin. Transitions with /F ZJ transitions.  =  zS.J are usually stronger than the corresponding  Chapter 2.  2.6  General Theoretical Considerations  30  Fine and Hyperfine Structures For molecules which involve one or more unpaired electrons, the rotational structure  of its spectrum may show splittings referred to as the fine and hyperfine structure (hyper fine structure arises only in molecules with a non-zero total nuclear spin). The resultant orbital angular momentum L and the spin angular momentum S interact with each other and with the molecular rotation R, resulting in fine structures of the spectrum. If the molecule further contains at least one nucleus with non-zero nuclear spin, the nuclear spin angular momentum I couples predominantly with S and/or L to produce additional structure in the spectrum, called the magnetic hyperfine structure. The nuclear spin angular momentum can generate hyperfine structure also in the spectrum of molecules without unpaired electrons, for example, through nuclear electric quadrupole interaction (provided that I1). In this section we will discuss the largest fine and hyperfine inter actions responsible for these effects. Details of such interactions are given by numerous authors. (See, for example, Refs. 2.6.1  ].)  Fine Structure Interactions  A molecule containing more than one unpaired electron can give rise to three types of fine structure interactions. These are the spin-orbit interaction, the spin-rotation interaction and the spin-spin interaction. The spin-orbit interaction arises as a result of the magnetic interaction between the magnetic moments associated with the spin and orbital motion of the electrons. The simplest expression for this interaction is given by  [-] AL• S  ,  (2.41)  Chapter 2. General Theoretical Considerations  31  where A is called the spin-orbit constant. For diatomic molecules for which L is well defined, i.e., molecules which closely approximate Hund’s case (a), Eq. 2.41 is written as SQ 11  where L±  =  A [Ls +  =  L ± iL and S  =  (LS_ + L_S+)]  (2.42)  S, + iS are the usual raising and lowering operators in  the molecule-fixed frame. The first term gives the first-order effect, while the second term connects electronic states which differ in A by +1. The latter part is combined with other Hamiltonian operators which are linear in L± to generate a large number of second-order terms, the most important contribution of which gives rise to the A-type doubling. For molecules in a 2 E state, the first order term is missing (since L 2fl  term, which connects a 2 E state with a  =  0), and the second  state, makes the leading contribution.  The second type of interaction, called the spin-rotation interaction, arises due to the magnetic interaction between the electron spin S and the nuclear rotation R angular momenta. The Hamiltonian for such an interaction is written as [33] HSR  -yR. S,  =  (2.43)  where y is the spin-rotation constant. For molecules following Hund’s case (a) coupling, this interaction is conveniently written as HSR  =  y(J  —  L  —  S) 5,  (2.44)  .  or HSR  =7  +  (J+S_ + J_S+)J  —  [is +  (LS_ + L_S+)j  —  . 2 yS  (2.45)  In contrast to L± and S which follow normal commutation rules, J follows anomalous commutation rules. As a result, the role of the J operator is similar to those of L and S operators, i.e., J+ is a lowering operator and J_ is a raising operator. For E states  Chapter 2. General Theoretical Considerations  (A=O), the matrix elements with zE  =  LA  =  32  0 for the second term in Eq. (2.44), i.e.,  L S, are zero. Therefore, the spin-rotation Hamiltonian for 7 SR 11  =  (J 7  —  S) S .  states is written as  N. S, 7  =  (2.46)  or HSR  =7  +  (J+S_ + J_S+)]  It is to be noted that many authors define HSR  =  —  . 2 -yS  (2.47)  N S even for states where A 7  0.  The difference between the two definitions is in the 7 L• S term. This is generally small, and is absorbed into the spin-orbit term which has the same form. (See also Footnote d of Table 5.3.) The third type of interaction, called the spin-spin interaction, occurs in molecules where the total electron spin S is larger than  .  The simplest example is for  state,  where the magnetic moments of the two electrons with parallel spins interact, causing the state to split  [2i.  The conventional way of writing the spin-spin interaction Hamiltonian  in simplified form is  [] ,X  [s  —  52]  ,  (2.48)  where ) is the spin-spin constant. Since, in this thesis work, all the electronic states studied have S= (which gives rise to doublet states), we will not consider this interaction any further. 2.6.2  Hyperfine Interactions  All internal interactions involving the nuclear spin angular momentum I are called the hyperfine interactions. When a molecule contains a nucleus with a finite spin, the nuclear spin interacts with the other angular momenta, causing additional splittings in  Chapter 2. General Theoretical Considerations  33  the spectra of the molecule, which generate the hyperfine structure. In most cases, the splittings caused by such intearctions are small and are resolved oniy with spectroscopic methods of high resolution. The most commonly encountered hyperfine interactions resulting in observable split tings in the spectra of molecules are the nuclear electric quadrupole hyperfine interaction and the magnetic dipole hyperfine interaction. In the case of open-shell molecules, the dominant interaction is usually magnetic, caused by the coupling between unpaired elec trons and nuclear spins in the molecule. In the case of closed-shell molecules, the electric quadrupole hyperfine interactions, i.e., the interaction of the electric quadrupole moment of the nucleus (I >  )  with the gradient of the electric field arising from an anisotropic  electron distribution, makes the dominant contribution to the hyperfine splittings. In the following section, we give in brief the structure of these hyperfine Hamiltonians. Detailed discussion of these interactions are given in Ref.  [, ,  A. Magnetic Hyperfine Interaction The nuclear spin magnetic interaction originates from the same types of mechanisms as the electron spin interaction, and basically consists of three dominant interactions. These interactions are between the magnetic moment of a spinning nucleus with (1) the spin magnetic moments of the electrons, (2) the orbital magnetic moments of the electron, and (3) the spin magnetic moments of other nuclei. These interactions are called the nuclear spin-electron spin interaction, the nuclear spin-electron orbit interaction, and the nuclear spin-nuclear spin interaction, respectively. Because the magnetic moments of nuclei are much smaller than those of electrons, nuclear spin-nuclear spin interaction is negligible in most cases. For HBr+ and HI+ the ground electronic state arises from a single unpaired electron (or hole) in the valence shell. This unpaired electron interacts significantly only with the Br (or I) nucleus; the interaction with the hydrogen nucleus was  Chapter 2. General Theoretical Considerations  34  found to be negligible. We give in brief the form of the magnetic hyperfine Hamiltonian only for such a case. Detailed theory for such interactions are given by Frosch and Foley  [j and by various other authors  [,  3].  The nuclear spin-electron spin interaction is very similar to the classical interaction between two dipoles. Classically, the interaction between two dipoles, rated by a distance r, is given by  sepa  [2J  =  where  and  r)]  —  ,  (2.49)  () is the unit vector from the position of one dipole to the other. For a nucleus  with spin angular momentum I and magnetic moment III  =  1j-  [2J, (2.50)  gNINI.  For an electron with spin angular momentum S and magnetic moment Ps  —ge[LBS  [2], (2.51)  .  The interaction Hamiltonian can be written:  1 H  =  —gegNpBpN  15 —  3(I.r)(S.r) 5 r  (2.52)  Here g is the electron g-factor, gj is the nuclear g-factor, 1 uB is the Bohr magneton, and IN  is the nuclear magneton. The parameters ge, IB and  ,UN  are taken to be positive.  In addition, there is an interaction similar to that between a nuclear magnetic moment and an s electron in atoms, which is usually called the Fermi contact interaction  [n].  Let  us consider the atomic case for a moment. It is only an approximation that the magnetic field from a nucleus is characteristic of a point dipole. If we imagine the nucleus to have non-zero radius, then the magnetic field it generates is characteristic of a loop current. At large distances this field is indistinguishable from a field generate by a point dipole.  Chapter 2.  General Theoretical Considerations  35  Usually, no electron, except an s electron, ever approaches the nucleus sufficiently closely for the difference to be significant, but since s electrons have non-zero probability density of being at the nucleus, for them the deviation from a point dipole field must be taken into account. Furthermore, the field close to the loop does not average to zero over a spherical distribution, and so this contribution to the total magnetic interaction does not vanish. The Hamiltonian for such an interaction is usually written as HFC  = ---gegN [LB[tN (r)I 6  .  S,  [au] (2.53)  where 6(r) is the Dirac 6-function which is defined such that it picks out the square of the amplitude of the electron wavefunction for the electron at the coordinate origin (i.e., the nucleus). The average value 6(r) is usually expressed as (6(r))  =  2 I’P(O)1  [,  ,  (2.54)  which represents the probability density of finding the electron at the nucleus.  For  diatomic molecules the electric field possesses only axial symmetry and no classification of states according to a total orbital angular momentum quantum number L (as is done in atoms) can be made. Thus, there can be no strict separation of the hyperfine interaction into characteristic s and non-s forms. Nevertheless, effects similar to the atomic s-state interaction are found. These are usually much larger for u-electrons than r or other electrons. The Hamiltonian for the nuclear spin-electron orbit interaction is very similar to the electron spin-electron orbit Hamiltonian, and is expressed as gNLBLN—--. HIL = 2  [] (2.55)  In a diatomic molecule where the nuclei lie on the molecular axis, the average value of L is just Ak, where k is a unit vector along the molecular axis. Then, Eq. (2.55) gives HIL  =  (2.56)  Chapter 2. General Theoretical Considerations  36  The magnetic hyperfine Hamiltonian representing the interaction of the nuclear spin magnetic moment with the electron spin and orbital magnetic moments can thus be written as [32] Hmhf = aA I k + b I. S + c (I. k)(S k)  where  [,  ,  (2.57)  37] a  b bF  and  c  gNBLN 2  c/3  —  () (3\  (2.58)  ,  (2.59)  ,  gegN[LBLN  gegNJBpN  2 I(0)I  (2.60)  ,  /3cos O 2 —1\ 3 r  \  /  (2.61)  where 0 is the angle between the molecular axis and the radius r from the nucleus to the electron. These co-ordinates specify a position within the electron charge distribution. The averaging implied by the angular brackets  ( )  is done over the electronic space  coordinates for the state being considered (since r is not a constant in a molecule). Expression (2.57) applies to each electron in the molecule. Of course, most of the electrons occupy orbits in pairs with oppositely directed spins so that the second and third parts of Eq. (2.57) cancel out, and the orbits are usually filled so that the orbital angular momentum A of most of the electrons cancel. Expression (2.57) need then only be applied to each of the unpaired electrons whose angular momentum is not cancelled. The constant a refer only to electrons that contribute to the total orbital angular momentum. This is indicated by the subscript 1 in the average defined in Eq. (2.58). Similar subscripts have been used for the electrons contributing to the spin part of the interaction. In order to evaluate the hyperfine energy levels, we need to consider the way the nuclear spin angular momentum I couples with the other angular momenta. In our case,  Chapter 2. General Theoretical Considerations  37 I  F R  z  A  Figure 2.4: Vector diagram of Hund’s coupling case (au). The resultant angular momen tum (exclusive of nuclear spin) J couples to the nuclear spin angular momentum I to give a total angular momentum F. this is best described by the Hund’s coupling case (a,3)  [] illustrated in Fig. 2.4.  In this  coupling case, J is coupled to I to give a total angular momentum F, i.e., J + I = F,  (2.62)  so that the quantum number F takes the values F = J+I, J+I—1,  IJ—’I  .  (2.63)  Since in case (a) coupling the electron spin angular momentum S precesses about the molecular axis k, it follows that S.k=E and I.S=(I. k)(S k). The magnetic hyperfine .  Hamiltonian in case (an) can then be written [32] as Hmhf = [aA +(b+c)E] Ik  (2.64)  Chapter 2. General Theoretical Considerations  38  In case (a,), the vector model gives (for matrix elements diagonal in J) [32] (I.J)(J•k) J(J+1)  —  and J.k  =  A +  =  l.  (2.66)  Thus, the hyperfine energy due to the magnetic interaction is [aA + (b + c)E]  W  J(J  1)’  (2.67)  where =  F(F + 1)  —  J(J+ 1)  —  •  For the  2111,2  (A=1;  =  —1/2) and the  2113/2  1(1 + 1)  (2.68)  (A=1; =+1/2) spin substates, Eq. 2.67  gives ) 2 W(i/  =  h  ()  J(J  (2.69)  1)  and ) 2 / 3 W( respectively. Here h±  =  =  h  ()  J(TJJJ)  ,  (2.70)  a ± (b + c). As can be seen from the above equations, the  magnetic hyperfine splittings decrease rapidly with increasing J. So far we have considered only those types of hyperfine interactions which would be identical for the two energy levels of a A-type doublet, i.e., for the e levels and the f levels. However, for 11 states certain additional hyperfine effects can occur which are different for the two A-doubled states  [2J.  Different hyperfine structure for two members of a A  doublet arises only from electron spin-nuclear spin interactions. In the previous section  Chapter 2.  General Theoretical Considerations  39  we were concerned only with the matrix elements diagonal in A to get the hyperfine energies (see Eq. 2.67). As a matter of fact, the matrix elements of hyperfine interactions also connect states with a difference in A of ±2 (and hence with differences in f). The  []  form of this interaction Hamiltonian is given by =  where 1±  ()  I ± iIi, and S  =  O (e2iI_S_ 2 gg[js1n + e_2I+S+)  (2.71)  S ± iS,,. Here, q is a spherical coordinate, and is related  =  with the molecule fixed x, y, z axis system by x  =  r sin8 cos4 and y  r sinO sinq. This  interaction is usually written as [9] =  d  (e21I_S._  + e_2iI+S+)  (2.72)  ,  with the constant d defined by d where  ( )  =  ()  (sm 8)  gegNpBpN  (2.73)  ,  denotes the average over the electronic space coordinates due to the electron  spin part of the Hamiltonian for the state being considered. The two A-doubled states actually involve equal mixtures of states with positive and negative values of A (See Sec. 2.4). Hence, in Hund’s case (a), matrix elements connecting states with a difference in not to a  2113/2  [2J, since it  of ±1 give first order (diagonal) effects for a  2111,2  state but  state. This type of hyperfine effect is also known as the hyperfine doubling  can remove the degeneracy from A-type doublets.  )  As mentioned earlier, the constant a contains  which strictly is to be averaged 1’ over electron or electrons which provide orbital angular momentum. The constants c and d involve similar averages, but over electrons which provide spin angular momentum. Usually spin and orbital momentum involve precisely the same electrons, in which case =  and hence, the three constants a, c, and d are related by c  =  3(a—d)  .  (2.74)  Chapter 2.  General Theoretical Considerations  40  It is to be noted that in most cases the above approximation fails. Such descrepancies have been explained as due to higher order effects such as exchange correlation and core polarization effects  [, 4].  B. Electric Quadrupole Hyperfine Interaction To a first approximation, a nucleus can be considered to be a collection of protons and neutrons. The protons and neutrons have spins of spin of 0,  ,  1,  .  •.  so that a nucleus can have a total  The electric charge within the nucleus is in general not distributed  in a spherically symmetric way except for nuclei with spins of IO and  where there is  a spherical charge distribution. For a nucleus with a spin I> 1, the charge distribution  has the shape of an oblate or prolate ellipsoid whose axis of symmetry coincides with the axis of spin, i.e., the nucleus has a quadrupolar (and possibly higher orders) charge distribution. An important hyperfine interaction occurs between the quadrupole moment of a nucleus and the electric field gradient at the nucleus caused by an anisotropic electron charge distribution. This interaction is the first non-vanishing term in the multipole expansion of the electrostatic interaction between a nucleus and the electrons which gives rise to hyperfine splittings. The classical interaction of a set of nuclear charge points qj, with a set of surrounding electrons e 3 is ç-  mult:po1e  where the distance  is the separation of the  3 qj e 3 r:  jt1  ,  electron from the  nucleon. If r, is  the coordinate of the nucleon relative to the center of the nucleus, and r 3 is the distance of the electron from the center of the nucleus, thell =  .  is given by (2.76)  Chapter 2. General Theoretical Considerations  If O 3 is the angle between r and r,, then  41  can be expressed in terms the distances r  and r 3 and the angle 8 jj as = r+r  3 cos 2rr  —  (2.77)  .  Therefore, Hmuitipoie  3 [r + r q e  =  —  o]  3 cos 2rr  1/2  (2.78) 2  = Since for nuclear coordinate r  .‘  1014  ]  —1/2  (2.79)  .  m while for electron coordinate r 3  i—’  10b0  m,  <<1. Using the expansion that, for y < 1, (1  —  2xy +  =  Fi(x) 1 y  (2.80)  where Pj(x) are the Legendre polynomials, we can write Eq. (2.79) as Hmuitpoie  ()  =  Pi(cos O)  (2.81)  .  Using the properties of Legendre Polynomials, this becomes Hmuitipoie  (±)Z  =  (_1)m (47r  1)  Y(O)  (2.82)  where Ym are the spherical harmonics. It is convenient to renormalize the Lm (8, q) 21+ 1  Y  =  Cm(O, ).  (2.83)  The substitution of Eq. (2.83) in Eq. (2.82) gives Hmuitipoie =  (_i)m 1=0 m=—1  [>  [  qirCL(8ii)]  .  (2.84)  Chapter 2. General Theoretical Considerations  42  The result is a sum of scalar products of tensor operators of rank 1, one referring to the electron coordinates, and the other to the nuclear coordinates. Let us consider the different expansion terms of Eq. (2.84). For 1=0, one obtains 0 H  =  (2.85)  .  j  i  For a set of charges e 3 the potential at a point in cgs units is given by V  =  3 r  ,  (2.86)  ,  (2.87)  and for a set of charges qj the total charge is Ze  =  where Z is the atomic number and e is the proton charge. Thus, 0 H  =  ZeV  (2.88)  .  If we consider the nucleus to be composed of a continuous charge distribution with charge density p(xyz), then 0 H  =  vfnuc p(xyz) dx dy dz,  (2.89)  which is the Coulombic monopole interaction between the nucleus and the surrounding electrons. This term is independent of nuclear size or shape and is usually incorporated into the electronic Hamiltonian. For 1=1, one obtains 1 H  =  qjejr: ii  2  j  +  siT’  O sin O  sin q).]  (2.90)  Chapter 2. General Theoretical Considerations  43  Looking at the sum over the nuclear charges we find that we must evaluate  and  q r cos 9  =  e z  q, r sin 0 cos  =  ex  q r sin 9 sin qfj  =  e y  j  =  f  =  p(xyz) z dx dy dz  ,  (2.91)  p(xyz) x dx dy dz  ,  (2.92)  p(xyz) y dx dy dz  =  (2.93)  .  These terms have the dimension of a dipole moment. We see that terms which involve any integrand of the form p(xyz)g (g  =  x, y, z;  n  =  odd integer) or any other odd  function of g will be zero when integrated over symmetric limits. The zero character of this 1=1 term means physically that the nucleus does not possess an electric dipole moment. For 1=2, five terms appear initially from the Eq. (2.84), given by 2 H  qejr 3 [ j 2e. n 9 2 j  =  (2cos 9  +  —  sin o)  e2i_)  (2cos o  —  +  cOs  0 sin 9  COS 0 sin  Oj  ei()  sin o)  + cos 01 sin 0 cos 0 sin 0 etj) +  0 2 sin  e_21i_i)]  (2.94)  .  In order to get some insight into the physical significance, let us examine the middle term arising from  1  C (6, qj) C (O, q). r 3 e 1 q  (2cos2o,  =  —  2o) (2cos20 3  E 1  qiejr  —  (3cos2o  sin28)  —  i) (3co820,  {qir(3cos2oi_1)} x Since r co.s 0 3  =  zj, the second factor in  {}  —  {ei  i) (3c0821)} .  (2.95)  gives the expression for the classical electric  field gradient tensor component in the z-direction:  (34)  =  _8yz)  0 —VE  ,  (2.96)  Chapter 2. General Theoretical Considerations  44  where E(xyz) and V(xyz), respectively, are the electric field and the electric potential due to the extra-nuclear charges. The first factor in  {}  on the right hand side of Eq. (2.95)  is basically a summation over the charges in the nucleus, denoted by qr (3cos2  —  q (3z  i)  Looking at this term we see that it will be non-zero if  —  r)  2 3z  Qo , 2 r  .  (2.97)  i.e., if the nucleus is not  spherically symmetric. Qo is a component of the nuclear quadrupole moment tensor and is a measure of the deviation of the nuclear charge distribution from spherical symmetry. The energy of interaction due to terms of the type (yE ) x Qo is the nuclear quadrupole 0 interaction energy. For nuclei with I 1, i.e., those with a quadrupole moment, the quadrupole Hamil tonian as derived from Eq. (2.84) is  HQ  [411 —eT ( 2 Q) : T (VE) 2  =  ,  (2.98)  qjr , ( 2 C q) 6  (2.99)  where  (Q) 2 eT  =  is the quadrupole tensor with e as the proton charge, and (VE) 2 T  =  —  (0 2 C , 3 3 e  qj)  3 r  (2.100)  is the electric field gradient tensor. By convention, the nuclear electric quadrupole mo ment operator is defined as  []  Q  =  -qjr(3cos2Oj_l)  .  (2.101)  The corresponding quantum mechanical observable, called the nuclear quadrupole mo ment, is defined again by convention as  Q  =  (I, m  =  IQI, mj  =  I),  (2.102)  Chapter 2. General Theoretical Considerations  45  where, regarding the z axis as that of spin quantization,  I is the state where the  ml =  component of I along z axis is a maximum. It can be easily shown that the quadrupole tensor operator is related to the conventional definition (see Eq. (2.97)) by T(Q)  =  eQ  (2.103)  ,  which gives  e(I, m  =  IT(Q)I, m 1  =  i)  =  eQ.  (2.104)  Similarly, the field gradient coupling constant is defined as [32]  qo where  =  (j, mj  0 (c)  mj  =  3 yje r  (3cos2O  —  (2.105)  ,  i)  (2.106)  .  Therefore (see Eq. (2.96))  T(VE)  qo  (2.107)  .  The experimentally measurable parameter is the quadrupole coupling constant given as  eQqo, with qo as the electric field gradient along the molecule-fixed z-axis (term arising from the T tensor component connecting states with /.iA=0). In a  2  electronic state,  matrix elements arising from the T 2 tensor components (which can connect states with LA  =  +2) are non-zero, and hence give rise to another measurable parameter given by  , with q as the electric field gradient normal to the molecule-fixed z-axis. 2 eQq  Chapter 3 General Experimental Considerations  3.1  Experimental Method  The molecular ions studied in this work are all produced in an alternating current (a.c.) glow discharge. Glow discharges produce weakly ionized plasmas and have many distinctive regions. For more descriptive information on glow discharges, the reader is referred to the Ph.D. thesis of W. C. Ho formed in a.c.  [4],  glow discharges are usually i0  and references therein. Molecular ions —  106 times less abundant than neutral  species. For example, in a discharge of a gas mixture of hydrogen and oxygen (mixing ratio of 112/02 is 8:1, total pressure is 1.8 torr), the observed current density,  j (=1/A)  is ‘.‘150 2 mA/cm [4]. One can then calculate the electron density from the relation,  =  V  eAvd  (3.1)  where Ne is the total number of electrons, V is the total discharge volume, I is the discharge current, e is the magnitude of the electron charge, A is the cross-sectional area of the discharge tube and vd is the electron drift velocity. With vd  7 x 106 cm/s [44],  the electron density  iO times less than  e  is estimated to be 1.3x10” cm , which is 3  the estimated number density of the gas (6x10’ 6 cm ). Eq. (3.1) is approximately 3 correct since the electrons, being thousands of times as mobile as molecular ions, carry most of the current.  46  Chapter 3. General Experimental Considerations  47  Conventional spectrometers such as grating or Fourier transform infrared spectrome ters usually cannot detect these elusive ionic species in direct absorption (without sophis ticated modulatioll methods) because molecular ions exist in low concentration and their signals are masked by signals of neutral molecules as well as the noise in the spectrome ter. A bright source and a highly sensitive and selective detection technique are therefore needed to study these ions. A brief description of the light source and the technique used to study the ions are given in the following sections. 3.1.1  Diode Laser Spectrometer  A vital part of the experiment for the study of molecular ions was a Spectra Physics Diode laser spectrometer, built around diode laser assembly and a grating monochro mator. Because of their high power density, low beam divergence, and high spectral purity (narrow laser linewidth), tunable lasers are chosen as the infrared sources. The high power is advantageous because the signal-to-noise ratio (SNR) increases with source power. The low beam divergence allows the radiation to pass through the sample cell many times (if necessary) without much attenuation in power. This increases the ab sorption length, and hence the SNR. The narrow linewidth allows us to resolve the fine and hyperfine structure in molecular transitions. We have used a tunable semiconductor diode laser in our study of the HBr+ and 111+ molecular ions. Such a diode laser provides the best combination of coverage and resolution in the mid-infrared (330-3000 cm’) region. Commercially made lead salt diode  lasers are widely available for application in the mid-infrared. A PbSi_Se compound is formed into a p-n junction by molecular beam epitaxy. Lasing action results from the  application of a bias voltage across the p-n junction. Electrons dropping from the n type conduction band to the p-type valence band emit photons which induce stimulated emission, which in turn is amplified in a planar resonator formed by the cleaved end faces  Chapter 3. General Experimental Considerations  48  of the small (.3— .5 mm) lead salt crystal. The energy of the photons is determined by the band gap energy, which has a gain profile of 100—150 cm’ in width. Detailed information on the working principle of diode lasers can be found in Ref.  [4].  In order to tune the laser frequency, the parameters which determine the band gap are varied. To produce changes in the laser frequency greater than 150 cm , one must 1 actually change the composition of the diode itself. The only safe way to do this is to acquire a new one. Diodes are housed in the cold head of a closed cycle refrigerator which operates just above liquid helium temperature’. Coarse tuning of the laser frequency is achieved by altering the cold head temperature (10—90 K), which essentially changes the resonator length of the diode. The fine scanning of the laser frequency is done by applying a current ramp to the p-n junction (typically between 100 and 500 mA). Because the resonator length is not synchronously tuned to the maximum of the gain profile, mode hops can occur and limit the continuous single mode tunability to —1—2 cm’. Usually, in most diodes, the two nearby laser modes do not overlap in frequency, even on changing the operating conditions of the diode. A typical useful frequency coverage of a single diode in its entire gain profile is --‘15—25%. In favourable cases, this can be as high as -‘50—60%. It was quite often found that the laser operated in a multimode fashion (i.e., more than one mode appeared to lase under the same operating condition). To filter out the unwanted modes from the mode of interest, the radiation from the diode laser was col limated and directed to a grating monochromator. The monochromator also provided a coarse (0.5 cm ) absolute frequency calibration. A schematic sketch of the monochroma 1 tor is shown in Fig. 3.1. It contained an interchangeable kinematically mounted grating, adjustable slits (30—2000 nm), a short focal length (f=2.5 cm) KBr beam collimating lnfrared laser diodes are now available which operate in a temperature range from 1 These diodes are primarily designed to work with liquid nitrogen cooled cold heads.  ‘-  20  —  120 K.  Chapter 3. General Experimental Considerations  49  CM)]  0  CM M  CM-7JS MC”  •  Coflimag .  II  fl  4- From HOC *ToH()C  Laser  mCell>  \\  Lens F,sitioner  I  9I  Infrared Laser Beam  Figure 3.1: Schematics of the diode laser monochromator. CR: Diode Laser Cold Head; CM: Concave Mirror; F : Filter; FM: Flip Mirror; G : Grating; HGC : Helium Gas Compressor; M : Plane Mirror; MC : Mechanical Chopper; S : Slit  Chapter 3. General Experimental Considerations  50  lens attached to an xyz lens positioner, a calibrated sine drive with readout proportional to wavelength, a mechanical chopper, beam transfer optics, and a flip mirror (also called a ‘shorting’ mirror).  Chapter 3. General Experimental Considerations  51  Aluminum Range  Teflon flange Metal screw  Gas Inlet  Water-Cooled Jacket Seal  Water  Stainless Steel Electrode  Pyrex cup  Figure 3.2: Design of a portion of the discharge cell showing electrode assembly 3.1.2  The Discharge Cell  The discharge cell was made of a 1 m long and 0.9 cm inner bore diameter pyrex tube with a water cooled jacket over the entire length. Fig. 3.2 shows a portion of the discharge cell used in this study. The electrodes, mounted axially at either ends of the discharge cell and inside pyrex bell-shaped cups, consisted of  inch stainless steel  U-shaped tubes through which cooling water was flowed. The electrodes were held in place in a teflon flange with  inch Swagelok ferrule fittings and 0-ring seals, and were  slightly off-center so that they did not block the laser radiation. The teflon flanges were  Chapter 3. General Experimental Considerations  52  themselves sealed with 0-rings to the electrode cups. A CaF 2 window (2 mm in thickness and 2.54 cm in diameter) was glued (with a regular 5-minute epoxy) to each end of the discharge cell with the help of pyrex tubing cut at the Brewster angle. The other end of the tubing was attached to the teflon flange with an 0-ring seal. 3.1.3  General Experimental Set-up  Fig. 3.3 shows a general layout of the experimental set-up for the study of the vibration-rotation transitions in HBr+ and 111+. Four infrared semiconductor diodes (Laser Photonics) were used to cover the range from 1975 cm 1 to 2360 cm’ (but not with continuous coverage). Typical output powers of these diodes were about 0.2 mW. The radiation from the diode was collimated by using a KBr lens LN of focal length f=2.5 cm. The beam passed through a beamsplitter BS which was virtually transparent in the mid-infrared and was 50% transmitting in the visible. The beamsplitter allowed us to superimpose the beam of a HeNe laser L (\=632.8 nm) on to the infrared beam for optical alignment purposes. Further collimation of the infrared beam was achieved by inserting another concave mirror CM just before the beam entered the discharge cell DC. The beam, after passing through the discharge cell, was redirected to the monochromator M. In addition to selecting a single resonator mode, the monochromator also served as a filter for the incoherent but modulated background plasma emission generated in the discharge. The output beam from the monochromator was then focussed down to a liquid-nitrogen cooled InSb detector D with a silicon filter F in front of the detector window. The silicon filter served to filter out the direct as well as any residual discharge plasma emission emerging from the monochromator. The output signal from the detector was pre-amplified and detected using a lock-in amplifier LIA tuned to the discharge  modulation frequency (22 kllz). The lock-in amplifier output was fed to a microVax computer C for data acquisition. The same computer was also used to control and tune  Chapter 3. General Experimental Considerations  53  Mft  DLCM  PA M  ‘  PS  A  To PG To pump  Figure 3.3: A general layout of the experimental set-up A : Voltage Amplifier; BS: Infrared Beam Splitter; C : tVax Computer; CM: Concave Mirror; D : Liquid Nitrogen Cooled InSb Detector; DC : Discharge Cell; DL : Infrared Diode Laser; DLCM : Diode Laser Control Module; EA : Confocal Etalon Assembly; F : Silicon Filter; G : Gas Mixture Inlet; I : Iris; L : HeNe laser (X=632.8 nm); LIA : Lock-In Amplifier; LN : KBr Lens; M : Plane Mirror; MC : Monochromator; PA : Pre-Amplifier; PS : PlasmaLoc Power Supply; W : Cooling Water Inlet  Chapter 3. General Experimental Considerations  54  the laser frequency. Spread through much of the frequency range of interest (>2200 cm ), there were 1 strong CO 2 absorption lines. The absorption of the laser power due to these CO 2 tran sitions was, in fact, one of the major problems in our experiment. Although the amount of CO 2 present in air is typically only 0.03%, many transitions were strong enough to absorb completely the output power of the diode laser before the radiation reached the detector. To reduce the associated strong atmospheric absorption, evacuated glass tubes with CaF 2 windows were inserted between optical components where possible, and the monochromator was flushed continuously with dry nitrogen gas. Where such an arrange ment was not possible, e.g., mirrors and other beam transfer optics, the portions were enclosed with a polythene wrap and flushed with nitrogen gas. With this procedure, we were able to remove about 80% of the CO 2 content from the path of the laser radiation. In fact, the remaining 20% was still enough for the very strong CO 2 transitions to black out the laser power completely (which, unfortunately, we had to live with). In the low frequency region, strong water lines were sometimes a nuisance. Fortunately, the water lines did not create much of a problem in our experiment, since the water lines are sparse and, in most cases, did not fall into the region where we expected to see ion lines. 3.1.4  Production of Ions  The discharge cell consisted of four gas inlet ports symmetrically distributed over its entire length (see Fig. 3.3). Two of these inlets were located on the bell cups (one on each end) and were used to introduce the gas mixture. Out of the other two inlets located in between, one was used to monitor the gas pressure inside the discharge cell, and the other was not utilized. The discharge was run with a PlasmaLoc II power supply (output power s200—500 W) operating at 22 kHz. The output voltage of the PlasmaLoc was increased further with a step-up transformer (RS 816) to about 2 kV rms, which was then  Chapter 3. General Experimental Considerations  55  applied to one of the water-cooled electrodes. The other electrode was kept at ground potential. The current amplitude was monitored with an oscilloscope by measuring the voltage drop across an 8  resistor placed in series with the discharge. Under normal  operating conditions, the rms current amplitude stayed between 100—200 mA. The cell was pumped out by a Direct-Torr rotary pump. The reactant gases were pre-mixed before introduction into the discharge cell by simple diffusion in a SwageLok union-cross fitting. All gases were carried to the discharge cell through  inch Polyflo  tubings which were connected to the gas inlet ports using Cajon fittings. The pressure in the cell was monitored with a Baratron gauge, specified for 0—10 torr range. The discharge was run in a continuous gas flow mode. One important feature of the gas delivery system was the insertion of small plugs of cotton wool into the Polyffo tubing just prior to the cell. This prevented discharge from running back to the gas regulators and pressure gauge head. The discharge also had a tendency to arc down the pump line. Although this did not seem to affect the amount of ion production, it surely did affect the stability of the discharge which, in turn, created a background pick-up noise in the lock-in amplifier and made the diode laser control module behave erratically. A similar trick of placing a cotton wool plug in the path of the pump line just after the discharge cell suppressed this discharge path. A. HBr+ ions In order to make HBr+ ions, two methods were applied. 100 mtorr HBr and 5 torr He was used.  At first, a mixture of  Due to very high vapour pressure of HBr  gas at room temperature (‘-..25 atm), it was necessary to cool down the gas bottle by immersing it into a dry ice-acetone cold bath in order to make possible fine adjustments in the pressure. Later, a mixture of 50 mtorr Br , 200 mtorr 112 and 5 torr He was em 2 ployed. Commercially available bromine was transferred to a pyrex reservoir and cooled  Chapter 3. General Experimental Considerations  56  down below its freezing point by immersing the reservoir into a dewar containing liquid nitrogen. The reservoir was then pumped out with the rotary pump to get rid of most of the air trapped inside it. Bromine was then allowed to melt and the process was repeated two to three times. Since the vapour pressure of Br 2 at room temperature is ‘—2O0 torr, a water-ice mixture cold bath was used to lower the vapour pressure. It was found necessary to let the bromine reservoir sit inside the cold bath for at least an hour before starting the discharge. This procedure helped in stabilizing the bromine pressure while the discharge was in operation. Due to slightly better signal-to-noise ratio of the observed transitions and the relative ease of controlling the pressure of the individual gases, the latter method was used to record most of the signals. B. HP ions Three different methods were employed to produce HI+ ions. The obvious method to start with was a discharge with a mixture of gas. At room temperature,  ‘2  ‘2  vapour and 112, with He as the buffer  is in a crystalline form with a vapour pressure of about  250 mtorr. Since, in the llBr+ work, we required only about 50 mtorr of Br 2 gas in the discharge, this method of making HI ions was thought to be the easiest one to handle. As it turned out later, this was not the case. The room temperature vapour pressure of  ‘2  was found to be insufficient for the molecules to mix efficiently with the relatively  high pressure H 2 (‘-.200 mtorr) and He (-.‘5 torr) gases required to run a stable discharge. Also, the  ‘2  vapour tended to crystallize at the surface of the water-cooled stainless steel  electrodes, resulting in pressure fluctuations inside the discharge cell. This, in turn, made the discharge very unstable, resulting in a huge varying d.c. offset in the lock-in amplifier due to electrical pick-up. The second method used to produce Hft ions was a discharge of methyl iodide 3 (C11 1 ) vapour with He (and 112 at a later stage). This method too was later abandoned since  Chapter 3. General Experimental Considerations  57  the cell tended to get blackened after running the discharge for about 30—40 minutes due to carbon build-up. The third method, used to record all the Hft transitions in this work, was a discharge with a mixture of a-IC1 (30 mtorr), 112 (150 mtorr) and He (5.5 torr). Commercial grade ICl, obtained from Aldrich (98% purity), is crystalline at room temperature (melting point  27° C) with a vapour pressure of about 25 torr. Hence, no cold bath was required  in this case. The IC1 vapour tended to freeze and subsequently block the pressure control valve used to regulate its flow. This problem was avoided by wrapping the control valve with a heating tape maintained at a temperature of about 60°C. In many instances, the composition of the individual gases in the mixture (including the buffer gas) was found to be critical in suppressing the formation of neutrals. Detection Technique  3.2  A very sensitive way to detect weak signals is to apply a modulation technique. The modulated signal can then be demodulated using a lock-in amplifier. Various modulation techniques are available. Some of the major modulation techniques are described in the following section. 3.2.1  Modulation Techniques in Molecular Spectroscopy  Two different widely-used modulation schemes will be considered. These are here referred to as source modulation and molecular modulation. Source modulation most often involves modulating the amplitude or the frequency of the radiation being absorbed. Molecular modulation is the modulation of the absorption coefficient through molecular properties such as density, population or energy levels.  Chapter 3.  General Experimental Considerations  58  Source Modulation  With amplitude modulation, the signal is most often detected at the frequency at which the radiation changes its amplitude (i.e., intensity). In practice, the modulation is square-wave, achieved by turning the radiation on and off either mechanically (using, for example, a mechanical chopper) or electronically (using an electronic switch), or sinusoidal (using, for example, a Pockel cell and a polarizer-analyzer arrangement). Frequency modulation is achieved by superimposing a waveform (usually square, si nusoidal or triangular wave) of frequency  f  (few kllz) on the carrier frequency of the  radiation. The signal is detected at either the first harmonic (if) or the second har monic (2f), or both, depending on the modulation waveform and the experimental re quirements. Frequency modulation was used in this experiment to record absorption of reference gases as well as etalon fringes. Molecular Modulation  Molecular modulation can be achieved in various ways. The velocity of the ions can be modulated by the application of the a.c. electric field in the discharge. This results in velocity-modulated signals. Depending on the type of the molecule, the energy levels can be modulated by a.c. electric or magnetic fields, resulting in Stark- or Zeeman modulated signals. Another method is to modulate the concentration of the molecules to obtain density-modulated signals. The detection of the molecular ions in this study was done by the technique of velocity modulation. This is described briefly in the next section. 3.2.2  Velocity Modulation  Velocity modulation is a very powerful and, by far, the most widely used modulation technique developed to detect signals of ionic species in absorption spectroscopy. This  General Experimental Considerations  Chapter 3.  59  technique was developed by Gudeman, Begemann, PfafF and Saykally described in a review article by Gudeman and Saykally  [n].  [4.]  in 1983 and  The essential feature of  this technique is that it exploits the Doppler shifts of the resonant frequency of ions to discriminate them from the overwhelmingly more abundant neutral species (about a factor of 106) that exist in the plasma. In electrical discharges, the velocity of the molecular ions can be decomposed into two parts  —  one due to the random, thermal motion (thermal velocity), and the other  due to the electric field of the discharge (field drift velocity). The thermal velocity gives the absorption a Doppler linewidth whereas the field drift velocity gives rise to a Doppler shift of the frequency of the absorption of an ion along with some Doppler broadening. When an a.c. voltage signal is applied to the electrodes, the ions in the plasma oscillate at the same frequency as the input signal frequency. As the laser is scanned and approaches a ro-vibrational transition of an ion, the absorption frequency is Doppler shifted in and out of resonance with the infrared radiation. This is analogous to frequency modulation where the absorption frequency is shifted in and out of resonance by changing the source frequency in a periodic fashion. Fig. 3.4 demonstrates the basic principle of velocity modulation spectroscopy. Let us consider that ions exist in the discharge cell and the potential difference between the two electrodes is zero. Also, we will assume that the ion can make a transition from a lower energy level to an upper energy level by absorbing a photon of frequency v , which 0 corresponds to the energy difference between the two states involved in the transition. When the laser frequency is tuned through the resonance, the ion will absorb the radiation at laser frequency vj=v, where the width of the transition corresponds mostly to the Doppler width. This is shown in case (i) of Fig. 3.4. Now, when a negative voltage is applied to the ‘hot’ electrode (case (ii)), the ion (considered to be positive here) will  Chapter 3. General Experimental Considerations  60  (i)  0 V  Vi  (ii)  Vi  (iii)  Vi  A-B] Velocity Modulated Signal —*  1 V  Figure 3.4: Basic principle of the velocity modulation spectroscopy. For simplicity, we have assumed that the formation and the temporal response of the positive ions are the same as that of the a.c. electric field.  Chapter 3. General Experimental Considerations  61  drifts towards the negative electrode’. If the drift velocity of the ion is vd, then the ion will absorb the radiation at a higher frequency (vo + Si’), where Si’ is the Doppler shift  (=+).  This is opposite in case (iii), where the application of a positive voltage to  the ‘hot’ electrode results to an absorption by the ion at a lower frequency (v 0 where Si’  —  Si’),  The velocity modulated signal at if the lock-in amplifier detects,  is the difference between these two signals (resulting from case (ii) and case (iii)), and the demodulated ion signal will appear as a superimposition of two lineshapes 1800 out of phase which are separated by twice the Doppler shift. One of the primary advantages of detecting the velocity modulated signal at if is its ability to discriminate the neutral species from the ions. Any neutral species will experience no such Doppler shifting of its absorption line due to the electric field. In fact, the signals arising from neutral species cannot be suppressed totally and are somewhat modulated due to asymmetries in the power generation and amplification systems used to run the discharge. These signals tend to have slightly asymmetric lineshapes and can be easily distinguished when demodulated at 2f by the lock-in amplifier because the neutral signals are now also density modulated (at 2f) and these appear orders of magnitude stronger than the ionic absorptions. It is a coincidence that the field drift velocities of ions in a.c. glow discharges are larger than or comparable to their thermal velocities. This fact is a key to the success of velocity modulation. An added bonus using this technique is that the charge of the ion can be easily determined because the if lineshape of the negative ions will be 180° out of phase with respect to that of the positive ions. 1111 general, the velocity modulated signal will be phase shifted from the applied a.c. electric field. But, for simplicity, we will assume that the formation and the temporal response of the ions are the same as that of the electric field.  Chapter 3. General Experimental Considerations 3.3  62  Spectral Recording and Calibration Absolute frequency markers were obtained by measuring known spectra of N 0, CO, 2  , 2 C0  [48.]  as well as OCS [49] reference gases. Measurements relative to the markers were  made using the transmission fringes of a confocal etalon whose free spectral range of 0.01 cm 1 was accurately determined for the region of interest. Because only a single infrared detector was available, measurement of the HBr+ and HI+ frequencies were made by recording four sequential scans of the same frequency region. First, a glass cell containing the reference gas was placed on the path of the laser beam and the absorption spectrum was recorded using laser frequency modulation. Second, the same frequency region was scanned to record the absorption due to ions using velocity modulation. This was followed by a third scan, which was a repeat of the first scan, to check the frequency reproducibility. The final scan recorded the transmission fringes of the confocal etalon for relative calibration. In most regions, more than one reference line was measured, usually with a single calibration gas, but occasionally with two. In a few cases, only a single reference line was available, and the free spectral range for an adjacent region was used for extrapolation. In most cases, all the four sequential scans were taken when the discharge was on. This reduced the peak intensities and increased the widths of the reference lines and the etalon fringes, but did not affect the line positions. Occasionally, especially when the discharge was not very stable, the two reference scans differed in frequency by as much as 0.01 cm’. Such differences were observed as an overall shift in one of the reference scans (which can be attributed to a shift in the starting position) as well as non-linearity in one scan with respect to the other (most probably due to discharge non-uniformity). This problem was severe with one of the diodes working in the low frequency region (1975 cm 1 to 2050 1 cm ) . This was because the tunability of this diode was about 5 times higher compared to the other three diodes.  Chapter 3. General Experimental Considerations  63  (These three diodes had a typical tuning rate of 0.04 cm’/mA). As a result, a small change in the diode current as a result of electrical pick-up from the unstable discharge led to a relatively large change in the frequency of the diode output. Most of such scans were recorded several times (usually 5 to 6 times) to estimate the average frequency positions of the ion lines as well as the reference markers. In recording the spectra, two different scanning modes were used: the wide mode and the narrow mode. The estimated uncertainty e in the frequency measurements depended on which of the two different scanning modes was used to record that spectra. In the fine structure rotational study, where frequency coverage was more important, the wide mode was used: the interval scanned was  “.-  0.6 cm’ and a higher scanning rate (typ  ically 0.12 cm’/min.) was used. For the resulting absolute frequency measurements, the estimated uncertainty e was  ‘  0.003 cm , as determined from the reproducibility of 1  the two different calibration scans (see above) for each region. Occasionally small shifts between these scans (as explained previously) required that this estimate be increased to . These values of e were upper limits to systematic effects and were some 1 0.005 cm what conservative. In studies of hyperfine structure and small A-doublet splittings, the narrow mode was used: the interval scanned was  --  0.2 cm’ at a somewhat lower scan  ning rate (typically 0.04 cm’/min.). For the resulting relative frequency measurements, the estimated uncertainty e was  ‘-‘.‘  0.001 cm’ as determined from the reproducibility of  frequency spacings in the ion spectra.  Chapter 4 Hyperfine Structure in HBr and HI  Laboratory spectroscopy of molecular ions was first carried out in the 1920’s, with measurements of the optical emission spectra of N, COP, and Ot [Q, i}. With the advent of tunable lasers, it has become possible to do higher sensitivity experiments using various spectroscopic techniques. The present chapter deals with the magnetic dipole and the electric quadrupole hyperfine structure observed in the spectra of HBr+ and 111+ molecular ions using the technique of velocity modulation. The chapter is divided into four sections. In the first section, the hyperfine energy of a molecule in a 211 electronic state is discussed. The second section deals with the general features of the observed hyperfine spectrum. The third section presents the data and the analysis for HBr+ and 111+. The results and their interpretation are discussed in the last section. In some cases, a section is divided into two sub-sections, one dealing with the HBr+ ions and the other with the Hft ions. 4.1  Hyperfine Energies The  2  ground electronic state of both, flBr+ and HI+arise from a single un  paired hole, thus exhibiting properties of an open shell molecule. There is a strong magnetic dipole hyperfine interaction. In addition, the halogen nucleus provides an elec tric quadrupole hyperfine interaction. These hydrogen halide ions act in many respect as a halogen atom perturbed by a neighbouring proton  [UI.  The proton has no electric  quadrupole moment and its magnetic interactions are negligible. The general form of 64  Chapter 4. Hyperfine Structure in HBr and Hft  65  magnetic dipole and electric quadrupole interaction Hamiltonians for such a system has been discussed previously in Chapter 2. The matrix elements of the hyperfine Hamiltonian for diatomic molecules in 211 electronic states have been treated previously by various authors (see Refs. [37, 52] and references therein). We mention here the matrix elements derived by Carrington et al. [2] (for zSA=O), and by Merer  []  (for LA  +2) using the signed quantum number  basis functions, e.g., those in Eq. (2.35). We shall consider for the moment only matrix elements which are diagonal in J. The work of Refs.  [] and [] will be used to obtain the  corresponding matrix elements of the Hamiltonian with the symmetrized basis functions. The matrix elements (diagonal in J) of the magnetic dipole ilamiltonian (HMD) and the electric quadrupole Hamiltonian (HEQ) in case (au) coupling using the signed quantum number basis functions, denoted by A S E; (A S E;  J; I F IHMDI A S ;  J; I F), are given by  J I F)  4J(Ji) (4.2)  (E +  1)}]1’2  (43)  J(J + 1)} [3R(F) {R(F) + 1} 4J(J + 1)1(1 + 1)] 81(21— 1)J(J+ 1)(2J 1)(2J+3)  (4.4)  [{J(J + 1)  —  (ASE; lJ; IFHEQASE; QJ; IF) —  =  E( + 1)}]h/2 ;  [(J + )(J +  {3ç2  (4.1)  ;  =  (ASs; J; IFIHMDAS+1; +1J; IF)  eQqo  [, ]  + 1) {S(S + 1)  f(1 ± 1)} {S(S + 1)  —  —  =  —  —  Chapter 4. Hyperfine Structure in HBr and HI  66  eQq [37?(F) {??(F) + 1} 4J(J + 1)1(1 + 1)] 2 16 1(21 1)J(J + 1)(2J 1)(2J +3) —  —  [{J(J + 1)  —  —  ± 1)} {J(J + 1)  —  (1 ± 1)(1 +  2)}]hi’2  (4.5)  where 7?.(F)  =  F(F+1)—I(I+1)—J(J+1)  The constailt h in Eq. (4.1) is related to the Frosch and Foley magnetic hyperfine param eters  []  a, b, and c as h  =  aA+(b+c).  (See Chapter 2, Sec. 2.6.2A for definitions of a, b, c, and d.) The phase convention used to derive these matrix elements is that defined by Condon and Shortley [5,  J.  As mentioned earlier (see Chapter 2, Sec. 2.4), for states other than a ‘E state, it is more convenient to use a symmetrized basis function. In the total parity labelling scheme, the symmetrized basis functions of rotational levels for a  211  electronic state in  terms of signed quantum number basis functions are given by 2S+1A,  JIF  ±)  =  [lAS E;J; IF)±(—1) —AS —E;—J; IF)],(4.6)  with S=1/2. Here, as usual, A,  and ! are the eigenvalues of the components along the  internuclear axis of L, S and J, respectively. In the first function on the right hand side, can be positive or negative while A> 0 and Q  0 with  =  A + E. The functions on  the left hand side of Eq. (4.6) form a symmetrized basis. Here A and  are understood  to be positive. With the help of Eqs. (4.1) to (4.5), one can easily obtain the matrix elements of the hyperfine Hamiltonian in the symmetrized basis of Eq. (4.6):  Chapter 4. Hyperfine Structure in HBr and HI  (2111/2  J IF + IHMD  2111,2,  J IF  (2113/2,  J IF + IHMDI  2113,2,  J IF  (2113/2,  JIF +llMDI  2111,2,  JIF+)  (2111,2,  J I F + IIEQ  JIF  2111,2  {h_ +  =  +)  b  =  +)  —eQqo  h  =  67  )J+ 1 (_  d  ( )} +  2J(J+ 1)’  3 I.J 2J(J + 1)  (4.7)  (4.8)  {(+) (J_)} 1\  1/2  jj  2J(J+1)  (4.9)  =  {31.J(21•J+1)—2J(J+1)I(I+1)} 16 1(21 1)J(J + 1)  (4.10)  —  (2113,2,  JIF ±HEQ —eQqo  (2113,2,  JIF+)  2113,2  =  {J(J+1)—}{3I.J(2I.J+1)—2J(J+1)I(I+1)} 4 1(21  J I F ± IHEQ  2111,2,  2 +(—1)eQq  JIF  —  ±)  1)J(J + 1)(2J  —  1)(2J +3)  (4.11)  =  (2J+1){31•J(21.J+1)—2J(J+1)I(I+1)} 32 1(21 1)J(J + 1) {(2J 1)(2J + 3)}1/2  (4.12)  —  —  where a + (b+c),  (4.13)  and I•J  =  F(F + 1)  —  1(1 + 1) 2  —  J(J + 1)  (4.14)  Chapter 4. Hyperfine Structure in HBr and Hft  68  For computation purposes, especially when dealing with matrix elements off-diagonal in J, it is more convenient to write the matrix elements in terms of 3j/6j symbols. A review of both magnetic dipole and electric quadrupole interactions and their matrix elements in Hirota  [aZI.  (2111/2  +1—  symmetrized basis functions for Hund’s coupling case (an) is given by  Representing (HMD + HEQ) by 11 hfs’ these matrix elements are given by [nj:  J I F ± Hhf 5  2111/2,  J’ I F  ±)  1 G(JJ’IF) (_i)J_ h_  =  /‘J  2  \  +  (2113/2  J I F ± Hhf 5  2113,2,  J’ I F  +  (2113/2,  J I F ± lHhfs  2111,2,  1J’’\  I _1o  (_i)J  ±)  =  I !I  =  d —  2/  eQqo  Q(JJ’IF)  (_i)J_  (_i)J_3eQq°  J’ I F  ±  (  1  I  1 2  \2  2 J’  J —  (_ ) 1 J_  I  ±  Q(JJ’IF)  IJ  )  G(JJ’IF)  PI\  2 —2  J)  2 J’’\  IJ —  eQq  )  J’’\  )  (4.15)  ;  1  h G(JJ’IF)  Q(JJ’IF)  J’  ;  (4.16)  1J’ .  (4.17)  Chapter 4. Hyperflne Structure in HBr and HI  69  In Eqs. (4.15), (4.16) and (4.17), G(JJ’IF)  =  Q(JJ’IF)  =  )J’+I+F 1 (_  )J’+I+F 1 (_  [1(1 + 1)(21 + 1)(2J + 1)(2J’ + 1)]  I  J F  J’  I  1  [(I + 1)(21 + 1)(21 + 3)(2J + 1)(2J’ +  (4.18)  J  J’  j  F  I  2 (4.19)  These follow the phase convention of Condon and Shortley [M, symbols are defined as in Ref.  [Ml.  3J.  The 3-j and 6-j  In the above matrix elements, a, b, c, and d are the  usual Frosch and Foley hyperfine parmeters, defined in Sec. 2.6.2A of Chapter 2, and h, follows the definition of Eq. (4.13). Three differences need to be mentioned when the above matrix elements are compared to those given in Ref. [7]. First, the + sign on the left hand side of Eq. 2.3.73(b) in Ref.  [j7J  has been changed to + in the matrix element represented here by Eq (4.17), to be  consistent with onr definition of q. The phase factor definitions of G(J’JIF) and Q(J’JIF) in Ref. a second misprint in Eq. 2.3.74(b) of Ref. should read [1(21  —  1)11/2 instead of [21  —  [7]  [7].  (_i)J’+  should read  which appears in the  (_i)J+I+F.  Third, there is  The denominator on the left hand side  111/2. The matrix elements given here are then  consistent with those one would obtain in the  +1—  symmetrized basis using the matrix  elements in the signed quantum number basis as given by Carrington et al.  [52]  and by  Merer [j]. From some points of view, it is easier to analyze hyperfine spectra using the  +/— sym  metrized scheme. However, the e/f scheme is considerably more convenient for vibration rotation and fine structure analyses. Hence, we decided to study both parts, i.e., the hyperfine structure as well as the vibrational-rotation and fine structure, using the e/f  Chapter 4. Hyperfine Structure in HBr and Hft  70  symmetrized scheme. In terms of e/f labelling, the symmetrized basis function of rotatio nal levels for a  211  electronic state in terms of signed quantum number basis functions is  given by , JIF) 2 2S+lAc  =  [IAS ;J; IF)±I—AS —;—J; IF)]  (4.20)  The matrix elements can then be easily obtained in the symmetrized basis labelled in the e/f scheme. These are given by (2111,2,  JIFflHhfs  2111,2  1  (_1)r G(JJ’IF)  J’IF)  Ij I  h  +  (2113,2  JIF  Hhfs  2113,2  ij’\  0  (_i)J  ±  d V’  Il  I  Q(JJ’IF)  (_ ) 1 J_  J’ I F  +  I  —1  (  J  JIF  Hhfs  2111,2,  J’ I F  ±  4 (_l)J_  )  I  J  ;  (4.21)  i J)  h G(JJ’IF)  aeQq° (_1)r Q(JJ’IF)  =  2 J’  —  —  (2113,2,  J’  2 J’’\  )  0  (4.22)  G(JJ’IF)  /1 ‘) 2 ieQq (_i)r Q(JJ’IF) I 4v6 1—2 \2  TI ,  (4.23)  2  where the definitions of G(JJ’IF) and Q(JJ’IF) are the same as given by Eqs. (4.18) and  Chapter 4. Hyperfine Structure in HBr and Hft  (4.19). In the above matrix elements, to LJ  =  0,+2 and to  =  ±1.  71  ( HI1 ) and (. Hhfs ) apply, respectively,  In general, the hyperfine parameters a, b, c and d involve a sum over all the electrons. However, if the hydrogen halide molecular ion can be considered to be a halogen atom perturbed by a proton, then the electronic structure can be viewed as approaching that of a closed shell atom with a single unpaired ‘hole’. In a case such as this, each of the summations over electrons reduces to a single term associated with the hole, and the expressions given in Eq. (2.58) to (2.61) in Chapter 2 should form a good approximation. Of course, higher order effects (such as exchange correlation  [4]) exist.  The resulting  corrections enter differently for the parameter a sensitive to the electron orbital angular momentum and for parameters b, c, and d sensitive to the electron spin. These differences are absorbed into the averages  ( )  entering in Eqs. (2.58) to (2.61) of Chapter 2. To  indicate this, the subscripts 1 and s have been added to the angular brackets. Fig. 4.1 shows the calculated hyperfine energy level patterns in the (v  =  1, J  =  0.5) and (v  =  2, J  =  2111,2  state for  1.5). The approximate energy spacings were calculated  using the constants determined in this work. The respective splittings of 2.02 cm and 3.97 cm’ shown in the central column are due to A-doubling. The left and the right columns of the Fig. 4.1 illustrate the e and f hyperfine patterns, respectively. The latter shows much wider spacing because of the hyperfine doubling due to the terms in d in Eq. (4.21). Notice also that the ordering of the F levels is reversed in going from the f to e A-doublets. The allowed hyperfine transitions with zF  =  O,±1  are shown; the hyperfine spectra for R(0.5)ee and R(O.5)ff are clearly very different.  Chapter 4. Hyperfine Structure in HBr and Hft  72  F F  v=2 J=1.5  3  0 1 2  2  3  1 0  2  1 2  0.220 cm’  1 Figure 4.1: The calculated hyperfine splittings for the rovibrational levels (v=1,J=O.5) and (v=2,J=1.5) in the 2111/2 spin substate of HBr. In the central column, the A-doubling alone is shown. On the left and right, the hyperfine splittings for T = e and f, respectively, are shown on a much expanded scale. The (v=2) levels are about 2278 cm’ above the (v=1) levels. The arrows indicate the allowed hyperfine transitions.  Chapter 4. Hyperfine Structure in HBr and Hft  73  General Features of the Spectrum  4.2 4.2.1  Hyperfine Structure in HBr  Between 1975 cm 1 and 2360 cm’, about 300 vibration-rotation transitions were Br+ 7 9 and HSlBr+. These belonged to the five measured for each of the isotopomers H vibrational bands with (v’ the spin substates  2111/2  —  and  v”)  =  2113/2.  (1  —  0), (2—1), (3—2), (4—3) and (5—4) of both  Here the single prime and the double prime label  the upper and lower levels, respectively. The richness of the spectrum is increased by the presence of two major bromine isotopes, 79 Br and 81 Br, each occurring with about 50% natural abundance. The low J lines of all the vibrational bands studied in this work show characteristic hyperfine splittings. These splittings arise in large part due to the magnetic dipole interaction and to a minor extent due to the electric quadrupole interaction. Because of the presence of two isotopomers, the hyperfine patterns appeared in pairs, with partners being separated from each other by approximately 0.3 cm . 1 Fig. 4.2 shows the hyperfine splittings for the  2113,2  Q(1.5) transition in the  (2—1) band of Br. 79 The A-doublets are not resolved; the (e’ H components for each (F’  —  —  f”) and (f’  —  e”)  F”) hyperfine transitions appear to be superimposed. The  spectrum was taken in the narrow mode (see Chapter 3, Sec. 3.3); the interval scanned was 0.24 cm . The three possible transitions with zF 1 the six possible transitions with LF  =  0 form the central line, while  0 form the satellites. The structure is symmetric  about the central line. The spacing between neighbouring lines is almost constant. The slight deviation from equal spacing is due to the bromine electric quadrupole interaction. lll Br+ isotopomer is 0.3226 cm’ lower The corresponding hyperfine pattern due to the 8 in frequency, and hence lies outside the range of this figure.  Chapter 4. Hyperfine Structure in HBr and HI  2258.73  2258.79  74  2258.85  2258.91  (cn1)  Figure 4.2: The hyperfine structure of the Q(l.5) transition in the (2’ 1”) vibrational band for the 2113,2 spin substate of Br. 79 The A-doubling between the (e’ f”) and H (f’ e”) components is too small to be resolved. The positions of the individual (F’ F”) hyperfine components are indicated. These positions are symmetric about the central line. The spectrum was recorded in 3 minutes with a time constant of 1 s and a discharge Br+ isotopomer is 1 power of 350 W. The corresponding hyperfine structure due to the HS -.0.323 cm’ lower in frequency, and is not shown here. —  —  —  —  Chapter 4. Hyperfine Structure in HBr and Hft  4.2.2  75  Hyperfine Structure in HP  Between 1995 cm and 2245 cm, more than 100 vibration-rotation transitions were measured. These belonged to the fundamental and the first two hot bands of the 2113/2  spin substate, and to the fundamental of the  2111/2  spin substate. As in HBr+,  all the low J lines studied in this work show characteristic hyperfine splittings. Unlike HBr, where the splittings arise mainly due to the magnetic dipole interaction, both the magnetic dipole interaction and the electric quadrupole interaction play major roles in the hyperfine splittings in Hft. This is shown in Fig. 4.3 which shows the observed hyperfine splittings (lower trace) for the  211,  Q(1.5) transition in the (1—0) band in  HI+. The A-doublets are too close to be resolved; the (e’ for each (F’  —  —  f”) and (f’  —  e”) components  F”) hyperfine transitions appear to be superimposed. The spectrum was  taken in the narrow mode (see Chapter 3, Sec. 3.3); the interval scanned was 0.24 cm’. As in HBr, the three possible transitions with LF six possible transitions with LF  =  0 form the central line, while the  0 form the satellites. The structure is symmetric  about the central line. As can be seen in this case, the spacings between neighbouring lines are very different from each other. This deviation from equal spacing is due to the iodine electric quadrupole interaction which is much larger than to that of bromine in HBr. The spectra display an interesting similarity between the P and R branches. If one assumes that the hyperfine constants do not change significantly for two consecutive vibrational levels (as is indeed the case in HBr and Hft) and further that the splittillg of the rotational levels due to A-doubling is negligible, then the hyperfine structure of the R(J) and P(J+1) transitions are expected to show similar splittings, except that the overall pattern is reversed in going from one case to the other. This occurs because the upper and the lower J involved in the R(J) vibration-rotation transition are same but  Chapter 4. Hyperfine Structure in HBr and Hft  2118.208  76  2118.308  2118.408  Frequency (cm ) 1 Figure 4.3: The calculated (upper trace) and the observed (lower trace) hyperfine struc ture of the Q(1.5) transition in the (1’ 0”) vibrational band for the 2113/2 spin substate of flI. The A-doubling between the (e’ f”) and (f’ e”) components in the observed spectrum is too small to be resolved. The positions of the individual (F’ F”) hyperfine components are indicated. These positions are symmetric about the central line. The spectrum was recorded in 3 minutes with a time constant of 1 s and a discharge power of 400 W. The calculated spectrum was obtained using the constants determined in this work. —  —  —  —  Chapter 4. Hyperfine Structure in HBr and Hft  77  opposite to those involved in the corresponding P(J+1) transition. An example of this is demonstrated in Fig. 4.4. Here, the observed hyperfine structures of the R(2.5) and the P(3.5) vibration-rotation trasitions belonging to the fundamental vibrational band of the  2113/2  spin substate of Hft are shown. They appear centered at 2159.8155 cm 1  and 2073.8401 cm , respectively. In each case, the A-doublets are not resolved; the 1 (e’  —  e”) and (f’  —  f”) components for each (F’  —  F”) hyperfine transitions appear to be  superimposed. Data and Analysis  4.3 4.3.1  HBr  Although hyperfine effects were observed in the recorded spectra to fairly high J (‘.‘lO.S), not all lines were used in the analysis of the hyperfine data. In the 2111/2  2113/2  and  spin substates, respectively, a total of 34 and 23 hyperfine transitions were entered  Br+, the corresponding numbers were 35 and 22. 1 79 For HS H into the data set for Br+. Lines were included from the P,  Q  and R branches. For each  2  113/2  transition, the two  A-doublets were not resolved in any of the hyperfine studies. The transitions for higher J values showed a slight broadening of the hyperfine lines due to A-doubling. However, since the broadening did not shift the centers of the lines, the relative spacings were not affected. Each unresolved doublet was counted as a single line in the hyperfine analysis. For each  2111/2  transition, the A-doublets were well separated and each was entered  individually into the data set. Each of the lines used in the analysis had good signal-to noise (20 in most cases) and was clearly resolved (except for the A-doubling mentioned above). The lines which were overlapped with other vibration-rotation transitions or blended with the neighbouring hyperfine components were not used to determine the hyperfine constants.  Chapter 4. Hyperfine Structure in HBr and Hft  78  Figure 4.4: The observed hyperfine structures of the R(2.5) (upper trace) and the P(3.5) (lower trace) vibration-rotation transitions in the (1’ 0”) vibrational band for the 211 spin substate of HIP, centered at 2159.8155 cm 1 and 2073.8401 cm , respectively. The 1 A-doubling between the (e’ e”) and (f’ f”) components in the observed spectra are not resolved. If the order of the hyperfine transitions in the lower trace is reversed while maintaining the frequency separations, the resufting pattern is very similar to that in the upper trace. —  —  —  Chapter 4. Hyperfine Structure in HBr and HI  79  In order to carry out the hyperfine analysis, preliminary values of the fine structure and rotational parameters were used. The resulting hyperfine analysis yielded hyperfine free frequencies for each vibration-rotation transition involved in the hyperfine study. These hyperfine-free frequencies were then used to refine the treatment of the fine struc ture/rotational problem. The process was then iterated. It converged rapidly. In the hyperfine analysis, the data for each isotopomer were studied separately. For discussion purposes, each vibration-rotation multiplet studied will be labelled (v”,k), where k is an integer running from one to the total number  of multiplets (including  both Q spin substates) measured for lower vibrational state v”. For each (v”,k), one hyperfine component was selected (usually the strongest) as a reference. Its displacement from the hyperfine-free frequency is denoted  6vl,k.  The data for each (v”,k) was entered  as a set of frequency differences for each hyperfine component from the reference line. The difference entered for the reference line itself was zero, of course. The data for Br+ 7 H 9 and flSlBr+ are shown in Table 4.1 and Table 4.2, respectively. A weighted least squares analysis was carried out. The estimated weights used are listed in Tables 4.1 and 4.2. Initially, the data for each v” were fit separately by varying a, (b+c), d and eQqo, as well as the  different frequency offsets Sv,,k. No significant  vibrational dependence was detected in the hyperfine parameters. Consequently, a global fit was performed to all the hyperfine transitions belonging to the different vibrational bands (and both  spin substates). The resulting values for the hyperfine parameters  are listed in Table 4.3. The associated differences between the observed and calculated frequencies are given in Table 4.1 and 4.2. The standard error of the fit (for both the isotopomers) was 0.0004 cm . Since this is less than the typical experimental uncertainty 1 0.001 cm , the fit is regarded as good. The values obtained for 1 determine the hyperfine-free frequencies.  were used to  Chapter 4. Hyperfine Structure in HBr and Hft  80  Table 4.1: Wavenumbers of the Observed Hyperfine Splittingsa in the X 79 H State of Br Branchb (1—0) P  J,  Electronic  Obs_Cald, e  F’  J”  r”  F”  Wtc  Obsd  1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 3.5 3.5 3.5  e e e e f f f f f f f f f  1 2 2 3 2 1 3 2 4 2 2 3 4  2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.0 1.0 1.0 1.0 1.0  0.0 14.5 24.4 53.5 0.0 67.5 120.3 219.2 0.0 35.1 0.0 16.4 37.4  —0.3 0.0 0.0 0.4 0.5 0.0 —0.3 —0.2 —0.1 0.1 0.6 0.0 —0.6  e e e e f f f f f  1/2 1/2 1/2 1/2  0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.5 1.5 2.5 2.5 2.5  f f f f  1 2 1 2 1 1 2 2 3 2 2 3 4  Q  1/2 1/2 1/2 1/2 1/2  0.5 0.5 0.5 0.5 0.5  e e f f f  2 1 1 2 2  0.5 0.5 0.5 0.5 0.5  f f e e e  2 2 2 1 2  1.0 1.0 0.8 0.8 0.8  —10.2 0.0 0.0 210.5 220.6  —0.2 0.2 —0.4 0.1 0.3  (2—1) R  1/2 1/2  1.5 1.5  f f  2 3  0.5 0.5  f f  2 2  1.0 1.0  —99.3 0.0  —0.5 0.0  (2—1)  1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2  T  211  a Only those transitions are listed which were well resolved with no or negligible overlap with the nearby lines. b Values in parentheses refer to vibrational quantum numbers (v’ v”). —  C  Relative weight used in the fit.  d Entries in this column are in the units of iO cm . 1 e The standard error of the fit is 0.0004 cm 1  Chapter 4. Hyperfine Structure in HBr and HI  Table 4.1 Branchb  81  Continued  —  J’  r’  F’  J”  TI’  F”  Wtc  Obsd  Obs_Cald, e  (2—1) R  1/2 1/2 1/2  1.5 1.5 1.5  f f f  0 1 2  0.5 0.5 0.5  f f f  1 1 1  1.0 1.0 1.0  18.2 53.8 121.4  —0.3 0.6 0.2  (1—0) P  3/2 3/2 3/2  e/f e/f e/f e/f  1 2 3  2.5 2.5 2.5 2.5  e/f e/f e/f e/f  2 3 4  1.0 1.0 1.0  0.0 13.7 41.5  3/2  1.5 1.5 1.5 1.5  3  1.0  77.5  0.4 0.2 —0.3 —0.4  3/2 3/2 3/2 3/2 3/2  1.5 1.5 1.5 1.5 1.5  e/f e/f e/f e/f e/f  2 1 3 2  f/e f/e f/e f/e f/e  3 2 3 1 2  1.0 1.0 3.0 1.0 1.0  0.0 23.5 65.4 106.4 128.9  —0.6 —0.2 0.4 0.1  3  1.5 1.5 1.5 1.5 1.5  3/2 3/2 3/2 3/2 3/2  1.5 1.5 1.5 1.5 1.5  e/f e/f e/f e/f e/f  2 1 3 2 3  1.5 1.5 1.5 1.5 1.5  f/e f/e f/e f/e f/e  3 2 3 1 2  1.0 1.0 3.0 1.0 1.0  0.0 23.3 64.2  0.1 0.4 0.1  105.1 128.4  —0.5 —0.2  3/2 3/2 3/2 3/2  2.5 2.5 2.5 2.5  e/f e/f e/f e/f  3 4  e/f e/f e/f e/f  3 3 2 1  1.0 1.0 1.0 1.0  0.0 36.8 64.9 79.1  —0.5 0.2  3 2  1.5 1.5 1.5 1.5  3/2 3/2 3/2 3/2  1.5 1.5 1.5 1.5  e/f e/f c/f c/f  2 1 0 3  1.5 1.5 1.5 1.5  f/e f/c f/c f/c  3 2 1 3  1.0 1.0 1.0 3.0  0.0 22.5 44.2 64.2  0.2 —0.4 0.2 0.0  (1—0)  (2—1)  Q  Q  (2—1) R  (3—2)  Q  3  —0.5  0.0 0.3  Chapter 4. Hyperfine Structure in HBr and Hft  Table 4.1 Branchb  —  82  Continued  J’  r’  F’  J”  r”  F”  Wtc  Obsd  Obs_Cald, e  (3—2)  Q  3/2 3/2 3/2  1.5 1.5 1.5  e/f e/f e/f  1 2 3  1.5 1.5 1.5  f/e f/e f/e  0 1 2  1.0 1.0 1.0  84.8 105.3 128.6  0.4 —0.2 0.0  (4—3)  Q  3/2 3/2 3/2 3/2 3/2  1.5 1.5 1.5 1.5 1.5  e/f e/f e/f e/f e/f  2 1 3 2 3  1.5 1.5 1.5 1.5 1.5  f/e f/e f/e f/e f/e  3 2 3 1 2  1.0 1.0 3.0 1.0 1.0  0.0 23.5 64.4 106.3 128.6  —0.1 0.3 —0.1 0.5 —0.3  (4—3) R 3/2 3/2 3/2 3/2  2.5 2.5 2.5 2.5  e/f e/f e/f e/f  3 4 3 2  1.5 1.5 1.5 1.5  e/f e/f e/f e/f  3 3 2 1  1.0 1.0 1.0 1.0  0.0 36.8 65.1 79.2  —0.6 0.1 0.1 0.4  Chapter 4. Hyperfine Structure in HBr and HI  83  Table 4.2: Wavenumbers of the Observed Hyperfine Splittings a in the X 211 Electronic 81 H State of Br Branchb  Obs_Cald, e  J’  r’  F’  J”  r”  F”  Wtc  Obsd  1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2  0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 2.5 2.5 2.5  e e e e f f f f f f f f  1 2 1 2 1 1 2 2 2 2  e e e e f f f f f f f f  1 2 2 3 2 1 3 2 1 2 3 4  2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.0 1.0 1.0  0.0 16.0 26.8 57.9 0.0 73.7 130.6 237.4 310.0  3 4  1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 3.5 3.5 3.5  0.0 18.2 41.1  —0.4 —0.1 0.1 0.4 —0.1 0.5 —0.1 0.1 —0.4 0.4 0.2 —0.5  Q  1/2 1/2 1/2 1/2 1/2 1/2  0.5 0.5 0.5 0.5 0.5 0.5  e e e f f f  2 1 2 2 1 2  0.5 0.5 0.5 0.5 0.5 0.5  f f f e e e  2 2 1 2 2 1  1.0 1.0 1.0 1.0 1.0 1.0  0.0 10.9 238.1 236.4 0.0 226.0  —0.4 —0.1 0.5 —0.3 0.4 —0.1  (2—1) R  1/2 1/2  1.5 1.5  f f  2 3  0.5 0.5  f f  2 2  1.0 1.0  0.0 107.2  —0.6 0.2  (1—0) P  (2—1)  a Only those transitions are listed which were well resolved with no or negligible overlap with the nearby lines. b Values in parentheses refer to vibrational quantum numbers (v’ —  c Relative weight used in the fit. d Entries in this column are in the units of 10 cm’. e The standard error of the fit is 0.0004 cm’.  Chapter 4. Hyperfine Structure in HBr and HI  Table 4.2 Branchb  —  84  Continued  J’  r’  F’  J”  r”  F”  Wtc  Obsd  Obs_Cald, e  (2—1) R  1/2 1/2  1.5 1.5  f f  1 2  0.5 0.5  f f  1 1  1.0 1.0  165.3 237.6  0.7 —0.1  (1—0)  Q  3/2 3/2 3/2 3/2 3/2 3/2 3/2  1.5 1.5 1.5 1.5 1.5 1.5 1.5  e/f e/f e/f e/f e/f e/f e/f  2 1 0 3 1 2 3  1.5 1.5 1.5 1.5 1.5 1.5 1.5  f/e f/e f/e f/e f/e f/e f/e  3 2 1 3 0 1 2  1.0 1.0 1.0 3.0 1.0 1.0 1.0  —68.3 —43.7 —21.5 0.0 22.4 45.0 69.4  0.5 0.6 0.1 —0.4 0.1 —0.1 —0.1  (2—1)  Q  3/2 3/2 3/2 3/2 3/2 3/2 3/2  1.5 1.5 1.5 1.5 1.5 1.5 1.5  e/f e/f e/f e/f e/f e/f e/f  2 1 0 3 1 2  f/e f/e f/e f/e f/e f/e f/e  3 2 1 3 0 1 2  1.0 1.0 1.0 3.0 1.0 1.0 1.0  —69.2 —44.6 —22.0 0.0 21.8 44.8 69.1  —0.1 0.1  3  1.5 1.5 1.5 1.5 1.5 1.5 1.5  (2—1) R 3/2 3/2 3/2 3/2  2.5 2.5 2.5 2.5  e/f e/f e/f e/f  3 4 3 2  1.5 1.5 1.5 1.5  e/f e/f e/f e/f  3 3 2 1  1.0 1.0 1.0 1.0  0.0 39.1 69.0 84.4  0.0 0.1 —0.2 0.1  (3—2) P  3/2 3/2 3/2  1.5 1.5 1.5  e/f e/f c/f  1 2 3  2.5 2.5 2.5  e/f c/f c/f  2 3 4  1.0 1.0 1.0  0.0 16.1 46.2  —0.6 0.3 0.3  3/2 3/2  1.5 1.5  c/f c/f  2 1  1.5 1.5  f/c f/c  3 2  1.0 1.0  —69.0 —44.4  0.0 0.2  (3—2)  Q  0.0 0.0 —0.1 0.1 0.0  Chapter 4. Hyperfine Structure in HBr and HI  Table 4.2 Branchb  —  85  Continued  J’  r’  F’  J”  r”  F”  Wtc  Obsd  Obs_Cald, e  (3—2)  Q  3/2 3/2 3/2 3/2 3/2  1.5 1.5 1.5 1.5 1.5  c/f c/f e/f c/f c/f  0 3 1 2 3  1.5 1.5 1.5 1.5 1.5  f/e f/e f/e f/e f/c  1 3 0 1 2  1.0 3.0 1.0 1.0 1.0  —21.9 0.0 22.0 44.7 70.0  —0.1 —0.2 —0.1 —0.2 0.7  (4—3)  Q  3/2 3/2 3/2  1.5 1.5  c/f c/f  f/c f/c f/c  1.0 1.0  —69.1 —45.0  0.1 —0.2  c/f  1.5 1.5 1.5  3 2  1.5  2 1 3  3  3.0  0.0  0.0  3/2 3/2 3/2 3/2  2.5 2.5 2.5 2.5  c/f c/f c/f c/f  3 4 3 2  1.5 1.5 1.5 1.5  c/f c/f c/f c/f  3 3 2 1  1.0 1.0 1.0 1.0  0.0 38.4 68.1 84.9  0.0 0.1 —0.2 0.1  (4—3) R  Chapter 4. Hyperfine Structure in HBr and HI  86  Table 4.3: Fitted and Derived Hyperfine Interaction Constantsa, b for HBr in the X 211 Electronic State Br+ 7 H 9  Br+ 8 H 1  Ratio c  65.81( 8)  70.92( 8)  —26.16(17)  —28.34(16)  1.0776(17) d 1.0833(93) d  52.73( 9)  56.75(10)  1.0762(26) d  52.350(31) e  56.485(27) e  1.07899(82) d  h_  78.89(14)  85.09(12)  )d 2 ( 10786 4  d  86.12(16)  92.80(14)  bf  1.0776(25) d  20.8(27)  29.6(23)  1.42(22) d  g  —47.0(28)  —57.9(23)  Parameter a b+c h  eQqo 2 eQq  4.77(38) )C 9 (i 20 s. )e 50 .i( 32  1.230(87) d  4.06(39)  0.85(11) h  )C 4 . 4 1 ( 4 6  0.854(44)11  )e 4 ( 6 . 22 7  0.70(18)  a Two equivalent models were used, one involving a and (b+c) and the other involving h and h_. b Except for the ratios, all the entries are in iO cm’. Quoted uncertainties in parentheses are one standard deviation and apply to the last digits of the values. C  Brj. 79 / parameter (H 1.0779358(2), from Ref. [].  Br+) 81 This refers to parameter(H  d For comparison,  =  e This value is obtained form Ref. [11].  f  This was held fixed at the value obtained in Ref. [111.  g This was derived from the values of (b+ c) and b. 11 For comparison,  =  0.8353822(10) in atomic bromine, from Ref.  [].  Chapter 4. Hyperfine Structure in HBr and HI  87  In these calculations, the matrix elements with LQ  =  +1 and those with J  =  +1 were treated by a simple numerical procedure. Initially, Hhfs was treated by first order perturbation theory. Once good approximations to the hyperfine constants were obtained, the frequencies were re-calculated from the eigenvalues of the 6 x 6 matrices which included the LSJ=O,±1 and the Q=O,±1 matrix elements formed using the values  [ii].  of b and eQq 2 determined by Lubic et al.  The differences between the perturbation  and diagonalization frequencies were then used to refine the hyperfine parameters. The effect of matrix elements with J  =  +2 due to eQqo was negligible.  As can be seen from Eqs. (4.13), (4.21) and (4.22), the fitting can be done by replacing a and (b+c) with the pair of parameters h  =  a + (b + c). A second fit using this  replacement was carried out. The resulting values for h+ and h_ are given in Table 4.3. The fit using h and h_ was essentially identical to that using a and (b+c) with regard to all the other parameters and the calculated frequencies. Although blended spectra were not used to determine the hyperfine constants, the stronger blended spectra were synthesized to check consistency with the results in Ta ble 4.3. Fig. 4.5 illustrates such a check for two vibration-rotation multiplets, namely the ( 1 )ee, if multiplet in the in the  2111/2  (2  —  2113/2  (1  —  81 and the R(l.5)ee multiplet H 0) band of Br  1) band of H Br. 7 9 The two fall in adjacent frequency regions; each  is partially resolved. In the stick spectrum for the  2113/2  multiplet, the average position  of the two unresolved A-doublets is shown for each hyperfine transition. However, the A-doubling is taken into account in the simulated spectrum. The lineshape assigned to each individual transition was constructed by summing two Gaussian curves differing in phase by ir and separated in frequency by the Doppler effect. The width of these Gaus sians and their Doppler splitting were adjusted to match the simulated spectrum to that observed. As can be seen from Fig. 4.5, the agreement is very good.  Chapter 4. Hyperfine Structure in HBr and HI  2291.92  2291.97  88  2292.02  2292.97  Frequency (cm ) 1 Figure 4.5: The hyperfine structure in two adjacent vibration-rotation transitions for HBr. The top plot shows the calculated stick spectrum. For the 2113/2 transitions, the A-doubling is not resolved; each stick is placed at the average frequency for the (e’ e”) and (f’ f”) components. The relative intensity of the two militiplets was estimated using the vibrational and rotational temperatures obtained in this work; see Sec. 5.7 of Chapter 5. The middle plot shows the simulated spectrum obtained by convolving the stick spectrum with a suitable lineshape as described in the text. The bottom plot shows the observed spectrum recorded in 150 s with a time constant of 1 s and a discharge power of 300 W. —  —  Chapter 4. Hyperfine Structure in HBr and Hft 4.3.2  89  HP  In the  2113/2  and  2111,2  spin substates, respectively, a total of 41 and 17 hyperfine  transitions were entered into the data set. Lines were included from the P, branches in the  2113,2  spin substate, and from the P and R branches in the  Q  and R  2111,2  spin  substate. The observed transitions in 111+ were much weaker compared to those of HBr+ (by at least an order of magnitude). Thus, some of the lines in the analysis had signalto-noise ratio of as low as 5, although the ratio was usually 15. Most of the hyperfine transitions used in the analysis were recorded at least three times. Hyperfine spacings measured from each of these recordings were then averaged and entered into the data set. As was the case in HBr+, for each  2113,2  transition, the two A-doublets were not resolved  in any of the hyperfine studies. They were counted as a single line in the hyperfine analysis. A procedure similar to that adopted in HBr was used in the analysis of the hyperfine data. Preliminary values of the fine structure and rotational parameters were used in hyperfine fitting. The resulting hyperfine analysis yielded hyperfine-free frequencies for each vibration-rotation transition involved in the hyperfine study, which were then used to refine the treatment of the fine structure/rotational problem. The process was then iterated until it converged. The hyperfine data for Hft along with the estimated weights and the differences between the observed and calculated values from the fit are listed in Table 4.4. The standard error of the fit was 0.0004 cm , which is well within the typical experimental 1 uncertainty  . Thus, the fit is regarded as good. The value of the constant 1 0.001 cm  b, which enters only as a higher order term in the hyperfine energy expression, was not determined. Unlike in HBr where the value of b is known (although with a large uncertainty) from laser magnetic resonance experiments on pure rotational transitions,  Chapter 4. Hyperfine Structure in HBr and Hft  90  Table 4.4: Wavenumbers of the Observed Hyperfine Splittings a in the X 211 Electronic State of HI Obs_Cald, e  Branchb  1  J’  r’  F’  J”  r”  F”  Wtc  Obsd  (1—0) P  3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2  1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.5  e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f  1 2 3 2 4 4  1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00  —81.2 —62.0 —33.8 —20.1 0.0 44.9 —23.9 —13.6 0.0 11.7 31.0  —0.1 0.0 —0.3 —0.6 0.4 0.5 0.0 —0.2 0.2  3.5 3.5  e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f  2 3 4 1  3 4 5 4 5  2.5 2.5 2.5 2.5 2.5 2.5 3.5 3.5 3.5  3/2 3/2 3/2 3/2 3/2 3/2 3/2  1.5 1.5 1.5 1.5 1.5 1.5 1.5  e/f e/f e/f e/f e/f e/f e/f  3 2 1 4 2 3 4  1.5 1.5 1.5 1.5 1.5 1.5 1.5  f/e f/e f/e f/e f/e f/e f/e  4 3 2 4 1 2  1.00 1.00 1.00 3.00 1.00 1.00 1.00  —77.6 —63.1 —44.5 0.0 44.7 63.3 78.0  0.3 0.3 0.1 0.0 0.0 —0.3 0.0  3/2 3/2  2.5 2.5  e/f e/f  4 2  1.5 1.5  e/f e/f  4  1.00 1.00  —44.4 —27.5  0.0 —0.4  (1—0)  Q  (1—0) R  5 4 4 5 6 4 5  3 3  0.6 —0.5  a Only those transitions are listed which were well resolved with no or negligible overlap with the nearby lines. b Values in parentheses refer to vibrational quantum numbers (v’ —  C  Relative weight used in the fit.  d Entries in this column are in the units of iO cm . 1 e The standard error of the fit is 0.0004 cm . 1  Chapter 4. Hyperfine Structure in HBr and flJ  Table 4.4 Branchb  Q  (1—0) R 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2  91  Continued  —  Obs_Cald, e  J’  r’  F’  J”  r”  F”  Wt’  Obsd  2.5 2.5  e/f e/f  0.0 19.4  3 2 5 4 6 5 4  e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f  1.00 1.00  e/f e/f e/f e/f e/f e/f e/f e/f  1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.5  4 2  2.5 2.5 2.5 3.5 3.5 3.5 3.5 3.5  5 1 4  3 2 1 5 4 5 4 3  1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00  34.1 62.0 81.4 —32.1 —12.2 0.0 13.2 23.9  —0.4 —0.1 0.6 0.0 0.3 —0.2 —0.7 0.2 0.2 0.5  (1—0) P  1/2 1/2 1/2 1/2 1/2  2.5 2.5 2.5 2.5 2.5  f f f f f  4 5 2 3 4  3.5 3.5 3.5 3.5 3.5  f f f f f  5 6 2 3 4  1.00 1.00 0.04 0.04 0.04  —17.4 0.0 13.8 34.1 60.4  0.4 —0.6 2.4 2.2 —0.1  (1—0) R  1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2  2.5 2.5 2.5 2.5 2.5 2.5  f f f f f f f f f f f f  4 3 5 1 4  1.5 1.5 1.5 1.5 1.5 1.5 2.5 2.5 2.5 2.5 2.5 2.5  f f f f f f f f f f f f  4 3 4 1 3 2  1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00  —116.0 —46.0 0.0 37.1 43.8 69.8 —97.5 —59.6 —31.6 —10.9 0.0 18.1  0.1 —0.6 0.0 0.3 0.0 0.3 0.2 0.3 —0.3 0.0 0.0 —0.3  3.5 3.5 3.5 3.5 3.5 3.5  3 5 4 3 2 6 5  5 4 3 2 5 4  Chapter 4. Hyperfine Structure in HBr and Hft  Table 4.4  —  92  Continued  Branchb  !  J’  r’  F’  J”  r”  F”  Wtc  Obsd  ObsCal’’ e  (2—1) P  3/2 3/2 3/2 3/2  2.5 2.5 2.5 2.5  e/f e/f e/f e/f  3 4 5 4  3.5 3.5 3.5 3.5  e/f e/f e/f e/f  4 5 6 4  1.00 1.00 1.00 1.00  —23.5 —13.0 0.0 11.3  0.0 0.1 0.0 0.0  3/2 3/2 3/2 3/2 3/2 3/2 3/2  1.5 1.5 1.5 1.5 1.5 1.5 1.5  e/f e/f e/f e/f e/f e/f e/f  3 2 1 4 2 3 4  1.5 1.5 1.5 1.5 1.5 1.5 1.5  f/e f/e f/e f/e f/e f/e f/e  4 3 2 4 1 2 3  1.00 1.00 1.00 3.00 1.00 1.00 1.00  —78.0 —63.7 —45.1 0.0 44.8 63.3 77.2  0.2 0.0 —0.2 0.2  (2—1)  Q  0.3 0.0 —0.6  Chapter 4. Hyperfine Structure in HBr and HI  93  this value is not known for 111+. Hence, we fixed b at zero and did not attempt to include the off-diagonal matrix elements in to our fit. Table 4.5 lists the hyperfine constants obtained for HIP. As in HBr, the blended spectra were not used to determine the hyperfine constants. These spectra were synthe sized to check consistency with the results in Table 4.5. Fig. 4.6 illustrates such a check for the partially hyperfine-resolved (1—0)  2113/2  .R( 2 S)ee  vibration-rotation transition.  In the stick diagram, the average position of the two unresolved A-doublet is shown for each hyperfine transition. The middle trace represents the hyperfine pattern simulated using the constants determined in this work. The A-doubling is taken into account in the simulated spectrum. The observed spectrum is shown in the bottom trace. As can be seen from this figure, the agreement between the simulated and the observed spec trum is very good. Fig. 4.3 shows a similar comparison for the (1—0) 2113/2 15 Q( )ef,fe vibration-rotation transition. In this case as well, the simulated spectrum (top trace) is a very good match to the observed hyperfine pattern. Results and Interpretation  4.4 4.4.1  HBr  The information now available on the hyperfine constants of HBr+ is summarized in Table 4.3. In addition to the current determinations, there are listed the results of Lubic et al. [ii] obtained in an LMR study of the paramagnetic 2113,2 spin substate. The present measurements of h+ and eQqo are in reasonable agreement with the more precisely determined values of Lubic and co-workers. The two measurements of h+ differ by 0.7%, an amount which is indeed ‘-.‘3 times the present error limit. However, in view of the differences in the two experiments and methods of analysis, a disagreement of this order is not felt to be serious. Because of the high precision inherent in the LMR method,  Chapter 4. Hyperfine Structure in HBr and Hft  94  Table 4.5: Fitted Hyperfine Interaction Constants a and Derived Averages over Functions of the Electron Space Coordinates for Hft in the X 211 Electronic State parameterb a  Value (10—s cm’) 72.47(48)  b+c  —42.71(96)  h  51.12( 7)  h_  93.83(96)  d  99.45(51)  eQqo  Parameter  () ( s  —23.77(57)  Value  (As)  136.95(84) 124.55(60)  a Quoted uncertainties in parentheses are one standard deviation and apply to the last digits of the values. b Two equivalent models were used, one involving a and (b+c) and the other involving h and h_.  Chapter 4. Hyperfine Structure in HBr and HI  2159.76  2159.80 2159.84 Frequency (cm’)  95  2159.88  Figure 4.6: The hyperfine structure in (1—0) 2113/2 R( .5)ee, of HIP. The top plot shows 2 the calculated stick spectrum. The A-doubling is not resolved; each stick is placed at the average frequency for the (e’ e”) and (f’ f”) components. The middle trace shows the simulated spectrum obtained by convolving the stick spectrum with a suitable lineshape as described in the text. The bottom trace shows the observed spectrum recorded in 150 s with a time constant of 1 s and a discharge power of 350 W. —  —  Chapter 4. Hyperfine Structure in HBr and Hft  96  this earlier study also yielded values of eQq 2 and b, although the error estimates for these were somewhat larger. Both of these constants are difficult to determine because they enter only through the  (z  +1) matrix elements; neither could be evaluated here.  From the LMR value of b and the current result for (b—i-c), the value for c listed in Table 4.3 was obtained. The sign of d is positive definite when Eq. (2.73) is a good approximation. However, in some molecules, this approximation fails. For example, in the 2 11g ground state of B0 , 2  d was found to be negative [51], a result attributed to relatively large core polarization effects [4]. In the current work, d is shown experimentally to be positive in. the ground state of HBr by making reference to the sign of the A-doubling parameter p. In general, the sign of d cannot be derived from a study of the hyperfine structure alone. If the sign of d is taken to be negative, then the hyperfine analysis yields an equivalent fit, but with the e-f level assignments interchanged, i.e., with each f level below the e counterpart. However, the e-f ordering here is known to have f higher (as shown in Fig. 4.1) from the fact that p is positive in HBr known [7,  ,  [, ,  9j. The sign of p in turn is determined once it is  9] that the dominant mixing of the  state (rather than a 2 E  211  ground state is with a distant 2 E  state). Thus the hyperfine analysis together with the positive  sign of p shows that d is positive. This result for the sign of d can then be used to investigate the electronic structure by means of Eqs. (2.58) to (2.61) and Eq. (2.73). The argument can also be reversed. If it is assumed that Eq. (2.73) is a good approximation, then it follows that the  211  state (rather than a 2 E  state).  ground state interacts predominantly with a distant  2-I-  Table 4.3 also presents a comparison between the results for the two different iso topomers. From Eqs. (2.58) to (2.61) and Eq. (2.73), clearly, for each of the magnetic Br+ and Br+ 1 79 should equal the H hyperfine parameters, the ratio of the values for HS ratio of the nuclear g-factors. As can be seen from Table 4.3, the agreement is very good  Chapter 4. Hyperfine Structure in HBr and HI  97  for a, (b+c) and d (as well as for h and h_). For b and c individually, the difference from the g-factor ratio is ‘-‘twice the error estimate. This discrepancy is perhaps not sur prising, given the difficulty in determining b even with LMR precision. For each of the Br+ and H 1 Br+ 7 9 should quadrupole coupling constants, the ratios of the values for HS equal the ratio of the quadrupole moments, assuming that the isotopic effects on the electric field gradient q are negligible. In footnote (k) of Table 4.3, the ratio is given for the atomic value of eQq in 81 Br to that in 79 Br. The agreement is good, although the precision is less than that obtained for the ratios of the magnetic parameters. Now that the hyperfine structure for both magnetic substates has been measured, further insight into the electron distribution in HBr can be obtained. From Eqs. (2.58) to (2.61) and Eq. (2.73) and the determinations given in Table 4.3, the average values of various functions of r and 8 have been calculated using the values of nuclear magnetic moment given in Ref.  [].  The results obtained are listed in Table 4.6. These averages  should be isotopically invariant to excellent approximation. In general, this is borne out in Table 4.6. For three of the functions, the isotopic change is —.twice the error limit, but each of these involves the LMR value of b. For the two functions independent of b, the averages are isotopically invariant to well within the error limits. Three different points regarding the electronic structure can be made from the results given in Table 4.6. First, the simplified single configuration picture is a good approxima tion. In this picture, the unpaired electron (actually a hole in this case) provides both the net spin and net orbital angular momentum, with the result that (1/r 1 ) 3 For HBr, the two differ by about 5%, i.e., by (5.2 + 1.3)  . 3 A  =  . 5 ) 3 (1/r  This indicates that there  is a small but significant contribution from the terms neglected in deriving the hyperfine energies. One possible source of this difference is core polarization effects  [, ffl].  Second, the dominant single configuration in HBr is that associated with a 4pir orbital. The non-zero value obtained for (O)2 indicates that some s character must be  Chapter 4. Hyperfine Structure in HBr and Hft  98  Table 4.6: Averagesa over Functions of the Electronic Space Coordinates for HBr+ in the X 211 Electronic State Br+ 7 H 9  HS B 1 r+  Averageb  )C 1 ( 9937 2  99.35(11) C  99.358(82)  106.3(14) d  (io)d 1028  104.11(85)  II,(0)I2  )e 3 ( 093 2  )e 2 ( 171 5  1.42(20)  2 9) (sin  (i 5 . 86 ) 6 9 f  86.56(13  86.57(10)  Parameter  K K  (  8 2 3cos  —51.5(16)  a All entries are in units of A . Quoted uncertainties in parentheses are one 3 standard deviation and apply to the last digits of the values. b Weighted average of the values of H Br+ 7 9 and Br+. 81 H C  Calculated from Eq. (2.58).  d Calculated from the relation (c/3) + d ). For comparison, 3 = gegN/ N (1/r / 3 the theoretical value of (1/r 8 for a 4p electron in Br atom is 102.9 A— ) 3 . See 3 Ref. [59]. Calculated from the relation (2b + h h_)/3 = 3 7r/3)gegN/ (0)I2. 8 ( N / In contrast, the theoretical value of ‘IJ(0) 2 for a 4s electron in Br atom is . See Ref. [59]. 3 165.1 A —  f  Calculated from Eq. (2.73).  g Calculated from Eq. (2.61).  Chapter 4. Hyperfine Structure in HBr and Hft  99  attributed to the electron distribution. However, for a 4s electron in a bromine atom, =  3 165.1A—  [],  which is —100 times larger than the value measured in HBr.  Thus the percentage s character in the electronic wavefunction is small. Further evidence O 1) : (sill: 8 3 2 for this conclusion is provided by the ratios : (3 cos For a  (.4)  —  pure pr orbital, these have been calculated to be 1: —0.40 : 0.80  [, ].  )•  From Table 4.6,  for H Br, 7 9 these ratios are 1: —0.44 : 0.81; for H Br, 8 1 they are 1: —0.52 : 0.84. The agreement for I _1), there 2 : (sm 8) is good. However, for : (3cos  (.4)  (.4)  is a discrepancy of from 10 to 30%, depending on the isotopomer. The difficulty again seems to be in the value of b. Third, the electron distribution in HBr is close to that of a bromine atom perturbed by a proton. For each isotopomer, the experimental determination of (1/r 5 lies within ) 3 4% of the value of 102.9  A  Br. A similar, but {j calculated for the free atom 79  less stringent test of the perturbed bromine model was presented earlier by Lubic and co-workers [II] using their value of eQq . It was argued that, in this model, eQq 2 2 should be the negative of the atomic value. Within the relatively large error in eQq , this is 2 indeed the case. Thus the dominant single configuration in HBr is a pr orbital that behaves very much like the 4p orbital of a Br atom. 4.4.2  HP  The hyperfine constants of HI+ are determined for the first time and are listed in Table 4.5. Along with the values of a and (b+c), there are listed the values of h and h_. As can be seen from the table, the precision of h+ is about an order of magnitude higher than that of h_. This is due to the fact that h and h_ are determined from the hyperfine transitions in the 312 11 the 2 and 2111/2  the  2111/2  spin substate, respectively. Since the  spin substate lies higher in energy by approximately three vibrational quanta of  2113/2  spin substate, the observed hyperfine transitions belonging to the  2111,2  spin  Chapter 4. Hyperfine Structure in HBr and Hft  100  substate were much weaker than the corresponding hyperfine transitions in the  2113,2  spin substate. Not oniy were the signal-to-noise poor for the transitions belonging to the  2111/2  spin substate, the amount of hyperfine data available for this state was also  limited. As mentioned earlier, our limited experimental accuracy did not permit us to determine the b and eQq 2 constants, which appear only as higher order terms in the hyperfine energy expressions. The magnitude of eQqo for I in HI+ is larger by a factor of 5 than that for Br in HBr+. Consequently, eQqo was determined to a higher accuracy in this case than in HBr. Comparison of various magnetic hyperfine constants of HBr+ and HI+ determined in this work shows that they are of similar magnitude. This, along with the definition of d given by Eq. (2.73), suggests that the sign of d in HI should be positive, as was in the case of HBr. This can also be concluded from the fact that the low lying electronic states in the hydrogen halide ion series are very similar, indicating that the dominant mixing of the  2fl  ground state is with a distant  2+  state (which is the first excited  electronic state in this series). Thus, with the assumption that this, in fact, is the case in HIP, the A-doubling parameter p should be positive, indicating that the f-levels are higher in energy than the corresponding e-levels. The fit to the Hft hyperfine data was performed under this assumption, which resulted to a positive value of d. Table 4.5 also lists the values of  (-)  and  for HIP. Since the hyperfine  parameters b and c could not be determined separately in this work, we were unable to 0 1) 3 (3cos2 evaluate the averages and as well as the parameter 2 KI(0)I ap —  (-i)  pearing in the Fermi-contact term (see Eq. (2.60)). A much higher resolution experiment must be performed before a more complete characterization of the electron distribution is possible.  Chapter 5 Rotational Energies and Fine Structure in HBr+ and 111+  The present chapter deals with the vibration-rotation transitions and the fine struc ture splittings observed in the spectra of HBr+ and 111+ molecular ions using the tech nique of velocity modulation. The chapter is divided into seven sections. The first section deals with the general features of the observed vibration-rotation spectrum. In the se  cond section, the vibration-rotation energies of molecules in a  211  electronic state and  the associated fine structure splittings are discussed in terms of the effective Hamiltonian appropriate to the “merged” model. As discussed in Chapter 1, this is the traditional approach developed for the case where  Ael  <<  We.  The data and the analysis of the  observed transitions in HBr+ using the merged model formulation are also presented as subsections. In HBr+, anomalies were observed in various fitted constants with this model. In order to explain such anomalies, the standard effective Hamiltonian of the  merged model was modified. This introduced the “split” model. The third section deals with matrix elements of the modified Harniltonian appropriate to the split model, and the analysis of the observed vibration-rotation transitions in HBr+ using this model. The molecular constants of HBr+ are presented in the fourth section of this chapter. The isotopic dependences of the constants as well as the harmonic and anharmonic potential constants are also discussed in this section. The fifth section deals with the data and the analysis of 111+ with the merged model. The molecular constants and the harmonic and anharmonic potential constants for HI+ are presented in sixth section. The last section deals with the estimates of the effective vibrational and rotational temperatures obtained 101  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  102  in this work. 5.1  General Features of the Spectrum The HBr and HI ions generated in glow discharges are highly excited vibrationally  and give a rich spectrum of vibration-rotation transitions in the X 2113/2 and X 2111,2 electronic states. In the case of HBr+, about 300 vibration-rotation transitions belonging to (v’  —  v”)=(l—O) to (5—4) vibrational bands were measured for each isotopomer in  a frequency region between 1975 cm’ and 2360 cm. The transitions appeared in pairs of almost equal intensity due to the presence of two isotopomers in approximately equal natural abundance. As expected, the lineshapes were derivative-like in form when recorded at the frequency of the discharge modulation. Under optimum conditions, the width az”, defined as the difference in frequency between the positive and negative peaks, was ‘-.0.006 cm. Fig. 5.1 shows a spectrum in which transitions from all five bands appear. It should be noted that the frequency scale in Fig. 5.1 is slightly non-linear. In addition, the laser power varies across the scan, so that the relative intensities of widely spaced features may not be reliable. As mentioned previously, each transition appears as a doublet; the H Br 7 9 component falls on the high frequency side. The splitting between isotopic partners varies from 0.22 cm to 0.36 cm depending on the vibrational band, the J value, and whether the transition belongs to a P,  Q  component is split due to A-doubling. For the  bands, the splitting is  2111,2  or an R branch. Each isotopic ‘—‘  2 cm , 1  which is so large that only one A-doublet appears in any particular scan. For example, see the (e-e) component of the P(3.5) transition in the (3—2) band in Fig. 5.1. For the  2113/2  bands, the splitting of the two A-doublets is much smaller. In the P and R branches, this splitting grows as J 2 and typically can be resolved for J  6.5. Examples of the A-doublets  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  R(9.5)  2 f l  103  (2 1) P(8.5)  (‘in  I  I  I  112 f 2 (32) l 3 ee  2 f l 2 f l  I  ) 5 . 2 (1 0) P(l  (4 3) R(l.5)  Figure 5.1: A portion of the spectrum of HBr showing transitions belonging to each of the five vibrational bands investigated in this work. The region is 1.55 cm’ wide. It was recorded in two parts with a total scan time of 9 minutes. The time constant was 1 s and discharge power was 350 W. The two frequencies listed are slightly higher than their counterparts in Table 5.2 because the latter have been corrected for hyperfine shifts. (See Sec. 4.3.1 of Chapter 4).  Chapter 5. Rotational Energies and Fine Structure in HBr and HI from the (2  —  1) and (5 —4) bands can be seen in Fig. 5.1. The (1  —  0)  104  2113,2  band shows  a rather different A-doubling. (See Sec. 5.2.2.) Each transition between eigenstates of 5 could be resolved eff really consists of a hyperfine multiplet. The splitting due to Hhf TT only at low values of J. See Sec. 4.2.1 of Chapter 4. In Fig. 5.1, the splitting shown in the (4  —  3)  2111/2  spectrum of Br 81 is due to hyperfine effects. A similar splitting H  Br+ 7 9 spectrum, but the low frequency member of the doublet is hidden occurs in the H under a much stronger line in the 2111,2  2  111/2  fundamental. The two frequencies listed for the  (3—2) 3 .P( 5)ee transitions are the experimentally observed frequencies of their line  centers, and are slightly higher than their counterparts in Table 5.2 because the latter have been corrected for hyperfine shifts. See Sec. 4.3.1 of Chapter 4. Fig. 5.2 shows three separate traces of spectrum of the  211,  P(9.5) transition be  Br+. 7 9 In each case, the longing to the fundamental and the first two hot bands of H frequency increases from the left to the right hand side of the trace. At the bottom of each trace are shown the transmission fringes of the confocal etalon with a free spectral range of -.‘O.Ol cm . These spectra have been plotted such that the frequency scales 1 are approximately the same. In Fig. 5.2, the transitions belonging to the (1—0) and the (2—1) vibrational bands as well as the associated etalon fringes appear to be slightly broader than their counterparts in the (3—2) band. This is produced by electrical pick-up generated by the discharge and detected by the diode laser control system. The level of this electrical pick-up was found to be sensitive to the discharge condition as well as to the cleanliness of the discharge cell. As can be seen from Fig. 5.2, the transitions belonging to the (2—1) and the (3—2) bands show a splitting of approximately 0.02 cm as a result of A-doubling. In contrast to this, the same transition belonging to the fundamental band shows no splitting. In fact, in all vibrational bands other than the fundamental band, the A-doublet splittings in the P-branch transitions were observed for J”  6.5. In the fundamental band, these  Chapter 5. Rotational Energies and Fine Structure in flBr and HI  Br 7 H 9  P(9.5)  (2-1)  (1-0) if  2  ee  ff  105  (3-2)  ee  Frequency  ft  (v’-vj  ee  0.01 cm  Figure 5.2: Portions of the spectrum of 2113/2 P(9.5) vibration-rotation transition of 79 belonging to (1—0), (2—1) and (3—2) vibrational bands. In each case, the fre H Br quency increases from the left to the right hand side of the spectrum. The bottom traces show the transmission fringes of the confocal etalon. In each case, the free spectral range is --0.01 cm’; the frequency scales are approximately the same for the three spectra.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI splittings were observed oniy for transitions with J”  106  15.5 (up to J”  18.5 which  =  is the maximum J we could study in this work with the available laser diodes). This anomaly arises from the fact that the v=0 vibrational level of isolated, whereas each of the other vibrational levels of the neighbouring interacting partner belonging to the  2111,2  2113,2  2113,2  spin substate is  spin substate has a  spin substate. See Fig. 1.3 and  Chapter 1. In a P-branch transition, the observed splitting is the difference between the A-doubling splittings of the upper and the lower rotational energy levels. The absence of observable splittings in the P-branch lines of the fundamental band up to J”  =  14.5 has  some interesting implications. The interaction of rotational energy levels belonging to the v=0 vibrational state of 2 111/2 with the rotational energy levels of the v=1 vibrational state of  2113/2  must be such that the splittings in the lower state  (2113,2  v=0, J) of these  P-branch transitions are almost equal to the splittings in the upper state  (2113,2  v=1,  J—1). In 111+, more than 100 vibration-rotation transitions were measured. These transi tions belonged to the (1—0), the (2—1) and the (3—2) vibrational bands of the spin substate and to the (1—0) vibrational band of the  2111,2  2113,2  spin substate of the ground  electronic state. As in HBr+, all the lines were derivative-like in form. Under optimum conditions, the width of these lines were —‘0.004 cm. Fig. 5.3 shows the observed A-doubling splittings in the spectra of R(9.5), R(10.5) and R(11.5) vibration-rotation transitions of the  211,  (1—0) band of Hft. Here the relative  intensities of the three transitions are not shown on the same scale. The frequency scale (abscissa) in each of the three plots are approximately the same. For all the transitions such as those in Fig. 5.3 where the two A-doublets are only partially resolved, the observed frequency of each of the doublet was taken as that of the average of its upper and lower peaks.  This assumption was shown to hold to good approximation with the aid of  computer simulations. As expected, the (e—e) A-doublet component in all the observed  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  R(9.5) ee  ff  2227.888 cm’  R(10.5) ee  ff  2236.052 cm’  107  R(11.5) ee  ff  2243.8 13 cm’  0.02 cm  Figure 5.3: Observed A-doubling splittings in three R-branch vibration-rotation transi tions belonging to the fundamental band of the 2113,2 spin substate in Hft. The rela tive intensities of the three transitions are not plotted to scale. These transitions were recorded with a time constant of 1 s and a discharge power of 400 W.  Chapter 5. Rotational Energies and Fine Structure in HBr and HJ splittings of the  2113/2  108  spin substate lie lower in frequency than the corresponding (f—f)  component. This is opposite to that in P-branch transitions where, in general, the (f—f) A-doublet component falls lower in frequency than the corresponding (e—e) component. In order to analyze the measured vibration-rotation transition frequencies of of HBr+, two different models were used. These are the merged model and the split model. In the  merged model, the usual effective Hamiltonian for a single vibronic state in the limit  Ad  <<We  was applied. The model was found to be insufficient to explain some of the  details in the observed spectra ofHBr+. In order to explain these anomalies, the split  was  model  developed. Here, the requirement that Ae <<  the merged model  was  W  was  removed. In 111+,  found to be sufficient to fit all the frequencies of the observed  transitions within the accuracy of our experiment. The next section deals with the structure of the Hamiltonian in the merged model and the analysis of the data using this model. Data and Analysis of HBr with the Merged Model  5.2 5.2.1  The Effective Hamiltonian  In the absence of external fields, the Hamiltonian of a freely vibrating, rotating diatomic molecule, neglecting hyperfine effects, can be written as H=HEv+HR+Hso+HSR+Hss.  [n]:  (5.1)  In Eq. (5.1), REV represents the non-relativistic Hamiltonian of the non-rotating mol ecule. The rotational Hamiltonian HR arises from the kinetic energy generated by the end-over-end rotation of the molecule. HSO,  HSR  and HSS arise from the magnetic in  teractions among L, S and J that produce the fine structure. The expression for the  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  109  Hamiltonian is then written as [27]: H  T+B(r)  =  [(J2J2)  (s2_S)] + (L2_L) +  +B(r) [(LS_ + L_S)  —  (J+L_ + J_L÷)  +A(r) {LZSZ +  (LS + L_S+)]  +7(r)  (J+S_ + J_S+)]  +  —  (J+S_ + J_S)]  (r)S + (r) [3s 7 2  —  —  52]  ,  (5.2)  where T represents the vibronic energy corresponding to the non-relativistic Hamiltonian, REV.  In Eq. (5.2), J  Jx+iJy, L  =  are so-called anomalous operators  =  [].  L± iLy and S  =  Sx±iSy.  Here, J and J_  (See Sec. 2.6.1 of Chapter 2.) The Hamiltonian  given by Eq. (5.2) represents the total Hamiltonian (exclusive of hyperfine interactions) containing all the major interaction terms. It should be noted that 7(r) in Eq. (5.2) is denoted by  7(1)  in Ref.  [i ]. (See also Footnote d of Table 5.3.) Since all the vibronic  states studied in this work arise from a single unpaired electron, the last term in Eq. (5.2), representing the spin-spin interaction, is zero. The Hamiltonian represented by Eq. (5.2) can be used to evaluate all the matrix elements representing the rovibronic energy levels of a 211 electronic state, interacting with a distant HBr and Hft, the dominant mixing of the 2+  211  2+  and/or a distant E  state. In  ground electronic state is with a distant  state. In setting up the matrix for the Hamiltonian represented by Eq. (5.2), the  basis functions can be written as a product of two factors. The first is the vibrational function  In, v)  (anharmonic, in general) for the electronic state n being considered. The  second factor is the electronic-rotational function for Hund’s coupling case (a) [1] in the e/f symmetrized scheme 2s+lA, J  [2]. =  This can be written as  [IA, 5, ;  ,  J) ±  I—A, S, —E; —Z, J)],  (5.3)  with S=1/2. The various quantum numbers used in the above expression have been described in previous chapters. In many cases, several of the quantum numbers labelling  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  110  the rovibronic basis functions will be suppressed; the notation v;,J, often be used. The index r  ) or Iv,1) will  e or f is used to label the two A-doublet energy levels.  In Eq. (5.3), e and f correspond to the upper and lower signs, respectively. The e-Ievels and f-levels have parities +(—l) and _(_i)’4, respectively, in accordance with the convention used by Brown et al.  [n].  See Eq. (2.39) of Chapter 2.  In order to derive the matrix elements’ representing the energy levels of a 211 state, interacting with a distant  2+  state, two Van Vleck transformations  on the flamiltonian matrix for the  2fl_2+  [, ] are performed  system. These are an electronic Van Vleck  transformation and a vibrational Van Vleck transformation. The electronic Van Vleck transformation is applied to the Hamiltonian of Eq. (5.2) to remove the leading terms in the mixing of the 211 state with the distant  2+  electronic state. This step introduces the  A-doubling constants, Pv and q [i 2J. The resulting Hamiltonian will be denoted as H. Second, the vibrational Van Vleck transformation is applied to H in order to remove the leading terms in the mixing produced by matrix elements within the states with  IV  211  manifold between  0 and z = 0, ±1. This step introduces the centrifugal distortion  constants. The two-step Van Vieck transformation produces an effective Hamiltonian eff for the 11  2jj  state whose eigenvalues are obtained by diagonalizing a 2 x 2 matrix:  ( Hffj) = (v; , J,  Hff v;  ci’, j, ),  (5.4)  where (i,j) = (1,1), (2,2), (1,2) correspond, in turn, to (,f’) = The convention is adopted that upper÷-*upper and lower-*1ower; that is, there are two separate sets of matrices, one for r = e and one for  T  = f.  In order to introduce the central difference between the merged and split models, it is necessary to consider the form of the vibrational Van Vieck transformation. As a A detailed procedure for setting up the matrix is given in Appendix A of this thesis. The derivation 1 of the matrix elements in the appendix is self-contained. The procedure can be followed without referring to the main body of the thesis. No attempt was made to maintain the internal consistency between the notation used in the appendix and that used in the main body.  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  result of this transformation being applied to H to second order  111  [n], the matrix elements  (i Heffj) include the correction term: (V;,J,THV;’,J,T)  = QI  x (v; l, J, r Here  u”v  {  1 v,O,J,r  —  1  v”,O”,J,r  HI v”; a”, J, T) (v”; !“, J, r  v;  V,1’,J,T  ‘,  J, T)  —  } (5.5)  .  is the diagonal matrix element of the Hamiltonian for level (v, f, J, r) after  the electronic Van Vieck transformation has been applied, but before the vibrational transformation has been introduced. For the purpose of illustration, consider the case where  =  Il’  =  .  Then the  correction can be written as (V;,J,TIHMV;,J,T)  {  (v;,J, r  ii  =  v”; , j, r)  +  Rt;j,  ‘  —  —  ‘r  }  (5.6)  The first sum involves matrix elements with different v, but the same !: i.e., from intrastack coupling . The second summation involves matrix elements with different 2 and different f: i.e., from interstack coupling . For v” 2  =  v + 1, the difference between  the energy denominators divided by their average has magnitude For molecules such as HF+, N and NO in which can be approximated by the average, [E,T  —  IAe/(’e + Ae/2)I.  IAe <<We, each of these denominators  E??,J,T]  (in which the 1 dependence is  neglected). The two summations in Eq. (5.6) can then be combined; this constitutes 1n the current work, the adjective “intrastack” will be used to refer to terms in Eq. (5.5) which 2 involve energy differences between levels which differ in v but not in Q. The adjective can be applied to the associated matrix element or coupling constant. The adjective “interstack” will be used in a corresponding way when the energy differences are between levels which differ both in v and in Q. By extension, the same adjectives will be used for H, the third order correction to the effective Hamiltonian which will be described later.  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft the merged model. For molecules such as SeD and HBr+ in which  112  Ael  W  but the  perturbation is non-resonant, the two summations in Eq. (5.6) cannot be merged. In fact, for the terms where v”  =  v + 1 in the sum in Eq. (5.6), the difference in the  denominators can be large compared with the average. (Remember Ae < 0 in HBr+.) This leads to the split model. Of course, if Ae  W  and the perturbation is resonant, the  vibrational Van Vleck transformation cannot be used and other methods are required. For HgH  []  and NSe  [i], diagonalization of the 2 x 2 matrix for the interacting partners  was carried out. As mentioned previously, the effective Hamiltonian “eff for the rotational-fine struc ture part of the problem takes the form of a 2x2 matrix representing the  (zM =  ±1)  interactions within a specific vibrational state. Following the pioneering papers of Hill and Van Vleck  [],  and of Mulliken and Christy  [fill,  this Hamiltonian for a 211 state  has been discussed extensively. For example, see Refs. [27,  fi21,  and references cited  therein. The matrix elements associated with the potentially significant parameters are given in Table 5.1 using the notation defined in connection with Eq. (5.4). In addition, z  (J +  1)2  and y  (z  —  1)1/2.  The largest parameter is the vibronic energy T; the associated matrix contains all the terms independent of both J and  .  Aside from T, the four lowest order parame  ters are the spin-orbit constant A, the rotational constant B, and the two A-doubling parameters p,, and q. The corresponding leading centrifugal distortion parameters are AD,,, D v PDv, and q,,. Generically, each of the four lowest order parameters will be rep resented by X,, and its leading centrifugal correction constant will be denoted XDv. For A,, and B,,, the next higher order centrifugal distortion constants are included, namely Ag,, and H,,, respectively. For completeness, the spin rotation parameter y,, has been listed. For each leading centrifugal distortion parameter XD,,, the matrix elements given in Table 5.1 can be obtained in a straightforward manner as the symmetrized product of  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  113  Table 5.1: The Effective Hamiltonian Matrix Elements for the Merged Modela, b, c Parameter  (Heff)ii  (hleff)22  (Heff)12  T  1  1  0  A  —1/2  1/2  0  ADV  —z  z—2  0  2 —2z  2 2(z—2)  B  z  z—2  D  2+y —(z ) 2  H,  2 + 3 z — 5z+2 3z  2 — 3 z + 5z—4 3z  Pv  (/+ 1)/2  0  — 1)/2  2 —(z  — 3y ) 2  —y 3 2y —y(3z — 2 5z+3)  y / 2 2  —(+/ + z)y/2  +/ + (z + 1)/2  y / 2 2  —(+/ + l)y/2  qDv  +2z— 1 2 +‘(3z— 1)+z  2 : 4 y / Fy  d 7  —1  0  +z/ + (3z  PDV  qv  y/2  a In this model the matrix elements have been derived with the assumption that the magnitude of the spin-orbit splitting is very small compared to the vibrational spacings, i.e., Al << We. b The upper and the lower signs in the matrix elements correspond to the e and the f energy levels, respectively. z  = (J +  1)2  ;  y  =  (z  — 1)1/2  d In Ref. [ia, ], this is denoted by y(’) + 7(2k. The second order term from the Van Vieck transformation, and is usually the dominant term.  7(2)  arises  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  114  the matrices for X and B. As an example, the method to obtain the matrix elements for the rotational centrifugal distortion constant D is outlined in Appendix A. For the next leading centrifugal distortion parameters, A and H, similar methods are used. The matrix for  is obtained from the symmetrized product of the B,, and AD,, ma  trices. The matrix for H,, is taken as the third power of the matrix for B,,. The leading centrifugal distortion parameters can be defined as: XD,, = x  (v, 2111 X(R) v’,  211) 211  —  Here (x is —1 for X=B, and is +1 otherwise. conform to the signs used traditionally  [1].  (v’, 2111 B(R) v, 211)  (5.7)  .  ‘—v’, 2fl  This factor allows definition (5.7) to  In Eq. (5.7), X(R) is the operator associated  with X expressed as a function of the internuclear separation R. The vibrational integral (ii,  211 is the limit as J—*O of 111X(R) Iv’, 211) will be denoted X,,,,,. The energy E 2  the energy E°,,JT introduced previously. All of the J-dependence in (i Heffj) has been included in the entries under (Heff)jj of Table 5.1 (i.e., the full Hamiltonian term is obtained by multiplying the parameter by the corresponding function of J). The definitions used here for all the parameters follow the conventions adopted by Brown et al.  [2J,  with one change. The present definitions of An,, and PDV must be  multiplied by a factor of 2 to agree with those in Ref.  [].  Here this factor of 2 has been  moved into the functions of J (given in Table 5.1) so that the definitions of As,, and pD can be put in the same form as that of D,,. The results given in Table 5.1 depend on the assumption that  Ael is small, first,  with respect to the dissociation energy Ve and, second, with respect to the vibrational frequency We. For HBr+, both  2111/2  and  2113,2  spin substates go to the same dissociation  [a].  Since  IAe/2)eI  is ‘-8%, the same interatomic potential can be used to a good approximation for  = 1/2  (2S and Br 3 ) limit; the dissociation products are H 112 ( ) 2 P . See Ref.  and l = 3/2; the same molecular parameters are used for both spin substates. In the  Chapter 5. Rotational Energies and Fine Structure in HBr and HI case of HBr+, Ae/WeI is  115  1. In spite of this, the conventional approach is used in the  merged model and, as discussed previously, the f’-dependence which should enter in the energy denominator of Eq. (5.7) has been dropped. This also affects, of course, the form of the matrix elements associated with the centrifugal distortion parameters. The energy splitting fe between the two A-doublets is of particular interest here. Explicitly, fe(’ f, J)  E(v; Q, J, f)  —  E(v; , J, e) ,  (5.8)  where E is the eigenvalue of 11 eff Although the data fitting was done by diagonalizing “eff’ the general behaviour of fe can be obtained directly from the diagonal matrix elements in the case of rotational energy levels belonging to the  2111/2  spin substate.  The splitting of rotational energy levels (A-doubling) arise due to the presence of the parity-dependent terms in the Hamiltonian. In the case of  2111,2  spin substate, the  parity-dependent diagonal matrix elements (neglecting distortion correction terms) are: (5.9)  Fj-v + qv.  Here the upper and the lower signs correspond to the e- and the f-levels, respectively. Therefore, the energy splitting between the f- and the e-level is given by  =  {+‘  +  = {p + 2q}  qv/}  (  +  -  -  .  (5.10)  The superscript m has been added to Life in order to show explicitly that the merged model is being used. As can be seen from Table 5.1, there are no major parity-dependent terms in the diagonal matrix elements representing the rotational energy levels of the  2113,2  spin  substate. The only parity-dependent term appears in the matrix element of qi, which  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  116  is negligible here. An approximate expression for the splitting of these energy levels can thus be obtained by treating the (/fZ perturbation theory. The  (z  =  +1) matrix elements with second order  +1) matrix elements can be obtained from Table 5.1.  Neglecting the distortion correction terms, these can be written as (511.) Using perturbation theory, the second order correction to the diagonal matrix elements is then given by  2 IHiJ  =  (i +  +  ±  2 q .  (5.12)  —  Here zE is the energy difference between the interacting levels of the 2113/2  2111/2  and the  spin substates. An approximate expression for E, including the major parity  dependent terms which are important in determining the splittings of the A-doublets, can be written as LE = A +  =  A (i +  (5.13)  .  Since p/ << 2A in our case, Eq. (5.12) gives 2 IHld  (  +  \/,_( 0 B,q —1) +  —1) +  = <  .  —  i)} (5.14)  Here, B, =  (5.15)  Since the third term on the right hand side of Eq. (5.15) is negligible, B can be approx imated by B = B +  —  .  (5.16)  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  117  To determine the splitting of the A-doublets, only the parity-dependent contribution to the energy shift needs to be considered. Therefore, retaining the parity-dependent terms in Eq. (5.14) and representing them as T  (i), we get  B +  =  +  (5.17)  Z)(Z_1).  In the above expression, the upper and the lower signs correspond to the e- and the f-level, respectively. In our case, the third term on the right hand side of Eq. (5.17) is negligible. Therefore, the approximate energy splitting between the f- and the e-level is given by  /(v; ,J)  =  LT(f)  {  =  where Yv  =  A/ [B + (Pv/ ) 4  —  —  zT(e)  E(v;,J,f)  +  —  E(v;1,J,e) ;  (5.18)  (7/2)]. Here again the superscript m on Life indicates  the use of the merged model. Of course, a similar splitting also effects the corresponding rotational energy level of the  2111,2  spin substate. This does not alter the general form of  Eq. (5.10) since the splittings in the than those in the corresponding  2111,2  2113/2  energy levels are orders of magnitude higher  energy levels.  As can be seen from Eqs. (5.10) and (5.18),  L$  can be written as the product of a  J-independent coefficient C(v; 1) and a function 0 f ( J), i.e., =  C(v;) fc(J)  (5.19)  .  Remember that the f-levels are higher in energy compared to their e-partners. HBr,  PVI  ‘--‘  300  q°j  and  negative. In Eq. (5.10), C(v;  C(v;  )  )YV  )  For  300; Pv is positive, while q and Y, are both  is dominated by Pv and is  cm’. In Eq. (5.18),  receive comparable contributions from the terms in p, and q; the two reinforce  and C(v;  )  7 x 10 cm . 1  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft 5.2.2  118  Merged Model Fit  For the rotational and fine structure analysis, the total number of vibration-rotation Br+ T 9 and HS Br+, respectively. 1 frequencies included in the data set was 295 and 284 for H For ll Br, these were distributed as follows: 69 in the (1—0) vibrational band, 72 in 79 the (2—1) band, 71 in the (3—2) band, 46 in the (4—3) band, and 37 in the (5—4) band. 81 the corresponding numbers are 66, 80, 65, 44, and 29. The highest J”-value H For Br+, measured  was determined in large part by the Hönl-London rotational line-strength  factor. For the Q-branch lines, jax05 and 7.5 for Q=1/2 and 3/2, respectively. For the P- and R-branch lines, Iax was typically —45.5. For the R-branch in particular, the selection of lines observed was limited by the frequency range covered by the diodes. The frequencies measured are listed in Table 5.2. When the two A-doublets in a  2113,2  transition were not clearly resolved, the mean frequency was used as a single entry. For low J multiplets with hyperfine splittings, the hyperfine-free frequency was entered in Table 5.2.  If the multiplet was clearly resolved, the hyperfine-free frequency was  determined in the analysis to obtain the hyperfine constants; see Sec. 4.3.1 of Chapter 4. If the multiplet was blended, the spectrum was synthesized using these constants; the hyperfine-free frequency was then deduced by matching the synthetic and experimental frequencies. See Fig. 4.5 of Chapter 4, for example. Since the transitions in the (1—0) information here on the isolated (v  2113,2  0, Q  =  band provides the only source of direct 3/2) state, every effort was made to study  the A-doubling in this band as thoroughly as possible. Using the notation introduced in connection with Eqs. (5.10), (5.18) and (5.19), the A-doubling splitting in the spectrum can be written: (J’) + (—i)”’C(v”,f) (J”) 0 LSz.’A(v;Q,J,J) = C(v”+1,) f 12 f  ,  (5.20)  with 2=3/2. In the Q-branch, the splittings in the upper and lower levels add. In spite  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  119  Table 5.2: Wavenumbers of the Observed Vibration-Rotation Transitions in the X 211 Electronic State of HBr Br+ 7 H 9 Brancha (1-0) 211  J,, r’ r” P 1.5 1.5 2.5 2.5 3.5 3.5 4.5 4.5 5.5 5.5 6.5 7.5 8.5 8.5 9.5 9.5 10.5 10.5  e f e f e f e f e f e f e f e f e f  e f e f e f e f e f e f e f e f e f  Obsb  C  2323.8695 2321.7890 2307.2132 2305.1022 2290.0976 2287.9599 2272.5306 2270.3633 2254.5136 2252.3256 2236.0654 2214.9550 2197.9025 2195.6427 2178.2056 2175.9258 2158.1088 2155.8090  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030  HS’Br+ Obs_Cald —0.0003 0.0002 —0.0008 —0.0015 0.0005 0.0012 0.0027 0.0007 —0.0016 0.0013 —0.0023 —0.0009 —0.0017 —0.0012 —0.0005 0.0006 —0.0004 0.0000  Obsb 2323.5224 2321.4450 2306.8763 2304.7626 2289.7662 2287.6242 2272.2004 2270.0352 2254.1948 2251.9997 2235.7504 2214.6457 2197.6035 2195.3424 2177.9090 2175.6287 2157.8208 2155.5226  Obs_Cald —0.0040 0.0011 0.0006 —0.0015 0.0021 —0.0005 0.0000 0.0008 0.0014 —0.0022 —0.0015 —0.0005 0.0024 0.0017 —0.0008 —0.0002 0.0009 0.0029  a Values in parentheses refer to vibrational quantum numbers (v’ b Observed frequency. If the two A-doublets were not resolved, the levels involved in the transitions are labelled r=e/f or f/e; the center frequency is listed and used in the fit. If hyperfine structure is observed, the hyperfine-free frequency is listed and used in the fit. —  C  HS B r+, the value of Estimated uncertainty in the frequency measurement. For 1 are similar; they are not listed.  d Difference between the observed frequency and that calculated from the best fit values of the parameters for the split model as listed in Tables 5.6 and 5.7.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  Table 5.2  —  Continued  Br+ 7 H 9 Brancha (1-0)  211,  J,, P11.5 11.5 12.5 12.5 13.5 13.5 17.5  (1-0) 211, P 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5 15.5 16.5 16.5 17.5 17.5 18.5 18.5  120  lH 8 Br+  r”  Obsb  C  f e f e f e  f e f e f e  2137.6224 2135.3063 2116.7573 2114.4192 2095.5211 2093.1665 2007.0312  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0050  —0.0002 0.0020 0.0019 —0.0014 0.0041 —0.0004 —0.0063  2137.3401 2135.0249 2116.4814 2114.1441 2095.2500 2092.8983  e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f e f e f e f e f  e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f e/f e f e f e f e f  2309.3174 2292.2846 2274.7975 2256.8715 2238.4987 2219.7100 2200.4930 2180.8744 2160.8570 2140.4493 2119.6584 2098.4985 2076.9777  0.0030 0.0030 0.0030 0.0050 0.0030 0.0100 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030  0.0004 0.0009 0.0003 0.0052 —0.0011 0.0035 —0.0027 —0.0018 —0.0004 0.0007 —0.0005 0.0005 0.0025  2308.9816 0.0037 2291.9498 —0.0001 2274.4677 —0.0013  2032.8775 2032.8883 2010.3249 2010.3390 1987.4465 1987.4638  0.0050 0.0005 0.0050 0.0006 0.0004 0.0050 0.0050 0.0006 0.0050 —0.0021 0.0050 —0.0020  r’  Obs_Cald  Obsb  2238.1819 2219.4013 2200.1919 2180.5751 2160.5642 2140.1632 2119.3820 2098.2297 2076.7148 2054.8415 2054.8505 2032.6300 2032.6418 2010.0865 2010.1007 1987.2160 1987.2337  Obs_Cald —0.0006 0.0025 0.0001 —0.0020 —0.0009 —0.0019  —0.0013 0.0053 0.0005 —0.0033 —0.0020 —0.0009 0.0004 0.0018 0.0022 0.0003 0.0010 0.0000 0.0005 0.0007 0.0003 —0.0026 —0.0029  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  Table 5.2  —  Continued  Br+ 7 H 9 Brancha  J,,  (1-0)  Q  (2-1)  P 3.5 3.5 4.5 4.5 5.5 5.5 6.5 6.5 7.5 7.5 8.5 8.5 9.5 9.5 10.5 10.5 12.5 13.5 14.5  r’  r”  1.5 f/e e/f 2.5 f/e e/f e 3.5 f f 3.5 e 4.5 f e 4.5 e f e 5.5 f f 5.5 e e 6.5 f f 6.5 e e f e f e f e f e f e f e f e f f e e  e f e f e f e f e f e f e f e f f e e  121  Br+ 8 H 1  Obsb  cC  2348.9588 2347.7504 2346.0645 2346.0570 2343.8986 2343.8801 2341.2527 2341.2202 2338.1289 2338.0777  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030  0.0023 0.0006 —0.0009 0.0007 —0.0004 —0.0008 —0.0009 —0.0018 —0.0011 —0.0018  2348.6051 2347.3989 2345.7152 2345.7062 2343.5497 2343.5319 2340.9054 2340.8742  0.0015 —0.0034 0.0014 0.0009 —0.0018 —0.0007 0.0012 —0.0009  2201.2128 2199.1031 2184.1339 2182.0000 2166.6174 2164.4570 2148.6706 2146.4836 2130.2986 2128.0883 2111.5110 2109.2808 2092.3226  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0050 0.0050 0.0050  0.0000 —0.0016 —0.0024 —0.0009 —0.0018 —0.0008 0.0003 —0.0005 0.0002 —0.0003 —0.0015 0.0005 0.0012  2200.9063 2198.7978 2183.8347 2181.7017 2166.3252 2164.1620 2148.3803 2146.1932 2130.0144 2127.8050 2111.2340 2109.0017 2092.0518 2089.7954 2072.4693  —0.0017 —0.0016 —0.0019 0.0007 0.0005 —0.0012 —0.0009 —0.0017 —0.0006 0.0000 —0.0010 —0.0004 0.0016 0.0000 —0.0003  2029.8452 2011.4467 1990.3733  0.0028 0.0017 0.0000  —  2072.7347 2070.4616 2030.1002 —  1990.6083  Obs_Cald  —  0.0004 0.0003 0.0008 —  0.0004  obsb  Obs_Cald  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  Table 5.2  —  Continued  Br+ T H 9 Brancha  J,,  ‘r’  r”  0.5 0.5  f e  e f  (2-1) 211, R 0.5 0.5 1.5 1.5 2.5 2.5 3.5 3.5 4.5 4.5 5.5 5.5  e f e f e f e f e f e f  e f e f e f e f e f e f  (2-1) 211,  (2-1)  2fl,  Q  P 3.5 e/f e/f 4.5 e/f e/f 5.5 e/f e/f e 6.5 e f 6.5 f e 7.5 e f 7.5 f e 8.5 e f 8.5 f e 9.5 e f 9.5 f 10.5 e e 10.5 f f 12.5 e e 12.5 f f 13.5 e e 13.5 f f 14.5 e e f 14.5 f  Obsb  122  cc  Br+ 8 H 1 Obs_Cald  obsb  Obs_Cald  2258.2123 0.0050 2254.2026 0.0030  0.0058 0.0016  2257.8906 2253.8789  0.0042 0.0016  2277.7305 0.0030 2279.6829 0.0030 2292.0154 0.0030 0.0030 0.0030 2307.6843 0.0030 2319.0818 0.0030 0.0030 2331.8436 0.0030 2333.6515 0.0030 2344.0829 0.0050 2345.8552 0.0030  0.0018 0.0009 0.0007 —0.0005 0.0011  2277.4015 2279.3537 2291.6804 2293.6029 2305.4663 2307.3467 2318.7348 2320.5897 2331.4991 2333.3073 2343.7348 2345.5042  0.0026 —0.0007 —0.0002 0.0013 0.0030 —0.0019 —0.0040 0.0028 0.0003 —0.0011 —0.0004 —0.0008  2203.8476 2186.8723 2169.4554 2151.6090 2151.5982 2133.3421 2133.3295 2114.6619 2114.6453 2095.5805 2095.5615 2076.1011 2076.0787 2036.0015 2035.9717  —0.0027 0.0020 0.0032 —0.0002 —0.0014 0.0002 0.0002 —0.0004 —0.0013 0.0013 0.0014 —0.0002 0.0000 0.0045 0.0046  2203.5430 2186.5681 2169.1602 2151.3202 2151.3093 2133.0573 2133.0448 2114.3826 2114.3660 2095.3101 2095.2906 2075.8384 2075.8155 2035.7458 2035.7156 2015.1448 2015.1103 1994.1792 1994.1411  0.0000 —0.0005 0.0041 0.0012 0.0000 —0.0004 —0.0003 —0.0017 —0.0025 0.0026 0.0023 0.0022 0.0021 0.0002 0.0006 0.0007 0.0008 —0.0049 —0.0042  —  —  —  —  —  0.0030 0.0030 0.0050 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0050 0.0050 0.0050 — 0.0050 1994.4167 0.0050 1994.3806 0.0050 —  —  —0.0005 —0.0017 —0.0007 0.0013  —  —  —0.0045 —0.0031  Chapter 5. Rotational Energies and Fine Structure in HBr and llI  Table 5.2  —  Continued  Br+ 7 H 9 Brancha (2-1)  211,  J, r’  TI’  Obsb  Q1.5f/e e/f 2.5f/e e/f e 3.5f f 3.5e e 4.5f f 4.5e e 5.5f f 5.5e e 6.5f f 6.5e 7.5f e f 7.5e  2258.8171 2257.6277 2255.9687 2255.9575 2253.8350 2253.8137 2251.2285 2251.1905 2248.1510 2248.0924 —  —  123  cC  Br+ 8 H 1 Obs_Cald  Obsb  Obs_Cald  0.0030 0.0007 0.0030 0.0002 0.0003 0.0030 0.0030 —0.0003 0.0030 0.0008 0.0007 0.0030 0.0030 0.0006 0.0030 —0.0004 0.0030 0.0008 0.0030 0.0012 0.0030 0.0030  2258.4945 2257.3018 2255.6466 2255.6355 2253.5122 2253.4902 2250.9056 2250.8706 2247.8335 2247.7753 2244.2852 2244.1995  0.0023 —0.0020 0.0010 0.0003 —0.0003 —0.0014 —0.0018 —0.0004 0.0022 0.0021 0.0003 0.0014  —  —  (2-1)  211,  R1.5e/f 2.5e/f 3.5e/f 4.5 e/f 5.5e 5.5f 6.5e 6.5f  e/f e/f e/f e/f e f e f  2296.0599 2309.7395 2322.9163 2335.5819 2347.7189 2347.7308 2359.3327 2359.3490  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030  —0.0003 —0.0008 —0.0001 0.0010 —0.0006 —0.0010 —0.0021 —0.0023  2295.7263 2309.4011 2322.5723 2335.2352 2347.3681 2347.3790 2358.9841 2358.9997  0.0014 0.0000 —0.0016 0.0000 —0.0032 —0.0042 —0.0002 —0.0007  (3-2)  211,  P1.5 e 1.5f 2.5e 2.5f 3.5e 3.5f  e f e f e f  2145.4093 2143.3893 2129.7239 2127.6710 2113.5842 2111.4998  0.0010 0.0030 0.0012 0.0030 0.0030 0.0000 0.0030 —0.0005 0.0030 —0.0003 0.0030 —0.0014  2145.1214 2143.1002 2129.4395 2127.3870 2113.3049 2111.2206  0.0014 0.0016 —0.0006 —0.0003 —0.0005 —0.0018  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  Table 5.2  —  Comtinued  Br+ 7 H 9 Brancha (3-2)  (3-2)  (3-2)  2ll  2H  Obsb  124  HS B 1 r+  J,,  r’  r”  p 4•5 4.5 5.5 5.5 10.5  e f e f f  e f e f f  2096.9982 0.0030 —0.0007 2094.8850 0.0030 —0.0009 0.0030 2077.8354 0.0030 0.0010 1986.3507 0.0050 0.0005  R 0.5 1.5 2.5 5.5 5.5 6.5 6.5 7.5 7.5 8.5 8.5 9.5  e e e e f e f e f e f e  e e e f e f e f e f e  2187.7275 2201.5466 2214.8658 2251.7753 2253.4954 2263.0315 2264.7158 2273.7522 2275.3923 2283.9293 2285.5255 2293.5591  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0050 0.0030  —0.0023 0.0004 —0.0010 0.0024 —0.0009 0.0014 0.0033 0.0003 —0.0003 —0.0014 —0.0037 —0.0000  P 2.5 3.5 4.5 5.5 6.5 6.5 8.5 8.5 9.5 9.5  e/f e/f e/f e/f e f e f e f  e/f e/f e/f e/f e f e f e f  2132.7450 2116.6996 2100.2099 2083.2954 2065.9584 2065.9458 2030.0377 2030.0166 2011.4717 2011.4474  0.0030 0.0030 0.0050 0.0030 0.0030 0.0030 0.0050 0.0050 0.0050 0.0050  0.0010 0.0000 —0.0039 0.0002 0.0000 —0.0004 —0.0012 —0.0011 —0.0065 —0.0042  C  —  Obs—Cal’  —  Obsb  Obs_Cald  2096.7252  0.0007  2079.7059 —0.0003 2077.5676 0.0004 1986.1125 —0.0013 2187.4246 —0.0015 2214.5541 —0.0011  2262.7086 0.0021 2264.3916 0.0018 2273.4265 0.0005 2275.0661 —0.0003 2283.6009 —0.0020 2293.2299  0.0003  2132.4578 2116.4184 2099.9356 2083.0254 2065.6942 2065.6794 2029.7819 2029.7632 2011.2243 2011.2006  —0.0003 —0.0002 —0.0024 0.0008 0.0013 —0.0019 —0.0021 —0.0009 —0.0042 —0.0031  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  Table 5.2  —  Continued  Br+ 7 H 9 Branch’  -‘  Obsb  C  1.5 f/e e/f 2.5 f/e e/f e 3.5 f f 3.5 e 4.5 f e f 4.5 e e 5.5 f f 5.5 e e 6.5 f f 6.5 e e 7.5 f f 7.5 e  2169.9891 2168.8098 2167.1731 2167.1613 2165.0631 2165.0424 2162.4872 2162.4508 2159.4466 2159.3878 2155.9373 2155.8505  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030  j,,  (3-2)  211,  Q  (3-2)  2,  R 2.5 e/f e/f 4.5 e/f e/f e 5.5 e f 5.5 f e 6.5 e f 6.5 f e 7.5 e f 7.5 f e 8.5 e f 8.5 f e 9.5 e f 9.5 f e 10.5 e 10.5 f f e 11.5 e f 11.5 f e 12.5 e  —  2244.2263 2255.9259 2255.9376 2267.1160 2267.1297 2277.7685 2277.7840 2287.8862 2287.9068 2297.4657 2297.4907 2306.4921 2306.5203 2314.9573 2314.9904 2322.8639  125  0.0050 0.0030 0.0030 0.0030 0.0050 0.0050 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030  HS B 1 r+ Obs_Cald 0.0013 —0.0026 0.0006 —0.0006 0.0000 0.0006 0.0000 0.0008 0.0009 0.0012 —0.0026 —0.0010 —  0.0005 —0.0016 0.0003 0.0043 0.0052 0.0011 0.0005 —0.0010 —0.0000 0.0017 0.0033 0.0016 0.0025 —0.0023 —0.0007 —0.0006  Obsb  ObsCal°  2169.6926 2168.5134 2166.8810 2166.8694 2164.7690 2164.7475 2162.1937 2162.1561 2159.1536 2159.0936 2155.6479 2155.5611  0.0019 —0.0025 0.0042 0.0032 0.0007 0.0003 0.0001 —0.0006 0.0001 —0.0012 —0.0012 —0.0004  2218.9658 2243.9105 2255.6058 2255.6170 2266.7890 2266.8029 2277.4429 2277.4598 2287.5589 2287.5785 2297.1362 2297.1607 2306.1601 2306.1890 2314.6285 2314.6628 2322.5283  —0.0030 0.0022 —0.0012 —0.0003 0.0003 0.0007 0.0009 0.0006 —0.0009 —0.0024 0.0016 0.0007 0.0008 —0.0003 0.0017 0.0011 —0.0019  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  Table 5.2  —  Continued  Br+ 7 H 9 Brancha  J,,  r’  r”  Obsb  126  C  HS B 1 r+ Obs_Cald  obsb  Obs_Cald  (3-2)  211312  R15.5 15.5 16.5 16.5  e f e f  e f e f  2343.1254 2343.1703 2348.7061 2348.7608  (43)  211,  P 4.5 5.5 6.5  e e f  e e f  2011.0670 0.0050 1994.5339 0.0050 1975.4283 0.0050  R 2.5 2.5 3.5 3.5 4.5 4.5 5.5 5.5 6.5 6.5  e f e f e f e f e f  e f e f e f e f e f  2125.1615 2126.9624 2137.5197 2139.2759 2149.3704 2151.0900 2160.7045 2162.3754 2171.5159 2173.1426  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030  0.0000 0.0015 —0.0008 —0.0029 —0.0009 0.0027 —0.0012 —0.0029 0.0000 —0.0012  2124.8829 2126.6790 2137.2341 2138.9892 2149.0810 2150.8078 2160.4126 2162.0831 2171.2219 2172.8467  0.0038 0.0008 —0.0008 —0.0032 —0.0015 0.0104 —0.0015 —0.0023 0.0004 —0.0016  R 7.5 7.5 8.5 8.5 9.5 9.5  e f e f e f  e f e f e f  2181.7935 2183.3776 2191.5316 2193.0663 2200.7217  0.0030 —0.0002 0.0030 0.0018 0.0030 0.0001 0.0030 —0.0001 0.0030 0.0005 0.0030  2181.4980 2183.0813  0.0011 0.0031  2192.7696 0.0021 2200.4198 —0.0002 2201.9060 —0.0023  0.0050 0.0050 0.0050 0.0050  2014.5604 0.0015 1998.1358 —0.0004 1981.2878 0.0003 1981.3000 0.0009  (43)  (43)  (4-3)  2111/2  2113,2  P 4.5 e/f e/f 5.5 e/f e/f e 6.5 e f 6.5 f  —  2014.8116 1998.3845 1981.5429 1981.5294  0.0030 0.0000 0.0030 —0.0038 0.0030 0.0007 0.0030 0.0022 0.0044 0.0026 0.0029  —  0.0035 0.0036 0.0041 0.0016  2010.8145 —0.0016 1994.2884 —0.0010 1975.1928 0.0054  —  —  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  Table 5.2  —  Continued  Br+ 7 H 9 Brancha (43)  211  Br+ 8 H 1  T’  Obsb  1.5 f/e e/f 2.5 f/e e/f e 3.5 f f 3.5 e e 5.5 f f 5.5 e  2082.4385 2081.2713 2079.6520 2079.6408 2075.0165 2074.9789  0.0030 0.0016 0.0030 —0.0030 0.0030 0.0002 0.0030 —0.0007 0.0030 0.0013 0.0030 —0.0003  2082.1685 0.0000 2081.0062 0.0002 2079.3835 —0.0003 2079.3732 —0.0004  J,,  Q  127  cC  Obs_Cald  Obsb  Obs_Cald  —  (43)  211  R 1.5 2.5 3.5 4.5 5.5 6.5 6.5 7.5 7.5  e/f e/f e/f e/f e/f e f e f  e/f e/f e/f e/f e/f e f e f  2117.3422 2130.1133 2142.3918 2154.1709 2165.4399 2176.1853 2176.1994 2186.4107 2186.4277  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030  —0.0004 —0.0009 —0.0010 0.0004 0.0003 0.0001 0.0004 —0.0008 —0.0014  2117.0644 2129.8318 2142.1047 2153.8823 2165.1455 2175.8894 2175.9018 2186.1153 2186.1308  (54)  211/  R 5.5 5.5 6.5 6.5 7.5 8.5 8.5 9.5 9.5 10.5 11.5  e f e f f e f e f e e  e f e f f e f e f e e  2070.8017 2072.4134 2081.1654 2082.7304 2092.5192 2100.2976 2101.7690 2109.0560 2110.4774 2117.2623 2124.9128  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030  0.0009 —0.0005 0.0010 —0.0002 0.0009 —0.0021 —0.0006 —0.0005 0.0003 —0.0005 0.0014  2070.5394 0.0019 2072.1521 0.0010 2080.8990 —0.0002 2082.4628 —0.0017  1996.1251 0.0050  0.0001  1995.8797 —0.0031  (54)  211/  Q  1.5 f/e e/f  —  0.0006 0.0002 —0.0018 0.0015 —0.0012 —0.0008 —0.0008 0.0009 0.0008  —  2100.0299 —0.0019 2101.4968 —0.0001 2110.2016 0.0006 2116.9921 —0.0016 2124.6443 0.0020  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  Table 5.2  —  Continued  Br+ 7 H 9 Brallcha (54)  (54)  211  211,  J,,  T  r”  R 1.5 e/f e/f 4.5 e/f e/f 5.5 e/f e/f e 6.5 e f 6.5 f R 7.5 7.5 8.5 8.5 9.5 9.5 10.5 10.5 11.5 11.5 12.5 12.5 13.5 13.5 14.5 14.5  e f e f e f e f e f e f e f e f  e f e f e f e f e f e f e f e f  Obsb  128  C  Br+ 1 HS Obs_Cald  2029.8759 2065.3640 2076.1962 2086.5107 2086.5221  0.0050 —0.0018 0.0030 —0.0006 0.0030 0.0004 0.0030 0.0008 0.0012 0.0030  2096.3087 2096.3246 2105.5841 2105.5998 2114.3204 2114.3384 2122.5193 2122.5407 2130.1700 2130.1944 2137.2652 2137.2949 2143.8020 2143.8337 2149.7626 2149.7962  0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0030 0.0050 0.0050 0.0030 0.0030  —0.0006 0.0017 0.0019 0.0011 —0.0010 —0.0024 —0.0005 —0.0014 —0.0002 —0.0010 —0.0003 0.0012 0.0032 0.0039 —0.0006 —0.0009  obsb —  Obs_Cald —  2065.1028 —0.0001 2075.9320 0.0007 2086.2429 —0.0001 2086.2545 0.0007 2096.0406 2096.0545 2105.3135 2105.3296 2114.0460 2114.0651 2122.2452 2122.2667 2129.8937 2129.9194  0.0005 0.0010 0.0024 0.0024 —0.0025 —0.0024 —0.0002 —0.0006 —0.0008 0.0001  —  —  2149.4854 2149.5182  —  0.0005 0.0000  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  129  of this, with C(v,) being so small in magnitude, the A-doubling in the Q-branch could only be resolved for J”  3.5. Since  ax =  6.5 for this branch, the information provided,  while very useful, was rather limited. In the P-branch, the splittings in the upper and lower levels subtract. As a result, if C(v”+l;  )  C(v”;  frequency than the corresponding (f—f) line (i.e.,  LVA  ),  the (e—e) line is higher in  is negative) and the magnitude  of the separation goes as J 2 (at high J). One would then expect to resolve the splitting for J>6.5. This behaviour was indeed observed for v”  >  0. The J-dependence of  /.S.VA  observed for the (2—1) band in H Br 7 9 and H Br 8 1 are shown in Figs. 5.4 and 5.5, respectively; all the other hot bands behaved in a similar manner. However, as is shown in Figs. 5.4 and 5.5, the A-doubling in the fundamental behaved quite differently. First, the lowest J”-value for which the splitting was resolved was high (15.5). Second, the sign of  VA  was shown to be positive (see below), i.e., the (e—e) line was lower in frequency  than the corresponding (f—f) line. Finally, the J-dependence (at least for the range where the splitting was measured) clearly does not go as J . This behaviour indicates 2 that C(2;  )  and C(1;  )  differ substantially and in just such a way that the two terms in  Eq. (5.20) almost cancel for J” 14.5. A non-linear least squares analysis of the frequencies in Table 5.2 was carried out for each isotopomer separately. The weights used were in the frequency measurement. For v  =  , 2 1/c  where e is the uncertainty  0, the rotational constant varied was B 0 itself.  For v  >  (B,,  B ) 0 . The same procedure was followed for all the other parameters used (except as  —  0, the parameter varied was (B , 1  —  Bo); B,, was then calculated from B 0 and  noted below). Because no transitions between the two spin substates were observed, the fit was insensitive to the absolute value of A,,. Hence A 0 was fixed 4 to the determination 1t should be noted that the expression for the energy difference between the two spin substates given 4 by Lubic e aL [fl] in Eq. (5) of their HC1 work (and subsequently used by them in the analysis of HBr data [ll]) i.e., 312 1 2 E( ) 1  —  112 1 2 E( ) 1  =  A + 2B  ,  ...  Coniinned on Page 13  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  0.05  •  Br 7 H 92 H  .01  .  P-branch (2-1)  (Merged Model Fit)  0.03  E  •  130  tz 2.5  6.5  10.5  J  14.5  18.5  Figure 5.4: The splitting between the two A-doublets in the P-branch of the (1’ —0”) and (2’ 1”) vibrational bands in the 2113/2 spin substate of Br 79 plotted as a function of H defined equal in to —A Eq. (5.20). The points with their error bars J”. The ordinate is represent the experimental data. The solid curves indicate the values calculated from the merged model using the constants in Tables 5.3. The horizontal dashed lines represent the lower limit for the magnitude of the resolvable splitting in the experiment. —  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  131  0.06  0.03  EC.) ci)  0.00  -0.03  2.5  6.5  10.5  J  14.5  18.5  Figure 5.5: The splitting between the two A-doublets in the P-branch of the (1’ 0”) and (2’ 1”) vibrational bands in the 2113/2 spin substate of Br 81 plotted as a function of H J”. The ordinate is equal to —A defined in Eq. (5.20). The points with their error bars represent the experimental data. The solid curves indicate the values calculated from the merged model using the constants in Tables 5.4. The horizontal dashed lines represent the lower limit for the magnitude of the resolvable splitting in the experiment. —  —  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft of Lubic et al. [11] and (A,,  —  132  ) was varied for v > 0. The errors obtained for A,, 0 A  for v > 0 then reflect oniy the relative uncertainties and not the absolute error in A . 0 The parameters AD,, and  ‘rny,,  are known  []  to be highly correlated. In principle, they  can be separated by using their isotopic dependence  79 H Unfortunately, for Br+  and HSlBr+, this dependence was so weak that the correlation could not be significantly reduced. In the final analysis, y,, was fixed at zero. The parameter  qD,,  could not be  determined and was held fixed at zero. Two higher order distortion parameters proved to be significant, namely At,, and H,,. For the latter, the v-dependence was not important and so all H,, were constrained to a single value. The best fit parameters obtained for Br+ are given in Tables 5.3 and 5.4, respectively. 1 Br+ 7 H 9 and HS Although the overall quality of the fit was good, there were three specific difficulties that showed the merged model to be inadequate. First, the vibrational dependence of AD,, was rather unusual; see Tables 5.3 and 5.4. Although smooth, it changes sign between  v  =  0 and 1, undergoes a clear minimum at intermediate v, and would appear to change  sign again for v> 5. This is demonstrated in Fig. 5.6 for the case of H Br. 8 1 Second, the vibrational dependence of q,, was unusual as well; see Tables 5.3 and 5.4. Fig. 5.7 Br+. 8 1 In this case, the parameter change in shows a plot of q,, as a function of v for H magnitude by “.40% between v  =  0 and 1, and by a further 12% between v  =  1 and 2.  Then it remains approximately constant. Third, the model produced systematic residuals for the A-doubling splitting  .VA  in the P-branch of the  2113,2  fundamental. This is  differs from that obtained here (see Table 5.3), i.e., 312 f 2 E( ) l  —  112 1 2 E( ) 1 = A  —  2B  by 4B. Since B is large (—.8 cm’), such a difference is of some concern when fixing A 0 to the value obtained by Lubic et al. [11] in our fit. However, the value of A 0 was obtained by Lubic et al. by using the optical data of Lebreton [9], and is very close to the value reported by Lebreton from his analysis. Since the expression for the energy difference between the two spin substates in the present work is the same as that of Lebreton [p], it is likely that there is a misprint in the expression given by Lubic et al.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  133  Table 5.3: Molecular Parametersa in the Merged Model for Br 79 in the X 211 Elec H tronic State  V  1 T+  0  2348.2339( 9)  1  B  Pt’  —2651.5910  7.96125(16)  2.0665(14)  2257.8748(14)  —2649.1780(16)  7.72144(18)  2.0410(14)  2  2168.8041(16)  —2646.3445(23)  7.48346(22)  2.0118(16)  3  2080.9818(21)  —2643.0428(27)  7.24697(21)  1.9769(17)  4  1994.3550(43)  —2639.2057(42)  7.01292(42)  1.9410(22)  —2634.7516(94)  6.78007(38)  1.8996(22)  5  A  —  d 104  io  v  q x 102  0  —0.768(30)  0.960( 62)  0.42(24)  3.5400( 67)  —6.69(69)  1  —1.062(32)  —0.747( 65)  1.07(25)  3.4976( 72)  —6.04(72)  2  —1.202(41)  —1.405( 71)  1.30(28)  3.4618( 87)  —5.79(81)  3  —1.187(39)  —1.430( 77)  1.13(31)  3.4298( 82)  —5.46(86)  4  —1.228(82)  —1.035(104)  0.82(46)  3.4132(104)  —5.41(89)  5  —1.179(72)  —0.284(137)  0.58(46)  3.3824(103)  —5.18(94)  AD x  x 106  x  PD  X  a All entries are in cm . Quoted uncertainties in parentheses are one standard 1 deviation and apply to the last digits of the values. b 0 A was held fixed at the value obtained in Ref. [ii]. For v> 0, the errors in A are relative only; they do not reflect the absolute error in A . 0 C  The values of -y were held fixed at 0.  d The values of H, were varied but constrained to a single value for all v; the best fit result is 9(2)x10 9 cm . 1  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  134  Table 5.4: Molecular Parametersa in the Merged Model for H Br 8 1 in the X 211 Elec tronic State  v  1 T+  0  2347.8825(10)  1  B  p  —2651.5894  7.95875(16)  2.0675(13)  2257.5516(15)  —2649.1741( 17)  7.71906(17)  2.0417(14)  2  2168.5078(16)  —2646.3430( 22)  7.48109(21)  2.0124(15)  3  2080.7128(21)  —2643.0413( 28)  7.24488(22)  1.9786(17)  4  1994.1151(49)  —2639.2022( 42)  7.01064(43)  1.9414(22)  —2634.7538(103)  6.77787(40)  1.9001(22)  5  A  —  d 104  io  v  q x 102  0  —0.754(28)  1.022( 66)  0.24(25)  3.5420( 79)  —7.31(60)  1  —1.050(30)  —0.727( 70)  0.98(27)  3.4984( 84)  —6.60(63)  2  —1.177(38)  —1.350( 75)  1.08(31)  3.4603( 99)  —6.71(72)  3  —1.198(40)  —1.382( 84)  1.00(35)  3.4283( 98)  —6.26(88)  4  —1.189(83)  —1.095(108)  1.23(47)  3.4078(115)  —6.04(85)  5  —1.143(73)  —0.286(150)  0.75(49)  3.3774(115)  —6.00(89)  AD  X  A  X  106  D  PDv X  a All entries are in cm . Quoted uncertainties in parentheses are one standard 1 deviation and apply to the last digits of the values.  b 0 A was held fixed at the value obtained in Ref. [ii]. For v> 0, the errors in A are relative only; they do not reflect the absolute error in A . 0 C  The values of -y were held fixed at 0.  d The values of H were varied but constrained to a single value for all v; the best fit result is 12(3)x 10 cm . 1  Chapter 5. Rotational Energies and Fine Structure in flBr and HI  135  0.002  Br 8 H 1 (Merged Model Fit)  0.001  0.000  -0.001  -0.002  0  1  2  3  4  5  V Figure 5.6: The variation of the effective spin-orbit distortion parameter AD with v in lfl 8 Br+ obtained from the fit using the merged model. The curve connecting the points is not a fit; it is intended only as a guide to the eye.  Chapter 5. Rotational Energies and Fine Structure in flBr and Hft  136  -0.006  Br 8 H 1 (Merged Model Fit)  -0.010  -0.0 14  I  I  0  1  I  2 V  I  I  3  4  5  Figure 5.7: The variation of the effective A-doubling parameter q with v in Br 81 H obtained from the fit using the merged model. The line connecting the points is intended only as a guide to the eye.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  137  illustrated for H Br 7 9 and H Br 8 1 in Figs. 5.4 and 5.5, respectively. The solid line shows the splitting predicted by the merged model. For J”  =  15.5 (see Fig. 5.5), the predicted  splitting is much smaller than the resolution limit of the experiment (indicated by the dashed lines), whereas the doublet is clearly resolved. For J”  18.5, the prediction is too  large by several times the error estimate e. These difficulties could not be removed within the framework of the merged model. For example, if the (e—e) and (f—f) identification of the four doublets just mentioned was interchanged, the resulting best fit predicted that the A-doubling in the P-branch should be resolvable for J7.5 (see Fig. 5.8), in clear contradiction to the observations. Fig. 5.8 also shows that merged model fit of the A-doublet splittings gets worse if the assignments of the e and the f levels are switched. The signature of the failure of the merged model lies in the results for the isolated levels for (v  = 0, Q = 3/2). The most dramatic change in q was between v = 0 and  1. See Fig. 5.7. Furthermore, the merged model fit the A-doubling splitting for all the vibrational bands other than the P-branch of the 5.5. For these reasons, the requirement that  2113/2  fundamental. See Figs. 5.4 and  IAe <<We was relaxed and the split model  was developed. 5.3  Analysis of HBr with the Split Model In order to treat properly the non-resonant interstack interactions , it must be rec 4  ognized that the approximation  Ike <<We entered the merged model primarily through  the leading distortion constants, namely ADD, D, and  PDV  For each of these constants,  the associated matrix elements were re-calculated starting from Eq. (5.5). For each X in turn (X  = A, B or p), the appropriate operator X(R) was entered in the first matrix  element factor on the right hand side, B(R) was entered in the second such factor, and See Footnote 2 on page 111. 4  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  138  0.05  Br 81 211 H 004  _,.—  P-branch  /  (Merged Mode’ Fit)  (2-1)  0.03  C.)  0.02  (1-0)  ::zzE 2.5  6.5  10.5  J  14.5  18.5  Figure 5.8: The merged model fit of the observed A-doublet splittings in 2113/2 P-branch of the (1’ 0”) and (2’ 1”) vibrational bands of Br 81 with the e and the f assignments H interchanged in the fundamental band relative to the assignments in Table 5.2. The horizontal dashed line represents the lower limit of resolvable splitting in the experiment. —  —  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  139  the resulting product was symmetrized. Throughout this process, care was taken to keep track of the dependence on v and Q in the energy denominators. As an example, if we take X(R)=B(R), the above procedure gives the split rotational centrifugal distortion matrix as _z2  (z__l)/Ei  (z-1)/f  _(Z_2)2  —(z—i)  + D, 112 *  + D:, 2 , 3  -(z-1)  where =  (5.21)  —  v, 2 fl  —  — —  =  L. o  —  V,V’  v’, 2 fl  V’,V  v’v v,1/2  (5.23)  —  v’v  v,3/2  —  v’,1/2  Here, D represents the intrastack interactions, whereas D, 112 and D, 312 represent the interstack interactions. In the definition of DV, the two sums containing (E 112 and  312 (E,  —  —  112 , 1 E )  312 in their denominators have been grouped together into a single sum , 1 E )  shown in Eq. (5.21). The a-dependence in this merged sum can be dropped since the potentials for the two fZ-substates are assumed to be equal. In the case of HBr+, only the Lv  =  +1 interstack interactions were found to be significant. Thus, in the split  model, retaining only the v  =  +1 terms for the interstack interaction, each distortion  Chapter 5. Rotational Energies and Fine Structure in HBr and H1  140  parameter XD,, of the merged model is, in general, replaced by three: 0 = XD -  *  2 XDU,l/  [J v’v ‘-iv, 2  *  i_It,,, 2fl  1 1 Xt,,t,_ Bt,_ , t,  —  —  v,1/2  2 / 3 XDt,,  —  =  Cx  1 1 Xt,,_ Bt,_ , t, v,3/2  +  v—1,3/2  —  v—1,1/2  —  (5.24)  ;  1 1 X,,,t,÷ Bt,+ , t, v,1/2  +  ,  Xv,+i Bt,+j,t, .  v,3/2  (5.25)  —  —  (5.26)  v+1,1/2  where Cx=—i for X=B and +1 otherwise. The parameters with the tilde and star in Eqs. (5.24) to (5.26) characterize the intrastack and interstack interactions, respectively. For each of these parameters, the energy denominators are the (J—*0) limits just as was the case in Eq. (5.7). The matrix elements for these nine split model parameters (3 each for X=B, A and p) are given in Table 5.5. For each of the three operators X(R) being considered, a single constant T was used to parametrize the various numerators in Eqs. (5.25) and (5.26) for all v. That is, one writes: Xt,,t,Bt,,t,  Tx .F(v, v’) ;  (5.27)  where all the dependence on v and v’ has been put into the function .F. It is straight forward to relate T to the equilibrium value XDe of XDt, (e.g., to relate TA to ADe). To begin with, X(R) is expanded to first order in  =  (R  —  Re)/Re, where R is the  equilibrium internuclear spacing: X(R)  =  Xe + X  .  (5.28)  Here Xe is the equilibrium value of parameter Xt, and X is the first derivative of X(R) with respect to  c  evaluated at R  =  Re. Both X and X are in wavenumbers. By setting  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  141  Table 5.5: The Effective Hamiltonian Matrix Elements for the Split Model” b, c Parameter  (Heff)ii  22 (Heff)  (Heff) 12  T  1  1  0  A  —1/2  1/2  0  ADV  —z  z—2  0  112 A,  0  0  312 A,  0  0  B  z  z—2  —y  Dv  2 —z  2 —(z—2)  3 y  112 D,  2 —y  0  312 D,  0  2 —y  Pv  (i/ + 1)/2  0  PDv  :FzV’+z  0  Pv,1/2  y / 2 2  0  Pv,3/2  0  y / 2 2  —(+2/ + z + 2)y/8  +/ + (z + 1)/2  y / 2 2  —(+\/ + l)y/2  —1  0  qv  7v  d  —y/4  (z  —  zy/2  —(z —  a In this model the condition that IAI << We (which led to the merged model) has been relaxed. As a result of this, each quartic distortion constant of the merged model (see Table 5.1) is replaced by three: one tilde parameter and two star parameters. These account, respectively, for intrastack and interstack interactions. The matrix for XDV of the merged model equals the sum of the matrices for XDV, of this model. The higher order parameters 12 and X, and fl in this model were treated differently; see text.  b The upper and the lower signs in the matrix elements correspond to the e and the f energy levels, respectively. C  z  =  (J +  )2  ;  y  =  (z —  1)1/2  d In Ref. [13, 62], this is denoted by 7(1) 7(2). The second order term + the Van Vieck transformation, and is usually the dominant term.  7(2)  arises from  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  =  q.  142  a change of variables is made to the dimensionless normal coordinate 5  Using the harmonic matrix elements of q, i.e., (vqv+1)  =  ( ) v+1 2  1/2  v)]  }h/2  (5.29)  it is easily shown that: =  x  { [(  +  +  (vi;  (v’  —  v) = +1.  By fitting the vibrational dependence of XD , to a power series in (v + 1 shown using the work in Ref.  ),  (5.30) it is easily  [2] that: 2 1 Xe’=() XDe.  (5.31)  Then X,’  Cx  ()  3/2  Xije  + (v +  /  ; jj  1/2 .  (5.32)  The final results for the factors in Eq. (5.27) are: =  ()3 —  Cx  De XDe;  (5.33)  with F(v, v’) = (v +  +  (v  —  v) .  (5.34)  The form of the dependence on v and v’ in Eqs. (5.30), (5.32) and (5.34) assumes that v’—v = +1. The dependence of X 2 on v reflects the fact that, as was mentioned in Sec. 5.1, / 3 the state with (v = 0, f = 3/2) is isolated: it has no nearby interacting partner from = 2r(pcv/h)’/ x, where p is the reduced mass, c is the speed of light, v is the frequency in units of 2 , h is the Planck’s constant and x is the displacement of the atoms from their equilibrium position. 1 cm  Chapter 5. Rotational Energies and Fine Structure in HBr and HP the  2111/2  143  spin substate. As can be seen from Eqs. (5.25) and (5.26), for each 1 X 12 and  , there is one dominant term with a near-resonant denominator, namely that for 312 X, level (v, 1 = 1/2) interacting with level (v + 1, ! = 3/2). There is one exception: for v=O, the term in X, 312 with the small energy denominator vanishes. The energy shifts produced by X 312 can cause the A-doubling in the 2113,2 ground vibrational state to differ significantly from that for the higher 2113,2 vibrational states. The parametrization of the split model is very similar to that of the merged model. intrastack constant XD replaces its merged model counterpart XD, and three Each 6 5 constants T are added, namely TA, interstack  T B ,  and T. The energy denominators  in Eqs. (5.25) and (5.26) could be calculated from the results for earlier iterations with sufficient accuracy since the contribution from these interstack terms were relatively small. At this initial stage, the higher order distortion constants,  and H, were  treated the same as they were in the merged model. Attempts to improve significantly on the best fit of the merged model at first were not successful. Using the equilibrium values of the molecular parameters obtained from the merged model, the three T were calculated from Eq. (5.33) (with XDe approximated by XDe)  and held fixed, while the remaining parameters were treated as were their counter  parts in the merged model. The overall quality of the fit got worse. The residuals in the best merged model fit did permit one interstack parameter to be floated. When TB was varied, for example, the original overall quality of fit was recovered, but TB tended to a value which was lower in magnitude by a factor of 3 to 4 than its theoretical estimate of  . The v-dependencies of AD and q, became more reasonable and the 2 cm  disagreement with the high J A-doubling measurements in the P-branch of the  2113,2  fundamental was reduced. However, the overall situation was unacceptable and did not improve when the calculation of the three Tx values from the equilibrium parameters See Footnote 2 on page 111. 6  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  144  was iterated. (In all iterations except the first, Xne was used in Eq. (5.33).) A different  type of refinement was clearly necessary. The third order Van Vieck transformation was then taken into account. The expres sion for third order Van Vleck transformation is given by (v;QHjv;1z’)  =  >  > > ,,  [7]  Ü v’; cz”) (v’;fz” E  (v;  0 — E, (E, ) (E, 011 01 — (v, 12  —  v, ci”) (v, ci”  (v, ci  1  K:”; 12” H v; 12’)  Ü v’ç ci”) (v”, 12” i v, ii’)  n) (Ev,cr — ) 2 — E.jit,ç111 E 0 , 1  fl v”v  QIHI  n”)  Ü v”, ci”) (v”, ci” fl v, ci”) (v, ci” U (no  V”v  —  no  ‘Ino  ‘—v”,O”) k’-v,Q  —  -,  ci’)  no  (5.35) where we have suppressed J and r from the wavefunctions and the energy denominatores for simplicity. As expected, the third order correction is fairly complicated and contains many terms. However, most of them are much smaller in magnitude than the smallest terms that need to be retained here. In the split model, we have incorporated only the most important terms; these are related to the third order rotational distortion parameter, H, of the merged model, and have both, a large numerator and a small denominator. The matrix elements of the effective Hamiltonian containing the leading terms of the third order correction to B using the split model are then given by I\V, .  1  ij() IL  ff  .  V,  — —  1 B÷ B+i,+ , 1  ‘  /  E 1 i\ °v,- — E°v+1,}) ‘E° \,  —  B  — E°v+14 , 1 B_  (Eo _ 3 E0 _Eo 1 1 i(Eo v, v—1,,j \ V, v—l,r  \  Z Z  —  z(z—1)  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  1 B+ B,+ , 1 —  (E  -  ) 13 E0÷  1 (E  —  ) 13 E0+  145  (z—1)(z—2); (5.36)  (v;Hv;)  =  v,  v—1,  B (ED  v,-  —  v,  v—1,  1 B_ B,_ , 1  (Eo E°v_ 1 ,J 1 \ t,  —  E°v—1,1  1 B+ B,+ , 1  (‘ +‘) -  /\V,i .  (3)  eff  21 V  —  .  —  I  2  1 2  (  B,v B,_ 1 f  IE° — 3 E°v—l,j ) v,  (z-1)(z-2)  z(z-1)  (z  -  1 )(z  -  2 ),  -  (5.37)  +  /  B,+i B , 1 + 3  IE° — 1 E°v+1, v,  1 B_ B,_ ,_ B_ 1 , 1  (EDv, \  (ED  —  ED  1 v—l,J  (ED 1 \ v,-  —  ED  v—1,  1 B+ B,+ ,+ B+ 1 1  V,r  (z  —  i>/J.  v+1,  (5.38) In each denominator, there is one small energy difference between interacting states. In each numerator, only the matrix elements of B(R) enter, as these have the largest magnitude. Furthermore, each numerator has one factor diagonal in v.  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  146  By introducing the third order Van Vieck transformation, the parameter H, was ‘split’, much as  was split earlier. The intrastack contribution 7 is characterized by  the constant H . (The explicit form for this term is not given here.) The interstack 0 6 arise from (v; ! contributions  IH v; ct).  To parametrize these matrix elements as  given in Eqs. (5.36) to (5.38), two steps were taken. First, each product of the form 1 was written as the product of TB and a linear function of v by using Eq. ± , 0 B (5.27) with X=B. Second, each diagonal matrix element of B(R) was written in terms of the equilibrium value Be of the rotational constant and its leading vibrational correction linear in v. This correction (proportional to  aB)  was introduced to test the importance  of a quadratic dependence on v in the numerators in the right hand side of Eqs. (5.36) to (5.38). This term in crB proved subsequently to be insignificant, but it was retained for completeness. The above-mentioned steps lead to the following matrix elements: = (v,H1v,)  v,  v,  v+1,  v+1,-  TB {Be_aB(v+)}  -  -  [TB  -  2  (  v,  (  —  v,  (v+1)]  —  (z-1)(z-2)  v+1,)  (5.39)  TB {BeaB(v1)} 2 v_14)  z(z -1)  -  {Be_aB(v+)}  —  See Footnote 2 on page 111. 7  ‘) (‘  (v)  —  (v) v_1,[)  (z-1)(z-2)  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  -  [ 1: v,  (  (v,H v,)  =  TB {Be  1  —  B  o,-  (v  +)  —  B(V  (E  -  v—1,  [TB {Be_ -  (v+)}(v)  +  o  3—i_i  v,  {Be -  v,  (  —  ii  (z  (o il_i 1 \ v,  1)(z -2)  (5.40)  1 (E  + 1)  (v+  —  ) 1 E+  2  (v)  €o .L.J  v—1,3  B(v+)}+ v,  -  v+1,)  ) 1 E 1  v—l,j  v+1,  +  ]  zz -1)  )} (v) + TB {BeB  TB {Be—aB(v—)}  (o li_i  v—1,  147  1)1  (z  -  1).  v+1,  (5.41)  By using Eqs. (5.39) to (5.41), the interstack terms arising from splitting H can be varied through TB; no new parameters were required. The full split model used here for HBr+ has now been developed. The estimates of the energy shifts due to the third order off-diagonal (in 1) matrix elements of the effective Hamiltonian, represented by Eq. (5.41), were down by a factor of about four relative to the diagonal terms of Eqs. (5.39) and (5.40). Inclusion of this off-diagonal term into the fitting routine failed to show any significant difference in the quality of the fit. This term was therefore not included in the final fit of our data. In addition, there are interstack terms that arise from the splitting of the third order distortion constant A. These terms also involve products with three matrix element  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  148  factors; two are of B(R) and the third is of A(R). However, these interstack terms proved to be insignificant, and the treatment for /  was returned to the merged form.  (3)  3\ 11 v; , the first two terms in Eq. (5.40) provide a higher order mechanism eff through which the (v = 0, Q = 3/2) state is affected by its lack of a nearby partner.  In v;  .  Both of these terms vanish for v=0, as they must, since the energy denominator is not defined for v=0. As was the case with the lower order mechanism associated with X, , 312 the energy shifts produced by the first two terms on the right hand side of Eq. (5.40) can cause the A-doubling in the (v (v > 0,  =  =  0, Q  =  3/2) state to behave differently from that for  3/2).  In the final round of fits, the intrastack constant H, was found to be negligible and it was fixed at zero. The third order distortion constant  (treated in the merged model)  did not show a significant v-dependence. All A were constrained to equal A , which 0 was then varied. The three interstack constants TA, T , and T were calculated by using 8  the derived equilibrium molecular constants of the merged model. Once the split model calculation was carried out, the three Tx were re-calculated and the procedure iterated. Convergence was very rapid. The overall quality of the fit was good. The best fit values of the parameters for Br+ 7 H 9 and 1 HS B r+ are given in Tables 5.6 and 5.7, respectively. The final values of the three T are listed as well. For both isotopomers, the differences between the observed and calculated frequencies are given in Table 5.2. For each isotopomer, the normalized 8  2 x  for the best fit was 0.3. The fact that this was less than unity by a factor of three  indicates that the random errors are somewhat less than the estimates e made for the experimental uncertainty. In light of the discussion in Sec. 3.3 of Chapter 3, this is not Normalized x 8 2  =  N  1  N  M  / obs  2 cal\ —  i=1  )  ,  where N is the number of data points, M  is the number  of parameters determined from those N data points; y5 and y’ are the observed and calculated transition frequencies, respectively; and o, is the associated uncertainty in the observed frequency. —  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  149  Table 5.6: Molecular Parametersa in the Split Modelb for Br 79 in the X H 11 Electronic 2 State  v  T 1 + 0 0  A  B  p  0  2348.2349( 8)  1 2 3 4 5  2257.8757(13) 2168.8062(13) 2080.9826(20) 1994.3559(25)  —2651.591 —2651.591(33) C —2649.1786(16) —2646.3447(22) —2643.0417(25) —2639.2034(33) —2634.7442(47)  7.95900(16) 7.95701(14) e 7.71885(17) 7.48075(21) 7.24423(20) 7.01024(42) 6.77742(37)  2.0644(13) 2.0443(24) 2.0344(14) 2.0032(15) 1.9678(16) 1.9321(22) 1.8910(22)  —  v  q x 102  0  —0.691(29) —0.689(10) —0.776(31) —0.824(40) —0.785(38) —0.820(81) —0.771(71)  1 2 3 4 5  AD x  d 103  2.512(46) 2.43(27) e, f 3.175(49) 4.023(55) 5.081(52) 6.319(55) 7.753(56)  x 3.5118(44) 3.516(27) C 3.4827(45) 3.4565(50) 3.4160(47) 3.3973(83) 3.3627(75)  Dv X  io  —5.82(63) —5.34(66) —5.11(75) —4.90(78) —5.14(82) —5.17(85)  a All entries are in cm . Quoted uncertainties in parentheses are one standard 1 deviation and apply to the last digits of the values. b In this model, the distortion parameters ADD, D and PDv of the merged model are split; their counterparts with tilde aild star take into account intrastack and interstack effects, respectively. The tilde parameters were varied. The star parameters were held fixed at the following values, all in cm 2 : TA = —5.4712; TB = 0.8566; = 0.1513. A was held fixed at the value obtained in Ref. [11]. For v> 0, the errors in A are relative only; they do not reflect the absolute error in A . 0 d The values of were held fixed at 0. The values of were varied (in merged form) but constrained to a single value for all v. The best fit value of 4.6(15) x iO cm’.  e This value is obtained from Ref. [11].  f  See discussion of  ADO  in the text.  is  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  150  Table 5.7: Molecular Parametersa in the Split Modelb for H Br 8 1 in the X 11 Electronic 2 State v  1 T+  A  B  Pv  0  2347.8833( 8)  1 2 3 4 5  2257.5524(13) 2168.5095(14) 2080.7141(20) 1994.1140(27)  —2651.589 —2651.589(30) e —2649.1770(16) —2646.3456(22) —2643.0435(25) —2639.2058(34) —2634.7475(48)  7.95646(15) 7.95460(13) e 7.71639(16) 7.47832(20) 7.24206(20) 7.00788(42) 6.77518(38)  2.0657(12) 2.0464(22) e 2.0357(13) 2.0045(14) 1.9703(16) 1.9329(22) 1.8922(21)  V  0 1 2 3 4 5  —  q  X  102  —0.675(28) —0.6828(93) e —0.754(30) —0.787(37) —0.778(39) —0.764(83) —0.719(72)  x  d 103  2.503(50) 2.34(25) e, f 3.189(52) 4.078(58) 5.156(56) 6.385(59) 7.816(59)  v  3.5085(47) 3.509(24) e 3.4784(48) 3.4518(54) 3.4127(52) 3.3914(80) 3.3579(74)  x iO  —6.36(56) —6.01(59) —6.23(67) —6.13(76) —5.97(77) —6.31(79)  a All entries are in cm’. Quoted uncertainties in parentheses are one standard deviation and apply to the last digits of the values. b In this model, the distortion parameters ADD, D and PDv of the merged model are split; their counterparts with tilde and star take into account intrastack and intestack effects, respectively. The tilde parameters were varied. The star parameters were held fixed at the following values, all in cm 2 : TA = —5.3797; TB = 0.8553; T = 0.1374. C  A was held fixed at the value obtained in Ref. [U]. For v> 0, the errors in A are 0 relative only; they do not reflect the absolute error in A . 0  d The values of were varied (in merged Yt’ were held fixed at 0. The values of form) but constrained to a single value for all v. The best fit value of is 5.9(17) x iO cm’. e This value is obtained from Ref. [11].  f  See discussion of  ADO in the text.  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  151  surprising. The final split model fit was clearly superior to its merged model counterpart. First, the vibrational dependence of AD 0 was smooth and monotonic; no sign changes were involved (see Tables 5.6 and 5.7). This is demonstrated in Fig. 5.9 for the case of Br. 81 H For comparison, the variation of AD with v obtained from fits using both the models  (merged and split) are shown. They are indicated by M and S in the figure. In the split model, the change of AD with v is somewhat larger than might be expected, but this is difficult to interpret because AD,, has absorbed the  j’,,  contribution. Second, the  vibrational dependence of q is much improved. See Tables 5.6 and 5.7. For Br, 81 the H values obtained from both the models are plotted in Fig. 5.10. The results of the split model shows that there is still an increase in magnitude of q,, from v=O to v=1, but now it is 12%, which is only twice the estimated error in constant (to within experimental error) for 1 J P-branch lines in the  2113/2  v  Ii  —  qo. Furthermore, q,, is  5. Third, the A-doubling for the high  fundamental is fit to well within the experimental error  with virtually no change in the quality of the fit of the A-doublings in the (2—1) band. See Figs. 5.11 and 5.12. The e and f assignments of the observed transitions in the fundamental band of the  2113,2  spin substate were again checked from the fit of the  split model. On switching the two assignments, the model predicted that the A-doubling splitting should be observable for J  10.5, a prediction which is in clear contradiction  with the experimental observations. Moreover, the fit showed a clear deviation from observed splittings (see Fig. 5.13). The improvements with the split model were accomplished using the calculated values of the interstack parameters. To make a check on how reliable the calculated interstack Br+ 8 1 transition frequencies by varying TB parameters were, a fit was performed on the H as one of the parameters. The fitted value of TB was 0.92(5) cm , which is very close to 2 the estimated value of 0.86 cm . The rest of the parameters as well as the quality of the 2  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  0.008  Br 8 H 1  -  0.006  152  -  0.004 S 0.002  0.000  -0.002  0  1  2  3  4  5  V Figure 5.9: The variation of the effective spin-orbit distortion parameter AD with v in 81 obtained from fits using the merged model (M) and the split model (S). The H Br curves connecting the points are not a fit; they are intended only as a guide to the eye.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  153  -0.006 S  ‘I  M  E  -0.010  -0.014  I  0  1  2 V  I  I  I  3  4  5  Figure 5.10: The variation of the effective A-doubling parameter q with v in Br 81 H obtained from fits using the merged model M and the split model S. The lines connecting the points are intended oniy as a guide to the eye.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  154  0.05  0.03  0.01  -0.01  -0.03 2.5  6.5  10.5  J  14.5  18.5  Figure 5.11: Split model fit of the splittings between the two A-doublets in the P-branch of the (1’—O”) and (2’—l”) vibrational bands in the 2113/2 spin substate of Br 79 plotted H as a function of J”. The ordinate is equal to defined in Eq. (5.20). The points with their error bars represent the experimental data. The solid curves indicate the values calculated from the split model using the constants in Tables 5.6. The horizontal dashed lines represent the lower limit for the magnitude of resolvable splitting in the experiment.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  155  0.06  0.03 C.)  0.00  -0.03  2.5  6.5  10.5  14.5  J  18.5  Figure 5.12: Split model fit of the splittings between the two A-doublets in the P-branch of the (1’—O”) and (2’—l”) vibrational bands in the 2113/2 spin substate of Br 81 plotted H as a function of J”. The ordinate is equal to —vA defined in Eq. (5.20). The points with their error bars represent the experimental data. The solid curves indicate the values calculated from the split model using the constants in Tables 5.7. The horizontal dashed lines represent the lower limit for the magnitude of resolvable splitting in the experiment.  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  156  0.05  0.04  _—‘  0.03  Q  0.02  0.01  0.00 2.5  6.5  10.5  J  14.5  18.5  Figure 5.13: The split model fit of the observed A-doublet splittings in 2113/2 P-branch of the (1’ 0”) and (2’ 1”) vibrational bands of Br 81 with the e and the f assign H ments interchanged in the fundamental bandrelative to the assignments in Table 5.2. The horizontal dashed line represents the lower limit for the resolvable splitting in the experiment. —  —  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  157  fit showed negligible difference between the two cases (i.e., TB fixed and varied). We were unable make the corresponding test for the other two calculated interstack parameters, i.e., TA and T, since the fit did not converge if either of these two parameters were varied. As mentioned earlier, the final fit was performed with all the three interstack parameters held fixed at their calculated values. Hence, in the split model, no new parameters were varied. Indeed, the number of parameters varied was reduced by six; H,, was eliminated and the six different Ag,, in Tables 5.3 and 5.4 were replaced by a single value which applied for all six vibrational levels. , Po, qo, 0 The current results for B  ADO  and D 0 for the ground vibrational state of  79 are compared in Table 5.6 with the values obtained by Lubic et al. H Br  [ij].  For  each of the two smaller parameters, the results from the two different experiments agree within the error. The value of  ADO  obtained in the current work cannot be compared  to the value obtained by Lubic et al. [ii] because the two results were determined by different methods. The parameters AD,, and ‘y,, are highly correlated [63]. Lubic et al. [11] fitted their data by fixing ‘yo to the unique perturber value. In the current work,  the values of  AD,,  were obtained by fixing 7,, to 0 in the fit. This makes the comparison  of the ADO values obtained from two different methods of fit difficult. For B 0 and P0, the relative precision is much higher; for each of these two, the disagreement is -‘8 times Br+ 8 1 is presented in Table 5.7. the expected error. The corresponding comparison for H The differences for the second isotopomer follows the same pattern as that of the first, suggesting that systematic effects are involved. However, the small discrepancies in B 0 and P0 are difficult to interpret because the two works are so different, both with respect to the theoretical model and with respect to the nature of the data set involved.  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  158  Molecular Constants for HBr+  5.4  In the split model, to this point, attention has been focussed on the three T parameters that characterize the interstack interactions. However, once the effect has been removed of these three and of the one sextic constant A (for which the v-dependence was negligible), it is possible to obtain more reliable equilibrium values for the eight prin cipal parameters that characterize each vibrational level. These are T , A 0 , B 0 , p 0 , 0  , 0 AD  , and 0 D  Dv  , 0 q  Aside from the vibronic energy T , the remaining seven will be here 0  generically represented by P. The quantity 0 T 1 + 0 in a power series in (v +  )  constants  WeZe  WeXe, WeYe  and  1 [T,÷  —  in terms of the harmonic frequency  ] has been expanded 0 T We  and the anharmonic  using Eq. (2.6) of Chapter 2. In order to use a common  notation for the seven different P , these parameters are here expanded in (v + 0  )  as  follows: 0 P where  —  p =  Pe +  p  v (v +  —1 if P=B, and  =  1  + v (v +  12  +  p  (v +  +  ...,  (5.42)  +1 otherwise. The signs of the coefficients in Eq. (5.42)  follow the usual conventions [1], although the symbols used in some cases do not. For each of the eight principal parameters, the expansion coefficients have been de T9 by performing a linear least squares fit to the values listed in Table H termined for Br+ 5.6. A similar calculation was done for Br 81 using Table 5.7. The results obtained H Br+ 7 HS B 9 and 1 r+ are given in Tables 5.8 and 5.9, respectively. In each case, a for H good fit was obtained and there was no difficulty in determining the minimum degree of polynomial to be used in the expansion. For A, the differences (A 0  —  ) with v > 0 0 A  were determined more accurately than A 0 itself. The differences were fit directly so that the error in A 0 enters only in Ae and does not affect cv, 1 A, or 3  ‘yA.  The parameter  q was the only one with a unusual v-dependence; see Fig. 5.10. The values of q, and  q 9 /  cvq  79 and Br H for Br 81 listed in Tables 5.8 and 5.9 characterize the best model H  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  Table 5.8: Equilibrium Molecular Constants a, b, in the Split Model Constant  Value  C  159  of the 2 X 1 1 Electronic State of H Br 7 9  Constant  Value  We  2439.8734(40)  Be  8.07978(22)  WeXe  46.1889(33)  crB  0.24204(21)  WeYe  0.2304(10)  B X 3 /  iO  9.56(38)  WeZe  0.00192(11)  X  iO  3.5276(39)  )d 2 _ 3 ( 652663 3  A  x 106  )d 3 ( 20609 4  aA  7A  3 ADe x i0 AD /AD  2.0780(18)  Pe  )d 1 ( 01597 9  A 3 /  —3.02(14)  x i0  —2.68(16)  x iO  —1.29(27)  2.253(61)  De x iO  —6.23(49)  x iO  4.69(51)  cr x 106  —2.2(16)  x iO  9.66(85)  q  X  iO  —6.53(42)  X  iO  9.5(39)  X  iO  1.40(73)  ),  a In the expansion in powers of (v + traditional symbols are used for the vibratio nal term values. However, for the remaining parameters, some non-traditional symbols are used. See Sec. 5.4 and Eq. (5.42) in particular.  b See Sec. 5.4 for a discussion of the expansion of q. C  All entries are in cm . Quoted uncertainties in parentheses are one standard devia 1 tion and apply to the last digits of the values.  d The error in A 0 is absorbed entirely into Ac and does not affect Sec. 5.4.  cA, 1 A 3  or 7A. See  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  Table 5.9: Equilibrium Molecular Constantsa, b, in the Split Model Constant  Value  C  160  of the 2 X 1 1 Electronic State of Br 81 H  Constant  Value  We  2439.4928(41)  WeXe  46.1740(34)  WeYe  0.2302(10)  B X 3 /  io  9.45(38)  WeZe  0.00190(11)  ije X  io  3.5239(42)  X  106  0.24190(20)  )d 2 _ 6 . 3 ( 652 62 0  Ae  )d 3 ( 20626 5  aA  2.0846(12)  X  iO  2.216(67)  x iO  5.18(55)  x 1O  9.09(91)  102  —2.64(15)  x iO  —1.35(26)  ><  )d 2 ( 001065 9  7A  —3.03(14)  Pe  )d 2 ( 01585 0  A 3 /  ADe  8.07715(21)  Be  x iO  —5.66(55)  x 106  —1.4(18)  X  iO  —6.32(40)  X  iO  _9.9(37)  q X 3 /  iO  De  q  1.56(71)  ),  a In the expansion in powers of (v traditional symbols are used for the vibratio + nal term values. However, for the remaining parameters, some non-traditional symbols are used. See Sec. 5.4 and Eq. (5.42) in particular. b See Sec. 5.4 for a discussion of the expansion of q. C  All entries are in cm . Quoted uncertainties in parentheses are one standard devia 1 tion and apply to the last digits of the values.  d The error in A 0 is absorbed entirely into Ae and does not affect Sec. 5.4.  aA,  or -y. See  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  161  for extrapolating q to higher v. However, their physical significance cannot be taken as  that associated with the standard coefficients of the de-perturbed form of q. For each isotopomer, the best current estimate of the equilibrium value of this de-perturbed  pa  rameter is the measurement of qo itself, with an error ‘—‘10%. It should be noted that the  values in Tables 5.8 and 5.9 for  and  We, WeXe, WeYe  tential that applies to both the (1  3/2) and (Q  =  WeZe  =  describe an effective average po  1/2) spin substates of the 211 ground  state. A similar statement applies for the parameters characterizing the expansion of B and D, and for the associated derived constants. For each of the better determined equilibrium constants, the observed isotopic depen dence has been compared with that expected on theoretical grounds [j]. The reduced mass of the HBr+ molecular ion is defined as: t(HBr) Here  me  =  (mH (mH  is the mass of the electron, while  me) mBr me) + mBr  (543)  —  —  mH  and mBr are the masses of the hydrogen  and bromine atoms, respectively. This definition is based on the model of the ion as a bromine atom perturbed by a proton, as described in Sec. 4.4 of Chapter 4 for HBr. Following Herzberg [1] the isotopic constant p is introduced, where:  p  =  [ (H79Brj  /i  2 (H81Brj]  (5.44)  -  From the known values [4] of the masses in Eq. (5.43), p =1.000 15563(85). 1 expected [1] that each of We,  (WeXe)12,  It is  and B’ 2 varies as prn’. From the results in Tables  5.8 and 5.9, it is found that: We  (H79Brj  [ee  (H79Brj  [Be (H79Brj  /  We  /WeXe  /  Be  (H81Brj  =  1.000 1560(23) ;  (5.45)  2 (H81Brj]’1’  =  1.000 161 (72) ;  (5.46)  2 (H81Brj]’1”  =  1.000 163 (27)  (5.47)  .  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  162  In each case, the agreement with the expected dependence on p is good. The equilibrium bond length Re has been calculated from the reduced mass p and the equilibrium rotational constant Be using their well known relationship and H Br, 8 1 respectively, it was found that Re = 1.448 366(19)  A  [1].  For Br 79 H  and 1.448 376(18)  A.  The two values are in good agreement, as is to be expected from the fact that Be was found to be proportional to  _hI2;  see Eq. (5.47). The average value of R = 1.448 371(13)  A.  Of course, this value of Re is model dependent. If HBr+ is taken to consist of a hydrogen atom (H) and a bromine ion (Br+), then the reduced mass of HBr+ is defined as (HBr)  =  mH (mBr mH  +  —  (mBr  me)  —  (5.48)  me)  A,  With this assumption, the average value of R becomes 1.447 981(13) 0.000 390(18)  A from the previous  a decrease of  value. In the absence of other systematic effects [32],  it is expected that the actual value of Re falls between these two values, but closer to that for the model where H is bonded to Br. The potential function can be expanded as a power series in cm’) =  0 a  2  1 [i + a  2 + a  2  +  3 a  [62,  +...]  where ao is the harmonic potential constant (in cm) and a , a 1 , 2  ]: ,  (5.49)  are dimensionless  anharmonic potential constants. The coefficients a can be expressed in terms of the expansion coefficients for T+ 1 and B in a straightforward manner [2]. The resulting expressions are given by 0 a  =  1 a  =  2 a  =  (5.50)  e —  [i +  23L  (5.51) +  a  ;  (5.52)  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  3 a  4 a  =  2 1[/3BWe  1 =  2  We  163  15(23)]  (Wey e  B  )  17  225  2  (5.53)  705 .  (5.54)  The notation used for the equilibrium constants in the above expressions follows the expansion given by Eq. (5.42) and the discussion preceeding this Equation. Eqs. (5.50) to (5.54) were then used to calculate a (n=0, 1, 2, 3, 4) for H Br 7 9 and H Br. 8 1 The values obtained are listed in Table 5.10. For each n, the a 1 determined are the same for the two isotopomers, as is to be expected on theoretical grounds  [2J.  The potential function can also be expanded as a power series in the normal coordinate  V()(_1)  = 4 2 f q 3 q 2+_f 3+_f 4+...  ,  (5.55)  where f 2 is the harmonic force constant and f , 4 3 f , ... are anharmonic force constants. The! sign in Eq. (5.55) denotes a factorial. Since q and q  () w  =  1/2  are related according to (5.56)  ,  comparison of Eq. (5.55) with Eq. (5.49) gives = 3 = f  4Be (18  (5.57)  We  1 ; 2a Be)”  a ; 4 = 24 Be 2 f  (5.59) 1/2  2B ” 3  ( —i) We  5 = f  120  6 = f  2 B 1440 —s- a 4  \  We  (5.58)  3 ; a  J  .  (5.60) (5.61)  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  164  Table 5.10: Harmonic and Anharmonic Expansion Coefficientsa, b for the Effective Po tential  Constant 0 a  Br+ 7 H 9  H81 Br+  184193.8(50)  184196.3(48)  1 a  —2.5077(13)  —2.5075(12)  2 a  4.0494(58)  4.0486(55)  3 a  —5.376(46)  —5.377(44)  4 a  6.73(48)  6.75(46)  a These constants are defined by Eqs. (5.50) to (5.54) (see Sec. 5.4) and are isotopically invariant, a 0 has units of cmi, whereas a , 1 are dimensionless.  ...,  b Quoted uncertainties in parentheses are one standard deviation and apply to the last digits of the values.  4 a  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  Here, f have units of cm 1 and are isotopically variant (f oc  ,L—”/4).  165  Using Eqs. (5.57)  to (5.61), the force constants were calculated for the two isotopomers, and are listed in Table 5.11. These constants can also be written in terms of isotopically invariant force constants, given by  [] / 2 t4ir  F = hcf  \fl/2 C[t We  )  h  (5.62)  79 and H H Br+ 8 1 are given in Table 5.11. As expected, these The values of F for Br+ values are the same for the two isotopomers. The dissociation energy for HBr+ in the  2113,2  spin substate of the ground electronic  state can be estimated from the current measurements using the expanded Morse poten tial of Dunham  [], given by 4 4 D(2+P J 5 ? 6+...) ?5+P +P U = 6  where D = ao/a and  = 1— €a1. The dimensionless isotopically invariant coefficients,  , P 4 P 5 and P , are defined as 6  =  (5.63)  ,  []  [1  (5.64)  ; —  5 P  = =  ; We (We Ye)  5aB  1  17  2 _  (5.65) 5.66  5.  From the value of a 1 in Table 5.10 and the results in Tables 5.8 and 5.9, the effective , P 4 P 5 and P 6 coefficients were determined. These refer to the average potential used for both the  2111,2  and  2113,2  spin substates. These values of P 5 and, P , P 4 6 were found  to equal 0.06058(31), 0.03020(74), and 0.02234(75), respectively. The values are the weighted averages for the two isotopomers. Using Eq. (5.63) with (R  —  Re)  —÷  oo, the re  sulting estimate for the effective dissociation energy V 0 is 3.89 eV. The dissociation energy  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  166  Table 5.11: Harmonic and Anharmonic Force Constantsa for the X 211 Electronic State of HBr Constantb  Br+ 7 H 9  HS B 1 r+  2 f  2439.8734(40)  2439.4928(41)  3 f  —1493.79(78)  —1493.35(74)  4 f  785.2(11)  784.8(11)  5 f  —424.2(36)  —424.1(35)  6 f  259(18)  260(18)  2 F  3.488396( 9)  3.488393( 9)  3 F  —18.1192(95)  —18.1181(90)  4 F  80.81(12)  80.79(11)  5 F  —370.3(32)  —370.4(30)  6 F  1920(136)  1927(130)  a Quoted uncertainties in parentheses are one standard deviation and apply to the last digits of the values.  b f (isotopically variant) and F (isotopically invariant) have 1 and mdyn/AI_1, respectively. units of cm  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  Do for the  ( = 3/2)  spin substate can then be estimated by (o + 1A /21) 0  167  =  4.06 eV.  This can be considered to be in reasonable agreement with the value of 3.893(3) eV ob tained by Haugh and Bayes [1J, particularly when it is recognized that only the lower  1/3 of the potential has been sampled here, and that the convergence of P is slow. 5.5  Data and Analysis of Hft with the Merged Model For the rotational and fine structure analysis of 111+, a total of 117 vibration-rotation  transition frequencies were included in the data set. Out of these, 73 transitions belonged to the (1—0) vibrational band, 36 belonged to the (2—1) vibrational band, and 8 belonged to the (3—2) vibrational band. As was in HBr, the highest J”-value measured  ax  determined in large part by the Hönl-London rotational line-strength factor. In the spin substate,  Tiax  was  2111/2  for the P- and R-branch lines are 5.5 and 12.5, respectively. Here,  no attempt was made to record the Q-branch transitions since they are expected to be quite weak. In the  211,  spin substate,  8.5 and 15.5, respectively. The  ax  ‘1ax  for the P-,  Q-  and R-branch lines are 8.5,  for the P-branch in both the spin substates was  limited by the frequency range covered by the diode. The frequencies of the vibrationrotation transitions measured are listed in Table 5.12. a  2113/2  When the two A-doublets in  transition were not resolved, the mean frequency was used as a single entry.  For low J multiplets with hyperfine splittings, the hyperfine free frequency was entered in Table 5.12. If the multiplet was clearly resolved, the hyperfine-free frequency was determined in the analysis to obtain the hyperfine constants; see Sec. 4.3.2 of Chapter 4. If the multiplet was blended, the spectrum was synthesized using these constants; the hyperfine-free frequency was then deduced by matching the synthetic and experimental frequencies. See Fig. 4.6 of Chapter 4, for example. The matrix elements of the Hamiltonian used here to fit the rotational and fine  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  168  Table 5.12: Wavenumbers of the Observed Vibration-Rotation Transitions in the X 211 Electronic State of HI  b 5  Brancha  J,,  T  (1—0) 211, P  3.5 3.5 4.5 5.5 5.5  e f f e f  e f f e f  2069.9351 2066.3044 2052.5909 2042.2596 2038.5387  0.0040 0.0020 0.0040 0.0020 0.0040  —0.0005 —0.0005 0.0039 0.0005 —0.0002  (1—0) 2111,2 R  1.5 1.5 2.5 2.5 3.5 3.5 4.5 4.5 5.5 5.5  e f e f e f e f e f  e f e f e f e f e f  2141.3911 2144.7231 2152.6564 2155.9348 2163.5496 2166.7758 2174.0697 2177.2384 2184.2080 2187.3146  0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0040  —0.0014 —0.0023 —0.0005 —0.0010 —0.0011 0.0014 0.0008 0.0022 0.0014 —0.0017  ObsC  Obs_Cale  a Values in parentheses refer to vibrational quantum numbers (v’-v”). b An entry under this column indicates that the measurement is of a A-doublet splitting in a transition belonging to the 2113,2 spin substate. + refers to (vh vi) and LI refers to (vi vh), where vh and vi are the frequencies of the higher and the lower A-doublet components, respectively. —  —  C  Observed frequency. If the two A-doublets were not resolved, the levels involved in the transitions are labelled T=e/f or f/e; the center frequency is listed and used in the fit. If hyperfine structure is observed, the hyperfine-free frequency is listed and used in the fit.  d Estimated uncertainty in the frequency measurement. e Difference between the observed frequency and the frequency calculated from the best fit values of the parameters for the merged model as listed in Table 5.14.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  Table 5.12 Brancha  J,,  r’  Sb  —  169  Continued  r”  ObsC  Obs_Cale  (1—0)  2111,2  R  6.5 6.5 7.5 7.5 8.5 9.5 9.5 12.5  e f e f e e f e  e f e f e e f e  2193.9580 2197.0099 2203.3226 2206.3114 2212.2850 2220.8526 2223.7177 2244.0952  0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020  —0.0007 0.0004 0.0024 0.0004 —0.0012 0.0008 —0.0010 —0.0010  (1—0)  2113,2  P  2.5 3.5 4.5 5.5 7.5 8.5  e/f e/f e/f e/f e/f e/f  e/f e/f e/f e/f e/f e/f  2087.1548 2073.8401 2060.1953 2046.2246 2017.3285 2002.4116  0.0020 0.0020 0.0040 0.0020 0.0020 0.0020  0.0010 0.0001 —0.0002 —0.0009 —0.0003 —0.0004  (1—0)  2113/2  Q  1.5 2.5 3.5 4.5 5.5 5.5 5.5 5.5 6.5 6.5 6.5 6.5 7.5 7.5 7.5 7.5  e/f e/f e/f e/f f  f/e f/e f/e f/e e  e  f  f  e  e  f  f  e  e  f  2118.3075 2117.4370 2116.2176 2114.6539 2112.7416 0.0066 2112.7350 —0.0066 2110.4844 0.0115 2110.4729 —0.0115 2107.8747 0.0162 2107.8585 —0.0162  0.0020 0.0020 0.0020 0.0020 0.0020 0.0010 0.0020 0.0010 0.0020 0.0010 0.0020 0.0010 0.0020 0.0010 0.0020 0.0010  0.0008 0.0003 —0.0010 0.0013 —0.0005 —0.0004 —0.0001 0.0004 0.0021 0.0003 0.0018 —0.0003 —0.0006 —0.0006 0.0000 0.0006  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  Table 5.12 Brancha (1—0)  2113,2  11 J  Q  (1—0) 2113/2 R  T’  Sb  T”  f  e  e  f  1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 9.5 9.5 9.5 10.5 10.5 10.5 10.5 11.5 11.5 11.5 11.5  e/f e/f e/f e/f e/f e/f e/f e/f e  e/f e/f e/f e/f e/f e/f e/f e/f e  f  f  e  e  f  f  e  e  f  f  8.5 8.5 8.5 8.5  —  170  Continued ObsC  Obs_Cale  2104.9219 0.0242 2104.8977 —0.0242  0.0040 0.0010 0.0040 0.0010  0.0006 0.0003 0.0003 —0.0003  2148.5896 2159.8155 2170.6737 2181.1653 2191.2804 2201.0142 2210.3636 2219.3235 2227.8848 —0.0055 2227.8903 0.0055 2236.0484 —0.0064 2236.0548 0.0064 2243.8089 —0.0074 2243.8163 0.0074  0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0010 0.0020 0.0010 0.0040 0.0010 0.0040 0.0010 0.0020 0.0010 0.0020 0.0010  0.0000 0.0001 —0.0020 —0.0004 0.0000 —0.0007 —0.0007 0.0000 —0.0002 0.0000 —0.0002 0.0000 —0.0004 0.0001 —0.0006 0.0000 0.0011 0.0002 0.0008 —0.0002  (2—1) 211  P  3.5  e/f  e/f  1995.8598  0.0040  —0.0014  (2—1) 211,  Q  1.5 3.5 4.5  e/f e/f e/f  f/e f/e f/e  2039.1116 2037.0194 2035.4512  0.0020 0.0020 0.0020  —0.0002 —0.0004 0.0000  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  Table 5.12 Brancha (2—1)  (2—1)  211  211,  Q  R  b 8  J,,  r’  5.5 5.5 5.5 5.5  e  f  f  e  3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 11.5 11.5 11.5 12.5 12.5 12.5 12.5 13.5 13.5 13.5 13.5 14.5 14.5 14.5 14.5 15.5 15.5 15.5 15.5  e/f e/f e/f  e/f e/f e/f  e/f e/f e/f e/f  e/f e/f e/f e/f e/f e  e/f e  z.S f  f  e  e  f  f  e  e  f  f  e  e  f  z.S  f  e  e  f  f  —  171  Continued ObsC  d  2033.5318 —0.0075 2033.5393 0.0075  0.0040 0.0010 0.0040 0.0010  0.0012 —0.0005 0.0017 0.0005  2089.9070 2100.0470 2109.8129 2119.1946 2128.1917 2136.7947 2145.0053 2152.8147 2160.2157 —0.0079 2160.2236 0.0079 2167.2103 —0.0089 2167.2192 0.0089 2173.7891 —0.0106 2173.7997 0.0106 2179.9476 —0.0119 2179.9595 0.0119 2185.6838 —0.0124 2185.6962 0.0124  0.0040 0.0040 0.0040 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0010 0.0020 0.0010 0.0020 0.0010 0.0020 0.0010 0.0020 0.0010 0.0020 0.0010 0.0040 0.0010 0.0040 0.0010 0.0040 0.0010 0.0040 0.0010  —0.0013 —0.0002 0.0027 0.0021 0.0025 —0.0005 —0.0004 —0.0011 —0.0011 —0.0003 —0.0008 0.0003 —0.0004 —0.0001 —0.0003 0.0001 —0.0003 —0.0005 0.0002 0.0005 —0.0005 —0.0004 —0.0001 0.0004 0.0020 0.0005 0.0014 —0.0005  Obs_Cale  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  Table 5.12  Brancha (3—2)  211  R  Sb  —  Continued  r”  ObsC  e/f e/f e  e/f e/f e  f  f  e  e  f  f  2037.3399 2062.0709 2076.5534 —0.0075 2076.5609 0.0075 2083.1795 —0.0082 2083.1877 0.0082  J,,  r’  6.5 9.5 11.5 11.5 11.5 11.5 12.5 12.5 12.5 12.5  172  Obs_Cale 0.0020 0.0020 0.0020 0.0010 0.0020 0.0010 0.0020 0.0010 0.0020 0.0010  —0.0015 0.0003 0.0010 0.0000 0.0009 0.0000 —0.0003 0.0005 —0.0008 —0.0005  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  173  structure data are given in Table 5.1. These are the matrix elements of the standard effective Hamiltonian of the merged model, derived with the assumption that Ael Ifl 111+, Ae is much larger than  We.  <<We.  Therefore, the interstack interaction between the  0 vibration-rotation levels is small. Due to a limited amount of data available to us, a fit to the merged model Hamiltonian was preferred and was found to be adequate for the flJ data. Therefore, no attempt was made to treat the potentials of the and  2111,2  2113,2  spin substates separately.  Compared to the merged model fit of the HBr+ data, a minor modification was intro duced into the fitting routine. In HBr+, the data set included the measured frequencies of both the A-doublet components of the transitions where the splittings were clearly resolved. Each of the components were given an estimated uncertainty of 0.0030 cm in the frequency measurement. Since the observed A-doublet splittings are very small in the transitions belonging to the  2113/2  spin substate, it is obvious that the accuracy of a  direct measurement of the A-doublet splitting is relatively high (better than 0.001 cm) compared to the absolute frequency measurement of the two A-doublets. Also, the pre cision to which the parameter q, is determined in a fit depends, to a large extent, on the accuracy of the splittings of the two A-doublets which enter into the data set. Therefore, in Hft, we decided to include the observed splittings of the  2113,2  A-doublets directly  into the data set. This was also necessary since the number of transitions observed with resolved A-doublets were very limited in this case. A non-linear least squares analysis of the frequencies in Table 5.12 was carried out by using the matrix elements of the merged model (see Table 5.1). The weights used were , where e is the uncertainty in the frequency measurement. At first, the data set 2 1/c included only the absolute frequencies of the two A-doublets. The associated uncertainty e in the frequency measurement is typically 0.0020 cm’, which is slightly less than that in HBr+. This is because most of the vibration-rotation transitions in 111+ were recorded at  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  174  least twice and the average value of the measured frequencies were entered into the data set. The fit converged rapidly. Although the parameter q was determined for all the four vibrational levels, the associated uncertainties were large (-.‘35%). In order to improve the precision of  q,  relative spacings of the two A-doublets (with f=0.0007 cm ) were 1  entered into the data set. The fit did not seem to converge. An attempt to fit the data by including the absolute frequency of one of the A-doublet component and the associated relative splitting for the other component showed quick convergence. Therefore, in the final fit the splittings were entered into the data set in a symmetric fashion in order to avoid any biasing of one of the A-doublet components.  Along with the absolute  frequencies of both the A-doublets, two relative splittings were entered into the data set with opposite signs. In Table 5.12 they are denoted by =  where  vh  —  Z)j  ;  =  and  —  and are defined as  Vh  and z-’ are the higher and the lower frequency A-doublet components, respec  tively. Since, each splitting was entered twice into the data set, the associated uncer tainty in the frequency measurement of the splitting was increased to 0.0010 cm . The 1 introduction of the splittings along with the absolute frequencies of both the A-doublet components did not increase the weight of any individual transition significantly since the relative weights given to the splittings were much higher than those given to the absolute frequencies of the two A-doublet components. In other words, the amount of information obtained by the parameter qv from the absolute frequencies of the two A doublet components (both of which have an uncertainty of 0.0020 cm’) during a fit is almost an order of magnitude less compared to the information obtained by it from the relative splittings (which have an uncertainty of 0.0007 cm ). 1 The overall quality of the fit was good. The normalized 9 See Footnote 8 on page 148. 9  2 x  value of the fit was  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  175  0.30, indicating that the estimated precision of measurement was conservative for most transitions. The best fit values of the parameters obtained for each vibrational level are listed in Table 5.13. An approach similar to that in HBr+ was taken to parameterize various 111+ molecular constants in the fit. For v  =  0, the rotational constant varied was B 0 itself. For v  the parameter varied was (B  —  Bo); B was then calculated from B 0 and (B,,  —  >  0,  B ) 0 .  The same procedure was followed for all the other parameters. Since our data set does not contain any transition between the two spin substates, the fit was insensitive to the . Therefore, A 0 parameter A 0 was fixed to the value obtained by Böwering et al. photoelectron spectroscopy. However, the parameter (A 1  —  [] using  ) was varied; this gave a 0 A  good estimate of the change in the value of A,, with v. The error in the value of A 1 reflects only the relative uncertainty with respect to the fixed value of A 0 and not the absolute error in A 0 itself. Since we did not record any transitions belonging to the (2—1) and the (3—2) vibrational bands of the  2111/2  spin substate, the values of 2 A 3 , A ,  AD2, AD3, P2  and p3 were extrapolated from the previous fit and were held fixed at those values. This procedure was repeated until convergence was achieved. Preliminary fits indicated that the values of pu,, for all v were approximately the same within their uncertainties and did not show any systematic trend. Therefore, in the final fit the values of  PDV  were replaced  by a single value which applied for all three vibrational levels. The best fit value of pD is given in Table 5.13. Since the parameters AD,, and -y,, are highly correlated [63], the values of  ,,  were fixed at zero in the fit.  Molecular Constants for 111+  5.6  As mentioned in the previous section, vibration-rotation transitions belonging to the 2111,2  spin substate were recorded only for the fundamental band. Therefore, it was  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  176  Table 5.13: Molecular Parametersa in the Merged Model for HI in the X 11 Electronic 2 State  v  T+i+  0 1 2  2116.0485( 5) 2036.8545(12) 1957.6539(51)  3  B  —  v  p  q x 102  0 1 2 3  3.4955(19) 3.4457(20) 33959 * 3.3460 *  —0.395(49) —0.420(58) —0.446(60) —0.466(65)  —5356.6 —5351.3886(9) —5346.1774 * —5340.9661 *  AD x  d 103  1.074(46) 1.995(47) 2.916 * 3.712 *  6.24305(26) 6.06777(30) 5.89220(31) 5.71614(34)  D x iO 2.0861(65) 2.0836(68) 2.0801(70) 2.0798(80)  a All entries are in cm’. Quoted uncertainties in parentheses are one standard deviation and apply to the last digits of the values. The values with an asterisk beside them are the extrapolated values from the fitted constants of the other vibrational levels. These values were held fixed during the fit (see text).  b 0 A is obtained from Ref. [] and held fixed at this value during the fit. The error in A 1 reflects only the uncertainty in the difference (A 1 A ). 0 —  C  The values of PDv were varied but constrained to a single value for all v; the best fit result is —9.47(63)x 10 cm.  d The values of Yv were held fixed at 0.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  177  necessary to fix the values of some of the parameters which could not be determined due to the lack of information available from the recorded transitions. In order to further improve the precision and to reduce the number of parameters held fixed during the fit to a possible minimum, a computer program was developed to fit the transition frequencies and the A-doublet splittings directly to the equilibrium parameters of Eqs. (2.6) and (5.42). Compared to HBr+, where the equilibrium molecular constants were determined by fitting the values of the parameters obtained for individual vibrational levels, the decision to perform a direct fit to the equilibrium molecular constants in 111+ was initiated for at least three reasons. First, unlike in HBr+, where some of the parameters showed somewhat strange behaviour with v (e.g., variation of q with v), in 111+, all the parameters showed a smooth and monotonic variation with v. Second, in 111+, the simple merged model was adequate to fit the, data. Third, although D and q showed a smooth and monotonic variation with v, the associated uncertainties in these two parameters were large compared to the variation itself. In each case, an attempt to fit the individual vibrational constants to the expansion given by Eq. (5.42) (as was done in HBrj did not succeed in determining the op parameter to reasonable accuracy. The parameters D  and  aq  were determined with an uncertainty which were larger than the values of the  parameters themselves. This is not so surprising when it is considered that the data set consisted of only four vibrational levels with very limited number of transitions in the (3—2) vibrational band. The equilibrium values obtained from a direct fit are listed in Table 5.14. The number of significant figures given for the values allows one to reproduce the calculated frequencies to better than 0.0001 cm”.  The normalized  2 x  value is 0.30, which is similar to  that obtained previously in the band-by-band fit. The residuals between the observed frequencies and the frequencies calculated from the fitted equilibrium parameters listed in Table 5.14 are given in Table 5.12. It should be noted that the number of parameters used  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  178  Table 5.14: Equilibrium Molecular Constantsa, 1 of the X 11 Electronic State of Hft 2 Constant  Value 2195.243451(842) 39.597374(313) 5359 _ c 2 .  We  WeXe  Ae  )d 8 ( 5211366 26  ADe x iO AD x io Be B 3 1 7B  De D  Pc PDe  q Qq  6.110(706) 9.209(141) 6.330697(158) 0.1753934(362) 1.6018(805) —6.616(107) 2.08627(448) —2.710(743) 3.52026(166) —4.9756(318) —9.403(582) —3.797(298) 2.552(650)  x i0 X iO X iO x x i0 x iO X iO X iO  ),  a In the expansion in powers of (v + traditional symbols are used for the vibratio nal term values. However, for the remaining parameters, some non-traditional sym bols are used. See Sec. 5.4 and Eq. (5.42) in particular. b All entries are in cm . Quoted uncertainties in parentheses are one standard devia 1 C  tion and apply to the last digits of the values. This is the best estimate from the available photoelectron data and the value of determined in this work. This value was held fixed during the fit.  d The error in  aA  cA  reflects only the relative error with respect to the fixed value of Ae.  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  179  in the direct fit (see Table 5.14) is significantly less than that used in the band-by-band fit (see Table 5.13). In Table 5.14, the value of Ae was determined from A 0 of Ref. [j and obtained from the difference (A 1  —  oA  ) of the values listed in Table 5.13. The value of Ae 0 A  was held fixed during the fit. As can be seen from Table 5.14, we were able to determine the  and  q  parameters by a direct fit of the data to the equilibrium parameters. An  attempt to determine the value of The value determined, normalized  2 x  WeYe  =  WeYe  was made by varying this parameter in the fit.  —0.00047(36) cm, was negligibly small. Moreover, the  value was improved only by 0.01. Therefore, the value of  WeYe was  fixed  to zero in the final fit. It should be noted that the values in Table 5.14 for  We  and  WeXe  describe an effective  average potential that applies to both the (1=3/2) and (1=1/2) spin substates of the  211  ground state. A similar statement applies for the parameters characterizing the expansion of B and D, and for associated derived constants. This treatment is similar to HBr+, except that the dissociation energies of the  2113/2  and  2111/2  spin substates in the case  of HI are estimated to differ by about 20% (see later in this section). In HBr this difference was about 8%. Nevertheless, due to the limited number of transitions recorded in 111+, a fit to an effective average potential was preferred in order to keep the number of parameters in the fit to a minimum. It is of interest to compare the present determinations of the vibrational constants with the results obtained recently by photoelectron spectroscopy. Table 5.15 shows such a comparison. Böwering et al. and  2111/2  []  fitted their photoelectron data by treating the  potentials separately as well as by using an effective average  211  2113/2  potential.  Zietkiewicz et al. [4.] fitted their photoelectron data using the former method alone. The results obtained by the two groups for the individual  2113/2  and  2111,2  potentials are listed  in the upper part of Table 5.15. For each of the spin substates, the two measurements of We differ by .—.‘5%. For  We Xe,  the values differ by almost a factor of 2. The lower part of Table  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  180  Table 5.15: Comparison of Molecular Constantsa of the X 11 Electronic State of HI 2 with Previous Determinations from Photoelectron Spectroscopy Constantb We  (2113/2)  WeXe We  (2113/2)  (2111/2)  WeXe  (2111/2)  Ref. [5]  Ref. [24]  2256.1 ± 3.4  2356.3 ± 54.4  58.6 ± 9.6 2248.2 ± 2.9 50.8 + 4.1  97.9 ± 8.7 2457.8 ± 138.7 103.3 + 37.4  Constant c  Ref. [5]  Current work  (211)  2252.7 ± 2.2  2195.24345 ± 0.00084  54.7 ± 5.2  39.59737 + 0.00031  6.9 ± 4.4  5.21137 + 0.00083  We  WeXe  (2fl)  a All entries are in cm. Quoted uncertainties in parentheses are one standard devia tion. b These constants refer to the 2113/2 and 2111/2 potentials treated separately in the fit. C  These constants refer to the effective average potential for the substates.  2113/2  and  2111/2  spin  d The definition of this constant is given by the expansion in the present work in Eq. (5.42). With this definition, the sign of aA is positive.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  181  [] and of the current work obtained by treating an effective average potential. Böwering et al. [] determined the  5.15 lists the constants of Böwering et gil. the two potentials as effective average of  We  and  WeXe  values for the X  2  state to be (2252.7 ± 2.2) cm 1  and (54.7 ± 5.2) cm’, respectively. These values differ significantly from the values of We =  (2195.24345±0.00084) cm 1 and WeXe  =  (39.59737±0.00031) cm 1 obtained in this  work. Thus, from the comparison in Table 5.15, it is clear that the photoelectron results are inconsistent with the current determinations. It appears that either the vibrational numbering of high v levels in the above two photoelectron works are misassigned, or the associated uncertainties in their measurements are somewhat underestimated. In our work, the vibrational numbering was verified by the following method. Consider the possibility that the lowest vibrational band observed is not the fundamental band but some hot band, say (v’ bands, i.e., (v’  —  ÷—  v”). We have certainly observed two consecutive vibrational  v”) and (v’+l  lower vibrational level (v”  —  —  v”+l). As a result, the constants for the next  1) can be extrapolated with reasonable accuracy. Keeping  this in mind, a fit to the observed transition frequencies was performed by increasing the v assignments of all the transitions by 1. The resulting equilibrium constants were then used to predict the vibration-rotation transition frequencies of the hypothetical (1—0) vibrational band.  An attempt to observe some of these transitions by scanning the  laser around the predicted frequencies proved unsuccessful. For example, the frequency predicted for the hypothetical (10)  2113,2  1 )ef,fe transition was 2191.9271 cm 55 Q(  with an expected uncertainty of ±0.05 cm . No transition was observed in the frequency 1 range from 2191.1 cm’ to 2193.1 cm’. Similar searches were done for the hypothetical transitions (1—0)  2113,2  )ef,fe and (1—0) 35 Q(  2113,2  ()ee,ff’ but without success.  The predicted frequencies for these two hypothetical lines were 2195.4088 cm’ and 2123.5000 cm , respectively. Prior to these check as well as immediately after the check, 1 it was confirmed that sufficient number of 111+ ions were being formed in the discharge by  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft recording the previously assigned (1—0)  centered at 2236.052 cm and the (1—0)  A-doublet resolved transitions  R(lO.5)ee  2113,2  2113,2  182  )ef, fe hyperfine resolved transitions 15 Q(  . The hypothetical (1—0) 1 centered at 2118.308 cm  2113/2  be stronger than the nearby observed (1—0)  Q(1.5) transition by at least a factor  2113,2  P(5.5) transition is expected to  of 4 if it is assumed that the vibrational and rotational temperatures were the same order as those obtained for HBr+. The laser power was verified to be approximately the same in the two regions involved. Since the  2113,2  Q(1.5) transition recorded as a  check under similar discharge conditions had a signal-to-noise ratio of about 30, and that the hypothetical  2113,2  P(5.5) transition was not observed in or nearby the predicted  region, it can be concluded that indeed the lowest vibrational band observed here is the fundamental band. The equilibrium bond length Re has been calculated from the reduced mass equilibrium rotational constant Be using their well known relationship mass was taken to be it(llhi  =  [1].  it  and the  The reduced  ——  (mH (mH  1 me) m me) + ml  In Eq. (5.67), me is the mass of the electron, while  mH  (5.67) and mi are the masses of the  hydrogen and iodine atoms, respectively. From the known values [4] of the masses in Eq. (5.67), the value of Re was found to equal 1.632356(20)  A.  This is, of course, model  dependent, depending, for example, on whether the ion behaves like H+I or 111+, or an admixture of the two. It appears that HI is the better model since the value of  (=  136.95(84)  A)  determined in this work is close to that of  (-i)  (=  127.68  (-i)  ) 3 A  calculated for a 5p electron in the iodine atom [9]. Also, Hft is expected to be similar to HBr+. The value of Re for HI+ is based on this model. However, evidence is considerably weaker in Hft than in HBr. The harmonic and anharmonic potential and force constants were determined using  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  183  the values of the equilibrium parameters listed in Table 5.14. The treatment was sim ilar to that of HBr (see Sec. 5.4). These results are listed in Table 5.16. From the 1 and the results in Table 5.16, the expanded Morse potential of Dunham were ob a, tained as an effective average for the (1=1/2) and (1=3/2) spin substates of the  211  ground state. The procedures used are outlined in Sec. 5.4 for HBr+. The dimensionless expansion coefficients P 4 and P 5 were found to equal 0.05037(12) and 0.01709(32), re spectively. Using these values, the effective dissociation energy 7 o is estimated to equal ). It should be kept in mind that this estimate is made from a very 1 3.59 eV (28,900 cm limited sampling of the potential, and hence, may be somewhat less accurate than the corresponding result for HBr+. Vibrational and Rotational Temperatures of the Ions  5.7  Although the plasma in the discharge is not in equilibrium, it is often possible to characterize the population distribution by vibrational and rotational temperatures. The two temperatures can be quite different because the characteristic times for energy transfer from vibration to translation  (‘—‘  10 s 5 ) can be orders of magnitude longer than  the corresponding times for rotation-translation transfer  (‘—‘  10 s 8 )  [7J.  By developing  a data base of these temperatures for different species under similar discharge condi tions, more insight can be obtained into the process by which energy exchanged between particles and between different degrees of freedom within the individual molecular ions  []. Since an extensive study of the vibration-rotation transitions in HBr+was carried out in the current work, the two temperatures were determined. They apply to an a.c. glow discharge operating at 400 W with small amounts of 112 and Br 2 added to 5 torr of He; see Sec. 3.1.4 of Chapter 3. The vibrational temperature T1b was estimated by comparing  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  184  Harmonic and Anharmonic Expansion Coefficientsa, b and Force 11 Effective Potential of HP 2 for the X  Table 5.16: Constantsb,  Coefficient 0 a  Value 190306.6(48)  1 a  —2.60116(35)  2 a  4.2876(16)  3 a 2 f  —5.872(14) 2195.24351(84)  3 f  —1300.98(18)  4 f  651.45(25)  5 f  —338.78(78)  2 F  2.8374692(37)  3 F  —13.5645(18)  4 F  54.790(21)  5 F  —229.84(53)  a These are isotopically invariant, a 0 has units of cm whereas , 1 a a are dimensionless. ...,  b Quoted uncertainties in parentheses are one standard deviation and apply to the last digits of the values. C  f (isotopically variant) and F (isotopically invariant) have units of cm and mdyn/A_1, respectively.  Chapter 5. Rotational Energies and Fine Structure in HBr and Hft  185  the intensities of the hyperfine transitions in the (1—0) and (2—1) bands as well as the intensities of the vibration-rotation transitions in the (1—0), (2—1), (3—2) and (5—4) bands. The pairs of lines used were: (1)  (1  0)  2111/2  79 H .5)ee in Br 2 P(  (2)  (1 —0)  2113/2  79 ll )ee,ff 2 ( 13 5 in Br  with  (2  (3)  (2  1)  2113/2  81 H )ee,ff in Br 8 ‘(  with  (5  (4)  (1  0)  2111,2  P(l ) 5 . 2 ff in H Br 8 1  —  —  —  with  with  (2  (3  —  —  1) —  2)  .S)ff in H 2 R( Br; 8 1  211  1)  2113/2  81 H )ee,ff in Br; 5 . 2 R(  4)  2113,2  81 H .S)ee,ff in Br; 9 R(  211,  81 H P(3.S)ee in Br.  In each comparison, the two lines of interest appeared on the same scan. The laser power in each scan was monitored and corrected for when determining the ratio of the observed peak intensities. Note that the second pair of transitions is the same as the first, except that the  2113,2  spin substate is involved and the A-doubling has not been  resolved. Since the first two pairs of transitions belong to relatively low rotational levels, they showed characteristic hyperfine splittings. One of the transitions in the last pair, i.e., (3  —  2)  2111,2  .S)ee, showed some broadening due to hyperfine effects, although 3 P(  the transition appeared to be a single line. Here we present, as an example, the expression for the ratio of the observed intensities for the first pair of lines mentioned above. Similar expressions can easily be obtained for other pairs as well.  (I(2—1)R(2.s) \I(1—o)P(2.5))Ob5  —  (f(2_1)R(2.s)  (vR(2.5)  f(1—o)p(2.5))cal 2.5)  N 5 . 1 2 (N 5 • 2 N  SR(2.5) SP(2.5)  —  —  N SP(35) 5 • N 3 2 5 5 . 1 1 N N R(1.5) (5.68)  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  186  In Eq. (5.68), (I(2_1)R(2.5) 1—O)P(2.5) 1 \  (5.69)  Jobs  represents the ratio of the peak intensities of the observed hyperfine transitions.  (f(2_1)R(2.5)) f(1—o)P(2.5)  (5.70)  cal  is the ratio of the calculated normalized’ 0 hyperfine intensities of the features being compared, neglecting the rotational line strength and the vibration-rotation populations. R(2.5) 11  and  vP(2.5)  are the transition frequencies of R(2.5) and P(2.5) lines, respectively.  N oc exp  {— } G(v) kTIb  (5.71)  refers to the vibrational population, where G(v) is the vibrational energy, k is the Boltz mann constant, and Tb is the vibrational temperature.  I F(J) N cx 1 g(J)expj— > ( ‘-‘-rotJ  (5.72)  refers to the rotational population, where g(J)=(2J+1) is the rotational degeneracy, F(J) is the rotational energy, and Trot is the rotational temperature. S is the normalized” Hönl-London factor  [1].  It is assumed that the transition dipole moments are the same  for both members of the pair of transitions being considered. ‘°The hyperfine intensities were calculated using the expressions given in Ref. [] and are normalized such that the sum of the intensities of the individual hyperfine components is a constant for any vibration rotation transition. The Hönl-London factor is normalized such that for any J, Sp(j) + SQ(J) + SR(J) 1 ‘  =  1.  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  187  In Eq. (5.68), the frequency dependence of the transitions can be neglected since they  are approximately the same for the pairs compared. Eq. (5.68) can then be written as /T  ‘(2—1)R(2.5)  \  \ kl—O)P(2.5)) ohs exp  1  exp  I  L.  =  (  ( ‘-‘R(2.5)  J (2—1)R(2.5)  f(i_o)P( . 2 s)) cal \ ‘ P(2.5) 5  0(1)  kT1b  —  /C*  I:  0(0) kTb  —  —  F(2.5)’L kTrot F(2.5) kTrot j  f  —  —  I exp 1 s — exp  f —  0(2) kTb 0(1) kTIb  —  —  F(3.5) kTrot F(1.5fl kTrot  f  5 73  where  —Q 2 (J+1)  R(J) 8  S(J)  —  =  In Eqs. (5.74) and (5.75), Q=1/2 for the  .  2111/2  (5.74) (5.75)  spin substate, of course.  Since most of the hyperfine transitions used in the determination of the vibrational temperature were at least partially blended with the nearby hyperfine components of the same vibration-rotation transition, the hyperfine patterns for each of these vibrationrotation transitions were simulated using the constants given in Table 4.3 of Chapter  4. The ratio of the calculated normalized intensities in Eq. (5.70) was determined by measuring the height from the bottom peak to the top peak of the simulated hyperfine patterns. Initially, T1b in Eq. (5.73) was numerically solved for various possible ratios of hyperfine component intensities by neglecting the rotational population factor. The value of Tb obtained from such an approximation was then used to determine the rotational temperature Trot (see later in this section). This was then used in Eq. (5.73) to refine the vibrational temperature. The procedure was repeated until convergence was achieved. It was then found that Tb = (3900 ± 400) K. This value reflects an average of the vibrational temperatures obtained from the analysis of all the four pairs  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  188  of transitions mentioned previously. Within the error limits, the intensities followed a Boltzmann distribution. To estimate the rotational temperature Trot, the intensities were measured for several transitions in the P-branch of the (1—0) band in the  spin substate. The values of  211  J” ran from 9.5 to 13.5. In each case, the (e—e) A-doublet component was studied. In all the transitions studied, the effect of hyperfine splitting in the vibration-rotation intensity was negligible. Both isotopomers were measured where possible. Since these transitions were spread over a range of about 85 cm , each transition was recorded separately 1 keeping the same gas mixture and the discharge power (400 W). The laser operated in a single mode while recording all the transitions used in the analysis of Trot. The laser power was monitored for each transition. The observed intensities were corrected for the variation in the laser power. The expression for the peak intensity of the (1—0) P(J) vibration-rotation transitions can be written as I(i-o)P(J)  CC  i’p(J)  0 N Sp(J) [Nv  —  1 SR(J_1)} 1 N_ N  (5.76)  .  The notation follows that described earlier for vibrational temperature measurement. Using the appropriate expressions for the vibrational and the rotational populations, Eq. (5.76) can be written as ‘(l—O)P(J)  CC  VP(J)  * Sp(J)  exp  I  j—  0(0) m -‘-vib 1  —  F(J) I m -‘-rot 1  —  exp  I  1—  0(1) 1  m  F(J  —  m Irot  1) ‘I  1  (5.77) where S(J) is defined by Eq. (5.75). By using the vibrational temperature obtained previously, the value of Trot was numerically determined for various possible ratios of the observed peak intensities. The intensities showed that a Boltzmann distribution forms  Chapter 5. Rotational Energies and Fine Structure in HBr and HI  189  an acceptable model. The average value obtained for Trot was (800±70) K. Because the rotational degrees of freedom relax much more rapidly than the vibrational degrees, it is not surprising that Trot is much lower than Similar temperature measurements have been made for a number of diatomic ions using a helium-dominated a.c. glow discharge such as that used here. For OH+, it was found that TIb=(4700±470) K  [].  Similar values of TIb have been obtained for CF,  ArH+, and C; see references cited by Rehfuss et al.  [].  These measurements are about  20% larger than the present determination of Tb in HBr. On the other hand, for DBr, it was found that TIb=(2300+200) K [68]. A similar value of TIb was obtained for DCft in a d.c. discharge; see Fig. 3 of Zeitz et al.  [].  In comparisons between results from  different laboratories, minor differences in discharge conditions (e.g., discharge power, gas mixture, e.t.c.) make it difficult to interpret small temperature differences. However, there is some indication that the vibrational relaxation is more rapid for the hydrogen halide ions than for the other systems.  Chapter 6 Discussion and Conclusions  The non-resonant interactions between the different spin substates  2111,2  and  2113/2  have been investigated in HBr+ for the case where the magnitude of the spin-orbit con stant Ae is the order of the harmonic vibrational frequency  We.  It has been shown that  the merged model developed for the limit Ael <<C-4e can reproduce most of the major fea tures of the vibrational spectra. However, several important detailed features cannot be properly accounted for, including the vibrational dependence of the A-doubling constant q and the distortion constant  Perhaps the most striking aspect of the limitations  of the merged model is its inability to reproduce the A-doubling splittings in the vibra tional fundamental at high J”. The split model which treats separately the interstack and intrastack contributions to the distortion effects shows that the matrix elements with (v  =  +1,  =  ±1) are responsible for the major deficiencies in the merged  model. However, a second order Van Vieck treatment of these matrix elements proved to be inadequate. Higher order terms were shown to be important, and were treated by a third order Van Vieck transformation. The equilibrium vibration-rotation constants, the harmonic and anharmonic force constants, and the equilibrium bond length were determined. The hyperfine transitions in the  2111,2  spin substate of HBr+ were observed for the  first time. From the analysis of the observed hyperfine transitions in the 2113,2  2111,2  and the  spin substates, the Frosch and Foley magnetic hyperfine constants and the electric  190  Chapter 6. Discussion and Conclusions  191  quadrupole constant were determined. This resulted to a more complete characteriza tion of the unpaired electron distribution in the HBr+ molecular ion than could not be obtained by studying the hyperfine transitions in the  2113/2  spin substate alone.  A high resolution vibration-rotation study was carried out for the first time of the X 1 2 1 electronic state of Hft. Unlike HBr, where Ae 2.5  We  .  ‘  We  ,  the value of  IAeI  in HI  Therefore, the interstack interactions are expected to be small. Indeed, no  anomalous behaviour was observed for A-doublet splittings in the transitions belonging to the fundamental and the first hot band of the  2113,2  spin substate. From the fit of the  observed transitions to the effective Hamiltonian of the merged model, the equilibrium vibration-rotation constants were determined to high precision. These constants were used to obtain the harmonic and the anharmonic force constants, and the equilibrium bond length. A study of the hyperfine transitions in HI belonging to both 2111,2  2113,2  and  spin substates was performed. The magnetic hyperfine constants and the electric  quadrupole constant were determined for 111+ for the first time. Let us consider the anomaly in the observed A-doublets in the fundamental band of the 2113,2  spin substate of HBr. At J”=18.5 in the P-branch, the merged and split models  differ with respect to the A-doubling splitting in the spectrum by only a few times the experimental error of 0.001 cmt See, for example, Figs. 5.5 and 5.12 of Chapter 5. For the hot bands, the two models agree. However, for Q=3/2, the eigenvalues E(v; 12, J, r) of the effective Hamiltonian are higher in the merged model than in the split model at J=18.5 by —0.35 cm for all v  5 for both r  =  e and f. A similar statement applies  for 12=1/2. Thus the small A-doubling discrepancies warn of a much larger error in the energies. In spite of the fact that the A-doubling effects in the spectrum are small, they pro vide perhaps the best method available for monitoring the importance of the interstack  Chapter 6. Discussion and Conclusions  192  distortion effects due to matrix elements with (Liv + 1, LJ ± 1). These effects are char acterized by the T-parameters TA, TB and T. The contribution of these parameters to the A-doubling energy splitting ‘fe defined in Eq. 5.8 of Chapter 5 is illustrated in Table 6.1 for Q=3/2 and J=18.5. For v  =  0, 1,.  ,  5, the values of ‘fe obtained from the full  split model are listed. Also given are the differences ’ 6 fe and the value  L1’F  (‘fe  —  FF)  between fe  obtained from the same model but with the three T-parameters fixed  at zero (i.e.,”turned off”). As expected, the magnitude of f 6 e is small for the isolated level with v=0. For v=1, 6LSfe is larger by a factor ‘-.‘5. For the higher values of v, it remains roughly constant at —‘0.14 cm’. It is useful to have a rough guide as to when these interstack perturbations become important. In the merged model, the leading terms in zfe(v; f, J) are given in Eq. (5.18) of Chapter 5 for 1=3/2. In the split model, the dominant contribution to A-doubling in the 2 312 f l spin substate from the interstack interactions is from the diagonal term corresponding to 3 D 12 (see Table 5.5). Let us denote this contribution by the symbol £. From Eq. (5.26) of Chapter 5, using Eqs. (5.32) and (5.33), this term can be written as £  TBV  = —  —  v—1,1/2  (—z + 1),  (6.1)  where we have considered only the leading term with a small energy denominator. For a given J, the terms in the energy denominator can be approximated by  2 i , 1 E_  (  =  We  =  We jV  +  1 —  WeXe  (  +  WeXe —  —  +  i’  ,i —  12  —  A  ;  (6.2)  A_ 1 F 2  (6.3)  where we have retained the major parity-dependent terms. The energy denominator of Eq. (6.1) can then be written as 312 E,  —  v-1,1/2  =  We  2  WeXe V  +  (A + A_ )+ 1  (6.4)  Chapter 6. Discussion and Conclusions  193  Table 6.1: Contributionsa of the Interstack Interaction to the A-doubling for Br 81 in H the 2, State at J=18.5  b  C  TBTermd  V  fe  0  0.441  0.020  0  1  0.535  0.099  0.129  2  0.559  0.128  0.150  3  0.548  0.137  0.146  4  0.528  0.139  0.136  5  0.493  0.138  0.124  fe 8  a All entries are in units of cm . 1  b This is defined in Eq. (5.8). It is calculated using the full split model with the parameters in Table 5.7. C  This is the increase that occurs in fe when the T-parameters are changed from zero to the values in Table 5.7.  d This is the contribution to fe from the term in TB in Eq. (6.9). It should approximate SLfe.  Chapter 6. Discussion and Conclusions  194  Since the change in the value of A with v is negligible compared to the magnitude of A itself, (A + A_ ) can very well be approximated by 2Ae. Therefore, 1 v,3/2  where S =  2  We  WeXe  Pv—ii/  (6.5)  + Ae. Substitution of Eq. (6.5) in Eq. (6.1) gives  V  = TBv{ +Pl}-’ 1  £ Since  {1 +  —  (6.6)  (z-i).  << 2S, one can write £  =  TBV{i  Pv-1V-}  (z-1).  (6.7)  Here £ is a function of e/f, with the upper and the lower signs corresponding to the e and the f energy levels, respectively. Thus, the contribution to the A-doubling splitting due to this interstack term is £(f)  fe  —  £(e)  =  TB Pv-i  V  (z  —  1).  (6.8)  This term is to be added to the leading contribution given in Eq. (5.18) of Chapter 5. Therefore, in the split model, the approximate expression for A-doubling in the rotational energy levels of the fe( 3/2, J)  =  2113,2  {()  spin substate is given by  + (Y + (We  2WeXe  V  +  Ae)2  } Q ) [(  +i)2i].  +  (6.9) This result takes into account only the T-effects arising from the second order Van Vleck transformation as it enters in the diagonal elements of the effective 2 x 2 Hamiltonian matrix. The second order off-diagonal contribution as well as the third order diagonal contribution are neglected. At J=18.5, the values of Lfe obtained from Eq. (6.9) differ in magnitude from those given by the full split model by less than 9% for 0  <  v  5.  Chapter 6. Discussion and Conclusions  195  In the earlier treatment of SeD, Brown and Fackerell [22] detected non-resonant per turbation between (v without the term in  =  0, !  WeXe.  =  1/2) and (1,3/2). Their analysis is equivalent to Eq. (6.9)  Of course, since only one vibrational band was involved, there  was no need to include the anharmonicity effects. The value of the TB-term in Eq. (6.9) is given in Table 6.1 as a function of v. It vanishes for v=0, and is approximately constant for 1  v  . If the 1 5 at 0.14 cm  contribution of the anharmonicity term in the denominator is neglected, then the TBterm would be linear in v. Thus the anharmonicity makes an important contribution. For each v, the value of TB-term in Eq. (6.9) should approximate the effect of turning on the T-parameters in the split model, i.e., should approximate the value of 5 zfe. As can be seen from Table 6.1, the agreement is surprisingly good. The contribution of the TB-term to LSfe for v> 0 is ‘-.‘25% of the total. The effect of the (Liv when  We  —  WeXe V 2  =  +1,  =  +1) matrix elements is clearly most important  + Ae is small. However, even when AC is small compared to  We,  the  contribution of the TB-term can be significant if the resolution of the experiment is high enough. With the completion of the current work on the vibration-rotation spectra of HBr+ and 111+, it is of interest to compare some of the properties of the HX neutrals and HX+ ions, where X  =  F, Cl, Br and I. Table 6.2 lists a comparison of various molecular  constants of the neutrals and the ions of the hydrogen halide series. The values without brackets and in brackets are for the hydrogen halide neutrals and their correspondillg ions, respectively. No attempt was made to verify that these are the best values to date available in the literature. The number of significant figures quoted for the values in Table 6.2 is considered to be sufficient for comparison purposes. The values of Re for the ions listed in Table 6.2 are calculated with the model H + X (X values of k (=F ) are calculated from Eq. (5.62). 2  F, Cl, Br, I). The  Chapter 6. Discussion and Conclusions  196  Table 6.2: Comparison of Equilibrium Molecular Constantsa of the X’E Electronic State of Hydrogen Halides and the X 11 Electronic State of their Ionic Counterpart. 2 HF (HF)  C1 3 H 5 C1) 35 (H  Br 81 H Brj 81 (H  HI (Hft)  4138.32 (3090.48)  2990.95 (2674.00)  2648.98 (2439.49)  2309.01 (2195.24)  89.88 (89.00)  52.82 (52.55)  45.22 (46.17)  39.64 (39.60)  Be (cm’)  20.956 (17.577)  10.593 ( 9.951)  8.465 (8.078)  6.426 (6.331)  (cm’)  0.798 (0.886)  0.307 (0.327)  0.233 (0.242)  0.169 (0.175)  Re (A)b  0.91681 (1.00131)  1.27455 (1.31542)  1.41444 (1.44837)  1.60916 (1.63236)  k (mdyn/A)  9.6569 (5.3829)  5.1636 (4.1247)  4.1167 (3.4884)  3.1406 (2.8375)  Ae/Be  (18)  (—65)  (—328)  (—847)  pe/qe  (—18)  (54)  (—300)  (—929)  We  (cm’)  WeXe  B  (cm’)  a These values are obtained from Ref. [62,69-72]. The results for Br 81 and Hft H were obtained from the current work. The values without brackets and in brackets are for the hydrogen halides and their corresponding ions, respectively. No attempt was made to verify that these are the best values available to date in the literature. The number of significant figures quoted for the values are considered to be sufficient for comparison purposes. 1? The values of Re for the ions are calculated with the model H X (X = F, Cl, Br, I). +  Chapter 6. Discussion and Conclusions  197  For both, neutrals and ions, the value of the equilibrium bond length Re, calculated with a model where the halogen atom is perturbed by a neighbouring proton, shows a large change on going from F to Cl and then increases more-or-less linearly with roughly similar differences between two consecutive species. This is also true for the harmonic ) listed for the ions. Fig. 6.1 shows a plot of the variation in 2 force constant k(= F the value of the harmonic force constant k with the equilibrium bond length Re for the neutrals and the ions of the hydrogen halide series. The value of force constant for the ions in this series is approximately a linear function of the bond length. The decreasing value of the force constant on going from HF+ to 111+ indicates a decrease in strength of the bond between the proton and the halogen atom, and this is reflected in the increasing bond length. This decrease in bond strength is much more rapid in the neutrals than in the ions. The two curves approach each other as the mass increases, indicating that the force constant is similar for the heavier neutral and the corresponding ionic species. The ratios of the equilibrium values of the spin-orbit constant and the rotational constant Ae/Be, and the A-doubling parameters pe/qe for the ions are listed in the last two rows of Table 6.2. The close proximity of the two ratios for all the hydrogen halide ions indicate that the dominant perturber of the ground electronic state of these ions is a single 2 E electronic state, which, in this case is the A 2 state [8, 9]. Table 6.2 also lists the major vibrational constants of the hydrogen halide neutrals and their respective ions. Comparison of the values of the anharmonic vibrational con stant  WeXe  shows that these values for the ions are almost the same as those for their  corresponding neutrals. This statement applies to HI and Hft as well as to the other members of the series, provided the current results for 111+ are used. On the other hand, all of the values of  WeXe  determined for HI+ using photoelectron techniques  are very different from that one would expect based on the similarity of the  [,  WeXe  24  ]  values  for the neutral and the ionic species listed in Table 6.2. (There is also a large variation  Chapter 6. Discussion and Conclusions  198  11.0HF  8.0  5.0  HI 2.0  0.9  1.3  Re  ()  1,7  Figure 6.1: A plot of the variation in the value of the harmonic force constant k with the equilibrium bond length Re for the neutrals and the ions of the hydrogen halide series. The curves connecting the points are intended only as a guide to the eye.  Chapter 6. Discussion and Conclusions  199  among the different photoelectric determinations of unlikely that  &eXe  wexe;  see Table 5.15.) It is highly  for HI is so different from that of HI, especially when compared to  other neutrals and ions in this series. A reliable value of  can easily be determined  from the observation of two consecutive low lying vibrational bands. Since we have ob served three consecutive vibrational bands, the value of WeXe determined in this work can be considered to be reliable to infrared accuracy. It is worth ending this chapter by making a remark and showing one more plot of a spectrum. When the project on 111+ was undertaken, not much information of the molecular constants was available for this molecule. The previously determined value of the fundamental band origin varied from 2000 cm 1 to 2200 cm’, with rather large uncertainties (within infrared standards). Moreover, the formation of sufficient number of ions in a discharge which can be detected in a laboratory, depends significantly on the gas mixture itself. Under these circumstances, one needs to search for ion signals by trial and error method. We started the search by choosing a frequency region where the diode output was stable, lased in a single mode, and had a reasonable amount of output power. I still feel excited to see the computer display of the very first trial scan recorded in search of Hft ions. This is shown in Fig. 6.2. More exciting was that not only did we observe an 111+ ion signal in the very first scan, we were also able to make a proper assignment of the observed transition from its characteristic hyperfine pattern. In fact, this is the hyperfine-resolved (1—0)  2113/2  Q (1. 5 )ef, fe  vibration-rotation transition in HIP, centered  at 2118.31 cm . Note that the value of the fundamental band origin is 2116.05 cm 1 . 1 The same transition recorded much later is shown in Fig. 4.3 of Chapter 4.  Chapter 6. Discussion and Conclusions  200  Figure 6.2: ‘In search of HI ion signal ‘... The result of the very first trial scan This is the hyperfine-resolved (10) 11 )ef fe vibration-rotation transition in Hft. 5 3/2Q(l. —  Bibliography  [1]  G. Herzberg, ‘Molecular Spectra and Molecular Structure  —  I. Spectra of Diatomic  Molecules’, 2nd Edition, Van Nostrand, New York, 1950.  []  H. J. Lempka, T. R. Passmore, and W. C. Price, F. R. S., Proc. Roy. Soc. A 304, 53-64 (1968).  []  A. Banichevich, R. 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Spectrosc. j, 332-336 (1976).  —  Appendix A Rotational Energy and Fine Structure in 211 Electronic State  In this Appendix, the procedure for deriving the matrix elements representing the rotational energy and the fine structure splitting of the energy levels in the 211 electronic state, interacting with a distant  2+  electronic state, is outlined. The derivation of the  matrix elements is self-contained. The procedure can be followed without referring to the main body of the thesis. No attempt was made to maintain the internal consistency between the notation used here and that used in the main body. For a freely vibrating-rotating diatomic molecule the Hamiltonian, in general, ne glecting the hyperfine interactions, can be written as H=HEV+HR+HSO+HSR+HSS, where  EV 11  (A.l)  is the non-relativistic (electronic and vibrational) Hamiltonian of the non  rotatrng molecule, HR is the rotational Hamiltonian arising from the the rotational motion of the nuclei; HSO is the spin-rotation Hamiltonian representing the interaction between the spin and the orbital angular momenta of the electrons; HSR is the spin-rotation Hamiltonian representing the interaction between the electron spin and the rotational angular momenta of the nuclei; and H s is the spin-spin Hamiltonian representing the 5 interaction between the spins of different electrons. In halogen acid ions, the ground electronic state, 2 llç, and the first excited electronic E+, arise from a single valence electron (or a hole). The spin-spin Hamiltonian, state, 2 55 can be ignored in these two states. H , 207  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State  In order to derive the matrix elements of the distant  2+  208  electronic state interacting with a  electronic state, the following procedure will be adopted:  • Choose a suitable unsymmetrized basis (signed quantum number basis) to define the coupling scheme of the various angular momenta and express the Hamiltonian in terms of the corresponding angular momentum operators. The matrix elements of all the operators appearing in the Hamiltonian can then be derived in this basis. • Construct the matrix to represent the energies of the vibration-rotation levels of E+ electronic state using the signed quantum number basis. At this stage, the 2 the 211  ,.2 +  interactions are neglected.  • Construct the matrix to represent the energies of the vibration-rotation levels of the 2ll electronic state using the signed quantum number basis. At this stage, the 211  2 +  interactions are neglected.  E+ states using the signed • Construct the interaction matrix between 2ll and 2 quantum number basis. • Choose a suitable symmetrized basis defined in terms of the signed quantum number basis. In the symmetrized basis, determine the matrix elements of the 211o and 2 + electronic states neglecting the 211 the E  2 +  interactions. Also, determine  2 + interaction in this same basis. the matrix elements of the 2ll E • Perform Van Vieck transformation (electronic and vibrational) to reduce an in finitely large supermatrix, representing all the possible interacting vibrational levels 2 + and 211o electronic states, to an effective 2x2 of E  211  matrix for a particular  vibrational level in question. This reduced matrix will then define the vibration rotation energy levels of the electronic state.  211o  electronic state interacting with the distant  2>+  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State  209  1. The Hamiltonian and its matrix elements in the signed quantum number basis  In the  2fl  ground electronic state of HBr+ and 111+ molecular ions, AA>> BJ, where  A, B and J are the spin-orbit coupling constant, the rotational constant and the rotatio nal quantum number, respectively. In this limit, Hund’s coupling case (a) best describes these states. Therefore, we choose the unsymmetrized wavefunctions appropriate to this coupling scheme as the basis set. This wavefunction will be represented by IASE; QJ), where A, E and  are the quantum numbers corresponding to the components of the  orbital angular momentum L, the spin angular momentum S and the total angular mo mentum J, respectively, along the internuclear axis. We will use this wavefunction also E+ electronic state (although Hund’s coupling case (b) is more for the first excited 2 suitable for this state). An expression for HR is given by Hougen HR  =  2 B(r)R  =  []  as  B(r) (R + 14)  (A.2)  ,  where R is the nuclear rotation angular momentum operator and B(r)  =  ) is 2 h/(4ircr  the radial part of the rotational energy operator, defined in terms of the internuclear distance r and the reduced mass  t.  The nuclear motion is necessarily in a plane that  contains the internuclear axis. Thus R is perpendicular to the z direction and R  =  0.  The total angular momentum excluding the nuclear spin is defined by J=R+L+S.  (A.3)  The rotational Hamiltonian can then be written as HR  =  B(r) [(Jx  —  Lx  —  2 + (Jy Sx)  —  Ly  —  ] 2 sy)  Appendix A. Rotational Energy and Fine Structure in  =  B(r)  + (L2  [(j2  —  —  +B(r) [(LS_ + L..S÷) where J±  Jx ± Jy,  L  =  L) + (S2  —  2  and  210  s)]  (J÷L_ + J_L÷)  L + iL  Electronic State  S  =  —  (J÷S_ + JS)]  ,  (A.4)  Sx ± iSy. Here, J and J_  follow the anomalous commutation rules. As a result, J+ is a lowering operator and J_ is a raising operator. The first three terms of HR have oniy diagonal matrix elements in the basis selected. This diagonal part of HR is the rotational energy of the IASE; QJ) basis function in the limit that coupling to other electronic states can be neglected. The final three terms of HR, which couple the orbital, spin and total angular momenta, are responsible for perturbations between different electronic states. The spin-orbit interaction in a 211 state is expressed by the Hamiltonian =  A(r)L S  =  A(r) [LZSZ +  .  (LS_ + L_S+)]  (A.5)  ,  where A(r) is the spin-orbit coupling constant. The Hamiltonian for spin-rotation interaction in a 211 state is given by HSR  =  7(r)R.S  =  ‘y(r)(J—L—S).S  =  7(r)  (J+S_ + J_S+)]  +  —  7(r)  [Ls +  (LS_ + L_S+)]  —  , 2 7(r)S (A.6)  where 7(r) is the spin-rotation coupling constant. For E states (A=0), the matrix ele ments of the second term, i.e., (7 r)L.S, are zero. Therefore, the spin-rotation Hamiltonian for E states is written as HSR  =  = 7(r) [JzSz +  (J÷S_ + J_S+)]  —  . 2 7(r)S  (A.7)  Appendix A. Rotational Energy and Fine Structure in  2]j  Electronic State  211  The Hamiltonian for spin-rotation interaction in a 211 state can also be expressed by Eq. (A.7) since the 7(r)L. S term is generally very small, and is absorbed into the spin-orbit term which has the same form. Expressions given by Eqs. (A.4), (A.5) and (A.7) can be combined together to re-write Eq. (A.1) (excluding Hss) as H  T + B(r)  [(j2 —  + (L2  +B(r) [(LS_ + LS) +A(r) [LZSZ + +  +y(r)  —  —  L) +  (s2  —  (J+L_ + J_L)  s)] —  (J+S_ + J_S-)] 1  (LS_ + L_S+)] (J+S + J_S+)]  —  , 2 -y(r)S  (A.8)  where T has been added to represent the vibronic energy corresponding to the nonrelativistic Hamiltonian, HEy. The Hamiltonian given by Eq. (A.8) represents the total Hamiltonian (exclusive of hyperfine interactions) containing all the major interactions of the  2  E+ electronic system. In deriving the matrix elements, we will represent the 2  expectation values of B(r), A(r) and 7(r) as B, A and y, respectively, with appropriate subscripts referring to their corresponding electronic states. Since these constants repre sent the radial part of their corresponding energy operators, their expectation values are usually determined separately for each vibrational level during a data fitting procedure. With the use of Hund’s coupling case (a) basis function, ASE; 1J), the matrix ele ments of the operators appearing in the expression of the Hamiltonian of Eq. (A.8) are given by (A S E; Q J (A 5  ç  j  j2 —  S2  —  J A S E; Il  =  J(J + 1)  S A S E;  =  S(S + 1)  (AS E; Q JL2 —LA SE; Il  =  (A.9)  —  2 E  (A.1O)  2 (L) ; L(L+ 1)—A  (A.11)  —  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State  212  (A±1SE; +1JIJL±j ASE; J)= [{L(L + 1) (A S ±1;  —  A(A ± 1)} {J(J + 1)  +1J JS±IAS E; [{S(S + 1)  —  JIL±SI  (A±1 S E+1;  [{L(L + 1)  J)  )}]hI’2 Q( + 1  (A.12)  =  E(E + 1)} {J(J + 1) A SE; 1 J)  —  —  ± 1)}]1’2 ;  —  (A.13)  =  A(A + 1)} {S(S + 1)  —  E(E + 1)}]1’2  .  (A.14)  For convenience, the it factor is moved into the coupling constants. These matrix elements 2 + and the can now be used to construct the energy level matrices for the E  2J  electronic  states using the signed quantum number basis functions. Since L is not a good quantum number for molecular systems, terms in the matrix elements containing (Lfl are usually incorporated into the vibronic energies  [].  While deriving the matrix elements, we will  treat these terms separately. At the final stage, these terms will be absorbed into effective vibronic term values.  2. Matrix elements of the 2+ electronic state diagonal in A  The Hamiltonian for the 2 E electronic state is given by H=HEV+HR+HSR.  We will represent  ASE; QJ)  =  ) by  0, +, +; +,  ). 112 2Et  The diagonal matrix  elements of the rotational Hamiltonian are given by (2E+ HR  2E+) =  BE  (  +  1)2  and the off-diagonal matrix elements by (2E+  HR  2E+) =  —BE  + BE KLI)E ,  (  +  ,  (A.15)  (A.16)  Appendix A. Rotational Energy and Fine Structure in  where BE  (2E  =  2  Electronic State  B(r)2,). In the definition of BE, L  =  =  —  213  0, ±1. To  the present level of approximation, the same numerical value for BE holds for all the allowed values of  and  ‘.  Similarly, we obtain the diagonal and the off-diagonal matrix  elements of the spin-rotation Hamiltonian as (2+  and  (2E+  where ‘y  =  HSR  HSR  2E+)  2E+)  1  = 7E  (A.17)  ,  =  (  +  (A.18)  ,  K2 ( 7 r)I2IE,). In the definition of 7E,  =  —  =  0, +1. To the  present level of approximation, the same numerical value for 7E holds for all the allowed values of  and 1’. Thus, the combined rotational and spin-rotation interaction matrix  for the  electronic state is given by  2>+  2v+ +1/2  E+1/2 2  TE  E-1/2 2  where z  (+  + BEZ  —  1)2  (BE  7E  —  2’+ —1/2  + BE (L)E  7E)  \/  —  (BE  TE + BEZ  —  (A.19) —  7E)  7E  + BE (LflE  and TE represents the vibronic energy arising from HEy.  Appendix A. Rotational Energy and Fine Structure in 3. Matrix elements of the  The Hamiltonian for the  214  electronic state diagonal in A  electronic state is given by  211  11=  Electronic State  2  REV  + HR + ilso + HSR  We will represent the basis function IAS; QJ) as follows 1 1 3 +1, +, +; +, J)  2  11+3/2);  1 1 1 \ +1, +, —; +, J)  2  11+1,2);  )  —1, +, +;  211);  1 1 3 \ —1, +, —; —, J)  2  11_3/2)  The diagonal and the non-zero off-diagonal matrix elements of the rotational Hamiltonian are given by (2  HR  2)  (2ll  HR  211,)  (211 HR  211,)  where B 11  =  (2ll±  =  11 [(J + B  =  11 B  (  )2  11 (Li) +B  +  (211  — 2] + B 11 (Li)  HR  211) =  (A.20)  ;  (A.21)  ;  11 —B  +  1)2  —  111/2 , (A.22)  B(r) 12111±o,)  The spin-orbit Hamiltonian has non-zero values only for the diagonal matrix elements. These matrix elements are given by (2ll  Ho 211±3,2)  =  An;  (A.23)  (2ll/  50 H  =  , 11 —A  (A.24)  211)  Appendix A. Rotational Energy and Fine Structure in where A 11  = (2n±I  A(r)  well as those of the form  2fl±).  211  Electronic State  The matrix elements of the form ( 11±cj H 2 0  ( l l±I HSO I 11±I HSO 1 2 K 11±w) and 2 2 11c’) with f 2  215  11c>, as 2 l 1’, are all  equal to zero. The diagonal and the non-zero off-diagonal matrix elements of the spin-rotation Hamiltonian are given by (2n (2  HSR 211) 2fl)  HSR  =  0 ;  (A.25)  7n ;  (A.26)  K2n±1,2 IISR 211±3/2)  where y  =  HSR 211±1,2)  [(j  =  (2ll±I 7(r) 2 T1±’) The matrix elements of the form  well as those of the form  211 HSO ( ±oI  T1I) 2  with  +  —  i]h/2  ,(A.27)  (2fl±I HSR 2ll),  as  are all equal to zero.  The combined matrix elements of the Hamiltonian for the  2ll  electronic state, using  the signed quantum number basis, can be written as  211,  211±,  11 T  —  211±3/2  11 + z A 11 + B B 11 (L) 11  211±3/2  —  (Bn  —  7H)  /J  —711  —  (Bri  —  yn)  11 + 1 T A 1 + Bn(z  —  /J 2) + 1 B 1 11 (LI)  (A.28) wherez=  . 2 (+)  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State 4  2  2+  216  interaction matrix elements  The operators in the rotational Hamiltonian, Rn, connecting the 2ll and the  2+  electronic states are J±L and L±S. Strictly speaking, one should include the vibrational part of the wavefunction for both the interacting states to derive the elements of the interaction matrix. We will represent these wavefunctions for 2ll± and states as 2ll±, vj-j) and  K  rj  respectively. Using Eq. (A.12), we obtain  yE),  JL±  2 / 1 2Et  electronic  v J±L 211±3,2,  (21,2  yE)  —  (  +  1  Vfl)  1/2  2 —  1  (A.29)  ,  and (211±1,2 VJ-J  JL±  2+1,2 ‘‘E)  (2+1,2  =  —  (  v JL 211±1,2,  +  Vfl)  (A.30)  ,  where = (2fl,  B(r)L  2E+,  yE)  (2E+,  v B(r)L_ 211, vfl)  (A.31)  .  The LS± operators appear in the rotational as well as the spin-orbit Hamiltonians. The matrix elements of these operators, using Eq. (A.14), are then given by  (211±1/2,  vu [B(r) + A(r)j LS  1/2’ VE)  (21,2, yz [B(r) + A(r)] LS =a+,  211  Vfl)  (A.32)  Appendix A. Rotational Energy and Fine Structure in  2  Electronic State  217  where  a and  = (2,  A(r)L  2+,  v)  = (2+,  vy A(r)L_  211,  11 v )  (A.33)  ,  is defined by Eq. (A.31).  The matrix elements of the  211  —  E system in the signed basis function can be 2  represented by a 6 x 6 matrix. Below we show in a tabular form the zero and the non-zero matrix elements of the 6 x 6 matrix representing such a system. For visual convenience, instead of writing the actual matrix elements, we present the zero and the non-zero matrix elements by  ‘Q’  and  ‘/‘,  respectively.  2,  211,  2111,2  2113,2  1 E 2  V  V  0  0  V  0  211,  V  V  0  0  V  V  211,  0  0  V  V  V  V  2 f l,  0  0  V  V  0  V  112 E 2 ÷  V  V  V  0  V  V  1 E 2  0  V  V  V  V  V  2 f l,  As can be seen from the structure of the above matrix elements, deriving an analytical expression to represent the energy levels of the f 2 J and the  2+  electronic states is  Appendix A. Rotational Energy and Fine Structure in  2  Electronic State  218  cumbersome. Of course, the energy levels can be calculated by diagonalizing the above 6 x 6 matrix, but the investigation of the J-dependences of these energy levels, especially the fine structure splittings which arise in the 2+  211o  state due to the interaction with the  state (called the A-doubling), is difficult. In the next section, we present a solution  to this problem where we define the ‘symmetrized’ basis functions for these two electronic states with the help of symmetry operations.  5.  Choice of symmetrized basis and the  2+/  2H matrix elements  Up to this point, wavefunctions with signed values of A,  or  have been used to  derive the matrix elements of the Hamiltonian, the eigenvalues of which can be obtained by diagonalizing the 6 x 6 matrix given in the previous section. In order to simplify the matrix, it is convenient to use properly symmetrized basis functions. This is achieved by using the o operator which corresponds to a reflection through the molecule-fixed xz plane (i.e., the plane containing the internuclear axis).  [n],  The a., operation, following the phase convention used in Ref. following transformation  [n]:  (a)  Orbital  (b)  Spin  av  (c)  Rotation :  a., J, 1)  where s=1 for a E  :  (_i)’ —A) ;  o, A) S,  leads to the  ) =  (A.34)  (_l)  IS, —E)  ;  (A.35)  (_i)J_  IJ, —Q)  ,  (A.36)  state and s=O for all other states. This power of s for a E state  appears in the orbital part to classify the A=O states according to their intrinsic E  or  symmetry. Thus, a a operation on a case (a) basis set gives aIA 5  J)  )A+S_E+J_+8 1 = (_  I—A S  —;— J)  .  (A.37)  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State  219  Eq. (A.37) can be slightly simplified to read: o IA S E; ! J) For a 1 E state (A  =  0, S  =  (—1) —A S —E;  =  —  J)  (A.38)  .  0), Eq. (A.38) gives:  a. 0 0 0; 0 J)  (—1) 0 0 0; 0 J)  (A.39)  Therefore, the eigenvalue of the parity operator o for a ‘E+ state is (_i)J and for a state is  (_i)1.  Thus, the sign change of the eigenfunction under the operation o  will alternate with the quantum number J. For a parity and levels with odd J have a a  ‘—‘  ‘—‘  state, levels with even J have a  parity. For a ‘E  parity and levels with odd J have a  For states other than a  lE+  ‘+‘  state, levels with even J have  parity.  state (where the electronic basis functions associated with  positive and negative signs of (A, E, Q) are degenerate), a o operation on the basis function (see Eq. (A.38)) changes the basis function itself. For example, for a 211+3,2 state,  H,  +, + +, j)  =  )J_4 _1, 1 (_ +, —;  Here, the basis function +1, +, +; +, —1,  —,  j)  .  (A.40)  ) on the left hand side and the basis function  —; —, J) on the right hand side are not identical. Hence, (_i)J_ is not  an eigenvalue and +1, +, +; +,  j)  is not an eigenvector of the parity operator u..  Therefore, for states other than a ‘E, it is necessary to use linear combinations of basis functions to obtain well-defined parities of the energy levels. Two methods are in common use for labelling the energy levels. These are the  +1—  2 + and 2 total parity scheme and the e/f symmetry scheme. For E 11ç states, which are of our interest here, the symmetrized basis functions in the total parity labelling scheme are given by 2+,  j  =  ±  [  2E+,)  )J_4 1 ± (_  2E+)j  ,  (A.4i)  Appendix A. Rotational Energy and Fine Structure in  211  Electronic State  220  and  ±  2 l l, j  [  2j)  )J_4 2ll)] 1 ± (_  ,  (A.42)  respectively. In Eqs. (A.41) and (A.42), it can easily be shown that the basis functions 2+,  J  ±) and  2ll,  J  ±) are the eigenfunctions of the a operator, each with the  eigenvalues of + 1. Following the definition of Brown et al.  [], the symmetrized basis functions for  E 2  and 211o electronic states in terms of e/f labels are given by 2+  =  H, j 2  =  [  2)  ±  )] 1 t 2 12  ,  (A.43)  and  =  [ ll+) ±  2ll_)]  .  (A.44)  The upper and the lower signs in the right hand side of Eqs. (A.43) to (A.44) refer to the e and the f symmetrized wavefunctions, respectively. As discussed by Brown et al. [j, it is often more convenient to use the e/f labels since it removes the (_i)J_ J-dependent factor from the definition of the symmetrized basis functions, making the label rotationindependent. We will use the e/f symmetrized basis functions defined by Eqs. (A.43) and (A.44) in the remaining part of the Appendix. It will also be assumed that the bra and ket labels which include the e/f labels refer to the symmetrized basis functions. The bra and ket notations without the e/f labels refer to the signed (unsymmetrized) basis functions. Matrix elements of the Hamiltonian with the e/f symmetrized basis functions can be written in terms of matrix elements derived using signed quantum number basis functions as  Appendix A. Rotational Energy and Fine Structure in  J  H  )  2+,  H  =  +  (  H  ) 112 2E+  2÷)  ±  211  +  (  Electronic State  112 (2Et  2+  H  H  221  2+1,2)  2)]  +(2ll+H 211)± (2flH 211)]  ,  (A.45)  (A.46)  ,  and J  (211  ;)  E+ H 2  (2+  J  ll, H 2  vH , 112 E 2 +  (2n,  vrj  H  ) 112 2E+  yE)  +  +  j (2_,  vj H  vrj H  (  2+1/2 VE)  21,2, yE)]  ,  (A.47)  In Eqs. (A.46) and (A.47), we have retained the vibrational part of the wavefunction only in the 2ll  2+  interaction matrix elements. It is assumed that the matrix  elements represented in Eqs. (A.45) and (A.46) (which are diagonal in A and 1) have their own vibrational dependences in their basis functions, and are not written here for simplicity. These vibrational dependences have explicitly been written only for the matrix elements representing the 211  _2  interaction (see Eq. (A.47). It is to be noted  that the above matrix elements are diagonal in e and f, i.e., the matrix elements of the form (e H f)  =  (f HI e)  =  0. Thus, the non-zero diagonal and off-diagonal (in 1) matrix  elements of the Hamiltonian with e/f symmetrized basis functions are given by (2113/2,  J  H  (2111,2,  , 2 / 1 JH211  2113/2,  J  )  Tn + 1 11 B A 1 + (z  2) + B 11 11 (L) ;  (A.48)  J)=Tn—An+Bnz+Bn(Lj) n 7 — 11 ;  (A.49)  =  —  Appendix A. Rotational Energy and Fine Structure in  (2111,2,  H  J  2113,2  ;)  = (2113,2  J  H  2111,2  211  J  Electronic State  =  —  (Bn  —  222  Lyn) v’i; (A.50)  (22,  J  H  2,2,  (2113,2,  J  H  2 2i-,,  (2111,2,  , J) 2 JH2,  =  j  T + BE(z +  = (2E+,  = (22,  ) + B (Li)E  H  2113,2  , 112 JH211  —  + ); (A.51)  J;)  =  —i3vi ;  J)  =  a+(1+). (A.53)  (A.52)  Since there are no non-zero matrix elements connecting e and f components, the over all 6 x 6 matrix can be factorized into two 3 x 3 blocks, one for the ‘e’ levels and the other for the ‘f’ levels. These matrices are then given in a tabular form by  11 2 ,e  2fJe  211  2+ e 1/2f  —An+Bnz— 1 T n 7  —  (B —  2+ e 1/2f  e  3/2f 11  yn) /Ei  ‘a+i9(1+/) 2  1+ 1 T A 1 + Bn(z —2)  —/3\/iE 7E(1+i/)  (A.54) Here, the vibronic term value TA is re-defined as T to incorporate the BA (Lj term into it. The upper and the lower signs in the matrix elements are for the e and the f  Appendix A. Rotational Energy and Fine Structure in  211  Electronic State  223  levels, respectively. It should be noted that the oniy difference between the matrices for the e and the f levels is in the diagonal matrix element of the  211 212  2+  interaction matrix elements and in the  state.  On diagonalizing this matrix, one obtains the energies for each vibration-rotation level corresponding to a single v and a single J value within a particular electronic state with no interaction with the energy levels of different electronic states and/or with different values of v of the same electronic state. In general, however, there is always some interaction with the energy levels of different electronic states and/or with different vibrational levels of the same electronic state. Because of the radial dependence of the operators appearing in the Hamiltonian, the molecule, as it rotates, couples neighbouring vibrational levels. Furthermore, as shown earlier, because of terms such as J L and L S, there are also off-diagonal matrix elements that couple different electronic states with the electronic state to be studied. Let us consider the energy levels within the vASJ block, where  is a specific elec  tronic state to be studied. The energy levels of the ivASJ block may be said to be perturbed by various interactions arising from the coupling with the other energy lev els. Occasionally, the perturbations can be ascribed to a small number of levels from the neighbouring blocks. Then the effect of such perturbations can be treated by di agonalizing the matrix of the interacting levels, provided one has a prior knowledge of the locations and symmetry of the perturbing levels. In most cases, the perturbations are small in magnitude and large in number. Moreover, they invoke many interacting levels from distant blocks whose energy positions are often poorly known. In such cases, an exact treatment is very difficult and one needs to formulate approximate procedures. Thus, in order to construct a generally applicable procedure to obtain a set of molecular constants from the observed spectra of diatomic molecules, these interactions are treated by incorporating their effects on the vASJ block of interest. This is accomplished using  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State an approximation, introduced by Van Vieck  [,  224  29], which is essentially an application  of the perturbation theory. 6. From a supermatrix to a reduced 2 x 2  2  matrix: An application of the  Van Vleck transformation In order to derive an effective Hamiltonian to represent the rovibronic energy levels of an electronic state of interest perturbed by distant electronic states, it is necessary to perform two contact transformations, namely, the electronic contact transformation and the vibrational contact transformation. The electronic contact transformation, also called the electronic Van Vleck transformation, uncouples different electronic states interacting with the electronic state of interest to a given order. (For example, if an off-diagonal matrix element is “removed” by a Van Vieck transformation, the coupling matrix element is reduced in magnitude by one order. Further reduction can be achieved, in principle, by applying the next order Van Vleck transformation.) This transforms the electronic part of the Hamiltonian so as to remove terms that give matrix elements off-diagonal in electronic states without altering the eigenvalues of the Hamiltonian to the order in question. Such a transformation is generally applicable only if the interacting electronic states lie far away from the electronic state of interest. The vibrational contact trans formation, also called the vibrational Van Vleck transformation, then performed on the resultant matrix elements of the Hamiltonian, uncouples the different vibrational states giving rise to an effective Hamiltonian for each vibronic state.  In our case, the Van Vieck transformations involve adding second and third order correction terms to the matrix elements of the 211 state in order to determine the ener gies of this state, perturbed by a distant 2 E state. The two transformations, i.e., the  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State  225  electronic and the vibrational Van Vieck transformations, can be applied simultaneously to the interacting electronic and vibrational states to derive an effective Hamiltonian representing the rovibronic energy levels of the 211 state.  Following Zare [nj, let the Hamiltonian be written as H  =  H° + AH’  (A.55)  ,  where A is an order sorting parameter. Here, it is assumed that the matrix elements of H° lie entirely within diagonal blocks (i.e., within the 2 x 2 211 matrices of interest) while the matrix elements of H’ may lie inside or outside diagonal blocks. Also, let the different energy levels within the vASJ block of interest be indexed by i,  j,  k,..., and let a,  /,  y,..., label levels of other interacting blocks. An application of a unitary transformation, lI  THT (also called the contact transformation), where T is a unitary matrix, causes  =  the matrix elements of H within the ,vASJ block to have the form =  ñ  (1) 2 +AH +A  fj (2) + A 3  (3)  +  (A.56)  Here, fl(O)  ) 2 fr  j)  =  (i H°  =  (i H’ j)  _  =  E  (A.57)  ;  (E+Ej)-E  (A.58) (i  IH’Ia> (a H’j);  (A.59)  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State  226  j)  (3) ii  (i  —  IH’ a) IE’  k4  a,3  —  H’I i3) (/3  (a V  ‘fi:’  L) :L:Ji  H’  1::’  —  i (1119 k) (k 119 a) (a H9j) + (i “9 a) (a I11’Ik) (k IH9i) 2 a,k (EkEa)(Ej Ea) (EkEa)(EjEa)  ( A 60 )  .  .  These represent the matrix elements of the Van Vleck transformed H (i.e., H), up to the third order. The bra(s) and ket(s) representing the matrix elements on the right hand sides of Eqs. (A.57) to (A.60) refer to the symmetrized basis functions. The e/f labels of the basis functions in these particular equations have been dropped for convenience. In order to demonstrate the use of Van Vieck transformation in reducing a supermatrix to a 2 x 2 211 matrix, we first consider the matrix elements arising from the f 2 l  2+  interaction (which gives the lambda-doubling terms, o, p and q). Later, we will consider an example of the centrifugal effects associated with the off-diagonal matrix elements of B(r), since B(r) has a fairly strong dependence on r. Using the second-order Van Vleck transformation given by Eq. (A.59), we obtain the matrix elements representing E+ interaction as 2 the 211o , (2) e H l 1 (f) 2  2  —  —  (211, v A(r)L 2 1 E, v+) (2+, v A(r)L_ 4 E°v(211i) —E°v’( E+) 2  2,  v)  ( 1 2 1,vB(r)L 2 ,v+)( E,v+ B(r)L_I 1 2 2 1,v) ( E°( E° fJi) 2  +  —  +  :  (211,  v  E+, v+) IA(r)L+I 2 1 1 2 v( ) 1  V  —  (2+,  v. IB(r)L_ 211, v)  ,i  E°v’( E+) 2  1 FVZ  (A.61) (e 2’  Vf)  2  —  —  (211, v  2 , v+) ( E+, v÷ IB(r)L_ 2 B(r)L+I E E°v(2H) E°v’( E+) 2 —  211, v)  y2  Appendix A. Rotational Energy and Fine Structure in  211  Electronic State  227  (A.62) u(2) “vI  2’  (e’  2  f)  —  =  —  u(2) “va  2’  —  1  —  2  1 ( 11,vIA(r)L+l 2  2-I-  I  \/2’ç’f  p0  I  p0  ‘v( f 2 lL)  IB(r)L_l  211,  v)  B(r)L_I  211,  v)  B(r)L_  211,  v)  B(r)L_I  211,  v)  y  —  2  ], v (2  B(r)L  p0  LJv(  -  VE+  fJ, V IA(r)L+I (2  -:  2+,  v+)  211t)  2+,  B(r)L  2v-I-  I  (‘F/)y  —  ( 2+  p0  (2fl)  -  p0  v+)  ) 3 ‘v(2H  i, v (2  ( 2+  p0  y  —  \12’c’+ J  —  I  ,VE+  E°v’( E+) 2  (‘+i)y, (A.63)  where z  =  (+)  2  y  =  and  represents the  nth  order correction to be  added to the 2 ( ] J, v, J 110 I 2 fJ’ v, J) matrix element. The summations on the right hand side of Eqs. (A.61) to (A.63) are to be done over all vibrational levels of the 2 E state, with v’  v. 2 E( f l) and  E(2E+) in  the energy denominators represent the un  perturbed energies of the (2ll, v) and 2 ( E , v) levels, respectively. In Eqs. (A.61) to (A.63), v appearing without a subscript refer to the vibrational levels of the 211 electronic state. It should be noted, in accordance with the convention introduced earlier, that the bra and ket labels on the right hand sides of Eqs. (A.61) to (A.63) refer to the signed (unsymmetrized) basis functions.  As mentioned earlier, the electronic Van Vleck transformation is applicable only for  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State  228  states perturbed by distant electronic states. Under this assumption, the energy differ fl 2 (E( ) ences 1  —  E°l(2E+))  and (E(2fla)  —  EI(2E+)) occurring in the denominators of  the above matrix elements are of similar magnitude (provided the spin-orbit interaction is small), and hence can be represented by a common denominator. Here, we will rep resent these denominators by (E( 211)  —  E°l( 2E+)). Therefore, these matrix elements can  be written as fli  (;) e (f)  uvv  =  o + pn (1+ (z  =  —  ()  \/)  + qn (1+  /)2;  (A.64)  1);  =  (A.65)  —piiv’J  —  qn (1 +  /)  (A.66)  where, 011  pn  qri  =  1  (211,  v A(r)L+l  2+,  UJv(211)  =  2  ( 1 2 1,vA(r)L 2  =  2  (211,  —  E°v(211)  —  ,  (A.67)  .LJv(2E+)  —  v B(r)L+ 2 E+, i4+)  are the A-doubling parameters.  v. A(r)L_I 211 v)  ,v+)( E 2 ,v+ IB(r)L_I 2 11,v) 11) 2 v(  v.  (2+,  v)  ,  (A.68)  ,  (A.69)  vl(2+)  (2+,  v.  E°vl( E+) 2  IB(r)L_I  Since the term containing  0  2,  v)  do not have any J  dependence, this term is usually absorbed in the term value.  For the centrifugal effects, we will consider only the second order effect of B(R) with itself, which leads to the rotational centrifugal distortion term D 11 for a particular vibrational state. Using the second-order Van Vleck transformation represented by Eq.  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State  229  (A.59), we obtain correction terms for the diagonal and off-diagonal matrix e’ements of  211 as e  >2  =  (211  vi B(r)  v’) (211, v’I B(r) ) EI( 2111 E( ) 2111  v’v  +>2  (2fl  11,v’) ( 11,v) vIB(r)I 2 11,v’IB(r) 2 2 E( f 2 l)  (211  v’v  +>2 ()f  ( 211  i B(r) I 2 1 u, vi B(r) 211, v’) ( 211, v  >2  +>2  (211  v’v  all  1)  (z  —  2)2  (A.71)  ;  2 , v) [1] vi B(r) 211, v’) ( 211, v’ B(r) I J ) 2111 fl) El( 2 E( 2  J, v) ( 211, v’ B(r) I 2 {_(z fl) 2 EI(  v B(r) 2 ]J, v’) flL) 2 E(  —  2)y]  —  2  2  fJ, v’) (211, v’I B(r) vi B(r) I 2 ) 2113 fl) El( 2 E(  I 211, v)  [_(z  —  2)y]  —  2  (A.72)  assume that the separation of the vibrational levels is significantly  large compared to the spin-orbit splitting of the 2ll state (i.e.,  are  (A.70)  —  (211  —  —  1);  v B(r) I 2, v’) (211, vi B(r) I 2 fJ, v) fl E( 2 E°( ) 4 fl) 2  v’v  (E( H 2 )  —  =  2  (211  differences  (z  —  v’v  For simplicity, if we  V)  Hla) 2 E( f )_El(  (211  v’v  E°l( f 2 l)  —  (±.)  =  —  viB(r)1 1 2 1,v’) 1 11,v’IB(r)1 2 ( 1,v) (z fl) 1 2 E( fl 2 El( )  v’v  ) 2 j(  2 I 2]J, v)  —  v’v  =  211,  E°I( f 2 l,))  IACI  <<‘e),  the energy  occurring in the denominators of Eqs. (A.70) to (A.72)  of similar magnitude. Under this assumption,  all  the terms in each of the matrix  Appendix A. Rotational Energy and Fine Structure in 211 Electronic State  230  elements represented by Eqs. (A.70) to (A.72) can be combined together and represented by a common denominator, say (E( 211)  — E°i(  211)),  with the terms summed over all  possible v’ levels. This we refer to as the “merged” model. In this model, the matrix elements can then be written as (211  1 HV 2 2  ()  =  2  2  ()  =  2  ()  2  vi B(r)  —  v( 211)  v’v  v’)  ,  v’( 211)  1( vi B(r) :11, v’) v( 211)  v’v  vi B(r)  =  2  (z2  v’( 211)  (2  0 H 2  —  .( H 2 )  v’v  —  + z -1);  (A.73)  -3 + 3);  (A.74)  211,  v)l2  v’( 1 2 1)  [_2(z  - i)]  (A.75)  or 1 H  (;)  =  H3  ()  =  3 H  ()  {— (z + z — i)] ; 11 [— (z — 3z + 3)] ; D 11 [2 (z — 1) /ET] () D  11 D  HI  =  (A.76) (A.77) (A.78)  .  Here, D 11 is the rotational centrifugal distortion constant, defined by -I-’  /2  c—’ \  iL,  V •nf .L)r)  E°,,( 2 fl)  \2 2ii , V  — E°  .,I(  211)  The negative sign in the definition of D - follows the usual convention 1  []J.  The matrix elements of all second-order distortion constants, i.e., D , 1 1 AD11, qDH,  pDn  and  can also be obtained in a simple way by taking the product of the B 11 matrix with  itself, or as the sum of the symmetric products of the B 11 matrix with other matrix of interest. For example, if [Bn] represents the matrix elements corresponding to the  Appendix A. Rotational Energy and Fine Structure in  2fl  Electronic State  231  parameter , 11 and [X B ] represents the matrix elements of the parameter of interest (X 11 =  A, p or q), then the matrix elements of the second order distortion constant of X, say  XDH,  can be obtained as [XD11]  =  ] [B 11 [X ] + [B 11 ] [X 11 ] 11  (A.80)  .  In order to demonstrate this, let us take the matrix elements of the B 11 matrix, represented by the pseudo-Hamiltonian matrix elements 1112 1121  =  11 B  _/Ei  1122  (z  —  2)  where , 11 1112, 1121 and 1122 correspond to the (211k H and  (2n  2113)  (A.81)  ,  (2H  2113),  (2113  2ll)  interaction matrix elements, respectively. As mentioned earlier, the  matrix elements corresponding to the second-order rotational centrifugal distortion con stant can be obtained by taking the product of the B 11 matrix with itself. Thus, 11 B  z  .\/Ei  (z—2)  z  11 B  -_/Ei  B  (z—2)  +z—1 2 z  —2(z—1)/J  _2(z_1)\/Ei  —3z+3 2 z  (A82)  Dividing by the energy differences between the interacting vibrational levels and sum ming over all the possible interacting vibrational states, and with the appropriate sign convention in the definition of D 11 [1], these matrix elements can then be written as 11 D  —  2+z (z  —  1)  2(z  —  1)/Ei  2(z—1)/J —(z —3z+3) 2  (A.83)  The definition of Dri in Eq. (A.83) is the same as that given by Eq. (A.79). These matrix elements are the same as those obtained earlier and given by Eqs. (A.76) to  Appendix A. Rotational Energy and Fine Structure in (A.78). The matrix elements of  211  2fl  Electronic State  232  state given in a tabular form by Eq. (A.54)) and  the A-doubling matrix elements of Eqs. (A.67) to (A.69), along with their distortions, constitute the effective Hamiltonian for a  211  electronic state perturbed by a distant  2  electronic state. All the paramaters in the effective 211 Hamiltonian correspond to a single vibrational level of the  2  electronic state. Therefore, the subscript 11 from the  parameters can be replaced by a subscript v, e.g., B, D, A,  ...,  which then represents  the parameter for a particular vibrational level v. A similar approach can be adopted to derive the matrix elements of the next higher order distortion constants, e.g., Hv,  etc. These matrix elements have been worked  out and all the major matrix elements of the effective Hamiltonian for a state in the merged model are listed in Table 5.1 of Chapter 5.  2  electronic  

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