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Intermolecular Forces, Exciton Splittings And Lattice Vibrations Of Crystalline Pyrazine

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Intermolecular Forces, Exciton Splittings And Lattice Vibrations Of Crystalline Pyrazine

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Content INTERMOLECULAR FORCES, EXCITON SPLITTINGS AND LATTICE VIBRATIONS OF CRYSTALLINE PYRAZINE by Lina Hsu A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Chemistry) January 19?1 HSU, Lina, 19^3- INTERMOLECULAR FORCES, EXCITON SPLITTINGS AND LATTICE VIBRATIONS OF CRYSTALLINE PYRAZINE. University of Southern California, Ph.D., 1971 Physics, spectroscopy University Microfilms, A X E R O X Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED UNIVERSITY O F SO U TH ERN CALIFORNIA T H E G R A D U A T E S C H O O L U N IV E R S IT Y PA R K L O S A N G E L E S , C A L IF O R N IA 9 0 0 0 7 This dissertation, written by .............Lina^Hsu.............. under the direction of Dissertation C om mittee, and approved by all its members, has been presented to and accepted by The Gradu ate School, in partial fulfillment of require ments of the degree of D O C T O R OF P H I L O S O P H Y CJLa Ajjj. v Dean B a te J a »... 19 71 DISSERTATION COMMITTEE ACKNOWLEDGMENTS It would not have been possible for me to learn spectroscopy and carry through this work without the help and encouragement from Professor David Dows to whom I am ienormously grateful, I am Indebted to my family for their everlasting encouragement across the miles through the period of the graduate studies, I wish to thank Chemistry Department of University of Southern California for teaching asslstantships and fellowships. Dean of the Graduate School Special Fellow ship is also appreciated. CONTENTS Page ACKNOWLEDGMENTS...................................... 11 LIST OF TABLES....................................... iv LIST OF FIGURES...................................... Vll Chapter I. INTRODUCTION................................. 1 II. THEORY....................................... 5 (A) INTRAMOLECULAR VIBRATIONAL SPECTRA....... 7 (B) LATTICE VIBRATIONS....................... 14 III. EXPERIMENT............................... .... 17 IV. RESULTS AND DISCUSSIONS...................... k l APPENDIX CARTESIAN DISPLACEMENT CALCULATIONS.......... 117 REFERENCES................................. 1^1 111 LIST OF TABLES Table Page 1. Vapor Pressures of Pyrazine................. 23 2. Potential Parameters for the “Best” set....................... 72 3. Deviations of the Calculated Molecular Position and Lattice Constants from the Observed Values......................... 72 Lattice Frequencies of Pyrazine............. 73 5. Normal Coordinates of Lattice Vibrations of Pyrazine for the "Equilibrium" Structure...... 7^ 6 . Number of the Nonbonded Atom-atom Contacts per Molecule for the "Observed" Structure........... 82 7. Calculated Factor Group Splittings for In-plane Modes of Pyrazine-d0 ............... 92 8. Calculated Factor Group Splittings for Out-of-plane Modes of Pyrazine-dg........... 93 9. Calculated Factor Group Splittings for In-plane Modes of Pyrazine-d^.......... 9^ 10. Calculated Factor Group Splittings for Out-of-plane Modes of Pyrazine-d^........... 93 11. Calculated Frequency Shifts for In plane Modes of Pyrazine-d0.................. 98 12. Calculated Frequency Shifts for Out- of-plane Modes of Pyrazine-dg,., ........... 99 13. Calculated Frequency Shifts for In plane Modes of Pyrazine-dj^.................. ^00 1^. Calculated Frequency Shifts for Out- iv Table Page of-plane Modes of Pyrazine-d^................. 101 15. Calculated Splittings and Frequency Shifts as Function of R for In-plane Modes of Pyrazine-d0 with the "Observed" Structure 103 16. Calculated Splittings and Frequency Shifts as Function of R for In-plane Modes of Pyrazine-d- with the "Equilibrium" Structure.................................... 10^4- 17. Calculated Splittings and Frequency Shifts as Function of R for In-plane Modes of Pyrazine-d/^ with the "Observed" Structure 105 18. Calculated Splittings and Frequency Shifts I as Function of R for the In-plane Modes of Pyrazine-d^ with the "Equilibrium" ; Structure ..... 106 ! 1 9. Calculated Factor Group Splittings for In plane Modes of Pyrazine-dg Using Set II of the Potentials 1 0 8 I 20. Calculated Factor Group Splittings for Out-of-plane Modes of Pyrazine-dQ Using Set II of the Potentials............ 109 21. Calculated Factor Group Splittings for In plane Modes of Pyrazine-dji, Using Set II of the Potentials. ..... 110 22. Calculated Factor Group Splittings for Out-of-plane Modes of Pyrazine-d^. Using Set II of the Potentials.................... Ill 23. Calculated Frequency Shifts for In-plane Modes of Pyrazine-d- Using Set II of the Potentials 112 j 2^» Calculated Frequency Shifts for Out-of- plane Modes of Pyrazlne-d0 Using Set II of the Potentials............................ 113 25. Calculated Frequency Shifts for In-plane j Modes of Pyrazine-dj, Using Set II of the j Potentials. .*«••«• 114 j 26. Calculated Frequency Shifts for Out-of plane Modes of Pyrazine-d^ Using Set II v Table Page of the Potentials........................... 115 27. Cartesian Displacements for In-plane Modes of Pyrazine-dQ......................... 121 28. Cartesian Displacements for In-plane Modes of Fyrazlne-d^........................ 125 2 9. Out-of-plane Force Constants for Pyrazine..................................... 132 30. Fundamental Frequencies for Out-of- plane Modes of Pyrazine-dQ and Pyrazlne-djj,.................................. 133 31. Cartesian Displacements for Out-of- plane Modes of Pyrazine-d0................... 13*H 32. Cartesian Displacements for Out-of- plane Modes of Pyrazine-d^................... 135 33. Cartesian Displacements for Fg Species of Adamantane........ 137; vl LIST OP FIGURES Figure Page 1. Apparatus for Measuring the Vapor Pressures ...................... 20 2. Vapor Pressures of Pyrazine as Function of Temperature ..................... 22 3. Polarized Infrared Spectrum for the Biu Fundamental of Pyrazine-do Near 1492 cm"1 ........................... 27 4-. Polarized Infrared Spectra for the Biu Fundamentals of Pyrazlne-do Near 1132 cm” 1 and Pyrazine-dij. Near 1020 cm” 1 ........................................ 29 5. Polarized Infrared Spectra for the Biu Fundamentals of Pyrazlne-dg Near 1021 cm” 1 and Pyrazlne-dk Near 868 cm” 1 ..................7.................... 31 6. Polarized Infrared Spectrum for the B3U Fundamental of Pyrazine-do Near 1426 cm” 1 .......................... 33 7. Polarized Infrared Spectra for the B3U Fundamentals of Pyrazine-do Near 1075 cm"1 and Pyrazine-dij, Near 836 cm” 1 ........................................ 35 8. Polarized Infrared Spectra for the b2u Fundamentals of Pyrazine-dp Near 800 cm” 1 and Pyrazine-di* Near 609 cm” 1 ................................... 37 9 . Polarized Infrared spectra for the b2u Fundamentals of Pyrazine-dp Near 4l4 cm"1 and Pyrazlne-dzj, Near 40^ OB-l .............7................ 39 10. Number of Nonbonded Atom-atom Contacts as Function of Summation Radius Cut off ......................................... 45 _______________________________ yll______________ __________________ Figure Page 11. Calculated Crystal Energy of Pyrazine as Function of Summation Radius ............ ^9 12a. Calculated Lattice Frequencies with Various A Values of N— H Interaction...... 55 12b, Calculated Crystal Energy and Frequency Deviation as Function of A Values of N-H Interaction......... 57 13a. Calculated Lattice Frequencies with Various B Values of N H Interaction...... 59 13b. Calculated Crystal Energy and Fre quency Deviation as Function of B Values of N-H Interaction ........... 6l 14-a. Calculated Lattice Frequencies with Various A Values of H H Interaction...... 63 l^b. Calculated Crystal Energy and Fre quency Deviation as Function of A Values of H-H Interaction ............ 65 15a. Calculated Lattice Frequencies with Various B Values of H H Interaction...... 67 15b. Calculated Crystal Energy and Fre quency Deviation as Function of B Values of H-H Interaction ................. 69 16. Calculated Lattice Frequencies vs. R for the "Observed" Structure ............. 76 17. Calculated Lattice Frequencies vs. R for the "Equilibrium" Structure .......... 78 18. Intermolecular Force Constants vs. R ........................................... 81 19. Calculated Lattice Frequencies vs. R Using Set I of the Potential for the "Observed" Structure ................... 85 20. Calculated Lattice Frequencies vs. R Using Set I of the Potential for the "Equilibrium" Structure ........ 87 vili CHAPTER I INTRODUCTION In heterocyclic molecular crystals, where weak hydrogen bonding Is expected, the nature of intermolecular Interactions has not been well known. This work Is dedicated to the study of the Intermolecular forces of jcrystalline pyrazine to give clues to tackling problems Involving crystals of similar molecules, A useful Intermolecular potential model with proper parameters is vital for this task. The potential energy summed over the molecules in the crystal gives the lattice energy. The second derivatives of the potential are the force constants required in the studies of the intermole cular Interactions. The details of these correlations are seen in Chapter II. The potential tested is to be Judged by its ability to predict lattice energy, Intramolecular vibrations and lattice vibrations of the crystal close to the experimental properties. The atom-atom pairwise potentials of the 6-exp type which closely explain the intermolecular interactions of aromatic hydrocarbons (1) were tested on pyrazine. The crystal structure observed by Wheatley (2) was refined to give minimum lattice energy for the potential assumed in order that the condition of zero net force and torque could be obeyed. Among the properties of lattice energy, Intramole cular vibrations and lattice vibrations, the first one gives the most direct measurement of the quality of the potential assumed. The calculated lattice energy is compared with the measured enthalpy of sublimation ( neglecting zero-point energy and entropy effects). No thermal data are available in the literature for pyrazinei vapor pressures were measured in this laboratory at a range of temperatures (see Chapter III) to give AHsul:) for comparing with the calculated crystal energy. The lattice vibrations of the molecular crystal yield the next most direct information concerning Inter molecular forces. Five llbratlonal bands in the Raman spectrum have been observed by Ito (3)i and two bands In the far-infrared spectrum by Gerbaux (4), These frequen cies were referred to give a further check on the potential ; model used when the lattice vibrations were calculated by the GF matrix method. The computation procedure is out lined in section B of Chapter II. The lattice energy and the lattice vibrations were evaluated for both the "observ- i edM and refined "equilibrium" structures for comparison. j The analysis of the intramolecular vibrational frequencies of molecular crystals also provides Important : i Information about the intermolecular Interactions. Dows I (5)* Mltra and Giellsse (6), Vedder and Hornig (7) and references cited by them have reviewed this subject. The theory developed is briefly mentioned in section A of Chapter II. The alterations of molecular vibrational spectra on going from gas phase to crystalline solid, namely, the band splittings and the frequency shifts, reveal how well the potential model of the Intermolecular interactions assumed suits the molecular crystal attacked. The calculated exclton splittings and frequency shifts were compared to spectra of polycrystalline samples ob served as described in Chapter III. Since the experimental ; results of the exclton splittings and frequency shifts are not complete, due to the failure of single crystal growing, ; the determination of potential parameters depends on the computations of the lattice energy and the lattice vibra tions. The resulting "best" potential was then used to calculate the exclton splittings and the frequency shifts. These changes of molecular spectra from gas to crystal states were also evaluated for both "observed" and refined "equilibrium" crystal structures for comparison. Difficulties Involved in selecting the "best" potential for this heterocyclic compound under variations j | of conditions, namely, various summation radius cut-offs, including (or not) the first derivatives of the potential in obtaining the intermolecular force constants (see | ;arguments in Chapter II) and various suitable "equilibrium" j 4 structures (such that the refined closest N-~-H Intermole cular distance does not get too short) are all displayed in Chapter IV. The comparison of all the calculated results to the observed values, If available, convinces us that the methods exploited in the computations are generally correct and that the intermolecular attractive part of the :force constants of hydrogen bonding plays an Important role. Details of comparison and discussions are also given in Chapter IV. CHAPTER II THEORY First of all, knowledge of the crystal structure is necessary for determining the selection rules for vibrational spectra in crystals. The gas phase spectrum is also required to have been treated well by normal coordinate analysis so that the solution is provided for the unperturbed case. The appendix of this dissertation is to contribute the required normal coordinates. The "molecular crystal" is one in which the mole cules preserve their individuality in the first approxima tion. This is because the interaction forces between molecules (intermolecular) are small in comparison with the forces acting between the atoms of a single molecule ( intramolecular). Therefore, all possible atomic vibrations in molecular crystals may be divided into two groups» 1, Internal vibrations of the atoms of a molecule relative to one another. In these vibrations, the center of gravity of the molecule is not displaced and there is no rotation of the molecule as a whole. The crystal with N molecules, each with n atoms, possesses N(3n-6) intramolecular vibrational degrees of freedom. 2. External vibrations, or lattice vibrations, which appear because of the rotational and translational degrees of freedom of the mole cule, There are 6n lattice degrees of freedom of the crystal, of the 6n modes, three acoustic frequencies are zero roots represent ing the translational motion of the crystal as a whole. The two groups of frequencies usually (and in particular for pyrazine) separate quite well. Due to the intermolecular interaction, the absorp- j tlon of light results in waves of excitation passing through the crystal as pseudo-particles with crystal momentum vector k. During the process of destruction of a photon and creation of an exclton, the crystal momentum must be conserved. In the visible or infrared region, the wavelength of a photon is very much longer than the unit cell dimension. Absorption of a photon thus leave the crystal pseudo-momentum essentially zero on this scale. With this selection rule, the motions in all the unit cells ; j of the crystal are in phase with one another. ' At times, the influence of the crystalline lattice on the internal vibrations of each molecule leads to a ; I change of selection rules and polarization, as well as to j j small displacements and splittings of the frequencies of ! i internal vibrations. All these changes are relatively J small (especially for vibrations with little or no change in dipole moment) relative to the internal vibration frequencies. The previously developed theory (5. 6 . 7. 16, 17) which accounts for the Intermolecular coupling and lattice vibrations of molecular crystals is briefly outlined here simply for convenience. The application to the nitrogen- heterocyclic compound, pyrazine, in detail, will be given in Chapter IV. (A) Intramolecular vibrational spectra In treating the real crystal, the "oriented gas" molecules (8) are considered to Interact according to some appropriate crystal potential through two first-order stages of refinement. The effects of the two refinements are of the same order of magnitude, and should be consider ed simultaneously. The two effects are (considering a single normal mode frequency)* (1) the static crystalline field on a single molecule at the origin site. This Involves a term in the second derivative of the inter- 8 2 V molecular potential V, Q g which will enter the equations for transition energies later on. Qjj is the normal coordinate of the origin molecule n, (2) the resonance interactions of exchanging energies between like vibrational energy levels with 9 ^ V neighboring molecules. This concerns terms " 9q~— ^Tq-* where ^ is the normal coordinate of the mth neighboring molecule including translationally equivalent and non equivalent ones. The type (1) effect plus the exchange process between translationally equivalent molecules governs the frequency shifts between gas and crystal spectra. The rest of the effects in (2) result in the formation of a ;set of N non-degenerate levels from each isolated molecule level if there are N molecules in the crystal. (In general there may be some degeneracy left if the crystal symmetry is high enough). The type (1) effect will also remove degeneracy present in the free molecule if the site symmetry is low enough. Second-order effects, in particular Interactions between different normal modes, are neglected in this work. Formal Theory It is natural to assume that the crystal potential energy is that of the isolated molecules plus a small perturbation V, and that a small additional energy term E* will have to be added to each energy. Applying perturbation theory to the isolated molecule k, which has exact solutions Ef e and from the Schrodinger equation Hk ^ - Ek for its vibrational energy levels, the energy levels of the crystal will then he the solutions of a Schrodinger equation ( ZHk + V ) $ = ( ZEk + E* )# (1) where Hk is the Hamiltonian operator for vibrational motion : separated by the Born-Oppenhelmer approximation (9)» $ is the wave function of the crystalline collection of oscil lators . i It is the subject of this work to learn about V by solving equation (1) with a potential model and compar ing the calculated E* with the observed value. The crystal potential has the full symmetry of the ! lattice, and the true wave functions for the crystalline states belong to representations of the space group of the lattice. If the "one-site exclton" approximation Is assumed, the Bloch-type function (5) can be used for the basic function to represent a one-quantum excitation with the energy spread over all the molecules occupying a particular set of translationally equivalent sites, and since we will be concerned only with transitions from the ground state in which a single exclton is created, the selection rule of k - 0 mentioned above applies. These i i treatments result In the number of states in the crystal being tremendously reduced so that in a further possible symmetry classification it will be sufficient to consider j 'the crystalline unit cell symmetry. This is because the j .10 motions within any unit cell are In phase with all other cells. The few possible symmetry Is the so-called factor group. The basis functions for solving equation (1) can now be constructed to transform under the representations of the factor group instead of the space group. This is not possible in general when the full k-dependence is retained. The factor group, or the unit cell group, is the group of all cosets of the translation subgroup of the space group plus the translation subgroup Itself as the l identity element. It is the group with operations trans- i i forming molecules among themselves within one unit cell (with glide planes and screw axes Included). The factor :group is always isomorphous with one of the crystallograph- I ic point groups. Using these basis functions and assuming the pair wise atom-atom potential, the energy difference between an excited-state and the ground state of the crystal is (10, 11) E 0 1 - E° - e* - e° + D + Ma j i where a indexes the representations of the factor group, ; £' and e° are the energies of the excited, <p* , and ground j j state, <£°, of the isolated molecule respectively. £ is I P P ] the basis wave function defined at the beginning of this section for the p^ moleculej is the wave function when X r 1 .U Q t ip molecule is excited. The terms D and M give the 11 crystalline correction on the Isolated molecule energy and take the forms where B is a unitary matrix for the transformation from the unperturbed normal coordinates of the molecules in the !unit cell to the symmetry coordinates of the factor group, iThe sums are over lattice sites, and t is the number of equivalent sites in a unit cell. The molecule p is on i site a, and r is on b, Vpr stands for the perturbation j potential between molecules p and r, if the perturbation potential V is expressed as a sum over pairs of molecules, V = | I I Vpr# Palrwlse assumption, D is of the type (1) effect, Ma is of type (2) and is composed of two ; parts, with the first summing over molecules translational ly equivalent to p, the other slimming over all other mole cules. Hence, D and the first part of Ha contribute shifts: ot to the vibrational energy, the second part of H causes j the factor group splittings as described before. Taking the usual assumption of the harmonic s oscillator wave function for < 4 in this vibrational i P i : ^ i analysis, and expanding the potential energy in terms of ; j 1 | the normal coordinates, to second order, equations (3) and (4-) "become 2 D = I (--.YpF ) ( h/ 8 TT 2 \> C ) ( 5) 3Qp ° 3 2 v = t X B * B (-----— — ) (h/ 8 ir 2 v c) (6 ) r aa otb 3q 9 o o where h and c are the fundamental constants and i > 0 Is the frequency in wavenumbers of the isolated molecule. Up to here, the frequency shifts and the factor group splitting of the vibrational level can be calculated as soon as the second derivative terms are obtained. The matrix element can be computed directly. In evaluating the derivative terms, the potential is expressed in the form consisting of central forces acting between the non-bonded atoms as Dows developed ( 5» 12, 13) vpr = i f Vip,jr <rij> where r^j is the interatomic distance between atom i and J of molecules p and r respectively. With this assumption on the potential, the derivative needed In equation (5) andj 13 } (7a) 3rl2 9«n 1 V__ v - - ;o = Z [<d V ) + a<*n8S » ” 1,3 arlJ2 ° 3ri 8Qn 9rJ S13m (!Zp£) ( A l , (Hi) (7b) j,olar1arj,'aQn M 3Qm 'J KfD} It is proved that i!fii = filLA. ari2 ru 3 2 j sin sin ^ cos 0 9ri9rJ riJ where e is the dihedral angle formed by the three vectors involved, Xij» and- ^ ls the an6le between r^, rij* Physically, 9rij and9rn are direction cosines »£i and can be expressed in any frame. Here for convenience in calculation, they are written In the crystal axis system. 9^i and 9—.1 are the cartesian displacements of the particular modes of molecules n and m. These will be defined and calculated in detail in the appendix, i Chapter IV will list the computed results of [ ' o t I equation (7)» and thus of D and M with specific potentialsj lA V. The percentage contribution of the second term in equa tion (?) (the "first derivative term") will also be given. (B) Lattice vibrational spectra These vibrations can for many purposes be divided into two typesi (1) translational motions in which the centers of mass of the molecules are displaced from their equilibrium positions, and (2) librations for which the centers of mass are fixed whereas the molecules undergo quasi- rotational motions. ; i For a centro-symmetric molecular site, there is complete separation of pseudo-translational and pseudo- rotational motions into different symmetries at l c - 0. The i symmetry coordinates formed from the translational displa cements belong to u-species whereas those formed from the librational displacements are even and belong to g-repre- sentatlons, since the rotational displacement preserves the i center of inversion whereas the translational displacement destroys the center of inversion. j The classical sources for discussion of lattice s i ■dynamics are Born and Huang (1*0, and Ziman (15)• j I ; I Formal Theory : ' i ' As in the method presented by Halmsley (16, 1?) for j jcalculating the lattice frequencies, we will consider only 15 the lattice motions which have a phase factor of unity. It is assumed that the coupling with internal motions is neg ligible, i.e., the molecules of the crystal vibrate as ri gid bodies around the equilibrium configurations. The ge neral theory was well treated by Walmsley (1 7), and Walms- ley and Pople (16), To treat organic molecules with many atoms, the calculations of the lattice vibrations were done by the GF matrix method. This leads to solving the conventional se- icular equation (18)i GF - El = 0 (8) where G is the inverse kinetic energy matrix and F is the potential energy matrix of the system, a . = * i - f f 2c2v2, c light velocity and v is in the wave number. The G and F matrices for the lattice vibrations are built in terms of their most convenient coordinates* G ma trix in molecular-fixed cartesian coordinates and F matrix in intermolecular "internal" coordinates. They are then transformed to the same coordinate system for solving the vibrational problem according to equation (8). Transformations into cartesian symmetry coordinates presented by Shlmanouchl et al. (19, 20) are given for ex ample * F ^ = U T* B* F ^ B T U' 16 a U G ^ U* where X = molecular-fixed cartesian coordinates Y = crystal-fixed cartesian coordinates S = cartesian symmetry coordinates R = intermolecular "internal" coordinates and S = U X R = B Y Y = T X The transformation matrices U, B and T are cons tructed from the symmetry and geometric properties of the crystal. The B matrix elements are the direction cosines of each atom position with respect to the crystal axes. Once F and G matrices are expressed on the same basis set, the problem left is to solve equation (8). An alternative transformation procedure bringing the G matrix to match the F matrix for the same coordinate system can also achieve the purpose of solving equation (8) . As a matter of fact, transforming G^x^ to G^R^ for solv-l ing the secular equation |g^R^F^R^-EA*|=0 is preferred in ! this work. I The details of computation with specific poten- | i i tlals V and the calculated lattice vibrations for pyrazine j iare covered in chapter IV. CHAPTER III EXPERIMENT The vapor pressures of pyrazine were measured for a range of temperatures to provide AHg^. The first section describes the work and furnishes the vapor pressure and AHSUb data. In the second section, the polarized Infrared spectra of crystalline pyrazine-dQ and pyrazine-d^ iwere taken and analyzed to give Information on intermole- cular Interactions. Heat of Sublimation The fundamental thermal properties of the sub stances provide useful data to correlate with crystal vibrations. For instance, the heat of sublimation can give a criterion for the lattice energy calculation with an intermolecular potential specified. No thermal data for pyrazine were found In the literature. The latent heat of sublimation was obtained here from the vapor pressure measurement described below. The vapor pressures of pyrazine was measured several times by warming and cooling the sample slowly through the temperature range of 2^-50°C. Pyrazine was obtained from the Aldrich Co. (melting point 5^-56°C). 18 The sample was purified In a vacuum line by successive sublimations and then trapped In a glass bulb which was used to measure the vapor pressure versus temperature. The glass bulb, connected to a U-shaped tube filled with mercury, was immersed in a water bath. The apparatus Is as shown In Fig. 1. A cathetometer was used to measure the: i mercury levels. The vapor pressure data were treated by the least squares method to fit the polynomial In P = A + B / T. P is the vapor pressure measured, T is the temperature in Kelvin degrees and A, B are the coefficients. Fig. 2 showsj the vapor pressures (In P) versus temperatures (l/T) for the combined results of four separate runs. The solid line shows the polynomial. In computing a table of vapor pressures for pyrazine and its heat of sublimation, the coefficients A and B were taken to be their average values from all the runs using the standard error of each run in the polynomial fitting as the weighting factors. Table 1 gives the vapor pressures, calculated from the polynomial with the averaged A and B coefficients, versus temperatures i in the region of 24-50°C. i In obtaining the heat of sublimation, the simplified Clapeyron-Clausius equation for a solid-vapor transition was used. I i : i d In P AHguv --=---° - (Clapeyron-Clausius equation) d(l/T) R Figure 1. Apparatus for measuring the vapor pressure. 20 ■ > TO VACUUM RUBBER TUBE TO VACUUM OR AIR WATER BATH MERCURY RESERVOIR SAMPLE GLASS VALVE Figure 2. Combined data of vapor pressures of four runs as function of l/T» T the temperature in Kelvin degrees. The solid line is the polynomial lnP=A+B/T fitted by the least squares method.„ A = 21.166 and B = -6.408 x 1 0 J with P in cm Hg. 21 Log P, 0.75 P in cm Hg O ■ ro L n U j o o 0 1 o o ro ro TABLE 1 VAPOR PRESSURES OF PYRAZINE AT TEMPERATURE RANGE OF 24-50°C TEMPERATURE (°C) PRESSURE (cm Hg, With Hg vapor corrected) 24 1.1009 25 1.1845 26 1.2739 27 1.3693 28 1 .4 7 1 2 29 1.5799 30 1.6959 31 1.8195 32 1.9512 33 2.0915 34 2.2409 35 2.3998 36 2 .5 6 8 9 37 2.7488 38 2.9399 39 3.1429 4-0 3*3586 4l 3.5875 42 3.8305 43 4.0882 44 4.3614 45 4.6510 46 4.9578 47 5.2828 48 5 .6 2 6 8 49 5.9910 50 6.3762 24 Therefore, AHg^ Is constant R times the slope of the poly nomial In P = A + B/T, i.e., coefficient B. The computed latent heat is 12.9 ± .1 Kcal/mole. It will he used to compare with the crystal energy as described In next chap ter. Spectra The Infrared spectra of pyrazine-dg, cls-pyrazine- dg and pyrazine-djj, have been reported by Callfano et al. (21) In the vapor and solid states. The dichroic behaviour of the bands was investigated on polycrystalline oriented samples with no temperatures specified, probably at the in strumental temperature. Since this work concerns the com parison between the experimental and the calculated split ting frequencies of the fundamental modes In crystal spec tra, the Infrared spectra of crystalline pyrazine-dQ and pyrazine-dij, were remeasured under high resolution and at various temperatures. A polycrystalline mass in which the microcrystals were all uniformly oriented In one direction was used due to the failure of single crystal growing. The quality of the polycrystalline layer was judged by uniformity of ex tinction In a microscope between crossed polarizer. Such a polycrystalline oriented layer, grown from the melt be tween two KBr windows by controlled cooling, was mounted iln an Air Products Joule-Thompson cryostat. The sample was annealed to minimize crystal strains and their spectral effects. The polycrystalline oriented layers have the dis advantage that the layer thickness is not uniform and the orientation is not always exact. With the aid of the assumption Califano made (21) (the (101) plane is coinci dent with that of the crystalline film), and with the Intensity ratio of the components of the split bands, the crystal axes were roughly located. The crystal symmetry is discussed in Chapter IV. The crystal axis notations for the polarization effect follows Califano's (21) and the correlation diagram (22) relating the molecular symmetry to crystal symmetry for illustrating the crystal selection rules is also given in his paper (21). The polarized spectra were obtained, at liquid H2 and a few higher temperatures as well as at room tempera ture, with a Perkin-Elmer model 521 spectrophotometer. A wire grid polarizer was used to provide polarization. The observed splitting frequencies are listed in the next chapter where they are compared with the calculated results. Bands with most illustrative polarization behav iour among several runs are shown in Figures 3 to 9* The bands of pyrazine-dQ in Figs. 3 "to 8 were from the same sample, Fig. 9 w&s of separate run. Those of pyrazine-d^ were all from the same run. In each figure, the absorp tions of both pyrazine-dQ and pyrazine-d^ are given unless ;that of pyrazine-d^ was not well observed. In each run, Figure 3 Polarized Infrared absorption of B]_u fundamental of pyrazine-do at room temperature. 0° polarized 9 0° polarized 26 27 Frequency (cm"1) 1510 1480 1450 / M 80 c u '60 40 20 Figure 4 Polarized infrared absorptions for fundamentals of pyrazine-dg and pyrazlne-di*.. pyrazine-dgt taken at room temperature 0° polarization 9 0° polarization pyrazine-dij.: taken at 150°K +60° polarization -30° polarization 28 Transmittance (%) Frequency {cm"1 ) 1 1 5 0 M O O Pyrazine - do 20 Transmittance (%) Frequency (cm"1 ) 1000 1050 80 60 40 Pyrazine - 64 ro vo Figure 5. Polarized Infrared absorptions for Biu fundamentals of pyrazine-dQ and pyrazlne-d^. pyrazlne-dg* taken at room temperature --— 0° polarization 9 0° polarization pyrazine-djj,: taken at 78°K — - +60° polarization -30° polarization 30 Transmittance (%) Frequency (cm-') 1050 1000 * ' v r ~ N , r 80 60 40 Pyrazine-do 20 Transmittance (%) Frequency (cm"1 ) 900 ■ 850 80 60 40 Pyrazine - d 4 20 V j O I - 1 Figure 6 Polarized Infrared absorption for fundamental of pyrazine-dQ, taken room temperature. 0° polarization 9 0° polarization P td 33 1450 Frequency (cm"1) 1400 80 60 F 40 20 Figure ?. Polarized infrared absorptions for Bo„ fundamentals of pyrazine-d0 and pyrazlne-d^. pyrazlne-dQi taken at room temperature 0° polarization 9 0° polarization pyrazine-d | j , i taken at 9 0°K --- +60° polarization -30° polarization 34 Transmittance (%) Frequency (crrr') 1090 1070 1050 1030 80 60 40 Pyrazine- 20 Transmittance (%) Frequency (cm-1) 860 840 820 800 80 60 Pyrazine-d4 Figure 8. Polarized infrared absorptions for B2r i fundamentals of pyrazlne-dQ and pyrazlne-dz*. pyrazine-dQi taken at room temperature 0° polarization 9 0° polarization pyrazlne-dj^i taken at 105°K — -- +6 0° polarization -3 0° polarization i [ r 36 Transmittance (%) Frequency (cm-1) 830 800 770 80 60 40 20 - I I I 1 \ | Pyrazi V zine - Transmittance (%) Frequency (cm -1 ) 620 • 600 580 80 60 Pyrazine - d4 40 do Figure 9. Polarized infrared absorptions for Bgu fundamentals of pyrazine-dQ and pyrazine-d^. pyrazine-dQ* taken at 25 5°K pyrazine-dj^i taken at 105°K — +6 0° polarization -30° polarization Transmittance (%) Frequency (cnrr1 ) 460 430 400. 801 1 I ! 1 I 1 1 60 40 Pyrazine -d o Transmittance (%) Frequency (cm-1) 4 3 0 410 390 370 60 - 40 20 V jJ NO 40 the "bands were recorded at two polarizer angles, 9 0° apart, which give the maximum (solid lines) and minimum (dashed lines) absorption intensities of 1492 cm~^ and 1022 cm~l for pyrazine-dQ and pyrazine-d/j, respectively. The temper atures that the bands were taken at are indicated in the graphs. Ito (3) has reported the Raman spectra of both ,internal and lattice vibrations and his data are used for comparison with some of the calculations. Recently the far-infrared spectrum of pyrazine has also been published W . CHAPTER IV RESULTS AND DISCUSSIONS Crystal Structure As we mentioned in Chapter II, knowledge of the crystal structure is of basic importance for studies in crystal states, Wheatley (2) has determined the molecular j orientations and unit cell dimensions of pyrazine. The 12 crystal is orthorhomblc, belonging to the space group Dg^ (Fnnm) with two molecules per unit cell. The molecules sit on sites having symmetry 2/m (Cg^)• The factor group is isomorphous with Dg^. The projection of the structure on (010) is given in the following picture for convenience. The shaded molecule is displaced by half a translation in the positive vertical direction, i.e. downwards, c — H — i 42 ; In calculating the Intermolecular distances of crystalline pyrazine, Wheatley’s atomic coordinates in the crystal were used*. The molecular dimensions after thermal motion corrections were also given in his paper along with the coefficients of the rotational and translational vi bration tensors for discussing the thermal motion in re lation to the packing of molecules In the crystal. The CH bond length calculated from the atomic coordinates is O 1.05 Aj after thermal motion corrections, It becomes I.09 o ! A. The latter value was used In the calculations of the normal coordinates of a free molecule In the appendix. Use of the original atom positions (and thus a CH bond O length of 1,05 A) In our crystal calculations is in consonance with Williams* conclusion (23) that the effec tive repulsion center for the H atom is closer to the C 0 atom than 1.09 A. A computer program was used to generate the atomic ; coordinates of the neighboring molecules and calculate their Intermolecular atom-atom distances from the origin O molecule. With a 5 A limit on non-bonded atom-atom ^Inconsistency occurs between the rotation matrices from the molecular principal axes to the crystal axis system for carbons and hydrogens. The atomic positions of hydrogens in the crystal were adjusted to stay In the same ; plane as carbons of the same molecule. The molecule is tilted 22° 3 1*, rather than 22° 3', around [a] to have the atomic coordinates listed by Wheatley. 43 : contacts, eight nearest neighboring molecules, two second nearest (along the b-axis), two third nearest (along the c-axls) and two fourth nearest (diagonally translationally equivalent, at (Oil)) neighboring molecules are included, comprising totally 408 non-bonded contacts. With varia tions of limit, the number of contacts per molecule changes as seen in Fig. 10 for both the "observed" structure and the "equilibrium" structure (described in next section). I i Nonbonded Potential Function When setting out to calculate the properties of a crystal theoretically, one starts with an assumption of a potential function. The nonbonded 6-exp pairwise potential; function with the form V = AR“6 + Bexp(-CH) (10) was exploited to account for the atom-atom Interactions. R is the interatomic distance. Six sets of the parameters A, B and C for different types of interactions H H, C--H i t c C, N H, N C and N N were required. Various : sets were tested for consistency with the crystal structure of pyrazine, its measured enthalpy of sublimation and the j observed lattice frequencies. The "best" set was sub- j i sequently used to calculate intramolecular vibrational j splittings. I i In testing the parameters for agreement with the i Figure 10, Number of nonbonded, atom-atom interactions as function of summation radius cut-off, H, using potential function V = A + B exp(-C R) In refining the crystal structure, where A B C NH.CH 139 9411 3.70 HH 36 4000 3.7^ CC,CN,NN 535 74460 3 .6 0 (denoted as set I later). 41+ 45 2000 Observed structure Equilibrium" structure . i j 1800 1600 1400 1200 1000 in 800 600 JCi 400 200 ' 4 9 3 5 7 8 6 R (A) 46 measured latent heat, two programs were usedi one computes the lattice energy by summing over the pairwise interac tions of the nonbonded contacts, A second program kindly provided by Dr. Donald E. Williams refines the crystal structure to agree with the potential, giving a minimum lattice energy for the parameter set. Williams' parameters derived from crystalline aromatic (1) and nonaromatic hydrocarbons (24) as well as their combined data were tested for their transferability to this type of molecule having weak hydrogen bonding. Since this work was exploring the use of the atom-atom type potential on a heterocyclic compound, the assumption was made of taking nitrogen the same as carbon in their interactions at first, with the idea that the parameters of the potential function chosen could be adjusted to be more suitable if the results were promising at all. The various parameter sets of Williams yield lattice energies in the range of -4.39 to -7.07 Kcal/mole, generally low compared with the observed AHgub. i It is worthwhile here to pick up one of Williams' potential parameter sets to see how lattice energy changes with various summation radius cut-offs, because the lattice| energy strongly depends on the number of contacts consider- : i ed. Williams' set IV of reference (1), listed below from ;data on aromatic hydrocarbons, was used, and the crystal o o 1 jenergies were computed in the range of 3 A to 12 A for j 47 both the "observed" structure and the refined "equilibrium" structure. (Notei the structure was refined separately at each radius.) Fig. 11 shows the energy results. The parameters In Kcal/mole and Angstrom units are A B C 36 4000 3.7^ 139 9411 3.70 535 74460 3 .6 0 | Hereafter, we shall refer this set as set I for convenience | 1 * As shown in Fig. 11, the crystal energy approaches o a constant value beyond 9 A. When nonbonded contacts O o within 5 A to 6 A are considered, only 65-85 %t respective- ! ly, of the observed heat of sublimation is accounted for. Williams* parameter sets, used in the structure refinement program, all tend to rotate the molecules to lengthen the closest H-— N distances, and the refined lattice constants deviate to positive values along the a and c axes, and negative along the b axis. The ideal case ! should have the lattice constants all slightly reduced from their observed values, because the calculation corresponds to a 0° K structure. Attempts have been made ; ; . j to find a set of parameters that would fulfill this ex- jpectation when lattice frequency computations were done to : l 1 Ihelp in better determining the parameters in the following I HH CH, NH CC, CN, NN Figure 11. Calculated crystal energies of pyrazine as function of summation radius cut-offs using set I of the potentials, for both the "observed" and refined "equilibrium" structures. 48 Crystal Energy o f Pyrazine (Kcai/mole) 4 9 Observed structure Equilibrium" structure -2 -4 -6 -7 -8 -9 - -10 -II 12 10 I I 8 9 6 7 R, (A) 50 section. Lattice Vibrations and Choice of Parameter Set Since the lattice frequencies can give very direct information on the intermolecular interactions, the poten tial parameters for this molecular crystal will be consi dered extensively in this section. Variations in the parameter sets were made to obtain a "best" set which was in general agreement with the lattice frequencies as well as with the crystal structure and AHSU^. Approximations involved in the calculation were considered in some detail. With the "best" set, the intermolecular coupling is then studied in the following section. Of the nine (6N-3> 3 for acoustic modes) k = 0 optical modes, six are libratlonal and Raman active (Ag, Bi . two B2g and two B^g), two are translational and in frared active (B2u and B^u), and one is translational and inactive in either spectrum (A^). In addition, three zero j frequency acoustic modes exist, belonging to translational species B^u, B2u and B^u. Dr. 0. Taddei (25) kindly provided us with his program for lattice frequencies which takes the product j of two matrices and of equation (8) in Chapter IIJ ! and diagonlzes the product matrix by Jacobi method. This overcomes troubles we ran into in performing sequences of transformations to factor the symmetry blocks as described 51 In equation (9) of Chapter II, The output of the program gives the lattice frequencies of all twelve modes, eigen vector solutions, crystal energy and the number of contacts per molecule. The last two values check the results ob tained in the previous sections. Input data include atomic coordinates of the origin molecule in the principal axis frame, the transformation matrix of molecular principal axes to the crystal frame, and unit cell dimensions, as ;well as the intermolecular potential parameters. The program does the Job of constructing and f R) f R) multiplying G' and F for the six degrees of freedom (R) ;of the two molecules in the unit cell. To build the F matrix is simply a procedure of taking second derivatives of the potential function used. In building the inverse kinetic energy matrix elements, it is simpler to construct them in a molecular-fixed cartesian frame, that is, the so-called principal axes. The translational vibrations have cartesian displacements equal to the inverse square root of total mass of a molecule for all atomsj the librational modes move inversely proportional to the square i root of the appropriate moment of Inertia of the molecule and proportional to the normal distance from the rotation I jaxis for each atom, j Since weak hydrogen bonding might be important in i this molecular crystal, we have emphasized variations in -potentials involving hydrogen. We choose graphic forms to display the calculated lattice frequencies for ranges of values of the attractive part (A values) and the repulsive part (B coefficients) of the H H and N H potentials. The exponent coefficients, C, of all types of intermolecu- lar Interactions, and the roles played by C H, C— -C, C n and N N are believed to be not very different from those in aromatic hydrocarbons and are held fixed through the calculations. They are assigned the following values, unless noted elsewhere. A B C NH, CH 3 .6 7 HH 3.7^ CC, NN, NC 535 74400 3 .6 0 (in units of Kcal/mole and Angstroms). The constants are essentially those of set I quoted earlier. For all the computed results to be given, the structural data were taken from the , , equilibrium, , structure as refined for the potential parameters in use by Williams* steepest descent procedure, unless Wheatley*s "observed" o ;structure was specified. The summation radii were 5 A for H H, 5-5 A for C H and N H and 6 A for C C, C N and N— -N. For each set of calculations, a corresponding diagram of calculated frequency deviations and crystal energies is also presented to give an idea of the roles 53 : the changing parameters play. The summation radius used should yield about 80 % of the observed (neglecting the difference between AE and AH), or -10,3 Kcal/mole, This value Is Indicated on the figures as a dashed line. The deviations D measure the quality of the lattice frequencies calculated using the tested potential para meters. They are the quantities Z | "^observed ^calculated! D = s r ' T j . L vobserved Five Raman bands observed at 4° K by Ito (3) and. two infrared active translational modes located at 77° K by Gerbaux (A) were taken for comparison. Figs. 12a to 15b show all these results. Symmetries are assigned from the form of the eigenvectors given by the calculation. The seven observed bands are drawn as energy levels at the very right of lattice frequency plots. The reason for using observed data at A-° K is because the "equilibrium" structure used in calculation corresponds to 0° K. Gerbaux' data at 77° K should show frequency increases if temperature is lowered. The lattice frequencies fluctuate ; in the region of tested parameters, but are drawn as I smooth lines for clarity. | ! Inspection of Figs. 12a through 15b shows that j ; | !increasing the attractive part of the N H interaction j ; I j gives better overall agreement of AHsul3 and lattice j Figure 12a. Calculated lattice frequencies with various A values of N---H Interaction. The rest of the parameters are A B C CH 139 9411 3.70 HH 36 4000 3.74 NH 9411 3.70 54 Frequency t/t (cm- 140 130 120 HO 100 90 80 70 60 50 40 30 20 120 140 160 180 200 220 240 260 280 300 A(N—H), Kcal/mole*A \JX Figure 12b. Calculated crystal energy and frequency deviation as function of A values of N H interaction. 56 i __ [-5 O -10 -12 - 2.5 c . 2 u- 0 260 280 300 180 200 240 160 140 120 Vji ■>3 Figure 13a. Calculated lattice frequencies of pyrazine varying B coefficient of N H interaction. Frequency V , (cm-1} 59 1 4 0 1 3 0 120 100 90 80 ( u 60 50 40 30 20 10,000 8000 6000 B (N — H), Kcai / mole i Figure 13b. Calculated crystal energy and frequency deviation varying B values for the N H interaction. 60 61 LlI LU £ -10 GO 10,000 8000 6000 B(N”*H) Kcal/mole I Figure l^a. Calculated lattice frequencies with various A values of H---H interaction. The rest of the parameters are A B C NH, CH 12^ 9^11 3.70 HH 27.3 ^000 3.7^ 62 Frequency V, (cm-1) 63 140 30 20 100 Au 90 80 70 / / X ■ ? * - / ■ ------ 60 50 /■ B2U 40 B3g 30 20 60 30 40 50 20 to A(H---H), Kcal/mole*A6 Figure 14b. Calculated crystal energy and frequency deviation as function of A values for the H H interaction. 64 A(H---H), Kcal / mole • o Crystal Energy E, Kcal/mole o r o o oj o o C J l O 01 01 o 0\ V J I Figure 15a. Calculated lattice frequencies with varia tions of B values of H H interaction. The rest of the parameters are the same as in Fig. l^a. 66 Figure 15a. Calculated lattice frequencies with varia tions of B values of H-— H interaction. The rest of the parameters are the same as In Fig. l^a. 66 1 3 0 120 NO 100 90 80 70 .60 50 40 30 20 10 B ig ■ Ag I I I I ______ 000 4000 6000 B (H-H), Kcal / biole Figure 15b. Calculated crystal energy and frequency deviation with variations of B values for the H-— H interaction. 68 B (H--H) Kcal /mote Crystal Energy E, Kcal/mole O SO 70 frequencies, though the effect on the latter is relatively mild. The crystal energy decreases 3*5 Kcal/mole in varying A of the N H interaction from 125 to 300 Kcal/ ° & mole • A . A smaller repulsive coefficient B for N H seems to improve the calculated lattice frequencies. This is reasonable for atoms involved in hydrogen bonding to attract each other rather than to repel compared, say, to C— H Interactions. Williams* value for the coefficients of the H H Interaction seems to be the best for this molecular crystal ' . The calculated lattice frequencies fluctuate compara tively more for H H than for N H parameter variation. Stronger repulsion of H H interaction tends to improve the B2g mode, though it is not favorable for B^g (the rotation around axis perpendicular to that passing through two nitrogens). Values of set I for H— H are used in later studies. In these calculations, frequencies of two B2g, one -f B3g and the B2u appear about 1 5 -^ 0 cm lower than observed; and they tangle with B3u in energy level diagrams. This is shown in Figs. 12a to 15b by the dashed correlation I I lines. The B^u frequencies agree fairly well with ob served values in most cases. Though it seems that no parameters of the 6-exp i i type potential function could raise the frequencies of B2g, j B3g and B2u to agree with observed values, yet fit the measured AHg^. the "best coefficients among all tested of H H and N H were picked (see Table 2) and combined with the rest of the parameters in set I for a "best" set, to be used In a complete run of all the properties studied and the exciton splittings (to be studied in next section). This set gives a calculated crystal energy of -11.32 Kcal/mole when the summation radius, R, was taken o to be 6 A. This is in good agreement with 85 % of AHg^ { o 11 Kcal/mole, appropriate to a 6 A radius). The "equili brium" crystal structure refined using set II is given in Table 3» Set II does make the unit cell parameters a bit smaller than observed, as expected. The calculated lattice frequencies and normal coordinates are listed In Tables b and 5. Lattice modes evaluated using the observed crystal structure are also listed for comparison. The refined structure raises all the calculated lattice modes from the frequencies of the "observed" structure using this potential parameter set. Since the calculated lattice modes depend upon the number of nonbonded Interactions considered, the quantita tive changes of the lattice frequencies with summation i radius, R, obtained from set II are given in Figs. 16 and ' ! 17. Results using both structures ("observed" and "equlll-! ! i brium") are shown for comparison. j L Fig. 16 shows the dependence of the lattice TABLE 2 POTENTIAL PARAMETERS IN KCAL/MOLE AND A* INTERACTION TYPES A B C N--H 230 7000 3.70 C--H 139 9^11 3.70 H--H 36 A000 3.7^ C--C , C--N, N--N 535 7*4460 3 .6 0 *From now on we shall refer to this set of parame ters as set II, TABLE 3 DEVIATIONS OF THE CALCULATED MOLECULAR POSITION AND LATTICE CONSTANTS FROM THEC OBSERVED VALUES (IN UNIT OF A, FOR R « 6a) MEAN DEVIATION MAX. DEVIATION PARAMETER n n o nm CARBON ATOMS J Aa Ab Ac LATTICE CONSTANTS -,0A -.121 -.126 TABLE A LATTICE USING SET II FREQUENCIES OF PYRAZINE CALCULATED 0 OF THE POTENTIAL FUNCTIONS (FOR R = 6A)* REFINED STRUCTURE OBSERVED STRUCTURE SYMMETRY SPECIES OBSERVED FREQUENCY CALCULATED FREQUENCY DEVIATION CALCULATED FREQUENCY DEVIATION Ag 126 138.3 +12 106.8 -19 Bls 134 1^7.9 +14 117.3 -17 B2g 86 56 70.6 41.3 -15 -15 57.1 2 9 .2 -29 -2 6 B3g 97 » • • 6 9 .8 31.5 -2? 5 6.6 7.2 -40 100.3 82.1 B2u 79 56.6 -22 46.3 -32 B3« 54 57-7 +4 48,7 -5 _ i ♦Entries are In cm . 74 TABLE 5 NORMAL COORDINATES OF LATTICE VIBRATIONS FOR "EQUILIBRIUM" STRUCTURE OF PYRAZINE CALCULATED USING SET II OF POTENTIAL FUNCTION (FOR R = 6A) FREQUENCY NORMAL COORDINATES* (cm-1) MOLECULE I MOLECULE II 3 1 . 5 +.4365Lf e +.5563Lc -.4364Lb -.5563LC 41.3 +.3954Lb -.5863Lc +.3954Lb -.5862LC 56.6 -,7071TC +.7071c 57.7 -.7071Tb -.7071Tb 6 9 . 8 +.5563L b -.4 3 6 5L c -•5564l b +.4364L c 70.6 -.5863Lb -.3953LC -.5862Lb -.3954lc 100,3 +.7071Ta -.7071Ta 138.3 -.7071La -.7071La 147.9 +.707lLa -.?071La *The symbols La, Ta, etc. refer to libratlon about, and translation along, the crystal a axis. The crystal axes were transformed from the molecular principal axis frame by the rotation matrix -.00309 .00035 -.9999 -.3833 .92364 .0 0 1 5 1 .92364 .3833 -.00272 Figure 16. Calculated lattice frequencies for the "observed" structure as function of summa tion radius cut-off, R, using set II po tential , The lattice frequency dependence upon R is discontinuous, but is drawn as smooth curves for simplicity. The experimental values are at the very right of the figure for comparison. The lowest frequency, B^g mode, decreases drastically as R increases and becomes imaginary near 8 A. r Frequency, (cm-1) >0 ro co C J l ro CD DO CD iQ iO -0 On Figure 17. Calculated lattice frequencies for the refined structure as function of summation radius cut-offs, R, using set II of poten tial function. (comments see Fig, 16.) 77 Frequency, (cm-1) 78 1 6 0 1 5 0 1 4 0 1 3 0 120 110 Au 100 90 8 0 70 60 50 02U 40 30 6 7 4 8 3 5 79 frequencies upon R using Wheatley's structural data. The . O calculated modes vary drastically between 2.6-3 A where strong positive N---H and H H force constants occur as O shown In Fig. 18. In the region of 3-^*2 A the negative region of N-— H interaction becomes dominant where the lattice frequencies attain their maximum values. Above . o 4-.2 A, the strong positive C G, C N and N N contribu tion has become negative along with the other negative force contributions from C H, H H and M H. Above o ;about ^ A the frequencies display slow changes. The numbers of contacts for the corresponding Interactions are given in Table 6. The dependence of the lattice modes on R for the refined "equilibrium" structure is quite different from that for the "observed" structure* see Fig, 17. The frequencies increase as R becomes larger. This is because the crystal unit cell dimensions reduce as R increases during the steepest descent refinement procedure. The o closest N H distance ranges from 2.668 to 2.500 A when o R takes on values from 3 to 8 A. From the results on the o "observed" structure, 6 A seems to be a good limit to cover all the Important contacts, but when the structure is refined to minimum lattice energy at each R, the , o frequencies seem to continue rising well above 6 A. The o computed lattice frequencies at 6 A were listed earlier :in Table ^ to compare with the corresponding results Figure 18. Intermolecular force constants as function of interatomic distances, R, using set II of potential function. The dashed line was calculated with B coefficient of C-— N Interaction equal to 1?3»300 Kcal/mole Instead. Q Force Constants, m dyne/Ax I03 CJl >0 C D 82 TABLE 6 NUMBER OP NONBONDED INTERACTIONS PER MOLECULE FOR OBSERVED STRUCTURE OBTAINED USING SET II R( A) N--H H--H C--H C---N N--N TOTAL 2.6 8 8 3.0 8 4 12 4.0 24 40 56 64 184 5.0 64 52 144 148 408 obtained from Wheatley's structure. Potential parameter set I gives also a similar comparison in lattice modes between "observed" and refined "equilibrium" structures with dependence upon R (see Figs. 19 and 20). The given calculated lattice modes in this section were computed with the first derivative term of the poten tial function eliminated (refer to section A of Chapter II)j Analysis shows that this term affects the calculated frequencies of the "observed" structure more strongly than 1 those of the refined one. It lowers the frequencies of the "observed" structure in a range of 1-18 cm-* and lowers: —l the results of refined structure by only 1-2 cm for most of the bands and about 10 cm-1 for the lowest two bands. Remarks on Lattice Frequency Calculation The 6-exp type atom-atom pairwise potential agrees with the crystal structure and heat of sublimation of pyrazine fairly well. The lattice frequencies can only be fitted moderately well, with two B2g, one B^g and the B2u too low. Ito (3) has done the lattice frequency calculation, ; ; i and seems to have better agreement with the observed values, j However, he has taken into account only the contacts within' o 3 A, which we have shown to be a "bad" range in lattice mode calculation. And the force constants he used for two Figure 19. Calculated lattice frequencies for the "observed" structure as a function of E using set I of potential function. (same comments as In Fig. 16.) 8^ Frequency* (cm-1) ~4 C P Figure 20. Calculated lattice frequencies for the refined structure as a function of R using set I of potential function. (see comments in Fig. 16.) 86 8? 140 130 120 100 Au 90 2U 70 L i- 60 3U 50 2 l l 40 30 20 O R, {A) H— -H Interactions can not be reasonably obtained from the same set of parameters of a 6-exp type potential function. Further, he added In the bending force constant of the N H-C hydrogen bond in his calculation. We checked on the importance of this force constant here. If the small bonding angle change Is expanded to second order to be expressed in the form of a C-— N central force, the poten- 2 tlal energy 2V = k(Afl) then becomes 2V = k(rCN/rCH rNH 2 2 ^ sin a) (ArCN) . k Is the force constant of the hydorgen bonding angle 9 (calculated to be 150° 29* for the "observ-i ed" structurej and 1^8° 1 6* for the "equilibrium" structure; ). r's are the interatomic distances or the bond length :of the subscripted atoms. From Ito*s data, k equals 0.007 O mdyne/ A at 293 K. Thus the bending force constant in central force form Is 0.0^8? mdyne/ A, about three times larger than that of the C N interaction obtained from our potential function. In order to reach such a large bending force constant, a potential with Increased B coefficient of C N interaction, about 2.5 times larger than that of set I and II (If A value keeps the same), Is ! needed. This will move the Intermolecular force range of j I C N interaction out farther in the crystal space as seen j ;in the dashed line of Fig. 18. At these large distances, | j i the contribution of the bending force constant to the j i lattice frequencies could be negligible without sacrificing! imuch accuracy. So, the consideration of N H-C bending, j Indeed, could be taken care of by the atom-atom pairwise potential, though It is not too Important, Thus, to use Ito's results as a standard to judge agreement of the calculated lattice frequencies with the experimental data using the 6-exp type potential will not be sensible, Intermolecular Coupling The symmetry correlation (22) of the dynamics of two pyrazine molecules in a unit cell having factor group isomorphous with shows that each molecular mode of the free molecule is split into two k = 0 modes in the crystal. All the split components of the molecular gerade modes are Raman active. Of the ungerade modes, both split components of a molecular B2u or B-^u mode are Infrared active. One component has its transition moment parallel to the b-axis, the other along c-axis. For A^ and B^u species, one compo nent is infrared active parallel to a-axis* the other will be inactive in both Raman and infrared. Thus an Au mode gains Infrared activity going from gas phase to crystal. The intermolecular potentials with parameters examined in the previous section are tested here for their compatibility in obtaining information on the Intermolecu lar forces in the crystal. Potential set I was used :primarily for detail studies of the factor group splittings land the frequency shifts under variations of summation 90 radius and structure {'•observed" or "equilibrium") for all the fundamentals. The calculated factor group splittings and frequency shifts using set II of the potentials (the "best" set) were listed later for comparison with the results obtained from set I. A computer program was written to calculate from the assumed potentials the derivatives in equation ( ? ) of Chapter II which effect the computations of the frequency shifts and splittings. The input data are atomic coordi nates of the origin and neighboring molecules of interest (output of a program previously described at the beginning section of this chapter, with Interatomic distances avail able) , cartesian displacements (provided In the appendix) and parameters of the interatomic potentials. The program does these Jobs* transforming cartesian displacements from molecular to crystal coordinates; computing direction cosines of angles between cartesian displacement and Interatomic vectors; evaluating the first and second derivatives of the potential, including processing the contact se- ! quence to sort out the interactions into six types; i combining the factors into products following i equation (7) in Chapter II, this requires a sortlngj 1 i procedure on Interactions between molecules which j ! j | are either translationally equivalent or not. 91 : This program ends up computing the lattice sum and provides output of two frequencies for each molecular mode. The doublet splittings were obtained by summing up the terms which belong to interactions between molecules not trans- latlonally equivalent, that is, not related by purely translational operations. Both in-plane and out-of-plane modes were studied, in spite of the lack of enough informa-: tion for the out-of-plane modes in the literature. The normal coordinates of both ln-plane and out-of-i plane modes of pyrazine-d0 and pyrazine-d^ are furnished in: ; i the appendix. The analysis was based on the reference (18). We used the valence force field of Scrocco et al.(26) for the ln-plane modes, and essentially duplicated their results. Difficulties arose while coming to treat the out- of-plane modes, since there are still uncertainties about their assignments. Modes with frequencies assigned for both pyrazine-dQ and pyrazine-dji,, were easily analyzed by means of the product rule. Other modes having only esti mated undeuterated frequencies were found by transferring force constants from benzene. Of course, adjustment had to be made for a few force constants to make observed and I calculated frequencies agree, j The calculated results of the frequency splittings j j and the observed splittings using parameter set I of the I j ^ potential are shown in Tables 7 to 10. The polarization ! ; behaviour of each split component is also given if avail- TABLE ? FACTOR GROUP SPLITTINGS FOR IN-PLANE MODES OF PYRAZINE-d0 CALCULATED USING SET I OF THE POTENTIAL FUNCTIONS© MOLECULAR APPROX. OBSERVED POLARIZ.* OBSERVED STRUCTURE REFINED STRUCTURE SPECIES FREQUENCY SPLITTING OF HIGH CALCULATED POLARIZ. CALCULATED POLARIZ. (D2h) COMPONENT SPLITTING OF HIGH* SPLITTING OF HIGH* As 3072 1577 1231 101*4- 618 .0*4-/.15 .12/.03 2.53/2. **9 9.21/8.92 9.39/9.12 I l g 4 A Blg .0*4-/. 01' .0 9 / .0 6 1. * 4- 6 /1 .**5 3.19/3.16 3.22/3.10 As{Big As AS is B2g 3071 1527 1368 6*4-2 .16/.03 1.28/1.11 3. 7^/3 • 8 * 4 - *^.23/3.79 > ?3s o2g 2g .08/.03 .*4-6/.*4-2 1.99/1.99 1.3V1.27 BZg r3s B2g Blu 3072 1*4-82 1132 1021 2 2 Blu** Blu .08/.03 5.01 A. 66 1.56/1.53 9.72/9.25 Au Blu .08/. 0 * 4 - 2.21/2 .1 2 • 7V.7A 3.39/3.27 S Blu B3u 3069 1*4-26 13*4-6 1075 ** 7 B 2u B 2u .1*4-/-- 1.11/1.23 l.lVl.39 1. 50/1 .1 2 b 2u b2u B2u _ m . 2u .0 ? / .0 2 .69/.71 .W.53 1.0 0 / . 8 8 b 2u b 2u b 2u 2u *The species of higher component of the doublet (also applies to Tables 8 to 10). | **Splittlng component of A^ symmetry almost disappears in the spectrum, @A11 numbers are in cm“l. TABLE 8 FACTOR GROUP SPLITTINGS FOR OUT-OF-PLANE MODES OF PYRAZINE-d, CALCULATED USING SET I OF THE POTENTIAL FUNCTIONS® 1 MOLECULAR UNPERTURBED OBSERVED POLARIZ. OBSERVED STRUCTURE REFINED STRUCTURE SPECIES (D 2 h ) FREQUENCY SPLITTING OF HIGH COMPONENT* CALCULATED SPLITTING POLARIZ. OF HIGH* CALCULATED SPLITTING POLARIZ. OF HIGH* B 1S 925a 3.76/3.13 As 1.4-3/1.22 As B 3g 751? 69 Q 3.65/3.37 2.25/2.31 b2s 2g 1.64-/1.48 1.13/1.11 b2s 2 g P°l 4 - 0 0 ^.35/3.62 21.87/18.30 *u 1.61/1.38 8.00/6.8? K \ B 2 u 8 0 0 ® 4 - 1 4 - 3 B 2u 2.01/1.88 .51/.36 ® 2 u b3 u .94-/. 8 6 .13/.09 b2 u B 3 u ^Frequency assumed by Califano, Table 7 of reference (21). Ito*s Raman spectrum, Table 2 of reference (3), °Spectrum taken In our laboratory, see Figures 8 and 9 in the previous chapter, ®A11 numbers are in cm . vo VJ TABLE 9 CALCULATED FACTOR GROUP SPLITTINGS FOR IN-PLANE MODES OF PYRAZINE-dk USING SET I OF THE POTENTIALS# MOLECULAR APPROXIMATE OBS. POLA OBSERVED STRUCTURE REFINED STRUCTURE SPECIES (°2h) FREQUENCY Calc. Califano® SPLIT. RIZA TION* CALCULATED SPLITTING POLARI ZATION* CALCULATED SPLITTING POLARI ZATION* Ag 2209 1532 1003 876 6l4 0.1/ 0.2 1.1/ 1.1 6.5/ 6.4 6.2/ 5*8 10.4/10.1 B1S ig Blg 0.0/0.0 0.4/0.4 2.2/2.2 2.7/2.6 3.6/3.^ Blg Blf As A RS Blg B2g 2284 1505 1036 616 0.3/ 0.2 1.4/ 1.2 2.1/ 2.3 4.2/ 3.8 B2g b3s B2g B2g 0.1/0.1 0.5/0.5 1.1/1.2 1.3/1.3 ®2g R3g b2s 2g Blu 2285 1363 999 853 2288 1370 1020 868 2 1 Blu Blu 0.1/ 0.0 3.3/ 3.1 5.4/ 5.4 3.0/ 2.6 Au AU Blu Blu 0.1/0 .0 1.3/1.2 1.8/1.8 0.8/0 .7 A* Au Blu Blu B3u 2279 1393 1248 836 2282 1350 1264 841 1 B2u 0.2/ 0.1 0.6/ 0.6 1.4/ 0.9 3.0/ 2.7 B2u b 2u b3u b 2u 0.1/0.0 0.2/0.2 0.4/0.3 1.6/1.5 b 2u b 2u b3u b 2u #A11 numbers are In cm"l. ©Califano's assignments (21) In this work. are given to compare with the frequencies calculated vO - p - TABLE 10 FACTOR GROUP SPLITTINGS FOR OUT-OF-PLANE MODES OF PYRAZINE-cL CALCULATED USING SET I OF THE POTENTIAL FUNCTIONS® ^ MOLECULAR APPROXIMATE OBSERVED POLARIZ. OBSERVED STRUCTURE REFINED STRUCTURE SPECIES (D2h) FREQUENCY (CALC.) SPLITTING OF HIGH COMPONENT* CALCULATED SPLITTING POLARIZ. OF HIGH* CALCULATED SPLITTING POLARIZ. OF HIGH* B1S 780a 2.1V1.70 As .78/. 64- As B2u 6°9b 4o*T 1 B 2u 1 .61/.58 . 6 5 / A ? b2u 3u .32/.30 .18/.12 b2u b3u I ^Califano (21) calculated It to be 72^ cm“ which is quite doubtful, | Bands observed in this laboratory, I ©All numbers are in cm-1. vo V_n able. Tables 7 and 8 show the splittings for pyrazine-d^. Tables 9 and 10 are for pyrazine-d^. The ln-plahe mode frequencies of pyrazine-d0 given are of Scrocco et al (26). Those of pyrazine-d^ were given both the calculated values and Califano's assignment (21) for comparison. The out- of-plane frequencies are referred to Table 30 in the appendix. Same frequencies will be given in further property calculations for identification. The calculated splittings contain two sets of numbers — those above the slash were obtained including the second term in equation :(7) of Chapter IIs the ones below were those neglecting this term in first derivatives of the potential function. Splittings for the "observed" and for the refined struc tures are listed in separate columns. The vibrational species of the molecule are derived for a molecular coordi nate system with y(B2u) being out-of-plane, z(®lu) going through two nitrogens and x(B^u) bisecting one C-C bond, perpendicular to both z and y. All of these calculations o include contributions from all contacts within 5 A. The refined crystal structure, i.e., the "equili brium" structure which gives the satisfactory minimized lattice energy, shows narrower calculated splittings of all the modes, In both pyrazine-dQ and pyrazine-d^, espe cially those widely split In "observed" crystal structure :calculations. The gerade modes are infrared Inactive, and Raman 97 active. No splittings in the Raman spectra are available. The mode near 13^6 cm“^ was observed only as a weak bump. The observed split component frequencies are listed later when the results calculated using set II of the potentials are displayed. For both pyrazine-dg and pyrazlne-d^, the split components for the refined structure do not change in polarization behaviour from those for the "observed" :structure. The first derivative of the potential function ! (appearing in the second term of equation (7) of Chapter II ) affects the splittings to less than 0,5 cm"*'1 * and should be reasonably neglected (12). Tables 11 to 1^ contain the computed frequency shifts for both pyrazlne-dQ and pyrazine-d^. Results for both ln-plane and out-of-plane modes on the "observed" structure as well as the "equilibrium" structure are all listed. Almost all the modes shift to higher frequencies except a few marked with stars. The term in the first derivative of the potential function contributes a nega tive amount to the frequency shifts. It seldom shifts the frequency toward higher wavenumber, and then only by a small amount. The out-of-plane modes, generally, shift J a larger amount than the in-plane modes. j The magnitudes of the shifts generally are larger j for the "observed" crystal structure than for the refined j ! structure} a behaviour comparable to that shown for the j TABLE 11 CALCULATED FREQUENCY SHIFTS FOR IN-PLANE MODES OF PYRAZINE-cL q USING SET I OF THE POTENTIALS® MOLECULAR SPECIES UNPERTURBED FREQUENCY OBSERVED STRUCTURE REFINED STRUCTURE av(HIGH) A^(LOW) ny(HIGH) A y (LOW) Ag 30?2 10.9/11.8 10.8/11.6 5.0/ 5.2 5.0/ 5.2 o 157? 1.3/ 1.5 1.2/ 1.4 0.8/ 0.9 0.7/ 0.8 1231 8.2/10.5 5.7/ 8.1 4.8/ 5.7 3.3/ ^.3 1014 9.8/10.0 0.6/ 1.1 3.5/ 3.6 0.3/ 0.5 618 10.2/10.3 0.8/ 1.3 3.6/ 3.7 0.4/ 0.6 B2g 3071 8.7/ 9.4 8.5/ 9.4 3.0/ 3.1 2.9/ 3.1 1527 1.6/ 1.6 0.3/ 0.5 0.6/ 0.6 0.1/ 0.2 1368 6.6/ 7.8 2.9/ 4.0 3.3/ 3.4 1.3/ 1.4 642 7.1/ 7*3 2.9/ 3.5 2.3/ 2.4 1.0/ 1.1 Blu 3072 10.8/11.7 10.8/11.6 5.1/ 5.2 5.0/ 5.2 1482 1.4/ 2.6 1.4/ 2.6 2.9/ 3-3 0.7/ 1.2 1132 4.5/ 5.7 2.9/ 4.2 3.4/ 3.8 2.7/ 3.1 1021 10.7/11.0 1.0/ 1.8 3.8/ 4.0 0.4/ 0.7 b3u 3069 8.7/ 9.5 8.6/ 9-5 3.0/ 3.1 2.9/ 3.1 J ** 1426 1.8/ 2.7 0.7/ 1.4 1.1/ 1.3 0.3/ 0.4 1346 1.0/ 1.3 0,2*/ 0.1+ 0.1/ 0.2 0.0/ 0.0 1075 9.2/ 9.5 7.7/ 8.4 4.2/ 4.1+ 3.0/ 3.0 ©All entries are in wave numbers. av’s are the calculated shifts from the unperturbed frequencies. The representations of the higher components are tabulated in Tables 7 to 10 correspondingly. Calculated shifts above the slashes were obtained including the first derivative of potential function, those below were calculated without the first derivative term. +The only case that the first derivative of the potential function gives positive contribution which is too small to be meaningful. TABLE 12 CALCULATED FREQUENCY SHIFTS FOR OUT-OF-PLANE MODES OF PYRAZINE-cU USING SET I OF THE POTENTIAL FUNCTIONS® MOLECULAR UNPERTURBED OBSERVED STRUCTURE REFINED STRUCTURE SPECIES FREQUENCY A a'(HIGH) A /'(L O W ) AHIGH) a v (low) Elg 925 4.89/7.20 1.14/4.09 2. 6/3 .6 1.2/2.4 B3g 751 698 27.0/33.29 30.5/30.1 36.14/39.1 34/36.9 2 0. 6 /21 27.1/27.4 19/19.5 26/2 6 .3 Au 950 400 4/6.4 36.9/47.2 0/2.8 16.4/30.6 1.5/2.5 23.7/28 0/1.2 1 6.1/2 1 .6 B2u 800 4l4 21.4/23.2 11/11.7 19.4/21.4 10.5/11.4 14.9/15.1 8.6/8.6 14/14.3 8.4/8.5 ®Same as In Table 11, the full notes also apply to Tables 13 and. l4. VO VO TABLE 13 CALCULATED FREQUENCY SHIFTS FOR IN-PLANE MODES OF PYRAZINE-d^ USING SET I OF THE POTENTIALS MOLECULAR SPECIES APPROXIMATE FREQUENCY® OBSERVED STRUCTURE REFINED STRUCTURE (HIGH) Ay(LOW) A^(HIGH) A^(LOW) A ~ 2209 6 .6/ 7.2 6 .5/7.0 3.1/3.3 3.1/3.2 1532 1 .6/ 1.7 0 .5/ 0.6 0 .7/ 0.7 0 .3/ 0 .3 1003 7.A/7 .7 1 .0/ 1.3 3.1/3.1 0.9/1.0 876 9 .0/ 10.6 2.9/A.9 A.A/5.1 1.7/2.5 6lA 10 .A/11.0 0. 5/1. 0 3 .8/3.9 0 .2/ 0 .5 B2g 228A 5 .2/ 5 .6 A.9/ 5 .A 1.8/1.9 1.7/1.8 1505 1.7/1.7 0.2/0.6 0.6/0.6 0. 1/ 0.1 1036 3 .8/A.5 1.7/2.3 1.9/2.0 0.8/0.8 616 7.3/7.A 3.1/A.7 2.A/2.5 1.0/1.2 Bi 2288 6.A/6.8 6 .3/ 6 .8 3.0/3.3 2.9/3.1 1U 1370 3.6/3.7 0.3/0.6 1.3/1.A 0.0/0.2 1020 6.9/7.2 1.6/1.9 3-1/3.1 1.2/1.3 868 6.8/8,2 3.8/5.6 3.3/3.9 2.5/3.2 B^u 2282 5.3/5.8 5-1/5.7 1.8/1.9 1.7/1.9 1350 0.6/1.1 0. 0/0. 1 0.2/0.2 0. 0/0.0 126 A 1.3/1.5 .2*/0.5 0.A/0.A 0.0/0.2 8Al 9.1/9.6 6.1/6.9 A.l/A.0+ 2.A/2.5 +Same as In Table 11. ©Frequencies of and B^u modes are from Califano (21), TABLE l4 CALCULATED FREQUENCY SHIFTS FOR OUT-OF-PLANE MODES OF PIRAZINE-d4 USING SET I OF THE POTENTIALS MOLECULAR UNPERTURBED OBSERVED STRUCTURE REFINED STRUCTURE SPECIES FREQUENCY A A'(HIGH) A (LOW) A^(HIGH) A^(LOW) Big 780 4.3/ 5.2 2.2/ 3.5 3.2/ 3.5 2.4/ 2.9 B2u 609 404 11.8/12.6 13-3/14.2 11.2/12.1 12.7/13.7 9.0/ 9.0 10.5/10.5 8.7/ 8.7 10.3/10.4 101 calculated splittings. The refined "equilibrium" structure seems to give much less first derivative (of potential function, the second term of equation (?)) contribution to both split ting and shifts, not to mention the smaller splittings and lower shifts. These comparisons are included to illus trate the consequence of studying the "equilibrium" struc ture using the atom-atom potential model. In order to assure that a 5 A limit is large enoughj for cutting off the neighbor contacts In the lattice sum, i a range of summation radii was tested on the in-plane modes; of pyrazlne-d0 and pyrazine-d^ for both "observed" and refined structures. The splittings and the corresponding shifts, calculated with the first derivative term Included, are tabulated In Tables 15 to 18. Numbers inside the parentheses are frequency shifts of the higher components of the normal modes. The species of these higher compo nents do not change through the range of testing. They are the same as Indicated in Tables 7 to 10. Some shifts i 0 suffer sudden drops, occurring only below a ^ A cut-off. i ° Sudden changes of splittings do not happen beyond the 4 A o limit either. Thus a 5 A summation radius should be con- ! i sidered as a large enough coverage. For some modes, the ! splittings and their shifts are not appreciably affected j j when increasing the limit. This is expected when the ! : i ;Interatomic distances that concern the molecular normal TABLE 15 CALCULATED SPLITTINGS AND FREQUENCY SHIFTS FOR IN-PLANE MODES OF PYRAZINE-d« AS A FUNCTION OF SUMMATION RADIUS® (OBSERVED STRUCTURE) MOLECULAR SPECIES UNPERTURBED FREQUENCY 5A 4 - «5A SUMMATION 4A RADIUS0 3.9A * 3.6A ' ■ .2Aa ^£T 3072 0.0 10.9) 0,0 10.9) 0.0 10.9) 0.0 10.9) 0,0 10.9) 0.0{ 10.5) o 157? 0.1 1.3) 0.2 1.4) 0.2 i.4) 0.0 1.2) 0.0 1.1) 0.4( 1.0) 1231 2.5 8.2) 2.5 8.4) 2.5 8.6) 2.5 8.5) 2 .5 8.3) 2.5( 7.3) 1014 9.2 9.8) 9.2 9.9) 9.1 9.8) 8.9 9.5) 8.9 9.4) 7.5( 7.9) 618 9.4 10.2) 9.5 10.3) 9.5 10.2) 9-5 10.0) 9.5 10.0) 8.3( 8.4) Bp 3071 0.2 8.7) 0.2 8.7) 0.2 8.8) 0.2 8.8) 0.1 8.7) 0.2( 8.3) 2g 1527 1.3 1.6) 1.3 1.6) 1.3 1.6) 1.4 1.6) 1.4 1.6) 1.2( 1.4) 1368 3.7 6.6) 3.8 6.8) 3.7 6.9) 3-7 6.9) 3 .7 6.7) 3-7( 5.9) 64-2 4-.2 7.1) 4-.2 7.2) 4.2 7.2) 3.6 6.5) 3 .6 6.4) 2.8( 4.9) B.. . 3072 0.1 10.8) 0.1 10.8) 0.1 10.9) 0.1 10.9) 0.1 10.9) 0.1(10.6) 1U 14-82 5.0 6.4-) 4-.5 6.4) 4.9 6.5) 4.9 6.4) 4.9 6.3) 5.0( 6.0) 1132 1.6 4-.5) 1.5 4.6) 1.5 4.6) 1.5 4.5) 1.5 4.4) l. 3 ( 3.3) 1021 9.7 10.7) 9.7 10.8) 9.7 10.8) 9.8 10.6) 9.8 10.5) 8,4( 9.1) Bo,. 3069 0.1 8.7) 0.1 8.8) 0.1 8.8) 0.1 8.8) 0.1 8.8) 0.1( 8.4) -Ju 14-26 1.1 1.8) 1.2 2.0) 1.2 2.0) 1.1 1.9) 1.0 1.8) 1.2( 1.3) 1346 1.1 1.0) 1.1 1.0) 1.1 1.0) 0.9 0.8) 0.9 0.8) 0.8( 0.6) 1075 1.5 9.2) 1.5 9.4) 1.4 9.4-) 1.3 9.2) 1.3 9.0) 1 - 7( 8.4) ^Numbers In the parentheses are the correspondingly calculated shifts for the higher components, simply listed for comparison. Entries are all in the unit of wave- number. This note applies to Tables 16 to 18 also, i^he computed results at 3*5A are practically the same as those obtained at 3-2A. 103 TABLE 15 CALCULATED SPLITTINGS AND FREQUENCY SHIFTS FOR IN-PLANE MODES OF PYRAZINE-cU AS A FUNCTION OF SUMMATION RADIUS® (OBSERVED STRUCTURE) MOLECULAR SPECIES UNPERTURBED FREQUENCY 5A K5A SUMMATION 4A — 1 . RADIUS0 3.9A * 5.6A . 2Aa AS 3072 0.0 10.9) 0.0 10.9) 0.0 10.9) 0.0 10.9) 0.0 10.9) 0.0 1 0.5) © 1577 0.1 1.3) 0.2 1.4) 0.2 1.4) 0.0 1.2) 0.0 1.1) 0.4 1.0) 1231 2 .5 8.2) 2.5 8.4) 2.5 8.6) 2.5 8.5) 2 .5 8.3) 2.5 7.3) 1014 9.2 9.8) 9.2 9.9) 9.1 9.8) 8.9 9-5) 8 .9 9.4) 7.5 7.9) 618 9 .4 10.2) 9.5 10.3) 9.5 10.2) 9.5 10.0) 9 .5 10.0) 8.3 8.4) Bo_ 3071 0.2 8.7) 0.2 8.7) 0.2 8.8) 0.2 8.8) 0.1 8.7) 0.2 8.3) 2g 1527 1-3 1.6) 1.3 1.6) 1.3 1.6) 1.4 1.6) 1.4 1.6) 1.2 1.4) 1368 3.7 6,6) 3.8 6.8) 3.7 6.9) 3.7 6.9) 3.7 6.7) 3.7 5.9) 642 4.2 7.1) 4.2 7.2) 4.2 7.2) 3.6 6.5) 3.6 6.4) 2.8 4.9) 3072 0.1 10.8) 0.1 10.8) 0,1 10.9) 0.1 10.9) 0.1 10.9) 0.1 10.6) 1U 1482 5.0 6.4) 4.5 6.4) 4.9 6.5) 4.9 6.4) 4 . 9 6.3) 5.0 6.0) 1132 1.6 4.5) 1.5 4.6) 1.5 4.6) 1.5 4.5) 1 .5 4.4) 1.3 3.3) 1021 9.7 10.7) 9.7 10.8) 9.7 10.8) 9.8 10.6) 9 .8 10.5) 8.4 9.1) 3069 0 .1 8.7) 0.1 8.8) 0.1 8.8) 0.1 8.8) 0.1 8.8) 0.1 8.4) 1426 1.1 1.8) 1.2 2.0) 1.2 2.0) 1.1 1.9) 1.0 1.8) 1.2 1.3) 1346 1.1 1.0) 1.1 1.0) 1.1 1.0) 0.9 0.8) 0 .9 0.8) 0.8 0.6) 1075 1-5 9.2) 1.5 9.4) 1.4 9.4) 1.3 9.2) 1.3 9.0) 1.7 8.4) ®Numbers In the parentheses are the correspondingly calculated shifts for the higher components, simply listed for comparison. Entries are all in the unit of wave- numter. This note applies to Tables 16 to 18 also. Q i^he computed results at 3.5A are practically the same as those obtained at 3.2A. 103 ! TABLE 16 CALCULATED SPLITTINGS AND FREQUENCY SHIFTS FOR IN-PLANE MODES OF PYRAZINE-& AS A FUNCTION OF SUMMATION RADIUS (EQUILIBRIUM STRUCTURE) MOLECULAR SPECIES UNPERTURBED FREQUENCY 5A SUMMATION RADIUS , 0 .0 4-.5A 4-A 3.8A 3.5A Ag 3072 0,0(5.0) 0.K5.D 0.0(5.1) 0.0(5.1) 0.0(5.1) O 1577 0.1(0.8) 0.2(0.9) 0.1(0.8) 0.2(0.7) 0.2(0.?) 1231 l.5(^.8) 1 *5(4*.9) 1.5(4-.9) 1.5(5-0) 1.5(4*.6) 1014- 3.2(3.5) 3.2(3.5) 3-1(3.4*) 2.7(3.0) 2.7(3.0) 618 3.2(3.6) 3.3(3.7) 3.3(3.6) 2.9(3.1) 2.9(3.1) 3071 0.1(3.0) 0.1(3.0) 0.1(3.1) 0.1(3.1) 0.1(3.0) 1527 0.5(0 .6 ) 0.4*(0.6) 0. 5(0. 6) 0.4*(0.5) 0,4-( 0.5) 1368 2.0(3.3) 2.0( 3 *4*) 2.0(3.4-) 2.0(3.4-) 2.0(3.1) 64-2 1.3(2.3) 1.3(2.3) 1.2(2.2) 1.0(1.7) 1.0(1.7) 3072 0.1(5.1) 0.1(5.1) 0.1(5.2) 0.1(5.1) 0.1(5.1) lu 14-82 2.2(2.9) 2.1(2.9) 2.1(2.9) 2.2(2. 9) 2.1(2.8) 1132 0.7(3.4*) 0.7( 3*4*) 0.7(3.5) 0.6(3.3) 0.7(3.2) 1021 3.4*(3.8) 3.4*( 3.8} 3.3(3.?) 2.9(3.3) 2.9(3.2) 3069 0.1(3.0). 0.1(3.0) 0.K3.1) 0.1(3.1) 0.1(3.0) 3u 1^*26 0.7(1.1) 0.7(1.2) 0.7(1.1) 0.8(1.1) 0.7(1.0) 134*6 0.5(0.1) 0,4-( 0.4-) 0A ( 0 A ) 0.4*(0.3) 0.M 0.3) 1075 1.0(4-.2) 1.0(4-.0) 0.9(4*.0) 1.0(3.9) 1.0(3.8) ^OX r TABLE 17 I i CALCULATED SPLITTINGS AND FREQUENCY SHIFTS FOR IN-PLANE MODES OF PYRAZINE-d^ AS A FUNCTION OF SUMMATION RADIUS (OBSERVED STRUCTURE) MOLECULAR SPECIES UNPERTURBED FREQUENCY Q 5A 4• 5A SUMMATION 4l RADIUS0 3.9A 3 .61 3 ,2Aa Ag 2209 0.1 6 . 6) 0.1( 6 . 6) 0.1 6. 6) 0.1{ 6.6) 0.1( 6. 6) 0.1( 6.3) o 1532 1.1 1. 6) 1 . 1 ( 1. 6) 1.1 1. 6) 1.2{ 1.5) 1.2( 1.5) 1.0( 1.1) 1003 6.5 7.4) 6.5( 7.5) 6.5 7.5) 6.3( 7.2) 6.3 ( 7.2) 5 • 0 ( 5.7) 876 6 .2 9.0) 6.1( 9.2) 6 .1 9.2) 6.1( 9. 6) 6.1( 9.0) 5.9< 8.3) 6l4 10.4 10.*4-) 1 0.5( 1 1.0) 1 0.5 10.9) 1 0.5(10.7) 1 0.5( 1 0. 6) 9-2( 9.1) B2g 2284 0.3 5.2) 0.3( 5.2) 0.3 5.3) 0.3( 5-3) 0.3( 5.2) 0 . 3 ( 5.0) 1505 1.4 1.7) 1.4( 1.7) 1.4 1.7) 1.5 ( 1.7) l.5( 1.7) 1.4( 1.5) 1036 2.1 3.8) 2.1( 3.9) 2 .0 3.9) 2.0£ 3.8) 2.0 ( 3.7) 2.1( 3.2) 616 *4-.2 7.3) 4.2( 7.3) 4.2 7.3) 3 • 6( 6.7) 3.6( 6.6) 2 ,8( 5.1) Blu 2285 0.1 6.4) 0.1 ( 6.4) 0.1 6.4) 0.1 ( 6.4) 0.1( 6.4) 0.1( 6.1) 1363 3.3 3.6) 3.2( 3.6) 3.2 3.5) 3.1( 3.4) 3.1( 3.4) 3.2( 3.2) 999 5*4 6.9) 5 M 7.0) 5.^ 7.0) 5 • 5 ( 7.0) 5 M 7.0) 4.3( 5.3) 853 3-0 6.8) 3.0C 7.0) 3.0 7.0) 3.0( 7.0) 3*0( 6.9) 3 - 0( 6.2) b3u 2279 0.2 5.3) 0.2( 5.4) 0.2 5.4) 0.2( 5.4) 0.2( 5.3) 0. 2( 4.8) J 1393 0.6 0.6) 0.6( 0.6) 0.6 0.6) o.3( 0.3) 0.3( 0.4) 0.2( 0,2) 12*4-8 1.4 1.3) 1.4( 1.3) 1.4 1.3) i.5( 1.2) l.5( 1.2) 1.1( 0.6) 836 3.0 9.1) 3.0( 9.3) 3.0 9.3) 2.9( 9-3) 2.9( 9.1) 3 .0 ( 8.4) ®The computed results at 3.5A are practically the same as those obtained at 3.2A. i - * o TABLE 18 CALCULATED SPLITTINGS AND FREQUENCY SHIFTS FOR IN-PLANE NODES OF PYRAZINE-d*. AS A FUNCTION OF SUMMATION RADIUS (EQUILIBRIUM STRUCTURE) MOLECULAR SPECIES APPROXIMATE FREQUENCY 0 5A 4.5A SUMMATION RADIUS 4a 3.8A 3.5A Arr 2209 --(3.1) 0.0(3.1 0.0(3.1) 0.0(3.0) — (3.0) o 1532 0X0.7) 0.4(0.? 0.4(0.7) 0.3(0.5) 0.3(0.5) 1003 2.2(3.1) 2.1(3.1 2.1(3.!) 1.7(2.6) 1.7(2.7) 876 2.7X4) 2.7(4.5 2.7(4.5) 2.7(4.5) 2.7(4.2) 6l4 3-6(3.8) 3.6(4.0 3.6(3.8) 3.3(3.4) 3.2(3.4) B2* 2284 0.1(1.8) 0.1(1.8 0.1(1.8) 0.1(1.8) 0.1(1.8) 1505 0.5(0.6) 0.5(0.6 0.5(0.6) 0.5(0.5) 0.5(0.5) 1036 1.1(1.9) 1.1(2.0 1.1(1.9) 1.1(1.9) 1.1(1.8) 6l6 1.3(2.4) 1.3(2.4 1.2(2.3) 1.0(1.7) 1.0(1.8) 2285 0. 1(3 .0) 0.1(3 .0 0.1(3.0) 0.1(3.0) 0.1(2.9) XU 1363 1.3(1.3) 1.2(1.3 1.2(1.3) 1.2(1.2) 1.2(1.2) 999 1.8(3.1) 1.9(3.1 1.8(3.1) 1.5(2.?) 1.5(2.7) 853 0.8(3.3) 0.8(3.4 0.8(3.4) 0.8(3.3) 0.7(3.2) B3u 2279 0.1(1.8) 0.1(1.9 0.1(1.9) 0.1(1.9) 0.1(1.8) 1393 0.2(0.2) 0.2(0.2 0.1(0.1) 0.1( — ) 0.1(0.1) 1248 0.4(0.4) 0.4(0.4 0.4(0.4) 0.4(0.2) 0.3(0.2) 836 1.6(4.1) 1.6(4.1 1.6(4.1) 1.6(4.1) 1.6(3.9) 106 10? motions have already been swept within the short limit. o For Instance, a 3*2 A limit is enough for the lattice sum In studying the C-H stretching mode, since the main Inter atomic reactions, H-— H and C H all come from close contacts. The refined "equilibrium" structure generally shows a milder response towards reducing the summation limit. The splittings and frequency shifts calculated using set II (the "best" set) of the potentials are given :In Tables 19 to 26 for comparison with those obtained from set I in Tables 7 to 1^. The data are presented in the same way as in Tables 7 to 14, and the explanations on the data are also to be referred to the notes on these earlier tables. The observed split component frequencies are given If available. The tabulated results were evaluated with o o a summation radius of 6 Aj those at 5 A are practically the same. Set II of the potential cut down most of the large splittings and frequency shifts of the "observed" struc ture and raised all the calculated splittings and shifts of the "equilibrium" structure from those of set I. This brought the refined structure to having wider splittings and larger shifts than the "observed" structure. More modes get positive contribution to the fre quency shifts from the first derivative of the potential in using set II than using set I. The first derivative TABLE 19 CALCULATED FACTOR GROUP SPLITTINGS FOR IN-PLANE MODES OF PYRAZINE-d0 USING SET II OF THE POTENTIALS® MOLECULAR APPROXIMATE OBSERVED POLA OBSERVED STRUCTURE REFINED STRUCTURE SPECIES (D2h) FREQUENCY FREQUENCY RIZA TION CALCULATED SPLITTING POLARI ZATION* CALCULATED SPLITTING POLARI ZATION* As 3072 15 77 1231 1014 618 0.0/0.1 0.1/0.1 4.1/3.9 4.4/4.4 3.6/3.6 Blg Big Afr Big 0,0/0 .2 0.1/0.1 4.8/4,5 6.4/6.4 5.5/5.5 Big A1S A® BS Blg 3 2g 3071 1527 1368 642 0.1/0 .0 0.4/0.4 3.V3.2 2.5/2.4 B2g/B3g B3g B2g B2g 0.1/0 .0 0.6/0 .6 4.1/3.9 3.7/3.5 B2g/B3g b2s B 2g Blu 3072 1482 1132 1021 1491 1489 1125 1022 1020 Blu K Blu *11 0.0/0.1 2.3/2 .2 2.2/2.1 2.9/3-0 Au/Blu Au Au Blu 0.0/0.1 3.1/3.0 2,7/2.4 4.9/4.9 Au/Blu Au Au Blu B3u 3069 1426 1346 10 75 1417 1413 1070 1063 b 2u b3u b 2u b3u 0.1/0.1 1.2/1.1 1.0/1.0 2.8/2.6 B2u /B3U b 2u B2u b 2u 0.1/0.1 1.4/1.3 1.4/1.4 3.1/2.9 b 2u b 2u b 2u b 2u *@Same as in Table 7, also apply to Tables 20 to 22. TABLE 20 CALCULATED FACTOR GROUP SPLITTINGS FOR OUT-OF-PLANE MODES OF PYRAZINE-d0 USING SET II OF THE POTENTIALS® MOLECULAR UNPERTURBED OBSERVED POLA OBSERVED STRUCTURE REFINED STRUCTURE SPECIES (°2h) . FREQUENCY FREQUENCY RIZA TION CALCULATED SPLITTING POLARI ZATION* CALCULATED SPLITTING POLARI ZATION* Big 925 2.0/ 1.9 Ag 2.4/ 2.1 AS b3s 751 698 3.8/ 3.5 2.4/ 2.4 5« 4.1/ 3.8 2.5/ 2.5 % 950 400 2.4/ 2.2 12.8/11.9 Au Au 2.8/ 2.6 15.5/13.9 k. B2u 800 ^ ' 1 1 4 - 799 796 b2u B3u 2.2/ 2.0 0.6/ 0.4 b2u b3u 2.4/ 2.2 0.8/ 0.5 B2u B3u ®A11 numbers are In cm-*. ■^Symmetry species are given for the higher components of the doublets. TABLE 21 CALCULATED FACTOR GROUP SPLITTINGS FOR IN-PLANE MODES OF PYRAZINE-d*,. USING SET II OF THE POTENTIALS MOLECULAR APPROXIMATE* OBSERVED POLA OBSERVED STRUCTURE REFINED STRUCTURE SPECIES (D2h) FREQUENCY FREQUENCY RIZA TION CALCULATED SPLITTING POLARI ZATION CALCULATED SPLITTING POLARI ZATION Ag 2209 1532 1003 876 6l4 0.1/0.1 0. 4/0.4 3.2/3.2 4.5/4.4 3.9/3.9 Blg 2* Ag BS is 0.1/0 .2 0.6/0 .6 4.7/4.6 5.9/5.7 6.0/6 .0 Blg ! l g aS Big B2g 2284 1505 1036 616 0.1/0 .0 o.4/0 .4 1.8/1.? 2.5/2.4 3 2g *p3s i z e 2g 0.1/0.0 0. 6/0 .5 2.1/2 .0 3. 6/3.4 b2® 5 8 Blu 2288 1370 1020 8 68 1023 1021 882 881 Blu flu 0,0/0 .0 0. 9/0 .9 2.4/2.4 1. 4/1.3 $ lu Blu 0.0/0.1 1.4/1.4 3.4/3.5 0.9/0.8 Au Au Blu Blu B3u 2282 1350 1264 84l 836 835 ?2u B3u 0.0/0.0 0.6/0.6 0.4/0.4 3.4/3.3 b2u b2u b3u b2u 0.0/0.0 0.8/0.8 0.6/0,6 4.0/3 .8 b 2u b 2u b3u B 2u ■^Frequencies of B^u and B^u modes are from Califano (21). TABLE 22 CALCULATED FACTOR GROUP SPLITTINGS FOR OUT-OF-PLANE_MODES OF PYRAZINE-dh USING SET II OF THE POTENTIALS® MOLECULAR APPROXIMATE OBSERVED POLA OBSERVED STRUCTURE REFINED STRUCTURE SPECIES (D2h) FREQUENCY FREQUENCY RIZA TION CALCULATED SPLITTING POLARI ZATION* CALCULATED SPLITTING POLARI ZATION* Blg 7 80 0.7/0.2 As 0.9/0.8 Ae B2u 609 404 609 608 * 1-03 403 B2u B3u 0.8/0.7 0.7/0.5 b2u B3u 0.8/0.8 1.0/0.7 b2u b3u ®A11 numbers are In cm"^. ^Symmetry species are given for the higher components of the doublets. TABLE 23 CALCULATED FREQUENCY SHIFTS FOR IN-PLANE MODES OF PYRAZINE-cU USING SET II OF THE POTENTIALS MOLECULAR SPECIES UNPERTURBED FREQUENCY OBSERVED STRUCTURE REFINED STRUCTURE ay (HIGH) AV (LOW) a^(HIGH) A^(LOW) As 3072 5.3/ 5.4 5.3/5.3 7.6/7.9 7.6/7.8 1577 1.0/ 1.1 1.0/1.0 1.6/1.7 1.4/1.6 1231 6.9/ 7.6 2.9/3.8 9.5/10.9 4.7/6.4 1014 4,7/ 4.7 0.3/0.4 6.9/7.1 0.5/0.8 618 4.3/ 4.3 0.7/0.7 6.5/6.8 1.1/1.3 B2g 3071 3.1/ 3 •1, 3.0/3.1 4.5/4.7 4.4/4.8 1527 0. 6/ 0.5 0.2/0.2 0.9/0,9 0.3/0.3 1368 4.8/ 4.4+ 1.4/1.2+ 6,6/6.5 2.5/2,6 642 4,0/ 3-9+ 1.4/1.4 5.6/5.7 2.0/2.3 Blu 3072 5.4/ 5-3+ 5.3/5.4 7.6/7.8 7.6/7.9 1482 3.3/ 3.6 1.0/1.4 4.7/5.4 1.6/2.4 1132 4.9/ 5.0 2.7/3.0 7.6/7.1 4.0/4.7 1021 4.0/ 4.2 1.1/1.3 6.2/6.7 1.4/1.9 b3u ?o6o 3.1/ 3*1. 3.1/3.1 4.5/4.7 4.5/4.8 1426 1.8/ 1.7+ 0.6/0.6 2.4/2.5 1.0/1.2 1346 0,8/ 0.7+ 0.2*/0.3*+ 1.1/1.2 .3*/0.3* 1075 5.7/ 5*3 3.0/2.7+ 8.0/7.6+ 4.9/4 ,7+ TABLE 24 CALCULATED FREQUENCY SHIFTS FOR OUT-OF OF PYRAZINE-d.0 USING SET II OF THE -PLANE MODES POTENTIALS MOLECULAR UNPERTURBED OBSERVED STRUCTURE REFINED STRUCTURE SPECIES FREQUENCY a/'(HIGH) A^(LOW) aV(HIGH) A/'(LOW) Blg 925 2.7/3.4 0.7/1.5 4.6/5.8 2.3/3.7 B3g 751 698 28.2/27.5* 24.6/24.1+ 32.9/32.2+ 30.6/29.8+ 38.2/38.1+ 46.6/46.7 34.3/34.4 44,3/44.3 AU 950 400 1.8/2.5 0.6/0.2 26.7/29.2 14.6/18.1 2.9/4,2 42.4/47.7 0.1/1.7 28.2/35.1 b2u 800 4l4 19.8/19.3* 17.7/17-3* 8.9/8.6+ 8.4/8.2+ 27.3/27.2+ 25.0/25.1 13.7/13.6+ 12,9/13.1 113 TABLE 25 CALCULATED FREQUENCY SHIFTS FOR IN-PLANE MODES OF PYRAZINE-d/j. USING SET II OF THE POTENTIAL FUNCTIONS MOLECULAR SPECIES APPROXIMATE FREQUENCYa OBSERVED STRUCTURE REFINED STRUCTURE a yThigh) AV (LOW) Ay (HIGH) Ay(LOW) A S 2209 3.3/3.3 3.2/3.2 4.7/4.9 4.6/4.7 1532 0.9/0.9^ 0.5/0.5 1.4/1.4 0.7/0.8 1003 4.3/4.l+ 1.0/1 .0 6.2/6.2 1.5/1.6 876 5.9/6.5 1.4/2.1 8.2/9.4 2.4/3.7 6l4 4.5/4.5 0.6/0.6 6.9/7.1 0.9/1.2 B2g 2284 1.9/1.9^ 1.8/1.9 2.8/2.9 2.7/2.9 1505 0.6/0.5 0.2/0.1+ 0.8/0.9 0.2/0.3 1036 2.7/2.5t 0.9/0.8 3.7/3.7 1.6/1.7 616 4.0/3,8 1.5/1.5 5.6/5.7 2.0/2.3 Blu 2288 3*2/3.2 3.1/3.2 4.5/4.6 4.5/4.7 1370 1.3/1.3 0.4/0.4 2.0/2.1 0.6/0.7 1020 4.0/4.0 1.6 / 1.5 5.7/5.8 2.3/2.3 868 4.2/4.6 2.7/3.^ 5.5/6.5 4.6/5.7 B3u 2282 2.0/2.0 2.0/2.1 2.8/2.9 2.8/3.O 1350 0.5/0.5 0.7/0.8 / 1264 0.6/0.6 0.3/0. 3^ 0.8/1.0^ 0.2/0.4 841 5.8/5.3+ 2.3 / 2.0 8.0/7.6 4.0/3 .8 frequencies of B]_u and B^u modes are from Califano (21) -t/TT TABLE 26 CALCULATED FREQUENCY SHIFTS FOR OUT-OF-PLANE MODES OF PYRAZINE-d^ USING SET II OF THE POTENTIAL FUNCTIONS MOLECULAR UNPERTURBED OBSERVED STRUCTURE REFINED STRUCTURE SPECIES FREQUENCY A/'(HIGH) A^(LOW) AV(HIGH) LOW) Blg 780 3.0/3.2 2.3/2.5 5.1/5.6 ^.2/4.8 B2u 609 ^■0^ 11.1/10.8+ 10.VlO.lt 10.9/10.2 10.2/10.1 15.9/15.9+ 1 6.8/16 .7 1 5.1/1 5.2 1 5.9/16.1 116 contribution has a stronger effect on the splittings and shifts of the refined structure with set II. The calculated magnitudes of exciton splittings do not agree well with the observed values; however, the signs (polarizations) are properly predicted in general and the observed widely split bands were predicted to be large to some extent, and the non-split and narrowly split bands were predicted to be small in the calculation. Thus, although these comparisons were based on only few split tings observed in the Infrared spectrum, the atom-atom 6-exp type potential used could be considered to be at least suitable for explaining the Intermolecular inter actions of this molecular crystal qualitatively, if not quantitatively. APPENDIX CARTESIAN DISPLACEMENT CALCULATIONS This part Is mainly installed to provide for the crystal studies of this work the unperturbed solutions of the molecular vibrational problem. Since no numerical cartesian displacements are available for pyrazlne in the literature, normal coordinate analyses were carried out for obtaining cartesian displacements. The normal coordi nate analyses were based on reference (18). The algebraic ■formulae for computing cartesian displacements are briefly outlined in Theory. Results are listed in the section of Calculations. Cartesian displacements of adamantane were also obtained in connection with another problem (reference paper submitted). Theory The normal vibrational analysis provides us with the normal coordinates Q. If cartesian displacement vectors g can be transformed to Qf the transformation matrix relating these two sets of coordinates will then give us the amplitudes of the atomic motions for each normal mode in term of a convenient orthogonal frame. 117 118 In building the relating matrix between the coor dinates Q and the steps performed are briefly summarized here algebraically. The displacement vectors { * can be transformed to the internal coordinates, by B matrix: 1 $ = B5 (1) The B matirx elements are determined by the geometry of the molecule. The normal coordinates Q are also related to the :internal coordinates 18 through the following relations: S = U*5 (2) s = L Q (3) The intermediate coordinates S are the symmetry coordinates. These are constructed in the analysis to factor the representations of different symmetry species, U matirx transforms internal coordinate frame to the intermediate symmetry coordinates which are then related to Q by the transformation matrix L. L matrix is part of the computer output when solving the secular determinant of normal coordinate analysis (not described here, but refer to reference (18)). The displacement vectors £ are therefore connected ; to the normal coordinates, Q, by relation (M combining j equations (1), (2) and (3)» I j B % = U* L Q (4) 119 F ro m the appendix of reference (18), w e obtain (L-1)+ = G -1 L . This is equal to L = G (L"1)+. T he relation between £ and Q thus becom es B K = U ’ G (L -1)+ Q . T he inverse kinetic energy matrix G can be constructed from the B matrixj G = U B M " 1 B ' U * w here M is the atomic m ass matrix. Therefore, B £ = U ' U B M " 1 B ' U * (L ”1)+ Q and £ = M "1 B ' U ' (L -1)+ Q , w here U ' U = E, the identity matrix, since the U matrix is orthogonal. T he cartesian displacements are thus obtained by computing this series of matrix multiplications for each normal m o d e Q . T he results for both pyrazine-d^ and pyrazine-d^ molecules are tabulated in Calculations. Those of adamantane, for the F 2 block only, are show n also in Calculations. 1 2 0 Calculations (1) In-plane modes of pyrazine-dQ and pyrazine-dj^i A s w as described earlier, the calculated normal frequencies of the in-plane m odes closely reproduce Scrocco*s spectra (26) using their valence force field. T he C N C angle of the molecular parameters should be 115° 58' instead of 115° B * * listed in reference (26). This is found out by a check of the numerical redundant coordinates. T he displacements in A are listed in Tables 2? and 2 8 for the ln-plane blocks A g, Bzg* ® 3 u * frequencies are the calculated values from Table 4 of reference (26) for pyrazine-d0. Those for pyrazine-d/j, are the calculated num bers from this w ork. T he lables of the atoms are as show n in the picture with axis system Indicated. z T N\ I 4 2 N j* A set of displacements calculated with this angle equal to 115° 8 * w as kindly provided by Dr. R . G . Snyder and it checked with our results using the sam e angle. J i I T A B L E 2? C A R T E S IA N D I S P L A C E M E N T S F O R IN -P L A N E M O D E S O F P Y R A Z IN E -d 0* 30?2 c m "^ 1014 -1 c m 618 A g c m -^ 1 2 3 1 „ -1 c m 1 5 7 7 c m “^ A T O M X z X z X z X z X z ! N 1 i 2 j .0000 .0000 -.0003 .0003 .0000 .0000 .1001 -.1001 .0000 .0000 -.1424 .1424 .0000 .0000 .0289 -.0289 .0000 .0000 -.0674 .0674 ! ci I -.0368 -.0368 .0368 .0368 -.0264 .0264 .0264 -.0264 .0975 .0975 -.0975 -.0975 .0603 -.0603 -.0603 .0603 .0897 .0897 -.0897 -.0897 -.0221 .0221 .0221 -.0221 c o 0 0 O O O O 0 0 0 0 0 r \ C r '» 0 ''\ ( r '\ OOOO .... I 1 -.0343 .0343 .0343 -.0343 -.0314 -.0314 .0314 .0314 .1218 -.1218 -.1218 .1218 H I i 2 I .3996 .3996 -.3996 '.3996 .2527 -.2527 -.2527 .2527 .0740 .0740 -.0740 -.0740 .1281 -.1281 -.1281 .1281 .0415 .0415 -.0415 -.0415 .0612 -.0612 -.0612 .0612 .2540 .2540 -.2540 -.2540 -.3904 .3904 .3904 -.3904 .1290 .1290 -.1290 -.1290 -.1074 .1074 .1074 -.1074 ( - * ■ PO TABLE 27 (CONTINUED) 1527 cm"l 6 4 2 -1 c m B 2g 3 0 7 1 c m -* 1368 c m -* ATOM X z X z X z X z N 1 2 -.1263 .1263 .0000 .0000 -.0699 .0699 .0000 .0000 -.0050 .0050 .0000 .0000 -.0531 .0531 .0000 .0000 Cl 2 3 4 .O 96O -.0960 -.0960 .0960 -.0398 -.0398 .0398 .0398 -.0839 .0839 .0839 -.0839 -.0975 -.0975 .0975 .0975 .0393 -.0393 -.0393 .0393 .0208 .0208 -.0208 -.0208 -.0101 .0101 .0101 -.0101 .0378 .0378 -.0378 -.0378 HI 2 I .0496 -.0496 -.0496 .0496 .0780 .0780 -.0780 -.0780 -.0980 .0980 .0980 -.0980 -.0862 -.0862 .0862 .0862 -.4019 .4019 .4019 -.4019 -.2504 -.2504 .2504 .2504 .2540 -.2540 -.2540 .2540 -.3817 -.3817 .3817 .3817 122 TABLE 27 (CONTINUED) B lu 1021 c m “^ 1132 c m "^ 3072 c m * * ^ 1482 c m -'* ' A T O M x z x z x z x z N 1 .0000 -.1299 2 .0000 -.1299 C l .0671 .0604 2 -.0671 ,0604 3 .0671 .0604 4 -.0671 ,06o4 H I .0048 .1835 2 -.0048 .1835 3 .0048 .1835 4 -.0048 .1835 .0000 .0402 .0000 .0000 .0402 .0000 .1000 -.0043 .0396 -.1000 -.004-3 -.0396 .1000 -.004-3 .0396 -.1000 -.0043 -.0396 .2558 -.2281 -.4018 -.2558 -.2281 .4018 .2558 -.2281 -.4018 -.2558 -.2281 .4018 -.0008 .0000 -.0687 -.0008 .0000 -.0687 .0214 -.0689 .0649 .0214 .0689 .0649 .0214 -.0689 .0649 .0214 .0689 .0649 -.2498 .1451 -.2956 -.2498 -.1451 -.2956 -.2498 .1451 -.2956 -.2498 -.1451 -.2956 TABLE 27 (CONTINUED) 3 0 6 9 c m -1 13^6 c m -1 ® 3 u 1075 c m -1 lif26 c m * 1 A T O M X z X z X z X z N 1 2 .00if2 .00if2 .0000 .0000 -.123if -.123if .0000 .0000 -.0779 -.0779 .0000 .0000 -.0if36 — • 0 4 -3 6 .0000 .0000 C l 2 3 if -.0361 -.0361 -.0361 -.0361 -.0255 .0255 -.0255 .0255 .06if5 .06if5 . 0 6 if5 .06if5 -.0781 .0781 -.0781 .0781 .053^ .0 5 3 * f .053^ .053^ .0678 -.0678 .0678 -.0678 .0051 .0051 .0051 .0051 .0975 -.0975 .0975 -.0975 H I 2 3 if .ifO O if .ifO O if .ifO O if .ifO O if .2537 -.2537 .2537 *.2537 .0892 .0892 .O892 .0892 -.1105 .1105 -.1105 .1105 -. 09if 3 -.09if3 -.09if3 -.09if3 .3292 -.3292 .3202 -.3292 .2ifl6 .2ifl6 .2ifl6 .2ifl6 -.2512 .2512 -.2512 .2512 ^Entries are In A , TABLE 28 CARTESIAN DISPLACEMENTS FOR IN-PLANE MODES OF FYBAZINE-di|* 2209 c n T * 1003 c m ”^ 61^ c m " ^ - 876 c m * * ^ 1 5 3 2 c m -1 L T O M X z X z X z X z X z N 1 .0000 .0001 .0000 .1 0 6 4 * .0000 -.1367 .0000 -.0270 .0000 -.0705 2 .0000 -.0001 .0000 -.1064- .0000 .1367 .0000 .0270 .0000 .0705 C l -.0520 - .0^50 .0903 .0 4 -9 4 - .0911 -.0193 .0008 -.0288 -.04-08 .1231 2 -.0520 .0^50 .0903 -.0 4 -9 4 - .0911 .0193 .0008 .0288 -.04-08 -.1231 3 .0520 .0^50 -.0903 -, 0 4 - 9 4 - -.0911 .0193 -.0008 .0288 .0 4 -0 8 -.1231 4 - .0520 -.04*50 -.0903 .0 4 -9 4 - -.0911 -.0193 -.0008 -.0288 .0 4 -0 8 .1231 D 1 .2596 .1680 .1 4 -5 3 .0198 .0390 .0769 .1 6 4 -6 -.2986 .0811 -.004-5 2 .2596 -.1680 .1 4 -5 3 -.0198 .0390 -.0769 .1 6 4 -6 .2986 .0811 ,0 0 4 -5 3 -.2596 -.1680 -.14-53 -.0198 -.0390 -.0769 -.164-6 .2986 -.0811 .0 0 4 -5 4 - -.2596 .1680 -.14-53 .0198 -.0390 .0769 -.164-6 -.2986 -.0811 -.00^5 TABLE 28 (CONTINUED) B2g 1 1505 c m -^ 6l6 c m ”'* ' 1 0 3 6 c m “^ 2284 c m * * ^ ITOM X z X z X z X z N 1 -.1295 .0000 -.0686 .0000 .0586 .0000 -.0170 .0000 2 .1295 .0000 • 0686 .0000 -.0586 .0000 .0170 .0000 C l .0851 -.0448 -.0824 -.0916 .0213 - .053? .0616 .0260 2 .0851 -.0448 .0824 -.0916 -.0213 -.0537 —.0616 .0260 3 -.0851 .0448 .0824 .0916 -.0213 .0537 -.0616 -.0260 4 -.0851 .0448 -.0824 .0916 .0213 .0537 .0616 -.0260 D1 .0827 .0617 -.0988 -.0869 -.1714 .2502 -.2630 -.1642 2 -.0827 .0617 .0988 -.0869 .1714 .2502 .2630 -.1642 3 -.0827 -.0617 .0988 .0869 .1714 -.2502 .2630 .1642 4 .0827 -.0617 -.0988 .0869 -.1714 -.2502 -.2630 .1642 126 TABLE 28 (CONTINUED) 999 cm” 1 CO H f a o Blu 1363 cm” 1 2285 cm" * 1 ATOM X z X z X z X z N1 2 .0000 .0000 -.0884 -.0884 .0000 .0000 .0833 .0833 .0000 .0000 .0951 .0951 .0000 .0000 -.0015 -.0015 Cl 2 3 4 .0995 -.0995 .0995 -.0995 .0478 .0478 .0478 .0478 .0102 - .0 1 0 2 .0102 - .0 1 0 2 -.0043 -.0043 -.0043 -.0043 .0844 -.0844 .0844 -.0844 -.0763 - .0 7 6 3 - .0 7 6 3 - .0 7 6 3 .0609 - .0 6 0 9 .0609 - .0 6 0 9 .0284 .0284 .0284 .0284 D1 2 3 4 .1543 -.1543 .15^3 -.1543 .0224 .0224 .0224 .0224 .1724 -.1724 .1724 -.1724 - .2 6 3 0 - .2 6 3 0 - .2 6 3 0 - .2 6 3 0 -.0117 .0117 -.0117 .011? .1237 .1237 .1237 .1237 -.2646 .2646 -.2646 .2646 -.1638 -.1638 -.1638 -.1638 127 TABLE 28 (CONTINUED) B3u 2279 -1 cm 1393 0 a 1 H1 1248 „ -1 cm 836 0 B 1 I - 1 ATOM X z X z X z X Z N1 .0121 .0000 -.0928 .0000 -.1197 .0000 .0265 .0000 2 .0121 .0000 -.0928 .0000 -.1197 .0000 .0265 .0000 Cl -.0512 -.0408 .0575 - .1 1 1 5 .0416 .0749 -.0358 - .0 3 3 7 2 -.0512 ,0408 .0575 .1115 . 04l6 -.0749 -.0358 .0337 3 -.0512 -.0408 .0575 - .1 1 1 5 .0416 .0749 -.0358 - .0 3 3 7 4 - .0 5 1 2 .04-08 .0575 .1115 .0416 -.0749 -.0358 .0337 D1 .2620 .1706 -.0196 - .0 0 2 9 .1678 -.0479 .1209 - .3 0 3 8 2 .2620 -.1706 -.0196 .0029 .1678 .0479 .1209 .3038 3 .2620 .1706 -.0196 - .0 0 2 9 .1678 -.0479 .1209 - .3 0 3 8 4 .2620 -.1706 -.0196 .0029 .1678 .0479 .1209 .3038 ^Entries are in A. 128 129 (2) Out-of-plane modes of pyrazine-d0 and pyrazine-d^i Among the studies of band assignments of pyrazlne, no definite conclusion has been drawn for out-of-plane modes, since only B2U is infrared active. Studies of pyrazine-d2 and pyrazlne-d^ still fail to give complete information for the assignments of species other than B2U« In this work, the crystal studies require data on carte sian displacements of the molecular vibrations. In order to analyze the out-of-plane crystalline spectra, the normal, mode calculations were needed. Fundamental frequencies of B2u species had been assigned in undeuterated and deuterated pyrazlne due to Infrared activity of this species. For normal modes of we assumed Callfano's assignment, the force constants in terms of symmetry coordinates of these two symmetry species were then no trouble to get. For B^g and A^ modes, transference from other molecules has to be made to obtain the force constants. Two frequencies, 751 cm*"^ and 698 cm~^ of Ito (3) were believed to belong to B j g in pyrazine-dQ. The pair at 950 cm-1 and 400 cm”- 1 ' was utilized for the Au species of pyrazine-dQ as assigned by Califano (21). With Just ; fundamentals from pyrazine-dQ for each species, only certain ranges of possible values of C-H bond out-of-plane bending force constants (in terms of symmetry coordinates) ■could be used while still keeping torsion force constants 130 of CCNC and NCCN as well as the bending-torsion force constants non-imaginary. Numbers between .5^9 and .65^ are good for B^g modes and .2^6 to 1 .1 5 7 for Au for the C-H constant in terms of symmetry coordinates. With the allowed C-H bending force constants assumed at first, torsion and bendlng-torsion force con stants (In terms of symmetry coordinates) were then obtain ed from pairs of two frequencies separately for B^g and Au j blocks. By doing so, the torsion and bending-torslon force constants strongly depend upon the values of C-H bending force constants assumed in fitting the observed frequencies. For B^g species, the allowed range of C-H bending force constant is small enough so that the calcu lated torsion and bending-torsion force constants should not be too far off, and consequently, the calculated cartesian displacements, exciton splittings and frequency shifts occurred in the crystal are expected to be reason ably good. For Au modes, the large range of C-H bending force constant makes the torsion and torsion-bending constants unjustifiably arbitrarily. Thus the listed calculated properties for A^ modes of pyrazine-dQ are * ! doubtful, especially the lower band at 400 cm which Is i mainly a torsion motion. In fact, the torsion and bending-j i torsion force constants could be checked by their consist- j :ency of values among all the out-of-plane symmetry blocks j ;if they are transformed to internal coordinates, involving 131 a "correction” for the redundancy conditions. However, it will be much less effort to achieve the check of the force constants once the fundamentals of pyrazlne-d^ are avail able, The analysis for B^g and Au modes of pyrazine-d^ was thus left out due to uncertainties of the force con stants solely determined from pyrazine-d0. Four force constants A, aQ, ap and am (in terms of Internal coordinates) of benzene (27) were transferred for |a trial calculation. Arbitrary adjustments were made on j ! jaQ and ap so that the symmetry-transformed force constants ; would agree with those of B2u and B^g computed above. The ;values are listed in Table 29. Numbers for benzene (27) are also given for comparison. Momentarily Sq was set at -.1268 and Bp at -.032. The computed results are tabulated in Table 30. On inspection of Tables 8 and 20, the calculated splittings for ^00 cm"1 of the Au species do look unusually; largej better C-H bending force constants than those of benzene apparently are needed for satisfactory calculations in the and A^ species. Since no experimental data on ! these two blocks are available, we are not too concerned about improving the results in this work, j Cartesian displacements were calcualted after the completion of normal coordinate analysis according to o Theory. Tables 31 and 32 show the results in A. Frequen- j i cles are the calculated values and the observed values if 132 TABLE 29 OUT-OF-PLANE FORCE CONSTANTS FOR PYRAZINE 0 (In the unit of mdyne/A) C-H BENDING OR BENDING-BENDING FORCE CONSTANTS USED . -j.. .............. . . 111 1 FORCE CONSTANTS FOR BENZENE* A .4434 .4434 ao - . 1 2 6 8 -.0689 £ L _ _ m .0024 .0024 ap -.0320 -.0202 ^Reference (27), TABLE 30 FUNDAMENTAL FREQUENCIES FOR OUT-OF-PLANE MODES OF PYRAZINE-dg AND PYRAZINE-d^ SPECIES PYRAZINE-do Califano Ito“ Obs. Calc. PYRAZINE-d^ CalifanoaObs. Calc. PRODUCT RULE Obs.Callfano Calc. Blg 925 (940) 925 (724) 780 1.28 1 .1 9 B3e 753 703 751 698 749 701 Au 950 400 950 400 b2u 786 417 800 4l4 800 4l4 597 609 400 404 610 404 1.35 1.37 1.34 Reference (21). “Reference (3). 133 TABLE 31 CARTESIAN DISPLACEMENTS FOR OUT-OF-PLANE MODES OF PYRAZINE-d0* ATOM Blg 925 cm-1 E 749 cm" s 3s 1 701 cm"1 *11 950 cm* 1 400 cm"1 b2u 800 cm"1 4l4 cm"1 N1 .0445 -.0295 .1653 .0462 -.0657 -.0349 .1615 2 -.0445 .0295 -.1653 .0462 -.0657 - .0 3 4 9 .1615 Cl -.110? - .0 3 6 8 -.1304 - .0 6 3 0 -.1614 -.095^ -.0685 2 . 110? - .1 0 6 6 .2605 - .0 5 6 7 .3317 .1361 -.1200 3 .1107 .0368 .1304 -.0630 -.1614 - .0 9 5 4 -.0685 4 -.1107 .1066 -.2605 - .0 5 6 7 .3317 .1361 -.1200 HI .3773 .4692 -.4261 .3918 -.5575 .39^5 -.0876 2 -.3773 .4692 -.4261 • 3918 -.5575 -.39^5 .0876 3 -.3773 -.4692 .4261 .3918 -.5575 .3945 -.0876 4 • 3373 -.4692 .4261 .3918 - • 5 5 7 5 -.3945 .0876 * Entries are in A. 135 TABLE 32 CARTESIAN DISPLACEMENTS FOR OUT-OF-PLANE MODES OF PYRAZINE-dij.* BlS B2u ATOM 780 cm- 1 609 cm“^ 4o4 cm-^ N1 .0527 - . 0 2 0 6 .1 7 1 0 2 -.0527 - . 0 2 0 6 .1 7 1 0 Cl -.1311 -.1218 -.0^45 2 .1311 .1^59 -.1551 3 .1311 - . 1 2 1 8 -.0445 4 -.1311 .1459 -.1551 D1 .2228 .2 2 7 6 -.0940 2 -.2228 - . 2 2 7 6 .0940 3 -.2228 .2 2 7 6 -.0940 4 .2 2 2 8 - . 2 2 7 6 .0 9 4 0 O ■ “■Entries are In A. 136 available. All the displacements are in y-directlon. (3) ?2 mocies adamantane, C^g F2 I s the only Infrared active species for this molecule, with its Td point group. Eleven modes belong to this symmetry. Frequencies are the computed values given for Identification. The numbering for the atoms and coordinate system are specified in the following pictures. viewing along axisi 0 carbon bonded to one H. Q carbon bonded to two H's. 3 ' 3-A o-K J s 2 1 I #a 20-2' c # _ S-O5 b# 4 ( '4—4 T T T( 5 4 y J - Force constants used were taken from a 36 parameter j ivalence force field derived by Snyder and Schachtschneider | :(28,29f30)» Table 33 gives the displacements in A. These j i \ results were used in interpretation of studies on the j I | [phase transition In crystalline adamantane (3l3L?__________ 1 : TABLE 33 | CARTESIAN DISPLACEMENTS FOR F2 SPECIES OF ADAMANTANE* - * - • " ' — — " ? ^ = ^ — ~ ~ ■■ - - - • ■ ■ - ■ ■■ ' » I 2912 cm”- * - 2856 cm"^ 2925 cm-^ ! | ATOM x y z x y z x y Ca .0000 -.0373 -.0268 .0000 t > .0000 .0373 - .0 2 6 8 .0000 c .0373 .0000 -.0268 .0030 d - .0 3 7 3 .0000 - .0 2 6 8 -.0030 1 .0000 .0000 .0065 .0000 2 - .0 0 1 7 .0 0 1 7 .0031 -.0010 3 1. .0017 .0017 .0031 .0010 4 - .0 0 1 7 - .0 0 1 7 .0031 -.0010 5 x .0017 - .0 0 1 7 .0031 .0010 6 .0000 .0000 .0065 .0000 Ha .0000 .3836 .2721 .0000 b .0000 -.3836 .2721 .0000 c - .3 8 3 6 .0000 .2721 - .0 3 3 5 d .3836 .0000 .2721 .0 3 3 5 1 - .0 2 9 5 .0000 -.0208 .3873 1* .0295 .0000 -.0208 -.3873 2 ,0010 .0022 -.0039 .0002 2* -.0022 -.0010 -.0039 .00*1-3 3 .0022 - .0 0 1 0 -.0039 -.0043 3’ -.0010 .0022 -.0039 -.0002 4 f . ■ -.0022 .0010 -.0039 .0043 .0010 -.0022 -.0039 .0002 5 -.0010 -.0022 -.0039 -.0002 5* y » .0022 .0010 -.0039 -.0043 6 .0000 .0295 -.0208 .0000 6* .0000 -.0295 -.0208 .0000 - .0 0 3 0 .0004 .0000 ,0010 -.0004 .0030 .0004 ,0000 -.0010 -.0004 .0000 .0004 -.0010 .0000 -.0004 .0000 .0004 .0010 .0000 -.0004 .0000 - .0 5 1 8 .0000 .0000 -.0014 .0001 - .0 0 0 5 .0240 -.0240 .0336 .0001 —•0005 -.0240 -.0240 .0336 -.0001 - .0 0 0 5 .0240 .0240 .0336 -.0001 - .0 0 0 5 -.0240 .0240 .0336 .0000 - .0 5 1 8 .0000 .0000 -.0014 .0335 .0197 .0000 -.0039 -.0032 - .0 3 3 5 .0197 .0000 .0039 - .0 0 3 2 .0000 .0197 .0039 .0000 - .0 0 3 2 .0000 .0197 -.0039 .0000 - .0 0 3 2 .0000 .2836 .0056 .0000 .0044 .0000 _ 1 _ .2836 -.0056 .0000 .0044 -.0043 .0028 -.0049 .2 696 -.1940 -.0002 .0028 -.2696 .0049 -.1940 -.0002 .0028 .2696 .0049 -.1940 - .0 0 4 3 .0028 .0049 .2696 -.1940 .0002 .0028 - .2 6 9 6 -.0049 -.1940 .0043 J. 1 .0028 -.0049 - .2 6 9 6 -.1940 .0043 .0028 .0049 - .2 6 9 6 -.1940 .0002 .0028 .2696 -.0049 -.19^0 -.3873 .2836 .0000 -.0056 .0044 G .3873 .2836 .0000 .0056 .0044 ^ TABLE 33 (CONTINUED) j 1302cm"l 8l6cm“^ lA57cm"*^ | ATOM x y z x y z x y z Ca .0000 .0323 -.0 7 1 2 .0000 b .0000 -.0323 -.0 7 1 2 .0000 s -.0 3 2 3 .0000 -.0 7 1 2 .0721 d .0323 .0000 -.0 7 1 2 -.0 7 2 1 1 .0000 .0000 .0692 .0000 2 .0058 - .0 0 5 8 .0017 .0320 3 i . - .0 0 5 8 - .0 0 5 8 .0017 - .0 3 2 0 A .0058 .0058 .0017 .0320 5 > - .0 0 5 8 .0058 .0017 -.0 3 2 0 6 .0000 .0000 .0692 .0000 Ha .0000 -.2200 .278A .0000 b .0000 .2200 .278A .0000 c .2200 .0000 .278A .1222 d -.2200 .0000 .278A -.1222 1 .0352 .0000 .0390 -.0190 1' -.0352 .0000 .0390 .0190 2 -.0?A? .0260 .0A9A -.1703 2* - .0 2 6 0 .0747 . 0A9A .0701 3 .0260 .07A7 .0A9A -.0701 3* .O7A7 .0260 .0A9A .1703 A - .0 2 6 0 -.07A7 .0A9A .0701 A* -.07A7 - .0 2 6 0 .0A9A -.1703 5 .07A7 - .0 2 6 0 ,0A9A .1703 5' .0260 -.07^7 . 0A9A -.0701 6 .0000 - .0 3 5 2 .0390 .0000 6* .0000 .0352 .0390 .0000 -.0 7 2 1 .0025 .0000 -.OlAO .0059 .0721 .0025 .0000 .OlAO .0059 .0000 .0025 .01A0 .0000 .0059 .0000 .0025 -.OlAO .0000 .0059 .0000 - .0 9 1 2 .0000 .0000 .OA93 - .0 3 2 0 .0AA1 -.0 0 5 1 .0051 .0033 - .0 3 2 0 .OAAl .0051 .0051 .0033 .0320 .OAAl -.0 0 5 1 -.0 0 5 1 .0033 .0320 .OAAl .0051 - .0 0 5 1 .0033 .0000 - .0 9 1 2 .0000 .0000 .OA93 -.1222 .06A0 .0000 - .0 0 5 1 -.0119 .1222 , 06A0 .0000 .0051 -.0119 .0000 .06A0 .0051 .0000 -.0119 .0000 .06A0 -.0 0 5 1 .0000 -.0119 .0000 -.0730 .3075 .0000 - .3 6 6 8 .0000 -.0730 - .3 0 7 5 .0000 -.3668 -.0 7 0 1 -.0013 .OlAA -.00A7 -.0121 .1703 -.0013 .00A7 -.OlAA -.0121 .1703 -.0013 -.00A7 -.OlAA -.0121 -.0 7 0 1 -.0013 -.OlAA -.00A7 -.0121 -.1 7 0 3 -.0013 .00A7 .OlAA -.0121 .0701 -.0013 .OlAA .00A7 -.0121 .0701 -.0013 -.OlAA .00A7 -.0121 -.1 7 0 3 -.0013 -.00A7 .OlAA -.0121 .0190 -.0730 .0000 -.3075 -.3668 -.0 1 9 0 -.0730 .0000 .3075 - .3 6 6 8 TABLE 33 (CONTINUED) 206cm"1 709cm"1 ATOM x y z x y .0062 -.0423 Ca .0000 b .0000 c - .0 0 6 2 d .0062 1 .0000 2 .0545 3 - .0 5 4 5 4 .0545 5 - .0 5 4 5 6 .0000 Ha .0000 b .0000 c -.0 0 5 1 d .0051 1 .0035 1* - .0 0 3 5 2 .2278 2* -.0002 3 .0002 3' - .2 2 7 8 4 -.0002 4* .2278 5 -.2278 5* .0002 6 .0000 6' .0000 -.0062 -.0423 .0000 -.0423 .0000 -.0423 .0000 -.0292 -.0545 .0429 - .0 5 4 5 .0429 .0545 .0429 .0545 .0429 .0000 -.0292 .0051 -.0407 - .0 0 5 1 -.0407 .0000 -.0407 .0000 -.0407 .0000 -.0342 .0000 -.0342 .0002 .1209 -.2278 .1209 -.2278 .1209 .0002 .1209 .2278 .1209 -.0002 .1209 -.0002 .1209 .2278 .1209 -.0035 -.0342 .0035 -.0342 .0000 -.0 5 1 5 .0000 .0515 .0515 .0000 - .0 5 1 5 .0000 .0000 .0000 .0540 - .0 5 4 0 -.0 5 4 0 -.0 5 4 0 .0540 .0540 - .0 5 4 0 .0540 .0000 .0000 .0000 .0119 .0000 -.0 1 1 9 - .0 1 1 9 .0000 .0119 .0000 .0094 ,0000 - .0 0 9 4 .0000 - .0 1 7 0 - .0 7 9 4 .0794 .0170 - .0 7 9 4 .0170 .0170 - .0 7 9 4 .0794 -.0 1 7 0 - .0 1 7 0 .0794 .0170 .0794 - .0 7 9 4 -.0 1 7 0 .0000 - .0 0 9 4 .0000 .0094 1110cm" 1 z X y z .0006 .0000 .0122 .0491 .0006 .0000 -.0122 .0491 .0 006 -.0122 .0000 .0491 .0006 .0122 .0000 .0491 .1141 .0000 .0000 - .0 3 5 2 .0444 .0078 -.0078 -.0 2 3 3 .0444 -.0078 -.0078 - .0 2 3 3 .0444 .0078 .0078 - .0 2 3 3 .0444 -.0078 .0078 - .0 2 3 3 .1141 .0000 .0000 - .0 3 5 2 .0953 .0000 -.1 2 1 9 .2514 .0953 .0000 .1219 .2514 .0953 .1219 .0000 .2514 .0953 -.1219 .0000 .2514 .1087 -.0141 .0000 -.0220 .1087 .0141 .0000 -.0220 .0782 .1546 - .1 0 5 4 -.1638 .0782 .1054 -.1 5 4 6 -.1638 .0782 -.1054 -.1546 -.1638 .0782 -.1546 -.1 0 5 4 -.1638 .0782 .1054 .1546 - .1 6 3 8 .0782 .1546 .1054 -.1 6 3 8 .0782 -.1546 .1054 -.1638 .0782 -.1054 .1546 -.1638 .1087 .0000 .0141 -.0220 .1087 .0000 -.0l4i -.0220 139 TABLE 33 (CONTINUED) ATOM X 1384-ciii~1 y z X 967cm“^ y z Ca .0000 -. 0441 -.0569 .0000 -.0 9 0 7 .014-6 b .0000 . 04-4-1 -.0569 .0000 .0907 .014-6 c .0441 .0000 -.0 5 6 9 .0907 .0000 .014-6 d -, 04-4-1 .0000 -.0569 - .0 9 0 7 .0000 .014-6 1 ,0000 .0000 .0308 .0000 ,0000 .0021 2 - .04-66 .04-66 .0650 -, 024-9 .024-9 -.04-25 3 i. .04-66 • 04-66 .0650 .024-9 .02^9 -.04-25 4 - - ,04-66 -.04-66 .0650 -.024-9 -.02^9 -.04-25 5 / ,04-66 -. 04-66 .0650 .0249 -, 024-9 -.04-25 6 .0000 .0000 .0308 .0000 .0000 .0021 Ha .0000 -.054-4- -.0816 .0000 -.1356 .0624- b .0000 .054-4- -.0816 .0000 .1356 .0624- c .054-4- .0000 -.0816 .1356 .0000 .0624- d -. 054-4- .0000 -.0816 -.1356 .0000 .0624- 1 -.0159 .0000 .0635 -.0196 .0000 .0301 1' .0159 .0000 .0535 .0196 .0000 .0301 2 .0828 -.0 9 1 6 -.1308 .1117 .14-33 ,114-1 2' .0916 -.0828 -.1308 -.14-33 -.1 1 1 7 .114-1 3 -.0 9 1 6 -.0828 -.1308 .14-33 - .1 1 1 7 .114-1 3’ I. -.0828 -.0916 -.1308 -.1117 • 14-33 .11^1 i, * .0916 .0828 -.1308 -.1^33 .1117 .114-1 4-* .0828 .0916 -.1308 .111? -.14-33 .114-1 5 -•0828 .0916 -.1308 -.1117 -.14-33 .114-1 5' -.0 9 1 6 .0828 -.1308 .14-33 .1117 .114-1 6 .0000 .0159 .0635 .0000 .0196 .0301 6' .0000 -.0159 .0635 .0000 -.0 1 9 6 .0301 *Entries are in A, REFERENCES 1. D. E. Williams, J. Chem. Phys. 4j>* 3770 (1 9 6 6). 2. P. J. Wheatley, Acta Cryst. 10, 182 (1957). 3. M. Ito and T. Shigeoka, J. Chem. Phys. 44, 1001 (1 9 6 6)., 4. X . Gerbaux and A. Hadni, J. Chem. Phys. 42,955 (1968). 5. D. A. Dows, Infrared Spectra of Molecular Crystals in "Physics and Chemistry of the Organic Solid State" Edited by D. Fox, M, Labes and A. Weissberger, Inter science Publishers, Inc., New York (19^5)* 6 . s. S. Mitra and P. J. Gielisse, "Progress in Infrared Spectroscopy" 2, Plenum Press, N. Y. (1964). 7. W. Vedder and D, F, Hornig, Infrared Spectra of Crystals In "Advances in Spectroscopy" 2, I89 (1961), Edited by H. W. Thompson, Interscience Publisher Inc., New York. 8. G. C. Pimentel, J. Chem. Phys. 12, 1536 (1951). 9. H. Eyring, J. Walter and G. E. Kimball, "Quantum Chemistry", John Wiley & Sons, Inc., New York (1944). 10. D. Fox and 0. Schnepp, J. Chem. Phys, 2^, 767 (1955). 11. D. S. McClure, Solid State Phys. 8, 1 (1959). 12. D. A. Dows, J. Chem, Phys. 32 , 1342 (i9 6 0). 13. D. A. Dows, J. Chem. Phys. 2836 (1 9 6 2). 1 14. M. Born and K. Huang, "Dynamical Theory of Crystal Lattices", Oxford University Press (1954). ______________ 141________ ___________ l42 ; 15. J. M. Ziman, "Electrons and Phonons", Clarendon Press, Oxford (i9 6 0). 16, s. H. Walmsley and J. A. Pople, Mol. Phys, 8, 3^5 (1964). I?. S. H. Walmsley, J. Chem. Phys. 48, 1438 (1 9 6 8). 18. E. B, Wilson, Jr., J. C. Declus and P. C. Cross, "Molecular Vibrations", McGraw-Hill Book Co., Inc. (1955). 1 19* T. Shimanouchl and M. Tsuboi, J. Chem. Phys. 1597 (1961). 20. I. Harada and T. Shimanouchl, J. Chem. Phys. 44, 2016 (1966). ; 21. s. Califano, G. Adembri and G. Sbrana, Spect, Acta 20, 385 (1964). 22. H. Winston and R, S. Halford, J. Chem. Phys. 1£, 607 (1949). 23. D. E. Williams, J. Chem. Phys. 4424 (1965). 24. D. E. Williams, J. Chem. Phys. 4£, 4680 (1967). 25. G. Taddei and E. GIglio, J. Chem. Phys. 2768 (1970;. ) . 26. M. Scrocco, C. di Lauro and S. Califano, Spect. Acta ; 21, 571 (1965). ! 27- E. J. O'Reilly, J. Chem. Phys. j?l, 2206 (1 9 6 9). j 28. R, G. Snyder and J. H. Schachtschneider, Spect. Acta ! 21, 169 (1965). 29. R. G. Snyder and J. H. Schachtschneider, Spect. Acta 1^3 12. 85 (1963). 30. J. H. Schachtsehneider and R. G. Snyder, Spect. Acta 12, 117 (1963). 31. P. Wu, L. Hsu and D. A. Dows, J. Chem. Phys. (in press).

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Creator Hsu, Lina (author)

Core Title Intermolecular Forces, Exciton Splittings And Lattice Vibrations Of Crystalline Pyrazine

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Degree Doctor of Philosophy

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Advisor Dows, David A. (committee chair), Beaudet, Robert A. (committee member), Spitzer, William G. (committee member)

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