IIIIIIIIIIIII I I “—L LIBRARY Michigan State University This is to certify that the thesis entitled A STUDY OF THE VIBRATIONAL SPECTRA OF THE PARTIALLY DEUTE RATED POLYCRYSTALL INE ETHYLENES presented by Richard G. Whitfield has been accepted towards fulfillment of the requirements for PH.D. Chemistry degree in fl '7 '7 [ZQCildénflQ Major professor Due September 26, 1977 0—7639 A STUDY OF THE VIBRATIONAL SPECTRA OF THE PARTIALLY DEUTERATED POLYCRYSTALLINE ETHYLENES By Richard G. Whitfield A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1977 ABSTRACT A STUDY OF THE VIBRATIONAL SPECTRA OF THE PARTIALLY—DEUTERATED POLYCRYSTALLINE ETHYLENES By Richard G. Whitfield The mid-infrared and Raman spectra of the partially- deuterated ethylenes: C2H3D, cis-02H2D2, trans-02H2D2, l,l-C H2D2 and C2HD3, and of dilute mixed crystals of the 2 partially—deuterated ethylenes in C2Hu and in one another have been observed in order to gain more insight into the intermolecular force field of ethylene and those of molecular crystals in general. Information concerning the retention of the symmetry of the parent (C2Hu) crystal as an "effective" symmetry in the partially-deuterated crys- tals has also been obtained. Various sets of atom-atom interaction potentials were used to calculatethe lattice frequencies of the 02H“ crystal using recent single crystal x-ray data. The best interaction potential was used to calculate the lattice frequencies of the partially-deuterated ethylenes, assuming the crystal structure and symmetry is unchanged by the deuteration. By comparing the observed and calculated Richard G. Whitfield frequencies, it was determined that the lattice modes of these crystals can be described in the virtual-crystal limit. Consistent with the virtual-crystal theory, mutual exclusion was retained for the lattice modes of these disordered solids and slight changes in the relative intensities of the observed librations were noted. An attempt to observe the missing optically active transla- tion of C2Hu in the far-infrared is described. It is suggested to be at 35 cm-l. The dilute-mixed crystal spectra were observed to determine the number of energetically-inequivalent or- ientations which exist for the partially-deuterated ethyl— enes in a lattice site; this reveals information concern— ing the "effective" site symmetry of the host, and was determined to be C1 in every case, consistent with the predictions of the virtual-crystal approximations. A comparison of the spectra of the partially-deuterated ethylenes in a 02H“ host and in each other indicates that the intermolecular interactions of the crystal are myopic; that is, the number of orientational components and the energies of these components are independent of the isotopic nature of the host crystal. Due to the "effective" Ci site symmetry, the internal can be treated modes of cis-ethylene-d and l,l-ethylene-d 2 2 in the pure crystal exciton formalism and are quite similar to the internal mode spectra of 02H“ and C2D“. However, Richard G. Whitfield the internal modes of ethylene-d1, trans—ethylene—d2 and ethylene-d3 must be treated as mixed-crystal modes. The coherent potential approximation was used to calcu— late the band structures of the internal modes of these "mixed" crystals, and the computational results were used to assist in the assignment of the components of the C2H3D’ trans-C2H2D2, and C2HD3 bands. The assignments of vu and v8 of 02HD3 have been switched. This assign- ment interchange is supported by observed static field energy shifts in the crystal, and by the calculated poten- tial energy distributions of these modes. Mutual exclu- sion was observed for the internal modes of trans-02H D 223 this further supports the concept of "effective" Ci site symmetry for the partially—deuterated ethylene crystals. to Pat to grow in our love and understanding of each other, our fellow man, and God. 11 "I (I) ACKNOWLEDGMENTS I want to first thank my wife, Pat, for her loving patience and understanding without which this work would have never been completed, and my son, Ricky, for keeping her distracted during the preparation of this Dissertation. I also want to thank my parents for their interest and understanding during the course of my education. The interest and advice of Dr. Leroi during my graduate career is greatly appreciated. I wish to acknowledge the financial support received from Michigan State University and the National Science Foundation. Finally, I wish to thank my fellow group members, students, faculty and staff of the Department of Chemistry for making my stay here a very pleasant and memorable experience. 111 [\0 on ad... v... Ono». 0" §“ Q“. ‘h TABLE OF CONTENTS Chapter Page LIST OF TABLES . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES. . . . . . . . . . . . . . . . . . ix I. INTRODUCTION . . . . . . . . . . . . . . . . 1 II. GENERAL THEORY . . . . . . . . . . . . . . . 5 A. The Vibrations of Molecular Crystals . . . . . . . . . . . . . . . . 5 B. The Crystal Structure of Ethylene . . . . . . . . . . . . . . . . 8 C. Theoretical Treatments of Molecular-Crystal Vibrations . . . . . . 1a Calculation of the Crystal- Vibrational Frequencies. . . . . . . . . 1A The Exciton Formalism. . . . . . . . . . 20 D. The Orientational Effect— Crystals of the Partially-Deuterated Ethylenes. . . . . . . . . . . . . . . 28 III. EXPERIMENTAL . . . . . . . . . . . . . . . . 3h IV. THE LATTICE VIBRATIONS OF THE PARTIALLY-DEUTERATED ETHYLENES . . . . . . . A1 A. Phonons in Disordered Solids . . . . . . H1 B. The Potential Function of Ethylene . . . U5 C. The Librations of the Partially— Deuterated Ethylenes . . . . . . . . . . 51 D. Conclusions. . . . . . . . . . . . . . . 6M V. PROBING THE SITE SYMMETRY OF THE PARTIALLY- DEUTERATED ETHYLENES VIA THE ORIENTATIONAL EFFECT O O O O O O O O O I O O O O O O O O O 66 iv Chapter Page A. Theory . . . . . . . . . . . . . . . . . 66 B. Experimental . . . . . . . . . . . . . . 69 C. Conclusions. . . . . . . . . . . . . . . 100 VI. THE INTERNAL MODES OF THE PARTIALLY- DEUTERATED ETHYLENES . . . . . . . . . . . . 102 A. Introduction . . . . . . . . . . . . . . 102 B. The Internal Modes of cis- -C2H2D2 and l, l— C H D . . . . . . 106 2 2 2 C. Excitons of Binary Mixed Molecular Crystals-The Coherent Potential Approximation. . . . . . . . . . . . . . 121 D. The Internal Modes of CZH3D, trans- C2H2D2 and C2HD3 O O O O O O O O O O E. Comparison of the Internal Modes of the Polycrystalline Ethylenes. . . . . . 155 F. Conclusions. . . . . . . . . . . . . . . 16A VII. DISCUSSION AND SUGGESTIONS FOR FUTURE WORK I O O O O O O O O O O O O O O O O O O O 166 APPENDIX A O O O O O O O O O O O O O O O 0 O O 170 APPENDIX B . . . . . . . . . . . . . 191 APPENDIX C o o o o o o o o o o o o o o o o o o o o 199 REFERENCES 0 O O O O O O O O O O O O O O O O O O O 2 O 2 Table 10 LIST OF TABLES a and orientation Lattice parameters matrixb for the ethylene crystal Structural parameters for free ethylene molecule and molecule in crystal Correlation table for solid ethylene . . . . . . . . . . . Partially-deuterated ethylenes- Analytical data. . . . . . . Nonbonded-potential parameters, where ViJ = Bexp(-Cr)—Ar-6 + Er-l. The observed and calculated lattice frequencies (cm-1) of solid ethylene The observed and calculated lattice frequencies (cm-1) of solid ethylene-d“. The calculated and observed libra- tional frequencies of the ethylenes. . The crystal normal coordinates of the librations of the isotopic ethylene crystals. . . . . . . . . . . . Perturbation strengths of the ethylene librations. . . . . . . . . . . vi Page ll 12 35 A6 “8 “9 56 58 62 Table Page 11 The number of orientational components for the partially-deuterated ethylenes in a C1 or Ci site . . . . . . . . . . . . 68 12 Observed frequencies of C2H3D funda- mentals in 1% 02H3D/C2Hu . . . . . . . . . 7O 13 Observed frequencies of cis-C2H2D2 fundamentals in 1% cis-C2H2D2/02Hu . . . . 71 1“ Observed frequencies of trans-C2H2D2 fundamentals in 1% trans-C2H2D2/C2Hu . . . 72 15 Observed frequencies of l,l-C2H2D2 fundamentals in 1% 1,1-02H2D2/02Hu . . . . 73 16 Observed frequencies of CBHD3 funda— mentals in 1% 02HD3/02HA . . . . . . . . . 7A 17 Observed frequencies of 02H3D funda- mentals in H% 02H3D/cis—02H2D2 . . . . . . 92 18 Observed frequencies of 02H3D funda— mentals in 1% 02H3D/trans-CZH2D2 . . . . . 93 19 Observed frequencies of C2H3D funda- mentals in A; C2H3D/1,1-C2H2D2 . . 9H 20 Observed frequencies of CZH3D funda- mentals in 1% 02H3D/02HD3. . . . . . . . . 95 21 Observed frequencies of 1,1-02H2D2 fundamentals in 1,1-C2H2D2/CZHD3 . . . . . 96 22 Observed frequencies of CZHD3 funda- mentals in 1% CZHDB/C2H3D. . . . . . . . . 97 vii Table Page 23 Calculated harmonic C-13 shifts (cm'l) from the C—12 frequencies . . . . . 104 2“ Internal fundamental frequencies -1 a (cm ) of cis-02H2D2 . . . . . . . . . . . 108 25 Internal fundamental frequencies -1 _ * (cm ) of 1,1 C2H2D2 . . . . . . . . . . . 110 26 Orientational Splittings (cm‘l) a of C2H3D, trans-C2H2D2 and 02HD3 . . . . . 129 27 Internal fundamental frequencies (cm'l) of c H D* . . . . . . . . . . . . . 132 2 3 28 Internal fundamental frequencies (cm-1) of trans-C H2D2*. . . . . . . . . . 13M 2 29 Internal fundamental frequencies (cm'l) of CZHD3* . . . . . . . . . . . . . 136 30 Potential-energy distribution and calculated frequencies of the out- of-plane vibrations of C2HD3 . . . . . . . 157 31 Gas-to-mixed crystal shifts of the l)a. ethylenes (cm- . . . . . . . . . . . . 160 32 Bandwidths ofthe ethylenes (cm-l). . . . . 163 viii LIST OF FIGURES Figure Page 1 The crystal structure of ethylene (from Reference 23) . . . . . . . . . . . 9 2 The structure of an arbitrary exciton band of ethylene. . . . . . . . . . . . . 26 3 The molecular structure and point group symmetries of the ethylenes . . . . . . . 29 A The orientations of C2H3D in 02H“ (C2HD3 in C2D“) o o o o o o o o o o o o o 31 5 The Raman lattice region of C2H3D, 1,1-C2H2D2 and C2HD3. The peaks marked with an asterisk are laser fluorescence lines. . . . . . . . . . . . 52 6 The Raman lattice region of cis- C2H2D2, 1,1-C2H2D2 and trans-CZHQDZ. The peak marked with an asterisk is a laser fluorescence line . . . . . . . . 53 7 The highest frequency Raman lattice modes of 1,1-C2H2D2 . . . . . . . . . . . 5H 8 The calculated librational frequencies versus the number of deuteriums . . . . . 60 9 The observed librational frequencies versus the number of deuteriums. The D2 frequencies are the mean of the ix Figure 10 ll 12 13 1A 15 16 17 Page cis-C2H2D2, 1,1-02H2D2 and trans- C2H2D2 values 0 o o o o o o o o o o o o o 61 Part of the infrared spectrum of CZH3D . . . . . . . . . . . . . . . . . . 76 Part of the infrared spectrum of 1% 02H3D/C2Hu; v7 and v12 of C2H3D o o o o o o o o o o o o o o o o o o 77 Part of the infrared spectrum of V10 Of CiS-C2H2D2 o o o o o o o o o o o o 78 Part of the infrared spectrum of 1% trans-C2H2D2/C2Hu; ”A and v12 of trans-C2H2D2o o o o o o o o o o o o o o o 79 Part of the Raman spectrum of 1% trans-C2H2D2/02Hu; v1 and v3 of trans-C2H2D2o o o o o o o o o o o o o o o 80 Part of the infrared spectrum of 1% 1,1‘C2H2D20 o o o o o o o o o o o o o o o 81 Part of the infrared spectrum of 1% Part of the Raman spectrum of Hz C2H3D. The peak marked with an asterisk is assigned as v3 of X Figures Page trans-C2H2D2. . . . . . . . . . . . . . . 8A 18 Part of the infrared spectrum of “% C2H3D/cis-C2H2D2; v1, vu and v3 of C2H3D. The peak marked with an asterisk is assigned as VA of trans-C2H2D2. . . . . . . . . . . . . . . 85 19 Part of the infrared spectrum of 1% C2H3D/trans-C2H2D2; v7 and v12 of C2H3D . . . . . . . . . . . . . . . . . . 86 20 Part of the infrared spectrum of U% CZHBD/1,1-C2H2D2; v1, v“ and v3 of C2H3D o o o o o o o o o o o o o o o o o o 87 21 Part of the Raman spectrum of AZ C2H3D o o o o o o o o o o o o o o o o o o 88 22 Part of the infrared spectrum of 1% C2H3D/02HD3; v7 and v12 of C2H3D o o o o o o o o o o o o o o o o o o 89 23 Part of the infrared spectrum of Of 1,1-C2H2D2 o o o o o o o o o o o o o o 90 2“ Part of the infrared spectrum of C2HD3 . . . . . . . . . . . . . . . . . . 91 xi Figure Page 25 Internal modes of cis-C2H2D2; v3 and v12. The peak marked with an asterisk is attributed to 0-13 sub- stituted molecules. . . . . . . . . . . . 11M 26 Internal modes of cis-C2H2D2; v10 and v9. The peak marked with an asterisk is attributed to 0-13 substituted molecules . . . . . . . . . . 116 27 Internal modes of 1,1-02H2D2; v1 and v2. The peak marked with an 12 2 C = asterisk is attributed to H 13CD 117 2 o o o o o o o o o o o o o o o o 28 Internal modes of 1,1-C2H2D2; VA and v10. The peak marked with an asterisk is attributed to H213C=120D2. . . 29 Mixed crystal exciton band struc- 118 ture. . . . . . . . . . . . . . . . . . . 122 30 Internal modes of 02H3D; v12 and The peak marked with an 12 _ 2 0‘ l3CHD . . . . . . . . . . . . . . . . . . 138 v10. asterisk is attributed to H 31 Internal modes of C2H3D; v3 and Va. 0 o o o o o o o o o o o o o o o o o o 139 32 Internal modes of trans-02H2D2; v1 and v12. . . . . . . . . . . . . . . . 1H0 xii Figure Page 33 Internal modes of trans-C2H2D2; V5 and V“ o o o o o o o o o o o o o o o o lLil 3A v8, trans-C2H2D2, calculated and observed Raman band. The peak marked with an asterisk is attributed to C-l3 substituted molecules. . . . . . . . 1U2 35 v7, trans-C2H2D2, calculated and observed infrared band. . . . . . . . . . 1&3 36 v1 region of C2HD3 in the infrared and Raman spectra . . . . . . . . . . . . 1A” 37 v6 region of C2HD3 in the infrared and Raman spectra . . . . . . . . . . . . 1A5 xiii Ill CHAPTER I INTRODUCTION It has been known for several decades that informa- tion concerning the symmetry and the intermolecular-force field of a molecular crystal can be obtained from a study 1-3 The amount of experi— of its vibrational spectrum. mental and theoretical work inthis area has increased tremendously since the first papers on the subject were published by Hornigu and by Winston and Halford5 in the late l9u0's. This increase was triggered by the avail- ability of laser sources for Raman spectroscopy and by the tremendous increase in the sophistication of far- infrared instrumentation. Much of the initial work in- volved attempts to develop a qualitative understanding of the vibrational spectra. The first attempt at a quantitative interpretation was reported in 1962 when Dows attempted to calculate the vibrational spectrum of the ethylene crystal.6 The early work in vibrational spectroscopy of molecular crystals has been reviewed by Dows7 (internal modes) and Schnepp8 (lattice modes). The more recent work in this area has been discussed by 10 Bailey,9 Pawley and by Schnepp and Jacobi.ll Many researchers have concentrated their efforts on the study of molecular crystals of simple aliphatic hydro— 12 l“ carbons, such as acetylene, ethylene13 and ethane, 15 and on the simple aromatic hydrocarbons, such as benzene 16 in order to develop a more quantita- and naphthalene, tive understanding of the intermolecular forces of the solids. These simple hydrocarbons are particularly attractive for study because they usually crystallize with two or four molecules per unit cell, and only three types of atom—atom interactions need be considered. This greatly simplifies the model calculations which are done to aid in the understanding of the vibrational spectrum of a molecular crystal. Crystalline ethylene has evoked the interest of molecular spectroscopists since the mid—infrared spectrum was reported by Dows in 1962,6’l7 who pointed out that the molecules in the unit cell could be oriented in two different ways in the crystal with the same resultant spectroscopic activity and pattern. Later the far- 18’19 and Raman20 infrared spectra of ethylene were re- ported, but no additional evidence for either crystal structure was indicated. In 1973, Elliott and Leroi reported the Raman spectrum of ethylene and ethylene-d“,l3 and were able to determine which intermolecular force field parameters for hydrocarbon crystals best fit the observed librational frequencies of ethylene. By means of oriented gas model intensity calculations, they de- termined that only one of Dows'two proposed crystal structures properly predicted the observed relative intensities. This determination has recently been 21 22’23 experi- confirmed by X-ray and neutron-diffraction ments. This study of the vibrational spectra of the partially- deuterated ethylene crystals; —d asym-d2, cis-d trans- l’ 2’ d2 and -d3, has been conducted to gain more insight into the intermolecular force field of ethylene and those of molecular crystals in general. Information concerning the retention of the symmetry of the parent (C2Hu) crystal as an "effective" symmetry in the partially-deuterated crystals has also been obtained. This provides some understanding of the effects of perturbations, due to disorder in the crystals, on the intermolecular—force fields of molecular crystals. The retention of symmetry enables interpretation of the spectra of some of the partially-deuterated ethylenes on the basis of pure- crystal theories, while a mixed-crystal treatment is required for the others. Duncan e_t_ a_1_. have recently measured the mid—infrared spectrum of polycrystalline ethylene-l,l—d2 to determine the 0-13 isotopic shifts.2u The C-13 shifts were used in a determination of the intramolecular force field of ethylene. This is the only previous vibrational study of a partially-deuterated ethylene crystal of which this author is aware. The only previous vibrational study on an entire series of isotopic crystals of which this author is aware involved the partially-deuterated benzene crystals; however, orientational and resonance effects precluded a general interpretation of the experimental spectra.25"27 J A 1" CHAPTER II GENERAL THEORY A. The Vibrations of Molecular Crystals The vibrational motions of molecular crystals are usually classified as either internal or external modes. The internal vibrations are essentially those of the free molecule, subject to solid state interactions. The external or lattice modes are entirely solid state vibrations, and are characterized by their low frequency. Lattice vibrations can be further classified into transla- tions, which arise from changes in the position of the centers of gravity of the molecules, or librations, which originate from the torsional motions of the molecules about their centers of gravity. This separation of the internal and external vibra- tions of the crystal can usually be accomplished because the intermolecular forces - the forces between the mole- cules in the crystal - are usually much weaker than the intramolecular forces - the forces between the atoms of a given molecule. This difference in the magnitude of the forces is often used as a criterion for defining a molecular crystal as opposed to an atomic crystal. Coupling between the internal and external vibrations can often be ignored if the lowest internal frequency is much higher than the highest lattice frequency. A molecular crystal has 3LpN degrees of freedom, where L is the number of unit cells, p is the number of molecules in the unit cell, and N is the number of atoms per molecule. Therefore, if the translational symmetry of the crystal lattice is ignored, the vibrational spectra would be expected to consist of 3pN bands each consisting of L closely spaced frequencies. If the translational symmetry of the crystal is accounted for, the k = 0 selection rule is introduced where k is the wave vector of a vibrational excitation traveling through the crystal with wave-like character. This selection rule greatly simplifies the optical spectrum since k must be zero for a molecular crystal vibration to be observed via infrared or Raman spectroscopy; this arises because the inter- action of a vibrational excitation with a photon requires a conservation of momentum. Therefore, the value of the wave vector of the vibrational excitation must be of the same order as that of the probing radiation. The vibrational excitations of a molecular crystal may be classified as either phonons or excitons. The lattice vibrations, which are delocalized excitations in the crystal, are referred to as phonons, and the internal vibrations, which are localized, are usually classified as excitons because they can be conveniently treated theoretically in the same formalism as electronic excitons. Since the phonons are delocalized, they are quite dependent on k whereas the localized excitons are only slightly dependent on k.1 In order that k = 0, the displacements of correspond- ing atoms in different unit cells must be precisely in phase and thus the k = 0 selection rule reduces the number of vibrational degrees of freedom that need be considered to 3pN. Three of these degrees of freedom are the acoustic modes, whose frequencies are zero at k = 0. (The acoustic modes correspond to the pure translation of the entire lattice in space.) The other degrees of freedom are accounted for by the librations (3p), transla- tions (3p-3) and the internal vibrations (3N-6)p, assuming a non-linear molecule. Therefore, if no degeneracies are present, the internal region would consist of 3N—6 vibrations, each showing p components usually referred to as Davydov or factor-group components. The symmetry of the crystal can be used to predict the number and activity of the vibrational modes of a molecular solid. Lattice dynamicists are often con- cerned with several types of symmetry groups in the crystal. These include the space group, the unit cell or factor-group, and the site-group symmetries of the lattice. The space group entirely describes the symmetry relations among the molecules in the lattice, including the translational symmetry. The space—group symmetry minus the translational symmetry is the unit—cell sym- metry which describes the symmetry relations of the molecules in the unit cell. The site group describes the symmetry of the static field in which the molecules 28’29 of the free- in the crystal interact. A correlation molecule symmetry with the site and unit cell symmetries provides information concerning the activity and number of Davydov components of the molecular-crystal modes. B. The Crystal Structure of Ethylene The crystal structure of ethylene has been the subject 31 of some question since an early X-ray investigation which reported the carbon positions and lattice parameters. 22,23 Recent neutron-diffraction studies on powdered samples of 02D“ and a single crystal X-ray21 study of C2Hu have determined solid ethylene to be monoclinic, under normal pressures, with space group P12l/n1 (Cgh) with two mole- cules per unit cell occupying sites of C symmetry. The 1 crystal structure of ethylene is shown in Figure 1. Table 1 lists the lattice parameters and the orientational matrix for an ethylene molecule centered at the origin and initially oriented with the C=C bond along the crystal- ographic x-axis and the hydrogens in the xz-plane. a, b and 0 correspond with Figure l. a,f3and y are the angles 13. Figure 1. The crystal structure of ethylene (from Ref— erence 23). Table 1. Lattice parametersa and orientation matrixb for the ethylene crystal. a = u.6263 b = 5.6203 c = n.067x a = 90° 8 = 9N.39° Y = 90° x y z a 0.8002 -0.5965 0.0665 b -0.5Hh8 -0.6752 0.u972 0 0.2517 0.U3DO 0.8651 aFrom Reference 21. bSee text. 10 between b and c, a and c, and between a and b, respectively. The orientation matrix arises from rotations about the x, y and z axes by angles of -57.3°, 36.6° and u.8°, respec- tively. The bond lengths, bond angles, and the changes caused by deuteration of the free molecule, and of the molecule in the crystal, are given in Table 2. H(1) and H(2) correspond to the hydrogens labeled 1 and 2 in Figure l. The 6's are the reductions in bond lengths caused by deuteration. An orientationally disordered phase (II) of solid ethylene, which occurs at high pressure, has recently been discovered by Ligthart23 using nuclear magnetic resonance spectroscopy. However, the current investi— gation is not concerned with the vibrational spectra of this phase. The correlation table for solid ethylene (phase I) is given in Table 3. Twelve doublets, six of which are Raman active and six of which are infrared active, are predicted for the internal modes. Six Raman-active librations and three infrared-active translations are expected in the lattice region. The vibrational spectra of ethylene and ethylene-d“ have been found to be con- sistent with these predictions, except that only two of the three infrared-active translations have been ob- served.l3’17’19 11 Table 2. Structural parameters for free ethylene molecule and molecule in crystal. Gasa Crystalb r(C=C) 1.338u3 1.31uK r(C-H(l)) 1.08701 0.9923 r(C-H(2)) 1.08703 1.0211 a(H(1)-C—H(2)) 117.37° 116.8° a(H(l)-C=C) 121.32° 121.2° 0(H(2)—c=0) 121.32° 122.0o 6r(C-H) [H-D] 0.00163 6r(C=C) [H-D] 0.000123 per D 00(H-c-H) [H2-D2] 0.0° aGas values and 5's from Reference 30. bFrom Reference 21. 12 smmm NH>.HH> mH 5mm smmm oa>.m> mH new sHmH A? sea 3» .wNmH mp m .m> w \ m <0 / Amoco: Hmsnmuch mnmo so awn dsopo monomm macho spam 9:090 anaeminwasomaoz .mconSpo vfiaom you wanna downwamunoo .m mamas 13 .mao>fipomdmmp .mpfi>fipom consumed cum cwsmm mmeHUCH mH use me smmfl Axev : : Assessoosuav mH gem Hma ANBV w x mmfl A my w «m mmH A me w m cm a ma A my Amoco: asshmpxmv cm H mm m o o 0 Q3090 pouomm dsonu opfim muons pcfiomianSomHoz .vmscfipcoo .m manna .L q1 1A 0. Theoretical Treatments of Molecular-Crystal Vibrations In the next two sections, theoretical descriptions of the vibrations of molecular crystals are discussed. The first section is a semiclassical treatment which follows closely the work of Taddei, 33 al.32 This was used to calculate the external frequencies of the ethylenes as discussed in Chapter IV. The second section will review the vibrational exciton formalism7’15’33’3u for pedagogical reasons. This will aid in the discussion of the experimental data for both pure and dilute-mixed crystals of the partially-deuterated ethylenes. A coherent-potential calculation will also be utilized for the internal modes of some of the partially-deuterated ethylenes. However, this is discussed in Appendix B and in Chapter VI along with the internal mode data. Calculation of the Crystal-Vibrational Frequencies The potential energy, V, of a molecular crystal is described as a sum of the intramolecular potential, V0 and the intermolecular potential, Ve, (1) The infinitesimal linear displacements of the atoms for 15 a crystal large enough to fulfill the cyclic boundary conditions are described in terms of the normal coordin- ates of the molecule and the external displacement co- ordinates. The external displacement coordinates are the Eckart-mass weighted translational and torsional coordinates. As usual in the vibrational problem, the potential energy is expanded in a Taylor series about the equi- librium configuration in terms of the internal and external coordinates, 2 01m V 1: [3NZ-6( 2V / 2 ) = 3 HQ Q 20.“ n 0 mm 0 3N 2 + 320 Slim (0 Ve/aQamBQva)oQamQBVm], (2) where u, v label the molecule in the unit cell, a, B the unit cell and 2, m, n the internal and external coordin- ates. The first (3N-6) coordinates are the internal co- ordinates for a non-linear molecule, where N is the number of atoms in the molecule, and the last 6 coordinates are the external coordinates. The force field of the free molecule may be assumed for V0 if the equilibrium configuration of the molecule is not deformed severely in the crystal. However, V0, in principle, should be the intramolecular potential (h r" h 16 relative to the molecule in the site, and uncoupled to the other molecules in the crystal. It should also be noted that the second derivative of V0 is with respect to the internal normal modes only. The form of V8 will be specified later. The second derivative of the intermolecular potential is with re- spect to both the internal normal coordinates and the external coordinates. The first derivatives of the potential have been assumed to be zero. These terms may be excluded only when the interaction terms between the intramolecular and intermolecular potentials are actually negligible and if Ve has a minimum for the observed crystal structure. The problem may now be simplified by expressing Equation (2) in terms of the symmetry coordinates va(k), which belong to the irreducible representation of the translational group. These symmetry coordinates are given as follows: Q (8) = If“2 2 Q um 8 exp(2nik°§8), (3) Bum where L is the number of unit cells in the crystal, :8 is a vector connecting the 8th cell with an arbitrary reference cell which will be labeled 1, and k is the wave vector. The potential energy in terms of these symmetry l7 coordinates becomes, _ l * V - 5 f E [g A£0u2(k)Qul(k) (A) where, FVm(k) = Z (a2xx/aQ ) Oexp(—2flik'r8) (5) luia Q80 m The frequencies of the crystal normal modes 0(k) can be calculated by solving the secular equation: vm - IF (k) - [1(k) - A 10 0 Gng - 0 (6) __ 22 - 22 v where 1(k) - An v (k) and Ag - Mn vg. The 0, s are the frequencies of the free molecule. The form of the intermolecular potential, Ve’ must be Specified in order to evaluate F:?(k). When only pair- wise interactions are considered, 1 st V = — , (7) e 2 “EB uzvv O. l-1 where V8V is the pairwise interaction between the nth on th th molecule in the a unit cell and the 0 molecule in 18 the 3th unit cell. Therefore, in terms of Equation (7), vm _ . 2 By B¢1 if u=v (8) 2 Br + Guv g g (3 le/anulanum)O' Bfil if u=T The molecule-molecule interactions are usually expressed as a sum of atom-atom interactions, 1 BVJ BVJ V =- Z X Z V’ (r ), (9) e 2 dB uv iJ aui aul where i labels the atoms belonging to molecule an: and J the atoms belonging to molecule 80. r231 is the inter- atomic distance between atoms 1 and J. In terms of the atom-atom potentials the second derivatives in Equation (8) are given by, 2 8v 2 3 V111 = X 8V11 3r1 . 8r11 anuzaQva 0 iJ 8r132 0 anul 3ri (10) p- - 33L..ifii+iy_i_i 1.3111 3111.311; aQva BPJ arid 0b anug 3P1 aQva BrJ 31" .l 19 M have been replaced aui where for simplicity, V531 and r by V1.j and r1J,respectively. Taddei gt al.32’35’36 have pointed out that traditionally the second term in the above equation has been ignored. They have calculated the lattice frequencies of benzene with and without this term. The inclusion of this term caused a 2% to 15% reduction in the calculated lattice frequencies, showing the effect of this term is not negligible. It has there- fore been included in the calculation of the ethylene lat- tice frequencies. The solution of the above secular equation (Equation (6)) will give the frequencies of the crystal normal modes over the entire Brillouin region. However, k se- lection rules make it possible to observe only the op- tically-active vibrations, that is vibrations with k = 0, via infrared spectroscopy and Raman scattering. It should be noted that k arises from the translational sym- metry of the crystal; therefore if the translational sym- metry is destroyed it may no longer be a "good" quantum number. The intermolecular potential is usually specified as the sum of molecule-molecule pairwise interactions which are expressed as a sum of atom-atom interactions. The atom-atom interaction potential for hydrocarbons is ex- pressed as, 2O -6 ij (11) V = A exp(-Br 13 ) - Cr iJ where riJ is the interatomic distance between atom i of molecule a and atom J of a molecule other than a in the crystal. A, B and C are the potential constants which are obtained by refinement to other crystal properties such as heats of sublimation and crystal lattice energies. The actual calculation of the crystal frequencies of the ethylenes were chum; using program TBON, a com- puter program written by Taddei and Bonadeo. The Exciton Formalism As stated above, this discussion of the exciton for— malism is included to aid in a qualitative discussion of the splittings and shifts of modes in the solid state in comparison to those of the free molecule. This will also assist in the explanation of the orientational effect in the next section. In the tight-binding limit, that is where the excited states are expressed in terms of localized excitations, the Hamiltonian for the excited states of a perfect crystal may be written as a sum of a one-site Hamiltonian, H0, and an intersite Hamiltonian, H1, 1 H=HO+H (12) 21 where, N 0 0 0 _ H — nil 0E1 Hna (13) and H1 = Z Vna n'a' (1“) nafin'a' ’ In Equations (13) and (in) n specifies the unit cell and a the site in the unit cell which is assumed to be interchange equivalent. Interchange equivalence indi- cates the existence of symmetry Operations in the unit cell group not included in the site group which inter- change the molecules in the unit cell. The Vna,n'a' terms represent the pairwise interactions of the molecules in the crystal. H0 may be considered to be the free-molecule Hamil- nu tonian, the Hamiltonian of the distorted molecule in some averaged field of all the other molecules, or the Hamil- tonian of the molecule whose nuclear framework has been distorted to match that of the molecule in the crystal. To facilitate the discussion of the orientational effect in the next section, Hg“ is choosen to be the Hamiltonian of the free molecule. The eigenfunctions of the Hamil- tonian are designated as 13a where f designates the excited vibrational state of the crystal. 22 The zeroth-order crystal states, ¢§a, represent a localized excitation on the molecule at site he, .. (15) It is assumed that the various f states do not mix or are premixed and that the localized excitation functions are orthogonal, f f' <¢na|¢nvav> = OnnvOaaiOff!‘ (16) The translational symmetry of the crystal is accounted for by forming extended exciton states, L r g —1/2 , ¢a(k) L nil exp(ik Bna)¢ f no (17) th where Bna indicates the position of the a molecule in the nth unit cell, = + T . (18) Here, rn is the vector from an arbitrary crystal origin th to the origin of the n unit cell and 10 is the vector from the origin of the unit cell to the molecule in the ath site. The k-dependent general matrix element is, 23 f f f f <¢a(k)lH|¢a.(k)> = (ef+D >aaa. + Laa.(k> (19) where f f O f _ f O f E = <¢nai nal¢na> - (WnaIHnalwna> (20) is the k-independent free molecule excitation energy, and f 0 f f l f _ I I ‘ 2 1i’nm‘i’n'm'IVnohn'ol'lwnozwn' a' D = <¢na H ¢na> na#n'a' (21) is the k independent contribution to the environmental shift. The k—dependent term has both a diagonal part, f L k = ex ik R -R (~) nafg'ai pt ~(~na ”n' a)]<¢nalvna,n'dl¢n'd> (22) which represents the interactions of the translationally- equivalent molecules, and off—diagonal parts, Laa'(k)= na,g'a exp[1k(Rn no- Rn' 'a')] (23) f <¢nal na,n'a'(l-éaa')l¢n' 'a'> which represent the interactions of the translationally- inequivalent molecules in the crystal. The energies for a particular value of k are found 2A by solving a c x 0 secular equation. For the ethylene crys- tal, where o = 2, the crystal energies are given by,34 Ef(k)i = cf + Df + %[L; (k) + Lg .(k)] (28) “(i—mtg (k)-L: .0012 Wlf‘akdk},)|21/2 where Ef(0)+ and Ef(0)- are the energies of the two Davy- dov components for the internal vibration f. For k = 0, = Lf Laa(K) L: a'(k): Lac’ Therefore, f: f f f f = + D + E E Lac '1: Laa'o (25) The terms of the right side of Equation (25) may be equated to the different contributions to the band struc- ture of ethylene. Df is the static field term and in- volves no resonance interaction among the molecules in the crystal. Lia is the resonance contribution caused by interaction of translationally-equivalent molecules and Lia, is the resonance contribution caused by inter- action of translationally-inequivalent molecules in the crystal. The static and resonance terms can be separated experimentally by observing the spectra of isotopic dilute mixed crystals. The complete separation of these terms is referred to as the ideal-mixed crystal; that is, there (I! J n) w 25 exist no resonance interactions between the molecules in the crystal. The ideal-mixed crystal is most nearly achieved if the following requirements are fulfilled: a) The guest is infinitely dilute; b) the guest and host differ only by isotopic sub- stitution; c) resonance interaction between the guest and host is negligible; d) the guest and host have the same dimensions and symmetry; e) the effects of isotopic substitution on the gas to ideal mixed crystal shift can be neglected. Items (c) and (e) can be more easily understood by referring to Figure 2, where the structure of an arbi- trary exciton band of ethylene is shown. A is the gas- to-ideal mixed crystal shift, + B. (26) B is referred to as the quasiresonance shift and ac- counts for any resonance interactions between the guest and host in the dilute-mixed crystal. The quasireson- ance interaction may be assumed to be negligible if the host and guest bands are well separated. C is the resonance shifting caused by interactions of 26 \\D/ /|\ .ocmamnpm mo pawn coufioxo upwnufinpm no mo chaposhpm one $328 % IH I. II oux mo com: o prmmpo omxaelawmoH Hopmmpo ooxfie oaaopomfi mpzafin .m mpswfim mow 27 translationally-equivalent molecules, and equals Lf . ad For ethylene, this term may be experimentally deter- mined by finding the difference between the mixed—crystal frequency and the mean of the g = O doublet assuming quasiresonance effects to be negligible. C is usually small for hydrocarbon-molecular crystals but it is not zero. T is one-half the exciton band width and is the same as Lia" It reflects the resonance contributions directly related to interactions of the translationally- inequivalent molecules in the crystal. Another type of splitting which may be observed in the spectra of molecular crystals is site-group, or static-field splitting. This is caused by the destruc- tion of molecular degeneracies when the gas phase mole- cular symmetry is lowered in the crystal. Because this is a static effect it must be accounted for in the Df term, and if this effect were included in Figure 2 there would be two A's, one for each of the site-group com- ponents. Ethylene has no degenerate vibrations; there- fore, site-group splitting need not be considered. How- ever, the orientational effect observed for some of the partially-deuterated ethylenes is very similar to site-group splitting. This will be discussed in detail in the next section. It should be noted that bands involving both site and factor-group effects cannot usually be treated as 28 a simple superposition of the two independently because once the degeneracy has been removed additional coupling terms not present in the degenerate case must now be considered. This will become quite obvious in the dis- cussion of the spectra of the partially-deuterated ethyl- enes. There the neat crystal spectra cannot be explained on the basis of a simple superposition of the orientational components and the Davydov components. D. The Orientational Effect-Crystals of the Partially- Deuterated Ethylenes Partial deuteration of ethylene destroys the D2h symmetry of the free molecule and this loss of symmetry has a marked effect on the symmetry of the molecular crystal. The molecular structures and point-group sym- metries of the partially-deuterated ethylenes are shown in Figure 3. The only partially-deuterated ethylene which retains a center of inversion is trans-ethylene-d2; therefore, mutual exclusion of the infrared and Raman bands can no longer be expected for these compounds except for trans-CaHeDz. Furthermore, mutual exclusion for the trans-ethylene-d2 crystal can only be expected if allowed by the site symmetry of the crystal. The altered symmetries of the free molecules intro- duce additional features to the spectra of dilute isotopic D\C H/ Figure 3. tr: / N 29 2h 2h 2v D\C ,_. C/D D/ \D D\C = C/H D/ \H ’> - .2 D \D The molecular structure and point-group sym- metries of the ethylenes. 30 mixed crystals of these compounds in comparison to mixed Cth/CZDH crystals. This arises because the guest may sit on a lattice site in different ways with possibly different intermolecular interactions. This is illustrat- ed in Figure H for a dilute—mixed crystal of C2H3D in 02H“ or C2HD in C2D“. If the nearest neighbor interactions 3 are examined for the C site of the host, it is seen that i the interactions in Figures ha and “b are identical as are those in Figures he and Nd. Therefore, although there exist four translationally-inequivalent orientations in the crystal, for a C1 site there are only two ener- getically—inequivalent orientations. Thus doublets are observed for the ethylene-d1 bands of a dilute-mixed crystal of 02H3D in 02H“ rather than singlets as found for example for the guest bands of a dilute mixed crystal 13’17’2u This added complexity is re— 26 1 of 02D" in 02H“. ferred to as the orientational effect. Bernstein reported orientational "splittings" of up to 5 cm- for dilute mixed crystals of partially-deuterated ben- zenes in C6H6 and C6D6 hosts. A group-theoretical procedure as described by Kopel- man37 may be used to determine the number of transla- tionally-inequivalent orientations and the number of energetically-inequivalent orientations which exist for a given isotope. The number of energetically-inequivalent orientations which exist for the five partially-deuterat- ed ethylenes in a C1 site are as follows: 02H3D(2), 31 (b) (a) The orientations of CZHBD in 02H“ (02HD3 in Figure U. 32 asym-02H2D2(l), cis- C H2D2 (l), trans— C MH 2(2), CZHD3(2). 2 It should be noted that the orientational effect is a static effect and is a function of both the site sym- metry of the host and the molecular symmetry of the guest. If two orientations are assumed (orientation A and orientation B), tWO Df terms must be considered in the exciton formalism, f Af l Af DA = <¢na|H l¢na > (27) and where the orientation of the molecule is indicated by the superscripts A and B. The orientational splitting is i and Dg. As mentioned previously, partial deuteration of ethyl- equivalent to the difference between D ene destroys the D2h symmetry of the free molecule and this loss of symmetry has a marked effect on the symmetry of the molecular crystal since the partially-deuterated molecules assume translationally-distinct orientations in the crystal. This results in a destruction of the translational as well as the C1 site symmetry of the crystal lattice. These crystals are disordered; however, it should be noted that if the substituents or the C=C 33 frame are ignored, the crystal structure is the same as that of C2Hu, assuming changes in the lattice parameters are negligible upon deuteration. Strictly, the theory discussed in the previous sec- tions does not apply to the partially-deuterated ethylene crystalsbecausethey lack translational symmetry. Never- theless, qualitative applicability remains. The useful- ness of the pure-crystal theory for the lattice and internal modes of the partially-deuterated ethylenes is discussed in detail in Chapters IV, V and VI. CHAPTER III EXPERIMENTAL The deuterated ethylenes were obtained from Merck, Sharp and Dohme of Canada with stated purities as shown in Table U. Research grade ethylene (C2Hu) was obtained from Mathesonznuihad a stated minimum purity of 99.98%. The mass spectrum of each compound was obtained using electron beam energies of 15 eV, 20 eV and 70 eV, but no additional information on impurities was determined. The samples were cooled using either an Air Products model CS-202 or GSA-202 cryogenic helium refrigerator system. Temperatures were measured with a gold (0.07% iron) versus chromel thermocouple attached to the cold block of the cooler, or with a hydrogen-vapor bulb thermometer for cooler model GSA-202. The temperature of the samples was controlled with either a Cryodial model ML 1&00 temperature controller or an Air Products temperature controller with calibrated platinum resistors for temperature sensing. The temperature of the samples was 15-25°K when the spectra were recorded. The higher temperature limit usually applied for the Raman experi- ments. The higher temperature is believed to be caused by the size of the Raman cell and the fact that the cell and shroud were not nickel plated. The lack of nickel 3U 35 Table A. Partially-deuterated ethylenes-Analytical data. C2H3D cis-C2H2D2 C HD G.C. Mass G.C. Mass G.C. Mass G.C. Mass G.C. Mass > 99% Anal: C2Hu < 1.57% C2H3D m 98.u3% CO2 < 0.2% > 99% Anal: C2H3D < 3.9% 02H2D2 m 98.83% Atom% D = 98.05% > 99% Anal: C2H3D < 1.17% 02H2D2 W 98.83% Atom % D = 99.ul% > 99% Anal: C2Hu < 0.72% C2H3D N “.065 C H D W 95.22% 2 2 2 > 99% Anal: 99.55 Atom % D 36 plating makes the cold block more susceptible to heating effects caused by surrounding radiation. The gas-phase samples were sprayed on to the cold— cell substrate for both the infrared and Raman experi- ments. The vapor-deposition technique was used because it allowed samples of different thicknesses to be studied in the infrared most conveniently. The spray-on tech- nique was used in the Raman to enable a truer comparison of the Raman spectra with the infrared data. The dilute mixtures of the partially-deuterated ethylenes in ethylene were made by mixing premeasured amounts of gas-phase samples of the guest and host. The amounts of gas-phase samples were measured using bulbs of different predetermined volumes and measuring the pressures of the gases with a mercury manometer. Very small bulbs were used for the partially-deuterated ethyl- enes, which were 1% of the mixture, to insure that the pressure of the gas exceeded the uncertainty of the manometrical reading. After the mixtures were condensed in a bulb, they were repeatedly frozen and warmed to room temperature to insure adequate mixing. The amount of mixture made was less than the total amounts of the pure samples available for the pure crystal experiment; therefore the back-pressure during the spray-on deposition was less. This was compensated for by using a needle valve to control the flow of the 37 samples. The needle valve was Opened further for the mixtures to insure a similar-flow rate. The Raman sample cell which was used has been des- cribed previously by Elliott and Leroi.l3 It is a closed cell which enabled evacuation independent to that of the shroud. The vacuum in the sample cell was at least as good as l x 10"5 Torr before the cell was cooled and the sample was deposited. The Raman samples were deposited at low temperature (~20°K) and then annealed at approxi- mately 80°K for an hour or more prior to recooling for spectroscopic observation. An Air Products cryogenic infrared cell was used for the infrared experiment. Cesium iodide windows served as the substrate and outer windows on the infrared vacuum shroud. Good thermal contact was achieved between the window and cell with indium wire. Indium sheets were used to insure good thermal contact between the infrared and Raman cells and the cold block of the cooler. For the infrared experiments the sample cell was not inde- pendent of the crystal and thus the vacuum in the shroud was critical. Therefore, a vacuum at least as good as l x 10-5 Torr was achieved before cooling the cell and the shroud was continuously pumped on throughout the experiment. The cesium iodide substrate was maintained at approximately 65°K during deposition of the infrared samples, and then cooled to low temperature. No 38 significant difference in sample quality was detected if the sample remained at 65°K for some time before cool- ing. Additional sample was then added when desired at low temperature, at a rate Just fast enough to prevent high scattering. The deposition rate was monitored manometrically and was usually about 2 mm Hg/min. Ap- proximately 0.75 liters of the pure samples at STP were sprayed-on to the CsI window which was round with a diameter of approximately 2 mm. The total volume of the mixtures deposited was approximately .25 liters. A Perkin-Elmer model 225 infrared grating spectro— photometer (H000 cm"1 - 200 cm'l) with 1 cm'1 resolution 1 and l - 2 cm'1 resolution below #50 cm"1 above “50 cm' was used to measure the mid-infrared spectra. The Raman spectrophotometer consisted ofaiJarrell-Ash model 25-100 double Czerny-Turner monochromator with a thermoelectric- ally cooled RCA 03103h photomultiplier tube. Both the Slusfi and #880 3 lines of a Spectra Physic model 16“ argon ion laser were used as exciting lines for all Raman experiments in order to distinguish laser-fluorescence lines. Baird-Atomic spike filters were also employed to help eliminate laser-plasma lines. In some cases the lattice spectra were also observed using the R765 3 line of the argon-ion laser as the exciting line to eliminate interference from laser-fluorescence lines. 1 The resolution in the Raman spectra was 1 cm’ or better 39 for the fundamentals and lattice modes,and the spectrom- eter calibration was checked using a neon lamp. Most of the reported infrared frequencies are believed to have an accuracy of :1 cm'l. However, for very broad bands in both the infrared and Raman the accuracy may 1. The accuracy of the Raman fundamentals 1 only be in cm- is believed to be :1 cm- as is that of the lattice modes except in cases where the bands are broad or poorly re- solved. In these cases the accuracy is believed to be :2 cm-l. The Raman overtones and combinations reported in Appendix A are believed to be accurate within 13 cm'l. An attempt was made to measure the far-infrared spec- trum of the ethylenes using a Digilab model FTS-l6 fourier transform far-infrared spectrometer. Crystalline quartz was used as the substrate and both polyethylene and Mylar (thickness 200 um) were used as outer windows for the infrared vacuum shroud. Instrumental limitations caused by the cryogenic cell, detector sensitivity and computer software and hardware capacities limited the success of these experi- ments in that only very noisy spectra with low signal- 1 to-noise ratio were obtained. However, the 73 cm" and 110 l 19 cm- translations of ethylene were observed, and the observed spectra suggested the existence of another mode at approximately 35 cm'l. This most certainly could be the as yet unobserved translation of the ethylene crystal 40 (see Chapter IV). Attempts to confirm this observation using a different beam splitter and different outer windows were unsuccessful. Due to the instrumental limitations, an attempt to observe the translations of all the ethylene crystals was abandoned. CHAPTER IV THE LATTICE VIBRATIONS OF THE PARTIALLY-DEUTERATED ETHYLENES A. Phonons in Disordered Solids As was discussed in Chapter II, the crystals of the partially-deuterated ethylenes are translationally dis— ordered, and therefore lack both translational and site symmetry. Since the crystals are no longer periodic, one might expect the k selection rules to be invalid, and the mutual exclusion of the lattice phonon modes to be destroyed due to the lack of C1 site symmetry. The study of phonons in disordered solids is cer- tainly not a new interest of physicists and chemists. The first papers on the subject were published by Lif- shitz in 19143.38 However, most of the past work has concentrated on atomic crystals, where only translational lattice vibrations occur. Molecules are the building blocks of a molecular crystal and therefore introduce rotational degrees of freedom which do not have to be considered in the atomic case. Much of the work on atomic crystals has been reviewed by Elliot, Krumhansl and Leath39 and by Baker and Sievers.”O Recently, phonons of heavily-doped isotopic mixed Al A2 molecular crystals have been investigated. It is be- lieved that the study of these simpler disordered systems will aid in the understanding of more complicated systems. The mixed crystals which have been studied include “1 durene-hlu: durene-dluu2 naphthalene-h8: naphthalene-d8, and benzene-h6: benzene—d6.”3 It has been observed that the phonon states of isotopic-mixed crystals are amalga- mated;uu that is, if x phonon bands are observed for the pure crystal then x bands will be observed for the crystal doped with its isotopic constituent of the same molecular symmetry. If the perturbation caused by the guest molecules is small (vide infra), the so-called virtual crystal limit39’u5 applies, and the mixed crystal is a substitutional dis- ordered system. In a substitutionally-disordered system the intermolecular force constant of the mixed crystals are the same as those of the pure crystal. In the virtual crystal limit the phonon energies of the mixed isotopic crystals can be expressed as fol- lows,u3’u6’u7 _ f 1/2 a - eA/(l-CBA ) , (29) where EA is the frequency of the pure host and CB is the concentration of the guest. Af is the perturbation strength of the band and is defined as (mi - ng)/n£, “3 where f designates the degree of freedom (that is, Rx’ Ry, Rz, Tx’ Ty, TZ) and n is the mass for translations or the moment of inertia for librations. If Af is very small, Equation (1) can be rewritten as f e = €A(l + % CBA ), (30) which predicts a linear dependence of the frequencies on the guest concentration. In a substitutionally-disordered solid k is a good quantum number, the phonons are delocalized, and the normal coordinates described in the mass - or moment-of- inertia-weighted coordinate system remain unchanged in going from the pure host crystal to the pure guest crystal. This leads to no change in the spectroscopic activity of the modes; thus if the pure crystal has centrosymmetric site symmetry, mutual exclusion would be retained for the isotopically-mixed crystals. No significant changes in the phonon band widths are predicted in the virtual crystal limit; however, the relative intensities of the phonon modes may be altered relative to the pure crystal if Af is different for each of the different degrees of freedom. For example, if the crystal has inversion sym- metry, the translations are usually infrared active and the librations are usually Raman active. For the mixed crystal the relative intensities of the translations uu would remain unchanged because Af is the same for each translation. The total Raman intensity would remain unchanged, but the relative intensities of the librations are expected to be altered, since Af is usually different for each rotation except for spherical and planar sym- metric tops. The crystals of the partially-deuterated ethylenes are comparable to isotopic-mixed crystals. The nearest neigh- bor atoms in both cases can be either hydrogens or deut- eriums, so the intermolecular interactions should be quite similar. The changes in energy of the phonons of the isotopic-mixed crystals are mass-dependent, so an analogy can be drawn between the percent deuteration of a partially- deuterated ethylene and the concentration of the guest in a C2Du:C2Hu mixed crystal. For example, the "effective mass" of the ethylene—dl crystal would be the same as that of a mixed crystal of 25% C2Du/C2Hu. Since in the virtual crystal limit the force constants and "effective" symmetry, that is, not the strict mathe- matical symmetry but the spectroscopically observed sym- metry, of the crystal remain unchanged, the phonon fre- quencies of the partially-deuterated crystals in this limit can be calculated assuming they are pure crystals. Therefore the semi-classical method of calculating the phonon frequencies (TBON) discussed in Chapter II would still apply. The crystal symmetry and potential function “5 remain the same for each ethylene; only the masses of the molecules are different. It is also assumed that the contributions to the normal coordinates from each degree of freedom; Rx’ Ry and R2 do not change much upon deutera- tion. This can be easily checked because program TBON also determines the eigenvectors. B. The Potential Function of Ethylene Elliott and Leroil3 tested several intermolecular potential functions for ethylene in order to determine which potential gave the best fit to the observed lattice frequencies. Since then, slightly better lattice param- eters have been obtained by neutron diffraction22’23 21 and single crystal x-ray experiments, and some new potential functions, which were not previously tested on ethylene, have appeared in the literature.32’u8'52 The best potential functions of Elliott and Leroi, as well as several new potential functions, have been used in the calculation of the lattice frequencies of ethylene. Table 5 lists the parameters of the potentials tested. Potential functions I, III and V were obtained by Williamsl'lg’50 by a least squares fit of observed and calculated crystal data of aromatic, both aromatic and aliphatic, and aliphatic hydrocarbons, respectively. These correspond to the potential functions labeled VI, 146 am 0 mm.m coo: wmm o «a... oooma a? o 3.: 88.. S SE». m: mmo.o ow.m h.~w=dh m.mza ~m0.0I Fm.m m.mHmzH n.2mn mmc.o zs.m n.mom~ «.02 HH> mm o ow.m >.d0bah m.HHm o Nw.m o.momo >.HHH o z~.m N.ONH~ =.=~ H> cm 0 om.m o.oomaw o.mom o he.m o.oooaH o.m~n o =5.m o.omm~ m.~m > mm o 936 06mm: 36mm 0 uooé odooo 3.2..." o unfim m.ow- mém PH cm 0 ow.m 0.0mmma c.00m o pc.m c.0hhm o.m~H o =>.m o.omw~ m.p~ HHH mm o Haw.m m.a~mmm NH.man o moo.m m.~h>m mo.omH o hmh.m m.u~:m a~.aa HH an 0 Sim c.0921. o.mmm 0 56 0.3.3 gun." a i..m 0609. 06m H m o m < m o m < u o m < do: one on: a-.. son .dnuu + unu upon: .uhouofiduda Huavcouoa couconcoz .m «Home A7 V and VII by Elliott and Leroi. Potential functions II and IV have been obtained by Taddei, g}; g_1_.32 by a least squares refinement of potential functions I and III to obtain the best agreement between the calculated and observed lattice frequencies of benzene. VI and VII have been obtained by least squares refinement between the observed and calculated crystal structures of both 52,A8 saturated and aromatic hydrocarbons, where VII has an added electrostatic contribution. Potential function VIII was obtained by Kitaigorodsky51 by fitting the crystal structures of various aromatic crystals. Each interaction potential was used to calculate the lattice frequencies of ethylene and the results compared to the observed values. The calculated and observed fre- quencies are tabulated in Table 6. The best agreements with the observed librational frequencies are obtained with potential functions I, III and V. The agreement for the translations is best with potential VII; however, an imaginary librational frequency is calculated with this potential, so it must be eliminated. The next best agree- ments with the observed translational frequencies are obtained with potentials I, III and V. To further examine which is the best interaction potential, the three best Potentials for C2HA were used to calculate the lattice frequencies of C2D”. The calculated and observed fre- quencies of C2D” are tabulated in Table 7. Again, the A8 Table 6. The observed and calculated lattice frequencies (cm’l) of solid ethylene. Potential Set Observed Librationsa I II III IV V VI VII VIII 73 68 5A 60 55 6A 57 78 no 90 83 68 73 68 75 69 85 55 97 9A 75 79 72 82 7A 9A 6A 11A 107 86 93 86 95 87 Imag. 68 167 163 127 132 120 133 120 102 109 177 175 139 1A2 l29 1A3 129 129 128 Translationsb 35° us as A6 uu us an 61 u3 73 61 52 55 52 56 52 67 53 110 86 73 79 73 82 75 93 68 aFrom Reference 13. bFrom Reference 19. CSee Chapter III and text. “9 Table 7. The observed and calculated lattice frequencies (cm'l) of solid ethylene-d“. Observed Potential Set Librationsa I III V 60 57 A9 53 75 71 63 6A 78 77 65 68 95 92 80 81 123 115 93 95 135 12a 101 102 Translationsb --- A5 A3 A5 69.5 57 52 53 10A 80 7h 77 aFrom Reference 13. bFrom Reference 19. 50 best agreement between the observed and calculated fre- quencies is obtained with potential I. Therefore poten- tial I, which was obtained by a least squares refinement to the crystal data of aromatic hydrocarbons, and is also the potential adopted by Elliott and Leroi, is the best potential function of the set for the calculation of the ethylene crystal lattice frequencies. With this potential the lowest translational fre- quency is calculated to be A8 cm'l. As mentioned in Chapter III, it is believed that a mode at 35 cm”1 was observed during the attempt to obtain the far-infrared spectrum of C2HA° This calculation shows that the third translation should be lower in frequency than the other 1 observa- two; supporting the possibility that the 35 cm- tion may be the "unobserved" translation of ethylene. However, it is of interest to point out that the other observed lattice frequencies of ethylene are higher than the calculated frequencies. The reverse is true 1 mode. for the 35 cm— The exciton splitting (k = 0 components) of the ethylene internal modes were also calculated using program TBON. However, all calculated-splittings were less than 1 cm-1 , therefore little meaning could be assigned to them. This certainly indicates that potential I is not the "correct" interaction potential of ethylene in that it only does a "good" Job of calculating the librational 51 frequencies, a poor Job of calculating the translations, and does not calculate the exciton splittings. C. The Librations of the Partially-Deuterated Ethylenes The Raman spectra in the lattice region of the partially- deuterated ethylenes are shown in Figures 5 and 6. At least five bands were observed for each molecule and a shoulder was resolved on the high-frequency side of the highest frequency bands of C2H3D,l,l—C2H2D and C2HD3 2 (see Figure 7). It is believed that a sixth band is present also for cis-CZH2D2 and trans-C2H2D2, which was unresolvable because of poorer sample quality. The rela- tive intensities of the bands are similar to those of ethylene and ethylene-d“,13 although minor differences do exist. Only two of the three expected infrared-active trans- lations of ethylene have been observed.19 The energies of these phonons are 73 cm'1 and 110 cm'1 for 02H“ and 69.5 cm-1 and 10A cm“1 for C2D“. The third optically- active translational mode has been calculated to be at lower frequency (see Table 6 and preceding section). Therefore, if any translations are to be observed in the Raman phonon spectra of the partially-deuterated ethylenes where they are not rigorously excluded, they are expected to fall between 110 cm'1 and 10A cm'1 or below 73 cm'l. 52 1,1-c H D , 2 2 2 1) (cm— Figure 5. The Raman lattice region of C2H3D, 1,1-C2H2D2 and C 2HD3. The peaks marked with an asterisk are laser—fluorescence lines. 53 cis-C2H2D2 * trans-C H D 2 2 2 L5 10 I5 L_U1 U1 (cm-1) Figure 6. The Raman lattice region of cis-C2H2D2, 1,1-C2H2D2 and trans-C2H2D2. The peak marked with an asterisk is a laser-fluorescence line. 5A .m a a? Q m m m OIH.H mo mmooe mofippma mocosuopm pmmnwfin one Eov AT 00H .5 opswam 55 Unfortunately these regions overlap the lattice libra— tional frequency range, so the observed bands cannot be assigned as librations or translations simply on the basis of their positions. The frequencies of the lattice modes of the partially- deuterated ethylenes were calculated using interaction potential function I. It was assumed that deuteration causes only a change in the mass of the molecules, and that the symmetry of the lattice remains unchanged. The calculated and observed librational frequencies are given in Table 8. The agreement between the calculated and observed values is quite satisfactory - as good as that for ethylene and ethylene-d”. The slight differences in the calculated C2H2D2 frequencies is caused by the place- ment of the deuteriums on the C=C frame. The bond-length changes caused by deuteration as listed in Table 2 were accounted for in the calculation; so depending on whether the nearest neighbor was a hydrogen or deuterium, the interatomic distance varied. This effect resulted in a varying of the calculated frequencies by less than a wave- number. Note that the frequencies were calculated assuming centrosymmetric site symmetry for each ethylene crystal; therefore mixing of the translational and rotational degrees of freedom was strictly forbidden. The crystal normal coordinates obtained from TBON for the librations 556 . m H QOBOLOQOK "=0de mm“ am“ ma" mma Amfl maH :7: a." --u mad sea Am” FAA mefl mm” mHH ama mm” mam mmfi man mm" maa mMH mmH mafi ema moa mm Na moH mm so” am moa mm eon mm «HA men aHH boa ms 55 am am 0a 40 mm mm om mm mm mm ha :0 ms H» as a» mm wk mm vs mo 05 pm as om mm am pm am an be He be ww no No me me ma we a..ao .ofluo .aao .oHau .ono .oHao .uno .oaao .nno .OHao .ebo .bHao .ebo .baeu ammo ozwo mnwmwuufl.~ mammwuuecubu ~o~z~0-oao ommmo ammo .nocoaanuo on» no nouocosaouu acceduounun ou>hmnno ecu otueflaeflao ene . m manna. 57 of the crystalline ethylenes are given in Table 9. None of the librational modes can be described as a pure rotation about a single principal axis, but the relative contributions of each rotation to each normal coordinate remain essentially unchanged in going from C2Hu to C2D“. The calculated and observed librational frequencies are plotted against the number of deuterium atoms on the molecule in Figures 8 and 9, respectively. A virtually linear dependence between the frequencies and number of deuteriums is predicted for the four lowest frequency vibrations in Figure 8; therefore the frequency depen- dence on the percentage deuteration should be described by Equation (30) for these bands. A deviation from lin— earity is predicted for the two highest frequency bands; thus Equation (29) should describe the curves for these. The perturbation strengths, Af's, of the calculated frequencies are given in Table 10 along with the experi- mentally observed Af's and those calculated from the moments-of—inertia of ethylene and ethylene-d“. The Af's of the calculated frequencies were determined by calculating the perturbation strength for each point with f Equation (29) and then finding the average A for the band. The experimental perturbation strengths were found by determining the slopes of the lines shown in Figure 9 for the five lowest frequency bands and using Equation f f (30) to determine A . A of the highest frequency band 58 Table 9. The crystal normal coordinates of the libra- tions of the isotopic ethylene crystals. Crystal Species “calc. Rx Ry c H A 83 -0.06 —0.22 2 u 2 9A 0.17 0.96 175 0.98 -0.18 B 68 —0.05 -0.96 . g 107 0.00 -o.27 -0. 163 -1.00 0.05 -0. C H D A 79 —0.06 -0.2A . 2 3 g 89 -0.19 -0.95 -0. 157 0.98 -0.20 B 65 0.06 0.96 -0 g 102 0.00 0.26 1A5 -1.00 0.06 -0 cis—C H D A 6 0.08 0.27 -0. 2 2 2 8 85 -0.20 -0.9A -0. 193 -0.98 0.21 -0. B 62 0.06 0.96 -0. g 98 0.00 0.26 133 1.00 -0.06 trans-C H 0 A 76 —0.07 -o.26 2 2 2 g 85 -0.20 -0.9A -0 Inn 0.98 -0.21 B 62 0.06 0.96 -0. 2 98 0.00 0.26 133 1.00 —0.06 1,1-c H D A 76 0.08 0.29 -0. 2 2 2 8 8A —0.19 -0.93 -0. 1A3 0.98 -0.21 B 61 0.06 0.97 -0. 8 98 0.01 -0.25 -0. 133 1.00 -0.06 59 Table 9. Continued. Crystal Species vcalc. R R R C2HD3 ‘ A 7A 0.09 0.31 -O.95 g 81 0.21 0.92 0.32 133 0.97 -0.23 0.02 B 59 -0.07 -0.97 0.25 2 95 0.00 -0.25 -0.97 123 -l.00 0.06 -0.02 C20“ A 71 -0.10 -0.33 0.9a 2 77 0.22 0.91 0.3a 12A 0.97 -o.2A 0.02 B 57 -0.07 -0.97 0.25 g 92 -0.01 0.25 0.97 115 1.00 -0.06 0.02 60 U one :9 .Ha mammo\ommmo «a mo Sappooam nonopMCH mom mHOH .ommmo one no ones .OH spawns ooom omom P _ sn-e63 77 mmmo RH mo Esnpooam Umsmpmcfi on» mo whom .HH opsmfim NH; one a; mammo\o NH> w? 78 oNQNmNO'mfiO Q0 OH? one >> .H> mammo\mmmmm01mfio RH mo ESLuOon oohmpwcfi esp mo ppmm .mH opswfim e9 OH 79 ma> one :9 mammo\mommm01mcmpp RH no Esppooqm ooumnmcfi on» no poem NH» mmm r .mommmouncono no 2> .mH madman 80 MO m .mommmouncvo > can H9 mammo\mammmoumcmpp RH mo Ednpooam cmEmm on» no ppmm .za opsmfim AHIEoV AHIEOV T J . J mema mmmfi mmmm mumm 81 .H .m m m o NH» m ouH.H e a mammo\mmmmmo-H.H RH so ssnoooan sameness one do pass .mH spawns H? N? NH) oema mama oi. oi. ommm oaom rllllL rlllllL F r. AHIEoV AHIEov AHIEoV 82 NH .m 9 one :9 NH» mmmH mmmH m .mom 0 no 9 mammo\mommo RH mo Esapooam oopmpmcH on» no ppmm .mH opstm me so mmw owe QHmm ommm _ 7. rIIIIlIIL HHIEoV AHIEoV 83 1500 cm'1 was considered the best place to observe the guest modes of the dilute-mixed crystal for this reason. The lack of overtones and combinations in this region facilitated the assignment of the guest modes. The guest bands are also less likely to be perturbed (quasiresonance) in this spectral region due to the lower density of host bands. The guest modes were most easily observed in the infrared, where thick samples could be studied. The uncertainty of the reported frequencies is expected to be higher for the Raman because wider slits were neces- sary to observe the guest modes. C2H3D impurity was present in all of the neat samples except C2HD3, so C2H3D was used as the probe species. The C2HD3 site symmetry was probed by the addition of 1% ethylene-dl impurity, and checked by observing the modes of the 1,1-02H2D2 impurity which was present in the neat sample. The site symmetry of the C2H3D crystal was probed by adding 1% ethylene-d3 impurity. Some of the guest modes of these dilute mixed crystals are displayed in Figures l7-2A. The displayed modes were selected where possible to enable comparison with the modes in C2Hu. The frequencies of all the guest modes are tabulated in Tables 17-22, and in Appendix A if the guest was an impurity in the neat sample. In the crystals where the ethylene-d1 is the probe, the guest modes are either doublets or singlets, and the ethylene-d3 8A .mommmoumssno no m9 mm oocmemm mH mepopmw mm anB cosmos xmoa one .Qmmmo mo NH m 9 new m9 ”mamm NH oano\Qmmmo a: mo Euppooam cwemm can no whom .NH onstm ia-eso a mNNH ommH 85 :9 mm nocmemw mH mepopmm cm an3 oomeE zoom one .Qmmmo mo m9 was :9 .H9 meNSNOImHo\Qmmmo a: mo Enhpooam copmpmcH on» no ppmm .mH mpstm a :9 i m 9 H9 ommH ommH mam ONOH moom mHom rlllllll. r . r _ AHIEoV AHIEov AHIEoV 86 .ommmo no NH9 one e9 mammmonmcmpp\ammmo RH no Esnuooam oopmnmcH on» mo pawn mHe omMH mozH com 1 AH .5 .ma magmas 0mm 87 .OmmNU .HO m? GEN 3? .H9 mmammNOIH.H\Qmmmo u: mo ESppoon oonmhm2H on» no phmm .om opstm :9 m9 ‘ a, f me H ommH mmm mHOH moom mH m p a b p _ AHIEoV HHIEoV AHIEOV 88 .mm m Q m o mo e9 ocm m9 mmammmolH.H\o mmo R: mo Esopooam :wEmm on» mo pnwm .Hm opstm AHIEoV AHIEov _ i. _ 74 com 0mm ommH ommH 89 no NH; one we ”mommo\om NH» .ommmo mmo RH no Esapooam oopmHMCH on» mo when .mm opstm mom omm soy 90 N H m N N N .NQNH.HNUI._H7_H MO NH? Use 9 . 9 m cm 0\ a mmolH.H mo Esppooam conwumcH map Ho ppmm .mm opstm NH» N.9 H9 mNMH mmmH man om» mmmm moom p . rIIIIL _II||||L AHnEov AHnsov HHusoV 91 mHm _ do w» one so mommmo\m Eov ammo RH mo Sappooam UmpmpmcH on» go pnmm .zm opstm mmm s9 92 Table 1?. Observed frequencies of 02H3D fundamentals in A% C2H3D/cis-C2H2D2. Infrared (cm'l) Raman (cm’l) Assignment 3011.2 3010.5 V1 1597.8 1595-“ 02 1286.1 1285.7 V3 1283.A 1282.8 1011.1 ”A 1003.9 30A7.5 05 112A.O v6 812.8 810.2 v7 809.6 2261.8 011 2259-3 1396.8 139A.7 v 12 93 Table 18. Observed frequencies of C2H3D fundamentals in a! Infrared (cm-1) Raman (cm-1) Assignment 1597.1 1598.7 02 813.2 0 810.1 7 1396.8 v12 9A Table 19. Observed frequencies of C2H3D fundamentals in A% C2H3D/1,1-C2H2D2. Infrared (cm’l) Raman (cm’l) Assignment 3009.8 3011. v1 1596.9 1596. 02 1285.0 1285. v 3 1282.8 1283. 1010.2 Vu 1002.9 30A8.0 30A8. V5 30AA.8 30AA. 813.2 810. ”7 810.0 730.9 9 729.5 10 2262.1 V 2260.2 11 1396.8 1396.6 0 95 Table 20. Observed frequencies of 02H3D fundamentals in 1% 02H3D/C2HD3. Infrared (cm-1) Assignment 1012.0 0“ 813.A 0 810.3 7 1396.0 912 96 Table 21. Observed frequencies of 1,1-02H2D2 fundamentals in 1,1-C2H2D2/C2HD3. Infrared (cm-1) Raman (cm-1) Assignment 2999.7 V1 1578.A v2 75A.A 07 2216.6 2217.5 v11 1378.6 1381.3 V12 97 Table 22. Observed frequencies of C2HD3 fundamentals in 1% C2HD3/C2H3D. Infrared (cm-1) Assignment 1039.8 93 769.8 VA 767.8 2320.9 V5 736.6 9 723.6 7 931.0 v8 921.“ 3031.3 09 2209.5 V11 98 modes of ethylene-d3/ethylene—d1 are also either doublets or singlets. For the ethylene—l,l-d2/CZHD3 only singlets were observed. In no case was a multiplet higher than that allowed by a C1 site observed. In the VA region of C2H3D in the A% C2H3D/cis-C2H2D2 infrared Spectrum a triplet is observed as shown in Fig- ure 18. The 1001.8 cm"1 peak has been assigned as an orientational component of Va of trans-C2H2D2. The other “A component of the trans-02H2D2 is believed to be hidden by v“ of cis-C2H2D2. Unfortunately this could not be compared with the Raman spectrum in this region because even for neat C2H D VA was not observed in the Raman (see 3 next chapter). However, VA of the trans—C2H2D2 is not expected to be Raman active, and the 1001.8 component was not observed either. (Many trans-C2H2D2 impurity modes were observed in the spectrum of neat cis-02H2D2.) Another triplet was observed in the 03 of C2H3D region in the C2H2D/cis-C2H2D2 Raman spectrum (see Fig- 1) ure 17). The low frequency component (1280.0 cm“ is assigned as an orientational component of 03 of trans- C2H2D2. frequency orientational component of 03 of C2H3D and the The middle component is comprised of the low high frequency component of 03 of the trans-C2H2D2 (note its greater intensity). These assignments are consistent with the spectra of C2H3D/02Hu and trans-C2H2D2/C2Hu. A doublet is observed in this region in the infrared. 99 This is consistent with the observed activity of the trans-C2H2D2 modes. The dilute-mixed crystal spectra clearly indicate an "effective" centrosymmetric site symmetry. The near— sightedness of the ethylenes can be further investigated by comparing the spectrum of the probe molecule in the partially—deuterated host with the spectrum of the probe in the C2Hu lattice, which has sites which are known to have C1 symmetry. The difference in the orientational splittings of v7 (CéHD3) in C2H3D and C2Hu hosts is thought to be caused by the presence of 010 of C2H3D between the orientational components of v7 (C2HD3) in the C2HD3/C2H3D crystal. Resonance interactions between the C2HD3 molecules and the C2H3D molecules is believed to cause the difference in orientational splittings of 1.6 cm'l. This is the only case where the difference in splittings from one host to another exceeds the experimental uncertainty. The following bands have shown a different number of orientational components in different hosts. V8 (C2HD3) - Shows two components in the 02H3D host, at 931.0 cm'1 and 921.A cm-l, but shows only one component, at 920.9 cm"1 in the C2Hu host. It is believed that the high frequency component in C2Hu host is hidden by the low frequency Davydov component of v7 (C2Hu) at 9A1.0 om‘l. 100 09 (C2HD3) - Shows two components in C2Hu at 303A.0 cm-1 and 3030.2 cm‘1 1 but only one component at 3031.3 cm" is observed in C2H3D host. Un— doubtedly the peak at 303A.0 cm-1 in the C2H3D neat crystal spectrum (see Appendix A) hides 1 the 303A.0 cm- component. 95 (C2H3D) — Shows two components in 02H“ at 30A7.8 cm.1 and 30A3.2 em”1 but only one component at 30A7.5 cm"1 is observed in the cis—C2H2D2 host. The 30A3.2 cm.l peak is obviously hidden in the cis-C2H2D2 host by 09 (cis—C2H2D2) at 30A3.2 cm-l. A comparison of the spectra of the mixed-partially deuterated ethylenes with those of the partially-deuterated ethylenes in C2Hu reveals the number and frequency of the orientational components are the same in most cases. This clearly indicates that the ethylenes are "nearsighted", that is the intermolecular interactions are very similar in all the ethylenes. C. Conclusions The following conclusions can be drawn from the ob— servations of the guest modes of the dilute-mixed crystals of the partially—deuterated ethylenes in 02H” host and 101 in each other. The "effective" site symmetry of the partially- deuterated ethylene crystals is C1. Therefore the ethylene molecules are nearsighted in that they cannot distinguish between hydrogens and deuteriums on neighboring molecules in the crystal. The intermolecular interactions are myopic; deuteration causes little change in the inter- molecular force field of the ethylene crystals. CHAPTER VI THE INTERNAL MODES OF THE PARTIALLY-DEUTERATED ETHYLENES A. Introduction It has been shown in the last two chapters that the crystals of the partially-deuterated ethylenes have an "effective" C1 site symmetry. This leads to the existence of only one energetically-distinguishable orientation of the molecules in the cis-C2H2D2 and 1,1-C2H2D2 crystals, and two energetically—inequivalent orientations in the C2H3D, trans-C2H2D2 and C2HD3 crystals (see Table 11). This chapter is therefore divided into six sections. Section B concerns the assignments of the internal modes of the cis-C2H2D2 and 1,1-C2H2D2 crystals; C describes the coherent potential approximation calculation used to com- pute the band splittings of the C2H3D, trans-C2H2D2 and C2HD3 crystals; D deals with the observed fundamental vibrations of these crystals; and in Section E a compari- son of all the internal mode data is discussed. A comment on the assignments of the observed fre- quencies is in order. The free molecules of the partially- deuterated ethylenes are of lower symmetry than the D2h symmetry of ethylene. Therefore, fundamentals, overtones and combinations which were inactive for the C2Hu and C2D“ 102 103 crystals may now become active, and Fermi resonance,53 which is observed only to a small extent in the ethylene 2“ may become more prevalent in the crystals of crystal, the partially-deuterated ethylenes due to the lower sym- metry of the molecules. The neat samples also contained some impurities as discussed in Chapter III, and natural abundance C-l3 impurity was present in all the samples. Thus the spectra of the neat partially-deuterated samples exhibit many bands and assignment of these is at best tedious. The spectra are most complicated in the higher frequency regions (>1500 cm'l) where the overtones and combinations appear, and therefore Fermi resonances become more likely. Bands due to C-13 substituted molecules, present in 2.2% natural abundance, may also complicate the C-12 bands through intermolecular Fermi resonance.13’26 The calculated C-l3 harmonic shifts from the free molecule C-l2 frequencies of the partially-deuterated ethylenes are given in Table 23. The shifts were calculated by 2A,5A Duncan using standard normal coordinate analysis procedures.55 The assignments given in this chapter were based on gas phase frequencies, relative intensities, calculated C-l3 shifts and observed dilute mixed crystal frequencies (see Chapter 5). Also, mutual exclusion was not observed except for trans-ethylene-d2, so a comparison of infrared and Raman data aided in the assignments. Only bands 10A Table 23. Calculated harmonic C—l3 shifts (cm’l) from the C-12 frequencies. a 12 =13 13 =l2 C2H3D H2 C CHD H2 C CHD V1 0.3 5-9 02 18.3 26.1 03 3.2 6.9 v“ 1.8 0.0 V5 9.5 0.6 V6 5.2 8.6 v7 8.0 0.2 08 0.8 9.2 09 0.3 12.8 010 1.9 0.2 V11 15.1 .A 012 10.7 .2 _ b 13 =l2 l2 =13 1,1 C2H2D2 H2 C CD2 H2 C CD2 V1 5.8 0.1 v2 5.7 13.3 V3 3-7 0.A v“ 0 0 05 0.0 16.7 V6 9.9 6.9 v7 0.3 10.7 V8 8.8 0.7 v9 11.6 0.1 V10 .1 2.3 v11 .9 12.1 .A 12.A 105 Table 23. Continued. 021103a H012C=13CD2 H013C=12002 01 3.8 1A.0 02 20.2 25.9 03 3.6 6.A vu 8.2 2.6 vs 17.8 0.A V6 A.9 5.6 v7 3.0 2.1 08 0.5 5.8 09 0.1 10.0 010 1.1 0.2 V11 9.7 1.A 012 6.8 1.A cis-C2H2Dg cis-HD12C=13CHD trans-C2H2Dg trans-HD12C=13CHD v1 8.2 v1 6.9 v2 23.9 92 23.5 v3 2.3 V3 7.5 “A 2.8 VA 1.0 V5 5.7 vs 10.0 06 8.6 v6 6.7 v7 2.9 v7 2.2 V8 A.8 VB 7.3 09 A.3 v9 0.1 v10 0.5 v10 0.9 v11 7.2 v11 8.6 v12 A.1 v12 1.1 aFrom Reference 5A. bFrom Reference 2A. 106 assigned as fundamentals are presented in the wavenumber tables in this chapter; however, all the frequencies observed for the neat crystals, and their relative intensi- ties are listed in Appendix A. B. The Internal Modes of cis-Cog2g2 and 1,1-C9fi2g2 An analogy can be drawn between the internal modes of the cis-C2H2D2, 1,1-C2H2D2, C2Hu and C2D” crystals because the internal mode spectrum is not complicated by the orientational effect. However, it must be noted that cis-ethylene-d2 and ethylene-l,l-d2 are still disordered because of the existence of the translationally-inequi- valent orientations. Therefore adherance to the k selec- tion rules may be questionable. Unfortunately there have been only a few studies of the internal modes of disordered molecular crystals.56'59 Most of these have dealt with isotopic mixed crystals whose AA (see next section), modes are in the separate-band limit and the goal has been to resolve disputes over the split— ings in the pure crystal spectrum. Little effort has been concentrated on the internal modes in the amalgamation limit1M (see next section). However, studies of the phonons of isotopic mixed molecular crystals have re- vealed the usefulness of k in the amalgamation limit (see Chapter IV). 107 The amalgamation limit is achieved when the energy difference between the host and guest vibration is much less than the bandwidth of the pure-crystal mode. Form- ally one can say that the energy difference between the orientational components of the cis-C2H2D2 and l,l’-C2H2D2 crystals is zero. That is, for each vibration all orienta- tions have only one energy, and thus the energy difference is zero. Zero is certainly less than the bandwidth of the vibration, and thus the internal vibrations of these crystals are in the amalgamation limit. Excitons are more localized than phonons, and thus the phonons of a crystal probe the periodicity of the lattice over a larger range than do the vibrational ex- citons. Therefore, if k is a "good quantum number" for the phonons of a disordered solid in the amalgamation limit it is certainly expected to be a "good quantum number" for the vibrational excitons of the disordered solid in this limit. If k selection rules still apply to these disordered solids, then the bands should show two Davydov components as do the bands of ethylene and ethylene-d”. A breakdown of k selection rules would lead to broad, structureless bands. The observed fundamental vibrational frequencies of cis-C2H2D2 and 1,1-C2H D2 are listed in Tables 2A and 25. 2 Two Davydov components were resolved for eight of the twelve cis-C2H2D2 fundamentals and nine of the twelve 108 Table 2A. Internal fundamental frequencies (cm-l) of cis- C2H2D2.a Mode Infrared Raman In C2Hub Gasc v1 2286.2 VS (1.9) 2287.1 vs (1.1) 2285.1 2299 2278.0 vw 2278.9e w 02 1569.0 m,asy (3.9) 1563.7 VW 1563. 1571 1560.3 vs 15A1.6e w 03 1218.0 w,bd 1219.6 vs 1212. 1218 1205.1 w,sp 1202.0 vs VA 991.5 m,asy (A.8) 989.0 sh 985.3 s 982.2 05 3038.0 VS (3.2) 3037.3 s,sh 3037. 305A 3035.9 5 3032.148 w 3032.3e w V6 103A.0 m,asy (A.5) 1036.7 m lOAA 1033.0 m 1026.5e w 07 8A9.A vs 850.8 vw (8.0) 853. 8A2 8A2.8 vs V8 766.6 m (5.6) 761.8 m (7.0) 763 09 30AA.6 vs (3.0) (30A7.5)d 30AA. 3059 30A3.2 vs (1.3) v10 663.1 m 661.3 s 661. 662 660.76 w 658.A s 658.8 sh Table 109 2A. Continued. Mode Infrared Raman In C2Hub Gasc v11 22u2.1 s,asy (A.9) 22A1.7 s,asy (2.2)22Al.8 225A 012 1338.6 vs 1335.0 13A2 1335.0 vs 1331.9e w,sh Key to Table: (61) (b) (c) (d) (e) v = very, s = strong, m = moderate, w = weak, sh = shoulder, bd = broad, sh = sharp, asy = asymmetric; the full width at half height is given in parenthesis for the unresolved Davydov doublets. Mean of observed infrared and Raman frequencies. See Reference 2A. See text. These frequencies attributed to C-l3 substituted molecules. 110 Table 25. Internal fundamental frequencies (cm-l) of 1,1-C2H2D2-* Mode Infrared Raman In C2Hub Gasc 01 3000.6 sh 2999.8 3017 2998.9 8 (3.2) 2999.3 vs 2993.2g w 2993.1g w V2 1577.9 m (5.2) 1575.9 vs (.8) 1579.2 1585 156A.1f w 15614.0f m 1552.5g w 1552.u8 m v3 102A.9 S,bd (17.0) 1029.1 vs 103A.7 1031 1017.5 vs VA 905.A vs 887 901.6 vs v5 2322.9 S (3.7) 2322.6 VS (1.6) 2322.8 2335 23011.0f w 2305.7f w v6 1139.0 VW (7.0) 11A2.A W 11A2. 1138.2 w v7 753-5 VS 752.8 s,asy (3.9)75A.8 750-5 7A9.1 VS 7A2.8f s 08 9A6.0 vs,asy(l3.0) 955.3 sh 9A2.A 951.5f sh 927.7 s 09 3080.2 5 3078.3 vs (2.3) 3077.5 309A 3078.A s 3066.Ag w 3067.82; vw 111 Table 25. Continued. Mode Infrared Raman In C211“b Gas0 “10 685.1 m 683.A vw (9.A) 683.A 687.A 682.6g w 680.1 s v11 2216.9 s (2.u) 2217.u vs (1.0) 2216.6 2230 22014.8f w 2205.5f w 912 1379.8 vs (8.8) 1382.6 3 1380.3 138A 1375.8 vs 1369.6f s 1371.0f m *Footnotes as in Table 2A except for f and g. f 12 = l3 l3 12 These frequencies attributed to H gThese frequencies attributed to H 112 1,1-C2H2D2 fundamentals. Only 9“ of ethylene-l,l—d2 and v12 of cis-ethylene-d2 were not observed in both the infrared and Raman spectra, although different relative intensities of the bands and of the components of the bands were observed in some cases in the infrared and Raman spectra. In general, the full widthaN:ha1f height of the peaks was larger in the infrared spectrum than in the Raman spectrum. This is attributed to a difference in sample quality and the fact that a larger portion of the solid is sampled by the infrared radiation, which thus detects imperfections in the sample that the Raman experiment does not detect. The full widths at half height of the infrared Davydov components were normally 1 and those in the Raman approximately 1-2 cm’l. 3-5 cm" In the infrared, the stronger the component the broader it tended to be, whereas in the Raman the very strong com- ponents tended to have instrumentally-narrow bandwidths. C-l3 isotopic shifts were observed for six of the twelve ciS-C2H2D2 fundamentals and nine of the twelve 1,1-C2H2D2 fundamentals. Both C-l3 shifts of 1,1-C2H2D2 were observed only for the v2 mode. The C-l3 isotope with the largest shift was generally the one which was observed. The full widths at half height of the C-13 components were usually approximately 1 cm-1 in the Raman and 2 cm'1 in the infrared. Selected bands of cis-02H2D2 and 1,1-02H2D2 are 113 shown in Figures 25-28. These bands exemplified the general features of the groups of bands discussed below. cis-C2H2D2 91 and V11 show strong singlets in both the infrared and Raman spectra; however 011 is asymmetric, suggesting the existence of two components. The 13C component of v11 was not observed even though the shift is calculated to be 7.2 cm-l. v2 shows one very strong and one very weak component in the Raman spectrum. In the infrared an asymmetric band of moderate intensity is observed. The band in the infrared is wide enough to encompass the two Raman com— ponents. v3, 05, v6, v7 and 012 show two Davydov components of similar relative intensity in either the infrared or Raman spectrum (see Figure 25). Seldom could the two components be resolved in both spectra; however, the bands were resolved in the infrared if they are active for ethylene in the infrared (v7, v12) and the same result was observed for the ethylene Raman active bands.(v3, v5, 96). v“ and 010 show triplets in the Raman and infrared spectra, respectively, where the lowest frequency peak is the most intense. Unfortunately, ”A was not observed in the matrix-isolated sample. The calculated C-13 shift 1 for v“ is 2.8 cm‘ If the peaks at 989.0 cm'1 and 11A .moHSooHoE oopszpmozn mHno on copdnHmppo mH mesopmo no an3 ooxpoe xood one .mH9 use m9 mmammmolmHo mo moooE Homeoch .mm opstm HoopohmchmH9 mmHH mmNH Eov AcoEomv m9 115 .moHSooHoE oopSpHquSm mHlo op oopspprpm mH mepopmo cm nqu ooxooe xeoa oze .m9 oco OH9 mmommmolmHo mo moooE Hospoch .mm opsmHm 116 53 ozom Aceeemv m9 J omom .ee eeseae HoonmAMCHv 0H9 117 .m mm on oeosoHAooe no xeaaeeee cm Qsz ooxpoE good one .m9 one 9 H mm OIH.H no woooE HocuopCH .em opstm AHIEoV HHIEov o.7 12 H 12 8.3 ome mmmm moom Acesemv m9 Acesemv H9 118 .moo H u omHmm op oouannupo mH menopne no nqu ooxnee neon one .o 9 one :9 ”mammmoleH mo moooe Honnoan .mm onstm AoonenmnHv OH9 AUmhmeHewflHV 3? mew mmm 0mm mHm . Eov 119 985.3 cm.1 are assigned as the Davydov components, then the 982.2 cm'1 peak has a shift of 5.0 cm"1 from the mean of the Davydov components - which is much larger than the calculated C—l3 shift. The larger shift can be ex- plained on the basis of a large gas to crystal field shift or as being due to an intermolecular Fermi resonance between the C-12 and C—13 molecules. The intensity of the 982.2 cm‘1 peak would support the intermolecular Fermi resonance argument. The v10 triplet (Figure 26) is assigned on the basis of the dilute mixed crystal data. v10 in C2Hu shows a peak at 661.3 cm’1 and the calculated 0-13 shift is 0.5 cm'l; therefore the peak at 660.7 cm"1 in the pure crystal is attributed to C-13. The intensity of this peak is weak, consistent with the expected C-13 intensities. V8 shows a single, relatively broad band in both the infrared and Raman spectra at 766.6 cm”1 and 761.8 cm’l, respectively. The difference between the observed in- frared and Raman frequencies is larger than the experi- mental uncertainty, suggesting that each may be a dif- ferent Davydov component. v9 shows a very strong component in the infrared spec— trum (Figure 26). In the Raman spectrum a very strong com- ponent at 30A3.2 cm"1 is observed and there is a very weak component at 30A7.5 cm'l. It is possible that the very weak feature is a Davydov component of v9, or an 120 orientational component of 05 of the C2H3D impurity which was present in the sample. l4—1-3-9-2321—2-2 vl, 05 and v11 are similar to 01 and V11 of cis-C2H2D2. They all show strong single components in the infrared and Raman spectra. However, a shoulder was observed in the Raman spectrum of 01 (Figure 27). This has been assigned as one of the Davydov components. v2 is the only vibration for which both of the C—13 shifts were observed. A strong narrow band was observed for 02 in the Raman spectrum (Figure 27), and a relatively broad, moderately intense peak was observed in the infrared spectrum. The C-l3 band shown in Figure 27 corresponds to 12 12 2 2. The H213C = CD2 band is observed at about 10 cm-1 lower energy with almost equivalent in- H C = 1300 tensity. v3, 0“, v7, 09 and v12 show two relatively strong Davy- dov components in either the infrared or the Raman spectrum. The two components were resolved in only one of the two types of experiments. Again, these were resolved in either the infrared or Raman spectrum depending on the activity of the modes for the 02H” crystal except for 012. v“ is shown in Figure 28. V6 is very weak in both the infrared and Raman spec- tra. The infrared band is quite broad, but the two Davydov components are resolved in the Raman spectrum. 121 98 and 9 both exhibit interesting triplets in 10 the Raman and infrared spectra, respectively (see Figure 28). Unfortunately, neither mode was observed in the 1 matrix isolation experiments. If the 955.3 cm- and 1 9A7.7 cm_ bands of V8 are assigned as the two Davydov components, the experimentally measured shift of the 951.5 cm"1 component from the mean of the Davydov components is zero. This would correspond to the calculated shift 12 _ 2 C tal field shift is small. Precisely the same argument 1 of 0.7 cm-1 for H 13CD2, assuming the gas to crys- holds for the 682.6 cm' peak of v10; however, in this case the weak central feature corresponds to H2130 = 12 CD2, C. Excitons of Binary Mixed Molecular Crystals - The Coherent Potential Approximation The molecules in the crystals of C2H3D, trans-C2H2D2, and C2HD3 can assume two energetically-inequivalent or- ientations in a Ci lattice site. Thus, these solids can be considered to be binary mixed crystals with 50% of the molecules having orientation "A" at energy 2A, and 50% of the molecules having orientation "B" at energy 68. Figure 29 shows the evolution of the mixed-crystal band. This diagram is similar to Figure 2; however, the B and C contributions to the gas—to-crystal shift have been 122 mHo9oH o u & .oLSpoznpm onon nOpHoxo Hopmmno ootz .mm onanm m¢l<r0)<60A+o(z)), (31) where CA is the concentration of component A at energy EA and CB is the concentration of component B at energy EB. A is the difference in energy of component A and component B (the orientational Splitting), and E is the mean single molecule excitation energy, E = CAeA + CBEB (32) fO(z) is a complex function related to the density of states of the pure crystal. The spectral density,“5 which can be considered to express the k components of the density of states and thus is intimately related to the optical properties of the crystal, is then expressed as for k = 0: Imo(E—iO+) 1 S(O,J,E) = v“ (33) [B-ELt -Reo(E-iO+)12+[Imo(E-10+)12 3 where tJ is the energy of the 3th Davydov component of the pure crystal. E-iO+ indicates that the density of states is nonvanishing in the energy range where the self energy is complex. A plot of S(O,j,E) versus E shows the calculated optical absorption for the binary-mixed crystal. The calculated optical absorption depends on the concentra- tion of the components, the energy difference of the 128 states of the two components, the pure crystal density of states, and the locations of the pure crystal Davy- dov components. For the crystals of C2H3D, trans-C2H2D2 and C2HD3 the concentrations of each component is assumed to be 50% and the A's are the observed orientational splittings which, due to the "nearsightedness" of the ethylenes, can be determined in any ethylene host. The experi- mentally determined orientational splittings of the ethylenes in a C2Hu host crystal are given in Table 26. The numerical frequencies of the modes are given in Chapter V, Tables 12-1A. The pure crystal density of states of the ethylenes are unknown and the locations of the pure crystal Davy- dov components are unmeasurable since a crystal with only one orientation of the molecules does not exist. A one-dimensional pure crystal density of states was assumed for the partially-deuterated ethylenes. It has 63 the form: 00(E) = v’1(T2-E2)‘1/2, (3A) where T is one—half the exciton bandwidth of the vibra- tion. This model density of states is not necessarily a good assumption for the ethylene crystals because the ethylene molecules are planar and therefore the 129 Table 26. Orientational splittings (cm-1) of C2H3D, trans- a C H D2 and 02HD3. 2 2 C H D trans-C2H D 2 3 2 2 2 3 V1 0 2.2 3.5 v2 0 1.7 2.3 v3 2.A 2.A 0 Va 8.0 11.0 2.2C 05 A.6 A.7 0 V6 -——- _..._ 0 07 3.3 7.A 11.2 - b,c 08 --- A.5 9.6 V9 -"" I405 307 010 1.2 2.2 0 v11 2.6 0 v12 0 3.5 2.6 aDetermined in 02H“ unless otherwise stated, 0 indicates unresolved, --- indicates unobserved. D From 1% C2HD3/C2H 3D 0It is likely that these assignments should be inter— changed (vide infra). 130 intermolecular interactions are expected to be at least two dimensional. However, semi—quantitative informa- tion concerning the band-structure dependence on the trap-depth of the "mixed" crystal can certainly be ob- tained, and this can be compared with the observed band structure to determine whether the CPA mixed crystal model can be applied to the internal modes of the C2H3D, trans-C2H2D2 and C2HD3 crystals. Therefore, the appli- cation of the coherent potential approximation to the crystals of the partially-deuterated ethylenes is not a test of the CPA, although the author is unaware of any previous applications to vibrational excitons. Rather it is assumed that the CPA model works, and the result of the calculations are used to assist in the assignment of the observed exciton components. The positions of the pure crystal exciton components are unknown and attempts to calculate these positions using program TBON with a variety of intermolecular potential functions have been unsuccessful. Therefore, the bandwidths of the pure crystal is used as a variable in the calculation of the mixed crystal exciton structure. The bandwidths were determined by obtaining the best fit between the observed and calculated exciton components. The CPA calculations were done using program MIX 68 which was obtained from Woodruff and Kopelman and revised for this application. The program is listed 131 in Appendix C. D. The Internal Modes of C2H3Ditrans—C2H2D2, and CQHD3 The frequencies of the observed internal fundamentals for crystalline C2H3D, trans-C2H D2 and C HD are listed 2 2 3 in Tables 27—29; selected bands are illustrated in Fig- ures 30-37. The bands of each crystal have been grouped together according to their orientational splittings in the following discussion. This classification is entirely arbitrary since the structure of each band in a group of bands with equivalent orientational splittings may vary depending on the bandwidth of the pure crystal Vibration. 92332 01, v2, and 912 show singlets in the C2Hu matrix so the orientational splitting is assumed to be very small and the bands are expected to be in the amalgamation limit. The observed neat crystal bands consist of a singlet for 02 and partially resolved doublets for 01 and 012 (see Figure 30). This is consistent with amalga- mation limit behavior. The bandwidths can be assumed to be the energy difference between the observed Davydov components for v and v12,and the full width at half 1 height for 02. As depicted in Figure 30, v10 shows a doublet in the infrared spectrum of the neat crystal. The observed 132 Table 27. Internal fundamental frequencies (cm-l) of C2H D.* 3 Mode Infrared Raman In C2Hub GasC V1 3011.1 S (3.3) 3010.8 Sh 3010.0 3002 3009.8 vs 30014.9f vw v2 1597.2 S (5.5) 1595.3 8 (.7) 1597.0 1607 1580.12 w 1579.1g w 1572.9h w 1571.0h w v3 1292.1 S 1285.3 1290 1285.0 m (12.6) 1282.9 1277.7 s vA 1017.6 w 1010.9 1000 1012.8 w 1002.9 1006.2 m 1002.2 m 1002.6 vw (8.0) v5 30A8.0 s 30A7.2 vs (1.2) 30A7.8 3061 30A5.l s 30A3.9 vs (1.6) 30A3.2 3037.5g vw 3037.3g vw V6 1123.6 w 1120.8 m (13.A) 1129 1090.8 bd,sh 97 809.6 bd 809.9 m (11.A) 812.5 808 805.8 asy 809.2 V8 957.3 Sh 993 950.2 s,sh 953.2 sh 9A5.2 s,asy 9A9.0 s 9A1.6g w,sh 133 Table 27. Continued. Mode Infrared Raman In C2Hub GasC V9 3080.1 vs 3080.A sh 3096.A 3079.8 sh 3079.6 vs 3069.1f w 3069.0f m v10 729.9 bd asy 729.1 w (6.8) 730.8 730 726.6 m 729.6 011 2262.6 3 2262.6 3 2262.A 2276 2260.2 sh 2259.8 8 2259.8 v12 1395.8 bd (7.9) 1395.3 sh 1395.6 1A02 1392.3 8 1385.55 w 1385.14g vw *Footnotes as in Table 2A except for f and g. f gThese frequencies attributed to H 2 These frequencies attributed to H213C=12CHD. 12C=13CHD. 13A Table 28. Internal fundamental frequencies (cm-1) of trans- C2H2D2.* Mode Infrared Raman In C2HA Gasc v1 2271.6 S (3.A) 2270. 228A 2268.9 3 (3%5) 2268. v2 1563.8 vw,sh 1565.4 1571 1562.0 S (2.2) 1563.7 15141.3e w 93 1289.5 m 1282. 1286 1282.3 w 1279. 1275.7 1273.58 w 0“ 1006.5 sh 1001. 988 1009.1 8 990. 991.0 s 985.A S vs 3030.2 s (2.3) 3030.6 30A5 3026.9 m 3025. 3025.2 m 3020.7e w V6 1005.0 m 100A 996.0 m 993.1 3 v7 732.9 s 731.5 725.2 725.6 s 72A.l 721.1 s 135 Table 28. Continued. Mode Infrared Raman In C2Hub Gasc V8 873.8 s 870. 86A 869.7 w 865. 86A.6 m 859.9e vw v9 3052.1 S (9.0) 3051. 3065 3098.0 S 3097. 30A2.9 w v10 671.6 s 672. 673 667.8 s 670. v11 2265.9 S 2266. 2273 2263.6 sh v12 1298.0 5 1295. 1299 1296.0 S 1291. l292.A 3 1290.6 5 “Footnotes as in Table 2A. 136 Table 29. Internal fundamental frequencies (cm—l) of C2HD .* 3 Mode Infrared Raman In C2Hub Gasc 01 2269.2 S 2269.6 S (1.9) 2270.1 2283 2266.3 S 2266.3 S (1.8) 2266.6 2255.5f vw 2255.3f w 92 1595.3 vw 1596.1 S 1596.3 1597 1592.6 w 1592.8 S 1599.0 v3 10A6.6 s 10A1.9 1095 1091.9 5 1039.2 VS (7.1) 1039.8 S 1032.8f w VA 767.6 vs (6.A) 769.5 sh 769.1 76A 765.9 s 766.9 v5 2320.2 8 (9.9) 2319.2 8 (3.0) 2320.1 2336 230A.0g w 2303.1g w v6 1002.8 W 1002.9 m 995.9 999 993.2 w,asy 996.5 w 989.5 w,sp 991.5 97 735.3 s 735.0 725 729.8 S 723.8 720.8 3 v8 932.0 vs 933.3 m (A.8) 931.0h 919 921.2 vs 923.A sh 921.17h 921.A s 916.1f w,sh Table 29. Continued. 137 Mode Infrared Raman In C2Hub GasC 09 3035.8 s 3036.6 5 (1.8) 3033.8 30A9 3032.1 5 3032.7 3 (1.7) 3030.1 010 628.8 w 627.5 627.5 N610 626.8 w 62A.6 s V11 2210.1 vs (2.0) 2211.A s (1.5) 2209.3 2222 2199.0g w 2201.112 w 912 1286.0 vs 1287.1 5 1286.2 1290 1283.0 vs 128A.3 s 1283.7 *Footnotes as in Table 2A except f, g and h. f These frequencies attributed to HD 130:12 hObserved in C H D matrix. 2 3 CD 2. 8These frequencies attributed to HD12C=13CD2. 138 . I N DmOMHIONH m on oopanhppm mH mecHopmw H As-eso ome mozH AcmEmmV NH? 139 AoononmnHv :9 ONNH omm ONOH F7 AH-soo .39 one m9 mm m mmo no moooE Honnoan AH- soy AflmEmmv m? .Hm enemae OOMH 1A0 HoonehmnHv NH9 mmNH .NH9 one H9 HNQNmNoImnenp mo moooE Hennoan IEoV AH omNN AneEemv H9 mOMH — Eov .Nm onstm owNN 1A1 .39 one m9 mNQNmNOImnepp mo moooE Hennoan .mm opstm HoonenmnHv :9 AA-eso omom mmom owe OHOH AseEemv m9 nov 192 ll Calculated Spectrum Observed Orientational Splitting Figure 3A. v8, trans-C2H2D2, calculated and observed Raman band. The peak marked with an asterisk is attributed to C-l3 substituted molecules. 193 L ’J Orientational ‘\\ .9’ Splitting Y I I Mean of Orientational u ' Components l 3: ' l I Calculated Spectrum (cm-l) ' 1 750 715 Figure 35. v7, trans-C2H2D2, calculated and observed infrared band. 1AA .enpooam nenem one oonepmnH one nH mommo mo nOHmon H9 HoonenmnHv Ae-sso H II. omNN mmNN omNN omNN r _ A Boy Aneeemv HI .wm onstm 195 .enpoonm neEem one oonenmnH on» nH QONo no nOHwop $9 .wm onanm AHIEOV mmm OHOH HoonepmnHV 0mm OHOH pl le-eso Aneeemv 1A6 orientational splitting is 1.2 cm’l. If a bandwidth of 2.2 cm"1 is assumed for 010, the observed Davydov com— ponent positions for the band are calculated. There- fore, this band is in the amalgamation limit. 03 and 011 both show doublets in the neat crystal spectra (see Figure 31, for example). The observed or- 1 1 ientational splittings are 2.A cm- and 2.6 cm' , respec- tively. If the bandwidth of 03 is assumed to be 12.8 cm-1 the observed Davydov components are calculated; therefore v3 is amalgamated. The orientational splitting and energy difference of the observed pure crystal com- ponents of v are approximately equal. Bands of this 11 nature are either close to or in the separate-band limit and the bandwidth is best approximated by the full width at half height of one of the two components, since theo— retically a quartet should be observed in this situation (see Figure 29). It is assumed that each experimental feature is an unresolved Davydov doublet. Unfortunately, the doublet observed for v11 in both the infrared and Raman spectra is not resolved very well, so the bandwidth is approximated by one half the total width at half height of the entire V11 band. The bandwidth approximated in this way is 2.6 cm-l. v5 and v7 have orientational splittings of A.6 cm"1 and 3.3 cm-l, respectively. V5 shows a relatively sharp doublet in the Raman spectrum of the pure crystal and 1A7 the observed splitting of this doublet is less than the orientational splitting. This band is then very similar to v11 and is presumed to be in the separate-band limit. The bandwidth is approximated by the mean of the full widths at half height of the observed pure crystal com— 1). ponents (1.A cm- 07 shows quite a broad band in both the infrared and Raman spectra. The two infrared com- ponents are barely resolved and a fit to these components 1. Four com- 1 leads to a calculated bandwidth of 0.8 cm- ponents are predicted since with a width of 0.8 cm- the band is in the separate band limit. The two un- observed but calculated components are assumed to con- tribute to the broadness of this band. VA has the largest orientational splitting (8.0 cm-l) of all the C2H3D bands. A quartet is observed in the neat crystal infrared spectrum (Figure 31). However, attempts to calculate this observed spectrum using 8.0 cm"1 as the orientational splitting were unsuccessful. If the orientational splitting is assumed to be 11.0 cm"1 the observed splitting pattern can be calculated, if the band- width is taken as 5.2 cm-l. The difference in the orien- tational splitting between pure C2H3D and the C2Hu host could be due to a dependence of the orientational split- ing on the nature of the host, a breakdown of the coherent potential approximation, or the assumption that the density of states of the band can be adequately described by a 1A8 one—dimensional model. The orientational splittings of the partially-deuterated ethylenes have been shown to be independent of the host (see Chapter V). Even where the guest modes are very close in energy to a host band, little change in the orientational splitting was observed. The cause of the difference in orientational splittings must therefore lie in the density of states function or the coherent-potential approximation. Assumption (5) (Appendix B) of the CPA states that the resonance inter— actions for each component and between the different com- ponents are equivalent. If this assumption breaks down, the effective-orientational splitting could increase due to resonance interaction between translationally-equiva- lent molecules (see Figure 2). v6, V8 and v9 were not observed in the C2Hu matrix; therefore it is difficult to classify these bands as either amalgamated or in the separate-band limit. 96 shows a rather weak band in both the infrared and Raman spectra. 08 shows a triplet in the Raman spectrum suggest- ing this band is in the separate-band limit. However, one of the shoulders on the high-frequency side of the 9A9.0 cm"1 component may be attributable to H213C=12CHD which has a calculated C-13 shift of 0.2 cm-l. 09 shows a doublet in the pure crystal spectrum so it could either be amalgamated or in the separate band limit, depending on the magnitude of the orientational splitting. 199 trans-C H D 2—2—2 91, v2, v3 and 910 have observed orientational split- 1 l 1 l ings of 2.2 cm- , 1.7 cm- , 2.9 cm- and 2.2 cm" , res- pectively. 01 can be assumed to be near the separate- band limit since the orientational splitting is approxi- mately equal to the energy difference between the observed pure crystal components (Figure 32). v2 is similar except that the high frequency neat crystal component is a weak shoulder rather than a component of near-equal intensity. 03 shows a triplet in the Raman spectrum for an orienta- tional splitting of 2.9 cm-1 , a bandwidth of approximately 19.0 cm'1 is needed for the outer components to have the observed energy difference. However, with such a band- width the band is expected to be amalgamated and a doublet not a triplet, should be observed. The middle peak at 1282.3 cm'1 does not correspond to the calculated C-l3 shift of 75 cm'l, and a weak peak is observed at 1273.5 cm.1 which is shifted 7.6 cm"1 from the mean of the or- ientational components which agrees with the calculated 1 peak suggests shift. The weak intensity of the 1282.3 cm- that the band is very close to being amalgamated. A doublet is observed for v10 in the pure crystal. If a bandwidth of 2.0 cm.1 is assumed, four components are calculated for this band. If it is assumed that the two observed components are unresolved doublets, v is in the 10 separate-band limit. 150 l, and 012 has an orientational splitting of 3.5 cm- a quartet is observed in the infrared spectrum of the pure crystal (see Figure 32). The band is similar to “A of C2H3D in that the pure crystal pattern cannot be cal- culated using the observed orientational splitting. However, if the orientational splitting is assumed to be l a bandwidth of 2.6 cm.1 gives the correct loca- 5.2 cm- tion of the four Davydov components. 05, V8 and v9 have orientational splittings of 9.7 -1 1 1 cm , 9.5 cm- and 9.5 cm- , respectively. v5 shows a triplet in the neat crystal spectrum (Figure 33); however, the energy difference between the outer components is nearly equal to the orientational splitting, suggesting that this band is near the separate band limit. The middle component might be due to C-l3, although (as with v3) the calculated shift does not agree with the observed 1 position and a weak peak at 3020.7 cm- which has a shift of 7.5 cm.1 from the mean of the orientational components is more likely the C-13 band. The calculated C-13 shift is 10 cm'l. The calculated band splitting and intensity pattern for v8, assuming a bandwidth of 5.0 cm-l, is shown in Figure 39 along with the observed Raman scatter- ing. A triplet is observed in the pure crystal and the middle component corresponds to one of the calculated center components. The other center component is assumed to be hidden by the 869.6 cm"1 band. This vibration 151 cannot be classified as amalgamated or in the separate- band limit since the orientational splitting is approxi- mately equal to the bandwidth. It is of interest that the CPA seems to work quite well here. 0 also shows a 9 triplet in the pure crystal infrared spectrum. However, the low frequency component is very weak, suggesting that it may be attributed to C-13; the calculated shift is 0.1 cm-l. The energy difference between the two strong components is approximately equal to the orientational splitting which classifies this band near the separate band limit. VA shows a quartet in the infrared spectrum of the pure crystal (Figure 33), as did VA of C2H3D. Again the orientational splitting must be increased - from 11.0 1 cm.1 to 18.7 cm- - to obtain calculated components near those of the observed band. The closest agreement was obtained assuming a bandwidth of 7.0 cm'l. 97 shows three strong components in the pure crystal spectrum. A quartet is calculated with CPA (Figure 35), to which three of the observed peaks correspond if the bandwidth is assumed to be 5.2 cm-l. However, the center component in the spectrum also corresponds to the cal- culated C-l3 shift from the mean of the Davydov components, so no definitive conclusion can be drawn. 911 shows only a single absorption in C2Hu, so the orientational splitting is assumed to be zero. A partially 152 resolved doublet is observed in the infrared of the pure crystal. Since the orientational splitting is zero, v is in the amalgamation limit and the bandwidth can 11 be closely approximated by the energy difference of the observed pure crystal components. V6 was not observed in the matrix, making it dif— ficult to classify this band. A triplet is observed in the Raman spectrum suggesting that V6 is in the separate band limit. However, the C-13 component was not observed and could be contributing to the triplet structure. 9.2923 v2, VA and 012 have orientational splittings of 2.3 cm’l, 2.2 cm-1 and 2.5 cm"1 respectively. A strong doub- let is observed in the pure crystal Raman spectrum for 02. If a bandwidth of 1.2 cm"1 is assumed, the CPA predicts a quartet which would correspond to the observed doublet if each observed component is assumed to be an unresolved doublet. A similar situation exists for VA if the bandwidth is assumed to be 1.6 cm-l. 012 shows a very strong doublet in the pure crystal and the energy difference between these observed bands is nearly equivalent to the orientational splitting. Thus, this band is near the separate-band limit, and each component is an unresolved doublet. 1 v1 and v have orientational splittings of 3.5 cm- 9 and 3.7 cm-l, respectively. A strong doublet is observed 153 ' in the pure crystal spectrum for 0 (Figure 36), and the l orientational splitting is approximately equal to the energy difference between the two bands of the observed doublet. Therefore 0 can be interpreted in the same 1 fashion as v is also quite similar, showing a 12' v9 strong doublet with a separation which is approximately equal to the orientational splitting. These bands are in the separate-band limit. 07 and v8 have orientational splittings of 11.2 cm-1 and 9.6 cm—l, respectively. 07 shows a strong triplet in the infrared spectrum of the pure crystal. If a band— width of A.0 cm“1 is assumed a quartet is predicted in which three of the calculated components correspond to the observed absorptions. The fourth component is assumed to be hidden by the 735.3 cm-1 component although it does not appear to be appreciably broader. A very-strong doub- let is observed for V8 in the infrared spectrum and a triplet is recorded in the Raman spectrum. The orienta- tional splitting is nearly equivalent to the splitting of the infrared components and the outer components of the Raman triplet. Thus, this band is in or near the separate- band limit with each of the two components an unresolved doublet. The center component of the Raman triplet may correspond to HD12C=13CD2 which has a calculated C-l3 shift of 0.5 cm-l; however, the observed shift from the 1 mean of the orientational component is 2.8 cm- The 15A difference in shifts may again be caused by Fermi reson- ance interaction. 93, 05, v6, 010 and 9 show no orientational splitting 11 in the C2Hu matrix. v3 exhibits a triplet in the pure crystal Raman spectrum. This suggests that either the mode is in the separate-band limit, or that one of the triplet peaks is due to HD12C=13CD2. 95 shows a strong singlet in the pure-crystal spectrum, which indicates that this band is likely amalgamated. V6 shows a triplet in both the infrared and Raman spectra (see Figure 37); however, no component corresponds to the calculated C-l3 shift. This suggests that 96 is in the separate band limit or that Fermi resonance exists between the C-12 and C-13 molecules. 910 also shows a triplet; however, 1 corresponds to here the center component at 626.8 cm- the calculated position of H012c=13CD2. v11 is a singlet like VB; therefore v11 is probably in the amalgamation limit. It is of interest to note that not nearly as many C-l3 components are observed for the C2H3D, trans-C2H2D2 and C2HD3 crystals as are observed for solid cis-C2H2D2 and 1,1-C2H2 2. This arises from the greater overall breadths of the C-12 bands of crystalline ethylene-d1, trans-ethylene-d2 and C2HD3, due to the orientational splitting. The full widths at half height of many of the C-12 components in these crystals are larger than those 155 of the corresponding cis—C H2D2 and 1,1—C2H2D2 fundamen— 2 tals, but the larger widths are usually accompanied by unresolved bands in the separate band limit. E. Comparison of the Internal Modes of the Polycrystalline Ethylenes A comparison of the band structures of the ethylenes may reveal information regarding possible perturbations of fundamental vibrations by other fundamentals, over- tones, combinations, or impurity vibrations of similar 29’26 However, it must be noted that the contribu- energy. tions of the various symmetry coordinates (that is, motions) to a particular fundamental vibration will change upon deuteration, so that a strict comparison is not pos- sible. For example, a normal coordinate calculation using Shimanouchi's program LSMA69 and Duncan's force field22 for ethylene reveals v1 to be 99% a symmetric C-H stretching motion, while for trans-ethylene-d2 v1 is comprised 99% of the symmetric C-H stretch and 99% of the asymmetric C-H stretch. These changes, however, are expected to have the most pronounced effect on the bandwidths of the vibrations which arise from dynamic interaction between the molecules, and less effect on the static interactions; that is, the orientational splittings and the A's - the gas to crystal field shifts — should be 156 less dependent on the potential energy distribution. The orientational splittings of the partially—deuterat- ed ethylenes are given in Table 26. The splittings ap— pear to be relatively constant for V1’ v2, v3, v5, v9, v10 and 012 if only the cases where non-zero splittings are observed are compared. An experimental orientational splitting of zero may indicate poor—sample quality rather than an absence of an orientational-energy difference. This is supported by the pure-crystal bands, which do not appear to be amalgamated although no orientational split— ing is observed. Therefore, it is considered reasonable to ignore the zeros in this comparison. The largest deviations from the norm for a given band appear for VA (C2HD3) and v7 (C2H3D), and a large dif- ference exists for V8 (trans-C2H2D2 and v8 (C2HD3). The orientational splittings of VA (C2HD3) and 98 (C2HD3) 1 1 are 2.2 cm- and 9.6 cm' , respectively. If these assign— ments were switched the agreement between the orienta- tional splittings of these C2HD3 bands and those of the corresponding vibrations of the other ethylenes would be more reasonable. The normal-coordinate calculation gives the potential-energy distribution of these modes listed in Table 30. The symmetry coordinates are taken from Kuchitsu, gt a_l_.70 and the force field is that of 29 Duncan. 3“ is the out-of-plane twisting motion of the ethylene molecule, and S7 and 38 are the plus and minus 157 Table 30. Potential-energy distribution and calculated frequencies of the out-of—plane vibrations of C2HD3. Calculated Frequencies (cm-l) Symmetry Coordinates 739 777 937 SA 39% 29% 37% S7 61% 13% 26% 38 0 63% 37% 158 linear combinations of the out-of—plane wags, respectively. In C2Hu there is no mixing of these modes and VA is purely the twisting motion while 07 is the motion arising from the plus linear combination of the wags and V8 is the minus linear combination. It can be seen from Table 30 1 that the 739 cm- band is 61% S7; therefore this is 07 as it has been traditionally assigned. However, the 777 cm"1 and 937 cm-1 bands have been assigned previously as v“ and v8, respectively by Arnett and Crawford,71 based on a less sophisticated force field calculation. The potential energy distribution given in Table 30 does not support these assignments. The mode at 777 cm-1 is 63% 88, and thus should be assigned as v8, leaving the 937 cm'"1 mode to be assigned as ”9’ If the v" and v8 assign— ments are switched the orientational splittings of C2HD3 agree more closely with those of the other ethylenes. It is of interest to note here that the orientational splittings of the out-of—plane vibrations tend to be larger than those of the in—plane motions of the ethylenes. This is especially true for “A and v The same trend 7. was observed for the benzenes by Bernstein.26 This is believed to be caused by the large vibrational ampli- tudes of the out-of—plane motions; although not always larger than the in-plane—motions, the amplitudes of the out-of-plane vibrations are never substantially smaller than the in-plane amplitudes. The large orientational 159 splittings seem to indicate that the static field is probed more by the out-of—plane motions than the in-plane- motions. A summary of the A's, the gas-to—mixed crystal shifts, for the ethylene is given in Table 31. A was calculated as e The shifts of C H D, trans-C2H2D2 2 3 and C HD3 were obtained by taking the mean of the orienta- gas’emixed crystal' 2 tional components as the mixed crystal energy. V1’ V5, 09 and v are the C-H(D) stretching vibrations of the 11 ethylenes. An examination of the A's for these vibra- tions reveals relatively large values (1313 cm-1) and relatively constant shifts for all the stretches. The shifts of the stretches involving deuterium motion rather than hydrogen motion tend to be smaller; implying the deuteriums do not probe the crystal field as much as the hydrogens. This is supported by calculated vibration— al amplitudes. v3, v6, v10 and v12 are the C-H(D) bending vibrations of the ethylenes. The shifts of these vibrations are smaller, and not nearly as constant as those of the stretches, although the values remain relatively constant for each vibration. No comment can be made concerning the hydrogen versus deuterium motions except that the deuterium shifts appear to be a little smaller if the C2HA and C2D” shifts of 93 and v12 are compared; however, this difference may not be real due to the experimental 160 .mpnoeanmee HenOHpHoenBo .nN one eH .mH moononomom Eopmn .nOHpoE Eanopsoo e HQV .nOHpoE OHnouonn mHHneEHnn e mopeoHonH Amve m.m H.m m.m s.m o.s :.o e.m NH9 s.ea noos.ea noos.e noos.ma nove.ea novm.sa s.os see e.s- m.sH- m.H 0.: s.o N.ou nu- 0H9 e.en Aeoe.en Aeoe.mfi Aeom.ea Amos.mn --- m.me a» H.m om.sn 0.3- in: in: in: m.H MW9 m.e- e.su e.mu m.eu :.HH- m.m- .1. s9 H.HH o.m nu- in- In: in: In: mt9 s.e nova.mn Aeoe.en gooe.ee leve.ee nevm.mn o.mn m5 e.ea- so.s- m.e- --- --- m.e- --- :5 H.: H.m a.s s.m- m.m a.m m.s m5 m.s m.H s.e e.m m.s o.OH o.m m9 m.se novs.se nooe.ss Aeoe.ea have.mn nevo.e- H.0m H5 enemo mommo memmmoueceno mommmonH.H memmmoueeo ommmo osmmo e.HHIEoV mononnuo on» no mpmHnm Hepwmno ooxHEIOpumeo .Hm oHneB 161 uncertainty. The only other comparison that can be drawn is between v7 and V8 and no obvious pattern has appeared; however it should be noted that the A's of all the out-of—plane motions are negative. Again the agree- ment between the values of the other ethylenes, and v“ and V8 of C HD3 is best if the assignment of these bands 2 are switched. A large deviation from a typical shift for a given vibration (if such a typical shift can be determined) usually indicates that the band in question is perturbed by Fermi resonance interaction with a nearby band. This is strikingly observed for V1 (C2H3D) and (1,1-C2H2D2), which are known to be perturbed in the gas phase. A 1 for these molecules are -8.0 cm- and 17.2 cm'l, respec- tively, and these shifts deviate from the general 01 shift (19.0 cm-l). Another example is v11 (C2Hu) which has the largest shift of the observed bands (96.9 cm'l). Once again this is known to be a Fermi resonanting 2“ The shift of v1 of 02H“ is 30.1 cm-l, suggest- ing that this band is also Fermi perturbed. This agrees band. with Elliott and Leroi's intermolecular Fermi resonance argument for this band in a C2D“ host.l3 Other bands which appear to have large deviations from typical A values include: 02 (C2H3D), 02 (C2HD3)’ v3 (1,1-C2H2D2), v5 (020“), v7 (cis-C2H2D2), v10 (C2HD3), 010 (C2Du), and v (trans-C2H2D2). The vlo's can be 11 162 eliminated from the preceding list because the uncertainty of the determined gas phase frequencies is quite high.2h It is not obvious how the other bands are being perturbed. The only cases for which there is an obvious nearby band is 02 (02HD3). 29” of C HD3 is observed as a doublet at 2 1531.0 cm"1 and 1527.6 chl (see Appendix A). The 1531.0 1 1 cm' component is strong while the 1527.6 cm- component is of moderate intensity. The unusual A value of 92 (C2HD3) and strong intensity of 20“ indicates a Fermi resonance interaction between these vibrations. It should be remembered that the A's of the different iso- topes are not strictly comparable because of the dif— fering potential energy distributions. This may account for the deviations observed for the other above-mentioned vibrations. The translational shifts of the ethylenes (C in Figure 2) are usually less than the experimental uncertainty, and therefore no other general comment can be made about them. The other resonance contribution to the exciton band structure is the bandwidth, that is, 2T in Figure 2. The bandwidths of the ethylenes are given in Table 32. In the cases where the Davydov doublets were not resolved the bandwidths were approximated by the full width at half height of the observed singlet, or the bandwidth determined with the CPA; otherwise the spac- ing between the Davydov components is taken as the 163 .eno .oeoeHsoHeo .oeeeer nHen we soon: HHsno o .munononEoo >ooz>eo noozpon mnHoenmo .am one NH .MH moononomom Eonme e.e om.e oe.e se.e oe.m oo.m s.m ea» m.H in- om.e oo.H om.m oe.m m.H HH9 H.s --- oo.e so.m es.s oe.e e.m one o.m oe.H no.3 om.H om.H nu- m.m m9 s.m ee.s oo.m se.e be.m --- a.a e5 o.e oe.s oe.m es.e oe.e oe.o e.s s9 m.m nu- -1- om.s os.m in- m.: o9 m.m nu- om.m oo.H os.H oe.H s.m m9 m.: oe.H oo.s oe.m om.e om.m m.m s9 m.eH -u- oo.sH oe.HH oe.sH om.mH m.mH m9 m.m om.H om.m no.0 oe.m os.o .H.s m9 m.H no.H om.m om.H pH.H oo.H. m.oH H9 o.momo moemo mommmouecene momemouH.H mememoueeo emmm memo . AHIEOV mwcmflhgpm QED rHO WCUUHNévcmm .Nm GHQMB 169 bandwidth. It is of interest that the bandwidths of the partially—deuterated ethylenes tend to be nearer to the C2D“ values than to the C2Hu bandwidths except for 95 and 96. However, the bandwidths of C2D“ are less than the C2HA values except for 05 and V6 also. No general increase or decrease in the bandwidths is observed with the addition of more deuterium atoms, nor are the values of the three d2-compounds particularly similar. The random behavior of the bandwidths is believed to reflect the changes in the motions which take place upon deuteration. F. Conclusions The following conclusions can be drawn from the ob- servation of the internal modes of the partially-deut- erated ethylenes. (a) k = 0 selection rules apply to the crystals of the partially-deuterated ethylenes. Thus the crystal bands of 1,1-C2H2D2 and cis—C2H2D2 are similar to the bands of C2Hu and C2D“, and mutual exclusion was observed for the internal modes of trans-C2H2D2. (b) The internal fundamentals of the crystals of C2H D, trans-C2H2D2 and C2HD3 can be interpreted as mixed 3 crystal bands. (c) The coherent-potential approximation appears to work within the approximations used here for the 165 vibrational modes of the partially-deuterated ethylenes; although it may have failed for particular bands. (d) The assignments of v“ and 98 of C2HD3 should be interchanged. (e) The observed gas-to-crystal shifts support previously assigned Fermi resonances including the inter- molecular Fermi resonance, assigned by Elliott and Leroi.l3 A new Fermi resonance has been observed between 20“ and 02 of C2HD3. CHAPTER VII DISCUSSION AND SUGGESTIONS FOR FUTURE WORK Probably the most interesting conclusion that can be drawn from this study is the existence of "effective" symmetry in the crystals of the partially-deuterated ethylenes. Although observed-lattice spectra of various isotopic mixed crystals have suggested the existence of "effective" symmetry in isotopically-disordered solids, no other study, of which this author is aware, has probed the site symmetry directly. This has been done by observ- ing the orientational splitting of the partially-deuterated ethylenes in one another. The observed-lattice fre- quencies and mutual exclusion of the lattice modes of the partially-deuterated ethylenes suggests an "effective" C1 site. This is confirmed by the observed orientational splittings and the mutual exclusion of the trans-ethylene— d2 fundamental internal modes. The lattice modes of the partially-deuterated ethylenes can be described in the virtual-crystal limit, implying that deuteration causes a purely mass effect on the ob- served energies. It would be of interest to study the lattice modes of mixed crystals of 02H” in C2D“, especially the 25%, 50% and 75% mixtures, and to compare these with the partially-deuterated ethylene lattice spectra. The 166 167 virtual crystal approximation predicts that these spectra would be the same as the ethylene-d1, ethylene-d2 and ethylene—d3 spectra, respectively. It would also be quite valuable to confirm the observation of the third optically active translation of C2Hu, and to observe the third optically active translation of C2D“. The transla- tions of mixed molecular crystals have been observed in very few cases; thus it would also be of interest to observe to far-infrared spectra of the partially-deuterated ethylenes and mixed crystals of C2Hu in C2D“. A better intermolecular interaction potential for ethylene is certainly required. In principle this can be obtained by a refinement of the potential parameters to the observed lattice frequencies, including both the librations and translations. In the same vein, the exciton splitting may be calculable with a refined po- tential; however, the intramolecular potential may need to be adjusted to suit the molecules in the crystal sites before this task is accomplished. Unfortunately, little specific information was ob- tained about the ethylene crystals from the internal mode spectra. Again, a better intermolecular potential func- tion may assist in giving more weight to the assignments of some of the bands of the partially-deuterated ethylenes; however an experimentally determined density of states function may be necessary to improve the assignments of 168 the 02H3D, trans-C2H2D2 and C2HD3 internal fundamentals. The coherent-potential approximation appears to have worked for the partially-deuterated ethylenes. However, one should be cautious in attempting to use this calcula- tion for the vibrational bands of mixed crystals in general, since the wave functions of different isotopes may be dif— ferent and this is not accounted for by the CPA. Certainly more testing of the CPA for application to the general- vibrational problem is necessary, but this should be done on systems with a density of states function and band- widths which are known, and on bands which show behavior from the amalgamation to the separate-band limit. Polarization studies of the internal modes of the partially-deuterated ethylenes may assist in a confirma- tion of the assignments made in this study and provide information concerning the symmetry of the internal modes. It would be of interest to establish what effect the "effective" crystal symmetry has on the symmetries of the internal modes. Obviously the observed activities gave little information concerning this problem, except possibly for the trans-ethylene-d2. Mixed-crystal studies may also assist in the confirma- tion of the assignments of this study. For example, if a guest band is well isolated from a host band, the Davy- dov splitting of the guest band varies with the concentra- tion of the host. If the crystal has a high concentration 169 of host molecules, the Davydov splitting is small, but it increases as the guest concentration is increased. Studies of this nature could undoubtedly distinguish C-12 Davy- dov components from C-l3 impurity components. APPENDICES APPENDIX A The following is a list of the observed frequencies (cm-l) and relative intensities of the bands in the internal spectral region of the neat, partially-deuterated ethylene crystals. 22532 Infrared Raman Assignment 3089.6 w v9 (C2Hu) 3080.1 vs 3080. sh 0 3079.8 sh 3079.6 vs 9 3069.1 w 3069.0 m 09 (H2l3c=12CHD) 3098.0 5 3097. vs 3095.1 8 3093.9 vs "5 3037.5 vw 3037. vw v5 (H212C=13CHD) 3039.0 vw 3033. vw 296+v7 3011.1 3 3010. sh V1 3009. vs 3009. vw v1 (H213C=120HD) 3000.6 vw v10+v11 2999.1 vw 2999. VW v1 (1,1-C2H2D2) 2996. vw V1 (C2Hu) 2979.1 m v11 (C2Hu) 2961.2 5 2960. s v2+vl2 170 171 Infrared Raman Assignment 2870.9 vw 2870.1 w v +v 2869.0 vw 2 3 2339.8 m 0 +0 2330.7 m 2332.5 m 8 12 2262.6 3 2262.6 5 0 2260.2 sh 2259.8 s 11 2122.8 vw 2122.2 w V9+v6’ vlo+v12 2015.5 m v3+v10 1997.5 w,sp 2v!4 1955.5 w V9+V8 1907-9 Siasy 208, v6+v7 1898.0 w V6+v10 1818.2 m Vu+V7 1769.0 W V7+V8 1620.0 m 2V7 1597.2 3 1595.3 3 v2 1591 8 vw l3 . v2 ( C of C2Hu)? 1580.1 w 1579.1 w v2 (H212C=13CHD) 1572.9 w 1571.0 w 92 (H213C=12CHD) 1969.5 w 20 10 172 Infrared Raman Assignment 1938.2 s 012 (C2Hu) 1935 1 w v (130 of c H ) ' l2 2 9 1395.8 bd 1395. sh 1392 v12 . 8 1385.5 w 1385. vw 012 (H212C=13CHD) 1338. w 93 (C2Hu) 1292. 3 1285.0 m VB 1277. 5 1123.6 w 1120. m V6 1090.8 bd, sh 1017.6 w 1006.2 m 1002.2 m 1002. vw 957. sh 950.2 s,sh 953. sh V8 995.2 s,asy 999. s 991.6 w,sh ”8 (H212C=13CHD) 809.6 bd 809. m ”7 805.8 asy 755.0 w 97 (l,l-C2H2D2) 729.9 bd,asy 729. w v 726.6 m 10 173 cis-CZHQD2 Infrared Raman Assignment 3126. vw vl+v7 3123.5 3 3121. s 2V2 3085.5 w 3082. m V11+V7 3081.5 m 3079.1 m 3052.0 w v1+v8 3097. 09,v5 (C2H3D) 3099.6 vs 3093. vs 3038.0 vs 3037. s,sh v 3035. s 5 3032.9 w 3032. w v5 (130) 3026.1 vw 3027. w v9 (130)? 3029. vw v5 (trans-02H2D2) 3011.2 vw 3010. w v1 (C2H3D) 3008. vw v11+v8 3001. vw vl (1,1-C2H2D2) 2961.9 vw 2960. vw v2+v12 (C2H3D) 2950. vw V1+V10 2886.5 8 2885. m v2+v12 2665.0 w 2667. vw 20 12 179 Infrared Raman Assignment 2595.9 W v2+v6 2555.9 vw v2+vu 2591. w 012+v3 2907. vw v (C D ) 2909. vw 3 2 2 2373. vw 2372. w v6+vl2 2339. VW V2+V8 2329. vw 2329. vw V9+V12 2286. vs 2287. vs vl 2278. vw 2278. w v1 (130) 2277. w 2V10+VA 2272. w v (trans-C H D ) 2269. w 1 2 2 2 2265. vw 011 (trans-C2H2D2) 2263. vw v3+v6 2261.8 w 22 v11 (C2H3D) 59. w 2292. s,asy 2291. s,asy v11 2213. s 2213. s v2+v10 2199. vw v3+vu 2065. vw v3+v7 2020. vw V6+Vu 175 Infrared Raman Assignment 1996.1 w,asy 1995.3 w V10+v12 1965.9 W v3+v8 1958.9 w 20“ 1906.0 w 2V8 (C2H3D)? 1881.0 w v6+v7 1868.6 vw V3+V10 1831.5 s,asy vu+v7 1799. w vu+v8 1700.9 vw 2v7 1:99;: :W 1691. w v6+v1° 1223:? 21:: 1612.1 m,asy v7+v8 1597.8 w 1595. m v2 (C2H3D) 1589. VW 2v8? 1569.0 m,asy 1563. vw v 1560. vs 2 1591. w v2 (13C) 1938.0 vw v8+v10 1396.8 3 1393. m v12 (C2H3D) 176 Infrared Raman Assignment 12 _13 1386. vw v12 (H2 C- CHD) 1338. vs 1335. vs v12 1331. w,sh 1333. w 012 (130) 1327. m 1325. m 2v10 1295. m 0 (trans—C H D ) 1292. m 12 2 2 2 1286. w 1285. m v3 (C2H3D) 1283. w 1282. m 1280. m 03 (trans-02H2D2) 1218. w,bd 1219. vs v. 1205. w,sp 1202.0 vs 3 1121. v6 (C2H3D) 1039. m,asy 1036. m v6 1033. m 1026. w 06(130) 1011. m v (C H D) 1003. m u 2 3 1001. m v“ (trans-C2H2D2) 997. vw ? 991. m,asy 989.0 sh 985. S V” 982. 177 Infrared Raman Assignment 952.1 s 952.9 w ”8 (1,1-C2H2D2), 887.0 vw ? 879.2 vw ? 899.9 vs 850.8 vw 07 892.8 vs 812.8 m 810.2 vw 809.6 m V7 (CZH3D) 766.6 m 761.8 m 08 753.1 vw v7 (1,1-C2H2D2) 790.0 vw v5 (C2H2) 735.8 vw 731.9 m 729 O m v7 (trans-C2H2D2) 702.0 vw v5 (C2HD) 687.0 vw ? 6 668.0 vw v10 (trans-C2H2D2)? 663.1 m 660.7 010(130) } v10 658.9 s 652.0 vw ? 556.9 w v5 C2D2 596. 2 w trans-023222 178 Infrared Raman Assignment 3190.0 w V8+V11 3129.1 m 2v2 3121.2 m 3081.0 vw 09 (C2H3D)? 3052.1 s 3098.0 5 v9 3092.9 w 3099. w 207+v2,v9 3030. 5 3026.9 m vs 3025. m 13 3020. w v5 ( C), 2v7+v2 3013. w vl+v7, 2v8+v3 2876. W 2V6+V8 2897.0 w v2+v12 2836. w v2+v3 2562.2 vw v2+vu 2553. w 203, v2+v6 2907.0 m 2v +v 2u02.0 w 8 10 2395.0 w Infrared Raman Assignment 2338.0 w 2v10+vu 2325. 05 (1,1—C2H2D2) 2299. m 2289. 2265. 2263. 2251. 2213. 2195. 1999. 1905. 1866. 1860. 1850. 1662. 1597. w,sh sh bd,asy 2271.6 2268.9 2252. 1961. 1798.2 w 1598.7 w V6+V12’ v2+v7 ”1 V11 V3+Vu, V3+V6 V2+V10 13 v2+V10 ( C)’ 3v7 v10+V12 v3+"10 ? V6+V8, ? ? VD+V8 2V8 ”6+”10 V2 (C2H3D) 180 Infrared Raman Assignment 1563.8 vw,sh ”2 1562.0 3 13 1591.3 w v2 ( C) 1396.8 m v12 (C2H3D) 1373.0 w,sh ? 1398.9 w 2le 1298.0 S 1296.0 8 v 1292.9 8 12 1290.6 5 1289.5 m 1282.3 w v3 1275.7 s 13 1273.5 v3 ( C) 1279.0 w v (130)9 1125.0 w 1098.0 w vu+v5 (C2D2 -° 1060.0 w 1006.5 sh 1009.1 S v“ 991.0 s 985.9 s 1005.7 m 1002.2 m V6 995.5 s 181 Infrared Raman Assignment 952.2 w “8 (1,1-C2H2D2), 08 (C2H3D) 873.8 s 869.7 w v8 869.6 m 859.9 w 08 (13c) 813. w v (C H D) 810. w 7 2 3 732. s 725.6 S v7 721.1 S 687. vw ? 671. s v 667. s 10 565.0 w 556.0 m v5 <02D2). “u (CZHD) 596. m 4—1 1" 929-2 3160.7 Sh v +v , 2v 3153. m 3151.0 m 8 11 2 3080. 3 3078.3 vs ”9 3078. s 3066. w 3067.8 vw 09 (H213C=12CD2) 3098. w 3097.9 w v5 (C2H3D) 3099. w 3093.9 W 182 Infrared Raman Assignment 3009.8 w 3011. V1 (02H3D) 3000.6 sh v1 2998. 5 2999.3 vs 2993. w 2993. w 01 (H213C=12002) 2988. vw 2989. w v7+vll 2973. vw v11 (C2H9) 2960. w 2960. w v2+vl2 (C2H3D) 2937. m 2938. v2+v12 2897. vw vlo+v12 2705. w 2706. vw v2+v6 2555. vw v2+v8 2512. vw V6+v12 2906. vw 03+012 2390. w v2+v7 2335. W V8+V12 2328. vw 2010+08 2322. 3 2322.6 vs v5 2309. w 2305.7 w 05 (H212C=13CD2) 2276. 183 Infrared Raman Assignment 2262 1 w 2261.6 w ' v (C H D) 2260.2 w 11 2 3 2251. m 2252. m V2+V10 2216. s 2217.9 vs 011 2209. w 2205. w v11 (H2 — 2 2066. vw 2068. vw 2v3 2059. vw 2059. vw 010+012 203a. VW Vu+V6 1906. s,bd 208 1839. m 1838. vw V9+v8 1818. m v6+v10 1798. vw v3+v7 1708. vw v7+v8 1596. w 1596. m 02 (C2H3D) 1577. m 1575. vs v2 12 13 1569. w 1569. m v2 (H2 C= CD2) 13 _12 1552. w 1552. m 02 (H2 C- CD2) 1505. S 2v7 1985. vw 207 (13C) 1937. m v12 (C2Hu), v7+v10 189 Infrared Raman Assignment 1396.8 8 1396.6 w 012 (C2H3D) 1379.8 1382.6 5 v 12 1375.8 vs 1369. 8 1371.0 m 012 (H212C=13CD2) 1368.9 m 2010 1335. w 1339.2 w 012 (cis-C2H2D2) 1295. vw v (trans-C H D ) 1291. vw 12 2 2 2 1285.0 m 1285.5 m v3 (C2H3D) 1282. m 1283.1 m 1139. vw 1192.9 w v 6 1138.2 w 1029. s,bd 1029.1 vs v 1017.5 vs 3 1010. s v (C H D) 1002.9 8 9 2 3 989. w ”9 (trans-C2H2D2) 996. vs,asy 955.3 sh 951.5 sh v8 (H212C=13CD2)} 08 997.7 s 921. w ? 905. vs v“ 901. vs 185 Infrared Raman Assignment 852. 813.2 810.0 753. 799. 792. 730- 729 o 729. 705. 701. 688. 685. 682. 680. 660. 656. 597. 556. 596. 531- 523. 010000 VW VW VW VW S 810.9 vw 752.8 s,asy V7 (cis-C2H2D2) v7 (C2H3D) "7 v (H 12C=13CD2) 7 2 010 (C2H3D) v7 (trans-C2H2D2) v (C2HD) 5 13 =12 V10 (H2 C CD2)}V10 ? v5 (C2D2) + v“ (C2HD) 186 9.28.123 Infrared Raman Assignment 3205. vw v6+v11 3198. W V1+V8 3085. w 3088. m 202 3081. s 3083. 3 3073.1 m 3076.0 8 V9+V5 3068.2 m 3071.3 m 3059. w 3060. vw vs+v6 3051. w v (trans-C H D ) 3096. vw 9 2 2 2 3093. vw v9 (cis-02H2D2) 3035. s 3036. 8 3032.1 5 3032. s v9 2999. w 01+07, v1 (1,1-C2H2D2) 2399. vw 2v7+v8 2390.9 m 2338.1 m v3+”12 2320.2 S 2319. S vs 2309 0 12 -13 . w 2303.1 w v5 (HD C- CD2) 2293. vw v6+v12 2285. vw 2287. vw v1 (cis—C2H2D2) 187 Infrared ‘Raman Assignment 2269.2 s 2269.6 s 2266.3 s 2266.3 s ”1 13 _12 2255. vw 2255. w vl (HD C- CD2) 2297. m 2vlo+v6 2291. 2216. 2210.1 2199. 2188. 2153. 1916. 1907. 1890. 1858. 1855. 1850. 1836. 1700. 1695. VW VS VW VW VW VW 2292. 2217. 2211. 2201. 2167. 1917. VW VW VW v11 (cis-C2H2D2) 011 (1,1-C2H2D2), v8"”12 011 12 13 211 (HD C= CD2) “11 (C2D9) v2+"10 3127 V6+V8 +0“ (1,1—C2H2D2) “3 3"10 2V8 V8+V9 (trans-C2H2D2) V3+Vu v6+v7 188 Infrared Raman Assignment 1688.2 vw V9+V8 1689.8 vw 1663.0 m v +v 1657.0 m 7 8 13 1635.9 vw v7+v8 ( C)? 1623.9 Sh + v 0 1618.8 w 6 10 1582.2 VW ? 1578.9 w v2 (1,1-C2H2D2) 1593.3 vw 1596.1 s v 1592.6 w 1592.8 3 2 1531.0 s 1530.8 m 2 v9 1527.6 m 1527.5 m 1522.8 vw 1519.0 sh v2 (020“), vu+v7 1512.2 m 1999.1 m 2v7 1991.1 m 1378.6 m 1381.3 w 012 (1,1-C2H2D2) 1368.9 vw 2v10 (1,1-C2H2D2) 1336.1 m 012 (cis-C2H2D2) 1295.1 m v (trans-C H D ) 1292.2 m 12 2 2 2 1293.0 m ? 189 Infrared Raman Assignment 1286.0 vs 1287.1 3 v12 1283.0 vs 1289. 3 1277.5 W,Sh V12 (13C)? 1275.5 w 1258.0 m 1262. vw 2le 1217. VW 03 (cis-C2H2D2) 1073.0 m 012 (C2Du) 1096. s 1091. 3 v3 1039.2 vs 1039. s 1032.8 w v3 (HD13C=12CD2) 1002.8 w 1002.9 m 993.2 w,asy 996. w v6 989.5 W,sp 991. s 982. vw v3 (CZD9) 952.2 m 959. vw V8 (1,1-C2H2D2), V8 (C2H3D) 932.0 vs 933. m 921.2 vs 923. sh V8 921. s 916.1 w,sh v8 (HD13C=12CD2) 852.9 m v7 (cis-02H2D2) 781. VW v8 (C2D9) 767.6 vs 769. sh v9 765. 190 Infrared Raman Assignment 735. s 729. s v7 720. s 660. vw V10 (cis-C2H2D2) 628. w 627.5 w 626. w 629. ”10 APPENDIX B Hoshen and Jortner have applied the coherent potential approximation to the electronic states of substitutionally disordered molecular crystals.62’63 The following discus- sion follows their treatment. The following assumptions are made for the binary mixed crystal: Assumption (1). The sites of the perfect empty lattice are substituted by the two components without any change in molecular orientation. Assumption (2). The tight-binding limit applies for the description of the excited states of the crystal; that is the excitations are localized. Assumption (3). Crystal field mixing effects between different states of the crystal are negligible. Assumption (9). The environmental shift terms, Df, for the two components are equal and concentration inde- pendent. Assumption (5). The intermolecular transfer matrix elements are invariant under isotopic substitution. (The Lia (g)'s). Assumption (6). The two components differ by only their gas-phase excitation energies and are characterized 191 192 by the same wave-functions. Assumption (7). Molecule-Molecule spatial correla- tions are neglected. The properties of a configuration- ally averaged system are considered. The Hamiltonian of the mixed crystal is written as: H = H + H (B—l) where H0 = X 1 Ina> E represent the wave functions in the tight-binding limit. J accounts for the intermolecular excitation transfer. H1 expresses the deviation of the crystal from the 193 virtual crystal as follows: H1 = A g §|na> gna = <1/(z-H)> = 1/(z-Heff(z)) (B-7) The above equation defines an effective Hamiltonian, 199 Heff’ which characterizes the properties of the configura- tionally—averaged crystal. The effective Hamiltonian can be expressed in terms of the virtual crystal Hamil- tonian and the complex self energy, {(2), with respeCt to the virtual crystal: Heff(z) = HO + {(2). (B-8) [(2) expresses the deviation of the mixed crystal from the virtual crystal; it describes the shift and broadening of the states of the virtual crystal. The poles of (z- Heff(z))-ldetermine the widths and positions of the con- figurationally averaged crystal states. It is now assumed that the local nature of the self- energy operator can be approximated as: A = oa(2) 6 (B-9) anaB’ where oa(z) is the self-energy at site a. If the sites are interchange equivalent, 00(2) = 0(2) (B-lO) for all a. V expresses the generalized perturbation resulting from the deviation of the mixed crystal from its 195 configurational averaged behavior as follows: eff = Z Vna’ (B'll) where the Vna are the local perturbations. Excitation scattering due to this deviation can be expressed as: T = V + VT, (B-12) and the relation between the mixed crystal Green's func- tion G(z) and the configurationally averaged Green's function is given by: G(z) = + T . (B-13) The exact condition for ignoring molecule-molecule spatial correlations is: = 0. (B-l9) This makes the coherent potential approximation self— consistent. If multiple scattering contributions to T can be neglected then Equation (B—l9) can be approxi- mated by: = 0, (B-15) 196 where tna is a molecular scattering matrix associated with site na, given by: tna = (1-Vna(z))-1Vna(z). (B-16) Thus, for a binary mixed crystal, _ -1 CAtA + CBtB — CA(1-VA) VA (B-l7) -1 _ + CB(1-VB) VB — 0, where VA = (ACA-o)|na>VB. (B-19) Therefore, 6(2) = (ACE-G(z))r°(z-ELo(z))(AcA+o(z)), (B—20) where f(z) is a complex function such that the density of 197 states is given as follows: 0(3) = n‘1 Imf(E-iO_1), (B-2l) f(z) is given by: f(z) = (N0 )‘1Tr = dE' 0(8')- (B—22) p —w z-E' The superscript zero in Equation (B-20) indicates the complex function is related to the pure crystal density of states. The spectral density, which can be considered to ex- press the 3 components of the density of states and thus is intimately connected to the optical properties of the crystals, may be expressed as: S(g,j,E) = n‘1 Im|g,d>, (B—23) where J labels the Davydov components. In terms of the self- energy, 0(2), the spectral density becomes: ImO(E-iO+) s(k.J.E) = < ‘1) .(B-29) ” 2 [E-E;tJ-Reo(E-10+)]2+[Imo(E-10+)l2 where tJ is the energy of the Jth Davydov component of the pure crystal. A plot of S(O,J,E) versus E reveals the absorption 198 pattern for the band of the binary-mixed crystal. The CPA is attractive because it does not have the limitations of the virtual crystal approximation and the average T-matrix approximation, and it requires a know- ledge of only the pure crystal density of states function and the position of the Davydov components of the pure crystal in order to calculate the mixed-crystal spectrum. APPENDIX C The following is a listing of program MIX: PROGRAM MIX(INPUT,OUTPUT,TAPE5=INPUT,TAPE6=0UTPUT) REAL INCH, IMSIG DATA D/lH,lH+,lH0/ 1000 FORMAT(7F10.2) 11 FORMAT('lCA=', F5.2,5X,'DEL=',F10.15X,'T='F10.1/'0') 1 READ(5,1000)CA,T,DEL,Zl,ZN,INCR,EPS IF(EOF(5)) 2000,2 2 WRITE(6,11)CA,DEL,T X=l./3. I=0 CB=1.0-CA Pl=T**2+(CA*DEL)**2-9.*CA*CB*DEL**2 01=CA*CB*(CB-CA)*(DEL**3)*2. R1=(CA*CB*(DEL**2))**2 10 P2=2*(Z1+CA*DEL) IF(P2.EQ.O.) GO TO 5 P=(Pl-Zl**2-2*Z1*CB*DEL)/P2 Q=Q1/P2 R=Rl/P2 A=(3.*Q-P**2)/3. B=(2.*(P**3)—9.*P*Q+27.*R)/27. CK=B*B/9.+(A**3)/27. IF(CK.GT.0.0)G0 TO 100 5 I = 1+1 E(I)=Zl SPEC(I)=0.0 Z1=Z1+INCR IF(Zl.GT.ZN) 00 TO 201 00 T0 10 199 100 50 201 101 51 109 200 A1=-B/2.+SQRT(CK) B1=-B/2.-SQRT(CK) IF(A1.LT.O.)AA=-(ABS(A1)**X) IF(B1.LT.0.)BB=-(ABS(B1)**X) IF(A1.GT.0.)AA=A1**X IF(Bl.GT.O.)BB=Bl**X IF(A1.EQ.O.)AA=0. IF(Bl.EQ.O.)BB=0. RESIC=0(AA+HB)/2.-P/3. IMSIG=ABS((AA-BB)*1.73205/2.) I=I+1 ALPHAXX=IMSIG/((Zl-EPS—T-RESIG)**2+IMSIG**2)/3.19159 BETAXXX=IMSIG/((Zl—EPS+T-RESIG)**2+IMSIG**2)/3.19159 SPEC(I)=ALPHAXX+BETAXXX F(I)=Z1 IF(I.GE.1000) GO TO 101 Z1=Z1+INCR 00 TO 10 FORMAT ('E=',FlO.l,5X,'SPECTRAL DENSITY=',E10.3) IF(I.EQ.O) GO TO 1 D0 51 J=1,I WRITE (6,50)E(J),SPEC(J) CONTINUE EMIN=E(1) YMAX=SPEC(1) DO 3 J=2,I IF(SPEC(J).LT.YMAX) 00 TO 3 YMAX=SPEC(J) CONTINUE WRITE (6,109)YMAX FORMAT(19H1 THE MAX Y VALUE IS, 2x,E15.7//) D0 52 J=1,I D0 9 KK=l,l2l YP(KK)=D(1) YP(1)=D(2) 105 52 2000 201 YC=(SPEC(J)/YMAX)*120.+.5 NC=IFIX(YC) NP=NC+1 YP(NP)=D(3) WRITE (6,105) E(J), (YP(K),K=1,121) FORMAT (1H,F6.2,2X,l2lA1) CONTINUE GO TO 1 STOP END REFERENCES 1. 10. ll. 12. 13. 19. REFERENCES G. Turrell, Infrared and Raman Spectra of Crystals, (Academic Press, New York, 1972)} P. M. A. Sherwood, The Vibrational Spectroscopy of Solids, (Cambridge University Press, Cambridge, 197277_—_ M. M. 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