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Photoelectron spectroscopy of some polyatomic molecules Chau, Foo-Tim 1975

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PHOTOELECTRON SPECTROSCOPY OF SOME POLYATOMIC MOLECULES by FOO-TIM CHAU B.Sc, The Chinese University of Hong Kong, 1970 M.Sc, The Chinese University of Hong Kong, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of CHEMISTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1975 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f QJXJIAWS ~>f y' lj T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e Oct , /">/.( ABSTRACT The photoelectron spectrum of a molecule displays the kinetic energy distribution of the ejected photoelectron by monochromatic radiation. From the fine structure on a photoelectron band, valuable information about the bonding properties of both the molecule and its cation, and the ionic geometry can be obtained. The work described in this thesis falls into two main parts. The first is concerned with the quantitative application of the Franck-Condon principle to group VI hydrides, nitrous oxide and dihaloethylenes, by which the geometries of their molecular ions were obtained. An iterative method was devised to facilitate the computational procedure. The second part of this thesis contains the results of photoelectron spectroscopic studies on several halogenated molecules, viz. fluorotribromo-methane, fluorotrichloromethane, 1,2-dichloro-, 1,2 dibromo-, and 1,2 diiodo-ethanes and their perfluoro derivatives, 1,2 bromochloroethane, 1,2-dibromo-1,1-difluoroethane, cis and trans 1,2 difluoroethylenes, and 1,2 dibromo-cyclohexane. One electron models including spin orbital coupling, and through bond and through space interactions, are applied to most of these molecules as well as the dichloro-, dibromo-, and diiodoethylenes. The NMR chemical shifts, nuclear quadrupole coupling constants, electronegativity of the halo-gen atom and the force constants of the molecules studied are discussed in the light of the calculated molecular orbital parameters. - i i -TABLE OF CONTENTS Page CHAPTER I: INTRODUCTION. 1 CHAPTER II: THEORETICAL ASPECTS OF PHOTOELECTRON SPECTRO-SCOPY 5 2.1 Interpretation of Photoelectron Spectra 5 2.1.1 General Comments 5 2.1.2 Koopmans1 Theorem 7 2.1.3 Franck-Condon Principle 9 2.1.4 Through Bond and Through Space Inter-' actions. 12 2.1.5 Spin Orbit Coupling 14 2.1.6 Jahn-Teller Splitting 16 2.2 Quantitative Application of Franck-Condon Principle in Photoelectron Spectroscopy 17 2.2.1 Method of Franck-Condon Factor Cal-culation 17 (a) General Description of the Method 17 (b) Method of Calculation 23 2.2.2 Iterative Method in Franck-Condon Factor Calculation 26 2.2.3 Normal Coordinate Analysis 32 (a) General Principles 32 (b) Molecular Force Field 34 (c) Force Constant Calculation 36 2.2.4 Change in Electronic Transition Moment 39 2.3 Experimental 41 - i i i -Page CHAPTER III: MOLECULAR CONSTANTS IN THE IONIC STATES OF SOME POLYATOMIC MOLECULES 46 3.1 Group VI Hydrides 46 3.1.1 Introduction 46 3.1.2 Gs^  and F_s matrices 47 3.1.3 Results and Discussion 51 (a) 2B 1 state of H 20 + and D20+ 51 2 (b) Bi state of the Molecular Ions, H2S, D2S, H2Se and H2Te 57 (c) 2Ax state of H20 and D20 58 3.2 Nitrous Oxide 59 3.2.1 Introduction 59 3.2.2 Results and Discussion 60 2 (a) Geometry of N2O in the X Tf and A- a £ * States 60 (b) Variation in Electronic Transition Moment in the X 'Z+ — » X a7T Transition 64 (c) Force Constants in the Molecule and the Ions of N20 64 3 . 3 Dihaloethylenes 65 3.3.1 Introduction 65 3.3.2 Method of Calculation 66 3.3.3 Results and Discussion 71 (a) Modified Urey-Bradley Force Constants of Dihaloethylenes 71 (b) Geometries of the Dihaloethylenes Molecular Ions 75 (c) Origin of the Geometrical Change on Ionization 77 - iv -Page CHAPTER IV: PHOTOELECTRON SPECTROSCOPY OF SOME HALOGENATED COMPOUNDS.. 81 4.1 Fluorotrifluoromethane and Fluorotribromo-methane 81 4.1.1 Introduction 81 4.1.2 Interpretation of the Spectra 82 4.1.3 Discussion 90 4.2 1,2 Dichloro-, 1,2 Dibromo- and 1,2 Diiodoethane 92 4.2.1 Introduction 92 4.2.2 Results and Discussion 99 (a) One Electron Model for the Lone Pair Orbitals of Trans 1,2-Dihaloethane ( C 2 h ) 99 (b) Relative Stability of Isomeric Trans and Gauche Ions 104 (c) Geometry of the C1CH CH-C1 Molecular Ions ....... 108 (d) Relation Between Observed Ionization Potentials and Some Physical Properties of XCH2CH2X 109 4.3 1,2 Dichloro-, 1,2 Dibromo- and 1,2 Diiodo-tetrafluoroethane, l,2-Dibromo-l,l-difluoro-ethane and 1,2 Bromochloroethane 120 4.3.1 Introduction 120 4.3.2 Method of Calculation 120 (a) One Electron Model for Trans 1,2 Dihalotetrafluoroethane (C z i x ) 120 (b) One Electron Model for Trans CH2BrCH2Cl and Trans CF2BrCH2Br (Cs) 126 4.3.3 Results and Discussion 131 (a) Interpretation of Spectra 131 (b) Orbital Energy of Gauche CF 2ICF 2I(C 2). 134 - V -Page 4.4 Gem and Cis and Trans 1,2 Dihaloethylenes 138 4.4.1 Introduction.... 138 4.4.2 Method of Calculation 140 (a) One Electron Model for the Dihalo-ethylenes 140 (b) One Electron Model for the Vinyl Halides 147 4.4.3 Results and Discussion 149 (a) Correlation between Huckel's Parameters of Dihaloethylenes and the Chemical Shift in Carbon-13 and Proton NMR 149 (b) Correlation between Huckel's Parameters of Dihaloethylenes and Electronegativity of Halogen 154 4.5 Cis and Trans 1,2 difluoroethylene 157 4.5.1 Interpretation of the Photoelectron Spectra of Cis and Trans 1,2 Difluoro-ethylene 157 4.5.2 One Electron Model for the Cis and Trans 1,2 Difluoroethylenes 165 4.6 1,2 Dibromocyclohexane 168 4.6.1 Introduction 168 4.6.2 Results and Discussion 170 (a) Interaction between Lone Pair Orbitals of Bromine Atoms in Diaxial 1,2 Dibromo-cyclohexane (C 2) 170 (b) Discussion on the Stability of Trans and Cis 1,2 Dibromocyclohexane 174 CHAPTER V: CONCLUSION 176 REFERENCES 181 APPENDIX I 196 APPENDIX II 208 APPENDIX III 213 - vi -LIST OF FIGURES Figure Page 1. Correlation between the Franck-Condon principle and the shape of PE bands for the removal of electrons from molecular orbitals of different bonding character 11 2. Plot of the weighted sum of squares deviation against the number of cycles for the 'I^—» A'TTu. transition of CS2 during the iterative process ( n w = -2, c = - 5.0 and dt = 0.5 x 10" 2 0 gm 1 / z cm) 30 3. Light source and 180° hemispherical analyzer unit 42 4. Scheme of the PE spectrometer 43 5. Internal coordinates of substituted ethylene 67 6. The PE spectrum of fluorotribromomethane 84 7. The second PE band of chloroform 87 8. The PE spectrum of f luorotrichloromethane 89 9. Correlation diagram for the first six highest occupied orbitals of CHX3 and CFX3 (X = Cl, Br). 2 1 g The data for CHCI3 and CHBr3 are from Dixon et al and Potts et a l 3 0 91 10. Plot of vertical IP's of (a) the ICL2 orbitals of CFX3 0 and CHX 3 3 0> 1 2 8, (b) the IOL2 orbitals of O P X 3 9 7 ' 2 2 3 and the orbitals of B X 3 3 0 and PX3 3 with X = F, CI, Br or I, against the Pauling electro-negativity of the halogen atom 93 11. The PE spectrum of dichloroethane 95 12. The PE spectrum of dibromoethane 96 13. The PE spectrum of diiodoethane 97 14. Effect of interaction on the molecular orbitals of XCH2CH2X (a) no perturbation; (b) through space interaction; (c) through bond interaction added; and (d) spin orbit coupling added 101 15. The lone pair orbitals of the halogen atoms in XCH2CH2X... 102 v i i -Figure - Page 16. Effect of stability of the trans and gauche con-formers and their ions on the relative magnitude of IP t and IPgi Case (a) where the gauche ion is more stable, and case (b), where the trans ion is more stable 107 17. Plot of the Pauling electronegativity of the halogen atom against (a) the first halogen lone pair, (b) the first sigma ionization potential, and (c) the second sigma ionization potentials of X C r ^ C r ^ X : A different scale for the IP's is used for different orbitals 110 18. Plot of the heat of formation ( AHj ) against (a) the first halogen lone pair, (b) the first sigma, and (c) the second sigma ionization potentials of XCH2CH2X. A different scale for IP's is used for the different orbitals 112 19. Plot of the C-X stretching force constant K c x against (a) the first sigma ionization potentials, and (b) the second sigma ionization potentials of XCH2CH2X. A dif-ferent scale for IP's is used for different orbitals.. 113 20. Plot of log Ycx against log IP of (a) the highest occupied and (b) the next highest occupied sigma orbitals of XCH2CH2X. (The values for r c x are ob-tained from ref. 150 ) 114 21. Plot some ionization potentials of (a) X2, (b) HX, (c) BrX (X = F, Cl, Br, I), (d) CY2, (e) OCY, (f) SCY and (g) H2Y (Y = 0, S, Se, Te) against the stretching force constants 116 22. Plot of some ionization potentials of (a) WH3 (W = N, P, As), (b) BX3, (c) CH3X, (d) C X 4 (X = F, Cl, Br, I), (e) YF6 (Y = S, Se, Te) and (f) HCCX against the stretching force constants 117 23. Plot of the logarithm of some ionization potentials of (a) X , (b) HX, (c) BrX (X = F, Cl, Br, I), (d) CY2, (e) OCY, (f) SCY and (g) H2Y (Y = 0, S, Se, Te) against the logarithm of bond lengths 118 24. Plot of the logarithm of some ionization potentials of (a) WH3 (W = N, P, As), (b) BX3, (c) CH3X, (d) CX4 (X = F, Cl, Br, I), (e) YF (Y = S, Se, Te), and (f) HCCX against the logarithm of bond lengths 119 25. The PE spectra of (a) 1,2 dichloro-, (b) 1,2 dibromo-, and (c) 1,2 diiodotetrafluoroethane 121 - v i i i -Figure Page 26. The PE spectra of (a) 1,2 dibromo-1,1-difluoroethane and (b) 1,2 bromochloroethane 122 27. Qualitative MO diagram of trans (CF 2X) 2, (a) no perturbation, (b) through space interaction, (c) through bond interaction added, and (d) spin orbit coupling added 124 28. Qualitative MO diagram of trans Cr^BrCr^Cl, ( a) no perturbation, (b) through space interaction, (c) through bond interaction added, and (d) spin orbit coupling added 127 29. Molecular orbitals of iodine atoms in the gauche form of CF2ICF21 135 30. Qualitative MO diagram of cis 1,2 and gem dihalo-ethylene (a) no perturbation, (b) through space interaction, (c) conjugative effect added, (d) through bond interaction added, and (e) spin orbit coupling added 141 31. Qualitative MO diagram of vinyl halide (a) no per-turbation, (b) conjugative effect, (c) through bond interaction added, and (d) spin orbit coupling added 148 32. The PE spectra of (a) cis and (b) trans 1,2 difluoro-ethylene 158 33. The first PE spectra of cis 1,2 difluoroethylene (a) the first band, and (b) the second and the third band 159 34. The PE spectra of trans 1,2 difluoroethylene (a) the first band, (b) the second and the third band, and (c) the fourth band 160 35. Qualitative MO diagram of cis 1,2 difluoroethylene (a) no perturbation, (b) through space interaction, (c) conjugative effect added, and (d) through bond interaction added 163 36. The PE spectrum of trans-1,2-dibromocyclohexane 169 37. Effects of interaction on molecular orbitals of trans-1,2 dibromocyclohexane (a) no perturbation, (b) through space interaction, (c) through bond interaction added, and (d) spin orbit coupling added 172 - ix -LIST OF TABLES Table Page 1. Calculated and observed overlap integral (010) for the aA, state of CFBr and the A *T\a state of CS2 31 2. Structural parameters of Group VI hydrides and their molecular ions in various states 48 3. The observed frequencies (cm~l) of H2O, H^ O*, D20 and D20 + in various states 49 4. Observed frequencies (cm"l) of H2S, D2S, H2Se and H2Te in the ground state and the 'B, states of the molecular ions 50 5. The Gs and Fs_ matrix elements of a bent symmetric triatomic molecule XY2 52 6. Valence force constants (mdyn/^ ) of Group VI hydrides and their molecular ions in various states 54 7. Distortion rotational constants (cm-*) and mean square amplitudes of vibrations (A 2) at 298° K of H20, D20, H2S and D2S and their molecular ions in various states 56 8. Calculated force constants (mdyn / 4 ), bond lengths ( A ) , observed vibrational frequencies (cm-l) and structural parameters of N2O in the X 'Z+ , and molecular ions in the X JIT and A 121t states 61 9. Calculated and observed Franck-Condon factors in the X *T7 and A *Z* states of N 20 + 62 10. Geometrical changes in dihaloethylenes upon ionization 72 O 11. Urey-Bradley force constants (mdy^/A ) of dihalo-ethylenes . 74 12. Predicted changes in geometry of dihaloethylenes upon ionization.. 76 13. Observed vertical ionization potentials in the photo-electron spectra of fluorotribromomethane and fluoro-trichloromethane 83 14. Franck-Condon factors of CFBr^ in the *A, ionic state 86 - X -Table Page 15. Observed and calculated vertical IP's (ev) of 1,2 dihaloethanes 98 16. Calculated MO parameters (ev) of (CH2X)2 105 17. Experimental IP's (ev) of (CF2C1)2, (CF Br) (CF 2I) 2, CF2BrCH2Br and CH BrCH2Cl 123 18. Calculated MO parameters (ev) of trans (CF 2X) 2 127 19. Calculated MO parameters (ev) of trans CF2BrCH2Br, - trans CH2BrCH2Cl and gauche (CF 2I) 2 130 20. Urey-Bradley force constants ( n \ d y r \ / / \ ) of trans and gauche (CH2C1)2 and (CH2Br)2 139 21. Calculated MO parameters (ev) of cis 1,2 dihalo-ethylenes C2H2X2 144 22. Calculated MO parameters (ev) of trans 1,2 dihalo-ethylenes C2H2X2 145 23. Calculated MO parameters (ev) of gem dihaloethy-lenes C2H2X2 146 24. Calculated MO parameters (ev) of vinyl halides C2H3X with £ = € x/2 • 150 25. Carbon NMR chemical shifts (ppm) Sc , bond angles and nuclear quadrupole coupling constants G*0.^  (Mc/s) of halogenated ethylenes 151 26. Ionization potentials and symmetric vibrational frequencies of the molecular ions of cis and trans 1,2 difluoroethylene 161 27. Observed and calculated vertical IP's (ev) of trans 1,2 dibromocyclohexane. 171 ACKNOWLEDGEMENTS I wish to express my gratitude to Professor CA. McDowell for his invaluable help, encouragement and guidance throughout the course of this work. I would also like to thank Professor D. C. Frost for his support and continuous interest in this work. I also wish to express my appreciation to Dr. C. E. Brion, Dr. R. J. Boyd, Dr. J. C. Bunzli, Dr. M. Chiang, Dr. D. P. Chong, Dr. G. Pouzard, Dr. S. T. Lee, Prof. W. C Lin, Dr. A. J. Merer, Dr. L. Weiler and Dr. N. P. C. Westwood for helpful discussions. Thanks are due to Drs.D. Solgadi and Y. Gounelle for providing me with the observed IP's of trans 1,2 dibromocyclohexane, Dr. K. Wittel for the Franck-Condon factors of the dihaloethylenes, and Dr. J. H. Calloman for a preprint of his work. I take this opportunity to acknowledge the skilful technical as-sistance of the staff of the Electronic, Glass and Mechanical Workshops of the Chemistry Department at UBC. I am especially grateful to Mr. E. Matter and Mr. C. McCafferty for maintenance of the photoelectron spectro-meter used in this. work. I am indebted to Dr. N. P. C Westwood for a careful reading of this manuscript. I am most grateful to my fiancee for her love and encouragement, and finally to my God of love for giving me this opportunity to study. - 1 -/ / CHAPTER I INTRODUCTION Irradiation of a molecule by monochromatic light with sufficiently high energy may produce ionization amongst other processes by the following event Mol + hi/ = Mol+ + e" (1.1) Measurement of the kinetic energy distribution of the ejected photoelectron yields the photoelectron spectrum of that molecule and hence also yields the ionization potentials (IP's) of the molecule studied. The kinetic energy E g of the ejected electrons in process (1.1) is related to the incident photon energy hv by the equation, E = hv - I? - A E . . - A E A E (1.2) e vib rot trans where AE .,, AE and AE^ are: respectively: the change in the vibrational, vib rot trans r ' b - 2 -rotational, and translational energy, involved in the transition. 1{ is the adiabatic IP which is the energy difference between the ground vibra-tional and rotational levels of the molecule and the corresponding states of the resulting ion; in contrast to the vertical IP, l\ , which corresponds to the most probable ionization transition from the ground state of the mole-cule. The translational energy of the ion is usually very small because con-, servation of momentum requires that the energy be almost completely transferred to the outgoing electrons. Under the present level of experimental resolution, 1 2 rotational energy is usually not resolved (the exceptions are , r^O and 3 HF where resolution better than 7 mev is required) and thus A E r Q t xs also neglected. Furthermore, the molecule is generally in the ground vibrational state. Therefore,, eqn. (1.2) is reduced to (1.3) where E*^ 1 S t n e vibrational energy of the ion. Depending upon the light source used to produce ionization, photo-electron spectroscopy can be classified into two categories. The first one involves the use of higher energy X-ray sources which enables one to study the core electrons. This technique is termed ESCA (Electron Spectroscopy 4 5 for Chemical Analysis) by K. Siegbahn and his collaborators ' and has been found to be a powerful analytical tool because of the sensitivity of the bind-ing energy to the chemical environment. It is, of course, a surface technique for solids, although recently there has been increasing use of gas phase ESCA. The second category, called molecular photoelecron spectroscopy^ or simply photoelectron spectroscopy (PES), involves the use of vacuum ultra-- 3 -violet radiation and is applicable only to the ionization of valence shell electrons. This kind of spectroscopic technique was developed independently by two groups of Russian and English scholars^ in the early 1960's. Since then, there has been an exponential growth of work in this field with the ap-11-29 pearance of a considerable number of review articles . So far, the He I resonance line (21.22 ev) is the most popularly used light source owing to its narrower line width and high intensity. However, there is a growing interest 30 in utilizing the He II resonance line (40.8 ev) . Other sources such as argon 31 32 (11.83 and 11.62 ev) and neon (16.85 and 16.67 ev) resonance lines ' have also been employed. This thesis is mainly concerned with PES using the He I radiation. It is well known that the ionization process is governed by the 33—36 Franck-Condon principle. Quantitative application of the principle en-ables one to determine the ionic geometry which is usually not attainable by optical spectroscopy. However, the computation of ionic geometries requires knowledge of the ionic frequencies as well as vibrational transition proba-bi l i t i e s . PES provides valuable information of this sort. This thesis will describe the current methods employed in the Franck-Condon factor calcula-tions, and application to group VI hydrides, nitrous oxide and dihaloethylenes. In addition, a Huckel molecular orbital treatment is devised to deduce some intrinsic molecular orbital parameters of a number of halogenated organic molecules such as 1,2 dihaloethanes, 1,2 dihalotetrafluoroethanes, 1,2 bromo-chloroethane, l,2-dibromo-2,2-difluoroethane, dihaloethylenes and 1,2 dibromo-cyclohexane, by making use of the experimental IP's derived from their PE 37 spectra (invoking Koopmans' Theorem ). The relation between the cal-- 4 -culated molecular o r b i t a l parameters and other p h y s i c a l constants, f o r instance, bond lengths, force constants, e l e c t r o n e g a t i v i t y of the halogen, and NMR chemical s h i f t s w i l l also be discussed. CHAPTER II THEORETICAL ASPECTS OF PHOTOELECTRON SPECTROSCOPY 2.1 Interpretation of Photoelectron Spectra 2.1.1 General Comments The PE spectrum of a molecule consists of several bands, each of which relates to an ionization potential of the molecule. The lowest energy band is usually associated with the ground ionic state, while the others 37 correspond to excited ionic states. According to Koopmans' theorem , (see below), the i."* vertical IP is equal to the negative of the i.* 1 orbital energy. Thus the PE spectrum is a direct display of the energies of the various molecular orbitals (MO's). This theorem has been used extensively in assign-ing the gross structure of the PE spectra. The vibrational fine structure, band shapes and ionic frequencies associated with each IP usually provide fruitful information about the bonding nature of the corresponding orbital, and are also helpful in assigning the correct orbital sequence. The interpretations of the band shape is largely based on the application of the Franck-Condon principle which will be discus-sed in Section 2.1.3. The assignment of the PE bands of a molecule is often accomplished with the aid of spectra from other related compounds^8 ^ . Resonance, induc-tive and hyperconjugation effects as well as the electronegative nature of the substituent groups are frequently utilized in the study. The perfluoro 41 42 effect ' may sometimes be used to differentiate the nonbinding, the TT and the 6 orbitals in planar molecules. Recently, the stretching force con-stants and also the logarithm of bond lengths are found to correlate linearly 43 44 with a large number of homologous series ' . The relationship obtained should also be helpful in assignment of spectra. In the assumption of the constancy of photoionization cross sections, the relative intensities of the PE bands reflect the degeneracy of the orbitals from which electrons are removed. In other words, degenerate orbitals are usually associated with higher intensity bands. However, this guideline should be used with caution because i t has been found to be invalid in some cases**'. 45 46 Experimentally ' , the intensity of a band related to an orbital with mainly s character, tends to increase with respect to that with mainly p character when higher energy light sources are used. Several theoretical investigations 47-53 using a plane wave approximation have been attempted to study the energy dependence of the photoionization cross sections and the bonding character of - 7 -orbitals, with some success. This approach may become a useful assignment criterion in the near future. Since PES is not a threshold technique, the spectra are generally not complicated by the autoionization process. However, because of its resonant nature, autoionization in PES can usually be identified by using . different light sources, and often leads to valuable information about the autoionizing states. It should be mentioned that the light source used in PES is not truly monochromatic. Each of the spurious lines adds its spectrum to the total and 54-57 errors in interpretation may occur . Under normal conditions, the 21.22 ev He I line of the helium resonance lamp is accompanied in low intensities by lines at 40.82 ev (He II), 23.1 ev (He IfJ), 12.09 ev (Lyman [3 ), 10.93 ev (NI), and 10.21 ev (Lyman c< ) 5 5 - 5 7 . 2.1.2 Koopmans' Theorem Koopmans' theorem, as mentioned before, states that the i orbital energy is related to the vertical IP, I* according to the equation, I i ' = -£? (2.1) with £ • = H.. + IJ U j i j - K i j f (2-2) HJL-L 1 S the expectation value of the one electron core hamiltonian correspond-ing to the orbital. J^j and K^ are known as the coulomb integral and exchange integral, respectively. Eqn. (2.1) can be derived from eqn. (2.2) by assuming that there is no reorganization e n e r g y ^ 8 a n d no alteration in correlation energy between the molecule and its cation. Hence, the validity of Koopmans' - 8 -theorem depends on the delicate balance between these two energy terms. Usually, the IP's calculated are too large. For closed shell molecules, cal-culations near Hartree-Fock accuracy of valence shell orbital energies are 59 often too high by-8%. Semi-empirical MO methods such as the CNDO/2 or 59 INDO methods usually give predicted IP's about 4 ev larger than the experi-mental values. Though the agreement is not perfect, in many cases, the cal-culated' result is sufficient to assign unambiguously PE bands to ionizations from specific MO's in the molecules. Despite the general success of Koopmans' theorem for closed shall molecules, its breakdown in predicting the correct ordering of the ionic states is not uncommon. In general, the theorem is more likely to f a i l when large orbital reorganization is involved in the ionization process. This includes the cases when the electron being removed is localized in one region of the molecule^, either f o r symmetry reasons, or by virtue of i t being a'lone pair'electron. In the same manner, molecules containing electron-rich atoms like fluorine*^ tend to deviate from the prediction of Koopmans' theorem. Caution should also be made in the case where two or more orbitals lie close together, leading to doubts in assigning the experimental IP's. Various theoretical methods have been attempted to reproduce the observed IP's. One approach is to carry out separate Hartree-Fock calcula-tions for the molecule and its cation and to obtain IP's by their difference. Other approaches^2 ^ involve the use of many-body techniques of second quan-70 71 tization ' by which the vertical IP's are directly related to the poles 72 of the one-particle Green's function . Very recently, Chong, Herring and 73 74 McWilliams ' of this Department have employed Rayleigh-Schrodinger per-turbation theory^ to estimate the correlation energy of a closed shell - 9 -molecule and also the correlation and reorganization energies of the cation. Basis sets of double or one and a half zeta Slater orbitals are used in the computation. The calculated IP's reproduce well the experimental values within ±0.5 ev. 2.1.3 Franck-Condon Principle The Franck-Condon principle states that an electronic transition takes place so rapidly that a vibrating molecule does not change its inter-nuclear geometry appreciably during the transition. The transition moment Prnn for a vibronic transition between the th th n vibrational level in the neutral ground state and the m vibrational 76 level of the ionic state can be expressed as, Pmn = <Ya(a')lM\Vz(GL)> (2.3). with M = £ ' > I X f j 4 > ( # , & > > (2.4) and m = U\. rr\L (2.5) in the Born-Oppenheimer approximation. The quantity with a line underneath denotes that the quantity is a matrix or column vector, -j^ represents the momentum operator summed over a l l elctrons n, ^  (Q) and ^ (c^. Q) are the total vibrational and electronic wavefunctions respectively. M is the electronic transition moment and the quantities q_ and Q designate the total electronic and normal vibrational coordinates of the molecule. If the electronic transition moment varies slowly with the nuclear configuration, the intensity of the vibrational components within a progression - 10 -will be proportional to the Franck-Condon factor (FCF), the square of the overlap integral ^KTJ'HA ) . In general, the FCF is appreciable only when the maxima or minima of the two wavefunctions lie on top of each other, i.e. at the same (element of matrix 0J and these maxima or minima for the higher vibrational levels appear near classical turning points of the motion. Therefore, the vertical electronic transitions are usually the most favorable. Fig. 1 shows qualitatively the influence of the Franck-Condon principle on the vibrational patterns and the band shapes observed in the PE spectrum of a diatomic molecule. The shaded area called the Franck-Condon region which can be defined in terms of the maximum and minimum internuclear distance where observable transitions can occur. The fine structures of a band are determined by the potential curve of the resultant ion, which is in turn determined by the equilibrium distance Ye and vibrational frequency V' of the ion. Ye may be greater, smaller or almost the same i f the ejected photoelectron came from a bonding, antibonding or nonbonding orbital. The band shape may be further complicated by dissociation or predissociation of the ion. The general features of a PE spectrum for transitions from the neutral ground state to different ionic states are described in Fig. 1. A further indication of the bonding character of the electron removed may be obtained by comparing the vibrational frequencies V ' in the ion with the corresponding frequencies l> in the neutral molecule. In diatomic molecules, these frequencies are related to the stretching force constant K by ( 2 . 6 ) - 11 -THE FRANCK-CONDON PRINCIPLE IN PHOTOELECTRON SPECTROSCOPY FIGURE 1 - 12 -Thus i f a nonbonding electron is removed, there w i l l be practically no change in force constant and frequency, whereas i f a bonding electron is removed, both quantities of the ion should be less than those of the parer molecule and vice versa for the removal of an antibonding electron. With regard to polyatomic molecules, the above arguments for deducing bonding character from band shape and vibrational frequency should be used with care since one is now dealing with multidimensional potential surfaces, and changes in bond angle as well as bond length are involved. Nevertheless, the same ideas as given above for the diatomic case may be u ;ed as a guide to bonding character. 2.1.4 Through Bond and Through Space Interactions Although the delocalized molecular orbitals obtained from a sel consistent-field calculation present a f a i r l y accurate picture of the ele • tronic structure of a molecule, i t is usually easier to understand and in pret a PE spectrum in terms of the more familiar localized orbitals. For instance, a halogen lone pair molecular orbital (LPMO) means that most of the electron density resides on the halogen atom, but there are also contributi <>ns from other atomic orbitals in the molecule. 77- 78 According to Hoffmann and his coworkers , localized orbital may interact with each other via two distinct symmetry-controlled mechanise;: (l) a through space interaction or (2) a through bond interaction. The former perturbation arises from the spatial overlap between the localize! orbitals and the magnitude dy between the I t h and orbitals can be - 13 -79 approximated as dij = k d Sy (£i Ej)'1 (2.7) 80 where Sg is the Mulliken overlap integral . £• and £j are the Coulomb energies for the I t h and j * orbitals respectively. The constant is set 81 equal to unity in this work In addition, localized orbitals can mix with each other indirectly through the intervening 6 bonds of the same symmetry. This through bond interaction between the i + h orbitals, U , and the (5" orbital, 6j , 82 may be described quantitatively by using the second order perturbation method in the approximation of zero differential overlap as, . £ i , I < U I H - | S J > | ' K h i H - i e ^ i ' with H' the perturbed hamiltonian. In this work, the second terms on the right hand side of eqns. (2.6 and 2.7) are parametized as S i and are determined from experimental IP's. Since S i is inversely proportional to the factor ( Ei. - £j ) , significant mixing between localized orbitals and 6 orbitals occurs only for a small energy gap between the two interacting orbitals. The concept of through bond and through space interactions has been used extensively in molecules with TT bonds , and with nitrogen atoms 88 89 ' . This observation in PE or ultraviolet spectroscopy is used in deducing the orbital sequence of the frontier occupied orbitals. So far, these interactions - 14 -are considered qualitatively only. In chapter four of this thesis, PE spectra of a series of halogenated compounds wi l l be given and both the through bond and through space interactions between the halogen LPMO's and € orbitals are treated quantitatively. Fruitful information is obtained on the bonding properties of the molecules studied. 2.1.5 Spin Orbit Coupling Consider a halide molecule RX (with X the halogen atom) in which the bond R - X coincides with an n-fold axis of the system (n > 3). Removal of an electron from a Tf type halogen LPMO w i l l give rise to an ion in a doublet spin state. Because of the spin orbit coupling interaction, the 'TT term is s p l i t into two levels *TTJ/z and * T T i ^ with an energy difference given by , £ CTT^) - £ CTT./J = A (2.10) The magnitude of the spli t t i n g A can be estimated by using the following 9 3 spin orbit operator H s o = I S(Yi) I* • S: (2.11) a l l eta rang with < a ) = — - — / d V ( T i M (2.i2) where e i s the electronic charge m is the mass of the electron e c is the velocity of light - 15 -Y\ is the distance of the i + h electron from the nucleus V 7 r t ) is the potential at the electron arising from the nucleus is the orbit operator for the l r t electron Si is the spin operator for the I electron If an electron is removed from the TT type LPMO's l \ (angular momentum quantum number), \ = + 1 or - 1 which is centered entirely on the halogen atom of RX, i.e. U = Px > then we should expect, according to . eqn. (2.10), that the corresponding band in the PE spectrum w i l l exhibit 37 two components with an energy gap (assuming Koopmans' theorem to hold), and IVz are the vertical IP's corresponding to the TT3/a and iTT,/^  states respectively. A i s equal to - x , which is 2/3 of the spin orbit coupling constant of X. If, on the other hand, the LPMO l\ where ionization takes place is not s t r i c t l y localized on X, but is a linear combination (in the zero differential overlap approximation) with 4*^ (X) being the basis atomic orbitals localized in the alkyl group. Then ^ in equn. (2.10) should now be given by, S = + Z C § k (2.15) Hence the spli t t i n g A between the two 2 TT components is a sensitive probe for the relative participation of the orbitals and 4\(K) in the LPMO I a . • - 16 -Recently, Heilbronner and his c o w o r k e r s ^ ' h a v e recorded the PE spectra of some alkyl bromides (some with low symmetry e.g. C ) and they found that the splittings A obtained from these spectra have the f u l l value of ^ in spite of appreciable mixing between orbitals of the alkyl group and the bromine atom. To explain these observations, a one electron model was proposed with the inclusion of conjugative interaction between orbitals of bromine and the alkyl group as well as spin orbit coupling between the LPMO's of bromine i t s e l f . Under the assumption that the ejected electron has spin (this yields the inverse order SCTTi*) > £(aTTa/») ), one obtains the following matrix elements: <P*|Hso|px> = <PalHSo|Pa> = 0 <Pzl HS0|Pa> = <Pz| H s o| Pa> = 0 <PjH So|P t> = 0 (2.16) <P*I H s o | Pa> = <Py|HSo|Px> = i*/2 The same spin orbit operator is used in the one electron models devised to interpret the PE spectra of halogenated compounds (in chapter 4). 2.1.6 Jahn-Teller Splitting 96 The Jahn-Teller theorem states that a non-linear molecule in a degenerate electronic state is unstable towards nuclear distortions which lower the molecular symmetry, and thereby remove the electronic degeneracy. 17 -When this effect is active in a PE spectrum, a band may be split into 97 several components, sometimes separated by as much as 1 ev . The effect is largest when the degenerate orbital concerned is strongly involved in the bonding and is minimal for degeneracy in nonbonding orbitals. However, there is no direct way of predicting the magnitude of this effect that may be observed. Quantitative treatment of the Jahn-Teller effect on PE spectra of methyl halides^, ammonia^ and methane*^ has been carried out to attain a better understanding of the structure of their respective cations. The corresponding effect in degenerate orbitals of linear molecules is called the Renner-Teller effect and has been observed in the case of water^* and hydrogen sulphide 2.2 Quantitative Application of the Franck-Condon Principle in Photoelectron  Spectroscopy 2.2.1 Method of Franck-Condon Factor Calculation (a) General Description of the Method The relative arrangement of the nuclei of a molecule at vibrational equilibrium is often simply called its structure. In general, the structure of an ion differs from its parent molecule, and the deviation therefrom depends upon the nature of the orbital involved in the ionization process. The geo-metry of neutral polyatomic molecules can be determined by various physical methods such as electron diffraction, ultraviolet, infrared, Raman and micro-wave spectroscopy. However, structural data are available for only a small number of ions 7 6 e.g. CO* COS*, N 0*. Rotational fine structure in the - 18 -vibronic bands of an ultraviolet absorption system occasionally yields information of this kind. Therefore, the Franck-Condon principle offers the only alternative method of obtaining information about the ionic geo-metry. Assuming the v a l i d i t y of the Born-Oppenheimer Approximation, and the constancy of the electronic transition moment, the vibrational transition probability or intensity I m a is given as, Imn = H a K ^ | t a > | 2 (2-17) The constant includes those contributions from the electronic transition 103 moment as well as the design of the PE spectrometer . In the harmonic oscillator approximation, each vibrational wavefunction ty (Q.) may be ex-pressed as a product of 3N-6 (for linear molecules 3N-5) Qi, Hermite ortho-82 gonal functions . N is the number of atoms in the molecule. Then the overlap integral in eqn. (2.17) can be separated into 3N-6 components b m n = < t y ; i % > = TTT 6 bn^ (2-18) W i t h b m ^ = <9m l(Qi)| G n i ( Q i ) > (2-19) To evaluate the integral b m i m > the coordinates of the ionic state must be expressed as functions of the coordinates in the ground state. 104 This can be accomplished by a linear transformation QL = J a + d (2.20) 19 where _T is a (3N-6) x (3N-6) square matrix and d. , a column vector of order 3N-6, having elements d.,. representing the separation of the origins of the two coordinates. A similar expression can be written in terms of the internal symmetry displacement coordinates or simply by the symmetry coordinates (section 2.2.3a) 5? and Sj_ as S_' = _S + AS (2.21) Upon employing the transformation from symmetry to normal coordinates, S _ ' = L i & ; S_ = Ls CL (2.22) we obtain 0,' = (Lk)"' U Q + (U)" ' AS (2.23) A comparison of eqns. (2.23) and (2.20) yields the definition 1 = (U)"' U (2-24) d = (Ls)" 'AS or AS = Ls. d (2.25) The off diagonal elements of matrix J_ are different from zero only when the I t h and j'"1 normal coordinates have the same symmetry unless changes in molecular symmetry occur upon ionization, cii is nonzero for totally symmetric vibrations only. From eqn. (2.25), a knowledge of d_ can be related to S', from which the change in internal coordinates (changes in interatomic distances or in the angles between chemical bonds or both) is in turn calculated. - 20 -The Ls_ and Ls' matrices are readily obtained through a normal coordinate analysis, (see section 2.2.3) or a force constant analysis using the vibrational frequencies in the corresponding state. However, the number of observed vibrational frequencies in the ionic state is usually not large enough to carry out a normal coordinate analysis. In such a case, we use Ls instead of Ls_ . Making use of the property of Hermite polynomials and relation (2.18), bmjn.j was expressed by Ansbacher*^ in the form, b (2.26) where a „ - W 2 — 1 ^ (2.27) and [ mj , rij ] denotes the smaller of the two integers rr\j and H J In this formula, H m i is the Hermite polynomial of degree mt , ^ = = ( UJ/i/ i') l i , Yj = oij dj and d* = ^ T T ^ J C / ^ w i t h K as Planck's constant. Now, the relative transition probabilities for overtones may be given as, b* j 0 _ 2 m j / z M ) m j / 1 - 3 / W ' O O ( m j i ) ' ( - ^ ) " H ^ C ^ O - ^ J (2.28) - 21 -according to eqns. (2.26) and (2.27), or l + |3j I! JJSL^LJI /3J + I 2 (2.29) according to Heilbronner and his coworkers 1 0 6. The summation extends only even over even (odd) indices I in the interval O L I ^ mj i f m.j is 82 (odd). With the aid of the recursion formula for the Hermite polynomials , i t can be shown that bm,*.,o _ fi. ft] Tj / m j v'fr _br^-i1o_ (2.30) bm j 0 ~ (l+pi)<mj + 0 * + l m j + i ' bmjo < + ti The Hermite polynomials rapidly become complicated as rYTj increases, but the recursion formula (2.28) allows one to calculate numerical values of bmjo / b o o UP t 0 a n v desired value of )Ttj with comparative ease. So far,, the treatment is restricted to a one-dimensional problem i.e. C u is determined separately. The method (hereafter referred to as me thod A) enjoys the popularity for the ease in computation. Using the 35 generating function technique, Rosenstock and his coworkers have developed a method to calculate FCF of overtones and combination bands of several vibrational frequencies at the same time. The derivation of this method (method B) is similar to the former one. Briefly, in the Born-Oppenheimer approximation, assuming constancy of the electronic transition moment, - 22 -harmonic o s c i l l a t o r approximation and l i n e a r r e l a t i o n s h i p (2.20), we get, I m I ( l Jf Ut" ( 2 m 2 V m ! n\) I m a (2.31) = i o e*p [(If A i t + jl B ) + ( uJ c Ut + y l D_) + u± J L IL^ where I, = ( det I f £ ' ) v f + ( det ( f £ ' J + n]"' 4 (2.32) x exp [-^dj _rd. + i r K I I r x + i i r ' r n'dj A = 2 JLt^Jl ' T + £)" ' T T r 1 4 - i . (2.33) B = - 2 r * [J (r rj_+ r r ' i T r - n d ( 2.34) c^= 2 irr j + rr' - 1 ( 2 . 3 5 ) D = -z (_!'£'Z + D Z. £.' 4L C 2 - 3 6 ) (2.37) - 23 -LC is dimension of matrices £_ , d_ etc. with value less or equal to 3N-6. _T is the diagonal matrix of frequencies 4TT1L^/h. . The super-script t indicates the Hermitian conjugate of a matrix (which is, for a real matrix, the transpose). T f and U f are the dummy variables for the ionic and neutral ground states respectively. The individual overlap integrals I ma are now obtained by expanding the right hand side of eqn. (2.31) and equating coefficients of appropriate powers of TV and \J± . General expressions for calculating FCF's of overtones, combination levels and hot bands are well documented in refs. 35 and are not repeated here. It should be mentioned that both methods A and B cannot be applied to ioniza-tion process with changes in the vibrational degrees of freedom between a molecule and its ion. (b) Method of Calculation The general procedure in FCF calculations with the application of method A is outlined below. 1. The intensities Imn of the fine structure components in a PE band are measured. In the approximation of the intensities they are assumed to be proportional to the heights of the fine structure maxima, i.e. under the implicit assumption of a constant half width of the components. Usually, the maximum FCF is arbitrarily set to be 1 unit and the other FCF's are scaled accordingly. 2. For each observed vibrational mode, the FCF's are evaluated at a series of d^ values or by means of a least square f i t method (section 2.2.2). The best d. value is then chosen on the basis of two citeria:- (i) the overall - 24 -qualitative features of the experimental FCF's from the experimental FCF's is a minimum. For those vibrational modes not observable in the spectrum, the corresponding d\'s are set equal to zero. 3. A normal coordinate analysis is performed on the ground state of the molecule studied, using vibrational frequencies determined from infrared and Raman spectroscopy. Hence the Ls_ matrix is obtained. 4. Through expression (2.25) and L_s used instead of Ls' , d_ is employed to calculate AS and hence the change in internal coordinates. Since the sign of each d^ is undetermined, there are 2*' possible ionic state models consistent with the observed FCF's, where n is the number of vibrational s modes of the same symmetry excited. If the geometry of the ionic state studied is known, the sign of Ar ; , the difference in bond lengths in the ionic state and the ground state, is simply chosen in accordance with obser-vation. However, this kind of information is not available for most poly-atomic molecules. Therefore, the following cirteria are usually used to determine the sign: (i) the change in a bond length during an ionization process is related in a reverse manner to the bonding character of the bond, and hence the orbital from which an electron is removed, (ii) the sign of AY<, agrees with that of f ^ _ r ^ > the difference in force constants or Vi - Vi , the difference in vibrational frequencies, and finally, ( i i i ) the change in bond distance parallels the difference in the overlap population, or bond order of the bond, between the neutral molecule and its cation. - 25 -With regard to a change in bond angle, there are no definite guidelines given in the literature for choosing the appropriate sign. This promotes us to propose a criterion to do so by making use of the concept 1 0 7 of anodal repulsive force . For instance, the highest occupied molecular orbital of gem dihaloethylene possesses C-C TT bonding but C-X antibonding character. The force f^ is attractive while is repulsive in the molecule. The interaction between X,. and is neglected for large internuclear dis-tances. If an electron is removed from this orbital, both f^ and f^ are weakened and hence the angle X^ C^ X^  is increased. The opposite is true for angle C^ C^ X^ . The angle changes should more or less depend on the over-lap population between nonbonded atoms. The CCH angle change cannot be predicted in this case because the coefficients of hydrogen atomic orbitals are zero. 5. If the number of ionic frequencies observed is not sufficient to carry out a force constant analysis, the structural parameters obtained from (4) are regarded as the geometry of the ion. If not, the structure obtained is - 26 -used to calculate Ls' and also a new AS . This process is repeated until the Ls' do not change significantly. Usually, one or two cycles are enough. Even for r^O in the state with geometry deviating largely from that of the neutral molecule, only three cycles are required to obtain a consistent Ls' matrix (section 3.1.3c). Determination of the ionic geometry using method B is mainly carried out by a FORTRAN program (appendix I) written for the IBM 370 computer. Input data required are the dimension JUL , frequencies V and v' , AS , Ls and Ls' matrices as well as the number of overtones and combination levels desired in the calculation, while the transition probabilities of overtones and combination bands are printed after the computation. The theoretical intensity distribution can be displayed graphically upon request. AS is varied in a trial and error manner until the criteria mentioned before are satisfied. The calculated result by using this program was checked against 108 that of Botter and Rosenstock with good agreement. 2.2.2 Iterative Method in Franck-Condon Factor Calculation In all the current FCF calculations (method A), evaluation of d^  from the observed overlap integrals for overtones 010 (square root of FCF) b m [ 0 is done in a trial and error manner. First a value of d^  is chosen arbitrarily and thus and are computed. A set of 010's is then eva-luated up to the desired number of nu and these are compared with the obser-ved 010's. tf\ is adjusted until agreement is reached. However, the treatment becomes more and more tedious to handle for increasing numbers of overtones observed, as well as for large changes in geometry upon ionization. In this - 27 -section, an iterative method is proposed by which the value of d^ will be refined automatically to match the observed 010's by a least squares f i t adjustment. Computation of d^ requires the knowledge of at least two observed 010's within a progression. If only two 010's are available experimentally, d^ can be determined uniquely. However, in many cases, the number of over-tones observed is greater than two and the least squares f i t technique is required to obtain a value of d^ that gives the best f i t between the obser-ved and calculated 010's. The general outline of the treatment used in this work is as follows:-(a) An estimated value of d., d.° is used to calculate a set of b , b. v i i mco' — Then these calculated 010's are compared with the observed ones b° b s , and the error vector Ab is defined as, A b = - b_ C2.38) The weighted sum of squares deviation is given as, Sd. = A b * W Ab_ (2-39) W is a diagonal matrix with elements giving the statistical weights of the observed 010's, W.. = b^.g . n w can be any real number and is fixed to i i be -2.0 throughout this work since the weighting factor enables one to f i t the observed 010's on a percentage basis. (b) Assuming that there is a linear relationship between Ab and Ad^, the correction for d., then we obtain, l A b = JP ACL (2.40) - 28 -where JJP is a column vector with element J P m . = d. bmio/cL cU , i.e. derivative of b with respect to d.. J P m . is related to b , ft. and X. in the m-o r 1 "'«• nii.o i i l following way:-J P = - J ^ L _ r b + 2 ^ ( r ^ l ) ' / z e x p [ - ] 7 M / . info 1 (^Lil) 2^ U O 1 + (3- ; ( L - | ) . (ITILZL). + i (2.41) from eqn. (2.29). Then Adj. is computed through the expression (2.42) JP_*_W JP Adi = TP. W.Ab (2-42) (c) A new d^ is obtained through the relation = + C a Adc (2-43) where C is an arbitrary scaling factor for damping purposes, and d. thus obtained is used over again until the desired number of reiterations are completed. A FORTRAN program (appendix II) based on the method described above has been written for the IBM 370 computer. The b°^ S, C & and d* comprise the input while the calculated b_, d^ and are printed out after each iteration. The least squares f i t method mentioned above has been tested for two cases. Transitions from the ground state to the A 1TTU. states of CS^, 29 and the A, state of CFBr^ where the corresponding 010's are given in ref. 11 and section 4.1 respectively. In these transitions, excitation of only one vibrational mode is observed. . The results of the calculations indicate that even for a poor 0 approximation of d^  (about 30-40% off from the best value of d^  for a cer-tain ion), it is always possible to know roughly where the best value of d. is by adjusting the factor C in expression (2.43). Then, a better x a approximation of d? can be made for the next calculation. Once a good estimation of d.° is made (within 90% of the best value of d. obtained), l l d^  will keep on increasing or decreasing with the value of decreasing first, passing through a minimum and then increasing very slowly afterward. Fig. 2 shows the general trend of the iterative process. The best value of d^  obtained for the 2A, state of CFBr^ and the A *7Ta state of CS^  are -20 '/2 1.8700 and 0.7349 x 10 gm cm respectively, and the calculated 010's for these molecules from these d^  values are shown in Table 1. Theoretically, A d s h o u l d approach to zero in the iterative process. However, in fact, this is not true, which is probably due to the validity of the linear rela-tion (2.41) as well as the accuracy of the intensity of the components within a progression which are overlapping with each other. Up to now, little attention has been paid by previous workers with regard to the criteria necessary in choosing a value of d^. Here, we have proposed the weighted sum of squares deviation as one of the criteria, since in such a treatment there is no discrimination on the low intensity components within a progression. - 30 -' 2 4 6 ~8 10 12 14 16 NO. OF CYCLES • Figure 2. Plot of the weighted sum of squares deviation against the number of cycles for the — * A I T T A transition of CS,, during the - 20 iterative process ( n w = -2, C = -5.0 and d* = 0.5 x 10 3. 1 gm cm). - 31 -Table 1. Calculated and Observed Overlap Integral (OIO) For the 'A, State of CFBr^ and the A 3TTu. state of CS2 CFBr CS. m. 3 2 1 : Obs. 0I0 a Cald. 010 Obs. 0I0b Cald. 010 0 — 0.15 0.65 0.55 1 0.42 0.41 0.93 0.98 2 0.59 0.62 1.00 1.00 3 0.79 0.80 0.87 0.89 4 0.92 0.94 0.70 0.70 5 1.00 1.00 0.53 0.51 6 0.98 0.99 0.35 0.34 7 0.90 0.91 0.22 0.21 8 0.77 0.79 9 0.64 0.65 10 0.53 0.51 Obs. OIO is estimated from Table 14. The maximum OIO is arbitrari set to be one unit. k Obs. OIO is obtained from ref. 11. - 32 -2.2.3 Normal Coordinate Analysis (a) General Principles A molecule or an ion may be considered as a system of atoms elastically coupled together. The forces between them are electrostatic in nature. When the nuclei are displaced from their equilibrium positions, restoring forces due to the potential field of the electrons and nuclei will act on the system. This kind of potential field is described in terms of the force constants associated with the restoring forces. Since force constants always provide valuable information about the bonding nature of the species studied, i t is desirable to obtain these constants from the observed properties such as vibrational frequencies. This is usually accomplished by calculating the frequencies, as-suming a suitable set of the force constants. If the agreement between the observed and calculated frequencies is satisfactory, this particular set of force constants is adopted as a representation of the potential energy of the system. The L_s matrix deduced gives the correlation between the symmetry coordinate and the normal coordinate. To evaluate the force constants, i t is necessary first to express both the potential and kinetic energies of the molecule in terms of some common coordinates. Changes in bond lengths and bond angles can be used to provide a set of displacement coordinates, the internal coordinates, R.. These coordinates are unaffected by translation or rotation of the 1 molecule as a whole. The corresponding force constants have a clearer physical meaning than those expressed in terms of other coordinates since they are characteristic of the bond stretchings and the angle deformations - 33 -involved. The number of internal coordinates is equal to or, if redundant coordinates exist, greater than 3N-6. In terms of the internal coordinates, the following secular 109 determinant can be obtained | Cr F - A | = 0 ( 2 . 4 4 ) G I L L - L A (2.45) or where ' fJ is the inverse kinetic energy matrix in internal coordinates F_ is the force constant matrix in internal coordinates A is a diagonal matrix with element 7vt X L = 4 TTA C* Vi (2.46) The Q_ matrix is a function of the molecular geometry only, and the methods 109 for constructing this matrix have been described in detail by Wilson et al and will not be given here. The time required to solve the vibrational secular determinant (2.45) increases rapidly with increasing order of the determinant. An ef-ficient way to reduce it is to use symmetry coordinates which transform as certain irreducible representations of the molecular point group. They are related to the internal coordinates by the orthonormal transformation matrix U_ S = UR (2.47) The projection operator**^ or the method given by Wilson et a l * ^ may be used to generate the symmetry coordinates. In terms of these coordinates, - 34 -the secular equations (2.44) and (2.45) are factored to the maximum extent and can be expressed as, Fs - A = 0 (2.48) o r G s . F s U = k A . C 2 - 4 9 ) where Gs_ a n d F_s are respectively the inverse kinetic energy matrix and the force constant matrix in the symmetry coordinates. 0_ is the null matrix. (b) Molecular Force Field As discussed in the previous section, the molecular force field of a polyatomic molecule is described by the F_ matrix i f internal coordi-nates are used. Such a force field is generally with no inherent assumptions at a l l i f there is no redundant coordinate and is called the general valence force field (GVFF). Methematically, i t is given by 2V = ZL f r . ( A T J 4 + I U j j 1 ^ ( ^ A o l i j ) ' + ^ i * j * K I M f (Am(r j kAo( j k) + N l f ( rcj Ao(jj) (r i l t A c U ) + (2.SO) where V is the potential energy function, r^ the bond length, o (_ the bond angle between the two adjacent i and j bonds and r ^ represents (r^r^) a # f^ and f o, are the stretching and bending force constants respectively while f , f , M~, N„, and P_ are the interaction force con-stants. In the most general case, a F^  matrix of order n will contain - 35 -n(n + l)/2 distinct force constants. For molecules possessing higher sym-metry than C. , i t contains a smaller number of independent force constants. exceeds the number of vibrational frequencies. Consequently, a unique set of force constants cannot be obtained. One of the methods to overcome this d i f f i c u l t y i s to assume the same set of force constants for the iso-topic isomers of the molecule. In the other approach, some of the inter-action force constants are neglected prejudicially. Since the factors determining which interaction force constants to be neglected are not simple, 109 various model force fields have therefore been proposed . Among these force fields, the Urey-Bradley force f i e l d 1 1 1 1 1 3 (UBFF) has been widely The number of general valence force constants of a molecule always used. The general form of the Urey-Bradley force f i e l d corrected to the second order is given by 112 V + 2^<j ( H ' -ij - Sj\ Fy + tjtj- F.J-J ( ^ A o G j )1 + ^ i< j t " tij tji Fij + s 3 Sjl ^ H A K J ) + (2.51) where = ( Tj - Y, C O S d . j ) / f l y (2.52) 9y = rt + r/ - 2 n fj c o s c ( ( j - 36 -K and H are the bond stretching and angle bending force constants res-pectively. F and F' are both the nonbonded repulsive force constants, and F' is usually taken as -0.1 F with the assumption that the repulsive 114 energy between nonbonded atoms i s proportional to l / r ^ The number of force constants in a Urey-Bradley force f i e l d i s , in general, much smaller than those in a general valence force f i e l d . In addition, the Urey-Bradley force constants (UBFC) have a clearer physical meaning and also they are often transferable from molecule to m o l e c u l e ^ , This latter property of the Urey-Bradley force constants i s highly useful in calculations for complex molecules. However, ignorance of the inter-actions between stretching vibrations as well as between bending vibrations in the force f i e l d sometimes causes d i f f i c u l t i e s in adjusting the force constants to f i t the observed frequencies. In such a case, i t is possible to improve the results by introducing more force constants^ (c) Force Constant Calculation The procedure in force constant calculations may be summarized as follows:-1. Choose the appropriate internal coordinates of the molecule or ion under investigation. 2 . Construct the proper symmetry coordinates so as to obtain the trans-formation matrix U. I f the number of internal coordinates exceeds the number of vibrational degrees of freedom, instead of discarding some of the internal coordinates, a l l these coordinates are s t i l l used to construct i the symmetry coordinates. The removal of the redundant symmetry coordinates - 37- -109 can be accomplished simply by discarding them a l l 109 3. Construct the G and F_ matrices by the standard method in the internal coordinate representation according to the symmetry of the molecule. 4 . Use the U_ matrix to transform £ and F_ to Gs_ and Fs. The problem now is to find a set of general valence force constants or Urey-Bradley force constants which satisfies, The elements of Ac<u. and AobS are the calculated and the observed values of the X's respectively. Various methods have been proposed to obtain such a set, e.g. Newton's method11"*, Herranz and Castanz"s method1'''6 and 117 118 a least squares f i t method ' . Among these the least squares f i t method has been widely used. Briefly the procedure of this method is outlined below. 1. A set of estimated force constants j£. is used to calculate a set of theoretical frequencies, the error vector A A (Aobs - A cat ) , and the Jacobian matrix J Z . The Jacobian matrix is defined 1 1^ as J Z = Ls Zs Ls (2.54) where Zs is defined by the relation 2. A linear relationship is assumed between the corrections to the init i a l force constants Asl and A A such that A A = J Z A $ ( 2- 5 6) - 38 -If the number of force constants m is smaller than the number of observed frequencies n, the first-order correction to the force constants 119 Aj> can be calculated from the normal equation AJ. = ( J Z * W J Z . ) " ' J Z W AA (2.57) where the weighting matrix W is usually taken to be A , which enables 118 one to f i t the observed frequencies on a percentage basis . If m = n, a unique solution will be obtained for eqn. (2.56). If m > n, a unique solution will not be possible. In such a case, some of the force constants have to be constrained, or simply set to zero so that the number of force constants refined is equal to or less than the number of observed frequencies. However, which force constants are to be constrained depends on the indivi-dual case. 3. The variations in 3l are used to give a new set of force constants = |o.d + C aAj& (2.58) The force constants are renewed after each iteration until the desired num-ber of iterations is completed or the desired accuracy is obtained. A program based on the method described above was written in 120 Hong Kong for an ICL 1904A computer and revised for the IBM 3 70 computer at the University of British Columbia. The Gs, Zs, A 0 b S and the assumed force constants JE comprise the input. The calculated frequencies and the force constants are printed out after each iteration, while the _Ls_ matrix and the Ls"1 matrix appear after the desired number of iterations. - 39 -2.2.4 Change in Electronic Transition Moment In the conventional FCF calculation mentioned in section 2.2.1, the electronic transition moment is assumed to vary slowly with the nuclear configuration. Therefore, the vibrational intensity distribution within a progression in a PE band is proportional to the FCF's. However, recent work on the PE spectra of the hydrogen molecule, and its deuterated deri-121-123 vatives show that the electronic transition moment is not constant. 123 A theoretical treatment with variation in vibrational cross section and with the use of a Morse potential function was employed to interpret the PE spectra of hydrogen and nitrogen molecules. However, the same approach cannot be applied to most polyatomic molecules owing to an insuf-ficient amount of data available in the ionic states. In the following paragraph, a simple method with the inclusion of variation in the electronic transition moment will be described. The electronic transition moment may be written approximately in terms of Q. to the first order^'* 2 4 as l M = n0 + l t ( 3 M / 3 a . ) o G U ( 2 - 5 9 ) with ( 3M/3di)o , the derivative of M with respect to the i. ^  normal coordinate and M„ the electronic transition moment at the equilibrium position. For instance, in a linear unsymmetric triatomic molecule, if only two of the bond stretching normal coordinates and Q are active in changing the transition moment, eqn. (2.59) becomes - 40 -and according to eqn. (2.3), the v i b r a t i o n a l i n t e n s i t y I i s m n = • r " i 0 1 < ©m, em'j i +1, a, + i.aj ea, e n , > i (2.61) with T ' - . ' ! £ U M . ( 2 . 6 2 ) The i n t e g r a l < Gm,! 9nj> i n expression (2.61) i s equal to uni t y f o r m3 = n 3 = 0. Using the harmonic o s c i l l a t o r wavefunction and the property of a Hermite polynomial, i t can be shown that < em, 9^1 i + T , Q , + T 2 Q 2 | Go, 9 0 2 > = M 0 (<9 m,| 0Ol> <9mJ6o,> + ^ <0m 1l9 1 |><0m 1l9o 1> + < Gm.i e0l> <e;je,a>j (2.63) where the subscript O2 means the v i b r a t i o n a l quantum of the second mode i s zero. The i n t e g r a l s i n eqn. (2.63) are evaluated using the expres-sions (2.29). - 41 -2.3 Experimental All the spectra in this thesis were taken on a spectrometer 125 described by A. Katrib which is shown schematically in Figs. 3 and 4. The He I 21.22 ev resonance radiation is generated by a low pres-sure microwave discharge in unpurified helium (Canadian Liquid Air). 126 The discharge takes place in a quartz tube within a resonant cavity (Fig. 3) and the power is supplied by a microtron-200 generator (Electro-Medical Supplies) with a maximum output of 200 watts at 2450 MHz. The radiation produced passes through a collimating capillary tube into the collision chamber. Differential pumping is applied between the quartz discharge tube and the collimating capillary to remove unwanted helium. The sample vapor is introduced through a Granville Philips leak valve to the collision chamber until the pressure in the whole system is about 3 x 10 6 torr. The photoelectrons produced by bombarding the mole-cule with the He I radiation pass through an exit hole at 90° to the photon beam. This hole is drilled in the center of the collision chamber. Under the collision chamber, a lens element (Fig. 3) with a cir-cular aperture is used in focussing the photoelectrons upon the entrance slit to the analyzer. This lens is electrically isolated from the chamber by a teflon washer and during operation applied voltage is varied until the best resolution and maximum intensity of the signal is attained. A 180° hemispherical electrostatic analyzer with a mean radius of 10 cm has been used in the present work. The capability of the hemi-42 -A - 180° Hemispherical Analyzer B - Vacuum Enclosure C - Channeltron M u l t i p l i e r D - C o l l i s i o n Chamber E - Lens F - Sample Gas Inl e t G - C o l l i m a t i n g C a p i l l a r y H - O-Ring Seal I - Boron N i t r i d e Construction J - He Pump Off K - Quartz Discharge Tube L - Microwave Cavity M - A i r Cooling N - Tuning Stub 0 - Microtherm Junction P - He Inl e t Q - Needle Valve Figure 3. Light source and 180° hemispherical analyzer u n i t . 1. 2. 3. 4. D . 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 21 " • 20 -23 22 180° Hemispherical Analyzer Vacuum Enclosure Collision Chamber Vacuum U.'V. Light Source He Cylinder Microwave Discharge Power Supply . Rotary fore pump Channeltron Multiplier Head Amplifier Argon Reservoir Diffusion Pump Rotary Fore Pump Rotary Fore Pump Scanning Potential for Collision Chamber Controls for Electron Lens System Controls for Energy Analyzer Ion Gauge Acceleration Voltage Pulse Amplifier 20. 21. 22. 23. Discriminator 24. Fabriteck 1000 Chan. Analyzer Chart Recorder 25. X-Y Plotter Ramp to Energy Analyzer Sample Inlet Figure 4. Scheme of the PE spectrometer. - 44 -spherical analyzer to accept a relatively large solid angle permits trans-mission of a relatively high electron flux. The electrode surfaces exposed to the electron trajectory are treated with Aquadaq in order to minimize any f i e l d distortion and to minimize stray scattered electrons. Theoreti-cally, the energy resolution of this type of analyzer can be approximated by where U)^ is the diameter of the entrance and exit apertures R is the radius of the electron path a AEi/2 is the energy spread at half maximum of the signal measured However, practically, the resolution is worse than the theoretical value owing to the undesired magnetic fields originating from various sources in and out of the laboratory and also from the earth's magnetic f i e l d in the region of the analyzer. Three pairs of Helmholtz coils mounted at right 127 128 angles to each other are used to reduce these fields. Other factors ' such as the Doppler effect arising from the motion of the molecules and electrons with respect to the stationary detector, surface charges on the walls of the c o l l i s i o n chamber and analyzer, s l i t widths, self-absorption in the light source and also the nature of the molecule under investigation also influence the resolution of the spectrometer. The best working reso-lution of the spectrometer on the argon doublet is of the order of 19 mev, but under normal operating conditions, and for the work described in this thesis, the resolution varied between 20 - 40 mev. - 45 -The detector chamber (fig. 4) consists of a circular lens with a central hole at earth potential. The ejected photoelectrons are accel-erated into a Mullard electron multiplier (Channeltron). Each electron produces a pulse, which is first preamplified and then passes to a EMI LA2 amplifier-discriminator and finally to a EMI RM1 rate meter. The output signal produced is porportional to the number of counts per second and can be plotted out directly on a chart recorder or accumultated in a Fabri-Tek model 1062 multichannel analyzer. The spectrum is scanned by varying the potential applied between the collision chamber and the analyzer. The scanning potential is obtained by amplifying a 4 volt ramp originating from the multichannel analyzer. The enclosure shown in Fig. 4 is evacuated by using rotary pumps (Duo-Seal model 1402). The main chamber is pumped by an o i l diffusion pump (C.E.C. type MCF 60) which is backed by a Duo-Seal rotary pump (model 1402). A Veeco ionization gauge RG 75K situated below the chamber is used to measure the pressure. The base pressure measured in the system is of the order of _7 5 x 10 torr. Finally, a l l the chemicals used in this study were commercially available, of the highest available purity, and were used as received. The IP scale was calibrated by comparison with IP's of argon, xenon and methyl iodide because these are well known. - 4 6 -CHAPTER III MOLECULAR CONSTANTS OF SOME POLYATOMIC MOLECULAR IONS 3 . 1 Group VI Hydrides 3 . 1 . 1 Introduction The group VI hydrides have long been a subject of discussion because of their importance in chemistry. Techniques such as electron 1 2 9 - 1 3 4 . _ . . 1 3 5 - 1 4 1 u * i «-impact ionization , photoionization and photoelectron 2 , 9 , 1 0 1 , 1 0 2 , 1 4 2 - 1 4 7 , , , . , , . f , spectroscopy have been employed in deducing useful information concerning the bonding nature of the states of several molecular ions. In recent years, there has been a growing - 47 -i n t e r e s t i n i n v e s t i g a t i o n of i o n i c s t a t e s x ^ u ' b y using o p t i c a l spectro-scopy. The better r e s o l u t i o n i n the l a t t e r method enables one to observe 148 fi n e structure , such as that a r i s i n g from c e n t r i f u g a l d i s t o r t i o n . Using data obtained from electron spectroscopy, molecular constants, e.g. geometries, force constants, c e n t r i f u g a l d i s t o r t i o n constants and mean square p a r a l l e l amplitudes of v i b r a t i o n s of the ions H^O* and D^ O i n t h e i r JB, and JA, states as well as H^S , D^^ , H 2 ^ e a n c * ^2^ e ^ n t' l e state can be evaluated and the constants obtained are compared with observed data i f a v a i l a b l e . 3.1.2 Gs and Fs Matrices A l l the molecules and ions under i n v e s t i g a t i o n are of C 2 y symmetry. ™ , 76,107,150,151 . . , . . 125,151-159 The s t r u c t u r a l parameters and observed frequencies of these species used i n the present c a l c u l a t i o n are given i n Tables 2 - 4 . The symmetry d i s t r i b u t i o n of the normal v i b r a t i o n s of a bent t r i -atomic XY 2 molecule i s given by P (vibration) = 2a, + b, (3.1) The symmetry coordinates used were^^ a, species: S, = (Ar, + Ar,)/f2 (3.2) S i = T A d b species: S i = (Ar, - Arx)/Jz (3.3) - 48 -Table 2. Structural Parameters of Group VI Hydrides and their Molecular Ions in Various States Molecule State Bond Length (A) Bond Angle (deg.) H20 V 0.9572 104.52 ¥ + V 0.995 109 \ 0.9577 166.1 D20 A i 0.9575 104.47 c c 1.023 109.9 D 1.024 113.1 D 20 + \* 0.995 109 X 0.9586 166.7 H2S V 1.3455 93.3 H2S+ \ 1.3676 94.6 D2S V 1.345 92.3 \ 1.364 92.5 H2Se X" 1.46 91 H2Se+ \ 1.48 91.3 H2Te v 1.7 89.5 H2Te+ 1.72 89.7 aRef. 151.. bRef. 108. ° r = 1.002 A and ok = 108.7 from rotational analysis (ref. 169). Ref. 150. Table 3 . The Observed Frequencies (cm-1) of H~0, H 0 + , D „ 0 and D 0 + in Various States H 2 0 D 2 0 D 2 0 +a A l c c D C V 2A 6  A l X A b  A l . c c D C 2 B l d 2A 6  A l 3 6 5 6 . 7 3179 3268 3195 3680 2671.5 2338 2381 2295 2 6 4 8 f vz ( a , ) 1 5 9 4 . 6 1407 1631 1400 975 1178.3 1041 1223 1015 715 3 7 5 5 . 8 3 0 6 0 g 2 9 5 8 g 3150 3 8 1 2 g 2788.1 2 2 4 7 g 2 1 7 5 g 2310 2 8 4 2 g The uncertainties in observed frequencies for the ions . are i estimated to be within + 50 cm \ except P3 , ± 100 cm * and U, for state of H 2 0 . -bRef. 151. CRef. 170. dRefs. 102 and 152. eRefs. 2 and 101. From product rule. sFrom normal coordinate analysis. a -1 2 Table 4. Observed Frequencies (cm ) of H2S, D2S, H2Se and F^Te in the Ground State and the B1 States of the Molecular Ions H 2 S ( 1 A 1 ) B H 2 S + ( 2 B 1 ) C D 2 S ( 1 A 1 ) B D ^ V B J ) 0 H 2 S e ( 1 A 1 ) D H 2Se +( 2B ) E H 2 T e ( 1 A 1 ) F H 2 T e + ( 2 B 1 ) E U,(CL,) 2614.6 2540 1896.4 1830 2344.5 2260 2000 2100 iJ^a.) 1182.7 1250 855.5 950 1034.2 861 U3(b,) 2666. 2185g 1926 1569g 2357.8 2000 ^he uncertainties in observed frequencies of a l l the ions are within t 40 cm except that of H„Te+ in 2B, -1 state ± 200 cm . bFrom refs. 76, 150-156. CRef. 125. dRefs. 157 and 158. e f s. Ref. 147. Ref. 159. 6From normal coordinate analysis. - 51 -The internal coordinates are defined as follows: r = X - Y bond distance and ol = Y-X-Y angle. The inverse kinetic energy matrix Gs_ as well as the force constant 109 matrix Fs_ in GVFF were set up by Wilson's GF formalism and are li s t e d in Table 5. They are the same as those given in ref. 160. f and f^, and are respectively the bond stretching and angle bending force constants while f and f are the bond stretch-bond stretch and bond stretch-angle r r rot bending interaction force constants respectively. JXK and J 1 Y are respectively the reciprocal of the atomic weights of the X and Y atoms. In the most general valence force f i e l d , there are totally four force constants f , f , , f and f . However, at most, only three r °< r r rd J frequencies can be observed for each hydride or i t s ion. To circumvent this d i f f i c u l t y , we adopt the conventional assumption that the force con-stants are the same for both the molecule (or ion) and i t s deuterated derivative. Normal coordinate analysis was carried out by the least squares f i t program mentioned in section 2.2.3 to obtain a set of refined force constants as well as the Ls_ matrices for the conventional FCF calculation. Method A is ut i l i z e d throughout this work. 3.1.3 Results and Discussion (a) 2E3, State of H 20 + and D 20 + The ground state configuration of group VI hydrides is represented a s , 1 1 (ia,)a (2a,)1 (ibJ* (3a,)1 ( l b , ) 1 'A, - 52 -Table 5. The Gs and Fs Matrix Elements of a Bent Symmetric Triatomic Molecule XY„ S y m ' Gs Matrix Fs Matrixa Species — A i i ^ 1 ^ 1 1 r r r G._ -J2 LL* S\nd F 2 f G22 2 [ ^ +>U<l-cos<*)] F 2 2 f^ b l G33 A <<-caSoO F 3 3 f r -a l n GVFF. - 53 -Ionization of an electron from the nonbonding I b, orbital w i l l leave the ion in J F j , state. The geometries of H^O* and D^ O* in this ionic state 108 161 have been discussed in detail by Rosenstock and his coworkers ' 108 and are listed in Table 2. By using the structural parameters and observed f r e q u e n c i e s ' ^ ' ^ 2 of the two ions (Table 3), a normal coordinate analysis was carried out and the calculated frequencies were found to con-verge to within 1% after five iterations. The force constants obtained are given in Table 6. In view of the large uncertainty in the measurement of vibrational frequencies by the electron spectroscopy technique, and also the anharmonic effect on these frequencies, force constant analysis of the two ions was again carried out by using the following sets of frequencies: (i) 50 cm * was added to every u' , but 100 cm ^ to every (hereafter referred to as set (1) ), and ( i i ) a value ( 6 J i . - i A ) was added to every V' (set 2) where O J i are the harmonic frequencies^"*" in the ground state. The result of the calculation shows that the valence force constants f and f d r * augment by 10% with a 5% increase in frequencies. Also the interaction force constants f and f are very sensitive to the frequencies used, rr rot. Though a unique set of force constants cannot be obtained because of the large uncertainties in observed ionic frequencies, f and f ^ are definitely smaller in the ions than in the molecules. This implies that the nonbonding lb, orbital possesses some 0-H bonding character. The off-diagonal elements of the Ls' matrix, just like the constants f and f are sensitive to the frequencies used. However, rr ra n the diagonal elements, which have values much greater than that of off-- 54 -Table 6. Valence Force Constants (mdyn/Aj of Group VI Hydrides and their Molecular Ions in Various States State £ r f rr f H20 \ 7.775 -0.081 0.712 0.122 C 5.490a .0.301 0.552 0.001 D 5.464a 0.688 0.731 0.010 \ 5.601 0.164 0.538 -0.038 Set 1 5.943 0.093 0.584 0.123 Set 2 6.180 0.174 0.571 0.042 \ 7.543a 0.577 0.251 0.000 H2S \ 4.038 -0.073 0.409 -0.052 H2S+ \ 3.24la 0.499 0.471 -0.049 H2Se \ 3.242 -0.055 0.345 -0.189 H2Se+ \ 3.021b H2Te \ 2.348 -0.002 0.225 0.123 H2Te+ \ 2.587b Estimated from Badger's rule ' From approximate method (see appendix IIIJ. - 55 -diagonal elements are essentially unchanged. In addition,the diagonal elements are quite inert to geometrical change. For example, in the 2A, state (section 3.1.3c), these quantities change by only 3 to 4% compared to those in the ground state even though there is a dramatic increase in bond angle in this state. This may explain the rapid convergence of the cycling process in method A. The constancy of the L_s matrix i s not unrea-sonable because the contributions of the normal coordinates to a symmetry coordinate are nearly the same in both the molecule and i t s cation. The structural parameters, Ls' matrix and the inverse force con-stants in symmetry coordinates obtained the last cycle, and also the ob-served frequencies are used to compute distortion rotational constants'*^2 and mean square amplitudes of v i b r a t i o n * ^ for the two ions. The results of the calculation are given in Table 7. The agreement between the calcu-148 lated and observed distortion constants for the H^ O ion is satisfactory in view of the fact that the fundemental frequencies instead of the harmonic frequencies are used. The distortion constants for the ion diffe r only slightly from the neutral molecule"*"^''''^. The mean square amplitudes of vibrations are found to be larger in both H^O and D^O ions than in their parent molecules'*" ^ . The greater f l e x i b i l i t y of the constituted atoms in the ions again indicates 0-H bonding character of the lb,orbital. 102 It has been pointed out by some authors that the 28, and the 168 170 C and D Rydberg states of water are very similar to one another. FCF calculations were carried out on these two Rydberg states of H^ O and D^ O. Unfortunately, divergence was obtained during the refinement of force constants. This is due to an unknown value for the y3' frequency. A set of force constants (Table 6) is chosen in such a way that values of Table 7 . Distortion Rotational Constants (cm ) and Mean Square Amplitudes of Vibrations (A 2) at 2 9 8 ° K of H 2 0 , D 2 0 , H S and D„S and their Molecular Ions in Various States H ^ V B ^ D 2 O + ( 2 B 1 ) D 2 O ( C ) H ^ V B ^ D 2 S + ( 2 B I ) D J 0 . 0 0 1 0 0 . 0 0 0 3 ( 0 . 0 0 0 8 ) A D R 0 . 0 3 4 8 0 . 0 1 0 7 ( 0 . 0 4 5 ) D J K - 0 . 0 0 4 5 - 0 . 0 0 1 3 ( 0 . 0 0 4 ) R5 0 . 0 0 1 1 0 . 0 0 0 3 R6 - 0 . 0 0 0 1 0 . 0 0 0 0 Sj 0 . 0 0 0 5 0 . 0 0 0 1 1*_ Y 0 . 0 0 5 6 1 0 . 0 0 4 1 0 1,*. T 0 . 0 1 5 7 6 0 . 0 1 1 3 6 0 . 0 0 0 2 0 . 0 0 0 4 0 . 0 0 0 1 0 . 0 0 9 6 0 . 0 0 2 0 0 . 0 0 0 5 - 0 . 0 0 1 0 - 0 . 0 0 0 7 - 0 . 0 0 0 2 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 O.'OOOl 0.0U735 0.00527 0.02030 0.01411 Experimental data from ref. 1 4 8 in parentheses. - 57 -171 172 f equal to that predicted from Badger's rule ' ; viz. 5.490 and 5.464 md*n//\ for the C and D states respectively. Badger's rule states that the logarithm of stretching force constants j - A 8 of a series of molecules i s related linearly to the logarithm of the corresponding bond length . The resultant geometry of D^ O in the C and D states is given in Table 2 and the calculated bond lengths of the C state differs appre- • 169 ciably from experimental values . The fundamental frequencies used instead of the harmonic frequencies in the calculation cannot account for such a large deviation. Therefore, method A including the variation in electronic transition moment (section 2.2.4) is applied to this state and the calculated structural parameters and intensities are the same as the experimental data 1 6^ with T 0 H and T W 0 M equal to 6.65 and 1.20 x 10^° - '/i -1 g cm respectively. The quantity T 0 H becomes smaller for lower values of the observed intensity used for nu = 1 (in this case, the error limit is + 0.2!.). Using the calculated Coriolis coupling c o n s t a n t s 1 6 3 , 1 6 4 (which describe interations between rotation and vibration) from this work and 169 173 observed g values of the C state of D 2 O , the inertia defect is found • i 169 to be 0.075 a n u A , in comparison to the experimental value , 0.074 amu A . Inertia defects of H^ O and D^ O in the B, states are predicted to be 0.089 and 0.117 amu.A respectively by assuming that the contribution from the electronic defect is negligible. (b) 2 8 , State of the Molecular Ions, H2S, D2S, H2Se and H2Te. Determination of the geometry of the H2S and D2S ions requires knowledge of d_ as well as _Ls_ and Ls' . The frequencies (Table 4) and inten-125 s i t i e s of peaks observed in the corresponding PE bands were employed to - 58 -compute d_. Ls_ and Ls" matrices were generated from a normal coordinate analysis. For the neutral molecules, convergence in calculated frequencies to within 1 cm 1 was obtained. However, this is not the case for the ions 171 172 again because is not available. So Badger's rule ' is used as a criterion for choosing a set of force constants which reproduce well the observed frequencies. The structural parameters (Table 2) obtained for the two ions are nearly the same as their parent molecules. Centri-fugal distortion constants and mean amplitudes of vibrations of the ions were evaluated and are lis t e d in Table 7. The ' 8 , states of H 2Se +and H 2Te + have been studied by PES 1 4 7. Only excitation of symmetric stretching frequencies was observed in the PE spectra. No electron spectra were reported for their deuterated deri-vatives. In this case, a force constant analysis for these ions is not applicable since the number of unknown force constants (even in the simple valence force f i e l d , SVFF, with only diagonal force constants being con-sidered) exceeds the number of observed frequencies. Therefore, Ls' is assumed to be equal to Ls_. Values of f gi.ven in Table 6 for H2Se and H2Te ions were approximated by the method described in appendix III. Owing to insufficient observed data for these two ions, no computation of dis-tortion constants or mean square amplitudes of vibration was carried out. (c) *A , State of the Molecular Ions H20 and D20 The PE spectra of H20 and D20 corresponding to the *A• state 101 2 147 have been obtained by Brundle et al , Asbrink et al and Potts et al 147 It has been mentioned by Potts et al that the vibrational structures of both ions displayed in the spectra dif f e r from those of other hydrides - 59 -of the same group by their rather constant spacings in a progression of 14' . This led the authors to conclude that these ions are probably linear in this state. Using the observed frequencies^'^\ and the e s t i -mated intensities of various peaks in the spectra, bond angles of the two ions are found to be close to 180° (Table 2). 108 It has been mentioned by Botter and Rosenstock that i f there is a large change in geometry of a molecule during ionization, there i s no longer a one to one correlation between the symmetry coordinates of an ion and i t s parent molecule, i.e. expression (2.21) is invalid, but they are related to each other by S' = Z * S + A S ( 3 - 4 ) where the off diagonal elements of the transformation matrix ZQ. measures the mixing of the symmetry coordinates. The matrix T. i s now given by, I = (Li)"' Z a Ls (3-5) From the result of the calculation, the off-diagonal elements of the Z q matrix, and hence 1_ have values comparable to diagonal elements. This indicates that mixing between symmetry coordinates of the same symmetry i s appreciable. 3.2 Nitrous Oxide 3.2.1 Introduction 174 The conventional FCF calculation (method B) has been applied to the X2TT and A * Z + states of the molecular ion of N^ O without success owing to the large discrepancy between observed and calculated FCF's. Recently, the two ionic states have been studied by using the optical - 60 -technique by which force constants j-; , vibrational frequencies v{ and geometries of the ions (both are found to be linear) are obtained. In light of the data available, we have reinvestigated the differences in bond lengths Alf; of N^ O between the ground states and the X ZTT and A'Z1" states by u t i l i z i n g the FCF's derived from i t s PE spectrum*''" 35 and the generating function method . However, only the experimental values in the A ZL* state can be reproduced. With regard to the X *TT state, the treatment is modified by including the variation of electronic transition moment. Both T N N and T N 0 in expression (2.62) are adjusted to give a value of d^ and, hence, Ay N f J , A T N 0 and calculated intensities that match well with the observed values. The L_s and Ls ' matrices and force constants (in GVFF) of both molecules and ions are generated from 175-178 a normal coordinate analysis by u t i l i z i n g the observed frequencies of Ni 0 and N^8 0 in various electronic states. The symmetry coordinates, Gs and Fs_ matrices used in the present calculation are the same given in ref. 160. 3.2.2 Results and Discussion (a) Geometry of N 20 + in the X 2TT and A 2 Z + States 179 Physical constants such as force constants , vibrational 175-178 frequencies adopted in the conventional FCF calculation, together 180 with the bond angle ck and bond length r^ and A r . in the ground state as well as the two ionic states are given in Table 8. Here j-m and fur, denote the stretching force constant of the N-N and N-0 bonds respectively, j - N N 0 the bending force constant and j - N N , N O t n e D O n d stretch-bond stretch interaction force constant. FCF's thus deduced using parameters given in Table 9 agree well with experimental values. Large - 6 1 -Table 8. Calculated Force Constants (mdyn/A), Bond lengths (A), Observed Vibrational Frequencies (cm and Structural Parameters of N^ O in the X , and the Molecular Ions in the X 'TT and A *I + States N20 N20 x • ! • K + X 'TT f , 18.0la 12.43 18.70 NN £... 11.33a 8.06 14.41 NO f n 0.486a 0.29 0.53 NNO fNN,NO 1 A 1 * 2 ' 2 8 -°-°6 U>, 2223.76b 1737.65° 2451.70° ^2 1284.91b 1126.47° 1345.52° L>i 588.77b 456.80° 614.10° ANNO 180 b 180 ° 180 ° obs d A rNN 0.0259° 0.0113 A r ° b S -0.0062° -0.0496° NO A r ° a l d 0.0174G 0.00686 NN rr (0.0259) A r ° a l d 0.00486 -0.03916 (-0.0062)f aRef. 179. bRef. 176. °Ref. 175. ^ r ^ - r. . e f From conventional FCF calculation. From method involving change in 20 electronic transition moment with and TN0 -1.704 and 0.515 x 10 qm^ ' respectively. - 6 2 -Table 9. Calculated and Observed Franck-Condon Factors in the X aTT and A ' l * States of N_0+ Vibrational X *TT A 2 L* Level Obs. Cald Cald Obs. Cald ra1 m2 m3 FCFa FCFa FCFC FCF FCF 0 0 0 0.91 0.91d 0.91d 0.749 0.749d 1 0 0 0.03 0.03 0.03 0.073 0.069 2 0 0 0.01 0.01 0.004 0.001 0 1 0 0.05 0.05 0.05 0.164 0.169 0 2 0 0.01 0.01 0.00 0.013 1 1 0 0.01 0.00 0.00 0.008 0.012 a b Ref. 11. From conventional FCF calculation (method B). From calculation including change in electronic transition moment with I N N and T N 0 -1.704 and 0.515 x 10 2 0 gm"'/2 cm 1 respectively. dAssigned to have value same as observed intensity. - 63 -discrepancies obtained between calculated and observed FCF's for the 174 X > TT transition by the previous worker are mainly due to the use of incorrect force constants and also neglect of variation in electronic transition moment as well (see discussion later). 33-35 Determination of Ar. by means of the current methods in FCF l calculations gives only the magnitude but not the sign of the change. For example, application of method B to the X'Z+ * X'TT transition gives two sets of parameters, with Ar N N and A r N 0 respectively, 0.0174 and 0.0048 A, and 0.0090 and -0.0244 A which can reproduce well the observed FCF's. Since some physical properties in the two ionic states studied are 175 known experimentally , i t is worthwhile to test the criteria given in section 2.2.1b. According to these criteria, the sign of AV"NN and A YH0 in the X 1 TT is predicted to be (i) positive and negative respectively from 59 a CND0/2 calculation , (ii) both positive from the variation in force constants or vibrational frequencies, and ( i i i ) same as (ii) in accordance 181 with the bond order change . Also, bond lengthening for both y N N and f N 0 in the A 2 Z + state in comparison with that of the ground state is ex-pected from the first two criteria. However, surprisingly, none of the above guidelines explains the experimental result satisfactorily. This is somewhat unexpected especially for the second criteria where the magnitude of the force constants derived should indicate the bond strength in both the ground and ionic states. - 64 -(b) Variation in Electronic Transition Moment in the X ' l * - * ^Transition The conventional FCF calculation on the X  XT\ state does not give the correct value of AT"N0 . The deviation cannot be explained by merely the anharmonicity effect. This leads us to reinvestigate the problem by introducing the change of electronic transition moment. The method used in computation has been described in section 2.2.4. Only one set of Tc's > -1.704 and 0.515 x 10^° g~ / l cm 1 for T N N and T N 0 respectively are used, which yields A rjs and the calculated intensities in good agreement with experimental data (Tables 8 and 9). The absolute value of T N N is found to be larger than that of T N O owing to the large value of one of the off-diagonal elements in the Ls 7 matrix and parallels the greater change in the NN bond length compared to the NO bond. It i s interesting to note that the magnitude of the transition probability for the l * * 1 vibrational mode with vibrational quantum number equal to two decreases with diminishing values of Tt but is rather insen-sitive to other I s . The physical significance of the sign of T i obtained is not clear, and l i t t l e work has been done on this aspect. (c) Force Constants in the Molecule and the I ons of ^ 0 182 It has been shown that the interaction force constant -f^^o is important in deducing correct bond stretching force constants for the neutral molecule. A normal coordinate analysis in GVFF gives values 18.01 ° 179 and 11.33 M(I-3n/A for f-NN and - f N 0 respectively while the same treatment in SVFF (this work) gives values 14.09 and 14.30  mdH n/% for fw and j-m respectively, which is obviously wrong. Also the off-diagonal - 65 -elements of the L_s_ matrix differ appreciably from those using GVFF. The quantity j - N N Q reflects the resonance effect between the NO and NN bonds. 183 The interaction force constants as mentioned by Jones may pro-vide valuable information about the bonding character of a species. The interaction displacement coordinate ( S N N ) N O , the change in N-N bond length to minimize the energy after a unit positive change in NO bond distance, is related to fun by f HH, NO = ~ (SNNJNO J~HH (3-6) The quantity ( S N l v J ) N 0 is found to be -0.08, -0.18 and 0.00 A for the X ' Z + , X 2TT and A *Z f states respectively. This indicates that the resonance effect i s greater in the X 1TT state, which agrees with 59 the result of a CND0/2 calculation that the highest occupied nonbonding TT orbital possesses weak NN bonding and NO antibonding character and the highest 6 orbital indicates strong NN and NO bonding character. 112 Application of the modified Urey-Bradley force f i e l d with six force constants, K N S , K N 0 , H N N O , f N N , N 0 , F and F' has been attempted to obtain more information about the potential energy surface in the two ionic states studied. However, no convergence in the force constant refine-ment was obtained owing to large correlations between them. 3.3 Dihaloethylenes 3.3.1 Introduction Today, the application of FCF calculations is usually limited to triatomic or tetraatomic molecules owing to the overlapping nature of the - 66 -PE bands of the molecule, the small number of ionic frequencies observed, the large uncertainty associated with these frequencies, the dimension of the matrices handled in the normal coordinate analysis, and also c r i t e r i a in choosing a reasonable set of calculated structural parameters. In this section, we report the FCF calculations on the geometry of a l l the gem and cis and trans 1,2 dihaloethylenes C^A^L^ molecular ions (except gem diiodo-ethylene) in some ionic states. The choice of calculated ionic geometry is based on the bonding properties of the neutral molecules. From the result obtained, the operation of various mechanisms proposed by Coulson 184 and Luz on structural changes upon ionization are discussed. In the course of this work, we found that most force constant 185-187 . , - .. 188-190 . ., . _ analyses with few exceptions on the m-plane vibration of dihaloethylenes are carried out on the assumption of the transferability of force constants among these molecules in either the Urey-Bradley force f i e l d or, in general valence force f i e l d . Some of the calculated force constants contradict each other 1^ 51^9, 190^ Therefore, force constants of a l l the dihaloethylenes are recalculated by using the modified Urey-Bradl f o r ce f i e l d to o b t a i n consistent results, and also different sets of force constants are used for different isomers of the same dihaloethylene. 3.3.2 Method of Calculation The gem, cis and trans 1,2 dihaloethylenes studied are found to be p l a n a r 1 " ' 0 , w i t h symmetry C^yy C^y and C2^ respectively. The symmetry distribution of the in-plane vibrations is 5 a , + 4b, for both cis and gem isomers, and is 5 a + 4 5 ^ for trans. A representative of each type of internal coordinate for a general, planar ethylene molecule is shown in Fig. 5 . Figure 5. Internal coordinates of substituted ethylene. - 68 -The symmetry coordinates used for cis, 1,2 difluoroethylenes in 196 the present calculation are similar to those given by Ziomek et al and are given as, a, : S, = jf (* r+ + ^ S 3 = ~ ( A ^ 3 + A4>* + A<j>5 + A $ 6 - 2 A d 3 + - 2 A c ( 5 t ) 5 4 = ( A r 3 + A r s ) 55 = (-A4>3 + A<f\ - A ^ s + A4>J = ^ ( M»3 + + A9> s t A4>6 + A W 3 + + A d * ) = c (3.7) S s = T| (Ar 4 - A r 6 ) 5 7 = j ± ( A r 3 - A r 5 ) 5 8 = jj= (A4>3 + A<P4 -A4^s -A4>6 - 2Ad3i + 2Ac/ 5<) 59 = (-A4>3 + A<K + A4>J - A4>«) S; = ^ ( A4>3 + A4\ - A<t>5 - A4>6 + Ac<J+ - Ad5<) -where r. and r. denote the C-H bond lengths and r„ and r. the C-F bond 4 6 6 3 4 lengths. With regard to trans 1,2 difluoroethylene, the symmetry coordinat used are identical to those of cis except that a, and b, are replaced by a^ and b^ respectively and also the subscripts 5 and 6 for the internal coordinates interchange. In this case, r^ and r^ are the C-H bond lengths while r_ and r. are the C-F bond lengths. 5 o - 69 -The symmetry coordinates used for gem difluoroethylene are 197 different from cis or trans 1,2 difluoroethylene and are in the form S, - Ag S+ = 7= ( A r s + A r & ) b 2 : Ss = j f (Ar 3 - A r + ) 5 7 = ^ ( A r y - A U 5 8 = j~ (A<t>3-A4>4) = (A4>3 - A4>6) Here r„ and r,. are the C-H bond distances and r r and r, the C-F bond 3 4 5 6 distances. As for dichloro-, dibromo- and diiodoethylenes, the symmetry coordinates adopted in the present calculation are the same as those given by expressions (3.7) and (3.8) except that S_, and Sg interchange with each other. The modified Urey-Bradley force field (MUBFF) employed in this work can be written as VM U B F F = V U E P F + V A (3-9) - 70 -The complete expression for V U B F P in terms of internal coordinates is , 198 given by 2 2 - ^ Kcx,(nAr c) + i ; ; ' 3 H C X L ( I ^ H A ^ ) 2 + H X jx 4 (Ys M ( A ^ J 1 + H X j X 6 (Y5 Y6) ( A c ^ ) * + FWAfc>* + F X s X 6 ( A ^ ) * t (3.10) 2 FX; 4^ ( ^ 4 A ^ 3 + ) + 2 FX;X6 A ^ 5 J + 2 cx;Xs ( ^ 3 5 + 2 cx;X6 A ^ ) / 198 with C and C as nonbonded repulsive force constants. C ' is related to C by a factor -0.1. The interaction potential function V A is 2 V A = K H H < A r C H ) ( A Y C M ) + K X X ( A r c x ) (Arc'x) + Z i + j e ( j (A<t>.KA4V 9 (rtYj)* (3 . i i ) where h H H , h x x and e.,j are the interaction force constants. The 186 199 addition of h H H and e.\j accords with other work ' and the second term in eqn. (3.11) should be important in dihaloethylenes owing to the conjugative effect between carbon TT orbitals and halogen lone pairs. - 71 -The inverse kinetic energy matrix Gs_ and the force constant matrix Fs_ using symmetry coordinates (3.7) and (3.8) are obtained by ^ , , _ , 109,200 standard methods The force constant calculation was carried out by the least squares 185-187 f i t program described earlier to reproduce observed frequencies ' 201-207 of the molecules and their deuterated derivatives. The i n i t i a l sets of force constants used in the iterative method were estimated from .185-189,197 previous work The FCF calculations are carried out using method A, and J^ s is used instead of Ls' since the ionic frequencies observed are not sufficient to perform a normal coordinate analysis. The FCF's of the molecules studied are derived from refs. 208 and 209 as well as from section 4.5. Some ionic geometries of dihaloethylenes obtained are shown in Table 10. It should be mentioned that the geometries derived are based on the planarity of the ions. The perturbations on FCF's arising from spin orbit coupling and variation in electronic transition moment is neglected in this treat-ment . 3.3.3 Results and Discussion (a) Modified Urey-Bradley Force Constants of Dihaloethylenes Convergence was obtained only for some of the dihaloethylenes. With regard to others, force constants were obtained in a t r i a l and error fashion to give a good frequency f i t (within 3.2%) as well as to agree with 171 172 Badger's rule ' qualitatively, i.e. shorter bond lengths are associated with greater stretching force constants. e H H and e X x in cis 1.2 dihalo-ethylenes and in gem dihaloethylenes are fixed to be zero for - 72 -Table 10. Geometrical Changes in Dihaloethylene upon Ionization Ionic ,?, State A rCH ( A j Ar c c(A) Ar c x(A) AoiHCX Ac< x c x A ^ H C H Cis 1,2 C 2H 2F 2 -0.U01 0.110 -0.055 7.2 -3.7 -3.6 5 7\, 0.004 0.022 0.027 -0.5 -0.7 1,3 3*B2 0,050 -0.016 0.005 2.5 0.2 -2.7 Cis 1,2 C2H2C12 2'6, 0.015 0.120 -0.001 2.3 -1.5 -0.8 or 0.012 0.081 -0.081 4.2 -0.2 -0.4 1 XAZ 0.002 0.025 0.052 -1.3 -0.8 2.1 Cis 1,2 C 2H 2Br 2 2'B, 0.015 0.076 -0.084 4.0 0.1 -4.-1 1 ^  0.001 0.013 0.039 -0.3 -0.6 0.9 Cis 1,2 C 2H 2I 2 2*B, 0.003 0.042 -0.055 1.2 0.2 -1.4 Trans 1,2 C H F 2*AU -0.004 0.133 -0.087 4.6 -1.4 -3.2 5*A9 -0.009 0.020 0.051 -4.9 7.4 -2.5 4*Btt 0.045 0.022 0.019 2.5 -1.9 -0.7 Trans 1,2 C2H2C12 2 aA u 0.009 0.202 -0.046 17.8 -10.0 -7.8 5*A9 0.007 0.028 0.033 2.3 -3.2 0.8 1 *B9 0-009 0.033 0.038 2.7 -3.7 1.0 Trans 1,2 C 2H 2Br 2 4 J8 a 0.004 0.004 0.049 2.1 -3.9 1.8 I JB3 0.005 0.004 0.062 1.9 -3.5 1.6 Trans 1,2 C H I 4-2Bu 0.004 0.005 0.027 1.5 -2.2 0.6 Gem C 2H 2F 2 2*6, 0.013 0.142 -0.033 -1.8 -3.0 3.7 6.1 or 0.017 0.128 -0.065 -2.4 -4.0 4.8 7.9 5*A, -0.056 0.013 -0.022 -0.9 9.2 1.7 18.5 Gem C2H2C12 2*6, -0.023 0.067 -0.057 -1.5 -8.8 3.0 -17.6 Gem C 2H 2Br 2 2*5, -0.011 0.071 -0.063 -1.7 -6.2 3.5 12.4 - 73 -convenience in calculation owing to their smallness in value and insigni-ficance in improving the frequency f i t . As for cis 1 , 2 diiodoethylene, . , . _• . 2 0 6 , 2 0 7 . . „ . . , . only vibrational frequencies of one isotopic species are observed. Therefore, only eleven force constants are determined with F C H a n d H C C H transferred from trans 1 ,2 diiodoethylene. The calculated force constants for a l l their dihaloethylenes are listed in Table 11. Using force constants given in Table 11, the in-plane vibrational modes of cis 1 , 2 Q^Q^l^ a r e predicted to be 2242, 1475, 779, 480 and 81 cm 1 for the a , species and 2212, 1064, 590 and 391 cm"1 for the b , species. 210 It has been mentioned that the introduction of interaction force constants between angles with one side as a C-C bond i s necessary to account for the higher energy of the b,g rocking mode and the lower energy of the 186 bzu_ mode of ethylene. In theirstudy of bromoethylenes , Scherer and Overend found that only the interaction between trans CH bending coordinates is important. However, this is not r e a l i s t i c judging by our results (Table 11) for a series of dihaloethylenes. The trans interaction cross term eij i s always found to be necessary to reproduce the observed frequencies. The absolute value of h Xx ' 1 S found to be smaller in the trans isomers than in the gem or cis isomers. This is related to a lesser amount of electron derealization between halogen atoms in the former isomer, and is also consistent with a smaller difference between the symmetric Us and the asymmetric Va% C-X stretching frequency observed in the trans 211 isomer . When Vs of the trans isomer is plotted against l^as , a good straight line is obtained which can be represented by the equation (in cm *) Vat. = 1.5 U$ + 5 4 5 3-12) Table 11. Urey-Bradley Force Constants (mdyn/A) of Dihaloethylenes KCC K C X K CH HCCX HXCII HXCX H CCH ^ C H FCX F X X FCH FHH FHX C HX CIIH • CXX h l |H e l l l i ^iT^ \\ \\ Cis 1,2 CHFCHF 7.312 S.688 4.911 0.485 0.074 0.191 N 0.600 0 .398 0 . 9 4 3 0.102 -0.1S9 -0.100 0.0 0 . 0 9 S 0.0 -0.42S Cis l , 2 a i C l C ! I C l 6.814 3.5S6 5.262 -0.172 0.258 0.410 1.604 -0.075 -0.225 0.064 0.156 -0.043 0.0 -0.069 0.0 O.C Cis 1,2 CKBrCHBr 6.481 2.648 5.638 0.179 0.417 0.486 1.625 -0.119 -0.878 0.079 -0.075 0.009 0.0 -0.026 0.0 0.550 Cis 1,2 c m G i l 6.402 2.021 4.502 0.102 0.004 0.240 0.418 0.537 0.299 0.005 0.151 Trans 1,2 CHFCHF 7.123 5.655 S.315 0 . 0 0 2 0.202 0.241 0.266 0.700 0.369 0.625 -0.278 0.105 -0.392 0 . 2 5 7 -0.302 Trans 1,2 DiClCHCl 6.903 3.511 5.416 -0.081 0.187 0.008 0.093 -0.624 0.398 0.624 -0.230 0.156 -0.5SS -0.039 0.056 Trans 1.2Ch3rCHBr 6.648 2.744 5.302 0.23S 0.542 0.356 0.986 0.649 -0.982 -0.156 0.OS9 -0.173 0.530 0.332 0.125 Trans 1,2 CHICHI 6.137 2.212 S.308 .0.133 0.538 0.239 1.168 0.537 -1.032 -0.137 0.113 0.261 0.009 -C.iS2 0.1-6 Z'jr. C-IFCIr 7.693 5.702 S.376 0.130 0.689 0.163 0.788 1.562 0.598 0.822 -0.871 -0.152 0.304 -0.017 0.0 -0.33; COT. CHC1CHC1 7.052 3.581 4.885 0.128 0.130 0.249 0.247 0.293 -1.924 -0.624 0.698 0.624 -0.397 0.0 0.0 -0.41S Gca CHBrCHBr 6.603 2.559 4.502 -0.036 0.447 0.126 0.183 0.402 -1.861 0.110 0.685 0.240 -0.328 0.191 0.0 -C.45! - 75 -(b) Geometries of the Dihaloethylene Molecular Ions The sign of cL determined from the F C F method A discussed pre-viously is not known because i t is derived from the square root of the F C F's. The choice of sign for bond lengths and bond angles is based on the bonding property of the neutral molecule (criter i a given in section 2.2.1 b). Table 12 l i s t s the overall change in bond length and angle of difluoro- and dibromoethylenes estimated from the results of a CND0/2 59 calculation . The bonding property of dibromo- and diiodoethylene should be similar to dichloroethylene. In Table 12, a positive sign means bond lengthening or angle widening. Those changes in internal coordinates which cannot be predicted from the c r i t e r i a are denoted by question marks. The geometric changes given in Table 10 are chosen to reproduce the expected sign varation given in Table 12 as close as possible. Some-times, two sets of parameters may f i t the same c r i t e r i a . In this case, both sets are listed. In general, the agreement i s good for the ground ionic states. The discrepancy in sign between calculated and predicted change in bond angle is probably due to modification of the repulsive force between nonbonding atoms during the alteration of bond distances. Ionization of an electron from the highest occupied TT orbital results in a large change in the C - C bond and sometimes in the C - X bond also. A r c c is largest for dichloroethylenes, and is smaller for dibromoethylenes, and then diiodoethylenes, while the opposite i s true for A r c x (except trans 1,2 dihaloethylenes). This agrees with the fact that the conjugative effect is more important, as well as 8c=c (sections 4.5 and 4.6) becoming larger for heavier halogen atoms in the halogenated ethylenes studied. A Tec °f difluoroethylenes is found to be the same or even less than that - 76 -Table 12. Predicted Changes3 in Geometry of Dihaloethylenes upon Ionization S°tate A rCH A 1CC ' A rCX A*HCX A+X A * H A^XCX A<< HCH Cis 1,2 C 2H 2F 2 2 JB, ? + - ? - ? 5 M , + + + ? + -3 a 6 i • + " + ? Cis 1,2 C2H2C12 2AB, ? + - ? - ? ? _ + ? - ? Trans 1,2 C 2H 2F 2 2'A ? + - i 5vC + + + + ? - + ? 4*8* Trans 1,2 C H Ci ? + - ? + 2 7V 5 " A , 4 * B ; + + ? l A 6 3 Gem C 2H 2F 2 + ?  + ? - + ? - ? + 2 *B, ? + - ? 5 V \ , + + - - " + + Gem C2H2C12 2'B, ? + - ? + ? aPositive sign means bond lengthening or angle widening. Question mark denotes that the change in internal coordinate cannot be estimated from the treatment of nodal repulsive force. - 77 -of dichloroethylenes. This is unexpected from the above reasoning for the other dihaloethylenes, and is probably due to reorganization of electron distribution in the compounds on ionization. At f i r s t glance, i t may be surprising to find that &YCH and A 4^ are nonzero in the ground ionic state. This arises from the nonvanishing value of the off-diagonal elements in the Ls_ matrix used in the calculation. The ionic frequency of the dihaloethylenes deduced from the PE band (refs. 208, 209 and section 4.5 also) is sometimes midway between the two ground state fundamentals, and i t is d i f f i c u l t to decide on the correct assignment. Results from the FCF calculation are useful in assigning the frequency. For example, the frequencies 1360, 992 and 1185 cm 1 respectively observed in the f i r s t , second and fourth PE band of cis 1,2 difluoroethylene are assigned to come from L»+ , l>+ , and U3' from the result of the FCF treatment. In the same manner, 1234 and 1125 cm 1 respectively obtained in the fine structure of the f i r s t and fourth PE band of trans 1,2 difluoro-ethylene are attributed to excitation of D's and V'^ modes. (c) Origin of the Geometrical Change on Ionization 184 Three mechanisms have been proposed by Coulson and Luz to describe the difference in geometry of in the ground neutral state and the lowest ionic state. They are (i) change in bond order, ( i i ) change in electrostatic interaction, and ( i i i ) change in repulsive exchange forces. The last mechanism is not clear since the available wavefunctions are not good enough. In the following paragraph, we are going to discuss the importance of mechanisms (i) and ( i i ) operating on the ground ionic state of difluoro- and dichloroethylenes. - 78 -In the dihaloethylenes, the highest occupied MO possesses C-C TT bonding and C-X antibonding character. Hence, the removal of an electron from this orbital weakens the C-C bond (lowers the bond order) but streng-thens the C-X bond. Quantitatively, the changes in C-C and C-Cl bonds are related to the bond order change ALL by A C ' = - 0.16 ALlcc / ( I + 0 . 2 4 AUcc) (3.13) A r c " | = - 0.205 A U c c , / ( 0 77 + 0.235 A Ilea) (3.14) 184 212 from equation ' originally derived for hydrocarbons. Thus AY"^1 is found to have a value 50% or less of A f c c from the FCF calculation for both difluoro- and dichloroethylenes given in Table 10. However, AT",1 s of cis, trans 1,2 and gem isomers are respectively -0.06, -0.06 and -0.05 A, compared with Ar c cjs -0.08 (or 0.00), -0.05 and -0.06 A. This indicates that mechanism (i) is important in altering the C-X bond rather than the C-C bond. Qualitatively, the bond order approach for bond angle changes is good for a l l the dihaloethylenes as mentioned before. However, a calculation on the bond order change in the C1CC1 angle in gem dichloroethylene gives A o s c ^ i 0.9° which is much smaller than that in Table 10, 3.0°. 184 In addition to mechanism (i) described, electrostatic forces between carbon and halogen atoms, or carbon and carbon, as well as halogen and halogen themselves in the ion may be active in the rearrangement of molecular coordinates (mechanism ( i i ) ) . Assuming the localization of the TT electron in the cai'bon skeleton, removal of an electron from the TT orbital leaves + e. charge on each carbon nucleus. Hence, there are coulombic attractive forces between carbon and halogen atoms. Using the same treatment as Coulson and Luz"^ 4 on C2CI4, A f "F' and Aol^f for gem dif luoroethylene have values of -0.03 A and 3.0 respectively, in good agreement with - 79 -o 0 -0.033 A and 3.7 from the FCF method. The C-F bond dipole moment used 213 is 1.43 Debyes from vinyl fluoride and the force constants are adopted from Table 11. In the case of gem dichloroethylene, the C-Cl dipole moment is taken to be 1.44 Debyes from C^H^Cl 2 1^. The values of &Tr°c\ and A O ( c i c c i obtained, -0.03 A and 1.34 deviate quite large from those 184 given in Table 10. As mentioned before , mechanism ( i i ) is predicted to be more important for C 2F^ than for C2C1^ and the result of our calculation supports this statement. Adopting the naive assumption that we have + ^  e charge on each carbon nucleus in the ion, then AC^ 1 is found to be about 50% of Af Cc • It seems that both mechanisms (i) and ( i i ) are active in C-C bond length changes. It should be mentioned that the two mechanisms predict that A r c c becomes larger for more electronegative halogen atoms in dihaloethylenes owing to the variation of the coefficient of the carbon atomic orbitals in the occupied TT orbital, as well as the diversity of the electron cloud in the C-C bond length. Because of the success of mechanisms (i) and ( i i ) in the estimation of structural changes in fluoroethylenes, a FCF calculation was carried out also on the ground ionic state of C 2F^ with FCF's derived from ref. 215 and the L_s_ matrix from ref. 216. On the assumption of the planarity of o o the ion, A1TCC , A1TCP and A d F C F are found to be 0.133 A, -0.062 A and 4.7° respectively. A r c * 1 is predicted to be 0.04 A from mechanism (i) while A r c " ' , ATcp and AdpcV are 0.04 A, -0.04 A and 4.8° respectively from mechanism ( i i ) . The force constants used are taken from ref. 217. - 80 -It seems that the widening of the fluorine-carbon-fluorine bond angle is mainly due to electrostatic forces between the carbon and fluorine atoms. - 8 1 -CHAPTER IV PHOTOELECTRON SPECTROSCOPY OF SOME HALOGENATED COMPOUNDS 4 .1 Fluorotrichloromethane and Fluorotribromomethane 4.1.1 Introduction The first four highest occupied MO's, a^, a^, e' and e" (C^v symmetry), of the trihalomethanes CHX^ , and their fluoro-substituted derivatives CFX^ (X = Cl,Br) are mainly contributed to from the formally nonbonding p orbitals of the halogen atom. The relative orderings of these orbitals have been the subject of several discussion. - 82 -Potts et a l " " suggested the electronic structures for chloroform and 4 4 2 2 4 2 4 2 bromoform to be (e") (e') ( a ^ (a ) and (e") (a ) (e') (a 2) res-pectively, in order of decreasing energy. To distinguish between the f i r s t two highest occupied e orbitals, e' and e " are chosen in such a way that they transform as different representations in the limit of a planar CX^ 4 2 4 2 group. A different assignment, ( e') (a^) (e") (a 2) , was given 218 by Dixon et al for both molecules. Recently, CNDO/2 calculations on the 219 chloromethanes were reported by Katsumata and Kimura and their results 4 4 4 2 gave the ordering of the orbitals as (e") (a 2) (e') (a^) . In planar 41 42 molecules, the perfluoro effect ' has been found to be active in lowering the 6 and TT orbital energies (within one ev for TT orbitals but greater than two ev for 6 orbitals) upon fluorine substitution of hydrogen in the molecules. A similar kind of effect may also operate on chloro- and bromoform with replacement of the hydrogen by a fluorine atom, in which large energy separations exist between 5 and lone pair orbitals. We have therefore measured the PE spectra of fluoro-substituted CHBr^ and CHC13, i.e. CFBr^ and CFC1 3, from which the energetic ordering of the nonbonding orbitals of the trihalo-methane can be deduced. The observed IP's for the two compounds and their assignment (which are j u s t i f i e d later) are given in Table 13. 4.1.2 Interpretation of the Spectra CFBr 3 The He I PE spectrum of CFBr^ is shown in Fig. 6. Bands with IP's below 13 ev are assigned as arising from the bromine lone pairs (a,,, e" ,e' and a^ orbitals in order of increasing IP) on the basis of the intensities of these bands, as well as for reasons given below. The f i r s t sharp peak in the spectrum reflects the nonbonding character of the a 0 orbital from - 83 -Table 13. Observed Vertical Ionization Potentials in the Photo-electron Spectra of Fluorotribromomethane and Fluoro-trichloromethane Orbital CFCl 3 a(ev) CFBr 3 a(ev) la 11.77 (1) 10.67 (1) 5e" 12.16 (1) 11.14 b(l) 4e» 12.95 (1) 11.81 b(l) 5a]L 13.46 (1) 12.38 (1) 3e» 15.04 (2) 13.96 (2) 4a : 18.44 (2) 17.59 (7) The reproductibility of the last d i g i t is shown inside the bracket. bMean of the fine structure components. - fr8 -- 85 -which the electron is removed. The IP of this orbital is shifted to a higher value by 0.2 ev with respect to bromoform. The fourth band (with vertical IP 12.38 ev) exhibits well resolved vibrational structure {Fig. 6) starting at 12.28 ev. The result of a FCF calculation (method A) using FCF's given in Table 14 indicates that the fine structure arises from a single progression of \j* rather than from a composite of and l>3 The C-F and C-Br bonds, and the BrCBr angle are found to be increased by o 0.11, 0.017 A, and 3 , respectively, during the ionization process. The second and third bands in the spectrum, relating to the loss of electrons from the e orbitals, consist of well-resolved doublets with separations of 0.25 and 0.21 ev respectively. The sp l i t t i n g of the f i r s t 30 30 band is 0.25 ev, compared to thatobserved for PBr^ , 0.2; C^Br^ , 0 . 1 4 ; 97 97 OPBr^ , 0.25 and SPBr^ , 0.22 ev. Single quantum excitations of V 3 (213 cm *) and V, (874 cm *) are also observed in the f i r s t as well as the second E band. 221 The result of a CNDO/BW calculation shows that there is a weak C-Br antibonding character in the e." orbital, and there is weak C-Br bonding character in the e ' orbital although these two highest occupied e orbitals are essentially nonbonding. The vibrational spacings in V3 indicate that the f i r s t E band is derived from the e." orbital. Recently we have measured the PE spectrum of chloroform, which 30 has been recorded by Potts et al with a lower resolution spectrometer. The right hand shoulder of the second E band (Fig. 7) exhibits two well * The vibrational modes Ut (C-F stretch), Vz (C-Br symmetric stretch), and 03 (C-Br^ symmetric bending) have values of 1 0 6 9 , 398 and 218 cm"l respectively in the neutral molecule 2 2 0. - 86 -Table 14. Franck-Condon Factors of CFBr^ in the Ionic State m^  Energy(ev) FCFC cald FCF Vibrational Spacings (mev) 0 12.251b 0.03 1 12.277 0,18 0.21 23 2 12.300 0.35 0.45 . 28 3 12.328 0.62 0.72 • 26 4 12.354 0.85 0.92 26 5 12.380 1.00 1.00 29 6 12.406 0.96 0.92 26 7 12.432 0.81 0.74 27 8 12.459 0.59 0.53 25 9 12.484 0.41 0.35 26 10 12.510 0.28 0.20 The intensity of a peak is assumed to be proportional to the height of the fine structure maximum. The value listed in the Table is chosen in such a way that the intensity of the highest peak is one unit. The value obtained by extrapolation. - 87 -Figure 7. The second PE band of chloroform. - 88 -resolved progressions i n Vz , the C-Br^ symmetric s t r e t c h i n g mode 76 with values s l i g h t l y less than that of the neutral molecule . The same r e s u l t was obtained f o r deuterated chloroform. This supports our assign-ment that the second PE band of CFBr^ i s r e l a t e d to the e.' o r b i t a l . The l a s t two PE bands of CFBr^ i n the 13-18 ev region are r e a d i l y assigned to the e' and CL, o r b i t a l s according to the MO c a l c u l a -t i o n . I t i s i n t e r e s t i n g to note that the Q.' o r b i t a l i s d e s t a b i l i z e d by about 0.8 ev with respect to the corresponding band of CHCl^. This r e f l e c t s the antibonding character of the &' o r b i t a l f o r the CF bond. The obser-vation p a r a l l e l s the r e s u l t of MO c a l c u l a t i o n s on t h i s molecule. CFC1 3 The PE spectrum of CFCl^ i s shown i n F i g . 8 and thus the IP's obtained are given i n Table 13. The r e l a t i v e areas of the f i r s t bands r e f l e c t the degeneracies of the l e v e l s involved and thus t h e i r ordering i s l i k e l y to be a, e, e, a. These o r b i t a l s contain a major c o n t r i b u t i o n from the c h l o r i n e p o r b i t a l s . The f i r s t band at 11.77 ev can r e a d i l y be assigned as a r i s i n g from the a^ nonbonding o r b i t a l . The next two bands are f a i r l y broad and asymmetrical with nonresolvable v i b r a t i o n a l f i n e struc-ture. The assignment of these two bands i s somewhat ambiguous. Assuming the o r b i t a l ordering to be the same as that of CFBr^, the experimental IP's at 12.16 and 12.96 ev correspond to the e" and e.' o r b i t a l s . However, the a l t e r n a t i v e assignment that the e' i s associated with a higher IP than e." cannot be d e f i n i t e l y r u l t e d out. In the fourth band, the v i b r a t i o n a l spac-ings « 1011, 641 and 303 cm * can be deduced. This suggested that a l l the three t o t a l l y symmetric v i b r a t i o n a l modes V{ , 1 ^ 2 and L>3 (1085, -1 222 535 and 350 cm r e s p e c t i v e l y i n the ground state ) are excited during - 90 -the ionization process. The last two bands are related to electron loss from the e ' and orbitals, and parallel to those in CHCl^ and CFBr^ mentioned previously. The e' orbital again shifts to lower energy with respect to CHCl^ presum-ably for the same reasons as we offered in explaining the downshift for the corresponding e.' orbital in CFBr^. The structure at 1 9 . 9 8 ev comes from low energy scattered electrons in the spectrometer i t s e l f . 4 . 1 . 3 Discussion From the analysis of the fluorotribromomethane spectrum, the e' orbital i s found to have a higher bonding energy than the e" orbital in this molecule. Following the assignment on the experimental IP's of CHBr^ 30 given by Potts et al , the <£' orbital i s stabilized by » 0 . 9 ev while the e" orbital is destabilized by & 0 . 7 ev from CHBr^ to CFBr^. The trends and the magnitudes of these shifts in energies in nonbonding orbital as influenced by the effect upon fluorine substitution of hydrogen seem to be too large in comparison to those shifts in the TT orbitals of planar 41 42 218 molecules ' . Because of this, Dixon et al's assignment on CHBr^ is preferred. When similar arguments are applied to CHCl^, the relative ordering of e ' and e." orbitals is found to agree with that proposed by 218 Dixon et al and by our work on chloroform. The correlation diagram (Fig. 9) shows that the energy levels of CFBr^ and CFCl^ have the same ordering with the former being uniformly shifted towards lower energies by « 1 ev. The same observation applies 97 to chloroform and bromoform, and phosphoryl chloride and bromide . This reflects the fact that the f i r s t five or six highest occupied orbitals are mainly built up from halogen p atomic orbitals. - 91 -3 a 1 18 H / "\ 3 a 1 3a-| / * 2 e ' 2 e ' 2 e ' H i > J 3 e ' / ' 3e- \ LJJ \ V , T / 4 a , 4 a , / V K - f f — 1 Q 2 \ 4 e " \ _ 3 e L l a 2 \ ' 4 a i %- 4 e" 10 H 1 l a 2 C H C l 3 C F C I 3 C F B r 3 C H B r 3 Figure 9. Correlation diagram for the f i r s t six highest occupied orbitals of CHX3 and CFH3 (X = Cl, Br). The data for CHCI3 and CIIBr3 are from Dixon et a l 2 1 8 and Potts et a l 3 0 . - 92 -In F i g . 10, the v e r t i c a l IP's of the non bonding o r b i t a l s l a 2 or IOJJ of CFX/ 0*, C H X 3 3 0 ' 2 1 8 , O P x / 7 ' 2 2 3 , B x / 0 and P x / 0 (X = F, C l , Br or I) are p l o t t e d against the Pauling e l e c t r o n e g a t i v i t y values of the halogen atoms. Straight l i n e s are obtained. The predicted IP's f o r l d 2 o r b i t a l s of CF 3I and 0 P I 3 > and f o r the I CX'Z o r b i t a l of P I 3 are 9.45, 10.28 and 9.36 ev r e s p e c t i v e l y . Following arguments s i m i l a r to those 224 by Kimura et at , the gradients of these l i n e s i n the pl o t s i n d i c a t e the contributions of halogen p atomic o r b i t a l s i n the t r i h a l o compounds. It i s i n t e r e s t i n g to note that the gradients i n the two series CHX 3 and 0PX 3 (Fig. 10) are nearly the same. It should be noted that our assignments of the f i r s t four nonbond-219 ing o r b i t a l s of CHC1 3 disagrees with that based on the CND0/2 c a l c u l a t i o n 219 However, i t i s well recognized that the r e s u l t i n g o r b i t a l sequences of the chloromethanes depend g r e a t l y on the s e l e c t i o n of parameter values f o r the chlorine atoms. The f a c t that no comparison was made between IP's of re l a t e d compounds may also lead to a d i f f e r e n t assignment. 225 While t h i s work was near completion, Doucet et a l presented a communication i n which they reported the PE spectrum of C F C l y Their data i s i n good agreement with ours, although they did not o f f e r a complete assignment of the observed IP's. 4.2 1,2 Dichloro-, 1,2 Dibromo- and 1,2 Diiodoethane 4.2.1 Introduction Compounds with a C-C sing l e bond such as the 1,2 dihaloethanes (CH 2X) 2 (X = C l , Br and I ) , e x i s t i n the vapor state as equilibrium mixtures of the gauche (C 2) and trans ( C ^ ) conformers because of r o t a t i o n about the * For CF^, the v e r t i c a l IP of the I Q 2 o r b i t a l i s considered the same as that of t, which under C 3 V symmetry s p l i t s into dz and €L . > L U Q_ 16 15 14 13 12 f-(a) <11 h o feioh LU > 9 r CFBr-/ - C H l 3 2.5 3.0 3.5 ELECTRONEGATIVITY CF 4 , / 4.0 > LU 16 15 14 13 12 f < 1 1 U 10 LU > 9 ( b ) B F 3 / O P B r 3 / / BCI 2.5 3.0 3.5 ELECTRONEGATIVITY Figure 10. (b) the IQj PR, J L 4 0 Plot of vertical IP's of (a) the i a a orbitals of CFX 3 3° and CHX3 3 0' 2 1 8, orbitals of 0PX3 ' and the 104 orbitals of BX 3 3 0 amd PX 3 3 0 with X = F, Cl, Br or I, against the Pauling electronegativity of the halogen atom. - 94 -C-C bond. Various physical methods such as infrared and Raman spectro-226-228 , .... . 229,230 . ^ 231,232 scopy , electron diffraction , microwave spectroscopy , 233 and X-ray diffraction have been used extensively to determine the mole-cular structures, populations and conformational energies of the conformers of the disubstituted ethanes. The trans isomers are generally found to be more stable than the gauche by 1.09 (77%)* and 1.68 kcal/mole (90%) for (CH 2C1) 2 and (CH 2Br) 2 respectively. Although no information of this sort is available for (CH 2I) 2, the s t a b i l i t y and population of the trans form of this molecule should be at least equal or even greater than that of (CH 2Br) 2 > in view of the larger nonbonded repulsions between the halogen • +u * -. -i 226-228 atoms in the former molecule In spite of the large amount of experimental data accumulated concerning the molecular structure of the dihaloethanes, l i t t l e work has been done on elucidation of the electronic structures, and the nature of the bonding in these molecules. In this section, we present the He I high resolution spectra of the 1,2-dihaloethanes, and discuss the interaction between the halogen LPMO's in these molecules, the enthalpy difference between the isomeric trans and gauche ions, as well as the geometry of the (CH 2C1) 2 ion in the f i r s t ionic state. Moreover, a correlation between IP's and certain physical properties w i l l be mentioned. The PE spectra of 1,2-dichloro-, 1,2-dibromo-, and 1,2-diiodoethane are shown in Figs. 11 - 13. The observed IP's are summarized in Table 15. Number in parenthesis refers to the concentration of the trans isomer in gas phase as determined by infrared spectroscopy*^, - S 6 -- 96 -- 98 -Table 15. Observed and Calculated Vertical IP's (ev) of 1,2 Dihalo-ethanesa (CH2C1)2 ( C H 2 B r^2 ( C H2 i : )2 Orbital Obs. Cald. b Obs. Cald. b Obs. Cald. b 11.22 11.22 10.57 10.58 9.50 9.49 11.39 11.39 10.63 10.63 9.56 9.55 11.55 11.55 10.96 10.95 10.08 10.08 11.83 11.83 11.08 11.07 10.26 10.27 13.68 13.68 12.61 12.60 11.50 11.51 14.39 14.39 13.84b 13.87 13.19 13.18 6,b9 15.28 15.28 14.88 14.85 14.69d 14.68 16.97d 16.87d 16.34d c l Experimental error within ± 0.02 ev. bThese IP's are calculated from values of parameters given in Table 16. c th fa is the i halogen LPMO. dExperimental error + 0.05 ev. - 99 4.2.2 Results and Discussion (a) One Electron Model for the Lone Pair O r b i t a l s of Trans 1,2- dihaloethanes (C^^) C l a s s i c a l l y , non-bonding electrons of halogen atoms in molecules can be considered to be l o c a l i z e d as lone p a i r s in the p o r b i t a l s . For 94 95 224 234 example, i n the ethyl halides EtX ' ' ' , the two highest occupied molecular o r b i t a l s are found to have an appreciable amount of halogen p character and may be regarded as LPMO's of the halogen atom. As the l o c a l c y l i n d r i c a l symmetry at the halogen atom i s disturbed to a n e g l i g i b l e 95 extent by, say, a methyl group , removal of electrons from these o r b i t a l s give r i s e to two intense sharp peaks i n the PE spectra, with separations 94 roughly equal to ^ x . In a trans dihaloethane, i f there i s no i n t e r -action between the LPMO's of the halogen atoms, i t s PE spectrum should be s i m i l a r to that of the corresponding ethyl h a l i d e . In f a c t , each of the two PE bands r e l a t i n g to i o n i z a t i o n from the LPMO's shows a double maximum (Fig. 11 - 13) with the separation being greater i n the one associated with the higher IP. This c l e a r l y i n d i c a t e s that there i s an appreciable amount of mixing between these LPMO's. The observation can be explained using a 44 one e l e c t r o n model s i m i l a r to that employed by B r o g l i and Heilbronner (section 2.1.5) f o r the case of the a l k y l bromides where both spin o r b i t coupling between the halogen LPMO's, and also the mixing between these o r b i t a l s and the 6 o r b i t a l s (section 2.1.4) are taken into consideration. 37 It i s , of course, assumed that Koopmans1 theorem can be applied i n these cases. Throughout t h i s work, the trans conformer i s considered to give r i s e to the main features i n the PE spectra. - 100 -Assuming that there is no interaction between the LPMO's of dihaloethane (Fig. 14, case a), then a l l these orbitals are degenerate. However, i f a direct spatial overlap between these LPMO's, that i s , a 77 78 through space interaction ' is present, the degeneracy of the orbital is removed (Fig. 14, case b). The four LPMO's w i l l combine with each other to give MO's of Qg, &u, b and b symmetry (Fig. 15) under the C 2f, point group. The through space parameter d which measures the degree of the interaction between the orbitals themselves i s a sensitive function of the X...X distance (« 3 A) and is estimated from expression (2.7). There is another kind of interaction called through bond inter-77 78 action ' by which LPMO's mix indirectly with each other through 6 MO's providing a l l of them belong to the same symmetry class. In other words, orbitals of the same symmetry repel one another. The degree of perturbati is inversely proportional to the energy separation of the two interacting orbitals. As a result of this through bond interaction, a l l LPMO's except [ 2 C L are destabilized, and the parameters Sa^, S b^ and S b u are used to describe the interaction. In view of the large energy difference between L^o^ and Qia^ (IP > 21 ev), the quantity S a u is small and may be assumed to be zero. It should be mentioned that the 6 orbital sequence given in Fig. 14 as well as in Table 15, namely, 64.^ > ^ 3ba > ^IH^ a m * 630.3 in order of increasing IP, i s chosen in the following way. According to the 59 221 CNDO calculations ' , the 6 MO ordering is 6 4 0 3 , 6 1 bg, S^ag, and <53ba . If this is true, then S b should be greater than S b because the through bond interaction is inversely proportional to the energy Figure 14. Effect of interaction on the molecular orbitals of XQ^Cl^X (a) no perturbation; (b) through space interaction; (c) through bond interaction added; and (d) spin orbit coupling added. Figure 15. The lone pair orbitals of the halogen atoms in XCH CH - 103 -difference between two interacting orbitals (section 2.1.4), in contrast to our observation (see discussion later). Because of this we put above 6ib g , this assignment, being confirmed by the following calculation on the relative magnitudes of the through bond and through space interactions. In addition to the through space and through bond interactions mentioned above, spin orbit coupling may also play an important part between 94 95 LPMO's,according to the investigations of Heilbronner and his coworkers ' (Fig. 14, case d). Using the spin orbit coupling operator described in section 2.1.5, the following secular equation corresponding to the MO model can be obtained: £ 0 i S/2 0 0 0 0 £ p - dx* - £ 0 0 0 0 £ p+d x*-£ 0 0 s D ' 9 0 0 0 Ep - d*x - £ 0 0 0 Sag 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 £b a - £ = 0 (4.1) for the interactions involving the l 5 0 . 3 , ( 4b u , l 2 b g , l 2 Q u » 6 ^ , <Slka and 6 3 b u . The through bond interaction for l 5 C L g and ^3(Xq  i s n o t included here. £p , £o<j . £b<j > a n d E b a a r e t n e energies of the unper-turbed LPMO's, 6 4 a g , 6,bj and $3bu respectively. ^ , the spin orbit coupling constant for the halogen atom is estimated from refs. 94 and 235, - 104 -and s/ , the i n t e r a c t i o n energy between U and S"^  i s r e l a t e d to approximately by expression ( 4 .2 ) with a pl u s s i g n f o r I s a g and S i " = S i ( S i + £ P - + d x x ) (4 .2 ) I-2.bc} a n c^ a m i n u s s i g n f ° r (4b u and l a a u T h e o r e t i c a l l y , a l l the seven parameters, £p , S^'s and £t's can be e x a c t l y s o l v e d through eqn. ( 4 .1 ) because there are a s u f f i c i e n t number of observed IP's known. However, the values o f these q u a n t i t i e s are a c t u a l l y obtained i n a t r i a l and e r r o r f a s h i o n because o f the high order o f the s e c u l a r determinant. Those l i s t e d i n Table 16 f o r d i c h l o r o - , dibromo-and diiodoethane are chosen i n such a way that they reproduce the experimental data w i t h i n the experimental e r r o r l i m i t (see Table 1 5 ) . The q u a n t i t y £ p f o r a dihaloethane possesses a gre a t e r value i n comparison w i t h t h a t o f the e t h y l h a l i d e £p [ ( I ^ ) a v from r e f . 2 3 4 ] . T h e i r d i f f e r e n c e £p'- £p represents the i n d u c t i v e s u b s t i t u e n t e f f e c t o f the halogen. (b) R e l a t i v e S t a b i l i t y o f Isomeric Trans and Gauche Ions Up to now, the PE sp e c t r a corresponding to LPMO's are i n t e r p r e t e d i n terms o f the tr a n s conformation only. This treatment i s at l e a s t j u s t i -f i e d f o r ( Q ^ B r ^ and (CH^l) 2 i n which the c o n c e n t r a t i o n of the gauche 226 228 isomer i s very low . On the high IP edge o f the PE band o f ( C ^ B r ^ at around 10 .6 ev, a small shoulder at about 870 cm * from the second maxi-mum i s observed, and t h i s i s probably due to e x c i t a t i o n o f the C-C s t r e t c h i n g or CH^ t w i s t i n g mode. S i m i l a r l y , peaks occuring on the r i g h t hand s i d e of the f i r s t band and a l s o on the second band of ( Q ^ I ^ (ca. 710 cm * and 986 cm * r e s p e c t i v e l y from the band center) could be a s s o c i a t e d w i t h - 105 -Table 16. Calculated MO Parameters (ev) a of (CH X) (CH2C1)2 (CH2Br)2 ^ 2 J h 0.070 0.330 0.630 Sxx 0.0003 0.0005 0.0006 0.007 0.011 0.012 -11.82 -10.99 -10.11 -11.01 -10.44 -9.63 £p - £ P 0.81 0.55 0.48 -1.05 -0.73 -0.77 Sb«. -0,96 -0,64 -0.91 Sbg -1.06 -0,93 -1.12 -13.09 -12,28 -11.07 £bu -13.95 -13.57 -12.77 £bg -15.02 -14.75 -14.50 Sa<j 0.59 0.33 0.43 S b a 0.37 0.15 0.28 0.32 0.22 0.27 Reproduced Observed IP's within + 0.03 ev. - 106 -excitations of C - C stretching and C H ^ deformation modes respectively, 235 or, may be due to the presence of C H ^ I , 9.6 and 10.2 ev. Two peaks at around 10.52 and 10.57 ev f a l l off in intensity after a period of time, and are unknown impurities. For ( Q ^ C I ^ J t n e ratio of the population 228 of the trans and gauche forms is about three to one in the vapor phase Therefore, among the dihaloethanes i t is most l i k e l y that this molecule wil l provide information about the s t a b i l i t y of the two isomeric ions, providing that the photoionization cross sections and energies of the LPMO's* are the same for both rotamers. The relative magnitudes of the IP's for the trans and gauche forms, namely IP and IP , respectively, depend on the enthalpy differences, t • g A H , and A H . of the isomers and their corresponding ions (Fig. 16). mol ion r to b The relationship between IP's and enthalpy differences is given by eqn. (4.3) when the gauche ion is more stable l P t " I P q = A H mot + A H i o n (4.3) and IP is always greater than IP irrespective of values of A H , t J b g m°l A H . ; or eqn. (4.4) when the trans ion is more stable ion n and I P t - IPq = A H m o l - A H i o a (4.4) In the PE spectrum of ( C ^ C l ^ (Fig. 11) no peak with an intensity about one-third that of the broad peak at 11.40 ev is observable on the low IP side. (A small peak at 12.6 ev comes from impurities (may be H^ O) The assumption is ju s t i f i e d by the fact that the predominantly through bond interaction is rather independent of C-C rotation^*78^ and the calculations indicate the energy increase due to a larger through space interaction in the gauche conformer is about 0.1 ev. trans gauche A b ion f gauche _ i A H trans gauche trans A H , mo| gauche trans o case (a) case (b) Figure 16. Effect of stability of the trans and gauche conformers and their ions on the relative magnitude of IP.^  and IP : Case (a) where the gauche ion is more stable, and Case (b), where the trans ion is more stable. - 108 -in the spectrometer.) This indicates that AH. is greater than AH r ion b mol even i f the difference, IP^ - IP , is not known. The stronger dipole-t g 6 v dipole interaction between the chlorine atom in the gauche ion than in the neutral gauche molecule agrees with this observation. The extra electro-static interaction energy for the gauche ion is estimated to be 0.30 ev by assuming that the energy is proportional to the reciprocal of the C1...C1 non-bonded distance, and also that the Cl atoms are very small and have charge + ^ e . Therefore, we conclude that the trans conformer i s s t i l l more stable in the ionic state as is the case in the neutral ground state. This is further supported by the work on 1,2-diiodotetrafluoroethane (section 4.3). (c) Geometry of the C1CH2CH2C1 Molecular Ions Among a l l the PE spectra obtained for the dihaloethanes, only that of the f i r s t band of (CH 2C1) 2 exhibits resolvable fine structure. Analysis of the vibrational spacings indicates that the V* (C-C stretching), and V6 (CCC1 deformation) are excited and have frequencies of 826 and 360 — 1 2 36 cm , respectively, as compared with those in parent molecule of 1052 and 300 cm J'. A FCF calculation to determine the ionic geometry of (CH 2C1) 2 has been carried out by using method A. Assuming the separability of the C-H vibrations from other deformations, the constructed symmetry coordinates 226 are S + = A g 5 5 = (AY + A r ' ) / 2 (4.5) 5 6 = (Ao< + A o O T/2 - 109 -where g and r are C-C and C-Cl bond lengths and c( is the CCC1 bond angle. The transition intensities of combination bands or overtones are simply taken to be proportional to the heights of the fine structure maxima. The changes in structural parameters of the C-C and C-Cl bonds and the CCC1 . angle are found to be, respectively 0.136 A, -0.068 A and -8° according to the c r i t e r i a mentioned in section 2.2.1b. These values seem to be quite large for a geometrical change of a nonbonding orbital. This may be due to, (i) the simplified symmetry coordinates used in the calculation, ( i i ) neglect of the perturbation in intensity due to spin orbit coupling and ( i i i ) overestimation of intensities of individual components in the progressions which are seriously overlapped by bands originating from the second highest occupied orbital of the trans conformer or the LPMO's of the gauche isomer, and so the errors on the estimated vibrational intensities are quite high. (d) Relation Between the Observed Ionization Potentials and Some Physical  Properties of XCH2CH2X. 237 234 In the halogen acids , and alkyl halides , the f i r s t IP's have been found to vary linearly with the electronegativities of the. halogen atoms. The gradient of these lines i s approximately related to 234 the magnitude of the contribution of the halogen p orbitals . The same kind of dependence is obtained in plotting the halogen electronegativities against the f i r s t IP's as well as the two highest occupied S MO's (mainly localized in the C-X bond) of the dihaloethanes (Fig. 17). The ionization potential of a molecule is a measure of the differ-ence between the heat of formation of the molecule and the corresponding T — i — i — — i — ! — i — i — i — i — i — i — i — i — IP(eV) Figure 17. Plot- of the Pauling electronegativity of the halogen atom against (a) the first halogen lone pair, (b) the first sigma ionization potential, and (c) the second sigma ionization potentials of XCl^ CI-^ X: A different scale for the IP's is used for different orbitals. - I l l -ion. Fig. 18 shows graphically the IP's corresponding to removal of electrons from the f i r s t three o u t e r orbitals plotted against the 238 heats of formation , AH^ which are deduced from the reaction scheme (4.6) CahUCgas) + X 2 (gas) = C 2 H 4 X 2 (gas) ( 4- 6) It is interesting to note that a linear relationship i s obtained for the heat of reaction AH of (4.6) and the f i r s t two 6 IP's. r Force constants have long been regarded as a measurement of the r i g i d i t y of a bond in a molecule. Ionization of an electron from the MO's which are localized in a certain bond, e.g. the C-X bond, wi l l change the energy of that bond. Hence the corresponding IP gives information about the C-X bond strength. Fig. 19 shows a plot of the f i r s t two €> IP's of the XCH2CH2X molecules plotted against the C-X stretching force con-228* stants KCx• A good approximation to a linear plot i s obtained. 171 172 According to Badger's rule ' , K c x is related to the C-X bond length r c x by eqn. (4.7) where a and b are constants. K c x = a r C x b or loq Kc* = log a - b log rcx (4.7) In light of the correlation obtained between the'IP's and K c x , the log (IP) of the f i r s t two 6 MO's is plotted against log r c x , and again straight lines are obtained (Fig. 20). In view of the correlation obtained for the observed IP's, force 239 * K c i is estimated from the C-I stretching frequency by using the approximate method^ 0 mentioned previously in the FCF calculation on (CH 2C1) 2. A + 2 0 -+ 10-CL> o OH a —10 H o 0 - - 2 0 -- 3 0 -- 4 0 -( C H 2 I ) 2 f ( C H 2 B r ) ^ N (CH2Br), (CH2C1)2 (CH?CI)?X.(c) { C H 2 C I ) 2 ^ (a) I.P(eV) Figure 18. Plot of the heat of formation (AH^) against (a) the first halogen lone pair, (b) the first sigma, and (c) the second sigma ionization potentials of XCH2CH A different scale for IP's is used for the different orbitals. f L - # — i 1 r ~ 1 1 1 1 r IP(eV) »• Figure 19. Plot of the C-X stretching force constant Kcx against (a) the first sigma ionization potentials, and (b) the second sigma ionization potentials of XCFLCH X. A different scale for IP's is used for different orbitals. Figure 20. Plot of log r c x against log IP of (a) the highest occupied and (b) the next highest occupied sigma orbitals of XCH2CH2X. (The values for VCx are obtained from ref. 150. ) - 115 -constants, and bond distances of the XCl^CH^X series, we have attempted to investigate whether the same kind of relationship holds for other 43 molecules. Figs. 21 - 24 show the plot of the stretching force constants ' 106 109 240-244 ' ' KAQ of the bond A-B of X2, HX and BrX (X = F, C l , Br, I), CY o J OCY, SCY and H0Y (Y = 0, S, Se, Te), WH7 (W = N, P, As), BX^, * 11 12 30 147 235 245-247CH3X, CX4, YF 6 and HCCX, against observed ^ i s 1 1 ' ! ^ ^ , " ' , ^ . ^ corresponding to the removal of electrons from orbitals mainly localized on the A-B bond. Good linear plots are observed in each case. 221 248 249 By plotting the result of CND0/BW calculations ' ' for the hydrogen halides (not shown here), we have observed a linear relationship between the calculated as well as the observed stretching force constants and the calculated total energy. A similar correlation exists between the force constants and a l l occupied valence orbital energies. This supports the experimental observations of Figs. 21 and 22, i.e. that the plot of IP versus force constant is linear. Thus, the exact similarity between the experimental and the theoretical plots (provided Koopmans' theorem holds) supports the vali d i t y of the correlation. At this stage, no obvious explana-tion can be offered to explain a l l the correlations observed in Figs. 21 -24. It should be mentioned that from the correlation of observed IP's with force constants as well as bond lengths for a series of molecules, IP's for compounds which are not available can thus be estimated. For instance, the fourth lowest IP of CI^ is predicted to be about 14 ev. * Second lowest IP (LIP) of SeF 6 and TeF^, and third LIP's of SF 6 > SeF and TeF were estimated from spectra in ref. 30. Figure 21. Plot of some ionization potentials of (a) X2, (b) HX, (c) BrX (X = F, Cl, Br, I), (d) CY2, (e) OCY, (f) SCY and (g) H2Y (Y = 0, S, Se, Te) against the stretching force constants. - 1 1 7 -Figure 22. Plot of some ionization potentials of (a) WH (W = N, P, As), (b) BX3, (c) C113X, (d) CX4 (X = F, Cl, Br, I), (e) YF^ (Y = S, Se, Te) and (f) HCCX against the stretching force constants. —i i 1 1 1 1 1 1 r L-#—1 1 1 1 T 1 1 1 , 1 — i f — | 1 1 , 1 1 1 1 1— 1.02 106 110 1.14 1.18 122 1.26 1.30 1.31 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.11 U 3 1.15 1.17 119 1.21 1.23 125 1 27 log IP • log IP • log IP • log IP log IP Figure 23. Plot of the logarithm of some ionization potentials of (a) x 2, (b) HX, (c) BrX (X = F, Cl, Br, I ) , (d) CY2, (e) OCY, (f) SCY and (g) H2Y (Y = 0, S, Se, Te) against the logarithm of bond lengths. - 119 -Fig. 24. Plot of the logarithm of some ionization potentials of (a) WH3 (W = N, P, As), (b) BX3 (c) CH3X, (d) CX4 (X = F, Cl, Br, I), (e) YF 6 (Y = S, Se, Te) and (f) HCCX against the logarithm of bond lengths. - 120 -4.3 1,2 Dichloro-, 1,2 Dibromo- and 1,2 Diiodotetrafluoroethane,  1,2-Dibromo-l,1-difluoroethane and 1,2-Bromochloroethane. 4.3.1 Introduction Rotational isomers of substituted ethanes such as (CF 2C1) 2, ^^2 B r^2' (CF 2I) 2, CF 2BrCH 2Br and CH2ClCH2Br have been studied by infrared and Raman 227,228,251-253 _ . ^ , . 254 , . spectroscopy , ultrasonic techniques and microwave spec-255 256 troscopy ' and the trans form of these compounds is found to be more stable than the gauche with conformational energy differences A H , ° O J mol of 0.44, 0.95, 1.03 and 1.43 Kcal/mole for ( C F 2 C 1 ) 2 2 5 6 , ( C F 2 B r ) 2 2 2 7 , o r 1 228 CF 2BrCH 2Br and CH 2ClCH 2Br respectively. The percentages of the trans forms of ( C F 2 C 1 ) 2 2 5 6 and CH 2ClCH 2Br 2 2 8 are respectively 52 and 85%, while those of (CF 2Br) 2 and CF 2BrCH 2Br are estimated to be 71 and 74% respectively 227 by considering the relation between ^ H r a o l a n ^ t n e population of the isomers. Even though such data is unavailable for (CF 2I) 2, the trans isomer should s t i l l be more stable than the gauche owing to both steric and electro-static effects in this molecule. The PE spectra of the five substituted ethanes are given in Figs. 25 and 26 and the observed IP's are summarized in Table 17. MO treatment similar to the one described in section 4.2 is employed and w i l l be mentioned below. Throughout this work, the trans isomer of a l l the molecules studied except (CF„C1) 9 is considered to give the main feature in their PE spectra. 4.3.2 Method of Calculation (a) One Electron Model for Trans 1,2 Dihalotetrafluoroethane (C2u) The four halogen LPMO's of trans (CF 2X) 2 are of the same energy i f they are free from any kind of perturbation (Fig. 27 case a). However, - 121 -11 12 13 14 15 16 17 18 19 2 0 9 10 11 12 13 14 15 16 17 18 19 2 0 I P ( E V ) • Figure 25. The PE spectra of (a) 1,2 dichloro-, (b) 1,2 dibromo-, and (c) 1,2 dLiodotetrafluoroethane. - 122 -— 1 1 1 1 I I I 1 I 1 1 z 10 11 12 13 14 1 5 1 6 17 18 1 9 2 0 LU h-I I I I I I I I I I 10 11 12 13 14 15 16 17 18 19 IP(EV) > Figure 26. The PF. spectra of (a) 1,2 dibromo-1,1-difl uoroethane and (b) 1,2 bromochloroethane. - 123 -Table 17. Experimental IP's of (CF2C1) , (CF2Br)2, (CF 2I) 2 > CF2BrCH2Br and CH2BrCH2Cl CCF2C1)2 (CF2Br)2 (CF 2I) 2 CF2BrCH2Br CH2BrCH2Cl 12.47b 11.44 10.11 10.86 10.65 12.82b 11.83 10.44 11.14 10.94 13.06 12.11 10.69 11.46 11.40 13.19 12.21 11.10 11.65 11.52 13.88 13.00b 12.02 12.96C 13.05 15.43 14.53 13.49 Sqa; 14.21C 6 2 0 " . 13.94 16.13 15.62 15.24 15.06 6 5 a . ' - 15.24C 16.81 15.99 15.67 6 5 a * . 15.51 16.85 17.54 16.61C 16.31 640.*+ 16.43 18.93 17.43° 17.13 6 3Q.* 18.63 6 3 a u 19.6 ? 18.66C 18.05 e 7 a - 19.88 5 * b u 19.37 Experimental error + 0.01 ev. Experimental error + 0.02 ev. Uncertainty not known. ( a ) ( b ) ( c ) ( d ) Figure 27. Q u a l i t a t i v e MO diagram of trans (CF^X)^, (a) no perturbation, (b) through space i n t e r a c t i o n , (c) through bond i n t e r a c t i o n added, and (d) spin o r b i t coupling added. - 125 -these orbitals are shifted to a different extent by both the through 77 78 sapce and the through bond interaction ' . According to a CNDO/2 59 calculation on (CF 2C1) 2, the f i r s t four highest occupied 6 orbitals are S 7 C i 3 , 6 6 b i A , 6 6 C L g and 6~4.bg i° order of increasing IP. Since only orbitals of the same symmetry can undergo through bond interactions, only the LPMO's i-sacj > ^7bu a n d ^5^s a r e considered to be destabilized by the amount, S a g , S b a and S b g respectively. To simplify the calculation, the influence on l-acig by €>60JJ is neglected. The through space interaction parameter d x x is evaluated from expression (2.7). In addition, the LPMO's can mix with each other through spin orbit coupling. Using the same spin orbit coupling operator described in section 2.1.5, the following secular equation can be set up in accord with the overall interactions for Uaq, > U b a > Isfag > i s a u > 57CLq> 6*b3 a n d 6 " 6 b u respectively. Sag , 2 bo, a n d £ a a a r e the unperturbed energies 0 i-S/2 0 0 0 0 £ P -dxx-S 0 L§/2 0 0 0 8 p + dx* -£ 0 0 Sbg 0 0 0 £p-dxx -£ 0 0 0 = 0 0 0 0 ca 9-£ 0 0 (4.8) 0 0 0 0 0 0 0 0 0 0 of 6~7clg , 64bg and 6 6 b u respectively. The value of in (CF 2X) 2 should be smaller than that in the corresponding (CH 2X) 2 owing to the mixing of the LPMO's with those of fluorine. In fact, a very small - 126 -s p l i t t i n g is observed in the f i r s t PE band of CF^CF^l . Since the number of unknown variables, £ p , Sag , Sba , Sb'^  , £a 3 , £b u a n d £ b g in eqn. (4.8) is equal to that of observed IP's, *5 is fixed to have values 0 , "?x/2. and . (This, therefore, includes the limits of <5 and the mean value.) The other parameters given in Table 18 are obtained in a t r i a l and error fashion to give good f i t with experimental data. (b) One Electron Model for Trans CH2BrCH2Cl and Trans CF 2BrCH 2Br (C s) Fig. 28 shows the qualitative MO diagram for trans CF^BrCF^Cl. In this molecule, the Coulomb energy of bromine £ B r and chlorine £c, are different with £ C ) < ^6r . The two pairs of doubly degenerate LPMO's (Fig. 28 case a) can combine to give orbitals [ 1Q:_ , I^ Q* , lsa' + a n ( i '. The subscripts + and - mean the in phase and out of phase combination of the LPMO's. The coordinate system used for this compound is the same as that for the dihaloethanes (Fig. 15). d-srci i s again estimated from expression ( 2 . 5 ) . 221 According to a CNDO/BW calculation on the molecule, the f i r s t four 6 IP's are related to 66Q/+ , S^al' » ^sa'. a n d with increasing IP. These S orbitals can interact with LPMO's of the same 77 78 symmetry ' . Since the energy gap between the LPMO's, and Ss-a'_ and 6 4 0 . + is large, only 6ga'+ a n ( i ^2.a. a r e considered to mix appreciably with the LPMO's. Under the operation of this perturbation, the orbital sequence of LPMO's in increasing IP is of symmetry (i) &'l t Q.L t d\ and a'; , (n) a':, a; , al and a"+ , or (m) a; _ a: , a'. a n d cT+ • Furthermore, these LPMO's interact with each other by spin orbit coupling, - 127 -Table 18. Calculated MO Parameters (ev) a of Trans (CF X) (CF2C1)2 (CF 2BrJ 2 (CF2 " 2 0.00,0.035,0.070 0.00 0.165 0.330 0.00 0.315 0.630 0.0003 0.0007 0.0007 0.0007 0.0010 0.0010 0.0010 0.008 0.017 0.017 0.017 0.022 0.022 0.022 -13.18 -12.20 -12,19 -12.13 -11.08 -11.04 -10.91 -0.70 -0.78 -0.78 -0.75 -0.95 -0.93 -0.84 -0.92 -0.94 -0.90 -0.76 -1.25 -1.19 -0.90 . -0.66 -0.52 -0.52 -0.26 -1.29 -1.24 -1.21 -13.18 -12.26 -12.26 -12.31 -11.07 -11.12 -11.35 Ebu. -15.05 -14.14 -14.18 -14.28 -12.83 -12.91 -13.17 -16.69 -15.92 -15.92 -15.97 -15.31 -15.34 -15.36 0.70 0.74 0.74 0.74 0.94 0.88 0.94 -Sba 0.38 0.39 0.35 0.39 0.66 0.59 0.66 0.12 0.07 0.07 0.07 0.36 0.33 0.36 Reproduce observed IP's within ± 0.02 ev. (a) ( b ) ( c ) ( d ) •B, = = ' 7 a l 1 d l4a°_ '8a: 7al . . . J C l '8a; '3a; '3a': ho 00 o cr: LU z: 6 >6a; >2a! 5a: Mai Figure 28. Qualitative MO diagram of trans CH^BrCH^Cl, (a) no perturbation, (b) through space interaction, (c) through bond interaction added, and (d) spin orbit coupling added. - 129 -and the coupling constants are different for chlorine <£Ci and <§8> 95 234 with <£B > >gCl . In ethyl bromide and chloride ' , their f i r s t PE bands give a sp l i t t i n g almost equal to of bromine and chlorine atoms. This reflects that the LPMO's of the halogen atom in the ethyl halides are only slightly perturbed by the 6 moiety. This should also be true in CH2BrCH2Cl. Therefore *§CI and *? 8 r are assigned to have the f u l l value of <§x , i.e. 0.07 and 0.33 ev respectively in the treatment. On the assumption that the election ejected from a LPMO has /3 spin only, then the secular determinant corresponding to the above MO model can be written as, €B l + d 8 r C I - £ 0 i ( C,1, § B r + C,j ?a )/a 0 0 0 0 £ - d -£ 0  Q 6 , c i £ s; 0 -L(cr,<§8,+ c,i?c,)/2 0 £ B t+ d B , c i - £ 0 0 s: 0 -uc,i ee.+ <S«e i)/2 0 £ c i - ^ B r c i - £ 0 0 0 s; o 0 0 0 o 5 : 0 0 £ai'-= 0 (4.9) for l 7 a r. , l 8 a. + , l+a»_ , l3a»+ , 6 6 Q; and 6"aa._ respectively. C l l and c^ 2 in the determinant come from the through space interaction. Since c i s found to be much greater than in the calculation, c ^ is simply set equal to unity and c ^ to zero. Table 19 l i s t s the calculated MO para-meters that reproduce the observed IP's within experimental error. Although there are three possible orderings in the LPMO's as mentioned previously, the eigenvectors of eqn. (4.9) indicate that they a l l give the same ordering of ( i i i ) ultimately. The calculated MO parameters are given in Table 19. Table 19. Calculated MO Parameters i [ev)a of Trans CF2BrCH2Br, Trans CH2BrCH2Cl and Gauche (CF2I) 2 Trans CF2BrCH2Br Trans CF^BrCF^Cl Gauche (CF 2 D 2 ^ 5 r H 0. .00 0.165 0, ,330 0. 33 0. .00 0. 315 0. 63 * B , P 0. .00 0.165 0, ,165 ^c , 0. 07 -11. .17 11. 17 -11. 19 0. ,001 0.001 0, ,001 0. 001 Sa. •0. ,19 0. 19 0. 22 ^ B r B r 0. .002 0.002 0, .001 0. 000 s b . 0. ,79 0. 79 0. 79 -11, ,46 -11.46 -11. .44 -11 . 37 Scu 0. ,91 0. 91 0. 94 -11. .65 -11.64 -11, .64 6c. -11 . 52 s ; - 1 , .14 -1.12 - 1 . .12 s ; ' - 0 . 95 s : -0. .95 -0.94 -0, .91 - 1 . 33 -13. • 71 -13.72 -13, .72 -12. 46 £a'- -12. .36 -12.37 -12. .41 -13. 24 s + 0. .50 0.49 0. .49 s + 0. 59 s _ 0. .60 0.59 0, .55 S - 0. 69 Reproduce observed IP's within experimental error. - 131 -With regard to C^BrCI^Br, the interactions between the LPMO's themselves, as well as the 6 orbitals are essentially the same as that in CH2BrCH2Cl. In CF2BrCH2Br, the Coulomb energy £ 8 r H for bromine with hydrogen attached to the same carbon is expected to be higher than cg r F for bromine with fluorine attached to the carbon, owng to the electron withdrawing nature of fluorine. In addition, ^ 8 is greater than ^srp for the same reason. Since both are not known for this molecule, "gBrH and § 6 are assigned to have values 0, § x/2 and (Table 19). 221 A CNDO/BW calculation on CF2BrCH2Br indicates that the two highest occupied MO's are G\oa<_ and 6qa'+ • T h e repulsive forces exerted by these orbitals destabilizes Lua'_ a n c* *-iza\ • Hence, there are also three possible orbital sequences in the LPMO's, the same as those given for CH2BrCH2Cl. However, in this case, It is difficult to determine the orderings owing to an appreciable amount of mixing between the orbitals. The set of parameters employed to reproduce the observed IP's is given in Table 19. 4.3.3 Results and Discussion (a) Interpretation of the Spectra (CF C l ) 2 , (CF 2Br) 2, (CF 2I) 2 and CF2BrCH2Br The He I PE spectra of these compounds (figs. 25 and 26a) are similar to each other. Each of the spectra consists of a high intensity band accompanied by seven or eight broad overlapping bands. These low intensity bands correspond to ionization of electrons from 6 orbitals. The assignment of these IP's (Table 17) is based on the relative intensity - 132 -of the PE bands, as well as the result of CND0/2J" or CNDO/BW*"''1 calcu-59 lations. The former MO calculations also indicates that 6*4.00 > » 63^ and 6'3Qu of (CF ^ X) 2 a r e mainly composed of fluorine LPMO's. The f i r s t PE band of each molecule derives from the four combina-tions of halogen LPMO's which are mixed with each other through the possible interactions mentioned previously. If the relative population of the trans isomer is very large compared to the gauche isomer in the vapor phase, this band w i l l give approximately four maxima. However, the spectrum may • become more complicated with an increasing population of the gauche form. In (^201)2* the concentration of the trans rotamer is only slightly higher than that of the gauche. The lowest IP band exhibits a double maxima with not well resolved structure instead of four distinct bands. The IP's given in Table 17 for these peaks are obtained from a band shape analysis which assumes that the trans isomer s t i l l gives the main features of the spectrum. However, the uncertainties associated with these two IP's are not known. With regard to (CF 2Br) 2 or C^BrCT^Br, the situation is less complicated owing to the lower concentration of the gauche form. The f i r s t PE band gives four maxima. However, i t is impossible to observe any peaks due to the gauche isomer,and these may be buried under those from the trans isomer. Among a l l the dihalotetrafluoroethanes studies, only (CF 2I) 2 gives well resolvable structure in the f i r s t PE band (Fig. 25c). The four sharp and intense peaks can readily be assigned to derive from the iodine LPMO's of the trans isomer. Each of these peaks is accompanied by a small peak - 133 -with IP's of 10.21, 10.55, 10.81 and 11.22 ev (experimental error ± 0.01 ev). A band shape analysis shows that the intensities of these peaks are almost constant with intensity 43 ± 4% of the main peaks. Excitation of vibrational modes cannot account for such a consistently high intensity on four separate LPMO's. This leads us to suggest that these peaks are mainly contributed from the gauche conformer of (CF 2I) 2. Assuming the constancy of the photoionization cross sections for both isomers, as well as the temperature of the c o l l i s i o n chamber (300°K), A H , is found to be 0.92 ± 0.05 Kcal/mole, compared to 0.95 Kcal/mole mol for (CF 2Br) 2. The magnitude of A H m Q l should be greater in the former 22 molecule than in the latter, owing to both steric and electrostatic effects However, the difference should not be large due to the electronegative nature of the fluorine atoms in these molecules . CH2BrCH2Cl The He I spectrum of CH2BrCH2Cl (Fig. 26 b) is almost the same as that of the dihaloethanes (Fig. 11 - 13). Only four peaks are observable in the region 12 - 18 ev. These can be attributed to ionization of electrons from 6 orbitals. The IP's of these orbitals are somewhere between those of 1,2 dichloro- and 1,2 dibromethanes (Table 15). The f i r s t PE band of this molecule is similar to that of (CF 2I) 2 and has resolvable fine structure. The four sharp major peaks come from the LPMO's of the trans isomer. The small peak at 10.50 ev cannot arise from the gauche form of this molecule, in accordance with the observation of a higher energy for this form in 1,2 dichloroethane (section 4.2.2b) and ( C F 2 I ) 2 > but rather from an impurity which may possibly be Br However, the assignment of - 134 -the peak at 10.79 ev is not so obvious. The intensity of this peak may be due partly to the excitation of CH^ mode or from the LPMO of the gauche form. On the assumption of the constancy of photoionization cross sections for both isomers, the intensity ratio of the PE bands of trans and gauche is calculated to be 1 to 0.2. In fact, a band shape analysis gives a ratio 1 to 0.3. (b) Orbital Energy of Gauche CF 2ICF 2I There are totally four LPMO's TTj , TTj" , TTg and TTg in the gauche form of (CF 2I) 2. They can combine spatially (Fig. 29) with each other to give, I I20.+ = Y ( IT^ + TTg + TT3' + TT8* ) ( IT, + Ha - TT3" - TTg ) (4.10) Lb. = \ ( TTj - Tfa - Ti;' + TTg) with 2JZ P: 3 . i p + ±2. p — 2  r*s Z • (4.11) 2/2 P; 2J3 3 Furthermore, some of these LPMO's are destabilized by mixing with 5 9 6 MO's of proper symmetry. A CNDO/2 calculation on the molecule, shows that the f i r s t three occupied 6" orbitals are 6",,a+ , 6"i0b_ and Figure 29. Molecular o r b i t a l s of halogen atoms in the gauche CF ICF I. form of - 136 -5ioa_ in order of increasing IP. Since i t is difficult to distinguish the 6 IP's of the trans and gauche conformers owing to the overlapping nature of the PE bands in the spectrum, a MO treatment on trans (C^X^ is not applicable in this case. Therefore, the through bond interaction parameter is considered as a diagonal element in the secular equation given below. In addition, when the spin orbit coupling is taken into account, the following secular equation can be set up 1.086 8P + Sa+- £ 0 0 LS/6 0 0.984- £ p+ Sa.-B 0 0 i<5/<$ o.9(4- £ p-£ 0 0 0 1.016 £p + Sk_-£ for li2a+> i i j a > ' • 1 2 0 + a n c * U b . respectively. The solutions of eqn. (4.12) with \t = 0.0, ^* /2 and are given in Table 19. It is interesting to note that £ P is smaller in the gauche isomer than in the trans (Tables 18 and 19). The same observation applies to cis and trans 1,2 diiodoethylene (section 4.4). It seems that the 'effective' electronegativity of the halogen increases with a larger through space interaction. This parallels the result of calculations on chloro- and 218 bromo methanes In 1,2 dichloroethane, the trans isomer is found to be more stable 227 than in both the gauche parent molecule and the molecular ion. However, the energy difference between the two isomeric ions AH. is not known. b J ion - 137 -In (CF^I)^, both peaks arising from LPMO's of the trans and gauche rotamers are discernible in the PE spectrum. If the energies of the fron-t i e r MO's of both isomers are the same, the trans ion is found to be more stable than the gauche by 0.14 ev ( & H m o j is equal to 0.04 ev from the previous discussion). Taking the difference in £p between trans and gauche isomers into account, AH. is estimated to be 0.06 ev, almost the same ion as A H ^. This may be due to the derealization of the positive charge over the whole cation by the fluorine atoms in ( C ^ I ^ - I n this respect, one would expect the difference AH. - AH . to be larger in 1,2 dichloro-r ion mol & ethane, or 1,2 bromochloroethane, than in the corresponding fluorinated 1,2 dihaloethane. In Tables 16 and 18, the quantity for dibromoethane is greater for increasing replacement of hydrogen by fluorine. The same observation is applicable to the other dihaloethanes, and their respective fluorinated derivaties. This is probably due to more higher lying 6 orbitals available for through bond interaction in the fluorinated compounds. This calculation indicates that the through bond interaction i s not the same for the two isomers of (CF2l)2> hut that the interaction is greater in the gauche form. If the inductive effect is predominant over other effects, and also i f no rehybridization occurs in the C-X bond of both isomers, the iodine NMR chemical shift is expected to be larger for the gauche form, while the nuclear quadrupole coupling constant is greater for the trans. Unfortu-nately, no such measurements have been made. In view of the quantity £ p obtained for both trans and gauche (CF I) , force constant calculations in a simple Urey-Bradley force - 138 -field" 1" 1^ are carried out on both isomers of (CI-^Cl)^ and (CH^Br)^ using frequencies given in ref. 236. In this treatment, only C-C and C-X stretching, as well as CCX bending vibrations are considered. Both the Gs_ and Fs_ matrices used are the same as those given in ref. 226. Different sets of force constants are used for the two conformers. The force constants given in Table 20 reproduce the observed frequencies within 5%. In general, the nonbonded repulsive force constant F increases with shorter interatomic distances between two nonbonded atoms. However, the constant is found to be smaller in the gauche form than in the trans for both (CH^Cl)^ a n d (CH^Br)^. This reflects a greater attractive interaction of the halogen. atoms in the gauche conformer. In ( ^ 2 1 ) 2 > Ep of the gauche form is lower in the two conformers. This implies that the electron in the iodine is more delocalized over the C-I bond in the gauche form i f we consider only the inductive effect. Therefore, the C-I bond should be stronger in the gauche form than in the trans. If the same situation occurs in the 1,2 dihaloethanes, one would expect a higher C-X stretching force constant in the gauche isomer. In fact, the calculated values of K Cx (table 20) parallels the above observation. 4.4 Dihaloethylenes 4.4.1 Introduction The electronic structures of the dihaloethylenes 02^^X2 have received considerable attention owing to their chemical and physical ore u u ^ A U ^ 208,209,215,258-262 properties. PES has been employed by several workers to elucidate the electronic structure of these molecules. From the IP's 37 obtained, and hence the orbital energies (assuming Koopmans' theorem - 139 -a • Table 20. Urey-Bradley Force Constants (mdyn/A) of Trans and Gauche (CH2C1)2 and (CH2Br)2 Trans Gauche Trans Gauche (CH2C1)2 ( C H2 C 1 )2 (-CH2Br:)2 ^ C H2 B r )2 K c c 2.99 3.21 2,93 2,96 K c x 2.71 2.28 1.97 1.93 H c c x 0.19 0.25 0.16 0.15 F x x 0.39 0.23 0.43 0.38 a KCC KCX' HCCX a n d F a r e t b e C _ C s t r e t c n » c _ x stretch, CCX bending and t and nonbonded repulsive force constants respectively. F = -0.1 F Y Y , A A AA - 140 -to hold), various parameters such as the Coulomb integrals £ p and £ c = c , resonance integrals between carbon and halogen fi> , as well as 208 261 the through bond and through space parameters have been deduced ' for the trans, cis 1,2 and also gem dihaloethylenes in the Huckel MO approximation. Inclusion of spin orbit coupling interaction i s found to be 209 necessary for the interpretation of the PE spectra of the iodoethylenes In this section, we again apply a one electron model, similar to that des-cribed in section 4.2, including spin orbit coupling, conjugative effects, as well as through bond and through space interactions, to a l l the dihalo-ethylenes except gem diiodoethylene and the fluoroethylenes. Qualitative correlation between the calculated MO parameters, the electronegativity of the halogen atom, the carbon and proton chemical shifts, Sc and £ H respectively, and the nuclear quadrupole coupling constant eiQ-9j of the ethylene derivatives studies* is discussed. Throughout this work, the assignment of the valence orbitals of these molecules is based on the work of Wittel and Bock . 4.4.2 Method of Calculation (a) One Electron Model for the Dihaloethylenes Fig. 30 shows schematically the mixing of carbon TT and 6> orbitals, and halogen LPMO's in the gem and cis 1,2 dihaloethylenes by various kinds of interactions. F i r s t , case b, where the degeneracy of the unper-78 turbed LPMO's is completely removed by the through space interaction which gives rise to the orbitals 14^ > L\a.i > i^t>, a n ( i ^ 5<X, with different energies. (The orbital symmetry used i s the same as that of Wittel and Figure 30. Qualitative MO diagram of cis 1,2 and gem dihaloethylene (a) no perturbation, (b) through space interaction, (c) conjugative effect added, (d) through bond interaction added, and (e) spin orbit coupling added. - 142 -Bock .) The qu a n t i t i e s d e and dir measure the through space energy s h i f t s of the o r b i t a l s and l s a , and l i a 2 and lib, , r e s p e c t i v e l y , from the unperturbed energy £p . In addition to the through space i n t e r a c t i o n s , conjugative e f f e c t s between the carbon TT o r b i t a l TT 2 b i and the LPMO lib, s h i f t the former o r b i t a l up, and the l a t t e r down by an amount ft' with l8 c=cl< I £ib, I • However, the reverse i s true f o r diiodoethylene where I 8c=c I > l£ib,l- The r e l a t i o n between [?> and fi' can be expressed mathematically as (3 = - ( 1/5'l (l(V| + I 8 P - £ C = J + cU);* 1 (4.13) where ft> i s the resonance i n t e g r a l . The operation of through bond i n t e r a c t i o n s on the lsa, and o r b i t a l s by the S o r b i t a l s of the same symmetry (the f i r s t four highest 208 occupied 6" MO's of c i s and gem isomers are of symmetry and CL, ) d e s t a b i l i z e s the two o r b i t a l s by energies So., and Sb^ r e s p e c t i v e l y . Further mixing between the LPMO's themselves with the same subscript 1 or 2 i n the symmetry representation i s allowed by the intr o d u c t i o n of spin o r b i t a l coupling. The coupling constant ^ i n dihaloethylenes should be smaller than that i n dihaloethanes owing to the i n t e r a c t i o n between the TT o r b i t a l s and the LPMO's i n the ethylene d e r i v a t i v e s . Consequently, the secular equation corresponding to a l l the perturbations mentioned above can be expressed as - 143 -£ c = c - / V - £ 0 0 0 £ p + d e t S b i - £ 0 iS/2 £P + dw-£. 0 0 0 0 0 0 0 0 0 0 0 (4.14) for TTab, > 14-bj, > Ua^ > lib, and r e s p e c t i v e l y . The x axis i s defined to be perpendicular to the molecular plane. The f a c t o r JZ i n the matrix elements with a r i s e s from the assumption of equal mixing between the TT and the LPMO's. S i m i l a r l y , a 5 x 5 secular determinant can be set up for trans 1,2 dihaloethylene i n c l u d i n g a l l kinds of i n t e r a c t i o n s mentioned previously f o r the c i s isomer. However, i n t h i s case, d s i s i d e n t i c a l to cl-rj- and also the symmetry representations b 2 > > <3.2 and b, i n eqn. (4.14) and F i g . 30 are replaced by &g , b a > bg and d a r e s p e c t i v e l y f o r the conversion to C 2 K symmetry. The four f r o n t i e r 6 o r b i t a l s are of bu. . _ ^ 208 and Qg symmetry only. In a secular equation such as (4.14), there are t o t a l l y s i x inde-pendent unknown v a r i a b l e s , £ c = c , Ep , [3' > SQ . , , Sb* and <^  with only f i v e IP's a v a i l a b l e . Since the value of should be i n the range of 0.0 to <$x , so i t i s constrained to have values 0, -€x/2 and <§x again. Then the other f i v e v a r i a b l e s are determined i n a t r i a l and error fashion to reproduce the observed IP's. In a l l cases except ^^-^Z w:*-t'1 ^ ~ ' the c a l c u l a t e d IP's f i t exactly the observed ones and the parameters used are given in Tables 21 - 23. It i s found that besides S;. , £ c=c > Ep , d g > dff and /3' are rather i n s e n s i t i v e to . So comparison of Table 21. Calculated MO Parameters ( e v ) a of Cis 1,2 Dihaloethylenes C H X 0.07 0.33 0.63 0.035 0.165 0.315 0.00 0.00 0.00 d 6 0.34 0.52 0.66 0.34 0.53 0.67 0.34 0.53 0.67 d w : 0.06 0.10 0.14 0.06 0.10 0.14 0.06 0.10 0.14 £ e. c -10.97 -10.79 -10.05 -10.97 -10.76 -9.89 -10.96 -10.76 -9.86 £ p -12.53 -11.60 -10.51 -12.53 -11.62 -10.66 -12.53 -11.63 -10.69 Sa, 0.87 0.89 1.11 0.87 0.92 1.22 0.87 0.93 1.25 S b z 0.55 0.30 0.03 0.55 0.35 0.37 0.55 0.37 0.42 |S' -1.17 -1.15 -1.07 -1.17 -1.13 -0.94 -1.16 -1.13 -0.92 3 -1.81 -1.54 -1.34 -1.80 -1.54 -1.32 -1.80 -1.54 -1.32 Reproduce observed IP's within experimental error. Reproduce observed IP's within + 0.08 ev. Table 22. Calculated MO Parameters(ev) a of Trans 1,2 Dihaloethylenes C H X C 2H 2C1 2 C 2H 2Br 2 C 2 H 2 I 2 b C 2 H 2 C 1 2 C 2H 2Br 2 C 2 H 2 I 2 C 2H 2C1 2 C 2 H 2 B r 2 C 2 H 2 J 2 0.07 0.33 0.63 0.035 0.165 0.315 0.00 0.00 0.00 0.004 0.007 0.008 0.004 0.007 0.008 0.004 0.007 0.008 -10.94 , -10.92 -10.51 -10.94 -10.88 -10.35 -10.94 -10.87 -10.30 -12.64 -11.52 -10.26 -12.64 -11.56 -10.41 -12.64 -11.58 -10.46 0.82 0.42 0.20 0.32 0.50 0.50 0.82 0.53 0.59 0.64 0.49 0.18 0.64 0.53 0.32 0.64 0.54 0.36 -1.14 -1.36 -1.33 -1.14 -1.33 -1.42 -1.14 -1.32 -1.38 /3 -1.80 -1.64 -1.45 -1.80 -1.64 -1.46 -1.80 -1.64 -1.46 Reproduce observed IP's within experimental error. Reproduce observed IP's within ± 0.04 ev. - 1 4 6 -T a b l e 23. C a l c u l a t e d MO P a r a m e t e r s (ev) o f Gem D i h a l o -e t h y l e n e s C _ H 0 X „ C 2 H 2 C 1 2 C 2 H 2 B r 2 C ^ C ^ . C ^ B ^ C 2 H 2 C 1 2 C 2 H 2 B r 2 0 . 07 0.33 0 . 035 0 .165 0 . 00 0 .00 0 .42 0.4U 0 .42 0 .40 U .42 0 .40 0 .17 0.17 0 .17 0 .17 0 .17 0 .17 -11 . 08 -10.88 -11 .07 -10 .85 -11 . 07 - 10 .84 - 12 .65 -11 .73 - 12 .65 -11 .76 -12 . 65 -11 .77 Sa, 0 .87 0. 90 0 .87 0 . 93 0 .87 0 . 94 Sb, 0 .61 0.57 0 . 61 0 . 63 0 . 61 0 . 65 /3' -1 . 08 -1.09 -1 . 07 -1 . 06 -1 .07 -1 . 06 ft -1 .74 -1.52 - 1 .74 -1 .51 - 1 .74 -1 .51 Re p r o d u c e o b s e r v e d I P ' s w i t h i n e x p e r i m e n t a l e r r o r . - 147 -these q u a n t i t i e s from molecule to molecule i s possible even though the exact value of ^ i s not known. (b) One Ele c t r o n Model f or the V i n y l Halides The bonding nature of the v i n y l halides (C 5) i s s i m i l a r to that of the dihaloethylenes i n some aspects. Therefore, i t i s desi r a b l e to compare parameters such as £ c = c > £p a n ( i between these two types of ethylene d e r i v a t i v e s . In the v i n y l halides (Fig. 31), one of the halogen LPMO's, {.io." can be combined with the TT o r b i t a l TT i a" (conjugative e f f e c t ) , while the other LPMO, 17a/ i n t e r a c t s with the remaining carbon and hydrogen framework. Furthermore, these LPMO's mix with each other by spin o r b i t coupling. The o v e r a l l i n t e r a c t i o n may be summarized i n the following secular determinant £ p + So,- ~ 8. -i<5/2j2 £ P - £ ft = 0 (4.15) for lia,' ' '••a." a n d TT2Q_" r e s p e c t i v e l y . Sa' measures the energy change of L 7 a- through the mixing of a . " 6 o r b i t a l s . The fa c t o r fz i n the determinant i s again due to the equal mixing of TT^ a." a n d I. to." • The number of unknown parameters £p , £c=c , ^ , (J and Sa' i n eqn. (4.15) exceeds the number of observed IP's. Therefore, *S i s assigned to have values of 0, "Sx/z and K , and {S i s varied from zero to a value which gives a reasonable physical r e s u l t . Then the other para-(a) (c) . (d) Figure 31. Q u a l i t a t i v e MO diagram of v i n y l halide (a) no perturbation, (b) conjugative e f f e c t , (c) through bond i n t e r a c t i o n added, and (d) spin o r b i t coupling added. -149 -meters are adjusted to give agreement between the c a l c u l a t e d and observed 208 o r b i t a l energies . The r e s u l t of the c a l c u l a t i o n i n d i c a t e s that £ p , £c=c ' ft a n c * So.' a r e rather i n s e n s i t i v e to the value of "5 . Also £ p increases but both £ c=c and Sa' decrease with diminishing values of (3 . Table 24 l i s t s the c a l c u l a t e d MO parameters with ^ = "fx/2 . It i s obvious that (l i n the v i n y l halides i s less than that i n the dihalo-ethylenes (Tables 21 - 23). The higher resonance energy i n the l a t t e r molecules i s probably due to an increase i n e l e c t r o n e g a t i v i t y of the carbon atom by one more electron withdrawing halogen atom, i n comparison to the former molecules. The choice of an appropriate /3 and hence other para-meters i s not obvious and w i l l be discussed l a t e r i n i n d i v i d u a l cases. 4.4.3 Results and Discussion (a) C o r r e l a t i o n between Huckel Parameters of Dihaloethylenes and the  Chemical S h i f t i n Carbon-13 and Proton NMR There i s a growing i n t e r e s t i n the a p p l i c a t i o n of carbon-13 NMR to many aspects of chemistry. From the NMR spectrum of a molecule, valuable information about the bonding properties of a p a r t i c u l a r carbon atom, and the electron d i s t r i b u t i o n within the atom can be deduced. For instance, the carbon chemical s h i f t Sc r e f l e c t s the s h i e l d i n g by the electrons of a carbon nucleus from i t s e l f , or from i t s neighbours. The chemical s h i f t s of some halogenated ethylenes are l i s t e d i n Table 25 for convenience i n comparison. The d i f f e r e n c e i n e l e c t r o n e g a t i v i t y between the halogen and carbon atoms i n halogenated ethylenes leads to an inductive e f f e c t which w i l l reduce the s h i e l d i n g of the carbon atom and thus increase the binding energy of the - 150 -T a b l e 24. C a l c u l a t e d MO P a r a m e t e r s ( e v ) 3 o f V i n y l H a l i d e s C 2 H 3 X w i t h ^ = S x12 /3 £c=c So.' C 2 H 3 C 1 -0.80 -10.39 -12.83 1.22 1.00 -10.55 -12.67 1.06 •1.20 -10.78 -12.44 0.83 1.32 -10.99 -12.23 0.62 •1.35 -11.05 -12.17 0.56 •1.40 -11.20 -12.02 0.41 C 2 H 3 B r -0.80 -10.17 -12.00 1.13 -1.00 -10.39 -11 .18 0.91 -1.20 -10.89 -11.28 0.41 -1.215 -11.09 -11.09 0.22 C 2 H 3 I -0.80 -9.73 -11.21 1.15 -1.00 -10.02 -10.91 0.86 -1.05 -10.16 -10.77 0.72 Reproduce o b s e r v e d IP's w i t h i n e x p e r i m e n t a l e r r o r - 151 -a b Table 25. Carbon NMR Chemical S h i f t s (ppm) £ c , bond angles and 2 c Nuclear Quadrupole Coupling Constants e Qq (Mc/s) of Halogenated Ethylenes c H ^ Gem Cis 1,2 Trans 1,2 2 3 C 2 H 2 X 2 C 2 H 2 X 2 C 2 H 2 X 2 Sc Cl 1 2 6 . l d 127. d 121.3 119.4 Br 114.7 d 97.0 d 116.4 109.4 I 85.3 d 96.5 79.4 4-CCX Cl 122.8° 123.8° 123° Br 124.1° 121° e 2Qq Cl 67.23 73.67 70.00 71.17 From ref.263. TMS i s used as i n t e r n a l standard. 'See r e f s . 150, 193-195. C R e f s . 266 and 267. The chemical s h i f t of carbon attached to halogen. - 152 -TT electrons of the carbon double bond. Conversely, the s h i e l d i n g of the halogen atom w i l l be increased, and thus the binding energy of the electrons associated with i t w i l l be decreased. The absolute magnitude of the parameter 6 c=c i n the dihalo-ethylenes (Tables 21-23) i s found to be gem > c i s > trans i n decreasing order. This i n d i c a t e s that the carbon atom i s more deshielded i n the gem isomer than i n the others providing that the bonding nature of the 6 bonds i s almost the same f o r these molecules. In f a c t , the observed chemical s h i f t s p a r a l l e l the above trend f o r £c=c . In the case of the dibromoethylenes, such agreement i s not obtained. Sc. i s predicted to be the greatest f o r trans and the l e a s t f o r c i s , i n 263 contrast to the observation c i s > gem > trans. According to a recent 264 t h e o r e t i c a l i n v e s t i g a t i o n on difluoroethylenes , diminution i n the s char-t acter along the C-F bond w i l l r e s u l t i n the widening of the CCF angle. This r e l a t i o n may probably hold f o r other dihaloethylenes also. The augmentation of s character i n the C-C 6 bond causes reduction i n the same o r b i t a l along the C-Br bond and hence an increase i n the s h i e l d i n g of the carbon atom and a deschielding of the hydrogen atom. I f t h i s e f f e c t predominates over that from TT electrons, the observed trend i n <£"<; can be reproduced with higher s character along the C-C bond of c i s than gem or trans. In f a c t , the CCBr a n g l e ^ ^ ' ^ ^ i s found to be 'greater i n c i s than i n trans by 3°. 265 The r e s u l t from proton NMR measurements , that the s h i e l d i n g of the proton i s less i n the c i s than i n the trans form further supports the explanation 193 194 offered. The constancy of the CCC1 angle ' i n the dichloroethylenes r e f l e c t s that only small v a r i a t i o n i n s character occurs i n the C-C 6" bond. - 15 3 -In diiodoethylenes, the c a l c u l a t e d value of £ c=c again f a i l s to give the correct ordering of Sc for c i s and trans isomers. This may s t i l l a r i s e from the d i f f e r e n t composition of the s o r b i t a l i n the 265 C-Br bond f o r the two isomers. The proton NMR data f o r these molecules also favors t h i s reasoning. A l l the halogens except f l u o r i n e have commonly occurring isotopes with a quadrupole moment. The nuclear quadrupole coupling constant e*Qcj obtained from radio-frequency spectroscopy f o r these halogens i n a molecule provides ample information about the e l e c t r o n i c structure of the halogen atom and i t s environment. Usually, the more the i o n i c character of a halogen atom or the higher the electron density around the halogen nucleus, the 266 267 lower the e*Qc^ value obtained. The e'Q.^ f ° r the chloroethylenes ' (see also Table 25) decreases from gem to trans and then c i s . This shows that the electron cloud i n chlorine i s more concentrated i n the c i s isomer, and the l e a s t concentrated i n the gem form. Then, the absolute magnitude of £ p f o r these isomer should be gem > trans > c i s . In f a c t , t h i s p a r a l l e l s our c a l c u l a t e d values of £ p (Tables 21-23). Since the NQR data i s incom-plete for bromo- and iodoethylenes, no such comparison can be made. However, the r e l a t i o n between the c a l c u l a t e d parameter £ P and the observed value €1Q.^ of these compounds may not be so straightforward owing to the v a r i a -t i o n i n composition of the s character along the C-X bond. It has been mentioned before that the choice of a reasonable set of £p , £c=c > /5 and So.' i s not easy and depends on the c r i t e r i a used. However, carbon MNR and NQR data are useful i n t h i s aspect. I f the 6 bonding i n the chloroethylenes i s s i m i l a r to each other, £c=c of - 154 -v i n y l chloride should be somewhere between -11.07 and -10.97 ev i n accord-ance with the observed chemical s h i f t s f o r c i s and trans 1,2 dichloroethylene and v i n y l c h l o r i d e . Then, /3 i s i n the range of -1.32 and -1.35 ev (Table 24), and the value of £p i s the smallest among the chloroethylenes studied 266 267 here. This i n consistent with an NQR study ' which gives the value of v i n y l c h l o r i d e less than a l l the dichloroethylenes. It has been mentioned that the d i f f e r e n c e o c - o c f o r the dihaloethylenes increases from a c h l o r i n e to iodine substituent. Considera-t i o n of both resonance and anisotropy e f f e c t s alone seems inadequate to 263 explain the d i f f e r e n c e . A t h e o r e t i c a l treatment of carbon chemical s h i f t s shows that the c o n t r i b u t i o n from paramagnetic s h i e l d i n g i s more important than that from diamagnetic s h i e l d i n g or anistropy e f f e c t . The paramagnetic s h i e l d i n g approximately depends on the energy gap between the highest occupied and the lowest unoccupied o r b i t a l . The decreasing energy d i f f e r e n c e between 268 the TT and TT* l e v e l s from dichloroethylene to diiodoethylene may at l e a s t o f f e r an explanation f o r part of the d i f f e r e n c e i n the observed chemi-c a l s h i f t . (b) C o r r e l a t i o n Between Huckel's Parameters i n pihaloethylene and E l e c t r o - n e g a t i v i t y of the Halogen. The replacement of a hydrogen atom by a halogen i n ethylene w i l l change the bonding property of the attached carbon. A halogen atom with a strong e l e c t r o n withdrawing e f f e c t gives a low value f o r £c=,c . Tables 21 - 23 show the general trend of the influence of t h i s inductive e f f e c t on the Coulomb i n t e g r a l £ c - c by d i f f e r e n t halogens. In Dewar's approach 269 270 ' , the influence of £ c = c by the e l e c t r o n e g a t i v i t y of the halogen can be expressed q u a n t i t a t i v e l y as - 15 5 -A £ c = c = S ( fVx) - f M ( C ) ] (4.16) where 8 i s the a u x i l i a r y inductive parameter. (x) i s the Mulliken e l e c t r o n e g a t i v i t y of the halogen X, with = 2.82 ^ P a u l i n^ . I f we cal c u l a t e A£c=c with & = 1/3 , Acr c=c i s found to be 0.43, 0.71 and 1.41 ev f o r the dibromo-, d i c h l o r o - and difluoroethylenes r e s p e c t i v e l y . Assuming that A£c=c i s equal to the di f f e r e n c e between £c=c of any halogen and that of iodine (which i s chosen as a reference), then the A £ ° ^ c ' s are 0.53, 0.59 and 1.79 f o r trans 1,2 dibromo-, d i c h l o r o - and difluoroethylene (section 4.5.2) r e s p e c t i v e l y . With regard to the c i s c abs isomers with bromine, chlorine and f l u o r i n e substituents, c c = c has values of 0.87, 1.08 and 2.33 ev r e s p e c t i v e l y . The large discrepancy between the cal c u l a t e d A 8 c=c a n d <^8c=Sc f ° r the c i s isomer i s due to the sur-p r i s i n g l y low value of £c=c obtained f o r the c i s diiodoethylene (the reference compound). In the Huckel MO treatment, the parameters Hp and [3 of a heteroatom are usu a l l y r e l a t e d to the standard value f o r £p and f30 , ^, 270 by the equation , P - K V / J . ( 4 - 1 7 ) •The constants Hp and kp r e f e r to C^li^X^. I f ethylene i s used as a 85 ° standard with /30 = -1.22 ev , and hence Sp = -9.29 ev evaluated from 271 the f i r s t IP of ethylene , then both k P and hp are found to be constant over a l l the isomers of a dihaloethylene f o r a p a r t i c u l a r halogen. The cal c u l a t e d values are 1.3, 1.5, 2.8, 1.9, 2.7 and 6.4 f o r k 8 r , KC| , kp , h 8 r > h Ci > and hp r e s p e c t i v e l y . In addition the r a t i o of - 156 -Ke , KC| a n d KP to one another i s almost equal to that of (>0 - - f M (O . 272 Recently, the PE spectrum of trans 1,2 dicyanoethylene has been obtained. Using the observed IP's and the above MO treatment ( = 0 for t h i s molecule), £ C a N , ^3 , S b u , Sag and 6 C = C are found to have values of -13.67, -0.74, 0.57, 0.89 and -11.89 ev r e s p e c t i v e l y with d K equal to zero. Owing to the higher electron withdrawing power of the cyano group i n comparison to the halogens (except f l u o r i n e ) , values of 8 c=c f o r the cyanoethylene studied are expected to be lower than that of the dihaloethylenes. The e l e c t r o n e g a t i v i t y of the cyano group evaluated through eqn. (4.17) i s 3.3, compared to 3.0 predicted from a group e l e c t r o n e g a t i v i t y 273 c a l c u l a t i o n . An extended Huckel c a l c u l a t i o n on the TT o r b i t a l of the CN r a d i c a l using expression (2.7) f o r the exchange i n t e g r a l gives the o r b i t a l energy of the highest occupied MO to be -14.84 ev. D e s t a b i l i z a t i o n of the TT o r b i t a l of the cyano group i n the dicyanoethylene (-13.67 ev) with respect to the CN r a d i c a l o r i g i n a t e s because of the d e r e a l i z a t i o n of the TT o r b i t a l onto the carbon skeleton. A p p l i c a t i o n of the same approach as above to trans 1,2 dimethyl-274 thioethylene with a n e g l i g i b l e c o n t r i b u t i o n of spin o r b i t coupling as well as through bond and through space i n t e r a c t i o n s gives -9.20, -1.49 and -9.50 ev for cTscHj > P> a n d £c=c r e s p e c t i v e l y . The magnitude of £ c = c obtained for the trans 1,2 dimethylthioethylene i s i n d i c a t i v e of the electron donating power of the -SCH^ group but the value seems to be too high. This i s probably because of our assumption that the through bond i n t e r a c t i o n i s unimportant i n t h i s case. 15 7 -4.5 Cis and Trans 1,2 Difluoroethylene 4.5.1 Interpretation of the Photoelectron Spectra of Cis and Trans  1,2 Difluoroethylene The He I PE spectra of c i s (I) and trans (II) 1,2 difluoroethylene are shown i n Figs. 32 - 34 and the derived IP's and associated v i b r a t i o n a l l e v e l s are given i n Table 26. The f i r s t two IP's of both I and II are 41 the same as those given by Brundle et a l The f r o n t i e r occupied o r b i t a l s of the fluoroethylenes studied are mainly of carbon IT bonding character and d e s t a b i l i z e d by i n t e r a c t i o n with one of the LPMOsof the f l u o r i n e atoms L | ( } (I) and L,o a (II) (conjugative e f f e c t ) . However, the electronegative nature of the f l u o r i n e atoms tends to withdraw e l e c t r o n density from t h e i r neighbours and thus the TT o r b i t a l i s s t a b i l i z e d (inductive e f f e c t ) . These two e f f e c t s work i n opposition to u *u T u i * i . i 208,215,259 , +, . 208,258 , each other. In chloroethylenes , bromoethylenes and 209 iodoethylenes , the conjugative e f f e c t dominates, r e s u l t i n g i n an o v e r a l l 271 d e s t a b i l i z a t i o n of the TT o r b i t a l r e l a t i v e to ethylene . However, the s i t u a t i o n may be d i f f e r e n t i n the fluoroethylenes owing to the e l e c t r o -n e g a t i v i t y of the f l u o r i n e atom. The f i r s t IP of the compounds studies, which corresponds to the removal of an electron from the TT o r b i t a l should provide information of t h i s sort. In ad d i t i o n , from the energy s h i f t s of the 41 42 6 o r b i t a l s of I and I I , the perfluoro e f f e c t ' which operates on the ethylene molecule may be investigated. The f i r s t PE band of I (Fig. 33a) with a v e r t i c a l IP of 10.42 ev and adiabatic IP 10.23 ev e x h i b i t s extensive v i b r a t i o n a l structure which can be inte r p r e t e d as being due to e x c i t a t i o n of Vx , i-4 and V5 modes with values of 1595, 1360 and 234 cm * r e s p e c t i v e l y . A reduction i n the C-C st r e t c h i n g frequency and an increase i n the C-F symmetric s t r e t c h i n g mode - 158 -f — I I I 1 I I I I I L I l I l I I I 1 1 I 1 I 1 9 10 11 12 13 14 15 16 17 18 19 20 21 IP(EV) • •'igure 32 . The PH spectra of cis, and (b) trans 1 , 2 difluoroethylcnc. - .15 9 -13.6 140 IP(EV) 14.4 14.8 Figure 33. The PE spectra of c i s 1,2 difluoroethylene (a) the f i r s t band, and (b) the second and the t h i r d band. 1 1 1 1 —I J—I I—• I—I 1 1 1 1 1 1 1 1 I I I I I I I I I 13.4 14.0 14.8 14.9 15.1 15.3 15.5 15.7 I P ( E V ) > Figure 34. The PE spectra of trans 1,2 difluoroethylene (a) the f i r s t band, (b) the second and the t h i r d band, and (c) the fourth band. - 161 -Table 26. Ionization Potentials and Symmetric V i b r a t i o n a l Frequencies of the Molecular Ions of Cis and Trans 1,2 Difluoroethylene -1 c ,, . ... , V i b r a t i o n a l Frequencies (cm ) _ , . ^  ., V e r t i c a l Adiabatic 1 -* O r b i t a l T r i (. . T n , . IP(ev) IP(ev) V, Vz L>+ Vs Cis 1,2 Ground C„H_F_ Neutral 3122 1716 1263 1015 237 z z z State TU, 10, .42 10.23 1595 1360 234 13. .81 13.81 1438 992 331 13, .99 13.99 1438 14. 17. .89 .10 14.89 2653 1185 l«b, j Ua, 1 18 .91 Trans A Ground 1,2 Neutral 3111 1694 1286 1123 548 C.H F State 112. 10 .41 10.21 1591 1234 533 13 .84 13.52 505 15 .11 15.11 2815 1125 Lb, 17 .08 »>«, , 18 .82 18.09 1791 920 l3b a » "The experimental error i s + 0.01 ev except the v e r t i c a l IP's of the f i r s t and the fourth band, ± 0.02 ev. The assignment of i o n i c frequencies i s based on the r e s u l t of the FCF c a l c u l a t i o n i n section 3.3. ° V, i s the C-H s t r e t c h i n g , ^ C-C s t r e t c h i n g , l>3 and Us angle bending and C-F s t r e t c h i n g mode. dRef. 185. - 162 -185 i n comparison to those of the neutral molecule (Table 26) i s consistent with the fact that the corresponding TT o r b i t a l possessC-C bonding and C-F antibonding character. With regard to II (the trans isomer), the f i r s t v e r t i c a l and adiabatic IP's are 10.41 and 10.21 ev r e s p e c t i v e l y , and there are three associated v i b r a t i o n a l modes of frequencies 1591, 1234 and 533 cm *. These may be assigned to the ^ , V3 and L>s v i b r a t i o n s i n accord with a FCF c a l c u l a t i o n i n section 3.3. In the region o f 17-20 ev i n the PE spectrum of e i t h e r I or II (Fig. 32), there are two bands with a higher i n t e n s i t y f o r the lower energy 59 one. According to a CND0/2 c a l c u l a t i o n , these bands are r e l a t e d to the f l u o r i n e LPMO's. These o r b i t a l s can combine with each other to give Ut>a > l i a z > L ib, a n d U a , > f o r 1 ( F ig- 35) and 1^ , l 3 b t A , l i 0 9 and I I da f ° r II- The symmetry notation used f o r I are the same as that f o r c i s 1,2 dihaloethylene i n the previous section. A l l these o r b i t a l s except l i b and i\cxz a r e s t a b i l i z e d to a d i f f e r e n t extent through e i t h e r mixing with 6 o r b i t a l s of the same symmetry, or with the carbon TT o r b i t a l . Therefore, the o r b i t a l that gives r i s e to the peak at around 17.10 ev f o r I or II i s assigned to ei t h e r liaz ( I ) o r 11bg (II) • The other PE band at about 19 ev i s a composite of two overlapping peaks with almost the same IP on the basis of i t s i n t e n s i t y and bandwidth compared to the other band (17.10 ev). One of these two peaks should be re l a t e d to the of I, or the l ( a i i of II which posses both C-C and 59 C-F bonding character from the r e s u l t s of a CND0/2 c a l c u l a t i o n . This i s i n accord with the observed frequency change i n the Vz mode of II (Table 26). Hence the second peak can r e a d i l y be assigned to correspond to ^a, ( I ) , ( a ) ( b ) ( c ) ( d ) p Q u a l i t a t i v e MO diagram of c i s 1,2 difluoroethylene (a) no perturbation, (b) through space i n t e r a c t i o n , (c) conjugative e f f e c t added, and (d) through bond i n t e r a c t i o n added. - 164 -o r ^ 3 b u (II) due t 0 a smaller through bond i n t e r a c t i o n exerted i n t h i s o r b i t a l compared to the other LPMO l 3 b z (I) or ( . 3 0 3 ( U ) ( s e e discussion l a t e r ) . Between 13 and 16 ev, the PE spectra of the two isomers each show two peaks with resolvable f i n e structure (Figs. 33b, 34b and 34c and 59 Table 26). CND0/2 c a l c u l a t i o n s on both isomers give three 6" o r b i t a l s 5 5 a , , 6 3 b ; L and 6"4t>2_ f o r I, and 6 4 a g , 6 5 Q ^ and 6 4 b a f o r II i n decreasing energy (these 6" o r b i t a l s have a high mixture of f l u o r i n e atomic o r b i t a l s ) with IP's lower than those o f d a 2 (I) and l i b 3 (II)-In I, the second PE band consists of two overlapping bands instead of only one. This i s e x e m p l i f i e d by both the r e s u l t s of a FCF c a l c u l a t i o n on t h i s state (section 3.3), and the r e l a t i v e i n t e n s i t y . The v e r t i c a l IP's are 59 13.81 and 13.99 ev f o r the two bands. An MO c a l c u l a t i o n i n d i c a t e s that the 650., possesses C-C and C-F bonding character. In view of the reduction i n the L>2 and ^4. modes observed i n the 13.81 ev band, i t i s assigned to come from ^>sa, • The remaining peaks with v e r t i c a l IP's 13.99 ( e x c i t a t i o n of V\ i s observed) and 14.89 ev ( e x c i t a t i o n of both U, and i s obser-ved) are simply r e l a t e d to the o r b i t a l s 6" 3 b ) and 64b, r e s p e c t i v e l y . In the case of I I , the assignment f o r the second and the t h i r d PE bands i s not so obvious even though the i n t e n s i t y of the second band i s high compared to that of the t h i r d one. It i s , therefore, d i f f i c u l t to d i s t i n g u i s h the IP's 64.0.3 and > which p o s s i b l y give r i s e to the second band, because only a si n g l e Vs progression i s observed. The t h i r d band, c o n s i s t i n g of Ut and U + progressions, should be r e l a t e d to i n 59 p a r a l l e l with the o r b i t a l sequence predicted from a MO c a l c u l a t i o n , as well as the c o r r e l a t i o n between the bonding nature of the C-H bond and the observed reduction i n the C-H st r e t c h i n g frequency. - 16 5 -According to the assignment given above, the small peak at 16.2 ev i n the PE spectrum of e i t h e r I or II may a r i s e from an impurity. However, t h i s i s not conclusive since the lower IP regions are 'clean'. It i s improbable f o r an impurity to have a f i r s t IP of about 16.2 ev. Also, no v a r i a t i o n i n i n t e n s i t y was observed i n t h i s peak for d i f f e r e n t scans of the spectrum. It i s probable that t h i s band corresponds e i t h e r to a f l u o r i n e lone p a i r or to a 6" o r b i t a l s h i f t e d up to 16.2 ev from one of the 67 o r b i t a l i n ethylene. The former p o s s i b i l i t y can be ruled out since the i n t e n s i t y of t h i s peak i s small (ca.one-third) of the other f l u o r i n e lone p a i r s . I f the l a t t e r s i t u a t i o n i s the case, then the band i n both the c i s and trans isomers at 14 ev i s due to only one i o n i z a t i o n . However, no d e f i n i t e assignment can be made at t h i s stage. 4.5.2 One E l e c t r o n Model f o r the Cis and Trans 1,2 Difluoroethylenes The four f l u o r i n e LPMO's of I can i n t e r a c t with each other s p a t i a l l y to give l i a , l , b ) , l^a., a n a " ' i b z with d i f f e r e n t energies (Fig. 35 case b). In addition, the conjugative e f f e c t d e s t a b i l i z e s the C-C TT o r b i t a l TT^ with an unperturbed energy £ c = c > but s t a b i l i z e s the l| L ) | by the same amount ft' . Furthermore, l i b z and 1 4 . ^ are 78 'repelled' by S o r b i t a l s of the same symmetry . Usually the greater the number of G o r b i t a l s a v a i l a b l e f o r i n t e r a c t i o n , as well as the smaller the energy gap between the 6 o r b i t a l and Li , the l a r g e r the energy s h i f t . Therefore, l+a, > ^ s considered to be less r e p e l l e d by the 6 moiety than (-2b2 does, i . e . S b l i s greater than SQ_ I - 16 6 -A Huckcl type ND c a l c u l a t i o n s i m i l a r to that f o r the dihalo-ethylenes was c a r r i e d out with ^ = 0 . In t h i s way, £ c = c , £ p , Sex, , ji' > 65-a, > ^ T T A N D ^ 6 are found to be -12.22, -17.10, 1.85, -1.80, -15.66, 0.01 and 0.04 ev r e s p e c t i v e l y , and reproduce the observed IP's exactly. The f a c t that only three o r b i t a l s with IP's lower than 17 ev are 2 08 found i n c i s 1,2 d i c h l o r o - and 1,2 dibromoethylene fur t h e r supports our assignment that there are three bands i n the region of 13-16 ev i n the PE spectrum of I. A s i m i l a r treatment i s applied to II and gives £ c=c > £p > S b u i , ( 3 ' , ( o 4 b u and d x x to be -12.14, -17.08, 1.73, -1.73, -16.84 and 0.00 ev r e s p e c t i v e l y . The resonance i n t e g r a l ft evaluated through expression 275 (4.13) i s found to be greater i n I than i n I I . T h i s agrees with a c a l c u l a t i o n on the mesomeric e f f e c t of f l u o r i n e which shows that the conjugative e f f e c t i s more important i n I. It i s i n t e r e s t i n g to note that the ' e f f e c t i v e ' e l e c t r o n e g a t i v i t y of the f l u o r i n e atom seems to be d i f f e r e n t f o r the two isomers. This i s probably due to a d i f f e r e n t degree of electron d e r e a l i z a t i o n between the f l u o r i n e lone p a i r s , carbon TT o r b i t a l s and the 6 o r b i t a l s . The f i r s t 271 IP of both isomers i s close to that of ethylene and t h i s i n d i c a t e s that the inductive e f f e c t and the conjugative e f f e c t are n e a r l y the same. The shortening of the C-C bond i n both I and II i n comparison to that of ethylene p a r a l l e l s the electronegative nature of the f l u o r i n e atom. The large d i f f e r e n c e i n e l e c t r o n e g a t i v i t y between the f l u o r i n e and carbon atoms reduces the s h i e l d i n g of the carbon atom and thus enhances the binding energy of the TT electrons of the C-C bond. Conversely, the s h i e l d i n g of the f l u o r i n e atom w i l l be increased and the binding energy of electrons - 1 67 -associated with i t w i l l be decreased. Information about the electron d i s t r i b u t i o n around the carbon and f l u o r i n e n u c l e i can be obtained from the chemical s h i f t s i n carbon and f l u o r i n e NMR. The chemical s h i f t s of I and II i n the f l u o r i n e NMR spectra are found to be -165.0 and -186.25 ppm r e s p e c t i v e l y . The lower value of <5p i n II than i n I r e f l e c t s the f a c t that the electron cloud i s more dense around the carbon nucleus i n the former molecule. In f a c t , the c a l c u l a t e d MO parameters £p agree with t h i s observation. The absolute value of 8 c=c i s f ° u n d to be greater i n I. I f the 6 framework i s nearly the same f o r both isomers, the chemical s h i f t 5c i n the carbon NMR should be greatest f o r I. However, the reverse i s 264 observed. A recent ab i n i t o c a l c u l a t i o n on the fluoroethylenes shows that the c o n t r i b u t i o n of the carbon s o r b i t a l along the C-C bond i s d i f f e r e n t i n these two molecules. The higher the s character involved i n the C-C bond, i . e . the more s h i e l d i n g around the carbon nucleus, the wider i s the CCF angle. 191 A c t u a l l y , the CCF angle i s found to be greater i n I than II . This implies that the e l e c t r o n density around the carbon nucleus i s greater i n I. Thus the observed trend i n Sc can be reproduced, providing that the e l e c t r o n i c e f f e c t a r i s i n g from the 5" o r b i t a l s predominates over that from the TT 277 o r b i t a l . The f a c t that the chemical s h i f t i n proton NMR i s greater i n II further supports the above argument. 215 The PE spectrum of gem difluoroethylene was reported, but no assignment was made on the observed IP's. We have remeasured the spectrum of t h i s molecule and the v e r t i c a l IP's of TT 2 b | , 65a., > ^ 4 b 2 » ^ 3 b 2 > I I Q 2 and L i b | are assigned to have values -10.64, -14.91, -15.75, -16.06, 59 -18.31 and -19.87 ev r e s p e c t i v e l y based on a CNDO/2 c a l c u l a t i o n , peak - 168 -i n t e n s i t i e s and a comparison with c i s 1,2 difluoroethylene. A p p l i c a t i o n of an MO treatment s i m i l a r to that f o r I gives 8p , ft' and 6 C = C as -18.37, -1.43 and -12.08 ev r e s p e c t i v e l y . The smallness of I £ c - c I i n the gem isomer compared with the other difluoroethylenes i s i n p a r a l l e l 278 with the l a r g e s t observed e l c t r o n s h i e l d i n g around the carbon n u c l e i of t h i s molecule. 4.6 1,2 Dibromocyclohexane 4.6.1 Introduction Trans 1,2 dibromocyclohexane may e x i s t i n two d i f f e r e n t forms with both bromine atoms at e i t h e r the equatorial p o s i t i o n (ee), or the d i a x i a l p o s i t i o n (aa). The conformational eq u i l i b r i u m between the isomers has 279-282 been well i n v e s t i g a t e d by various methods , and the aa conformer i s found to be more stable than the ee despite the non-bonded repulsions between the a x i a l bromines and the a x i a l hydrogens. The population of the aa and ee forms i n the vapor phase has been 281 studied using electron d i f f r a c t i o n . Both conformers were found to e x i s t i n nearly equal amounts. Recently, however, by using the d i l u t e s o l u t i o n 282 method, Ul'yanova and h i s coworkers estimate the mole f r a c t i o n of the aa form i n the gas phase to be 0.95. The r e s u l t of these l a t t e r authors may be more r e l i a b l e i n view of the technique used at that (1946) f o r measurement of the i n t e n s i t i e s i n electron d i f f r a c t i o n experiments. In t h i s section, we report the He I PE spectrum of 1,2 dibromocyclohexane (Fig. 36), and, - 169 -- 170 -assuming the v a l i d i t y of Koopmans' theorem, the i n t e r p r e t a t i o n of the spectrum i s based on the assumption that the aa conformers of t h i s molecule exists almost e x c l u s i v e l y i n the vapor state. The observed IP's are given i n Table 27. 4.6.2 Results and Discussion (a) Interaction between the Lone Pair O r b i t a l s of Bromine Atoms in D i a x i a l  1,2 Dibromocyclohexane (C^) F i g . 37 shows schematically the s p l i t t i n g of the bromine LPMO's by spin o r b i t coupling, and the through bond and through space i n t e r a c t i o n s . In case a, there i s no perturbation operating on a l l four LPMO's, namely p x and py which are degenerate o r b i t a l s . The x axis i s taken to be perpendicular to the BrC^C^Br plane while the y axis i s perpendicular to both the x and z (along C-Br bond) axes. However, the degeneracy i s p a r t l y removed with the i n t r o d u c t i o n of a through space i n t e r a c t i o n by which these o r b i t a l s combine to give Li3ctx , li2ay > '• l lby a n d '•'obx ° ^ a » a > D a n d D symmetry under the point group. l a t implies that the lone p a i r o r b i t a l of a symmetry i s contributed from the L ^ ( p x or p,j ) atomic o r b i t a l . Idi. = (i-, + 11)/f2 a n d lbi = { ~ L i ) / J 2 • with j of i _ . the number of the halogen atom. The bromine LPMO's can further mix with each other through o r b i t a l s of the same symmetry (case c ) . Thus, the degeneracy i s completely removed and a l l these o r b i t a l s are d e s t a b i l i z e d to d i f f e r e n t extents. 59 According to the r e s u l t of CND0/2 c a l c u l a t i o n s on d i a x i a l 1,2 dich l o r o -cyclohexane, the p„ and p^ o r b i t a l s of the chlorine atoms combine mainly with the px and o r b i t a l s of the cyclohexane. In view of the s i m i l a r - 171 -Table 27. Observed and Calculated V e r t i c a l IP's (ev) of Trans 1,2 Dibromocyclohexane Molecular Obs. I P a Obs. IP Cald. IP Cald. IP O r b i t a l (This Work) (ref. 283) (Set 1) (Set 2) 10.06 10.02 10.05 10.05 10.19 10.20 10.20 10.41 10.42 10.40 10.42 A 10.65 10.66 10.64 10.65 11.21 11.21 11.21 11.56 11.64 11.59 12.46 12.50 12.53 13.75 13.74 67b 14.63 6 8a 15.24 <57a 15.70 6<b 16.74 17.75 Experimental e r r o r i s within ± 0.01 ev f o r the f i r s t four IP's and within ± 0.03 ev for a l l other IP's. I t i l 7< i represents the i bromine LPMO. (a) (b) (c) ( d ) A 2d Sby > '13a x '12a y 'Hby, '10b x x c5 cr L J L U }11a y 6 1 0 a x  6 9 ^ 6 8 b y 'X to Figure 37. E f f e c t s of i n t e r a c t i o n on molecular o r b i t a l s of trans-1,2 dibromocyclohexane (a) no perturbation, (b) through space i n t e r a c t i o n , (c) through bond i n t e r a c t i o n added, and (d) spin o r b i t coupling added. - 173 -of the bonding nature between t h i s molecule, and d i a x i a l 1,2 dibromo-cyclohexane, only a LPMO with appropriate symmetry and d i r e c t i o n (x or y axis) i s allowed to combine with a p a r t i c u l a r o r b i t a l . The presence of spin o r b i t coupling enables LPMO's of the same symmetry to mix with one another. Using the operator described i n section 2.1.5, the following secular equation corresponding to the o v e r a l l i n t e r -action can be Sp~d- £ 0 i§/2 0 0 0 0 S a y 0 £ p-+d -£ 0 ' i*/2 0 0 0 0 £ P + d-£ 0 s ; 0 0 0 0 -i.2/2. 0 £ P-d- £ 0 Sa x 0 0 0 0 0 £ b > -£ 0 0 0 0 0 0 Sa x 0 £a»-£ 0 0 0 S b y 0 0 0 0 0 0 0 0 0 0 0 £ a a - £ obtained f o r l l i a y , L,,b y , l | 0 b x , l,ia% , 6 ~ 8 b x , 6 ~ , o Q x , 6"q B y and 6 n a y r e s p e c t i v e l y . £ p , £ b x , < £ a x , £ b y and £ a y denote the energies of the unperturbed lone p a i r s , S 8 b x , 6, 0 Q x , S'qby a n d 6"May r e s p e c t i v e l y . ^ the spin o r b i t coupling constant of bromine i s taken to be 0.33 ev (section 4.2.2a). In the secular equation (4.18), the number of unknown parameters ( t o t a l l y nine) exceeds the number of observed q u a n t i t i e s . So i t i s impossible to get a unique s o l u t i o n f o r the determinant. Assuming that S b x i s equal to zero, £ p , Say , So.* and S by are found to be -10.67, 0.41, 0.53 and 0.27 ev r e s p e c t i v e l y (set 1). Increasing S b x by 0.02 ev, £ p and S a y - 174 -remain the same, while -Sq.x and Sby increase to 0.59 and 0.29 ev (set 2). The calculated IP's f o r the f i r s t eight highest occupied o r b i t a l s from set 1 and set 2 are l i s t e d i n Table 27. The values of the parameters deduced from set 1 may be considered as the lower bound f o r these q u a n t i t i e s , and they are found to be greater than those of 1,2 dibromoethane (Table 16). Thus the through bond i n t e r a c t i o n has a greater influence i n 1,2 dibromo-cyclohexane than i n 1,2 dibromoethane owing to a smaller energy gap between the 6 MO's and the LPMO's i n the former molecule. In other words, the more paths a v a i l a b l e f o r through bond i n t e r a c t i o n , the stronger the i n t e r a c t i o n between the <o MO and the LPMO. (b) Discussion on the S t a b i l i t y of Trans and Cis 1,2 Dibromocyclohexane Recently, PE spectra of a s e r i e s of trans and c i s 1,2 d i s u b s t i t u t e d cyclohexanes CgH^gBrX with X = F, C l and Br, have been reported by Botter 283 et a l . From the sign and magnitude of the di f f e r e n c e -CP-irans - IPcis of 283 t h i s s e r i e s , the authors concluded that the trans ion of C,H BrX i s 6 io l e s s stable than the c i s ion. In t h i s work, d i f f e r e n t conclusions are drawn fo r 1,2 dibromocyclohexane. The non-bonded distance between the two bromine atoms i n trans ee 1,2 dibromocyclohexane i s shorter than that i n the aa form of trans 1,2 dibromocyclohexane, and, hence, the through space i n t e r a c t i o n i s stronger i n the former molecule. However, the dif f e r e n c e i s small, only 0.01 ev as estimated through expression (2.5). By considering the enthalpy d i f f e r e n c e between the c i s and trans conformers i n the neutral ground state (~~ 0.08 283 ev ) and t h e i r f i r s t lowest IP, the trans ion i s found to be more stable - 17 5 -by 0.05 ev on the assumption of the i n s e n s i t i v i t y of the through bond i n t e r a c t i o n to the p o s i t i o n of the bromine atoms. This i s not unreasonable because of a stronger di p o l e - d i p o l e i n t e r a c t i o n i n the c i s form compared to the trans. The He I PE spectra of trans 1,2 dichlorocyclohexane has also been recorded. However, the band corresponding to the chlorine LPMO's i s not resolvable owing to the overlapping of bands a r i s i n g from both aa and ee conformers as well as the small energy gap between these LPMO's. - 1 7 6-CHAPTER V CONCLUSION The r e s u l t s described i n t h i s thesis have demonstrated the usefulness of photoelectron spectroscopy f o r the measurement of i o n i z a -37 t i o n p o t e n t i a l s , and hence v i a Koopmans' theorem , the o r b i t a l energies of many molecules e x h i b i t i n g c i s - t r a n s isomerism or various conformers. The assignment of the PE spectra of the molecules studied i s 59 221 based on the r e s u l t s of CNDO c a l c u l a t i o n s ' and Franck-Condon f a c t o r c a l c u l a t i o n s , band shapes, v i b r a t i o n a l f i n e structures and the associated i o n i c frequencies, r e l a t i v e i n t e n s i t i e s of PE bands, and comparison with other r e l a t e d molecules. In the substituted ethanes studied, t h e i r PE spectra are complicated by the existence of two stable conformers, the trans and the gauche forms i n the vapor phase. The i n t e r p r e t a t i o n of the spectra i s based on the r e l a t i v e population of the two isomers from other spectroscopic techniques, e.g. i n f r a r e d and Raman spectroscopies as well -17 7-as the constancy of the photionization cross sections f or these rotamers. In a l l the ethanes studies, the trans form i s found to be more stable than the gauche and has a higher concentration i n the vapor phase. Hence the main features of the spectra of these substituted ethanes are assumed to derive from the trans form. In t h i s manner, the f r o n t i e r o r b i t a l s of the gauche diiodotetrafluoroethane are observed. However, determination of the conformational energy d i f f e r e n c e between the trans and gauche ions cannot be estimated with high accuracy owing to the presence of v i b r a t i o n a l f i n e structure on bands of the trans isomer. In general, the trans ion i s found to be more stable than the gauche even though a q u a n t i t a t i v e r e s u l t has not been obtained. \ From the IP's obtained from a s e r i e s of halogenated organic mole-cules, some molecular o r b i t a l properties such as the Coulomb energies, reso-nance energies, and through bond and through space i n t e r a c t i o n parameters have been deduced using one e l e c t r o n models with the i n c l u s i o n of spin o r b i t coupling and through bond and through space i n t e r a c t i o n s between halogen lone p a i r o r b i t a l s . This treatment enables one to determine the through bond and through space i n t e r a c t i o n s q u a n t i t a t i v e l y , such parameters not being previously a v a i l a b l e . Naturally, the method can also be applied to organic molecules containing heteroatoms other than a halogen, e.g. nitrogen, oxygen or s u l f u r and hence one may deduce valuable information about the bonding properties i n such molecules. The r e l a t i o n between o r b i t a l energies, MO parameters from the above treatment, force constants, bond lengths, nuclear quadrupole coupling constants, NMR chemical s h i f t s and e l e c t r o n e g a t i v i t y of the halogen atom i s discussed - 178 -for the dihaloethanes, dihaloethylenes and halotrifluoromethanes. It has been found that the c a l c u l a t e d MO parameters are useful i n explaining some physical p r o p e r t i e s , such as NMR chemical s h i f t s , nuclear quadrupole coupl-ing constants and bonding properties of dihaloethylenes. It would be f r u i t -f u l to apply a s i m i l a r MO treatment to the halogenated benzenes to know more about the nature of the ortho e f f e c t and the carbon and halogen NMR chemical s h i f t s . PE bands of molecules sometimes e x h i b i t v i b r a t i o n a l f i n e structure from which the v i b r a t i o n a l t r a n s i t i o n p r o b a b i l i t i e s can be evaluated. From these t r a n s i t i o n p r o b a b i l i t i e s , the i o n i c geometries of molecules may be determined by means of two current methods, the generating function method 33 and the method developed by Coon et a l . These methods have been applied to the group VI hydrides, nitrous oxide, dihaloethylenes, 1,2 dichloroethane and bromofluoromethane. The r e s u l t s show that these methods are powerful for determination of the i o n i c geometry and h e l p f u l i n assigning i o n i c frequencies observed i n a PE band e.g. i n the c i s and trans 1,2 d i f l u o r o -ethylenes. Also, the r e s u l t s of c a l c u l a t i o n s give a better p i c t u r e of the changes i n bonding properties during an i o n i z a t i o n process. Although the technique has not been widely employed by PE spectroscopists, i t w i l l be used extensively i n the near future when easier methods f o r FCF c a l c u l a t i o n s are developed. C e r t a i n l y the least squares f i t technique described i n 2 section 2.2.2 s i m p l i f i e s the procedure of c a l c u l a t i o n . Recently, Dr. Chong of t h i s Department has developed a s i m i l a r treatment which gives promising r e s u l t s . In most current work, including the work described i n t h i s t h e s i s , the v i b r a t i o n a l t r a n s i t i o n p r o b a b i l i t i e s during the i o n i z a t i o n process are assumed to be proportional to the height of the corresponding components - 179 -i n the PE band. This implies that the band widths at h a l f height are constant within the progression. This approximation i s good only when the v i b r a t i o n a l components are well separated from each other. It becomes more d i f f i c u l t to use when the components are s e r i o u s l y overlapped by one another, e.g. t h i s s i t u a t i o n a r i s e s to some extent in the dihaloethylenes where many v i b r a t i o n a l modes may be excited. In t h i s case, r e s o l u t i o n of these components using a Gaussian or Lorentzian band shape an a l y s i s should be e f f e c t e d to obtain a better estimation of the r e l a t i v e i n t e n s i t i e s . Ap-285-287 p l i c a t i o n of a deconvolution technique may further improve the date 288 obtained. The transmission f a c t o r of the p a r t i c u l a r spectrometer should also be considered. The above approaches are rather unexplored, and i t would be well worthwhile to improve the accuracy of the v i b r a t i o n a l i n t e n s i t y measurements by these mathematical techniques. Recently, t h e o r e t i c a l c a l c u l a t i o n s using Rayleigh-Schrodinger 289 290 perturbation theory and Green's function techniques have been c a r r i e d out to reproduce the v i b r a t i o n a l structure of the PE spectra of diatomic and triatomic molecules. The t h e o r e t i c a l r e s u l t s are i n s a t i s f a c t o r y agree-ment with the experimental spectra. In p r i n c i p l e , the treatment can be applied to other polyatomic molecules with more than three atoms. However, i t i s unfortunate that the time for computation increases r a p i d l y with the number of atoms i n the molecule under consideration. In summary, the work discussed i n t h i s t h e s i s has i l l u s t r a t e d the p o t e n t i a l of the methods described herein (the method of FCF c a l c u l a t i o n , and the Huckel MO treatment) to obtain information about the bonding proper-t i e s , o r b i t a l energies, i o n i c geometries, through bond and through space - 1 8 0 -i n t e r a c t i o n s of molecules. In p a r t i c u l a r , the extension of FCF c a l c u l a -tions to larger polyatomic systems w i l l further a s s i s t the i n v e s t i g a t i o n of the molecular parameters of molecules under consideration. - 181 -REFERENCES L. Asbrink, Chem. Phys. Lett. ]_, 549 (1970). L.Asbrink and J . W. Rabalais, Chem. Phys. Lett. 1_2, 182 (1971). T. E. H. Walker, P. M. Dehmer, and J . Berkowitz, J . Chem. Phys. 59, 4292 (1973). K. Siegbahn, C. Nordling, A. Fahlman, R. Nordberg, K. Harmin, J . Hedman, G. Johansson, T. Bergmark, S. Karlsson, I. Lindgren, and B. 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P H Y S . 4 4 , P L 2 7 1 ( 1 9 6 6 ) NO R L S ( I , J ) R L P ( I » J ) F F ( I ) F P ( I ) R S ( I ) T A U ( I , J ) T A ' J P ( I t J ) R J ( I , J ) R K { I , J ) R J T J { I , J ) R J T J I ( I , J ) R E ( I , J ) A ( I , J ) B ( 1 ) C ( I , J ) • ( I ) c ( I , J ) I O V E R < I ) M O V E R ( I ) I C J M 8 ( I ) T P C ( I t J ) R E M E M B E R T H E I C 3 M B 3 ( I ) S O I I ) FS0{ I ) D I M E N S I O N 0 I M E N S ! O N D I V t N S I J N Q I M E N S I O N D1 M E N S I O N D I ME MS I O N D I M E N S I O N O F L S M A T R I X I \| I N P U T I M ROW L S M A T R I X I N I N P U T I N R O N F R E Q U E N C Y I I F R E Q U E N C Y I N T H E L S , F R E Q U E N C Y M A T R I C E S • J R O U N D S T A T E { A M U * * 1 - 1 / 2 ) ) ( = 3*1-6) U P P E R S T A T E M A X ^ I O G R O U N D S T A T E { I N OP E X C I T E D S T A T E ( = E M - 6 ) C M * * - 1 ) M A X = 1 0 ( = 3 N - 6 ) : X C I T E D S T A T E ( I N C M * * - L ) M A X = l O ( = 3 N - 6 ) C H A N G E I N S T R U C T U R A L P A R A M £ T 2 R S I N I N T E R N A L S Y M M E T R Y C O O R D I N A T E S ( I N A N G S T R O M ) T A U M A T R I X I N L O W E R S T A T E T A U M A T R I X I N U P P E R S T A T E J M A T R I X NI U N I T K M A T R I X A M U * * ( 1 / 2 ) * A ( J T ) ( T A U P ) ( J ) + T A U I N V E R S E O F M A T R I X I D E N T I T Y M W P . I X ( = S « - S ) A M U ^ - ^ - l A * * - 2 S A M E U N I T R J T J { I , J ) 3 N - 6 3 N - 6 A M A T R I X 3 N - 6 B M A T R I X 3 N - 6 C M A T R I X 3 M - 6 X 3 N - 6 D M A T R I X 3 N - 6 E M A T R I X 3 N - 6 X 3 N - 6 A R R A Y W I T H E L E M E N T S ( O R D E R 0 N O C A L N • W I L L B E D O N 1 1 C A L N o W I L L B E D O N E O N M A X I M I U M N J . I U V E R ( I ) I F E Q U A L T O N O ) O N O V E R T O N E O V E R T O N E OF O V E R T O N E C A L O . F O R T H E C O R R E S P O N D I N G I T I S N O N Z E R O . I N P U T N U M B E R O F V C O M B I N A T I O N O R D E R I N G O F A R R A Y W I T H ' 0 E Q U A L T O N O ) O N C O M B I N A T I O N B A N D O F 1 ) A R R A Y W I T H E L E M E N T S ( O R D E R 0 N J C A L M . W I L L B E D O N E 1 C A L N . W I L L B E D O N E M A T R I X W I T H E L E M E N T S A S T R A N S I T I O N P R O B A B I L I T Y B A N D O F I A N D J T H M O D E O V E R T O N E I N S U C H W A Y ( 1 , 1 ) , ( 2 , 1 ) , ( 3 , 10. O F E L E M E N T S = ND N O . C A L N . W I L L B E O O N E O N E X C I T A T I O N O F T H R E E V I B R A T I O N A L M O D E S 1 C A L N o W I L L B E D O N G A R R A Y T O S T J R E A L L p C F • S F O R P L O T A R R A Y T O S T O R E A L L F R E Q U E N C I E S F O R P L O T T I T L E (2.0) , P L S ( 1 0 , 10) , R L P ( 1 0 , 1 0 ) , F F ( 10) , F P ( 10) , L ( 15) -,:S ( 1 0 ) , TALK 1 0 , 10 ) , T A U P { 1 0 , 10) , R L P I ( 1 0 , 10 ) , R I ( 2 2 5 ) fM15) R J ( 1 0 , 1 0 ) , K K ( 1 0 ) , R W ( 10 , 1 0 ) , R J T J I 1 0 , 1 0 ) , R X ( 1 0 , 10) R J T J I ( 1 0 , 10) , R E ( 1 0 , L O ) , A ( 1 0 , 10) , R P ( 1 0 , 10) , i i ( 1 0 ) C ( 1 0 , 1 0 ) , D ( 1 0 ) , F ( 1 0 , 1 0 ) , F H 2 0 ) , T P C ( 4 , 4 ) , T P T ( 2 , 1 , 1 ) I O V E R ( 10) , M ) V E R ( 1 0 ) , I C O M B ( 1 0 ) , N I N ( 10 ) , T C 31 ?» 1,1) D I M E N S I O N I C O M a ? ( 1 0 ) , R I ( 1 0 ) , S ) ( 5 0 0 ) , F S 0 ! 3 0 0 ) C I N P U T D A T A 1 3=5 !W = 6 37: A D (I 'R, 1 0 ) T I T L E 1 0 F O R M A T ( 2 0 A 4 ) W R I T E ! I W , 2 0 ) T I T L E 2 0 F 3 R M A T { 1111 , / / / , 1 0 X , 2 0 A 4 ) W R I T E ! I W , 3 0 ) T I T L E 3 0 F 1 R M A T ! / / , 1 G X , 2 0 A 4 ) R E A D ! I R , 4 0 ) N O 4 0 F O R M A T ( 1 5 ) 0 3 6 0 I = 1 , N D 6 0 R E A D ! I R , 5 0 ) ( R L S ( I » J ) » J -" 1 , N D ) 5 0 F O R M A T ( 1 0 F 8 . 4 ) D G 7 0 1 = 1 , N O 7 0 R E A D ! I R , 5 0 ) ( R L P ( I » J ) , J = 1 » N D ) R : A 0 ( I R , 8 0 ) ( F F ( I ) » 1 = 1 , N D ) 8 0 F O R M A T { 1 OF 3 , 2 ) R E A D ! I A , 9 0 ) ! F P ( I ) , 1 = 1 , N D ) R E A D ( I R , 9 5 ) ( I O V E R ! I ) , 1 = 1 , M O ) R E A D ( I R , 8 5 ) ( M O V E R ! I ) , 1 = 1 , N D ) R E A D ! I R , 8 5 ) ( ICDM>3< I ) , T = 1 , N U ) R E A D ( I R , 3 5 ) ( I C G M B 3 { 1 ) , 1 = 1 , N D ) 8 5 F O R M A T ! 1 0 1 5 ) W R I T F ( I W , 9 0 ) ( I , F F ( I ) , I = 1 , N D ) 9 0 F O R M A T ( / / / / , 5 X , • F U N D A M E N T A L F R E Q U E N C I E S I N L O W E R S T A T E ( I N C M * * -2 ' / / , 1 0 ( 2 X , I 2 , F 8 . 2 ) ) W R I T E ! I W , 1 0 0 ) ( I » F P ( I) , I = 1 , N D ) 1 0 0 F O R M A T ( / / , 5 X » • F U N D A M E N T A L F R E Q U E N C I E S I N E X C I T E D S T A T E I I N C M * * -2 ' / / 1 0 ( 2 X , I 2 , F 8 . 2 ) ) R E A 0 ( I R , 1 1 0 ) ( R S < I ) , 1 = 1 , N D ) 1 1 0 F O R M A T ( 1 0 F 8 . 5) W R I T E ! I W , 1 2 0 ) I R S ! I ) , 1 = 1 , N O ) 1 2 0 F O R M A T ! / / / , 5 X , ' C H A N G E I N S T R U C T U R A L P A R A M E T E R S ( = S P - S ) * / / , 5 X , 2 1 0 F 1 0 . 5) W R I T E ! I W , 1 3 0 ) 1 3 0 F 0 R M A T { / / / , 4 X , • L M A T R I X I N G R O U N D S T A T E ' / ) D O 1 4 0 I = l , N D 1 4 0 W R I T E ( I W , 1 6 2 ) ( R L S ( I , J ) , J = 1 , N O ) W R I T E ! I W , 1 5 0 ) 1 5 0 F O R M A T ! / / / , 4 X , ' L M A T R I X I N E X C I T E D S T A T E ' / ) D O 1 6 0 1 = 1 , N D 1 6 0 W R I T E ! I k , 1 6 2 ) I R L P ( I , J ) , J = 1 » N O ) 1 6 2 F O R M A T ( 2 X , 1 0 F 8 » 4 ) C C E V A L U A T I O N O F T A U A'-JD T A U P M A T R I C E S DO 1 7 0 I = 1 , N D D O 1 7 0 J = 1 , N 0 T A U ! I , J ) = 0 . 0 T A U P ( I , J ) = 0 o 0 1 7 0 C O N T I N U E 0 0 1 8 0 1 = 1 , N O T A U ! I , I ) = 0 . 0 2 9 o 8 1 * F F ( I ) T A U P ! 1 , 1 ) = 0 . 0 2 9 6 3 1 * F P ( I ) 1 3 0 C O N T I N U E I K = 1 D O 1 9 0 J = 1 , N D - 198 -C C C ' D O 1 9 0 1 = 1 , N O R I ( I K ) = R L P ( I , J ) I K = I K + i 1 9 0 C O N T I N U E C A L L M I N V i R I , N O , D D , L , M ) I K = 1 DO 2 0 0 J = 1 , N 0 D O 2 0 0 1 = 1 , N O R L P I ( I , J ) = R I ( I K ) I K = I K + 1 2 00 C O N T I N U E C C E V A L U A T I O N O F J A N D K M A T R I C E S DO 2 2 0 1 = 1 , N D DO 2 2 0 J = 1 , N D R J ( I , J ) = 0 . 0 D O 2 1 0 K = 1 , N 0 R J ( I , J ) = R J ( I , J ) + R L P I ( I , K ) * R L S ( K , J ) 2 1 0 C O N T I N U E 2 2 0 C O N T I N U E D O 2 4 0 1 = 1 , N O R K ( I ) = 0 . 0 D O 2 3 0 K = 1 , N D R K ( I ) = R K { D + R L P I ( I , K ) * R S ( K ) 230 C O N T I N U E 2 4 0 C O N T I N U E C C E V A L U A T I O N O F A , e , C , D A N D £ M A T R I C E S DO 2 7 0 1 = 1 , N D 0 0 2 7 0 J = 1 , N 0 R W l I t J ) = 0 o 0 DO 2 6 0 N N = 1 , N D 0 0 2 5 0 K = 1 , N 0 R W ( I , J ) = R W U » J ) + R J ( N N , I ) * T A U > { N N , K ) * R J ( K , J ) 2 5 0 C O N T I N U E 2 6 0 C O N T I N U E 2 7 0 C O N T I N U E C D O 2 8 0 1 = 1 , N D D O 2 8 0 J = 1 , N D R J T J ( I , J ) = R W < I , J ) + T A U ( I , J ) 2 8 0 C O N T I N U E I K = 1 D O 2 9 0 J = 1 , N D D O 2 9 0 1 = 1 , N D R I { I K ) = R J T J ( I , J ) I K = I K + 1 2 9 0 C O N T I N U E C A L L M I N V t R I , N D , D D , L , M ) 1 K = 1 D O 3 0 0 J = l , N D DO 3 0 0 1 = 1 , N O R J T J K I , J ) = R I ( I K ) I K = I K + 1 3 0 0 C O N T I N U E C C C A L C U L A T I O N O F T R A N S I T I O N P R O B A B I L I T Y O F O V E R T O N E S r C NSO=I CALL ABC O F ( N O , R J , T A'. IP , T VJ , R J T J I , F<K , A , B , C , 0 , E ) 00 7 30 1=1,NO I F ( I OVER{I ).EQ.0) GO T O 7 30 MM = MOV E F ( I ) +1 1 1 = 1-1 CALL I N T S T K C , J,MM, I, F I ) WRITE! IW,720 ) I, FP( I ) ,MJVER( I ) 720 F O R M A T ( / / / / , 5 X , « FRE )P ( ' » 12 »• ) = • , F 10 . 2 / 5X , 1 NO. DF JV ERTONES ( FROM 0 2 TO MM = ', 13//,5X, ' V ,3X, ' TRANSITION P RO R A BI LI S FOR OVERTONES'/) 00 721 J=1,MM J1=J-1 721 WRIT£( IW, 722 ) J1 , F I ( J ) 723 F 0 ^ A T { 4 X , I 2 , 1 4 X , F 1 0 . 4 ) DO 722 J=1,MM S C ( N S O ) = F I ( J ) FSO(NSO) = F P ( I )~*FLOAT( J) 7 22 NSu=NSO+l 730 CONTINUE C C CALCULATION OF TRANSITION PROBABILITIES OF COMBINATION BANDS MD1=ND-1 DO 76 0 I=1,ND1 1 F( ICOMB( I ). EQ .0 ) GO TO 760 J = I + 1 740 CONTINUE IF( I C O M B t J J . E Q . O ) GO TO 750 CALL COMB IN ( C D , I , J,TPC) W R I T E ( IW , 1 0 1 0 ) I , F P ( I ) , J , F P ( J ) 1010 FORMAT(////,5X,'TRANSITION PROBABILITY OF COMBINATION BANDS WITH*/ 25X,'EXCITATION OF TWO VIB. MODES•/5X,•FR2QP(• , I 2 , ' ) =',F10.4,5X, 3•FREQP( • , I 2 »') = 1 ,F 10.4//5X,•I« ,2X , • J * ,3X,•TRANSI TI ON P R O B A B I L I T Y 4S OF COM3. BANDS 1/) DO 1015 N=l,4 1015 WR ITE(IW, 10 20 ) N,TPC(N,1> 1020 F O R M A T ( 4 X , I 2 , 2 X , » 1 » , 4 X , F 1 0 . 4 ) DO 1030 N=2,3 1030 WRITEtIW,1035) N,TPCtN,2) 1035 F 0 R M A T ( 4 X , I 2 , 2 X , , 2 « , 4 X , F 1 0 . 4 ) DO 745 LL=1,4 SO(NSO)=TPC(LL,1) F S O ( N S O ) = F » ( J ) + F P ( I ) + F L O A T ( L L ) 745 NS0=NS0+1 DO 747 LL=2,3 S C ( N S 0 ) = T P C ( L L t 2 ) F S D ( N S O ) = 2 „ * F P < J ) + F P ( I ) * F L 0 A T ( L L ) 74 7 NS0=NS0*1 750 CONTINUE I F ( J . E Q . N D ) GO TO 760 J = J + 1 GO TO 740 760 CONTINUE C C CALCULATION OF TRANSITION PROBABILITIES OF THREE VIB. MODES C EXCITED AT THE SAME TIME NO 2-NO-2 DO 900 1 = 1,MO 2 c c r - 200 -IF( IC1MB3( I J . E 0 . 0 ) G J TO 9J0 J=I + 1 0 10 CONTINUE I F { I C0'•'B3 ( J ) o tQo 0 ) GO TO 920 K = J+1 9 30 CONTINUE IF{ I C O M B 3 { K ) . E Q » 0 ) GO TO 940 CALL CJM3TH(C,0, I , J,K,TPT) 00 9 34 L L = i , 2 S 0 ( N S 1 ) = T P T t L L , l , l ) FS J(MSO)=FP( J ) + F P ( K ) <-FP(I)*FLOAT(LL) 934 NS0=NS0+1 WRITE(!W,201J) I ,FP( I ) , J , F P { J ) , K , F P ( K ) 2010 FORMAT!////, 5X, 'TRANSITION PROBABILITIES OF COMBINATION WITHV5X, 2'EXCITATION OF THREE V U „ .-10 0 E S ' / 5 X , ' F R E 0 P ( ' , 12, ' ) = « , F 1 J„ 2 , 5X , = • ,F10.2//5X, ' I',2 X,•J', 3'FREOPC ,12,' ) =• ,F10.2,5X,«FR| G P ( ' , 1 2 , • ) 2020 940 920 900 4 2 X,•K' ,3 X, 'TRANSITION PROBABILITIES'/) WRITE( IW,2020) T P T ( 1 , 1 , 1 ) , T P T ( 2 , 1, 1) FORM AT{5X,•1' ,2X ,'1' ,2X , •1• ,10X,F10.4/5X , '2' ,2X, « I • , 2X, • 1' , 10X, 2F10.4//) CONTINUE IFIK.EQ.ND) GO TO 920 K = K + 1 GO TO 930 CONTINUE IF(J. E 0 „ N D 1 ) GO TO 900 J = J + 1 GO TO 910 CONTINUE NORMALIZATION OF CALCULATED FCF NS0=NS0-1 8IG=S0(1) G R = F S O l l ) 00 1068 JJ=1,NS0 I F { S 0 ( J J ) . L E . B I G ) GO TO 1066 B I G = S O l J J ) 10 66 CONTINUE 106 8 CONTINUE 00 1073 JJ=l,NSO I F ( F S O ( J J J . L E . G R ) GO TO 1072 GR=FSO(JJ) 1072 CONTINUE 10 73 CONTINUE DO 1074 JJ=1,NS0 S O ( J J ) = S J < J J ) / B I G 10 74 CONTINUE XMAX=GR+1000. CALL A LSCAH-1000, CALL ALGRAF(FSO,SO,NSO,-3) CALL PLOTND STOP END ,XMAX,OoO,1.2) C C C c r SUBROUTINE TO EVALUATE A,B,C,0 AND ND DIMENSION or MATRICES E MATRICES (JN-6 ) - 201 -C C c C R J J M A T R I X C T A U P T A U ' M A T R I A F I R E X C I T E D S T A T E C T A U T A U M A T R I X F I R G R O U N D S T A T E C R J T J I I N V E R S I O N M \ T : U X O F J T J C R K D M A T R I X A , B , C , D , E O U T P U T E D M A T R I C E S r SUBROUTINE ABC DE (NO, R J , T AU P» T AU , R J T J I , RK » A , B , C , 0 , E ) D I MENS ION RW( 10,10) ,RJ{ LO, LO),T AUP(I 0 , L O ) , R J T J ( L O , L O ) , T A U ( 10 , 10) DIMENSION R E ( 1 0 , 1 0 ) , R J T J I ( 1 0 , 1 0 ) , A ( 1 0 , 1 0 ) , R R ( 1 0 , 1 0 ) , R Z ( 1 0 ) D I MENS I CN RX( 10, 10) , RK. (10) , B ( 10 ) ,NIN ( 10 ) ,C I 10,10 ) , D( 10 ) , E ( 10 ,10) C EVALUATION OF A MATRIX M V< ID IW = 6 00 310 1=1 , NO 00 310 J=1,ND R E ( I , J ) = 0 o 0 310 CONTINUE DO 3 20 I=1,ND RE( I , I )=loO 320 CONTINUE 00 3 70 1 = 1 f MO DO 370 J=1,N0 RW(I,J)=0.0 00 360 L=1,ND 00 350 N=l,ND DO 340 M=1,ND DO 330 K=1,ND RW(I,J)= RW{I» J)+SQRT(TAUP(I,L) )*RJ<L,N)*RJTJI(N,M)*RJ(K,M)*S0RT17A 2 U P ( K , J ) ) 330 CONTINUE 340 CONTINUE 3 50 CONTINUE 360 CONTINUE 3 70 CONTINUE 00 3 80 1=1,MO 00 330 J=1,N0 A ( I , J ) = 2 . 0 * R W ( I , J ) - R E ( I , J ) 380 CONTINUE 00 382 1=1,10 3 32 NIN( I ) = I WRITE(IW t 390 ) ( N IN { I ) , I = 1,NO) 390 F 0 RMA T ( / / / , 5 X , 'A MATRI X • ,5X,•A MATRI X•//7X , 1 0 ( I 2,9X ) / ) 00 400 1=1,NO 400 WRITEtIW,410) N I N(I ) , ( A { I , J ) , J = 1,NO) 4 LO FORMAT (13, LOF 11.6) r C EVALUATION OF B MATRIX N0*1 DO 450 I=1,ND 00 450 J=1,N0 R U I , J)=0.0 00 440 N=1,ND 00 430 M=1,ND DO 420 K=1,ND Rw ( I , J ) =" W ( I , J ) + RJ( I ,N ) "RJ T J I (N,M) *R J( K , M) TAUP ( K, J) 420 CONTINUE 4 30 CONTINUE 440 CONTINUE - 202 -C C c 450 CONTINUE 00 460 1=1,NO 00 460 J=1,ND RR ( I , J)=RW{I,J)-RE{ I , J) 460 CONTINUE DO 490 I=1,ND R Z ( I ) = 0 . 0 DO 480 M=1,ND DO 470 K=1,ND RZ( I )=RZ(I)+SQRT(TAUP(I ,M) )*RR(M fK)*RKIK) 4 70 CONTINUE 480 CONTINUE 490 CONTINUE DO 500 1=1,NO B( I )=-2.0*RZ(I ) 500 CONTINUE C C EVALUATION OF C MATRIX ND*ND DO 530 1=1,ND DO 530 J=1,ND RWlI,J)=0.0 DO 520 M=1,ND DO 510 K=1,ND RW ( I , J ) = R W { I , J ) « - S Q R T ( T A U ( I , M ) ) * R J T J I ( M , K ) * S Q R T ( T A U ( K , J ) J 510 CONTINUE 520 CONTINUE 530 CONTINUE DO 540 1=1,ND DO 540 J=1,ND C ( I» J ) =2 o 0*RW { I , J ) -RE ( I , J ) 540 CONTINUE WRITElIW,550) ( N I N ( I ) t I = l , N D ) 550 FORMAT(///,5X, 'B MATRIX',5X,• B MATRI X•/4X,10{I 2,9X)) WRITE(IW,552) <B( I),1=1,ND) 552 FORMAT(10F11. 6) WRITE(IW, 560) (N IN( I ) »1=1,ND) 560 F0RMAT(///,5X,'C MATRI X* , 5 X , * C MAT RI X • / 7X , 10 ( I 2 , 9X ) ) DO 570 1=1,ND 570 WRITElIW,410) NIN( I ) , ( C ( I , J ) , J = 1 , N D ) C C EVALUATION OF 0 MATRIX ND*1 DO 620 1=1,ND R Z ( I ) = 0 . 0 DO 610 1=1,ND 00 600 N=1,N0 00 5 90 M=1,ND DO 580 K=1,ND R Z ( I ) = R Z ( I )+SQRT(TAU{ I ,L ) ) * R J T J I ( L , N ) * R J ( M , N)*TAUP(M,K)*RK(K ) 530 CONTINUE 590 CONTINUE 600 CONTINUE 610 CONTINUE 620 CONTINUE DO 630 1=1,ND D(I )=-2.0*RZ( I ) 630 CONTINUE WRITE(IW,640) ( N I N ( I ) , I = i , N D ) - 203 -640 FORMAT!///,5X, «0 MAT\I X• ,SX, 1D MATRI X•/4X, 10( I 2 , 9X) ) WRITE! IW,5'>2 ) (0(1 ) , 1 = 1 ,00) C C EVALUATION OF E MATRIX N0 * N 0 DO 680 1=1,ND 00 6Q0 J=1,N0 RW(I,J)= 0o 0 0 0 6 70 N=1,N0 DO 660 M=1,ND DO 650 K=1,N0 RW(I ,J) =RW( I » J ) + S Q R T ( T A U ( I , N ) ) * R J T J I ( N , M ) * R J ( K , M ) * S 0 R T ( T A U P ( K , J ) ) 6 50 CONTINUE 660 CONTINUE 67 0 CONTINUE 680 CONTINUE DO 600 1=1,ND DO 690 J=1,N0 690 E( I , J )=4oO^RW(I,J) WRITE(IW,700) (N IN(I ),1 = 1,00) 700 FORMAT!///, 5X, 'E M A T •< I X • , 5 X , ' E MATR I X • / 7X , 10 { I 2 , 9X ) ) 00 710 1=1,NO 710 WR ITE ( IW,4 10) N IN(I) » ( E (I , J) » J = 1 » N 0 ) RETURN END C c c FUNCTION FACT!I) I F ! I.GT.O) GO TO 5 FACT=1. GO TO 20 5 IFACT=1 DO 10 N=l,I IFACT=IFACT*N 10 CONTINUE FACT = FLOAT(I FACT) 2 0 CONTINUE RETURN END r C C C SUBROUTINE TO CALCULATE TRANSITION PROBABILITY OF OVERTONES C METHOD USED BASED ON T.E.SHARP AND H. M. OS EN STOCK C J.CHEM.PHYS. 41,P3454!1964) SEE ALSO CORRECTION C C ( I , J ) AND D ( I ) ARE IN PUT ED MATR ICES SUBROUTINE I NT SI N ( C , 0 , M Z » N , F I ) C M=MAXIMIUM NO. OF 0VERT')N ; CAL ' D. C N=N TH VIBRATIONAL MODE C F I ( I ) = TRANSITION PROBABILITY 1UTPUTED DIMENSI ON C(10,10) ,0!10) , F K 20) F I ( 1 ) = 1 . F I ( 2 ) = U)(N)*«2)/2« F I ( 3 ) = ( 2 <, *C ( N , N ) +0 ( N ) **2 ) + *2/ 8 . F I ( 4) = ( 6. *n(N)*C(N,N)+ 0(N) ** 3 ) * *2/4 B. F I ( 5 ) = ( 1 2 . ' M C ( N , N ) * " 2 ) H 2 . M D ( , J ) - « * 2 ) * C ( M , M ) + 0(N ) «*4) **2/334. - 204 -r r F I ( 6 J = ( 60. *P (N ) M C ( N , N) * <?. ) f 20. - I 0 ( 0 ) '< - 3 ) **C ( N , N) + 0 ( N ) * "5 ) *«2 / 3 S40„ f-1 { 7 ) = ( 1 2 0 o * ( C (N » N ) « 1) +120. •( 0( M) *~2 ) * < C (0, N ) **2) + 2 3 0. * ( 0 ( N ) **4) *C ( N,N)+D ( 'J) '6 ) 2 / 46 0 30. I F(MZ.LE.7) G ) TO 20 IK = 8 5 FFI=0. IK1=IK-1 MX=IKl/2+l F I { I K ) = 0 . 00 10 1=1,MX IQ=I-1 -10= IK 1-2*10 FFI= F F : + ( C ( N , N ) * * I Q ) K.N) k"*MQ ) / ( F ACT ( I U ) * F A C T ( M « ) ) 10 CONTINUE FI ( IK)=FFI'<FFI *FACT( I K 1) / ( 2.**IK1) IF( IK.EQ.MZ) GO TO 20 IK=IK+1 GO TO 5 20 CONTINUE RETURN END C r C C SUBROUTINE TO CALCULATE THE TRANSITION PROBABILITIES OF C COMBINATION 3ANDS SUBROUTINE COMB!N(C,D,I,J,TPC) C C AND D MATRICES FROM A AIN PROGRAM C I ANO J REFER TO ITH AND JTH MODE OF VIBRATIONS THAT GIVE C RISE TO OBSERVABLE COMBINATION BANDS C T P C ( I , J ) MATRIX WITH ELEMENT AS TRANSITION PROBABILITY OF C COMBINATION BANDS DI MENS I ON C ( 1 0 , 1 0 ) , D ( 1 0 ) , T P C ( 4 , 4 ) T0C<1»1)=<2.O*C( I , J ) + D U ) * D ( J) ) * * 2 M . O TPC{2fl)=( 2+o0*D{ I ) * C ( I, J)+2„0*D( J)'<C( 1,1 ) + ( D U ) * * 2 ) - s 0 ( J ) )*«2/16.0 TPC13,1 ) = ( 12.*C( I, I )*C ( I, J ) +6.0 MD( I )**2 )*C< I , J ) +12. H) ( I ) * D ( J)*C{ I ?, I ) + (D{ I ) **3) *L){ J) ) **2/96.0 TPC(4,1) = ( 1 2 . * D ( J »*(C(I» I ) * * 2 ) + 4 8 . * D ( I ) * C ( I, I ) * C ( I , J ) + 1 2 . * D ( J ) * ( D t 21 ) * * 2 ) * C ( I , I ) + 8 . 0 * ( 0 ( I ) * * 3 ) * C ( I »J ) + ( 0 ( I ) * * 4 ) * D ( J ) ) ^ 2 / 763. TPC{2,2)=(4.0*C( I, I ) * C ( J»J )+3.0*(C( I , J ) **2) + 2. 0 * ( 0 ( I ) * - / 2 ) * C ( J , J)+2 2 . 0 * < D ( J ) * * 2 ) * C ( I ,1 )+8.0*0 ( I ) *0( J) *C ( I , J ) +( 0{ I )*D( J ) ) * * 2 ) **2/64. T P C ( 3 t 2 ) = ( 2 4 . 0 * D ( J ) * C ( I, I ) *C ( I , J ) + 12 . *D ( I ) *C ( I , I ) *C ( I , J ) + 2 4o 0"':0 ( I ) 2 * ( C ( I , J ) * * 2 ) + 6 . 0 * 0 ( I ) * ( D ( J ) * * 2 ) * C ( I , I ) + 2 . 0 * ( 0 ( I ) * * 3 ) * C ( J , J ) + 12.0*( 3 0 ( I ) * * 2 ) - ' 0 ( J ) * C ( I, J ) + ( 0 ( I ) **3 ) •* ( 0 { J ) 2 ) ) * "<2 / 3 34. RETURN END C SUBROUTINE TO CALCULATED TRANSITION PROBABILITIES OF C EXCITATION OF THREE VIB. MODES SUBROUTINE C ilMBTHt C , 0 , I , J , K, TPT ) C C AND- D MATRICES INPUTiD FROM M \ 10 PROGRAM C I , J AND K REFER TO I,TH, J TH AND K TH MODE OF V I B . THAT GIVE C RISE TO OBSERVABLE COMBINATION BAN^S C T P T ( I , J , K ) MATRIX GIVES THE TRAOSITIJN PROBABILITIES OF C COMBINATION BANDS DIMENSI CM C ( 1 0 , 1 0 ) , 0 1 1 0 ) , 7 P T ( 2 , 1 , 1 ) TPT( 1 , 1 , 1 ) - ( 2. 0*0 ( I ) =C ( J , !<.) +2.0*0 ( J ) * c < I t K ) +2.0 *',){ K ) *C ( I , J ) + ? J ( I ) * 0 ( J )*0(K) )**2/3. TPT ( 2, 1 , 1 ) = < 4oO*C( I , I ) *C ( J ,K ) +0.0 •'«C ( I , J ) *C ( I ,K ) +4. 0* (D ( I ) * * 2 ) * - 205 -2C ( J i K ) + 4 o 0 M) ( I )"»){ J ) -<C( I T K ) *4.0*D< I ) Q (K ) *C( I » J ) *• ( 'J ( I ) -< '2) -0 ( J ) * 3D(K) )**2/*2.0 RETURN END C SUBR0U TINE MINV C PURPOSE C INVERT A MATRIX 0 USAGE C CALL MINV(A , N , 0 » L , M ) C DESCRIPTION OF PARAMETERS C A - INPUT MATRIX, DESTROYED IN COMPUTATION AND REPLACED OY C RESULTANT INVERSE C N - ORDER OF MATRIX A C D - RESULTANT DETERMINANT C L - WORK VECTOR JF LENGTH N C M - WORK V E C T J 3 OF LENGTH N r C REMARKS C MATRIX A MUST BE A GENERAL MATRIX C SUBROUTINE AND FUNCTION SUBPROGRAMS REQUIRED C NOME C METHOD C THE STANDARD GAUSS-JORDAN METHOD IS USED* THE DETERMINANT C IS ALSO CALCULATED. A DETERMINANT UF ZERO INDICATES THAT C THE MATRIX IS SINGULAR C C c c SU3ROUT INE MINV(A , N,0,L,M) DIMENSION A(225) ,L(15) ,M(15) D=l.O NK = -N 00 80 K=1,N NK=NK+N L ( K ) =K M(K)=K KK=NK+K BIGA=A(KK) DO 20 J=K,N IZ=N*(J-1J DO 20 I=K,N IJ= I Z + I X = A B S ( B I G A > - A B S ( A ( I J ) ) 10 I F ( X ) 15,20,20 15 BIGA=A(IJ) L ( K ) = I M(KJ = J 20 CON TIMUE C C INTERCHANGE ROWS r J=L(KI I F ( J - K ) 3 5,35,25 25 KI=K-N DO 30 I=1,N K I = K I + N HOLD = -A(K I ) - 206 -r c JI=KI-K+J A ( KI ) = A < J I ) 30 A(JI1=H0LD C C ' INTERCHANGE COLUMNS r 35 I=M(K) I F ( ! - K ) 4 5 , 4 5 , 3 8 33 JP = N * U - l ) 00 40 J= L , N JK=NK+J J I = JP+J HOLD=-A(JK) A ( J K ) = A ( J I ) 40 A ( J I ) = H 0 L D C C DIVIDE COLUMN BY MINUS PIVOT (VALUE OF PIVOT ELEMENT IS C CONTAIND IM BI GA C 45 I F ( B I G A ) 4 8 , 4 6 , 4 8 46 D=0.0 RETURN 48 DO 55 1=1»N I F { I - K ) 5 0 , 5 5 , 5 0 50 IK=NK+I A U K ) =A( l K ) / ( - B I G A ) 55 CONTINUE C C REDUCED MATRIX C DO 65 1=1,N IK=NK+I HOLD=A(IK) IJ=I-N DO 65 J=1,N I J = I J + N I F ( I-K)60,65,60 60 I F ( J - K ) 6 2 , 6 5 , 6 2 62 KJ=IJ-I+K AlIJ)=HOLD*A(KJ)+ A( I J ) 65 CONTINUE C C DIVIDE ROW BY PIVOT r KJ=K-N 00 75 J=1,N KJ=KJ+N I F ( J - K ) 7 0 , 7 5 , 7 0 70 A ( K J ) = A ( K J l / B I G A 75 CONTINUE C C PRODUCT OF PIVOTS C D=D*BIGA C C REPLACE PIVOT BY RECIPROCAL - 207 -C C c A{KK)=le0/3IGA 80 CONTINUE C C FINAL ROW AND COLUMN INTERCHANGE r K = K 100 K={K-1) IF{K) 150,150, 105 105 I=L(K) I F ( I - K ) 120, 120, 103 108 JQ=N*(K-1) JR=N*<1-1) 00 110 J = l , N JK=JQ+J HOLD=A(JK) JI=JR+J A(JK. ) = - A ( J I ) 110 A{JI)=HOLD 120 J=M(K) I F U - K )100, 100, 125 125 KI=K-N 00 130 1=1,N KI=KI+N H0LD=A(KI) JI=KI-K«-J A ( K I ) = - A { J I ) 130 A(JI)=HOLO GO TO 100 150 RETURN END $ S I G - 208 -r r C r r c c r C c r C C /•* r r C C r C r r r c r C r C C C C C C C APPENDIX II COMPUT C 0 "J RO I REF IN<-F.T. C THIS P NUMBER 0 F C F ( I CFCF<I P ( I ) OF < I ) A J ( I ) ND MAX PP N S E T N C S C A L S D L N C N NW F F F P S D D D X N F B E T A PROCRAM FOR CALCULATION OF D, DISPLACEMEN T OF F, DURING ELECTRONIC TRANSITION OR IONIZATION -p NAT" M2NT O C r) By HAU CHEM DEPT ' RPOR/-M CAN H *' i * L . , V, 19 NO AL 0 JAC ABIAN METH 10. BC 6TH,JUNE,1974 VIBRATIONAL P R 0G R E S SI 0 N UP TO OUVITUM FCF( I ) OBSERVED oI 0 CALCULATED JIG WEIGHING MATRIX ERROR MATRIX O F C F ( I ) -JACOB I AN MATRIX NO. OF PRIORESS ION CALCULATED. = V + 1 wA X. 19 Nle OF L U G E S T FCF V(MAX)+1 POWER FOR P ( I ) MA TRIX < 1 FCF WITH LOW VALUE IS EMPHASISED = 0 ALL FCF'S GIVE SAME WEIGHT 1 < FCF WITH HIGH VALUE IS EMPHASISED NO. IF S":TS CALCULATED NO. OF CYCLES OESIRED DURING ITERATION 1 FOR OILY TWO FCF'S INPUTEO SCALING FACTOR. THE D E SIR c 0 UPPER LIMIT FIR EL = ME? INTEGER FOR CHECKING NO. OF CYCLE 0 READ PP,SCAL FTC. AGAIN 1 READ FROM THE VERY BEGINNING FREQUENCY IN GROUND STATE FREQUENCY IN EXCITED STATE WEIGHTED STANDARD DEVIATION OF TH? INPUT VALUE OF D VIBRATIONAL QUANTUM NO. FIXED AS STANDARD 0 ALL THE FCF'S WILL BE CALCULATE-) WITHOUT ITERATION = FF/FP SAME AS PRO IN HE IL3 RONNE R'S NOTATION DIFFERENT FROM SMITH AND WARSOP'S NOT AT 10' ITS OF ERROR MATRIX DURING ITERATION ERROR MATRIX T I T L E ( 2 0 ) , 1 F C F ( 4 0 ) , C F C F ( 4 0 ) , NWQ5) ,AJ(20) ,BJ(20) P(20) , D F ( 2 0 ) ,A M(20),CM(20) NS ET ) 01 MENS ION DIMENSION IR = 5 IW = 6 N T = 1 READ(I^,1) NSET 1 FORMAT(1515) R E A D(IR,1 ) ( N W ( I ) , I = l , 5 READ(IR, 10) T I T L E 10 FORMAT(20A4) WPITF(IW,20) T I T L E 20 FORMAT ( IH L ,///, 1 OX, 20A4/7X , ' F P A NC K-C ONDO N C ALC'JL AT I ON ' /7X, • FRANC K 2CONDON CALCULATION'/) READ(I=,1) ND,MA X ND1=NQ-1 MAXl=MAX-l WRITf(!W,50) NO 1 , MAX 1 50 FORMAT(///,5X,•NT. np OBSERVED FCF = «,I3/5X,'Ni 2CF =•,I 3) REA0(IR,00) (OFCF( I ) , I = 1,N0) Of- THE MA XI MI 0' 60 FORMAT(10F 3.4) NORMAL! Z AT TON OF 0<3S ERV Hi) FCF A MO Wfi K i l l I NG MATRIX IF(N0.L2.16) GO TO 430 NOX=ND ND = 16 '+30 DMA X= 0FC F ('1 AX ) DO 8 5 !=1,ND 85 OFCF(I)=OFCF( I) / D M A X ESTIMATION OF GAMA PROM OBSERVED O F C F ( l ) AND 0 F C P ( 2 ) READ(I^,90) FF,FP 90 FORMAT(iOF ' 2.2 ) BET A=Ff-/FP WRITE( IW , 6 1 0 ) FF,FP,BET A 610 FORMAT( / , 5X » ' FREQUENCY IN OR 00ND STATE = ' , P 9 0 2 / 5X , •FREQU ENCY IM E 2 C I T E D S T A T E = • , F9„ 2 /5X , META = ' , F 9 0 6, 2X,' SAME AS PRO IN HEILBRON 2ER NOTATION') 98 IF(OFCF{ 1 ) .r.Q.0. ) GO TO <U0 GAMA=(OFCF(2)/OFCF(1) )*<l.+B'TAJ / t 1 „ 41 42 1 3 6* BET A ) DELTA = 0. 5*= GAMA 9 10 BETA 1 = B E T A f l . CX=-2.0*BETA/BETA1 DO 710 M=1,N0 MM=M-1 710 AM { M) = ( R ET A* ''0 o 2 5 ) *S QRT { F A CT ( MM ) ) / S ORT { B ET A1 * { 2 . ** ( M M- 1 ) ) ) 93 READ(! ! ?,95) NC,PP,SCAL 9 5 FORMAT(I 5,2F5.2) RE AD(IR,411) DDX,NF 4 11 FORMAT(F9.6,16) IF(DDXoEQ.O.) GO TO 420 G A MA = D 0 X* S Q R T ( F P * 0 • 017 S 7 7 ) DELTA=0.5 *GAMA 420 CONTINUE IF(MF.GT.O) GO TO 422 ND=NDX GO TO 440 422 CONTINUE WRITE( IW,96 ) PP,SCAL 96 FORMAT(5XPOWER FOR WEIGHING MATRIX F5.2/5X, 'SCALING FACTOR = 2,F5.2) WRITE(!W,8 70) NF 870 F0RMAT(5X,'NF =•,13) MCN= 1 DO 86 !=1,ND IF ( O F C F ( D . G T . O . ) Gl TO 81 P { I ) = 0.0 GO TO 86 81 PIT) =OPCF(I)**PP 86 CONTINUE SUM=0.0 DO 37 I=1,ND 87 SUM = SUM*-P( ! ) DO 09 I=1,ND 89 P(I)=P{T)/SUM C A L C U L A T r AND NORMALIZE FCF FROM GAMA OBTAINED ABOVE 97 CFCF( 1 ) = I. C F C F { 2 ) = 1 • 414 2 1 3 6 * B 2 T A * G \M A / ( 1. «- b 2 T A ) [F( Nfi.LT. 3) GO TO 152 440 DD 100 M - ? , M O M1=M-1 M2=M-? 100 C F C F ( M ) = C F C F ( M l ) * ( 1. 4 142 1 3 6* a -T A* G A 1 A / ( { 1. + !3 - T A ) * S 0 'R T ( F L ° A T ( M) ) ) + 2S0»T( FLOAT ( (M-l ) / M) )* (CFCF {M2J/CFCF ( 01 ) ) * (( 1 . - f E T A ) / ( l.+fJETA) ) ) C I F ( N F . : o . O ) GO TO 450 DMA=OFCF ( N F)/CFCF(NFJ 00 11.0 M--I .NO 110 CFCF(M)=CFCF{M)*DMA C C CALCULATION OF ER*0 3 MATRIX USING F0?.M0LA B Y CHAD DO 102 1=1,NO 102 OFl I) = T F C F I I ) - C F C F { I) 450 OM=CFCF(MAX) DO 103 1=1,NO 103 C F C F ( I ) = C F C F ( I ) / D M IF(NFoGToO) GO TO 460 W RITE ( I V!, 4 7 0 ) 470 F 0 R M A T ( / / , 5 X, 'V 1 , 6 X,'0 rt S FCF«,6X,•CALC FCF'/) 00 480 1=1 ,NO N 1 = 1-1 4 80 WRITE(IW,490 ) N1,0FCF ( I ) , C F C F ( I ) 4 90 F 0°M AT ( 4X » ! 2 » 5X » F3 • 4 , 5X , p 8 • 4 ) GO TO 170 460 IF1BETA+1.) 104,104,200 104 00 120 1=1,ND 1 1=1 - 1 120 AJ ( I )=CFCF( I ) * ( 2 , * AM! I ) * GAM A + 2. * I '*8B kC FC F I 11 ) /C F C F ( 1 ) *C M ( I ) / 2CM(I 1 )*EXP (AM( I)-AM{ I I ) ) ) SJ = 0 , 0 DO 800 1=1,NO IF(OFCF( D . F Q . O . ) GO TO 3 0 0 SJ = SJ+P(I )*AJ( I )*AJ ( I ) 8 00 CONTINUE DO 820 1 = 1 ,MO 020 AJ{ I ) = P( I ) * A J ( I ) / S J WRITE(IW,130) 130 FORMAL {//, 5X, ' V ,6X, "OBS FCF • , 6 X , • C ALC FCF' , 6 X , 'JACOB! A, 0 * , 6 X , 2 ' P ( I ) • , 3 X , ' N O . OF CYCLE'/) DO 140 1=1,NO N1=I-1 140 WRITE!IW,150) M l , O F C F ( I ) , C F C F ( I) ,AJ{ I ) , P ( I ) , NCN 150 F O R M A T { 4 X , I 2 , 5 X , F 8 . 4 , 5 X , F 0 . 4 , 5 X , F 3 . 4 , 5 X , F 3 . 4 , 6 X , I 2 , 2 F 1 2 . 4 ) SOn= 0 . DO 920 1=1,NO I F ( O F C F ( I J . E Q . O . ) GO TO 92 0 S D D = S D O + D F ( 1 ) * ) F I ! ) * P ( I ) 9 20 CONTINUE SO=SORT(SDO) WRITE! IW, 9 30) SD 930 F0RMA T(5X,'STD D E V I A TI IN =«,F9.6) 152 0 0=GAMA/SORT!FP *0.0!7877) WRITE! IW,155) GAM A,00, OFCF( 1 ) ,OFCF{2 ) L5r- FORMAT!//, 5X, • GA MA - • , FO. 6 , I OX , • 0 = • , F 9 . 6 , • L0*«-20 G M * v +0.5 'C M'*;" 21'/5X, 'OFCF! 1 ) = ' ,F Bo 4,5 X, 'Of C F! 2) =«,Ffl.4) - 2 1 1 -r I F ( M C N . G ~ . N C ) G . l T 0 170 NCN=NCN+ 1 0GAMA=0.0 0 0 1 6 0 T=1.MD I C ( O F C F ( I ) . E O . O o ) GO TO 160 0 G A M A = 0 G A M A • Q F ( I ) * A J ( I ) 160 C O N T I N U E GAM A= GAM A «-S C AL *O G A MA GO TO 97 r C B E T A G R R A T 2 ? THAN 1, HE IL BRONNE ? ' S FOR-IULA I S A)OPT ' " . n f{~i C H E M C AC""A VOL 5 4 , P 5 8 ( 1 9 7 1 ) 200 D E L T A = 0 . 5 * ( i A M A BX=-4„ 'B " T A * D E L T A / B E T A l DO 2 1 0 M=1,ND MM=M-1 C A L L SOMAT{BETA,M M,0ELTA,SUM, SUX ) A C = A M { M ) * - XP ( C X " ( D E L T A ^ "2 ) ) 2 1 0 AJ (,M) = BX *AC*SUX+AC --SUM/DELTA SS=0.0 DO 8 1 0 I=1,ND IF(CIFC«=< I ) o E O o O o ) GO TO B I O SS = SS + P ( I )'<AJ( I >*AJ( I ) 310 C ONT!NO R DO 820 1=1,MD 8 3 0 A J ( I ) = P ( I ) * A J ( I ) / S S W R I T E ( IW , 1 3 0 ) DO 5 70 1=1,ND N1=I-1 5 7 0 W R I T F . ( IW, 1 5 0 ) N1, 0 FC F ( I ) , C FC F ( I ) , A J ( I ) , P { I ) , NCN GAMA = 2 o " D E L T A D D = G A M A / S Q R T ( F P * 0 . 0 1 7 3 7 7 ) 500=0 . DO 5 6 0 1=1,NO I F ( O F C F ( D.EO.O. ) GO TO 560 SOD = SDO*-DF( I ) *DF ( I ) * P ( I ) 5 6 0 C O N T I N U E S D = S Q R T ( S O D ) W R I T E ( T W , 6 0 0 ) G A M A , D 0,S D 6 0 0 F 0 R M A T ( / / , 5 X , 'GAMA = • , F 9 . 6 ,10X, • 0 = * , F 9 . 6 , » !0' :'--20 GM* * i-O. 5 *C M*<: 2 1 ' / 5 X , , S T D D E V I A T I O N = ' , F 9 . 6 ) I F ( N C N . G T . N C ) GO TO 170 NCN=NCN+1 D D C L T A = Oo DO 5 5 0 1=1,NO I F ( O F C F ( I J o E Q . O o ) GO TO 550 D 0 R L T A = D D E L T A f D F ( I ) * A J ( I) 5 50 C O N T I N U E 0 E L T A = 0 EL T A + S C AL * 0 D EL T A GAMA = 2.*D:-LTA GO TO 97 17 0 C O N T I N U E MT= N T+1 I F ( N T . G T . N S F T ) GO TO 180 I F ( N W ( N T ) . G T . O ) GO TO 5 GO TO 9 3 I BO C O N T I N U E - 212 -C STOP END r r FUNCTI Kl FACT( I ) IF ( I . G T . O ) GO TO 5 F\CT=1„ GO TO 20 5 IFACT=1 DO 10 N = l , I IFACT=IFACT*N 10 CONTINUE F A C T = F L 0 A T ( I F ACT) 20 CONTINUE RETURN END C C SUB TOUTING USEO TO CALCULATE SUMATION OF HERMITE POLYNOMIALS 5UBR OUT INE SUM AT{BET A » M , 0 1 L T A , S U M , S U X ) DD=-4. ''<OELTAASQRT { BETA) / ( 1 . + BET A ) IFIM.GT.O) GO TO 3 SUM=0. SUX=1. GO TO 60 3 CONTINUE IF{MOD(M,2).GT.O) G1 TO 40 MX=M/2 SUX=( ( ( BET A - l . )/ (3-T A + 1. ) ) **MX ) / F ACT ( MX ) SUM=0. 00 30 IP=2,M,2 MP=(M-Tp)/2 SUX=SUX + ( DD**I P) *( ( ( BETA-1. ) /( BETA + 1. ) ) **MP) / { FACT ( T P) F ACT ( MP ) ) 30 SUM = SUM + I P * ( D D * * I P ) ' M { ( B E T A - 1 0 ) / { B E T A + l o ) ) * * M P ) / ( F A C T ( I P - 1 ) * F A C T ( J P ) ) GO TO 60 40 SUM=0. SUX=0. 00 50 IP=1,M,2 MP=(M-IP)/2 S U X = S U X + ( O D * * I P ) * ( ( ( B ^ T A - l . ) / { B E T A + 1 . ) ) * * M " ) / ( P A C T ( ! P ) « F A C T ( M P ) ) 50 SUM = SUM + I p*(DO** IP ) * ( { ( B'ET A- 1 „ ) / { BET A+ 1. ) )**MP ) / ( F AC T ( I P-1 ) * F ACT ( UP) ) 60 CONTINUE RETURN END $ S I G - 213 -APPENDIX I I I . Approximation Method for Evaluation of Force Constants of Ions. The Fs_, Ls_ and A, matrices i n the ground state are r e l a t e d to one another by A = L l fk Li (A1) In the i o n i c s t a t e , a s i m i l a r r e l a t i o n i s obtained Upon subtracting eqn. (A2) by eqn. ( A l ) , expanding and regrouping, one obtains A A = ( U F ? + Ls A F S + A L s F S + A L S A P S ) A U + (Ls + A L S ) A F S Ls + ALs F s Ls (A3) where AA = A l - A_ , AFs = f$_ - h , and ALs = U - U_ Usually A Ls i s very small compared to A F 5 . By neglecting terms with A Ls , a simple r e l a t i o n i s a r r i v e d at f o r AA and AFs A A = Ls. A F s LS_ (A4) or AFs = <Ls.)"' A A (U)"' (A5) - 214 -The expression (A5) provides a method to c a l c u l a t e Fs' i f Fs, Ls, A and _A_' are known. In order to check the v a l i d i t y of the method, c a l c u l a t i o n s were performed on both the 3B, and 2A, states of F^O. Values of fr and -f A thus obtained are, r e s p e c t i v e l y , 5.725 and 0.551 ^ n / ^ f o r the 2B, state and 7.690 and 0.273 m c * y n / A f o r the JA, state which agree well with those from the rigorous method (Table 6). It should be noted that the approximation method enables one to estimate values f o r the force constants of an ion even though the geo-metry i s unknown. I f the number of observed frequencies i s les s than that of the force constants to be determined, computation of -fr or or both, s t i l l can be done by assuming that the i n t e r a c t i o n force constants and the di f f e r e n c e i n the unobserved frequency A\; are zero. The numerical values of fr f o r H^Se and H^Te i n the JB, state were obtained i n t h i s way. 

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