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The infrared electronic spectrum of FeO Taylor, Alan William 1986

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THE INFRARED ELECTRONIC SPECTRUM OF FeO by ALAN WILLIAM TAYLOR B.Sc. (Hon.). University of Victoria, 1978 M.Sc., University of Victoria, 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Chemistry We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1986 c Alan William Taylor, 1986 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Chemistry The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date February 21. 1986 DE-6(3/81) ii ABSTRACT The near-infrared electronic system of FeO has been recorded in emission at high resolution with the 1-meter FT spectrometer at Kitt Peak National Observatory. The system has been found to consist of two overlapping band systems lying in the region 7 000 - 14 000 cm:1 These share a common lower state and this is the same lower state as that of the orange system. This state, which has vibrational constants (in cm"1) a>e = 880.4148, a>exe = 4.632 15, and w£ye = 5.55 x 10:" has been established as the ground state of the molecule by the matrix isolation experiments of Green, Reedy, and Kay (J. Mol. Spectrosc. 1979, 78, 257-266) and has been identified as a 5Aj state in earlier studies done in this laboratory on the orange system. The present work has proven this identity, and least-squares data reduction has yielded the ground state constants: B g = 0.518 721 cm;1 a e = 0.003 825 cm;1 7 e = -4.8xl0"7 an;1 D e = 7.210 x 10"7 cm;1 0 e = 1.51 x 10"9 cm;1 r g = 0.161 64 nm, and A = -94.9 cm:1 These were based on combined data from the infrared and orange systems for the levels v = 0-3, together with six microwave frequencies obtained by Endo. Saito, and Hirota (Ap. J. 1984, 278, L131-L132). Both the 5A 0 and 5A, substates are found to show small A-doublings, with that in 5A 0 being the larger (~0.3 cm"1). In contrast to all the known excited states of FeO, the ground state is well-behaved, but even this state possesses one small rotational perturbation. This is observed only in one parity component of the J = 15 level of the v = 2 level of the X 5A 2 substate and is probably caused by the low-lying 7Z state predicted by Krauss and Stevens (J. Chem. Phys. 1985, 82, 5584-5596). The upper level of the infrared system consists of two overlapping states about 10 200 cm"1 above the ground state and separated by only about 213 cm"1 at their v = 0 levels. The upper of these, a 5 0| state, has the approximate values AG 1 / 2 =* 627 cm-1 and A 0 =* -217 cm;1 and for the lower s * j state AG 1 / 2 593 cm"1 and A 0 -47 cm:1 These are very rough estimates only, as a result of the iii vast number of rotational perturbations present. Effective rotational constants for the individual substates have been determined rather than a global set of constants, again because of the perturbations. Amongst the perturbing states, at least one multiplet L state is present and there may be as many as three. Evidence is also presented for the existence of a 3 $ state. 4 iv TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES vii LIST OF FIGURES viii ACKNOWLEDGEMENTS x AUTHOR'S PREFACE xi CHAPTER I. INTRODUCTION 1 l.A. Historical Survey 2 LB. Other Work Done in This Laboratory 15 I.B.1. Characterization of the ground state 15 I.B.2. The orange system 18 I. B.3. Ground state spin-orbit intervals 21 I.C. Reasons for studying FeO 25 I. D. Some Very Recent Studies Based on the Results of the Present Work 30 CHAPTER II. EXPERIMENTAL DETAILS 35 II. A. Introduction 35 II.B. Description of the Source 37 II.C. Methods of Detection 38 II. C.l. Grating spectra 38 II.C.2. Laser-induced fluorescence 38 I1.C.3. Fourier transform spectroscopy 39 II. D. The Spectra 40 CHAPTER III. THEORY 44 III. A. Introduction 44 III.B. The General Molecular Hamiltonian 47 III.C. Separation of Electronic and Nuclear Motions — The Born-Oppenheimer Approximation 49 V TABLE OF CONTENTS (cont.) III.D. Separation of the Vibrational and Rotational Motions 52 III.E. Electron Spin 55 III.E1. Spin-orbit interaction 56 III.E2. Spin-rotation interaction 59 III.E.3. Spin-spin interaction 61 III.F. Angular Momenta and Hund's Coupling Cases 66 III.F.l. Angular momenta 66 III.F.2. Hund's coupling cases 69 III.F.3. Angular momentum basis functions 73 III.G. Symmetry, Parity, and Lambda-doubling 75 III.H. Effective Hamiltonian and Matrix Elements 82 III.H.l. The Van Vleck transformation 83 III.H.2. Construction of the effective hamiltonian and matrix elements in a case (a) basis 88 III.H.2.a. Lambda-doubling in a 5 A state 99 III.I. Intensities 110 III.J. Perturbations 120 III. K.. Electron Configurations 130 CHAPTER IV. THE GROUND STATE: X 5Aj 134 IV. A. Introduction 134 IV. B. Lambda- doubling 141 IV.C. Data Reduction 145 IV.D. A Ground State Perturbation 151 CHAPTER V. THE 5 n / 5 * COMPLEX NEAR 10000 cm"1 157 V. A. Introduction 157 V.B. Perturbations 161 V.B.I. Evidence for a 3 $ state 163 vi TABLE OF CONTENTS (conL) V.B.2. Perturbation in s * , , v = 0 at J=44 187 V.B.3. Complex perturbation in 5 $ 2 , v = 0 189 V.B.4. Complex perturbation in 5 $ 3 , v = 0 195 V.B.5. Complex perurbations in 5n 0, v = 0 and nearby substates 210 V.B.5.a. Avoided crossing between 5 § > f t , v = l and 5 I1 0 , v = 0: a test of the ground state spin-orbit intervals 218 V.B.6. Perturbations in 5n 0 , v = l 219 V.B.7. Perturbation in 5 * 3 , v = l at J - 18.5 221 V.C. Lambda- doubling 225 V.C.1. Lambda-doubling in the 5II state 225 V.C.2. Lambda-doubling in the 5 $ state.. 230 V. D. Data Reduction 234 CHAPTER VI. CONCLUDING REMARKS 240 VI. A. Energy Levels of FeO: The Current Picture 240 VLB. For the Future 243 REFERENCES 246 APPENDIX I. Line Frequencies for the Orange System 257 APPENDIX II. Line Frequencies for the 5II - X 5 A System 267 APPENDIX III. Line Frequencies for the 5 * - X 5 A System 312 APPENDIX IV. Term Values for the X 5 A State 360 APPENDIX V. How the Weighting Was Done for the Ground State Fit 370 APPENDIX VI. The Data Used for the Ground State Least-squares Fit and the Residuals 372 APPENDIX VII. Term Values for the 5n State 403 APPENDIX VIII. Term Values for the 54> State 412 vii LIST OF TABLES I. Millimeter-wave pure-rotational transitions of FeO 34 II. Angular momenta present in a diatomic molecule possessing no nuclear spin.. 67 III. Total parity and e/f labels in the ft = 0 components of odd-multiplicity I states 81 IV. Effective rotational and spin hamiltonian for A states 105 V. Effective matrix elements for a 5II state in case (a) coupling 106 VI. Effective matrix elements for a 5 A state in case (a) coupling 107 VII. Effective matrix elements for a 54> state in case (a) coupling 108 VIII. Effective matrix elements for a 3 $ state in case (a) coupling 109 IX. Line strength factors in a case (a) basis 119 X. Operators responsible for perturbations and their selection rules 122 XI. Effective spectroscopic constants for the X 5Aj ground state of "FeO 147 XII. Least-squares fitting: 5 $ „ , v = 0 and perturbing state, with aossing point at J <* 57 166 XIII. Least-squares fitting: 5 $ 3 ,v=0 and perturbing state, with crossing point at J =* 47 170 XIV. Least-squares fitting: 54>, v = 0 — "unperturbed" regions 184 XV. Least-squares fitting: 5 $ „ , v = 0 and perturbing state, with aossing point at J 44 188 XVI. Least-squares fitting: 5<t>2> v = 0 and perturbing state, with crossing point at J * 35 190 XVII. Relative intensities in the perturbed 5 $ 2 substate 194 XVIII. Spin-orbit matrix elements for a 5 I I a / 7 Z^ interaction 200 XIX. Spin-orbit and L-uncoupling elements for a SU&/SL*^ interaction 203 XX. Least-squares fitting: 5n 0, v=l and perturbing fl = 0 state, with crossing point at J * 21 222 XXI. Least-squares fitting: 54>3, v=l and perturbing state, with crossing point at J * 18.5 223 XXII. Effective rotational constants for the 5II state 235 XXIII. Effective rotational constants for the 5 $ state 237 viii LIST OF FIGURES 1. Interaction between the fl' - 2 and fl' =3 substates of the 585- and 592-nm bands . 20 2. Rotationally resolved fluorescence of FeO excited at 17 080.131 cm"1 23 3. Heads of the fl' =1 band at 561.4 nm and of the fl' =0 bands near 562.4 nm 42 4. Vector diagram for Hund's coupling case (a) 72 5. Schematic representation of two perturbing states 124 6. Homogeneous and heterogeneous perturbations 127 7. Energy level diagram for TiO, showing the orbital occupancy for the ground state 131 8. Fourier transform ir spectrum of FeO in the region 9365 - 9420 cm;1 showing the heads of the fl = 5-4 and 4-3 subbands of the 5 * - 5A (0.1) band 135 9. J-number determination for the 5 $ 5 - X s A „ (0,0) subband 137 10. Overview of the infrared spectrum of FeO 140 11. A portion of the 5 * - X 5A (0,1) band near 9090 cm"1 showing A-doubling in both the fl = 0 and fl = 1 levels of the ground electronic state 142 12. Two portions of the 5 $ 3 - X 5 A 2 (0,2) subband near 8450 cm"1 showing a ground state perturbation at J" = 15 152 13. Part of the 5II - X 5A (1,0) band near 11 000 cm"1 158 14. The 5n/54> complex near 10 000 cm"1 160 15. The energies of the 5II and 54> states plotted against J(J-tT) 162 16. Extrapolated high vibrational levels of the X 5 A state superimposed on the energy level diagram of Fig. 15 178 17. The complex perturbation in 5II 3 , v=0 196 18. Possible states responsible for the perturbation in 5n 3 ,v=0 198 19. The 5II 0 , v=0, 5n 3 , v = l , and 5 #, ,v=l substates 211 20. Detail of one perturbation in the upper parity component of 5II 0 , v=0 213 21. Probable explanation for the perturbations in 5II 0 , v=0 and 5 $ f l , v=l (schematic) 215 ix LIST OF FIGURES (cont) 22. Pieces of a I state or states superimposed on the energy level diagram of Fig. 19 (schematic; 217 23. Perturbations in 5 n 0 > v=l 220 24. A-doubling in 5n_, 226 25. A-doubling in 5 n 0 227 26. A-doubling in 5Yl- .228 27. A-doubling in 5 * , 232 28. Large scale plot of the energy levels of the 5 $ , , v = 2 level, showing irregularities in both parity components 233 29. The low-lying electronic states of FeO 241 X ACKNOWLEDGEMENTS First and foremost, I sincerely wish to thank Dr. Anthony J. Merer for suggesting this project, for many helpful comments and suggestions throughout my time here, for helping with the task of assigning the lines, for answering my many questions, for patience, and — by being such an expert in his field — for serving as a role model; . . . I hope that someday I may know so much! I wish to thank Dr. Walter J. Balfour for first introducing me to the subject of high resolution electronic molecular spectroscopy. I shall always be indebted to Dr. Allan S-C. Cheung for all his assistance in integrating me into the FeO experiments that were already going on when I arrived here and for all his subsequent help. My thanks also go to Dr. Marjatta Lyyra for all her laser-induced-fluorescence work and to Mr. Nelson Lee for his assistance in assigning lines in the infrared system. I also thank Mr. Rob Hubbard of the Kitt Peak National Observatory staff for his very capable technical help in recording the infrared system. And, more recently, I thank Dr. Ulf Sassenberg for many useful and interesting discussions. I wish to thank my parents for endless support and encouragement and, well, for just being my parents! Financial assistance from the Natural Sciences and Engineering Research Council of Canada in the form of a Postgraduate Scholarship during much of my stay here is gratefully acknowledged. xi AUTHOR'S PREFACE The work reported in this thesis was part of an ongoing study into the electronic spectrum of FeO being conducted in this laboratory and begun prior to the author's arrival. Thus, while the present thesis deals with the electronic infrared system of FeO, this cannot be considered in isolation from the work on the orange system, which formed part of the subject matter of the thesis by Cheung (Ph.D., The University of British Columbia, 1981). Since the two projects were going on simultaneously for awhile, and in particular since the orange and infrared systems involve a common lower state, the two projects were interactive in nature, with information from one contributing to the analysis of the other and vice versa. In light of this, although described in the section "Other Work Done in This Laboratory" (Section I.B), the author was partly involved with the work in which the spin-orbit intervals of the ground state were determined. On the other hand, tables of rotational lines of the orange system have been included in this thesis (Appendix I), although the author was only slightly involved in their determination. Justifications for this inclusion are (1) for completeness (since most of the line positions were not included in the thesis of Cheung) and (2) the fact that orange-system data were included in the ground state least-squares fit that forms part of this thesis. CHAPTER I INTRODUCTION Iron monoxide, FeO. . . at first glance, a simple molecule: diatomic, consisting of two commonly occurring elements, relatively easy to obtain by several methods. Yet FeO possesses an electronic spectrum so complex it has repeatedly defied analysis, including attempts by a number of notable spectroscopists. The present chapter summarizes these early investigations into the spectrum of FeO and culminates with a description of the recent cracking of the ground state problem in this lab by Cheung and Merer. The following chapters present a continuation of the investigation into the electronic spectrum of FeO, namely the analysis of the infrared system. 1 2 LA. Historical Survey Over the years, the spectrum of FeO has been observed in numerous laboratories, not always by design. Since the spectrum can be obtained simply by striking an iron arc in air, it had probably been observed on a number of occasions before it was actually identified. Early references to the spectrum of FeO have been summarized by Kayser (1) and by Eder and Valenta (2). The most detailed of these early studies was that reported by Domek (3) in 1910. The emission spectrum obtained from an iron arc was found to exhibit a weak band in the green plus two bands in the yellow and red beginning respectively at 578.974 and 618.066 nm and both degrading towards the red. The latter two "bands" actually consisted of a large number of narrower bands and close-lying lines, the positions of which were tabulated. Comparison with the grating spectra obtained in this laboratory as part of the present study shows that individual rotational lines were resolved but that only the more intense lines were reported. This was considerably higher resolution work than that of many of the later studies (to be described shortly). No vibrational or rotational analysis was done, however, since this was prior to the development of modem quantum mechanical theory and its application to spectroscopy. Howell and Rochester (4) made prism spectrographic observations of FeO in an arc flame during the early 1930s but gave no details of the spectrum or its analysis. Richardson (5) also photographed the spectrum of an iron-arc flame and, in 1934, published the band-head positions of six red-degraded bands, together with the wavelengths of the fourteen strongest lines in the strongest band (head: 578.965 nm), but was unable to provide an analysis. Comparison with the present grating spectra shows that two of the fourteen lines (those with X^j. = 582.2182 nm and 582.8916 nm) are absent or very weak in our spectra. 3 Based upon absorption work with a prism spectrograph, Trivedi (6), in 1935, reported a continuous region of absorption, with a long wavelength limit of 250 nm. There is some doubt as to whether FeO was actually present, since the visible bands were not seen, though this may have been due to the experimental conditions, as explained by Trivedi. The emission spectrum of FeO is also obtained, usually as an impurity, when CO is burned in air or oxygen if the carbon monoxide has been stored in steel cylinders. This is due to the formation of a small amount of iron pentacarbonyl. Not surprisingly, the spectrum is also obtained if iron pentacarbonyl is introduced deliberately into a flame, and this is one of the favorite sources that has been used over the years. This relationship between FeO, CO, and Fe(CO)5 has been mentioned by Gaydon in several publications (7,8,9). In fact, work on the spectrum of FeO formed part of Gaydon's PhD thesis in 1937 (10). He observed, at low resolution, a number of strong emission bands in the orange and near-infrared regions. This work was apparently not published, however (other than as the above-mentioned thesis), until the appearance of the (now well-known) book by Pearse and Gaydon, The Identification of Molecular Spectra (11). Initially unaware of the work done by Gaydon, during the early 1940s Rosen also studied the spectrum of FeO, again at low resolution, this time by means of exploding wires as well as with an arc (12,13). During 1940 he studied the yellow region of the spectrum in collaboration with Delsemme (14) and later, with Malet (1942) (15), extended the investigation to the blue and photographic infrared. These investigators found that band systems observed in the orange and blue regions of the spectrum appeared to have the same lower state vibrational frequency (w0 = 875 cm;1 u)0\o = 5 cm"1)1 and thus probably possessed a common lower level, which they believed likely to be the ground state. They also observed bands in the near-infrared 'Standard spectroscopic notation is used throughout this thesis (16). 4 region, most of which they classified into a system having a lower state vibrational frequency of 955 cm;1 which does not agree with more recent work (this thesis), though they did recognize that some of the infrared bands belonged to a series having the 875 cm"1 frequency. They classified the orange bands into two systems, which they called A and B, with the A system being further subdivided into A- and A 2 . The blue system was labelled C and that part of the infrared system having a lower state frequency of 955 cm;1 D. These designations, especially A and B, are still . frequently referred to today, though another subdivision of the A system, introduced by Pearse and Gay don (11), viz. Aj and A^, is sometimes used. In 1952, Bass and Benedict (17), although not specifically looking for FeO, did in fact record its near-infrared spectrum while doing studies on the combustion of carbon monoxide containing a small quantity of iron pentacarbonyl. The spectrum was observed as an intense band system extending from 7 000 to 15 000 cm:1 Although the resolution was very low, Bass and Benedict were able to analyze the vibrational structure, and their analysis was basically correct Their assignments of the bands as (2,0), (1,0), (0,0), (0,1), and (0,2) were correct for what is shown in the present work to be the 5II - X 5A system. They did not, however, realize that there were two overlapping systems, so some of their sequence band labels were not correct In a compilation of metallic oxide spectra by Gatterer, Junkes, and Salpeter (18), Rosen revised much of the existing data on FeO but was still not satisfied with the proposed analysis of the band systems. He suggested that further work needed to be done. During the course of flash photolysis experiments involving explosions of amyl nitrite, n- heptane, and oxygen mixtures containing ferrocene or iron pentacarbonyl additives, Callear and Norrish, in 1959-60 (19) observed not only the orange system (in emission and absorption) and the infrared system (in emission) but also a diffuse ultraviolet system (in absorption) between 241 and 243 nm. They reported the latter 5 system as a new feature of FeO, though we point out that there is a slight possibility that this was the continuous region mentioned previously as having been reported by Trivedi (6). This diffuse absorption band was seen again (at 241 nm) about 1966 by Callear and Oldman (20), who speculated that this may be a Rydberg type of transition and that it probably involves a low-lying state of FeO, but not the ground state. No further work has been done on this feature, and its true nature remains undetermined. In 1962, Bass, Kuebler, and Nelson (21) recorded the FeO spectrum in absorption in selected parts of the visible and near UV regions using a flash heating technique. They observed some of the orange system bands and interpreted their presence in the absorption spectrum as indicating that their lower state was the ground state of the molecule, in agreement with Rosen's earlier interpretation. They did not, however, detect any absorption due to the blue system. This implied a different lower state for this system, a different conclusion to that arrived at by Rosen. They also obtained the diffuse absorption feature of Callear and Norrish at approximately 242 nm. In 1966, Dhumwad and Narasimham (22) recorded the orange system at higher resolution than in any of the previous investigations, using an arc at low pressure. Their analysis showed that the system possessed simple R and P branches with no A-doubling. An attempt was made to analyze several of the bands in terms of a 1 Z - transition. However, this analysis was refuted by Barrow and Senior (23) three years later on the following grounds: (a) The ground states of Fe and O are respectively 5 D and 3 P. These cannot give singlet molecular states. (b) Comparison with other first-row transition metal oxides shows that the derived ground state internuclear distance is much too large: 6 VALUES OF r e (nm) (from ref. (23)) ScO TiO VO CrO MnO FeO 0.1668 0.1620 0.1589 0.1627 70.197? Barrow and Senior recorded the orange system using similar experimental conditions to those of Dhumwad and Narasimham. Their new analysis yielded the much more reasonable value of r e = 0.1626 nm. They obtained the ground state constants (in cm"1): C J £ = 880.53, o)exe = 4.63, B £ = 0.512 71, a e = 0.003 76. Their interpretation was that the lines analyzed probably belonged to the fl' = 0-S2" = 0 part of a multiplet L - Z transition, with the states most likely being 5 Z or 7 I . West and Broida (24), in 1974, studied the visible and infrared systems at relatively low resolution in chemiluminescent flames. They observed some weak new bands in the orange region and a "new"2 group of weak bands between 500 and 540 nm which they called the "Green System." Some of the new orange bands were found to be separated by the ground state vibrational spacing, o>g « 880 cm;1 and the green system had upper and lower state vibrational constants the same as those given by Malet and Rosen for the blue system, although it did not appear to be connected to this system nor to the orange system when looking at the spectrum. They also did some laser-induced fluorescence and radiative lifetime studies on the orange system and obtained values of u>t = 875.8 cm"1 and wexe = 4.6 cm"1 for the ground state. This u e value was in rather poor agreement with the value of Barrow and Senior (880.53 cm"1), a point which we will return to shortly. Although they determined some additional spectroscopic constants, they were unable to settle the question of the natures of the states involved — in particular, was the ground state 5 1, 7 1 , or what? Even 3II had once been proposed (by Lagerqvist and Huldt (25)). 'Possibly this was the weak green band mentioned previously as having been reported much earlier by Domek (3). 7 Shortly after this, Montano, Barrett, and Shanfield (26), while performing low-temperature Mossbauer studies on iron monomers and dimers in inert gas matrices, happened to obtain a six-line spectrum which they attributed to FeO. They found a large positive internal magnetic field at the Fe nucleus ((3.52±0.06)xl07 A-m"1 in an argon matrix). This was indicative of a high spin multiplicity for the ground state (27), but unfortunately did not remove the ambiguity as to what this state was since most of the candidates for ground state were of high multiplicity. Engelking and Lineberger (27) studied FeO by means of photoelectron spectrometry of FeO" in 1976, with mixed success. They obtained a ground state vibrational frequency of 970±60 cm;1 which was in poor agreement with the previously accepted value of u>t = 875-880 cm:1 They also observed another state lying 3990±100 cm"1 higher. Their interpretation was as follows: The state with a vibrational frequency of 970 cm"1 was the true ground state (which they therefore labelled X), and the previously accepted ground state with a vibrational frequency of -880 cm"1 was the excited state at 3990 cm"1 (which they labelled X'). Calling upon the theoretical predictions by Bagus and Preston (28) and Walch and Goddard (29), they assigned the X state as a 5 A state and the X' state as a 5 Z + state. Influenced by Engelking and Lineberger, DeVore and Gallaher (30), in 1978, attempted to obtain the infrared vibrational spectrum at ~970 cm:1 They claimed to have found this band at 943 cm;1 but this author finds the line assignments in their published spectrum unconvincing. Weltner (31,32) attempted, unsuccessfully, to observe the ground state via ESR in a low-temperature matrix. As discussed by Weltner, this negative result was evidence that the ground state was an orbitally degenerate state, such as Engelking and Lineberger's 5 A state. 8 Kay, Bartelt, and Byler (33) conducted flash heating-kinetic spectroscopy studies of gaseous FeO and obtained high-resolution electronic absorption spectra of the orange B system. Their measured rotational constants and band separations were in good agreement with the emission values of Barrow and Senior. The ease with which these bands were obtained in absorption in this and the previous absorption studies provided some supporting evidence for the lower state of these transitions being the ground state, as had been assumed prior to the work of Engelking and Lineberger. So we see that as recently as 1978, despite the efforts of many people over the course of many years, not even the ground state, let alone any of the excited states, had yet been established. The questions still to be answered for the ground state were: (i) Which of the observed states was the ground state? and (ii) What was the identity of this state? The choice of candidates had essentially been narrowed down to two: the state seen as the lower state in a number of studies having a vibrational frequency of ~880 cm"1 and labelled as either 5 Z or 7L by Barrow and Senior, versus the state with a vibrational frequency of ~970 cm"1 seen by Engelking and Lineberger and labelled by them as 5A. In an effort to clarify the situation, Green, Reedy, and Kay (34), in 1978, recorded the infrared vibrational spectrum of FeO in an Ar matrix at 14 K. They obtained values of cj g = 880 cm"1 and ^ e x e = 3.5 cm:1 That the species being studied was indeed FeO was established beyond a doubt by the identification of peaks due not only to "Fe"0 but also to the other three isotopomers 5 4Fe uO, 5*Fe l ,0, and "Fe^O, with relative absorbances in agreement with the natural abundances of the isotopes of iron and the known oxygen isotopic composition of the reaction mixture. Any shift in vibrational frequency due to the matrix would not be expected to exceed about 1% of the observed value (34) and would probably be much less than this. 9 No state other than the ground state has ever been shown to be populated in matrix-isolated molecules near absolute zero (34). Hence this study definitively established the ground state of FeO as having a vibrational frequency of ~880 cm:1 The conclusion was that the ground state was the lower state of the orange A and B systems and possibly of the blue, green, and infrared systems also. (FeO had actually been seen previously (1977) in a low-temperature matrix by Abramowitz and Acquista (35) during the course of studies on Fe02. They observed a frequency of 873 cm"1 in agreement with the A G 1 / 2 value of Green, Reedy, and Kay. Of the two studies, that of the latter group is considered the definitive study on the subject due to the isotopic work performed.) Since there cannot be a state lower than the ground state, the photodetachment spectrum of Engelking and Lineberger must now be reinterpreted. The discrepancy between 970 and 880 cm"1 leads to some uncertainty as to whether it was actually the FeO"-*-FeO system that was seen at all. Assuming that it was, it may have been an excited electronic state of FeO that was produced by the laser photodetachment process, as suggested by Green, Reedy, and Kay (34). In this thesis we will assume that it was the ground state that was observed and attribute the discrepancy to experimental error or to interpretational error. One possible cause for this could be that the vibrational peaks (v' ^ , v" ^ ) v v FeO FeO" assigned by Engelking and Lineberger as "(1,0)" and "(2,0)" may actually have contributions • from transitions involving v" _ =1 and 2. We point out that the FeO" energy difference between the peaks labelled as (0,1) and "(2,0)" corresponds very closely to three FeO vibrational intervals (using the correct ground state interval, rather than the value of Engelking and Lineberger) and the separation between (0,2) and "(2,0)" corresponds to four times this interval. Therefore, the peak labelled as "(2,0)" may actually be a superposition of the transitions (3,1) and (4,2). Similarly the peak labelled as "(1,0)" may actually be a superposition of (2,1), (3,2), and (1,0). While 10 the populations of the FeOr v = 1 and 2 levels appear small from the intensities of the (0,1) and (0,2) bands, the Franck-Condon factors may favor the sequence bands, since the FeO" vibration frequency is smaller than that of FeO. Another anomaly visible in the spectrum published by Engelking and Lineberger, though not commented on by them, is that the peak labelled as "(2,0)" appears to have extra intensity relative to the other bands in the sequence "(0,0)", "(1,0)", "(2,0)", and (not labelled) "(3,0)." A Franck-Condon envelope for such a sequence cannot have any extra bumps in it Here again, the reinterpretation given in the preceding paragraph could account for this. Yet another explanation will be presented later in this thesis (Section IV.D) While the foregoing interpretation is admittedly pure speculation — it could be tested by redoing the experiment at higher resolution or at a different temperature — we have dwelt on this because, assuming that it was indeed the ground state of FeO that was seen, the work of Engelking and Lineberger is important in that it contains the only observation of the state at ~3990 cm:1 Although there is no experimental evidence as to the nature of this state, it is currently assumed to be a 5 I + state on theoretical grounds (see Section IV.D). We point out, however, that in view of the uncertainty surrounding the ground state vibrational frequency, the value of 3990 cm"1 could easily be in error by an amount larger than the quoted ±100 cm:1 We will have cause to return to the work of Engelking and Lineberger later in this thesis in connection with a new 7 I + state found during the course of the present work. Meanwhile, work on FeO continued in several laboratories around the world. McDonald (36), in 1979, photographed the orange systems in emission with a 3.4 meter Ebert spectrograph. He confirmed the analysis by Barrow and Senior of the orange Ay bands and rotationally analyzed several other bands all having the same lower state as the Ay bands but new upper state levels. 11 At about this same time, Harris and Barrow (37) also recorded the orange system on a 3.4 meter Ebert spectrograph (Jarrell-Ash). They extended the analysis of Barrow and Senior and determined that the bands involved in that earlier study belonged to a common, but highly perturbed, upper state level. They additionally analyzed' bands belonging to three other, also highly perturbed, upper state levels. All the transitions analyzed were parallel ones, and no Q branches were found. The bands were shown to belong to a lower state vibrational progression having a frequency of 880 cm;1 thus leading to the conclusion, in view of the work of Green, Reedy, and Kay, that the orange system lower state is the ground state. They obtained the following values for the ground state (in cm"1): u)Q = 880.61, coexe = 4.643, B e = 0.512 72, a £ = 0.003 760. Harris and Barrow were able to account for the poor agreement of the West and Broida vibration frequency value (we = 875.8 cm"1). The latter investigators had neglected to allow for the effect of the rotational J-dependence on the vibrational intervals. (In 1980, Trkula (38) also started work on FeO, as did Lindgren and Sassenberg (39) in 1981 or 2. Trkula obtained low-resolution spectra that showed the orange and blue systems, but did not show the gTeen system, and he started some laser-induced fluorescence work on the orange system. This study by Trkula was terminated, however, upon his learning of the state of advancement of the corresponding studies being conducted in this lab. Lindgren and Sassenberg obtained spectra of the 550 - 620-nm region, but these were found to be similar to those already published by Harris and Barrow and by this lab, so no further analysis was done.) Although the matrix-isolation study of Green, Reedy, and Kay had established which of the observed states was the ground state of FeO, neither this nor the more recent gas-phase studies were able to characterize the state involved (i.e. identify its 12 symmetry or multiplicity). For example, the orange-system bands observed by Harris and Barrow were all parallel-polarized, with simple R- and P-branch structure and no Q lines detected, but such structure would be consistent with any of the most commonly proposed transitions for these bands: 5 1 - 5 L , 5A - 5A, 7L - 7L. Several theoreticians have conducted ab initio investigations into the electronic structure of FeO. One study by Bagus and Preston (28) predicted three low-lying states: 5 2, 5 n , and 5A. The study concluded that the ground state is not *L* thus leaving 5 n and 5 A as possibilities. Walch and Goddard (29) obtained similar results. Based on the work of these two groups, Engelking and Lineberger (27) gave a state ordering 5A, SL* 5 n , ... . Michels (40), on the other hand, predicted that the ground state is 5 Z + . His assignments for the upper states of the A and B orange systems were 5 Z + and 5 n , respectively, and for the infrared system, 5 I I . The theoreticians, being unable to come to agreement amongst themselves, were unable to help in characterizing the ground state. The lack of a low-temperature, matrix-isolated ESR spectrum would seem to eliminate Z states, but more positive evidence was really needed to establish the identity. Such was the state of affairs when work was begun in this laboratory. This work will be the subject of the next section and the following chapters; but first, brief mention will be made of a number of other FeO-related studies. Various estimates of the dissociation energy of FeO have been made. Gaydon, in the first edition of his book. Dissociation Energies and Spectra of Diatomic Molecules (1947) (41a), gave a linear Birge-Sponer (42) extrapolation value of 480 kJ«mol"1 (5.0 eV) for the ground state: FeO ( g ) ==* Fe ( g ) . + 0 ( g ) AH0° = D0° = 480 kJ-mol"1 ...(I.A.1) with the recommendation that this value be reduced by 20% to 390 kJ'mol"1 (4±1 eV) to take into account the fact that deviations from linearity frequently occur. 13 By the third edition of this book (1968) (41(b)), the recommended value had been revised to 410 kJ-mol"1 (4.3±0.5 eV) on the basis of a number of other studies that had been done by this time. More recently (1971), Balducci, de Maria, Guido, and Piacente (43) obtained a value of D0° = 406±13 kJ-mol"1 during mass spectrometric measurements of the vaporization of Apollo 12 lunar samples. The reaction studied was FeO ( g ) ^ Fe ( g ) + I 0 j ( g ) ...(IA2) and the AH0° obtained by second- and third-law calculations together with the accurately known dissociation energy of 02 yielded the above value. Jensen and Jones (1973) (44) obtained the value D0° = 402±20 kJ«mor 1 from spectrophotometric studies of iron-containing flames by second-law analysis only. The reaction studied was Fe(g) + 0 H(g) ^ F e ° ( g ) + H(g) "•(I-A-3> By means of high temperature mass spectrometric measurements of the components of an effusion beam, Hildenbrand (1975) (45) determined, by third-law analysis, the value D0° = 405±13 kJ«mor 1 . This time, the reaction looked at was Fe(g) + ° ' ( g > - F e 0(g) + °(g) -< I A - 4 ) While these various recent values for D0°(FeO) may appear to be in fairly good agreement with each other, as pointed out by Hildenbrand (45), caution must be exercised when interpreting the results obtained from equilibrium measurements. The values depend significantly on the particular molecular and spectroscopic constants used in the analysis. For example, Hildenbrand recalculated the results of Balducci et al. (see above) using the constants that he had employed in his analysis and obtained D 0° = 427 kJ'mol"1, considerably different from the value of 406 kJ'mol"1 obtained 14 by Balducci et al. The spectroscopic constants of FeO used by Hildenbrand were those of Barrow and Senior (23). Thus, although thermodynamic studies such as these hold the potential of yielding a much more accurate value for the dissociation energy than a simple Birge-Sponer type of extrapolation, they still require accurate spectroscopic values. For completeness, we briefly mention several other specialized studies on FeO not directly related to the present work. Intensity measurements of the ground state vibrational band were made during the early 1970s by von Rosenberg and Wray (46), who additionally studied the kinetics of reaction (I.A.4), as did also Fontijn and Kurzius (47) around the same time. Also at about this time, absorption coefficients for the infrared vibration-rotation spectrum were calculated by Fissan and Sulzmann (48). 15 LB. Other Work Done in This Laboratory (49,50,51,52) I.B.1. Characterization of the ground state Work was begun in this laboratory on the electronic spectrum of FeO in September, 1979. High resolution spectra of the region 550 - 630 nm were photographed, following which a laser-induced fluorescence study of a number of bands of the orange system was performed. This provided the breakthrough which finally allowed for the rotational analysis of the ground state and identified the ground state as being a 5 A state. At the time work was started, the lower level of the orange system had been established as the ground state of the molecule by the matrix isolation study of Green, Reedy, and Kay (34), but the classification of this state was still undecided upon. The possible candidates, supported by various theoretical studies, were SL* 7L* or 5A. The unravelling of the problem proceeded as follows. The band first studied was the 582-nm band, since it occurred in a comparatively uncrowded region of the spectrum. This band, which was one of the bands also studied by Harris and Barrow (37), exhibits a number of small rotational perturbations in which the lines appear doubled, with two lines of equal intensity. Such a pair of lines could be produced either by an exact coincidence of perturbed and perturbing levels or by A-doubling. The existence of several exact coincidences would seem rather unlikely, so A-doubling was the more probable explanation. The A-doublet splitting had been too small to resolve in the spectra of Dhumwad and Narasimham (22) or those of Harris and Barrow or (at other J values) in the current grating spectra, but a sub-Doppler intermodulated fluorescence spectrum of the unperturbed line R(15) at a resolution of about 75 MHz showed that A-doubling was indeed present, there being two equally intense components having a separation of about 120 MHz (51). Lower J lines, however, such as R(10), did not show any resolvable A-doubling. Thus A-doubling 16 was present but was small and J-dependent Since A-doubling cannot occur in L states, which are non-degenerate, the existence of A-doubling indicated that the states involved were not £ states. Harris and Barrow showed that the 558-nm band, which was the (0,0) band of Delsemme and Rosen's (14) A, system, possessed a common lower level with the 582-nm band (which, incidentally belongs to Delsemme and Rosen's B system). The 558-nm band had been studied previously by Barrow and Senior (23), who had postulated that the states involved had £ 2 = 0 . Since, to first approximation, A-doublet splitting is given by [J(J+l)]p for J2 = 0 a constant, J-independent A-doublet splitting would be expected, with any J-dependence being small relative to this. This was not what was observed, a fact which would imply that either the constant A-doubling term in the lower state was being cancelled by an equal constant term in the upper state (not very likely) or that 0 0.3 Q lines are forbidden for S2' =0 = 0 transitions in Hund's coupling cases (a) and (c) and for Z - L transitions in case (b) but are allowed for other transitions (16(a)). Although Barrow and co-workers (23,37) had not found any Q lines, armed with the above new evidence that 0 may not have the value 0 and since the existence of A-doubling showed that the states were not £ states, a search was made for Q lines in the grating spectra. A branch was indeed found and its numbering established by plotting the line positions against n(n+l), where n was an arbitrary ninning number, and choosing the best straight line. The analysis of the Q-branch was confirmed with rotationally resolved laser-induced fluorescence experiments. The lowest-J Q line observed in the grating spectrum was Q(4). This was confirmed as being the lowest-J line by the laser-induced fluorescence work: while excitation of higher-J Q lines gave R-, Q-, and P-branch emission, excitation of the 3Ref. (49) seems to imply that the very existence of A-doubling is the reason why 0 * 0. This is not correct. The real reason is as presented here. 17 Q(4) line yielded only Q(4) and P(5) emission (no R(3) emission). This meant that « fl- = Q" = 4. Several of the other bands studied by Harris and Barrow were also studied and all were shown to possess Q branches and to have J = 4 first Q lines. Some of these bands form a lower-state vibrational progression. As shown by Harris and Barrow and confirmed here, the vibrational constants are in close agreement with the matrix-isolated values of Green et al. (34): Gas: o)e = 880.61 cm"1 wex£ = 4.64 cm"1 Matrix: a>e = 880.02 cm -1 <J ex £ = 3.47 cm:1 Therefore the ground state of FeO contains an 0 = 4 spin-orbit component Of the three possible candidates for the ground state ( 5Z* 7 1* or 5 A), only 5 A is consistent with the presence of A-doubling and only 5 A possesses an ft = 4 component The 0 values for a 5A state run from 0 to 4, whereas the highest ft values for 7L* and 5 Z + are 3 and 2, respectively. Thus the ground state of FeO was finally established as being a 5Aj state. All subsequent work performed in this laboratory has yielded no information inconsistent with this assignment and in fact has firmly established it as being correct LB.2. The orange system All of the bands (actually subbands) studied in the work described in the previous section involved ft' = ft" = 4 levels as did, it is now known, all subbands which had ever been rotationally analyzed by previous investigators (22,23,24,37). Subbands involving other spin components must also be present Following submission of the foregoing work for publication (49), continuing laser-induced fluorescence experiments on the orange system did eventually find, not one, but several subbands involving each of the ft values 0, 1, 2, and 3. 18 The prominence of the fl = 4 subbands results from the fact that at this high an fl value the A-doubling is usually not resolved, at least not until high J values, so that the lines are effectively twice as strong as those of other subbands where the A-doubling causes resolved splittings often even at low J values. In the case of absorption experiments, the fl = 4 subbands are additionally enhanced, especially for v" = 0, due to the Maxwell-Boltzmann distribution that governs the populations of the levels. Since the effective temperature of the molecules involved in the excitation spectra obtained in this lab was roughly 350 K, the 0" = 4 substate — being the lowest in energy of the five inverted 5 A substates, which have a spin-orbit separation of about 190 cm"1 (see next section) — is heavily favored over the other substates. A total of 34 subbands (including two "FeO isotopic subbands), lying in the wavelength regions 558-564 and 579-623 nm were rotationally analyzed in this laboratory. These involve over 20 fl' substates lying between 16 350 and 18 550 cm:1 Every one of these substates is extensively perturbed, and the rotational analysis of most would not have been possible without the use of rotationally resolved laser-induced fluorescence. The measured line positions of the orange system are presented in Appendix I. A detailed description of the individual subbands and corresponding upper state energy levels has been given in ref. (51) and will not be repeated here. Instead we describe only the overall pattern which emerged from this study. The upper state of the orange system is a 5Aj state, together with a large number of extra fl substates. The 5A state has a B value of about 0.475 cm;1 corresponding to a bond length of 0.169 nm. Some of the "extra" substates may form another 5Aj state, but this is much less certain. When the upper energy levels are plotted against J(J+1), many of the substates show marked curvatures to their plots over and above that caused by centrifugal distortioa Such curvatures indicate interaction with fairly distant perturbing 19 states via large interaction matrix elements, and the matrix elements are rotation-dependent and/or the zero-order energy separations are rotation-dependent In some cases, the curvature is probably due to spin-uncoupling, with the "fairly distant perturbing states" just being other 0 substates of the same vibrational level of the same case (a) state, acting via the -2BJ-S operator,4 although it is not always clear which of the observed (or unobserved) levels are the other 0 components. Other instances of curvature are due to avoided crossings, with the interactions being either homogeneous or heterogeneous in nature. An example of the latter effect is shown in Fig. 1, in which the rotation-dependent nature of the interaction between the £2 = 2 and Jl = 3 upper levels of the 592- and 585-nm bands is clearly seen. The large difference between the B values of these substates (as determined at low J before the "onset" of the interaction) indicates that the substates belong to two different electronic states. In this case the interaction is via the -2BJ«L operator. This type of perturbation has proved to be extremely useful in that the resultant mixing of the upper state levels of different 8' values has allowed the determination of the spin-orbit coupling of the ground state, as will be discussed in the next section. In addition to the perturbations causing large overall curvatures, numerous small perturbations are seen throughout the upper states of the orange system in the form of local avoided crossings. From arguments based on the anomalous vibrational intensity patterns observed, the conclusion is reached in ref. (51) that many of the observed levels probably would not, in isolation, possess transition moments to the ground state. They obtain their oscillator strengths by various interaction mechanisms. In view of the large energy separations between these levels and the levels from which they borrow intensity, the interaction matrix elements in a case (a) basis must be very large (hundreds of cm"1 'Vectors are denoted by a symbol in this thesis. 500 1000 1500 J(J + 1) Fig. 1. Interaction between the £2' = 2 and S21 =3 substates of the 585- and 592-nm bands. Upper state energy levels are plotted as a function of J(J+1) after having been scaled by subtraction of the quantity 0.428J(J+1) - 10"6 J2(J+1)2. The separation between the two components has been reduced by ~12 cm:1 A number of small local avoided crossings and splitting apart of the A-doublet components are apparent in addition to the large interaction between the two substates. 21 units). This suggests that electrostatic interactions are present, in addition to the more common spin-orbit interactions. Electrostatic interactions have similarly been proposed to explain some of the effects in the MnO spectrum (53). Yet another feature is the presence of numerous branch fragments that can be followed for only a few J values and that have non-Boltzmann intensity distributions, yet which bear no simple relation to a pattern of avoided crossings. These also appear to result from levels which have no zero-order transition moments of their own, and which obtain small transition moments by spin-orbit or some other form of homogeneous mixing with comparatively distant states having the same fl' value. In this case, the oscillator strengths picked up are small. The observed intensities vary erratically with J because of interactions with other such states that have also picked up small transition moments by similar means, but which have different B' values. Where the transition moments happen to reinforce for several J values, a "branch fragment" appears. Such effects commonly occur in perturbed polyatomic spectra (e.g. CS2 (54,55)), but have apparently not been encountered before in diatomic molecules. In short, the upper states of the orange system are extremely complex. The density and magnitude of the rotational perturbations are quite exceptional for a diatomic molecule and are more comparable with the notorious visible system of N0 2 (56,57,58). Although the FeO orange system upper states basically belong to Hund's coupling case (a), they show strong tendencies toward Hund's case (c) coupling. LB.3. Ground state spin-orbit intervals As mentioned in the last section and shown in Fig. 1, the upper ievels of the 592- and 585-nm bands, with fl' = 2 and 3 respectively, perturb each other. The spin-orbit components belong to different electronic states, and the rotational-electronic matrix element is 22 <v A ft = 3 J I-2BJ - L l v1 A-lft=2J> = - < v A | B L + | v1 A- l> [J(J+1) - 6] 1 / 2 ...(I.A.1) where we find |-<vA |BL + | v' A-l>| = 0.374 cm:1 The result of this interaction is that the substates mix with each other. In the absence of such mixing, transitions to the ground state would be purely parallel transitions (Aft = 0) and there would be no way of accurately determining the separations between the ground substate levels. As a result of the mixing, however, Aft = ±1 transitions can occur. That is, because the upper state wavefunctions are really mixtures of ft' = 2 and ft1 = 3 wavefunctions, transitions to both ft" = 2 and ft" = 3 levels of the ground state can occur from a single upper level. The difference gives directly the ground state spin-orbit separation between the ft" = 2 and ft" = 3 substates. Figure 2 shows the resolved laser-induced fluorescence obtained upon excitation of the R(20) line of the 585-nm band. Four fluorescence lines result, arranged in two pairs separated by about 192 cm:1 One pair, the excited R(20) line together with P(22) has a separation (ground state rotational combination difference, AjF") showing that this pair has ft" = 3. The other two, another R(20), P(22) pair, likewise has ft" = 2. Laser excitation of either member of the latter pair gives the same resolved fluorescence pattern. The intensity of the P3(22) line, where the subscript denotes ft" = 3, is greater than that of the P2(22) line (Fig. 2). (The R(20) line intensities cannot be directly compared because of laser light scattering at the excitation frequency.) This indicates that the upper state wavefunction has more 0' = 3 character than ft' = 2 character. Beyond J = 30, the states become essentially completely mixed, so that four equally intense lines are obtained in the resolved fluorescence spectra. The weak 587-nm band, which has an upper level that is primarily ft' = 3, also gives ft" = 2 spin-orbit satellites. The interacting ft' = 2 level is again probably 23 Spin-orbit satellites from the 5845 A band R120) laser P(22) R(20) P(22) n ' = 3 a'-2 17080.131 cm-' 17035.988 16887.633 16843.230 Fig. 2. Rotationally resolved fluorescence of FeO excited at 17 080.131 cm:1 Because of perturbations in the excited state the fl' = 3 level emits to both the fl" = 2 and 3 components of the ground state, permitting determination of the fl = 2-3 spin-orbit interval. 24 the upper level of the 592-nm band. Altogether, it has been possible to measure the spin-orbit separations between the J2 = 2 and J2 = 3 substates of the v = 0 level of the ground state for almost every J value between 11 and 34. Likewise, the 0" = 1-2 separation has been measured from spin-orbit satellites induced by an unseen ft' = 1 level in the $2' = 2 upper level of the 582-nm band. Unfortunately, this separation has only been measured for one J value (J = 21) and even for this one, the lines are weak and blended. The £2" = 1-2 interval is therefore not as well determined as the 2-3 interval. Weak spin-orbit satellites occur fairly commonly in the resolved fluorescence of many of the bands of the orange system. Since the upper state spin-orbit separations are very large and since the spin-orbit satellites are often seen at quite low J values, these must arise from rotational-electronic interactions between different electronic states, caused by the -2BJ • L operator, rather than from spin-uncoupling effects within a single electronic state. The frequency of occurrence of these satellites indicates that such interactions are widespread. Unfortunately, other than in those cases already mentioned, the extra lines resulting from these interactions are too weak to identify in the laser excitation spectra, so accurate measurements of the line positions have not been possible. Thus only the $2" = 2-3 and (less accurately) 1-2 intervals of the v = 0 level have been measured. This does, however, provide enough information to determine the major spin-orbit parameters, not only for the ground state v = 0 level, but for the other observed vibrational levels (v = 1-3) as well (see Section IV.C). 25 I.C. Reasons for studying FeO Prior to the work done in this lab, not even the symmetry classification of the ground state was known, let alone that of any of the excited states. Over and above the challenge of being the first group to establish such a seemingly fundamental property as the classification of the ground state, the information is itself very useful in diverse ways. Comparison of the properties of FeO with those of other transition metal oxides yields information regarding trends across the periodic table and down the Fe, Ru, Os column — such trends as the change in ground-state bond length and the relative energy ordering of the 4sa and 3d5 molecular orbitals (Section III.K) as one progresses across the first-row transition metal oxides, and the effects of shielding as the d orbitals become progressively filled. As described previously (Section LA), theoreticians have been unable from purely ab initio arguments to correctly predict the energy ordering of the states of FeO. Experimental determination of the correct energy ordering should help the theoreticians refine their basis sets. FeO possesses such esoteric states as s4>, 5 A, 5II, 5 1 , and 7Z. Such states are comparatively rare and little has so far been published about them — e.g. the rotational matrix elements for 5 A and 5 $ states are not available in the literature, and had to be worked out (Section III.H). Of the three types of states with which the present -thesis is mainly concerned, only examples of 5II states have ever been reported and rotationally analyzed and these only in one molecule (CrO — also, incidentally, studied in this lab (59,60)); to the author's knowledge, 5 A and 5 $ states have never been reported, although there is indirect evidence for the presence of a 5 A state in CrO (60). It has previously been indicated (Section LA) that accurate moments of inertia and vibration frequencies are required for the most reliable third-law treatments of the 26 high temperature equilibria from which dissociation energies are derived. Moreover, the ground state and any nearby states must be well-characterized in order to establish the electronic partition function. The importance of this is illustrated by TiO (61). The ground state could be 'I* 3Z"; or 3 A according to various qualitative arguments. The corresponding electronic degeneracy factors, if only one of these states contributes, would be 1, 3, or 6, respectively, with corresponding contributions to the free energy (at 2500 K) of 0, 23, or 38 kJ«mol"1, respectively. The difference between these values cannot be ignored when calculating an accurate value for the dissociation energy. In actual fact, the ground state is known to be 3 A (62), but a 1 A and a 1 I * state lie very close to the ground state and contributions from all three states must be considered. In FeO, if the ground state were a 5 Z + state as once thought, the electronic partition function, q e for this state in isolation would be 5, whereas for the actual 5 A ground state it is 10 (ignoring the spin-orbit intervals; 7.8 taking them into account, at 2000 K). A 5 1 state at 3990 cm"1 would have q e = 0.3 at 2000 K (a typical temperature in the mass spectrometric methods used for FeO dissociation energy determinations). We show in this thesis that there is also another low-lying state, a 7 Z state. For this, q e = 1.5 at 2000 K, assuming this state to have its v = 0 level at 2100 cm:1 This state therefore has a significant population relative to the ground state, and possibly even higher than indicated here if the level detected has v > 0. Thus, a thorough description of the low-lying states is of importance to the accurate determination of the dissociation energy. An application where accurate thermochemical data (such as dissociation energies and ionization potentials) are required is upper atmospheric research. Chemiluminescence and resonance fluorescence of FeO were studied by Best and co-workers (63) upon release of iron pentacarbonyl from a rocket into the atomic oxygen-rich region of the upper atmosphere. The formation of FeO from Fe(CO)5 plus atomic oxygen and its destruction to Fe and 0 2 were studied. 27 An example of a more direct application of spectroscopic information to "atmospheric research" was provided by the detection of FeO, amongst other species, in a meteor as the meteor passed through the atmosphere. Ceplecha (64) was able to attribute 157 lines to the FeO molecule." Individual rotational lines or groups of lines were identified by comparison with the tables published by Domek (3). In fact, FeO turned out to be the third brightest molecular species visible in the whole meteor spectrum and the second brightest, behind only N 2, in the continuum part of the emission. The meteor was of cometary origin, and thus the information derived is of use not only in studying the composition of meteors but of comets also. The published line positions resulting from work in this laboratory should be of use in future studies of this sort Proceeding farther from the earth's surface, FeO is potentially of great astrophysical importance because of the high cosmic abundances of both Fe and O. Optical absorption spectra have shown the presence of the metal oxides TiO, VO, CrO, YO, ZrO, and others in the atmospheres of M stars (65). Since FeO is formed at relatively high temperatures in the laboratory it could be abundant in the atmospheres of late-type stars and/or circumstellar shells. Although TiO and CaO have not been found in molecular clouds despite radio-wavelength searches (66), gas-phase Fe is more abundant than Ti and Ca in interstellar clouds, so FeO is a potential species to be found there. Again, such searches are dependent on laboratory-obtained line positions by which to identify the astrophysically-observed frequencies. Delsemme and Rosen (14) have pointed out the close agreement between the position of the R-head of the 579-nm band and one of the previously unidentified lines in the solar spectrum. This line is listed at 578.9767 nm in the Revision of Rowland's Preliminary Table of Solar Spectrum Wave-lengths (1928) (67) and also appears as a weak line at 578.976 nm in the more recent atlas of Delbouille, Roland, 28 and Neven (68). The first "line" of the yellow band measured by Domek (3) occurs at 578.974 nm, which is in very good agreement with the solar line. The higher resolution grating spectra obtained in this laboratory show that the R-head of the 579-nm band actually occurs at 578.963 nm. (Richardson (5) reported 578.965 nm, in good agreement with our value.) There are, however, several other intense lines to the long wavelength side of this so that Domek's "line" is really a blend. We therefore conclude that the R-head (actually several lines) of the 579-nm band is a possible assignment for this solar spectrum line. Richardson (5) compared the wavelengths of fourteen of the strongest lines in the 579-nm band (not in the region of the head) with the line positions present in the spectrum of a sunspot Although good matches were found for twelve of the fourteen lines, the other two were conspicuously absent in the sunspot spectra.5 The conclusion was that FeO was either absent from the sunspot or not present in sufficient abundance to be detected. Lest the reader be left with the impression that a study of the spectrum of FeO is of use only to theoreticians and star-gazers, the following examples illustrate "down-to-earth" applications. Garger (69) studied the spectrum of the Bessemer flame and found that lines for FeO and atomic Fe predominate in the spectrum during the second half of the fusion operation and that the spectrum could be divided into 5 periods, with each period having characteristic lines. This research could lead to insight into mechanisms occurring in this industrially important process. It is known that a number of transition metal compounds act as antiknock agents in spark ignition systems. Ferrocene and iron pentacarbonyl are two such compounds which have been studied in this regard by Callear and Norrish (70,19) and by Erhard (71,72). These investigators performed flash photolysis experiments in which 5 These are not the same two lines reported in Section I.A as being absent in our grating spectra. 29 ferrocene or iron pentacarbonyl were used as additives to various fuel mixtures in the presence of oxygen. Spectroscopic observations of the resultant explosions detected, amongst other species, the presence of FeO. In fact, the FeO visible and infrared emission spectum occurs with extraordinary brilliance under these conditions, and it was even speculated (70,19) that this FeO emission could be responsible for the antiknock behavior of these compounds: the emission could affect the temperature of the burning gas by radiative cooling and/or propagating centers of preignition could be deactivated by electronic excitation of FeO. While this mechanism has been cast into doubt by the later work of Erhard, both groups of workers used the spectrum of FeO extensively to monitor the course of the explosions and thus helped advance mankind's knowledge of how to prevent knocking in spark ignition engines. Finally, we point out that the infrared system, the analysis of which forms the basis of this thesis, has never been studied previously at high resolution. This alone is justification for the project! 30 LD. Some Very Recent Studies Based on the Results of the Present Work In Section LA the history of research on FeO was traced prior to the beginning of work in this laboratory. In Section I.C an attempt was made to justify the present study. We now continue the historical review and present some very recent studies which have been done or are being done or will be done. All these studies are based in part on information obtained in this laboratory. The work has already proved useful; . . . what better justification than this? While working out the matrix elements for a 5A state for the first time (Section III.H), the problem of how to handle the A-doubling in the X 5 A state came up. This led to the question of A-doubling in A states in general. In the past, A-doubling in A states has often been considered too small to. be of any significance, since it would be a fourth-order effect It has been resolved on previous occasions, but only in a very few molecules. Those that the author is aware of are: in the A 2 A state of PtH (73), in several 2A states of NiH (74), in the X 6 A 1 / 2 state of FeCl (75), in a "A - *A transition of FeH (76), and, reported very recently, in the A 2A state of CH (77). At the time of the early stages of the current project no detailed treatment of A-doubling in A states had yet been published. (Recently a partial treatment for the case of a 6A state has been performed by Delaval and Schamps (75), and more recently still Brazier and Brown (77) have dealt with the 2A case.) Early in the present work, A-doubling in the 0 = 0 component of the X 5A state was handled in a phenomenological manner by means of the terms t-jfo + OjJ(J+l)] in the diagonal positon of the energy matrix. However, upon finding A-doubling in the $2 = 1 substate as well, it was decided that such an approach was no longer adequate. Clearly the entire phenomenon of A-doubling in A states needed to be studied. This provided a theoretical project for Cheung and Merer, the results of which are to be published shortly (78); this paper will undoubtedly become the definitive work on the subject of A-doubling in A states. 31 As stated in previous sections, theoreticians have been unable to correctly predict the energy-ordering of the lower states of FeO. This statement still stands, although the situation is improving. Krauss and Stevens (79) have very recently performed multiconfiguration self-consistent-Field (MC-SCF) calculations for FeO and RuO. In the case of FeO, five states are calculated to lie within 0.1 eV of each other, of which 7 L * is the predicted ground state — the order, from lowest in energy to highest, is 7 Z + < 5n < 5 $ < 5A < 5 Z + . While unable to predict the correct ground state, the complete active space (CAS) MC-SCF wavefunctions provide a qualitatively correct description of the electronic structure as shown by comparison of the calculated spin-orbit coupling constants, A, with the experimental values determined in this laboratory (in cm"1): 5 A 5TJ 5 $ calculated: -84 - 203 -40 experimental: -95 —217 —47. Agreement is also fair for the ground state bond length: calculated: r g = 0.168 nm experimental: r e = 0.162 nm but poor for the vibrational frequency: calculated: cj g = 681 cm"1 o>exe = 6.3 cm"1 experimental: we = 880 cm"1 wexe = 4.6 cm:1 The agreement is better for RuO. The calculated natural orbitals and their occupancies (i.e. electron configurations) for the various states are useful for understanding the bonding in FeO, and, given the experimental fact that the 5 A state is the ground state, Krauss and Stevens were able to formulate an aufbau for the ground states of most of the first and second row transition metal oxides in terms of these orbitals. 32 In the last section, the importance of a correct experimental description of the lower states of a molecule for the accurate determination of such thermodynamic properties as the dissociation energy was discussed. Most of the previous determinations of the FeO dissociation energy have assumed a 5 Z ground state. Armed with the information provided by this laboratory that the ground state is actually a 5 A state, Smoes and Drowart (80) have recently made a new determination of the dissociation energy by the mass spectrometric Knudsen cell method: D 0°(FeO)= 403±8 kJ-mol:1 This value should be more reliable than any of the previous determinations. The authors have critically compared this value with the literature data and have explained a number of discrepancies. We point out, however, that the low-lying state tentatively identified as a 7 Z state, discovered during the course of the present work (Section IV.D), was not allowed for in the determination of the above D 0° value, since the existence of this state had not at that time been published The foregoing value for D 0 °, while being an improvement over the previous values in the literature, may therefore not yet be the ultimate determination. The potential astrophysical importance of FeO due to the high cosmic abundances of both Fe and O was indicated in the previous section. This prompted a search by Merer, Walmsley, and Churchwell (65) for FeO in a number of astrophysical environments based on line frequencies predicted by the work of this laboratory. The pure rotational lines searched for and their predicted rest frequencies are as follows6: 5 A , v = 0 J = 5-4 153.141 GHz V = 1 J = 5-4 152.014 GHz 5 A 3 V = 0 J = 5-4 154.058 GHz 'Later work (Endo, Saito, and Hirota (81) and this thesis) has refined the frequency values slightly to 153.135, 152.013, and 154.060 GHz, respectively. The conclusions of ref. (65) are not altered. 33 with 3a errors of ±0.005 GHz. Although a selection of molecular clouds, stars, supernova remnants, and planetary nebulae were searched, the results were negative in all cases. The search was not in vain, however, as the absence of FeO also carries information. In the Orion nebula, for example, at most 2 x 10"7 of the cosmic abundance of iron can be in the form of FeO, otherwise it would have been detected. One cause for the lack of FeO could be condensation of interstellar iron onto grain surfaces, thus depleting the amount available in the gas phase. The very low value for the upper limit on the abundance of FeO implies that such depletion must be very efficient. Other possible reasons for the low abundance of FeO are discussed in the paper by Merer, Walmsley, and Churchwell (65). P. Feldman (82) has recently expressed interest in conducting further astronomical searches for FeO (and other metal oxides) at other frequencies and has requested predicted frequencies from this laboratory. The values obtained in this thesis have been forwarded to him. Meanwhile back on earth, Endo, Saito, and Hirota (81) have detected and measured six pure rotational lines of FeO in the millimeter-wave region, observations made possible by the predictions of this work. The measured frequencies are listed in Table I and, because of their high accuracy, have been included in the latest least-squares Fits presented in this thesis (Section IV.C). Cheung and Radford (83) and Evenson (84) are currently looking at the far-infrared laser magnetic resonance (LMR) spectrum of FeO. Again, line-frequency predictions from the present work have been communicated to both groups. These will show where to look and will help in the interpretation of their spectra. TABLE L Millimeter-wave pure-rotational transitions of FeOa J' J" ft Frequency'3 5 4 4 153.135 259 (55) 6 5 4 183.757 163 (23) 5 4 • 3 154.059 726 (3D 6 5 3 184.867 223 (14) 5 4 2 154.948 453 (19) 6 <— 5 2 185.932 444 (20) (a) Measurements by Endo, Saito, and Hirota (80). (b) Values in GHz; numbers in parentheses are the standard deviations of the measurements in units of the last two figures. CHAPTER II EXPERIMENTAL DETAILS II.A. Introduction A variety of methods of obtaining FeO have been successfully employed in the past In emission work, Rosen and co-workers (12,13,14,15) used arcs and the method of exploding wires to study the visible and near infrared regions. Arcs were also employed by Domek (3) (visible), by Dhumwad and Bass (85) (orange system), by Dhumwad and Narasimham (22) (orange), and by Barrow and Senior (23) (orange). Low-pressure chemiluminescent flames obtained by reacting Fe, evaporated from a furnace, with 03, N20, N02, or discharged 02, in a flowing inert gas allowed West and Broida (24) to study the infrared, orange, and blue systems at a relatively low temperature (700 K). They also produced FeO at atmospheric pressure by seeding an oxygen-acetylene flame with Fe(CO)5. Bass and Benedict (17) obtained the near infrared spectrum during combustion studies of CO containing a small quantity of iron pentacarbonyl. McDonald (36) used an electrodeless discharge through a mixture of Fe(CO)5, 02, and Ar (orange system). Trkula (38) has recently looked at the orange and blue regions by sputtering an Fe cathode; a mixture of 5% 0 3 in 0 2 was used and a bright yellow "flame" was obtained. In absorption work, Green, Reedy, and Kay (34) used a hollow-cathode sputtering technique to study the vibrational infrared spectrum of matrix-isolated FeO. Bass, Kuebler, and Nelson (21) studied the orange system and the diffuse feature at ~241 nm by flash heating metallic Fe in the presence of 02. Callear and Norrish (70,19) conducted emission and absorption flash photolysis experiments (orange system and 241-nm feature) in which ferrocene or iron pentacarbonyl were used as additives to amyl nitrite and n- heptane fuel explosions in the presence of oxygen. 35 36 Similar experiments using pentyl nitrite plus hydrogen or isoamyl nitrite and hydrogen as fuel were performed by Erhard (71,72). Mixtures of Fe(CO)s in AT were flash photolyzed by Callear and Oldman (20) to obtain the 241-nm feature in absorption. Lindgren and Sassenberg (39) conducted flash-photolysis experiments involving the reaction of Fe(CO)5 with N20 (whether in absorption or emission is not stated). Very recently, Endo et al. (81) have studied the millimeter-wave region by means of a DC glow discharge in a mixture of ferrocene and oxygen in a 1-meter long free-space absorption cell. Some more exotic sources have been employed to obtain FeO for various specialized studies such as the shock tube experiments on Fe(CO)5 + 0 2 in Ar by von Rosenberg and Wray (46) and the tubular fast-flow reactor used by Fontijn and Kurzius (47) for a kinetic study of the reaction Fe + 0 2 —•* FeO + O. Even vaporization of Apollo 12 lunar samples has been used as a source of FeO (by Balducci el al. (43) for a thermodynamic study). None of the sources are completely ideal for spectroscopic work. Many produced strong atomic Fe lines. Dhumwad and co-workers (85,22) experimented with various sources, including a hollow cathode discharge and found that a low-pressure arc (—50 mm Hg) was best at suppressing atomic lines. Barrow and co-workers (23,37) also tried low-pressure arcs, but the lines obtained were weak and broad and occurred against a continuous background A water-cooled composite-wall hollow cathode discharge of the type described by Bacis (86) was found to produce better spectra (orange system), but its behaviour was described by Harris and Barrow (37) as "capricious." 37 n.B. Description of the Source Gaseous FeO molecules were produced in a flow system by passing a mixture of ferrocene, oxygen, and a noble gas through a 2450-MHz electrodeless discharge at low pressure. The experimental arrangement was as follows. Ferrocene (dicyclopentadienyliron, Fe(C}H4)2) was heated to between 65 and 80°C in a wide-bore pyrex tube wrapped with heating tape, and the vapor was entrained in a stream of noble gas. Argon was used for the visible region; helium for the near-infrared. A small amount of 0 2 was added, and the mixture (at a total pressure of a few mm Hg) was pumped rapidly through a microwave discharge cavity, with the discharge operated at a power of 120 W. A long orange-white "flame" resulted, the orange color being due to the orange emission system of FeO. Solid rust-like products tended to accumulate on the walls of the apparatus, and if allowed to become excessive resulted in the discharge becoming unstable, presumably because of interference with the transmission of the microwave power. However, with experience it was found possible to run the discharge continuously for several hours, with only occasional "tweaking" of conditions. 38 H.C. Methods of Detection n.C.l. Grating spectra1 "Survey" emission spectra covering the region 550-630 nm were obtained with a 7-meter Ebert- mounted plane grating spectrograph and photographed on Kodak Ila-D plates. Calibration spectra were provided by a neon-filled iron hollow cathode lamp, for which the wavelengths have been listed by Crosswhite (87). The spectra were measured on a Grant Instruments Co. oscilloscope-setting comparator, and reduced to vacuum wavenumbers using a four-term polynomial. D.C.2. Laser-induced fluorescence2 The tail of the "flame" of the microwave discharge was pumped across a cube-shaped metal fluorescence cell fitted with Brewster-angle windows and light baffles. The laser beam was sent through this tail, and the resulting fluorescence was observed at right angles to the beam and to the stream of molecules. The laser system consisted of a Coherent Inc. model CR 599-21 cw tunable dye laser, operating with rhodamine 560 or 590, and pumped by a model CR-10 argon ion laser. In some of the later experiments the blue-green lines from a model CR 3000K krypton ion laser were used to pump a model CR 699-21 ring dye laser. Laser powers of up to 450 mW were used. Non-laser-induced FeO emission (from the microwave discharge) was present as a background, but this was weak in comparison with the laser-induced fluorescence and could be almost entirely suppressed by chopping the laser beam and using phase-sensitive detection. Laser excitation spectra were obtained for large regions of the orange system (see Section I.B.2). Additionally, sub-Doppler intermodulated fluorescence spectra (88) were recorded for certain regions of the 582-nm band. With both types of spectra, JWork done by R. M. Gordon and A J. Merer. 'Work done by A. S-C. Cheung, A. M. Lyyra, A. J. Merer, and the author. 39 molecular iodine fluorescence lines were recorded simultaneously for calibration. Wavenumbers from the iodine spectrum atlas of Gerstenkorn and Luc (89a) were used, but with 0.0056 cm"1 subtracted from each listed value as a calibration correction (89b). Markers for interpolation between the iodine lines were provided by a Tropel 299-MHz free-spectral-range fixed length semiconfocal Fabry-Perot interferometer. The FeO fluorescence was detected through a sharp-cut yellow filter with an RCA C31025C photomultiplier tube. For the sub-Doppler spectra, the sum of the chopper frequencies was selected with a narrow-band electrical filter, and a Princeton Applied Research model 128A lock-in-amplifier was used to extract the intermodulated signal. Fluorescence from the calibration I2 cell was detected by an RCA 1P28 photomultiplier tube operated at -870 V DC. Resolved fluorescence spectra were obtained using a Spex 0.75-meter scanning monochromator in first order. ILC.3. Fourier transform spectroscopy Fourier transform spectra covering the near-infrared region 4 000 - 14 000 cm"1 were recorded at Kitt Peak National Observatory near Tucson, Arizona, using the 1-meter FT spectrometer constructed by Dr. J. W. Brault for the McMath Solar Telescope. The bright center of the microwave discharge was focused directly into the aperture of the spectrometer. Suitable wavelength cutoff Filters (CS5-56 and RG715) were used, and the emission was detected with a liquid nitrogen-cooled indium antimonide detector. The resolving power of the spectrometer was set to just over 800 000. A total of 46 interferograms, each of which took 6 minutes to record, were co-added for the final spectrum. A quintic window and apodization were applied to the transform. 40 E.D. The Spectra The complexity of the orange system spectrum has already been described in Section I.B.2. In many regions, the spectrum as recorded on the grating photographs appears very dense, with numerous branch fragments overlapping each other. Very few bandheads stand out, and the only obvious progressions are the two corresponding to Delsemme and Rosen's A, and A 2 systems, which begin at 558 and 561 nm, respectively. The analysis of most regions was only made possible by extensive rotationally resolved fluorescence studies and the very accurate A2F" combination differences obtained from the laser excitation spectra, together with a simultaneous analysis of the infrared system, which involves the same lower state. The infrared system, analysis of which is the main subject of this thesis, will be described more fully in Chapters IV and V. Suffice it to say for now that this region of the spectrum is also very complex, with numerous perturbations, and in many regions involves extensive overlap of branches belonging to various vibrational levels of various substates of two separate band systems. The analysis of this region was, in its initial stages, somewhat more tractable than that of the orange region, especially with the availability of accurate A2F" combination differences obtained from the orange system. However, the extensive perturbations and the pileup of branch structure in certain areas still made analysis of this system a challenging problem, as will become evident in later chapters. As previously mentioned, reaction by-products tended to build up on the walls of the discharge cell and then made the discharge susceptible to unstable operation. Continued running under these conditions often gave mainly CO spectra. It was possible to obtain CO-free grating spectra with good FeO intensity if recording times were kept to under one hour. Operating times much longer than one hour were used when doing the laser experiments and recording the Fourier transform spectra, due to experience gained in 41 maintaining optimal running conditions. It was usually possible to avoid CO emission by observing the color of the discharge with and without a hand-held spectroscope, although admittedly one band due to CO has been discovered in the FT spectra. This band, the (2,0) Asundi band, was identified from line positions calculated from the molecular constants and matrix elements published by Effantin et al. (90), and, once identified, caused only minor difficulties in the FeO analysis. Atomic Fe lines were present in the grating spectra. These were often absent in the laser spectra, however, and therefore caused no gTeat problem: where a portion of the grating spectrum was "blanked out" by an atomic emission line, the laser could often "see through" to excite the FeO spectrum. A notable example is provided by the strong Fe I emission line at 17 802.46 cm;1 which blanks out about 3 cm"1 of the grating spectrum but is entirely absent in the laser excitation spectrum because it does not involve the 5 D ground level of iron (Fig. 3). Atomic lines present in the FT spectra were of greater consequence, particularly the very intense ones, not merely because of the areas blanked out by the lines themselves, but, more significantly, because of "ringing" and loss of signal-to-noise in the surrounding regions. One particularly bad region was caused by the extremely intense O I lines at 11 836.17 and 11 836.32 cm"1: the effect extends to some 70 cm"1 to either side of these lines. The FT spectra were obtained in the form of tables of intensity versus wavenumber, which were subsequently plotted out Line positions were extracted with a third-degree polynomial fitting program. In cases where this method failed, due to blending, a simple three-point comparison was used to find the intensity maxima. The spacing between data points was 0.013 608 cm;1 corresponding to three or four data points within the Doppler profile of each molecular line. The Doppler-limited full-width-at-half-maximum for unblended lines is 0.03 cm:1 and the experimental accuracy 42 Fig. 3. Heads of the 0' =1 band at 561.4 nm and of the 0' =0 bands near 562.4 nm. The laser excitation spectra are shown as stick diagrams above the photographic prints of the discharge emission spectra, with stick height roughly proportional to intensity. Note the difference between the two types of spectra in the regions of the strong Fe lines. 43 of the positions of unblended lines should be better than 0.007 cm:1 3 The accuracy of the laser excitation line positions is roughly 0.002 cm"1 for unblended lines of the orange system. '0.007 cm"1 = one-half the spacing between data points. CHAPTER m THEORY JJ1.A. Introduction The energy levels of any atomic or molecular system are given by solution of the time-independent Schrodinger equation H^ = EvJ/ ...(III.A.1) where H is the hamiltonian operator, ^ is the wavefunction, and E the desired energy. Unfortunately, this equation can only be solved exactly for the hydrogen atom, so already in only the second sentence of this chapter we must start considering the use of approximate methods if we are to have any hope of understanding our experimental results! The first approximation, due originally to Born and Oppenheimer (1), is to consider the motions of the electrons and nuclei as occurring independently of each other. The result is that the Schrodinger equation can be divided into separate electronic and nuclear parts, as will be shown in Section m.C. The next step (Section III.D), which again involves approximation, is to continue this division process and separate the nuclear motion into vibrational and rotational parts. The hamiltonian used in Sections IU.C and D neglects the spin of the electrons. Spin, which arises as a relativistic effect, is then introduced in Section E To improve upon the approximations made in the above separations, an effective hamiltonian can be constructed (Section H) that operates on the rotational wavefunctions within a given vibronic1 level as does the hamiltonian derived from the 'The term "vibrational level" is used to refer to the vibrational levels of a specific electronic state, whereas the term "vibronic level" is used to refer to any vibrational level regardless of which electronic state it belongs to, the electronic state then being included within the vibronic state labelling informatioa 44 45 Born-Oppenheimer and rotation-vibration separation methods, but which, unlike the latter hamiltonian, takes into account interactions with energetically distant-lying states. This method does not take into account interactions with close-lying states, however; these effects are treated as perturbations — the topic of Section J. Since 5 A and 54> states have never previously been identified, their energy matrix elements have not been derived before. Also, there is only one known example of a molecule possessing a 5 n state where a detailed rotational analysis has been attempted, namely CrO (2,3). Hence much of this chapter is devoted to the derivation of the effective hamiltonian and matrix elements for these high multiplicity states, together with sufficient background theory to carry out this derivation. During the course of the foregoing procedures, the eigenfunction \p is expanded in terms of a complete set of basis functions 0j, + = Z c ^ i ...(HI.A.2) where the Cj's are constants. In Sections C and D, the basis functions are the Born-Oppenheimer basis functions (defined by eqn. (III.C.4)). In Section H, the effective hamiltonian is derived in terms of a different basis. In fact, a choice of several bases exists for this purpose and the form of the effective hamiltonian is dependent on the choice made. Each of these bases consists of the eigenfunctions of a certain set of commuting angular momentum operators corresponding to one of Hund's coupling cases, as will be discussed in Section F. Once a suitable basis is chosen, the energies of the rotational levels are obtained by writing the matrix H of the hamiltonian H and diagonalizing it Lambda-doubling is discussed in a general way in Section G, then for the specific case of a 5 A state in Section H.2.a. This topic is of particular interest in that observations of A-doubling in states with |A| > 1 have been comparatively rare. 46 While a study of intensities in the spectrum of FeO was.not a primary objective of this project, a derivation of the intensity formulas is still useful since it demonstrates the origin of the. selection rules operating in the spectrum. Also, the intensity formulas are useful in a qualitative sense for understanding the relative branch intensities and the relative line intensities within a branch. Section I (for intensity!) thus covers this subject One cannot go far in studying any part of the spectrum of FeO before one encounters perturbations — lots and lots of them! Hence this chapter would not be complete without a section on this topic. Section J provides a general introduction to the subject of perturbations. Finally, in Section K, single-configuration representations of electronic structures are used to view FeO in terms of the more general picture of the first-row transition metal oxides considered as a series. 47 DIB. The General Molecular Hamiltonian The non-relativistic hamiltonian operator for a molecule consisting of N nuclei and n electrons is h 2 N V A H 0 = - ^ Z —^ (kinetic energy of nuclei) 2 A M A 2 n Z Vr (kinetic energy of electrons) h - V T-7 2 2me a a + V(q,Q) (potential energy) ...(III.B.1) where M A is the mass of nucleus A; m e is the mass of an electron; n is Planck's constant divided by 2ir; and V 2 ("del squared") is the Laplacian operator, which is, in Cartesian coordinates, v 2 •  + &  + - ( n L R 2 ) and in spherical polar coordinates, V R 2 9R^K 9R ; R 2 s i n e d d ( S i n U dd} R 2 s i n 2 0 d<p2 „.(ni.B.3) The potential V (q, Q), which depends on the positions both of the electrons (q) and of the nuclei (Q), consists of three terms, N N Z A Z B e 2 V(q,-Q) = Z Z ——-— (nuclear-nuclear repulsion) A B>A rAB N n z A e 2 - Z Z —— (electron-nuclear attraction) A a r aA n n 2 + Z Z —s— (electron-electron repulsion) ...(IU.B.4) a b>a r b a where e is the charge of an electron, ZAe is the charge on nucleus A, and r x v is the distance from particle x to particle y. 48 A relativistic treatment (4) yields a hamiltonian in agreement with the above but with a number of additional terms. Most of these extra terms describe effects that are too small to be detected by the vast majority of spectroscopic methods, but two are of importance and must be added to the non-relativistic hamiltonian H 0 : H = H 0 + H s + H h f s ...(ffl.B.5) These new terms involve electron spin (Hg) and nuclear hyperfine structure (Hjjfs). In the case of iron oxide, both Fe and O have nuclear spins of zero, hence nothing further will be said about H f^g. The electron spin hamiltonian H s , however, will be considered in detail in Section ffl.E The hamiltonian H 0 is written in a space-fixed (= laboratory-fixed) coordinate system. We are not interested in the kinetic energy due to translation of the molecules, because in free space this is not quantized. The translational motion can be separated off by shifting to a coordinate system parallel to the laboratory-fixed system, but with its origin at the center of mass of some particular molecule (5a,6). This is very nearly an exact separation for a low-pressure gas in the absence of externally applied electromagnetic fields, since intermolecular forces are too weak under these conditions to perturb the free translational motions to any significant extent 49 UI.C. Separation of Electronic and Nuclear Motions — The Born-Oppenheimer Approximation The mass of a. proton is about 1840 times that of an electron. As a result of their smaller masses, electrons move much faster than nuclei, and this fact provides the foundation for the Born-Oppenheimer separation of electronic and nuclear motions. The electrons are treated as if they move in a field of fixed nuclei and are able to adapt themselves instantaneously to successive nuclear configurations as the nuclei move. The original derivation by Born and Oppenheimer (1) is rather long and complicated and will not be described here. Born later published a simpler, yet more general, derivation (7,8), and this is the treatment we follow. A more detailed derivation, including relativistic terms, has been given by Bunker (9). The Schrodinger equation for the non-relativistic hamiltonian is Ho* = \ - § Z 7 r ' o?T 2 V + V ( q , Q ) U = Etf ...(III.C.1) I 2 A m A 2 M e a a * In the limit of infinitely heavy nuclei, the hamiltonian, H 0 , becomes H_ = - Z V a 2 + V ( q , Q ) ...(III.C.2) e 2 me a a which can equivalent^  be regarded as the electronic hamiltonian for the case of fixed nuclei. Born (7,8) assumes that the electronic problem has been solved for each fixed value of Q, H e ^ ( q , Q ) = E*(Q) i j£ (q ,Q) ...(III.C.3) where (q, Q ) is the electronic wavefunction for the k^1 state and is a member of the complete set of normalized orthogonal electronic wavefunctions. He then tries an expansion of the total wavefunction i / > (q, Q ) , tf(q,Q) = Z ^ ( q , Q ) ^ ( Q ) ...(III.C.4) 50 as a solution to the Schrodinger equation for the complete system, where ^ (Q) depends only on the nuclear positions. Substitution of this into eqa (III.C.1), left-multiplication with ^g*(q,Q), and integration over the electronic coordinates q, selects out the k^1 electronic state and yields as the final result2 |H„ + U k(Q)} (Q) + C = Ek<//k(Q) ...(III.C.5) where Ti2 N V H„ - - | J ^  ...an.C6) U k (Q) = E k (Q) + 5 ' £ r 7 - / [ V A ^ ( q , Q ) ] 2 d q ...(UI.C.7) 2 A MA and C stands for the cross-terms C = Z { / ^ ( q , Q ) H n ^ ( q f Q ) d q i*k I e " e - "h2 Z j ^ - ; ^ ( q , Q ) V A ^ ( q , Q ) d q V A | ^ ( Q ) ...(ffl.C.8) Neglect of the cross-terms constitutes the Born-Oppenheimer approximation. The energy obtained by solution of the resulting equation is really the total energy of the k* state, but because of the way the calculations are actually carried out in practice this is regarded as only the nuclear energy, E£. This comes about as follows. The electronic energy Ek(Q) obtained from eqn. (UI.C.3) varies with changing values of the nuclear coordinates represented by Q to form what is called a potential well. In practice, only that value corresponding to the bottom of the potential well is called the "electronic energy" of the k m state. The variation in E^Q) with Q is then incorporated into the "nuclear energy" (actually into the vibrational part of the "nuclear energy") via the U^Q) operator. That is, the electronic energy function Eg(Q) Cactually Uk(Q) — see next paragraph) plays the part of potential energy for 'The steps involved are given in more detail by Schutte (10). 51 the nuclei. The zero of the "nuclear energy" scale is then set to coincide with the bottom of the potential well for the k1*1 state. To sum up, provided that the cross-terms can be omitted, the nuclear motion has been separated from the electronic motions. The nuclei are found to move against a potential U (Q) which depends on the positions of the nuclei. This potential is almost equal to the energy determined from the electronic Schrodinger equation as a function of Q to which is added a small correction term (the second term on the right-hand side of eqn. (III.C.7)). This term, which contributes to the electronic isotope shift, was not actually derived in the original Born-Oppenheimer treatment.3 It is therefore often overlooked in discussions about the electronic/nuclear separation. However, it should be included, especially for light molecules. The cross-terms neglected in the Born-Oppenheimer approximation couple the electronic and vibrational motions in different electronic states. If, in specific cases, the electronic wavefunctions ^g(q.Q) can be solved for, the magnitude of these terms can be determined. In general, it can be seen if these terms are treated by perturbation theory that, provided the energy separation between the k^ state and any other electronic state is large, the effect of these terms will be small. In the other extreme, if the symmetry is high enough to permit two electronic states to become degenerate, the Born-Oppenheimer approximation breaks down4 and the phenomena known as the Renner-Teller effect (for linear molecules) or the Jahn-Teller effect (for symmetric top molecules) occur (11,12). 'Often the phrase "Born-Oppenheimer approximation" is used to refer to the approximation resulting from the neglect of the second term on the right-hand side of eqn. (III.C.7) in addition to the neglect of the cross-terms C, whereas "adiabatic approximation" is used to refer to the neglect of only the cross-terms. 4The term labelled [ c ] * 0 in this situation, since the stationary states are complex, rather than real. 52 m.D. Separation of the Vibrational and Rotational Motions The separation of the nuclear motion into vibrational and rotational parts, for polyatomic molecules, is not a trivial process due to the presence of cross-terms between the two motions (13,14). For diatomic molecules, the process is considerably simpler since there are no such cross-terms to couple the vibrational and rotational motions. The separation for diatomic molecules follows directly from the fact that the vibrational and rotational motions of the nuclei must be orthogonal. Vibrational motion can only occur in a direction along the internuclear axis — i.e. stretching or compression of the internuclear distance. Motions of either or both nuclei in any other direction(s) must automatically produce a translational or rotational motion of the molecule as a whole. The derivation of the energy level expressions for the individual vibrational and rotational parts is well known (5b,15) and is not reproduced here. Experimentalists usually express the vibrational energies in the form (16a) G(v) = w e(v+i) - cj ex e(v+i) 2 + « ey e ( v + ± " ) 3 + . . . ...(III.D.1) where G(v) is the "energy" (in cm"1 )5 of the v m vibrational level and v is the vibrational quantum number. If only the first term in the series is retained, the expression describes the energy levels of a harmonic oscillator; the additional terms are anharmonic terms. The vibrational frequency u>t and the anharmonicity constants wexe, o>eye, ... are treated as empirical constants, which are obtained by fitting the observed vibrational spacings (extrapolated if necessary to zero rotation) to eqn. (III.D.4). If the internuclear distance R is assumed to be fixed (the "rigid rotor approximation"), the solution to the rotational Schrodinger equation is Obtained by dividing energies in joules by he. 53 F ( J ) = o 2 h r , 2 J ( J + 1 > ...(III.D.2) where F(J) is the rotational "energy" (in cm"1) of the 3^ rotational level, J is the total angular momentum quantum number, c is the velocity of light (in cm'S"1), and n is the reduced mass of the molecule. In reality, the molecule is not a rigid rotator. To correct for this, eqn. (III.D.2) is modified in two ways. First, because the molecule is vibrating, it is necessary to average over the coordinate R, which we do by defining the rotational constant B v such that B v = elfel ; * v R T M r -(m.D.3) Since the value of B varies with vibrational level, as denoted by the subscript v, it is usual to express this dependence on v as a power series, B v = B e - a e(v+i) + 7 e (v+3) 2 + 6 e ( v + l ) 3 + . . . ...(III.D.4) where B_ = — t h e value of B at the bottom of the potential well (where e 8ir zCMR| R = R e, the equilibrium bond length), and a e , fe, 6e, ... are vibration-rotation interaction constants. Another type of vibration-rotation interaction arises from the centrifugal force that results when the molecule rotates. This occurs even for v = 0 and has the effect of stretching the bond. This is taken into account by means of additional terms,' F V ( J ) = B yJ(J+1) - D V [J (J+1) ] 2 + H V [J (J+1) ] 3 + L V [ . J ( J + 1 ) ] » + . . . ...(UI.D.5) Usually it is only necessary to use one centrifugal distortion term, the D v term. Its power series expansion is D v = D e + /3 e(v+i) + ... ...(m.D.6) 'Some authors use a "-" sign in front of the L y term. 54 In summary, as a result of the separations described in this and the previous section, the total energy for a diatomic molecule (neglecting spin for now — see next section) can be written as the sum of electronic, vibrational, and rotational parts, E k v J = Eg ( R e ) + G ( k ) ( v ) + F ^ k ) ( J ) ...(m.D.7) where the electronic energy (now in cm"1) is defined as that value of EgXQ) occurring at the bottom of the potential well, the vibrational energy scale has its zero value at this same point, and the rotational energies are defined relative to the appropriate vibrational level. 5 5 III.E. Electron Spin While not present in classical theory, the concept of spin arises when relativity theory is applied to the wave equation for the electron (or nucleus) (17). This explains a number of properties not accounted for with the non-relativistic hamiltonian. For a molecule such as FeO with no nuclear spin, the additional features of interest are: (i) spin-orbit coupling; (ii) spin-rotation coupling; and (iii) spin-spin coupling. i.e. H s = H s o + H s r + H s s ...(IILE1) An electron possesses a magnetic moment u, which, according to the relativistic theory, is given by U = ...(III.E2) where g is the gyromagnetic ratio (= magnetogyric ratio = 2.002 3); Mg, the Bohr magneton, stands for the collection of constants = 9.274 x 10" 2 4 J«T _ 1 (where -e is the charge on the electron,7 -1.602 x 10'19 C, and mg is the mass of the electron, 9.110 x 10"31 kg); and s is the spin angular momentum' of the electron. The interactions of this electron spin magnetic moment with (i) the magnetic field generated by the orbital motions of the electrons; (ii) the magnetic field generated by the rotation of the nuclei; and (iii) the spin magnetic moments of the other electrons give rise to the three coupling terms listed above. 7The convention e = +|e| is used here. "Angular momenta will be defined and discussed in Section III.F. 56 D3.E.1. Spin-orbit interaction We first consider a single atom possessing one electron with non-zero orbital angular momentum. The positive charge of the nucleus produces an electric field E. Thinking classically, an observer positioned on the electron would experience a magnetic field B due to the apparent motion of the nuclear electric field, 5 = ~ v e n x E = -V E x v e n ...(III.E3) where v e n is the velocity of the electron relative to the nucleus. Since the electric field is the gradient of the potential V, and since the spherically symmetrical field arising from the point charge of the nucleus corresponds to a central field potential V = V(ren), we can write E = - grad V = - fen) j f f l - ...(III.E.4) \ ery U l e n where r e n is the position vector of the electron from the nucleus and r e n its corresponding magnitude. Then 2 - " c T T — (gfM £ e n x Sen -CHLE5) c r en y U I e n / Classically the hamiltonian for the interaction of a magnetic field and a magnetic moment is given by H = - u>B ...(IL7..E6) Upon substitution of the relativistic expression for M (eqn. (HI.E2)) and the expression for B from (LTJ.E5), this becomes We now decompose the vector v e n , the velocity of the electron relative to the nucleus, into components corresponding to the individual velocities of the electron and 57 nucleus in a laboratory-fixed coordinate system. Unlike in classical mechanics, however, v e^n is not simply the vector difference of the electron velocity v g and the nuclear velocity vR. There is an additional relativistic correction (a factor of j) that arises from a phenomenon known as Thomas precession (18). As discussed by Jackson (19), this occurs because the electron rotates with respect to the laboratory frame of reference as a result of the acceleration experienced by the electron as it moves in the Coulombic field of the nuclei and other electrons. With this correction the vector v g n becomes Zen " he " Zn -0ILR8) so that H • " E ^ ( d f ^ ) • < * • . - ! „ > • • ...(III.E.9) The coordinate system was specified above to be a laboratory-fixed frame, but it can actually be chosen to be a frame moving with the molecule, with origin at the center of mass for example, provided that it is regarded as instantaneously space fixed (20). For an atom with more than one electron, the interaction energies are assumed to be additive.' Likewise, in a molecule, each electron is assumed to interact with every nucleus, and each such interaction contributes a term of the form of eqn. (III.E9), i.e. 'These procedures are not strictly correct For example, the interaction of M with the magnetic fields resulting from the relative motion of the electric fields of trie other electrons has not been taken into consideration. To account for this, Van Vleck (21) includes an additional term, itself similar to eqn. (III.E10) but for the other electrons. Here, we allow for this in an approximate way when we introduce (see eqn. (III.E14)). (Note that eqn. (37) of Van Vleck's paper (21) contains several errors. A corrected version has been given by Kayama and Baird (20); this version also contains misprints (two of the subscript K's in their eqn. (28) should be k's).) 58 By the distributive principle, this breaks up into two terras, + E Z _ L _ | ^ ^ M r e n x v n . s _ ...(III.E.11) t i c 2 e n r e n ^ dreny ~en ~n ~e where the first term has been multiplied and divided by the mass of the electron. It can now be recognized what the two terms represent The second term, which contains v^ n, involves the motion of the nuclei — specifically, for diatomic molecules, the rotation of the nuclei.10 This is the spin-rotation interaction, which will be discussed later. The first term is the spin-orbit interaction, H S Q , as becomes clear if this term is rewritten as H=o - - 1 1 ^ (^j i e n • » . -OB*** where the substitution i e n = £en x meZen ...(III.E.13) has been made. l e n is the orbital angular momentum of electron e about nucleus a So far we have not allowed for the effect of the charges on the other electrons. This can be done satisfactorily by taking the potentials V e n as Coulomb potentials in which "effective'' nuclear charges Zejj-ne describe the screening effect of the electrons: Zpff n e V e n = A ' ...(HI.K14) where e 0 is the permittivity of free space. Then ^ e n = _ f e f f ^ n f ( m R 1 5 ) d r e n 4 7 r e 0 r | n "Diatomic molecules possess no vibrational angular momentum, so there is no spin-vibration interaction. 59 g i v i n g 9^1 z e f f n H so = 7 2 2 a f l P n - S P ...(III.E16) This can be rewritten as (22)u H s o " § a e ( r ) i e ' £ e ...(III.E17) by summing over the nuclei, where , 2 a e ^ r ^ i e 47re 0c 2ti 2 n r | n ~ e n ...(III.E18) The relation between 1_, the total orbital angular momentum of electron e, and the l e n ' s , the orbital angular momenta of electron e with respect to each nucleus n, is deliberately left vague because of the lack of a common origin for the l e n ' s , a point which has been addressed by Veseth (23) for diatomic molecules. The hamiltonian given by eqn. (III.E17) is called the "microscopic spin-orbit hamiltonian." A simplification of this equation that is often used in practice, but which involves further approximations, will be presented in Section III.H.2. m.E.2. Spin-rotation interaction The second term in eqn. (III.E11) gives the spin-rotation interaction. As with the spin-orbit interaction, the potentials V e n are approximated as Coulomb potentials for each nucleus, with the nuclear charges being partially shielded by the electrons, H s r = -~A ^ - 2 * : 2 2 3 f r e n x v n * s e ...(III.E19) s r 4 7 r e 0 c 2 h e n r | n ~ e n ~n ~e Following Van Vleck (21), we write the nuclear velocities as "Note that l e , l e n , and s each contain Ti. The units of are joules (or cm"1 if the expressions are divided~by he). 60 v n = . g x r n ...(III.E20) where r n is the position vector of nucleus n relative to the center of mass and CJ is the angular velocity of the nuclei, with the nuclei being regarded as forming a rigid rotator.13 The rotational angular momentum R is the moment of inertia multiplied by the angular velocity R = la ...(III.E21) We therefore obtain H - - 9 " B 6 sr  A 2 r v ^ e a f , n [ r e n x (R x r n ) ] - s e ...(III.E22) which, upon use of the identity a x (b x c) = (c-a)b - (b-a)c ...(III.E23) becomes _ _ 9"B e v v z e f f ,n r r e H „ r = -A 5T*: ^ £ — 3 — L ( r n • r _ n ;R J • s_ | g H R• £ e n ) r e n ] • s e ...(m.E24) 4jrc 0 c 2 Iri w . e n In fact, at least for diatomic molecules, only the first term contributes to the spin-rotation interaction of a given state. (The second term, which is very small, connects different electronic states.) The "microscopic form" of the spin-rotation interaction hamiltonian is then (24,25,26) H s r = R- | b e ( r ) s e ...(HI.E25) 1JThe motion of the nuclei due to vibration can be ignored, since positive and negative contributions cancel in the mean. The variation of the bond lengths due to anharmonicity and centrifugal distortion will be discussed in Section in.H.2. 61 where b e = "7 ^TTZ 2 a r n ' r ^ n ...(III.E26) e 47re 0c 2Iri n r | n ~ n ~ e n As with the spin-orbit interaction, a simplified version of this expression is usually used in practice (see Section III.H.2). III.E.3. Spin-spin interaction Each spinning electron behaves as a tiny bar magnet According to classical electromagnetic field theory, the magnetic field at a point i that is a distance r^  from an electron at point j is (27a) S i = V x ^ i ...(III.E27) where is the magnetic vector potential,13 which is related to the magnetic dipole moment £j of the electron at j according to (27b) if-i x £ i i A; = — ~ J , ~ 1 J ...(III.E28) where u0 is the permeability of free space. Eqn. (III.E6) gives the hamiltonian for the interaction of the magnetic moment of an electron located at point i with the magnetic field due to the electron at j, namely H s s , i j = _ i£i ' S i 47Tti2 ~ i V X S-: X =41 ...(IJJ.E29) where the magnetic moments have been written in terms of the spin angular momenta using eqn. (III.E2). The relation vf—!—] = ...(III.E30) Vrij/ r i j "In SI units, ki has units of Wb«m"1 J. 62 then gives , 2 , , 2 " o 9 T 1 1 H s s , i j = + - 4 ^ ~ Si'r X ~3 X V J 7 T T r 2 „ 2 M o 9 M B " r -l 1 M o 9 2 M B 2 s = + ^ 4 ^ [ (£i'Sj>V 2 - ( S i ' V M j S j - V ) ] ^ ...(III.E31) where the relationship V x V x a = V(V«a) - V 2 a ...(III.E32) has also been used.14 Now let [a] = (S i ' S - j i V 2 — ] — „.(in.E33) and [b] = ( si • V) ( • V ) — ...(m.E34) Consider the [ b ] term. This will become undefined for = 0 because of the r^  in the denominator. We can handle this problem by dividing [ b ] into two parts, [b] = [b]r^Q + [ b ] r = 0 ...(III.E35) The first part is easily treated, as follows: [b3r#0 - £i-[*<5j-V>7J-r] 14Note that eqn. (ITI.E32) is not a general vector relation (28). It holds only in cartesian coordinates and then only for the individual cartesian components. If V x V x a = b then for the x-component, b x is ^-(V-a) - V 2 a x = b x 3 x ~ x x with similar equations for the y- and z-components. The right-hand side of eqn. (III.E32) is a symbolic shorthand for these three equations. 63 - . i - ^ . V ) ^ ] " - £ i - [ ( ^ J * V ) { ^ i J ) l ...(HI.E.36) 1 i j J Then by the usual rule for differentiating a product, t ^ r ^ O = - S i - [ s j ' r l - " 3£ij r4~ f 1 1 1 S i * S i ( S i - T i j ) ( S J T : J ) = - ~ x ? ~ J + 3 ~ i - 1 J s ~J - 1 J ...(III.E37) r i j r i j The second part of [b] is more difficult to treat — but, not impossible! Any use to which the hamiltonian will be put will involve multiplication by some function (e.g. the square of the electronic wavefunction g^Vg) then integration over the spatial coordinates. For the [b] r _Q part, this can be done by carrying out the integration over a small spherical volume of space of radius e surrounding the origin, then letting e go to zero — i.e. lim /[b]f(r)dr . Now ( s ;«V)( s . :«V) can be rewritten as ( s i . V ) ( s j . V ) = s i x s j x £ 2 + s i x s j y £ JL + (HI.E38) where only the first two of nine terms have been shown. After operating on ' r i j with this expression, multiplying by f(r), and then integrating over the region bounded by r = e, when the limit as e-*0 is taken, the cross-terms (such as the second term on the right-hand side of eqn. (III.E38)) will become infinite. These can be ignored, since we will only be interested in energy differences; the infinities, which are constant, will cancel. Thus we are left with l^irn [ / [ b ] £ ( r ) d r ] - l^im [ / ( s i x s j x £ \ + s i y S j y ^ + s i z s j * ^ )7pf(r)dT] ...(m.E.39) 64 By symmetry, the value of the integral does not depend on the absolute spatial orientation of the vectors s ; and S:, but only on their relative orientation. Therefore, ~ i ~ j sixsjx' siysjy ^  sizsjz c a n e a c n ^ e r e P l a c e d hy the average value 3 E l E j ' lim [ j [ b ] f ( r ) d T l = \ ( • ) lim [/V 2— ]—t (r ) d r l ...(III.E40) e - 0 U J 3 - 1 ~1 e - * 0 L e r j j J where the definition of V 2 (eqn. (III.B.2)) has been employed. Since all uses to which the hamiltonian will be put involve integrals of this type, we can effectively set t b ^ r = o - [i(£i-£j>v2FH « -(IIIE41) This is of the same form as the [ a ] term, which we now look at A consequence of Gauss's divergence theorem is that (29) V 2 — = - 4JT8 ( r j * ) „.(in.E42) rij where 5(rjj) is the Dirac delta function, which is defined to be zero everywhere except at r^  = 0. This definition is consistent with the way in which the [ b ] r _ n term is to be evaluated (i.e. lim ). Thus the [ a ] and [ b ]. n terms have the e — 0 1 u same form and can be combined, giving * [<«i-2j>'i3 - 3 ( s i . r i j ) ( s j . r i j ) r ^ ] M 0 j . . . T O where the [ ] r ^ g indicates that any integrals involving this part of the hamiltonian are to be evaluated excluding the infinitesimally small region about ry = 0. In summary, the hamiltonian as given in eqn. (III.E31), when present in integrals evaluated over all spatial coordinates, will yield values of infinity — actually, infinity plus additional finite terms. The derivation following eqn. (III.E31) has projected out these finite terms (by neglecting the infinite parts, a legitimate procedure 65 since they cancel upon taking differences . . .spectral lines being energy differences). The first term in eqn. (III.E.43) is a contact term — i.e. the two electrons are in contact with each other, since ry = 0. The contact interaction was first worked out quantitatively by Fermi (30), using the relativistic Dirac equation. It can, however, be treated using purely classical theory, as has been shown by Ferrell (31).15 The derivation of the spin-spin interaction presented here is a "middle-ground" treatment in the sense that the spin was assumed to come from relativity theory, then classical theory was used to derive the hamiltonian (29). That part of the derivation dealing with the contact term is a modified version of that given by Bethe and Salpeter (33). Eqn. (III.E43) describes the interaction of two electrons. For more than two electrons, we sum over the electrons, being careful not to count twice, + [<Si *Bj >ri j " 3<£i-Eij><£j-£i:J>rIj]rJ ~(ni.E44) This equation applies equally well to atoms and to molecules. The first term, the contact term, gives a constant contribution to the energy of a given electronic state, so in practice it is included with the potential energy functon U (^Q) obtained during the derivation of the Born-Oppenheimer approximation — i.e. this part of H s s is to be included as an additional term in eqn. (III.C.7). Only the remaining part, the dipolar term, is conventionally regarded as giving rise to the electron spin-spin interaction. 1 5 See also Levine (32) for a clearer explanation of Ferrell's work. The derivations by Fermi and by Fenell were actually for the case of interacting electron and nuclear spins, rather than for electron and electron spins as here. The arguments and forms of the equations are the same, however. The contact interaction in the electron-nuclear case is called the Fermi contact interaction. 66 II1.F. Angular Momenta and Hund's Coupling Cases The last several sections have derived the various parts of the hamiltonian operator. In this section some basic properties of angular momenta will be presented, following which the types of angular momenta that occur in diatomic molecules and the ways in which they couple will be considered. UI.F.1. Angular momenta A quantum mechanical operator A is an angular momentum operator if its space-fixed cartesian components obey the commutation rules (34a,22) [ A I r A j ] = i i i L e I J R A K ...(m.F.l) K where + 1 if UK are a cyclic permutation of XYZ 0 if any two coordinates are repeated -1 if UK are an anticyclic permutation of XYZ ...(III.F.2) eIJK It is often convenient to refer to a molecule-fixed coordinate system (denoted by small x,y,z, as opposed to capital X.Y.Z for space-fixed components). When referred to molecule-fixed axes, some angular momenta (1^ , L, S, J^ a) H obey commutation rules analogous to those in space-fixed coordinates, namely [ A i f A j ] = iti I e i j K A K ...(III.F.3) while others (R, F, J , N, O) obey anomalous rules (22) [ A i f A j ] <4 j k A K ..(m.F.4) "For definitions of the angular momentum symbols, see Table II. TABLE II. Angular momenta present in a diatomic molecule possessing no nuclear spin. O r b i t a l a n g . m o m . o f t h e i*"* 1 e l e c t r o n S p i n o f t h e e l e c t r o n T o t a l a n g . m o m . o f t h e i f c ^ e l e c t r o n T o t a l e l e c t r o n i c o r b i t a l a n g . m o m . T o t a l e l e c t r o n s p i n R o t a t i o n o f t h e n u c l e a r f r a m e w o r k T o t a l a n g . m o m . T o t a l a n g . m o m . e x c l u d i n g e l e c t r o n s p i n T o t a l a n g . m o m . e x c l u d i n g e l e c t r o n o r b i t a l a n g . m o m • T o t a l e l e c t r o n i c a n g . m o m . T o t a l a t o m i c e l e c t r o n o r b i t a l a n g . m o m . ( e . g . f o r a t o m A ) T o t a l a t o m i c e l e c t r o n s p i n ( e . g . f o r a t o m A ) T o t a l a t o m i c e l e c t r o n i c a n g . m o m . ( e . g . f o r a t o m A ) T o t a l a n g u l a r m o m e n t u m V e c t o r Q u a n t u m n o . 1 h 1 L S = Z s. . - l l R J = R + L + S K = R + L = J - S 0 = R + S = J - L J = L + S . A . . A l l ! A = I ? A i L S J N P r o j e c t i o n o n m o l e c u l a r a x i s V e c t o r p r o j e c t i o n 1 1 2 1 Z 1 Z N 2 (E L z ) J (i J ) a z z Q u a n t u m n o . a X. + a. i 1 A £ H • A+ Z A ( a ) ( b ) X ^ = 0 , 1 , 2 , 3 , . . . a r e s i g n i f i e d b y a , TT , 6 , <(>,. A = 0 , 1 , 2 , 3 , . . . a r e s i g n i f i e d b y I , n , A , 4 > , . —] 68 An angular momentum basis function |AaM A > is simultaneously an eigenfunction of three operators: (i) A 2 . . .the square of A itself (ii) A^.. .the projection of A onto the molecular (z) axis (iii) kj. • .the projection of A onto the space-fixed Z axis. The eigenvalues are h 2A(A+l), "he, and tiM^, respectively, where A, o, and M A are quantum numbers and are also used as labels for the basis function, i.e. |AoM A>. All matrix elements of A.2, k-%. Ay. and A^ can be derived from the commutation rules (eqn. (III.F.1)) without ever specifying the explicit functional form of |AoM A > or the differential operator form of A (35a) <A'a 'M A |A 2 |AaM A > = + h 2 A (A+1 ) 8 A . A S a , a f i M A M A <A' a' M A | A2 | AaMA> = + h M A8 A, A 8 f l , ...(III.F.6) i (g - 0 ) <A'a 'M A |A ± |AaM A > = +he M A M A ± 1 [A(A+1 ) - M A ( M A ± 1 ) ] 1 / 2 * 6 A ' A 6 a ' a 5 M A M A ± 1 -(III-FJ) For the molecule-fixed components of angular momenta that obey the normal commutation rules (eqn. (III.F.3)), we have the analogous matrix elements <A'a 'M A |A 2 |AaM A > = + l ia6 A . A 8 a , a 5 M A M A ...(m.F.8) <A'a 'M A |A ± |AaM A > = + he' ( < t > a ~ 4 > a ± 1 * [ A ( A+1 )-a( a± 1 ) ] 1 / 2 • « A ' A 5 a ' a ± 1 5 M A M A For the molecule-fixed components of angular momenta that obey the anomalous commutation rules (eqn. (III.F.4)), eqn. (III.F.8) still holds, but 69 <A'a'M^|A ± |AaM A > = + r i e l ( 0 a 1 ] [ A (A+1 )-o( a? 1 ) ] 1 / 2 • 5 A ' A 6 a ' a T l 6 M A M A ~(m-F-10> In the foregoing equations, ladder operators have been used; these are defined as A ± = A x ± i A y ...(III.F.11) and A * = A x ± i A y ...(III.F.12) ^MA* ^MA±1' 9 X 6 arbitrary phase factors of the basis functions. For space-fixed components, the convention of Condon and Short!ey (35(a)) is generally adopted. By this convention, the 2A+1 |AcM A > basis functions (with M A = -A, -A+l +A) all have the same phase factor, obtained by requiring all matrix elements of A^ = \ (A+ + A_) to be real and positive. Unfortunately, for molecule-fixed components, a standard phase convention is not in universal use, though such a standard convention has been proposed by Brown and Howard (36) (see also Lefebvre- Brion and Field (22)). For this work we choose e a a - ' = e a a + ' = + 1 ...(in.F.13) JJLF.2. Hund's coupling cases Table II serves to define the various types of angular momenta present in diatomic molecules, though only certain ones will be relevant in any particular case. The angular momenta present in a diatomic molecule are described as "coupled" according to the relative strengths of their various interactions. Hund (37) distinguished four such types of coupling, labeled (a) to (d). A fifth type, case (e), not discussed by Hund, was introduced by Mulliken (38). A number of sub-categories 70 have been described (36)17: cases (b') and (d') and the close-nuclei and far-nuclei forms of case (c). Yet further sub-categories arise when nuclear spin is present (42). All of these cases and sub-categories thereof are limiting situations. Any given state of any given molecule may be best described by one or another of these limits or may be somewhere in between. By "best described by" we mean the basis set is the one in which the hamiltonian matrix is the most nearly diagonal — see next section. Within a particular state, conditions may change from being closer to one limiting coupling case to being closer to another as rotation increases (i.e. as J or N increases). Since the states of FeO studied in this thesis exhibit behavior typical of the case (a) situation, the present discussion is limited to Hund's case (a) coupling. For the other cases, the reader is referred to such general texts as Herzberg (16), Townes and Schawlow (43), or King (44a). Case (a) coupling occurs when the energy separations between the components of a ^S+l^ multiplet state resulting from the diagonal elements of H ^ are much larger than the off-diagonal elements of the S-uncoupling operator BJ^S*. The matrix elements involved will become clearer after Sections III.F.3 and III.H; for now they are just stated without explanation: | <T}ASZflJM | H S Q 177ASinjM>-<77ASZ ' fl' JM | H s o | TJASZ ' fl' JM> | = | AA | ...(III.F.14) |<T?ASZflJM|BJ~S+| 7?ASZ'fl' JM> | = BJi/2S ...(III.F.15) where Z' = Z - 1 ...(III.F.16) fl' = fl - 1 17Kopp and Hougen (39) defined a case (a') for Z states, but the prime is not really necessary (see, for example, Refs. (40) or (41)). 71 Thus case (a) occurs when |AA| » BJ/2S ...(III.F.17) In principle, the limit of pure case (a) is approached more and more closely as | A. | -—<*> or as B-*•(). However, in practice, if the spin-orbit interaction becomes too large, it becomes difficult to recognize the multiplet components as belonging to the same state and the case (c) model becomes more appropriate. Thus the best examples of case (a) occur when A is moderate in value and B is small. Note also the J-dependence in eqn. (III.F.17); as a consequence, as J increases case (a) becomes a less good description and eventually case (b) may become a better one. In Fig. 4 the angular momenta present in case (a) are represented as vectors. Both the total electronic orbital angular momentum L and the total electron spin S are coupled to, and precess rapidly about, the molecular axis, with constant projections and Sz and associated quantum numbers A and Z, respectively. The total angular momentum, in the absence of nuclear spin, is J = R + L + S, and its component along the molecular axis is represented by the quantum number ft = A + Z. The quantum number Z can take the 2S+1 values Z = S, S-1 , . . . , -S ...(ITI.F.18) The total angular momentum J cannot be smaller than its component and so, for a given ft, J has values J = |ft| , |ft+l| , |0+2| f . . . ...(ITI.F.19) Since S = ZSj, where Sj = ±\, and since ft = A+ Z; S, Z, ft, and J will all be integral or half-integral depending on whether there are an even or an odd number of electrons present in the molecule, respectively, and this will hold for all states of the molecule. The multiplicities, 2S+1, of the states must correspondingly be odd or even With a total of 34 electrons, all states of FeO must have odd multiplicity and the values of S, Z, ft, and J will all be integral. 72 Fig. 4. Vector diagram for Hund's coupling case (a). 73 H1.F.3. Angular momentum basis functions If two (or more) operators commute, that is, if AB0 - Bk<t> = 0 ...(III.F.20) for all functions <t>, or symbolically, if [A,B] = AB - BA = 0 ...(III.F.21) then a complete set of functions must exist that are eigenfunctions of the two (or more) operators simultaneously. While there are only a few operators that commute with the exact hamiltonian H (two examples being J 2 and J z), there are many that commute with parts of H — notably, and of most importance here, a number of angular momentum operators. In Section (III.H), H will be partitioned into two parts Hund's coupling cases each correspond to a set of mutually commuting angular momentum operators. The partitioning indicated by eqn. (IU.F.22) can be done in such a manner that H° commutes with the set of angular momentum operators of a given coupling case. Therefore, there must exist simultaneous eigenfunctions of H° and the angular momenta of that coupling case. Since the actual functional form of an angular momentum eigenfunction does not have to be specified (Section III.F.1), these eigenfunctions can be designated simply by listing the quantum numbers associated with the angular momenta of the particular coupling case. These are called good quantum numbers for H°, since H° possesses only diagonal matrix elements in this basis set The choice of basis set is, in principle, completely arbitrary, provided the basis set forms a complete set of functions. However, usually one set will be more H = H° + H' .(ni.F.22) 74 appropriate or convenient than the others. In this thesis, a case (a) basis set will always be used, because the states of FeO being studied exhibit good case (a) behavior (although one could argue that they are tending towards case (c) as a result of the large number of perturbations present), and because the matrix elements of H r and H g 0 can be evaluated in this basis using simple ladder operator techniques (see Section III.H) as a result of the presence of the maximum number of molecule-fixed z-components of the angular momenta (A, Z, and fl) and the absence of intermediate angular momenta (such as N in case (b), £ a in case (c)). The good quantum numbers in Hund's case (a) are A, S, Z, fl, J, and M. The basis states are therefore written as |nvASZflJM>, where nv (sometimes written as TJ) designates some particular vibronic state — n is a label standing for the electronic state and v is the vibrational quantum number. M, the quantum number associated with the projection of £ on the space-fixed Z axis, has 2J+1 values: -J , -J+l +J. In the absence of external electric or magnetic fields, each J level has a degeneracy of 2J+1 and M does not need to be specified in the basis set labels. The basis states can be factorized as follows |nvASZflJM> = |nA; v; SZ; flJM> = |nA>|v>|SZ>|flJM> ...(HI.F.23) where the parts have the meanings | nA> electronic |v> vibrational (=radial) | SZ> spin |flJM> rotational (=angular) non-rotational basis • rotational (total) basis 75 III.G. Symmetry, Parity, and Lambda-doubling The concepts of symmetry and parity form an integral part of any discussion of A-doubling, so we begin with these. Parity refers to the behavior of the complete wavefunction (apart from translation) when the signs of the space-fixed Cartesian ic components of all particles are reversed. The parity operator or inversion operator E can be defined by its behavior on the space-fixed coordinates of the i1*1 particle as18 E * ( X i , Y i , Z i ) = ( - X i r - y i f - Z i ) ...(III.G.1) Another operator of importance here is the reflection operator in a plane containing the molecular axis, o v ( x z ) (46b) a v ( x z ) ( x i ' Y i ' z i ) = ( X i ' - Y i r z i ) ...(III.G.2) a v ( x z ) ( x i ' Y i ' z i ) = ( - X i ^ Y ^ - Z i ) ...(m.G.3) If the total molecular wavefunction $ is represented by a case (a) basis function | TJASZJ2JM> with signed values of A, Z, and fl, and if | T?ASZ> represents the corresponding nonrotating molecule basis function, and if | J?A> represents the electronic orbital part of the wavefunction then it can be shown that the actions of these operators are as follows (45,22) E* 17?ASZflJM> = (-1 ) J ~ S + S|T } f - A , S , - I , - n , J , M > ...(III.G.4) c v ( x z ) |i?ASZ> = (-1 ) A + Z ~ S + S | T ? , - A , S , - Z > ...(III.G.5) < M x z ) U A > = ( - D A + S h , - A > ...(III.G.6) "Some authors use a different symbol: E* (Larsson (45)) = I (Hougen (46a)) = / (King (44b)) = (Zare et al. (47)) = P (Landau and Ufshitz (35b)). Note that the space-fixed inversion operator is different from the molecule-fixed inversion operator. 76 where (47) 1 for Z" states s = ...(III.G.7) 0 for Z + states and (by arbitrary choice) for all other states See Larsson (45) and Lefebvre-Brion and Field (22) for discussions about the phase choices implicit in these equations. The two operators are related as follows (45): E * | rjASZflJM> = a v ( x 2 ) | TJASZ>C 2 ( y ) |&JM> ...(III.G.8) where C-2(y) rotates the molecular frame around the y-axis by an angle of ir. The eigenvalues of these operators must be ±1, as follows immediately from the fact that two successive operations must return the original eigenfunction (E*) 2 | TjASZflJM> = p2|T?ASZOJM> = | T?ASZfiJM> p = ±1 ...(III.G.9) U v ( X 2 ) ) 2 |*JA> = p| | TJA> = | T?A> - ~ p e = ± 1 ...(HI.G.10) p (= + or - ; or "even" or "odd") is called the parity of the total wavefunction, and p e (= + or -) is called the symmetry of the electronic wavefunction, where we are considering only the spatial (molecule-fixed) coordinates of the electronic wavefunction (not spin). Corresponding to the two eigenvalues of E are two eigenfunctions, which can be distinguished by means of + and - signs denoting their parity, E*\l/* = +\{/* # ...(HI.G.11) E i / / " = - i / r and similarly for the <^v(xz) eigenfunctions, a v(xz)*e = ...(III.G.12) a v(xz)*e = 77 States with |A| ^  1, if completely isolated from all other states, are doubly degenerate because there are two values, ±|A|. As can be seen from eqns. (III.G.4)-(G.6), for |A| ^ 1 the case (a) basis functions, or parts thereof, are not eigenfunctions of the inversion or reflection operators, since they are themselves changed by the actions of the operators. However, we are free to take linear combinations of the basis functions, since by the definition of a complete basis set, any linear combination of basis functions must also be a member of the set. Hence we can choose as our basis set those linear combinations that are eigenfunctions of it E and of o r v ( x z ) so long as they are linearly independent As can easily be ic checked by the use of eqns. (III.G.4) or (G.6), these are, for E , |T7ASLJ2JMp> = -J=|j T?ASZnjM> ± 177,-A, S, ~Z, - f i , J ,M>J ...(III.G.13) and for o v ( x z ) , |7?Ape> = ^[l*?A> ± |T?,-A>] „ . ( i n . G . 1 4 ) where p = ± (-1 ) J _ S + S ...(IH.G.15) p e = ± ( - 1 ) A + S ...(ITI.G.16) The basis set formed by taking the linear combinations of case (a) basis functions as in eqn. (III.G.13) is called a parity basis, since the resulting basis functions have a definite parity. These parity basis functions correspond to the eigenfunctions called \p* and \p~ in eqn. (HI.G.11), i.e. 7?ASZ:njM+> = \JJ* ...(III.G.17) T?ASZGJM-> = The corresponding electronic wavefunctions likewise have definite symmetries as seen 78 from eqn. (III.G.14), where |TJA+> = ^ ; ...(III.G.18) | 7?A-> = <//" Note that a total wavefunction having positive (= even) parity, for example, does not necessarily belong with an electronic wavefunction having positive symmetry. In fact, each 0 substate of a given electronic state has two components corresponding to the symmetries + and - , and each of these has associated with it a set of rotational levels whose parities (even or odd) alternate with J. For a given value of J, the two symmetry components have opposite parities, where the parity of a rotational level is that of the corresponding total wavefunction. The symmetry components of an electronic state can be denoted in the state symbol by means of a superscript + or - , e.g. 5A* or 5A". 1 9 For Z states, A = 0, so there is no double degeneracy. Z states normally belong to Hund's case (b), for which E * 17?,A=0,SJN> = (-1 ) N + S 17},A=0,SJN> ...(HI.G.19) Thus the case (b) functions themselves form a parity basis without needing to take linear combinations. Since there is no degeneracy in A, a given Z state will be either a Z + state or a Z" state. If a Z state is represented by a case (a) basis, eqn. (III.G.13) still applies, with A = -A = 0. The hamiltonian is invariant under inversion (i.e. H commutes with E ); hence, according to group theory, nonzero matrix elements can only occur between rotational levels having the same parity, i.e. selection rule: + —* + - — - ...(III.G.20) "Herzberg gives a different definition of + and - on page 239 of his book (16). However, in two other places (pp. 217 and 237) he gives the same definition as used here. King (44c,b) also defines the + and - symmetry labels as used in this thesis. Some authors (e.g. Kovacs (48a) and Merer (49)) define + and - as p e = ± instead of as in eqn. (HI.G.16); this definition is not based on a v . 79 Another selection rule is (see Section III.J) AJ = 0 ...(III.G.21) Since + and - parity levels so not occur in pairs in Z states as they do in non-Z states, interaction of a non-Z state with a Z state or states results in a lifting of the degeneracy in the non-Z state — i.e. A-doubling occurs. Specifically for odd-multiplicity states, as in FeO, it is the 0 = 0 component of the Z state(s) that causes A-doubling, since the other R components occur as parity pairs. It is convenient to distinguish the two A-doubled members of each rotational level by means of a label other than the symmetry label ±, since this is too easily confused with the parity symbol ± with which each level is also labelled. Various conventions have been used in the past; the modern labelling scheme uses the letters e and f, defined as follows (50): levels with parity ± ( - l ) J ~ a are f levels ...(in.G.22) where •5 for even-multiplicity states a = ...(IH.G.23) 0 for odd-multiplicity states The definitions of e and f are such that for a given non-Z state all e levels have electronic wavefunctions with the same symmetry (+ or -) and all f levels have electronic wavefunctions with the same symmetry but opposite to that of the e levels (- or +)." When the hamiltonian matrix is written in a parity basis, it factorizes "Whether the e levels correspond to an electronic wavefunction symmetry of + or of - depends on the values of A and S. This can be determined from the simultaneous use of eqns. (IILG.13), (G.14), (G.15), (G.16), and (G.22). For example, for a 5A state, these equations yield 5 A + ~ e; 5 A ~ ~ f. 80 into two submatrices — one of symmetry e and one of symmetry f.21 The selection rules for interactions between levels become, in this nomenclature, e — e f — f ...(III.G.24) and, as before, AJ = 0 ...(III.G.25) In situations where it is not known which level of a parity-doublet pair is e and which f, the labels a and b are used for the levels lying lower and upper in energy, respectively (50). Since the 0 = 0 components of Z states are of particular interest (see above), the parities and e/f labels for these are given in Table III. For other S2 components and for even multiplicity Z states, eqns. (III.G.13) and (G.15) are used to determine the total parities just as for non-Z states. The matrix elements responsible for the interactions that lead to A-doubling will be discussed in Section III.H.2.a, where the specific case of A-doubling in a 5A state will be treated in terms of an effective hamiltonian. 2'Sometimes in the literature the word "parity" is used loosely to refer to the e/f labels, or even to the + / - symmetries, instead of just in its restrictive meaning used in this section — i.e. the behavior with respect to E of the total wavefunction. 81 TABLE HI. Total parity and e/f labels in the fl = 0 components of odd-multiplicity Z states.2 + t o t a l p a r i t y - t o t a l p a r i t y e l e v e l s f l e v e l s V . X. % e t c . even J o d d J a l l J -V , 3Z+, 5 I ~ , e t c . o d d J even J - a l l J Based on L e f e b v r e - B r i o n and F i e l d (22). 82 HI.H. Effective Hamiltonian and Matrix Elements During the analysis of a spectrum, a process of data reduction must of necessity occur. Starting with often as many as thousands of measurements of line frequencies, the object is to mathematically reduce these to only a certain few parameters ("constants"), without loss of information, and in the process to enhance understanding by emphasizing physical interpretations. If enough terms are included, experimental data can always, in principle, be fitted to a mathematical formula of some sort, but such a process is only worthwhile if the terms can be interpreted to convey physically useful information. Since the nuclei and electrons in a molecule can arrange themselves in essentially an infinite number of ways, a complete mathematical description of a molecule could, in principle, contain an infinite number of terms — obviously not a desirable nor physically enlightening approach to take. The key to making the situation more tractable is the separation into individual electronic, vibrational, and rotational sub-problems described in Sections III.C and D. While these sub-problems still involve infinite series (see eqns. (III.D.1) and (D.4)), in practice, since the terms in the series become successively smaller, only a few are actually needed. The problem is that the separations do involve approximations. In reality, interactions between the various motions do exist and for high resolution work cannot be completely neglected. The solution is to manipulate the hamiltonian in such a way that the separation into individual electronic, vibrational, and rotational parts becomes an acceptable approximation. This can be accomplished by creating an effective hamiltonian that operates only within the rotational sub-space of a given vibronic state. A number of methods of doing this have been developed — see, for example, Miller (51), Soliverez (52), and Brown and co-workers (53,54). These usually start by dividing the exact hamiltonian into two parts 83 H = H° + H' ...(III.H.1) in such a manner that the matrix representation of H° is completely diagonal in the basis set being used and that any off-diagonal matrix elements will come from H'. Some form of perturbation theory treatment is then used to bring the effects of H' onto the diagonal positions or into diagonal blocks. Ordinary nondegenerate perturbation theory is not appropriate for this purpose since it does not apply to close-lying energy levels such as the different spin-levels of a multiplet state. Degenerate perturbation theory can be used (51,52) but is very complicated in practice. An alternative method, derived initially by Van Vleck (55) and later described in more detail by several other authors (56,57,58,22), is simpler to use than degenerate perturbation theory. This method applies a form of nondegenerate perturbation theory to situations where there are close-lying levels (such as the spin levels of a multiplet state or several close-lying vibronic levels), and separates these levels from all distant-lying levels. Unlike ordinary nondegenerate perturbation theory, which only introduces energy correction terms into on-diagonal matrix positions, Van Vleck's method yields both on- and off-diagonal elements within the group of close-lying levels." 1T1.H.1. The Van Vleck transformation Following Lowdin (59), the infinite basis set is partitioned into two classes: class 1 includes the one or more (but a finite number of) basis functions whose associated energies lie in the energy region of interest; class 2 consists of the infinite number of remaining basis functions, which lie' at remote energies to the class 1 functions. "This will be seen in eqns. (III.H.18)-(H.22) below: there are matrix elements for a * b as well as for a = b. 84 The Van Vleck method consists of applying a specific unitary transformation, a type of contact transformation called a Van Vleck transformation, to the original exact hamiltonian matrix, H, to produce a new, still exact, matrix, H, in which class 1-class 2 elements are smaller by a factor of [ H 1 2 / ( E ^ - E§) ] 2 " than the corresponding elements in H. That is, TtHT = H ..(III.H.2) or, in terms of the submatrices, H 2 2 H 1 2 H 2 1 H(i VAN VLECK TRANSFORMATION H 2 2 H 1 2 H 2 1 H 11 ...(III.H.3) Since the matrix elements in H 1 2 (=H21) are smaller than the original H 1 2 (=H21) elements, it is a better approximation to neglect these elements than to neglect the original off-diagonal elements (as would be done in the Born-Oppenheimer approximation). Thus, to a good approximation, an effective hamiltonian matrix can be defined as ..(HI.H.4) that is, only the class 1 part of H is retained. Expressions will now be derived for the matrix elements of H n following Kemble (57), Wollrab (58), and Lefebvre-Brion and Field (22). We start by writing the matrix representation of eqn. (III.H.1) with a perturbation parameter X inserted. H = H° + XH' ...(III.H.5) where X can have values between 0 and 1, and the second term, XH', is regarded as a perturbation. The situation for X = 0 corresponds to the unperturbed problem (the "The symbols will be defined shortly. 85 class 1 states in isolation), and when X = 1, the corresponding hamiltonian is the exact hamiltonian for the complete system (class 1 + class 2). As used here, X is an order-sorting parameter. The matrix after transformation can be expanded as a power series in X, H = H < 0 ) + X H ( 1 ) + X 2 H < 2 ) + . . . ...(IH.H.6) It is the aim of the Van Vleck method to eliminate the First-order terms in X that connect classes 1 and 2. This is accomplished as follows. Let T be a unitary matrix, which can always be written in terms of a hermitian matrix J5 as follows, then expanded, T = e i X i = 1 + iXS - | 2 S 2 - - ^ 3 S 3 + . . . ...(ni.H.7) 2 o Performing the unitary transformation indicated by eqn. (III.H.2) H = H ( 0 ) + XH ( 1 > + X 2 H ( 2 > + . . . = (1 - iXS - | 2 S 2 + ...) (H° + XH' ) (1 + iXS - - | 2 S 2 - ...) ...(III.H.8) terms in equal powers of X can be equated, X°: H ( 0 ) = H° ...(in.H.9) X 1 : H < 1 ) = H' + i ( H ° S - SH°) ...(III.H.10) X 2 : H ( 2 > = i (H'S - SH') + SH°S - ^ ( H ° S 2 - S 2 H ° ) ...(IIJ..H.11) The functional form of the elements in S has so far not been specified, other than that it be hermitian. We are free to construct S so that T will transform H into the desired approximately block-diagonal form. Denoting the individual elements within each class by Roman letters (a,b) for class 1 and by Greek letters (a,/3) for 86 class 2, S is constrained so that no mixing can occur among the class 1 functions, sab " 0 ..(III.H.12) or among the class 2 functions, S a 0 = 0 ...(III.H.13) for all values of a and b (a * b) and of a and 0 (a * 0). In order to force the first order elements of H off-diagonal in class to vanish, it follows that, for any values of a and a, . _ i H a a *aa E° - E° a a ...(IH.H.14) where E a = <a|H°|a> = H a a E° = <a|H°|a> = H ° a ..(DLH.15) This can easily be checked by substituting eqn. (III.H.14) into the appropriate element of (HI.H.10), which we look at first: H (D aa = H a a + i[(H°S) a„ - ( S H ° ) a J — —'aa aa-= HAa + ^ " a a ^ a " S a a H a o ] ,(m.H.16) where use has been made of eqns. (ffl.H.12) and (H.13), together with the diagonal nature of H° (within the classes as well as between classes). Substitution of eqns. (III.H.14) and (H.15) gives H a a ~ H a a E a H a a H a a E a ' = H a a " H a a 87 - ° ...(IH.H.17) which is the desired result As a consequence, the lowest order class 1-class 2 interaction terms in H occur in the second-order term H < 2 ) . and these do not contribute to the energies until the fourth order (56). The Van Vleck transformation is completely defined by eqns. (III.H.9) - (H.14) and similar equations for higher orders. The class 1 block matrix elements H n are H ab = H (O) ab + H ab + H ab + H ab ..(III.H.18) where H (0) ab < a | H ° | b > = E ° 6 a b ..(ffl.H.19) H (i) ab 5& H (3) ab <a|H'|b> = H a b a L a,0 H aa H ab + HkaEab E o _ Eg -...(ffl.H.20) ,(m.H.2i) HaaHa/3H^b (E° - E°)(EJ - Eg)_ 2 ^ a, c H a c H c a H o b H a a H a c H c b ( E o _ E o ) ( E o _ E o ) (E° - E ° ) ( E ° - E ° ) ...(in.H.22) where 8^ is the Kronecker delta. Notice that the zeroth- and first-order terms for the (a,b) element are the same as the (a,b) element of the exact hamiltonian, i.e. Hajj. The higher-order terms fold into the class 1 submatrix the effects that were originally off-diagonal in class. The second-order term is often rewritten in terms of an averaged energy denominator 88 which differs from the version given in eqn. (III.H.21) by a negligible amount if |E° - Eg | « 3 2 b - E° ...(III.H.24) Furthermore, when several of the class 2 levels are close in energy, it is sometimes convenient to similarly use an average energy value E° for these levels. This will be done in writing eqn. (III.H.41) in the next section. In a similar manner to eqn. (III.H.24), this approximation will be good if the energy separation between the class 2 levels involved is much smaller than the class 1-class 2 separation. The summations over the class 2 levels in the above equations are for all spin substates of all vibronic levels of class 2, regardless of how near or far in energy they lie and regardless of symmetry species, including ionization and dissociation continua (which are integrated over rather than summed over). In summary, the Van Vleck transformation has eliminated the first-order off-diagonal terms connecting the class 1 levels with all distant states. The information has not been lost but has been "folded" into the class 1 block of H. The remaining class 1-class 2 off-diagonal elements, which are very small, are then neglected; only that part of the transformed hamiltonian acting within the class 1 block is retained, and this is called an effective hamiltonian. Note that the effective hamiltonian is really defined in terms of its matrix representation (eqn. (III.H.4)); thus its form is specific to the particular basis set being used. m.H.2. Construction of the effective hamiltonian and matrix elements in a case (a) basis Brown et al. (54) and Brown and Merer (60) support the concept of an effective hamiltonian involving a minimum number of experimental fitting parameters. Certain (small) interactions are neglected in the construction of the effective hamiltonian; their effects are then incorporated into the other parameters when a fit to 89 the experimental data is performed. Thus the physical interpretation of the parameters differs somewhat from what it would be if the parameters were the exact parameters. This is a disadvantage but is done in the interests of practicality. For the reasons stated at the end of Section III.F.3, a case (a) basis | TJASZJJ2> is chosen. We write the effective hamiltonian for the rotational levels of any given multiplet state as H E » . H ; " + H;J£* H|I*+ H ; j f * ^ where the terms are the vibronic, rotational, spin-orbit, spin-spin, spin-rotation, centrifugal distortion, and lambda-doubling terms, respectively. Zare et al. (47) have given expressions for most of the terms in the effective hamiltonian, and these were derived in a manner similar to that used here. Some of their signs are different from ours, however, due to a different phase choice." The vibronic hamiltonian is of little interest here and will be handled in a phenomenological manner by including T 0 terms in the diagonal positions of the energy matrix to represent the vibronic origins. The rotational hamiltonian is given by" H r = B(r )R 2 = B ( r ) ( J - L - S) • (J - L - S) ...(IH.H.26) where B < R > = Q—2^ 2 ...(IU.H.27) The constants have been defined previously (cf. eq. (III.D.2)) except that the "They chose e'^a ~ ^a±P = -1 in eqn. (III.F.9) for S 1, in contrast with our choice of +1 (eqn. (III.F.13)). "Some authors (e.g. Hougen (46) write H r = B(r)(R 2 +R 2 ) to explicitly show that R 2 = 0 in a diatomic molecule. However, if R | is included, as here, all contributions from it will cancel each other out anyway. Thus both versions of H r are completely equivalent. 90 internuclear separation is now represented by small r to avoid confusion with R. Eqn. (III.H.26) can be expanded to give H r = B ( r ) [ j 2 + S 2 + L | - 2 J Z L 2 - 2 J Z S Z + 2 L Z S Z ] + B ( r ) [ L 2 + L y ] - B ( r ) [ 2 ( J x S x + J y S y ) - 2 ( L x S x + L y S y ) + 2 ( J x L x + J y L y ) ] ...(III.H.28) The term on the second line of eqn. (III.H.28) possesses non-zero matrix elements that are diagonal in all quantum numbers associated with the case (a) basis, including A, but they can be on- or off-diagonal in electronic state. It can be seen how this arises by rewriting this term in ladder operator form (cf. eqn. (IH.F.12)) [ L 2 + L 2 ] = ^ [ L + L ~ + L " L + ] ...(III.H.29) The ladder operators ladder up and down in A, but the final state, although having the same value of A as the initial state, may be different. Those matrix elements diagonal in electronic state are often represented by the symbol BV<L2> (46c) B v<ASZnj |L 2 + L2|ASIflJ> = B v<ASZfiJ |L 2 - L2|ASZJ2J> = B y<L 2> ...(III.H.30) where the vibrational level has been integrated over (eqn. (III.D.3)). It can be shown that, to a good approximation, this term is constant for a given electronic state eff (46c,22). We therefore neglect it entirely from H r , which is equivalent to saying that we include it with the electronic energy (where it, incidentally, causes part of the electronic isotope shift (9)). Those elements off-diagonal in electronic state will presumably also contribute an approximately constant shift to a given electronic state; in any case, their effects will be small, so they are ignored also. All the terms in the first line of eqn. (III.H.28) are diagonal in all quantum numbers within a given vibronic state, and furthermore, there are no off-diagonal 91 elements between different electronic states, — e.g. for J 2 <n v ' A ' S ' Z ' O ' J ' M ' | B ( r ) J 2 | n v A S Z O J M > = B v , V J ( J+1)6 n ' n 6 A ' A 5 S ' S 6 Z , I 6 n , n 6 J ' J 5 M ' M ...(III.H.31) where B v , v = <v'|B(r)|v> = h ...(III.H.32) 8TT 2 These terms are therefore assigned to H ° for the purpose of performing a Van Vleck transformation. (For the moment, the existence of non-zero matrix elements of these terms between different vibrational levels of the same electronic state (cf. eqn. (III.H.32)) will be ignored.) The terms from the third line, which are off-diagonal in at least one quantum number, then form H ' H° = B ( r ) [ j 2 + S 2 + L 2 - 2 J Z L Z - 2 J 2 S 2 + 2 L Z S Z ] ...(IJI.H.33) <n ,v ,A ,S ,Z±1 ,fi±1 , J |H£. | nvASZJft> = - B V [ J ( J + D - n ( n ± i ) 3 1 / 2 [ s ( s + i ) - z ( z ± i ) ] V 2 ...(m.H.36) where the vibrational level has again been integrated over (eqn. (m.D.3)). The diagonal elements (eqn. (III.H.35)) have come from the terms in (III.H.33) by means of eqns. (III.F.5) and (F.8), while the off-diagonal elements (III.H.36) have arisen from the -2B(r)(J xS x + J y S y ) operator with the aid of eqns. (m.F.9), (F.10), (F.12), and (F.13). The latter operator, which can be rewritten as - 2 B ( r ) ( J x S x + J y S y ) = - B ( r ) ( J + Z " + J " Z + ) „.(in.H.37) H'r = - 2B(r) [ ( J X S X + J y S y ) - ( L X S X + L y S y ) + ( J X L X + J y L y ) ] ...(III.H.34) The matrix elements of H r for a given vibronic state are, through first order, < n v A S Z j n | H r ° |nvASZjn> = B y [J(J+1 ) + S(S+1) - SI2 - Z 2 ] ...(in.H.35) 92 mixes together the components within the multiplet state and is called the S- uncoupling operator; it is responsible for the transition from Hund's case (a) to case (b) as J increases. The remaining two terms in eqn. (IU.H.34) only produce non-zero matrix elements that are off-diagonal in A and these therefore do not contribute in first order. Second- and higher-order terms arising from matrix elements connecting the multiplet vibronic state of interest with other, distant vibronic states could be included eff in H r . However, we choose not to do so. Their effects will be included as, or absorbed into, other parts of the total effective hamiltonian. Second- and higher-order contributions from the terms in eqn. (III.H.33) and from the S-uncoupling operator in eff eqn. (III.H.34) provide a centrifugal distortion correction to H r . This will be included as part of H c j . The operator 2B(r)(LxSx + L y S y ) = B(rXL+Z- + L~L*) has the same form as part of the spin-orbit hamiltonian," namely A(r)(LxSx + L y S y ) (see below), so that only their sum, [A(r) + 2B(r)](LxSx •+• L y S y ) , will be determinable. Since for eff case (a) states A(r) is larger then B(r), the combined operators are included with Hgg, eff where they contribute in second and higher orders. They also contribute to H T jy The operator -2B(r)(JxLx + J y L y ) = ~B(r)(J+L~ + J~L +) is known as the L-uncoupling operator, and it is responsible for a gradual transition from case (a) to case (d) as J increases. It possesses non-zero matrix elements between electronic states eff differing by one unit in A and will therefore be assigned to H T jy In the foregoing discussion, it has actually been the effective rotational matrix elements that we have been deriving (cf. eqn. (III.H.4)). The effective rotational hamiltonian then follows: it is that hamiltonian which when operating solely within "There is a slight difference if one considers the microscopic forms of the operators as is done in calculating the higher-order spin-orbit terms (22) but this is ignored here since A(r)»B(r) for FeO and since the difference will be absorbed by the effective nature of the parameters. 93 the vibronic state of interest produces matrix elements that are the same as the effective matrix elements. The effective rotational hamiltonian is H ^ f f = B v [ J 2 + S 2 + L 2 - 2 J Z L Z - 2 J Z S Z + 2 L Z S Z - (J + S" + J*S + ) ] ...(III.H.38) for a particular vibrational level of the multiplet state under consideration. Sometimes this is written as H ® f f = B V R 2 = B y ( J - L - S ) 2 ...(III.H.39) but then one must remember that certain parts of this are to be thrown away, as discussed above. We next consider that part of the centrifugal distortion hamiltonian HC(j arising eff from H r , and use it to illustrate how second- and higheT-order contributions to H are handled. As a result of the r-dependence of B(r), B v , v * 0 ...(JJI.H.40) (cf. eqn. (ni.H.32)). Thus all the terms in H r , except those off-diagonal in A (i.e. except the last two terms in eqn. (ffl.H.34)), cause the vibrational levels within a given electronic state to couple together. If the vibrational spacings are large compared with the splittings between the multiplet components, then the matrix elements off-diagonal in v can be treated by the Van Vleck transformation method. They give rise to second- and higher-order contributions to the matrix elements acting within the vibrational level of interest We must now repartition H r such that all those terms that previously formed H° (eqn. (III.H.33)) are now included with H£., leaving H° = 0." From eqn. (UI.H.23) "Actually H£ and H .^ could have been defined this way in the first place instead of as in eqns. (ffl.H.33) and (34), and the same results would have been obtained. It is conventionally done as here, however, to emphasize the usefulness of Hund's coupling cases. 94 Hi 2^ = I <nvASZafiaJ | H^ . | nv' ASZ' fi' jxnv' ASZ' fl' J | H^ . | nvASZbObJ> v'*v E n v ~ Env' 'Z' ...(III.H.41) where E ^ is the average value of the energies of all the multiplet components of vibrational level v, and E^t has the corresponding meaning for level v'. Using these average energy values is a good approximation provided that E ^ - E ^ . is large compared with the multiplet separations. As per Section III.D, eqn. (III.H.41) can be separated into a product of radial (i.e. vibrational) and angular (i.e. rotational) parts, <nv|B(r) |nv'xnv' |B(r) |nv> £J(2) _ H a b " Z y'*v Env ~ Env' . I" Z ( <ASZ afl aJ | R 21 ASZ' fl1JXASZ * fl' J |R 2 | ASZbflbJ> 1 ...(III.H.42) where we have written R 2 for convenience, but it must be remembered that certain parts are to be neglected. Following Zare et al. (47) (see also Albritton et al. (61)) we let the first factor in this equation define - D v , the centrifugal distortion constant for the v m vibrational level," <nv IB (r) I nv' x n v ' I B (r) I nv> D v = - Z ! L ! ...(III.H.43) W v E nv E nv ' so that" " D could alternatively be called B Q by analogy with Ajy, \jy, etc. D is more usual. eff "When working out the matrix elements of H c ( ^ r , the angular part of eqn. (III.H.42) or (H.44), i.e. Z ^ < Z a « a | R 2 | Z' £2* ><Z' fl' | R 2 | Z bfi f a>, is conveniently calculated by writing down the matrix of H r for any vibrational level but with the common factor B Y omitted, then squaring it This automatically carries out the summation Z . It works because, as a result of the factorization into 0' Z' radial and angular parts, the angular parts are the same for all values of v', including v' = v. The summation over v' in the foregoing equations, in principle, really includes integration over the vibrational continuum. However, as the | v ' > states differ progressively in quantum number from the |v> state, the increasing difference in number of nodes in the two states results in a rapid decrease in magnitude of the nondiagonal matrix elements < v|B ( r ) | v ' > , so that, except for states near the dissociation limit, the contribution from the continuum is seldom needed (61). 95 "ab = - D v < A S 2 a R a J l £ a | A S Z b R b J > ...(III.H.44) Higher-order centrifugal distortion terms can be included as needed, though they seldom are. These are worked out in a similar manner to the second-order term (61). The effective centifugal distortion hamiltonian for rotation is then H c " r " - D v S " + H v S 6 + • • • ...(III.H.45) The microscopic forms of the spin-orbit, spin-rotation, and spin-spin hamiltonians have been given earlier in eqns. (III.E17) and (E18); (E25) and (E26); and (E44); respectively. However, these forms are not practical for the purpose of data reduction. It is traditional to replace them by phenomenological expressions involving a limited number of adjustable parameters: H S Q = A(r )L-S ...(IILH.46) H s r = 7(r)R-S or H s r = 7<r)N-S ....(III.H.47) H s s = § X ( r ) ( 3 S 2 - S 2 ) ...(III.H.48) where L = E 1_ ...(III.H.49) e e and S = I s e ...(in.H.50) ~ e Two forms of the spin-rotation hamiltonian have been given in eqn. (III.H.47). Both forms are in common use in the literature. For case (b) states, the form in N is perhaps more convenient (47,22), but for case (a) states the choice is arbitrary. Since N = R + L, the two forms differ by the amount 7(r)L«S, which is of the same form as the spin-orbit hamiltonian, A(r)L«S. Therefore the choice of form used for H s r affects the value of A, but since usually A » 7, this difference can 96 normally be ignored (except for very light molecules like H 2 (62)). The 7(r)R«S version has been used in this thesis. By means of the Van Vleck transformation method, the effective spin-orbit hamiltonian is found to have the following diagonal contributions: eff(°> = 0 ...(III.H.51a) e f f (i) H s o = A V L 2 S 2 „.(III.H.51b) H l o f ( 2 > = f V 3 S z - S 2 ) ...(III.H.51C) Hso f ( 3 > = r?vLzSz[SI " i ( 3 § 2 " 1 ) 1 ...(III.H.51d) HIof(fl> = r s M 3 5 S z ~ 3 0 § 2 S | + 25S| - 6S 2 + 3 S « ] ...(III.H.51e) As discussed earlier, these terms contain small contributions from the 2B(r)(LxSx + LySy) part of H r . There are also even smaller contributions from -7(r)(L xS x + LySy) if the R«S version of Hgj. is used — see below. It may seem excessive to include terms as high as fourth order in pertubation theory. However, for quintet states, with five substates and four spin-orbit intervals, it is necessary, in principle, to use five parameters to give their positions. These can be written either as five term origins TJJ, as has been done by Cheung, Zyrnicki, and Merer (63), or as one vibronic term origin T 0 plus four parameters characterizing the four spin-orbit intervals. The latter method has been used here; hence the necessity of going to fourth order. The spin-rotation hamiltonian expands out as H s r = 7 ( r ) ( J - L - S)-S = 7 ( r ) [ J 2 S 2 - L 2 S 2 • S 2 + ( J X S X + JySy) - ( L X S X + LySy) ] ...(m.H.52) from which we get 97 e f f < ° > Hg£ L = 0 „.(III.H.53a) H s r f < 1 > = ">V[jzsz " L z s z " £ 2 + i(J + S ~ + J"S + ) ] ...(HI.H.53b) Higher-order contributions are usually not needed, since 7 is usually small in comparison with A. In any case, higher-order contributions from -7(r)(L xS x + LyS v ) can be assumed to be included in with the corresponding eff higher-order terms of H S Q . Comparison of eqns. (III.H.48) and (H.51c) will show that the (first-order) spin-spin hamiltonian has exactly the same form as the second-order spin-orbit hamiltonian. The two are 100% correlated and are experimentally indistinguishable. The parameter X v therefore contains contributions from both interactions. Since the second-order spin-orbit contribution is usually the larger, we set H | g f = 0 ...(III.H.54) eff and let the spin-spin interaction be included with H^. In states of high multiplicity, the term in X has the effect of making the spin-orbit intervals unequal — an effect which may be considered as the early stages of a transition to case (c) coupling. Until now, when calculating second- and higher-order contributions to the effective hamiltonian, we have considered each part of the hamiltonian in isolation from all other parts — e.g. H r by itself or H g 0 by itself. Really, the total hamiltonian must be employed when calculating higher-order terms, and if this is done, cross-terms between the various parts of the hamiltonian occur. We illustrate this by considering two parts of the hamiltonian together, namely H = H r + H s o ...(III.H.55) and writing, in shorthand form, the second-order correction, 98 <Hr + H _ n x H r + H_n> g(2) _ £ r so r so AE« <H r> 2 + < H , . x H r > + < H r x H c - > + <HC_>2 = I — — - — A g - ^ ^ — ...(III.H.56) The first and last terms in the final line are the second-order centrifugal distortion correction to H r and the second-order spin-orbit correction, both of which have already been considered. The middle two cross-terms are new, however, and these <Hr><Hr> together, by analogy with Z — , are called the centrifugal distortion correction A E to H s 0 . The two terms are not identical, because H s o and H r do not commute; hence both terms have to be retained. The r-dependent part can be taken out as a common factor and is given the symbol A D v „ „ <nv|A(r) I n v ' x n v ' |B(r) |nv> A D = 2 Z ! L _—! ! ...(III.H.57) v'*v E nv " E n v ' eff The factor 2 has been inserted in order to write H c ( i s o in the form of an average, e f f (L-S)R 2 + R 2 ( L - S ) H c d , s o = A D , v 2 ~ ~ ~ ...(IH.H.58) where the parts off-diagonal in A are to be neglected. (From an alternative but eff equivalent point of view, the form of H c ( j s o as an average is necessary to ensure that the energy matrix is hermitian (63).) In the same manner we obtain eff (R-S)R 2 + R 2 (R-S) H c d , s r = 7 D , V 2 ~ ~ ~ ...(IU.H.59) e f f , (3S 2 - S 2 ) R 2 + R 2 ( 3 S 2 - S 2 ) H c d , s s = !^D,v ~ ~ ~ 2 ~ — -.(III.H.60) There are also other cross-terms, such as that between and H s r but these are ignored; their effects will be absorbed into the various effective parameters. One such cross-term that should be specifically mentioned is that arising from (H^ + H r). eff When calculating H^gQ, only those parts of and H r diagonal in A were used. 99 If the off-diagonal parts are now considered, we find that the second-order terms arising from the product of matrix elements involving the J + L~ and L +S~ operators are of the same form as those from 7J+S~. In fact, except for H 2 , these second-order effects are larger than the first-order H s r contribution (22). However, since they are experimentally indistinguishable from the true spin-rotation contribution, we retain the name "spin-rotation" coupling constant for 7V. In the same manner in which the vibrational dependences of B v and D v were expressed in the form of a power series (eqns. (III.D.4) and (D.6)), so also can this be done for the other parameters. For example, for Ay (25) A v = A e - a A (v+i) + . . . ...(III.H.61) where a A is a vibration-rotation interaction constant Tables will be presented summarizing the effective hamiltonian and giving the matrix elements for a number of states at the end of the following section dealing With Hjjy HI.H.2.a. Lambda-doubling in a 5A state Lambda-doubling arises because of interactions with Z states. There are only two types of interaction terms that have arisen in the previous section that can connect different electronic states, namely terms involving the operators (i) L + S" + L~S + and (ii) L + J " + L " J + . The first type arises primarily from Hgg, with a small contribution from H r and smaller still contributions from elsewhere.30 The second type arises primarily from H r . These both involve the ladder operators L 7 which ladder up or down in A by only one unit at a time. Therefore, in order for A-doubling to occur in a A state, the A state must interact with the Z state(s) via an intermediate II state(s). This 3 "Again, small differences at the microscopic level are being ignored. 100 means that the lowest order of perturbation theory in which the interaction between a A and a Z state can occur is fourth order. That part of the expression for the fourth-order perturbation term of relevance has the form ~ ( „ = ^ < 5 A 1 H S O + H R |5 n><5 n| H S O + H R | 5 z x 5 z | H S O + H R l 5 n x 5 n ) H S O + H R | 5A > — energy denominator ...(III.H.62) where the operators are represented by their primary sources in this simplified equation and where the specific substates within the various states have not been shown. The effect of the ladder operators can be represented schematically as We could go from A = -2 — - - 1 -•• 0 -1 -2, for example, but because of the symmetry of the energy matrix, this will produce a matrix element equal to that from A = -2 - ^ - 1 0 — + 1 +2, so we need only consider the latter type. The IrS* and LrJ* operators can occur in various combinations within this scheme. For example, four applications of the l^S* operator alone, i.e. = < 5 A | H S O | 5 n > < 5 n | H s o l * Z > < 5 Z | H s o 1 5 n > < 5 n [ H S O | 5 A > energy denominator produces a term in the effective energy matrix for the 5 A state that is J-independent The four applications of the part of this operator produces a selection rule A Z = ± 4 and this can only occur in a 5 A state in the $2 = 0 part of the matrix — e.g. the (A=-2, Z = 2 , O=0; A=+2, Z = - 2 , $2=0) element Other combinations of the LrS* and L * ^ operators produce J-dependent terms in other locations in the matrix. 101 The effect of this treatment is to sum over the interactions involving all distant Z states (Z + states and Z~ states) and thereby to produce effective matrix elements lying only within the rotational manifold of the 5 A state. The resulting matrix is, however, for each value of J, a 10x10 matrix, because there are elements off-diagonal in A in the region connecting the +|A| and - |A| blocks. This is then subjected to a Wang transformation, H e f f _ TT-1 H e f f n —sym —sym— Hsym where, for a 10x10 matrix, the Wang symmetrizing matrix is ...(III.H.63) U = U~ 1 = - i -Hsym Hsym ^ -1 1 -1 1 -1 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 ...(JII.H.64) with zeros in all other positions. The effect of applying a Wang transformation is to factorize the 10x10 matrix into two 5x5 blocks — one block having symmetry e, the other f. Thus the Wang transformation produces that matrix which would have resulted if the matrix had been written directly in a parity basis. The advantage of originally deriving the matrix in a non-parity basis (sometimes called a signed basis) is that interactions with all (distant) Z states can easily be summed over, without regard to symmetry. It can readily be shown that for all the effective matrix elements derived in the previous section (47) < n v A S Z ' J J 2 ' ± | H e f f |nvASZJS2±> = <nvASZ' Jft' | H e f f |nvASZJ«> ..(HI.H.65) 102 so the matrix elements derived previously still apply. The effect of the Wang transformation has been to fold into the two blocks effective terms dealing with the A-doubling. The two blocks differ only in the signs of the A-doubling parameters. The number of parameters needed to adequately describe the A-doubling depends on the multiplicity of the state involved. For a 'II state, one parameter is needed. This has traditionally been called q. For a 2II state, two are needed: q and p (64). For triplet and higher-multiplicity II states, three A-doubling parameters are needed (60): q, p, and o.31 A subscript II can be appended to each symbol to distinguish from the parameters for A states. In A states, again one parameter is needed for singlet states, two for doublets, and so on. This time a maximum of five parameters are required, so a 5 A state is the lowest-multiplicity state in which all five make their appearance. Due to the scarcity of observations on A-doubling in A states, a standard nomenclature has not yet been established. Rather than break from the tradition established with II states, we adopt the following names (65). The one parameter in a 1 A state is called q^, the two in a 2 A state are called q^ and p^, and so on. For quintet and higher-multiplicity A states, the five parameters are called q^, p ,^ oA, nA , and mA. A A A The A-doubling parameters in n states have historically been defined in a case (b) basis. Likewise we define the above A-state parameters in a case (b) basis. In case (a), only a single parameter is required for any given matrix element, but these parameters appear as linear combinations of the case (b) parameters. With a string of up to five parameters defining each single constant, the nomenclature becomes rather clumsy. We therefore introduce the following new symbols for use in A states in case (a) (65): "Mulliken and Christy's (64) "o" has a different definition from that of o used here. Their o is not a A-doubling parameter. See Brown and Merer (60). 103 case (a) symbol corresponding combinations of case (b) symbols «A = q A o A = oA + 3pA + 6qA n A = n A + 2oA + 3pA + 4qA ffiA = m A + n A + oA + p A + q A We then suggest a similar convention be adopted for n states in case (a): case (a) symbol corresponding combinations of case (b) symbols «n s in ? n = P n