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The infrared spectrum and structure of crystalline barium formate Morrow, Barry Albert 1962

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THE INFRARED SPECTRUM AND STRUCTURE OF CRYSTALLINE BARIUM FORMATE by BARRY ALBERT MORROW B.Sc. University of British Columbia, 196l A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Chemistry We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1962 In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia, Vancouver 8, Canada. y Date ABSTRACT The absorption of polarized infrared radiation by single crystals of barium formate has been recorded between liOOO and £00 cm~\ Factor group splitting has been observed for molecular modes and for some of the overtones and combinations. Lattice mode frequencies are inferred from combinations with molecular modes. The infrared spectrum of polycrystalline lead and sodium formate, and of aqueous barium formate has been similarly recorded and compared with that of crystalline barium formate. The nuclear magnetic resonance spectrum of barium formate has been recorded, and a discrepancy between the theoretical and experimental second moment indicates that a previously proposed crystal structure is incorrect. However, an X-ray and infrared study of a single crystal confirm that the factor group of barium formate is P2]_2i2]_ and that there is no degradation of the crystal symmetry due to a possible distortion of the C-H bond of the formate ion. It is concluded that the coordinates of the carbon and oxygen atoms are incorrect in the original crystal structure determination. The nuclear magnetic resonance spectrum of lead formate also indicates that the crystal structure of this compound is in error, but no attempt is made to elucidate the correct structure. i ACKNOWLEDGEMENT I sincerely wish to thank Dr. K. B. Harvey for his encouragement and assistance in carrying out this study, and Mr. J. A. Donnan of the Geology Department for donating the materials for grinding and polishing the crystals. The help of the n.m.r. group for recording the n.m.r. spectra, Mr. N. Camerman for assisting with the x-ray analysis, and Mr. R. Muehlchen for the construction of the polarizor i s also appreciated. Finally, I would thank the National Research Council of Canada for a Bursary. i i i Contents Page Acknowledgment i Abstract i i Contents i i i List of Tables iv List of Figures v Chapter 1: Introduction 1 1-1. The Problem 1 1- 2. Methods of Investigation 3 Chapter 2: Experimental Work 5 2- 1. Materials investigated spectroscopically 5 2-2. Growth of single crystals of barium formate 5 2-3. Grinding 9 2-ii. Apparatus 11 2- 5 . Spectra 12 Chapter 3: Theory 19 3- 1. General theory of vibrations in^crystals 19 3-2. Application of symmetry 21 3-3. Crystal symmetry 2 5 3- U. Analysis of vibrations 26 Chapter k' Discussion 29 ii-1. Infrared spectra 29 4- 2. N.M.R. results hO ii-3. Crystal structure of barium formate J|2 I4-4. Conclusion llll Appendix: Factor group analysis of sodium formate 46 Bibliography 1*7 iv List of Tables page Table 1: Infrared Spectra?. 12 2; The Infrared Absorptions of Barium Formate 16 3 : Summary of previously Reported Raman Absorptions and of Observed Infrared Absorptions of Sodium Formate and Barium Formate-d^ 18 k'. Character Table and Factor Group Analysis of Barium Formate Under 222 30 5: Lattice Modes of Barium Formate 35 6 : Calculated C3^ Frequency Shifts of Formate Ion 38 7: Product Rule Frequency Shifts of C1^ Formate Ion 39 8: Factor Group Analysis of Barium Formate Under 2 1*3 V List of Figures-To follow page Figure 1: Infrared, spectrum of polycrystalline barium formate 13 2'r Infrared spectrum of polycrystalline barium formate-d^ 13 3: Infrared spectrum of barium formate single crystal (be), 2600-1000 cm"1 13 1*: Infrared spectrum of barium formate single crystal (ac), 2600-1000 cm"1 13 Polarized infrared spectrum of barium formate single crystal, 3300-2^ 00 cm"1 Hi 6t Polarized infrared spectrum of barium formate single crystal, 2000-1000 cm-1 Hi 7: Polarized infrared spectrum of barium formate single crystal, 800-71*0 cm"1 Hi 8r Infrared spectra of barium formate, 800-7UO cm"1 (a) ca single crystal (b) cb single crystal (c) polycrystalline in KBr pellet 1$ 9t Infrared spectra, 3200-2600 cm"1 (a) Aqueous solution of barium formate (b) Polycrystalline sodium formate in KBr (c) Polycrystalline lead formate in KBr 15> 10: (a) Crystal structure of barium formate projected on (001) (b) Projection of the positions of the protons of barium formate on (001) 29 CHAPTER 1: INTRODUCTION 1-1 The problem Investigations of molecular motions by means of infrared and Raman spectroscopy have yielded a vast amount of information about the forces which hold atoms together to form molecules. However, the study of the vibrations of single molecules is limited to the gas phase, where interactions with other molecules are generally small. Procedures for determining force constants and fundamental frequencies of free molecules are well developed (1). It is only relatively recently that procedures shave been developed for analysing the motions of molecules in condensed systems, and in particular, in crystals. The experimental difficulties in obtaining extremely thin single crystals (30 to 100 microns) deterred earlier workers, at least while there was a store of substances which could be conveniently studied as vapors. Some work was done on the Raman spectra of single crystals around 19aO, where much thicker crystals can be used than in the infrared technique, but the very slow progress in Raman spectrometer instrumentation has discouraged extensive work in this field. When the experimental difficulties can be overcome, much valuable information can be obtained concerning the nature of crystalline forces. The most important difference between the spectrum of a substance in the vapor and crystalline state is that in the latter, many of the corresponding vapor phase frequencies are split into doublets and triplets, or may even be broadened into a "band" with considerable fine structure., The nature of this splitting will be discussed later. When a molecule having a certain degree of symmetry is built into a crystal having a lower degree of symmetry, some of the degenerate modes may 2 be split in the infrared spectrum of the solid. Since the orientation of a molecule in a crystal is fixed, and i f the crystal structure is known from X-ray data, the use of polarized radiation in recording the spectrum of the solid may aid in the vibrational assignments. Conversely, i f the crystal structure is not known, the infrared spectrum of the crystal may point to the correct crystal structure. This technique for determining crystal symmetry is particularly useful for small molecules which have low melting points, and for hydrogen containing compounds where X-ray methods may not be applicable. Halford and Hornig have presented several papers in the last decade illustrating the use of infrared spectroscopy is determining the crystal structure of several small molecules. (2 - 5 ) . A salt of the formate ion was chosen for this study since the modes of vibration of an XYZ2 type molecule are well characterized (6). The reasons for the choice of the barium salt are outlined in Chapter 2. The main purpose of this study was to test the theories developed by Bhagavantam and Venkatarayudu (7,8), and Halford (9) for the treatment of vibrationsia crystals, and in particular, in a more complicated system than these workers have worked with. Of further interest in the case of barium formate is a discrepancy between the data obtained on the crystal structure from X-ray analysis (11), and that obtained from nuclear magnetic resonance studies on single crystals. (10) X-ray data give- no information on the positions of the protons of the formate ionj however, using an average C-H bond length of 1.09 A, the position of the proton can be estimated in the crystal, assuming the C-H bond is along the line bisecting the 0C0 angle. Saito et. al, (10) investigated single crystals-of barium formate by N.M.R. and did not find agreement between 3; the predicted line shape and the experimental line shape. These workers concluded that the crystal structure data must be in error. This conclusion is further substantiated by the fact that the calculated distance between the barium ion and." its nearest hydrogen is only 1.3 A using the X-ray data, a value less than the sum of the radius of the barium ion and hydrogen, which is 3.2 A. Hence, i t was hoped that our investigation of the infrared spectrum of barium formate could throw some light on this discrepancy. As far as could be found from extensive searching of the literature by this author, no report has ever been publishedion the infrared spectrum of any inorganic formate. However, studies have been carried out on the Raman spectrum of several formates. Gupta (16) reported the Raman spectrum of sodium and cadmium formate in aqueous solution, and of cadmium, barium and lead formates as crystals. But, very few frequencies were reported, most of them being of doubtful accuracy. Lecomte (17,18) has reported the Raman spectrum of aqueous sodium and barium formate, with incorrect assignments, and does not report a l l the frequencies. Fonteyne (19,20) did a very accurate Raman study of aqueous sodium formate and sodium formate-d-^, made a complete vibrational assignment, and calculated the force constants of the formate ion, using a Urey-Bradley potential function. This latter work will be referred to in the discussion, as i t is the only known vibrational study of any accuracy of the formate ion in either solution or crystal. 1-2 (a) Methods of Investigation The infrared spectra of polycrystalline barium formate in KBr, of single crystals of barium formate with polarized radiation, of a solution of barium formate i n water, and of deuterated barium formate in KBr were recorded. X-ray photographs with a precession camera were taken to determine the space group of crystalline to barium formate. Nuclear magnetic resonance studies were carried out on polycrystalline barium formate and lead formate to determine the second moment. The infrared spectrum of sodium formate and of lead formate were recorded for comparison with that of barium formate. The crystal symmetry of lead formate is reported to be the same as that of barium formate, and therefore should have a similar infrared spectrum. 5 CHAPTER 2: EXPERIMENTAL WORK 2-1 Materials investigated spectroscopically Barium formate Barium formate-d-|  Sodium formate  Sodium formate-d-^ Lead formate (a) Prepared by the neutralization of formic acid with barium carbonate, filtered, and recrystall-ized three times from solution. (b) Bios Laboratories, reagent grade. In both cases there was a slight contamination of insoluble material, probably barium carbonate, which was removed from solution by filtration. Prepared as above (a) with formic acid-d-^. Baker and Adamson reagent grade. Merck Chemical Company Prepared by crystallization from a solution of sodium formate and lead nitrate, and recrystallized twice. 2-2 Growth of Single Crystals of barium formate A salt of the formate ion was chosen for this study since i t is a small ion whose vibrational spectrum is well characterized. It was desirable to have a salt which is neither a hydrate when crystallized from solution, nor hygroscopic at room temperature. Only the barium, calcium and lead salts had these desirable properties. The simplest method of growing large single crystals from solution is by slow evaporation of a saturated solution of the material being considered. With this procedure, a small seed crystal is usually mounted in the solution, and growth proceeds with evaporation, Lead 6 formate was then eliminated as it is relatively insoluble in water at room temperature. However, calcium and barium seed crystals were grown very easilyj when a one cm. layer of saturated solution was placed in a 7 cm. (diameter) evaporating dish covered with a f i l t e r paper, small seeds up to two ram. in length appeared in two to three days. The crystals of calcium formate were quite irregular, and it was felt that these would be much more difficult to work with than the symmetrical bisphenoids of barium formate. Hence, the choice of barium formate for this study. Several methods were tried in order to grow a large single crystal, one to two centimeters in length, from the small seeds. In the first attempts, seeds were suspended by thread, hair, or thin platinum wire, into saturated solutions of barium formate, in a 150 ml. beaker. The "support" was tied around a small seed, (1-2 mm. in length) and was suspended from a glass rod placed across the top of the beaker. A paper "tent" was placed over the beaker to keep dust particles out., Fairly large (1-1.5 cm.) but often imperfect crystals resulted; the rate of evaporation was probably too great in an open 150 ml. beaker. Further complications arose from the great clusters of smaller crystals that grew along the support, often growing into the main crystal itself. In order to obtain a suitable crystal by this method, the main crystal would have to protrude at least 10 to 15 mm. on each side of the support, so that there would be a clear section to grind. This was not realized and the method was discarded. Since i t appeared desirable to slow down the rate of evaporation, f i f t y ml. conical flasks were used in further trials. A seed was placed on the bottomeof a flask, containing about one cm. of saturated solution, and the flask was placed in a cupboard where the dust circulation from 7 the air was minimized. About one out of five crystals grew to an appreciable size (1 to l.S> cm. in length) before a mass of smaller seeds grew on the bottom of the flask. However, these larger crystals were not too well developed on the bottom face, and always had many flaws. The large crystals from the above method grew in about three weeks. However, i t was noted that very frequently when a small seed was placed in such a flask, after two to seven days the crystal would have perhaps quadrupled in volume being six or seven mm. in length, by 2mm. wide. These medium size seeds were usually flawless, and were used in further experiments to be described. When reference is made to a medium seed, i t will mean one of the type described in this paragraph. An unsuccessful attempt was made to grow large crystals by a slow cooling method. From the solubility-temperature data for barium formate, it was estimated that about two grams of material should deposit from one hundred ml. of saturated solution in cooling from forty to thirty degrees C. A medium seed was cemented vertically to the end of a small coil of plastic coated copper wire ("CIL" Cement was used) and was placed in a jar (13 cm. high, 7 cm. in diameter) containing four hundred ml. of saturated solution at h£ degrees C. The jar was placed in a thermostatically controlled bath at the same temperature and a rubber stopper with a small hole to allow a glass stirring rod to be inserted was placed over the jar. The solution in the jar was stirred at seven to ten r.p.m., while the temperature of the bath was lowered 0.2 to 0.U degree C. per day until the temperature was around 30 degrees C. Several attempts were made to grow crystals by this method, but without success. Only slight growth occurred in most cases. 8 Evaporation procedures were again resorted to. Two methods proved very successful and w i l l be described. Both entailed the ve r t i c a l mounting of a medium seed on the end of a c o i l of wire, i n the manner described in the last paragraph. In one method, the c o i l of wire with the seed was placed in a f i f t y ml. conical flask, f i l l e d nearly to the top with saturated solution. This gave a very small surface area, and hence, very slow evaporation. After three to six weeks a large crystal (15 to 20 mm. above the end of the wire) grew, which could be broken off at the wire by twisting the crystal slowly. About two out of five crystals grown by this method were completely flawless, and were perfect for grinding. Even seme of the others with flaws were usable. In the second method, the c o i l with the seed was placed in a jar paper covered the top, with a small hole for a st i r r e r , turning at about seven to nine r.p.m. Due to the large volume of solution, the rate of supersaturation of the solution was slow, even though the rate of evaporation was fast. Hence, after ten to twelve days a large single crystal grew from the seed:, one specimen was 25 mm. long, but the others were stopped before this size was reached. The crystals grown by this method frequently had several flaws, but often the top centimeter of the crystal was flawless, and could be broken off from the rest-, of the crystal. Again, about two containing about four hundred ml. of saturated solution. A f i l t e r 9 out of five trials were successful. 2-3 Grinding In order to study the infrared absorption spectrum of a single crystal, the crystal generally must be around forty to f i f t y microns in thickness. Thinner crystals would be desirable, but this is usually mechanically unfeasible. For this work, the crystal had to have a cross section of 10 mm. x 1.5 mm. in order completely to cover the light beam. Hence, it was necessary to start with a crystal at least one cm. in length, usually being four to five mm. wide-. About a dozen suitable crystals were grown, most of them being fifteen to twenty mm. in length. Grinding was done on a plate glass lap, with a suitable abrasive and lubricant. In barium formate, the crystal was ground perpendicular to the a or b axis, (the crystal axes are those designated by Groth (22), and are shown below) until the maximum cross5 section was obtained. Hence, grinding proceeded from two edges, to the center of the crystal. external shape of the barium formate crystal k end view b GRINDING 1st stage 2nd stage For the first stage, the crystal was held by hand and was ground by slowly moving i t back and forth, over the face of the lap, with slight i 10 pressure applied. Kerosene was used as a lubricant at a l l times. Initially a carborundum abrasive #600 was employed; this cut very quickly and was used until the crystal was to within ten or fifteen percent of the size desired. Then a finer bauxite abrasive #950 was used for the remainder of the grinding, giving a flat, relatively smooth surface. Finally, the fresh surface was polished on a piece of silk cloth held over a flat glass plate. A drop of kerosene was initiall y used, but was done away with for the final polishing. At this stage the other half of the crystal could be ground. The polished face was cemented with Radio Service Cement to a sodium chloride plate. Care was taken to see that the entire surface was cemented down in order to eliminate the possibility of chipping when the crystal became very thin. This particular cement was used as i t could be applied in a reasonably thin layer, and is very soluble in acetone, which was used to remove the crystal from the NaCl plate. The reason for using kerosene as a lubricant now becomes clear. Both water soluble barium formate and the cement are insoluble in kerosene. Other liquids have this property, but a desirable lubricant for this-work should not have too high a viscosity or volatility. The grinding with the two abrasives then proceeded in the same manner as described. When the crystal was down to about 150 microns thickness, grinding proceeded wery slowly with only slight pressure on the plate,, as fracturing easily occurred. Care was taken to ensure that the thickness was as uniform as possible. The grinding continued until the crystal was the-iiesired thickness (I4O to 50 microns); i t was deemed thin enough when the strongest absorption bands of the formate ion were resolvable in the infrared (ignoring the absorption due to the cement). The thin section 11 was polished slightly on the silk cloth, but frequently severe fragmentation occurred, and crude polishing only was possible for this surface. Uniformity of thickness and extinction was checked under the polarizing microscope. The thin crystal section was removed from the NaCl plate by dissolving off the cement in acetone. This usually required one to three days. Shattering occurred i f the crystal was "pushed" to remove i t from the plate. Instead, the acetone solution was swirled periodically, and eventually the crystal become free, moving slightly over the surface of the NaCl plate. The crystal was then transferred carefully to a brass plate having a narrow s l i t for use in the spectrometer. The transfer was done in the acetone solution by placing the brass plate at the same level as the NaCl plate, and slowly pushing the crystal with a wooden toothpick over to the slit.. By means of the polarizing microscope, one principal axis of the crystal was alligned parallel to the s l i t - The crystal was cemented to the brass plate with an acetone insoluble epoxy resin cement, and was washed by dipping in acetone. The polarizer was constructed by mounting two sets of three l / 2 mm. thick silver chloride plates in the form of a "V", so that light passed through both stacks of plates at Brewsters angle to each set. It was similar to that described by Charney (12). 2-ii Apparatus A l l spectra were run on a Perkin-Elmer dual grating spectrophotometer, model U21. The n.m.r., spectrometer used was a Varian Associates Vli200/1}300B 12 instrument operating at 1*0 megacycles per second and 9395 gauss for proton resonance. For the X-ray analysis, precession photographs were taken of the hO/, h l i , OkJ?, and lkJ? zones of a single crystal of barium formate. 2-5 Spectra Table 1 liststhe infrared spectra shown in Fig. 1-9. If the spectrum was recorded using polarized' light, the direction of polariz-ation is indicated as being along axis a, b, or c. A crystal section that was ground along the a orb axis is listed as a be or ac type crystal respectively. TABLE 1  Infrared Spectra Fig. No. Description Polarization 1 Ba(CH02)2 -polycrystalline in KBr 2 Ba(CD02)2, -polycrystalline in KBr 3 Ba(CH02)2 -single crystal, be li ti -single crystal, ac 5 II -single crystal, 3300-2500 cm"1 a,b,c 6 ti- -single crystal, 2000-1000 cm"1 a,bj c 7 tt -single crystal, 800—71*0 cm_1 a,b,c 8 it -single crystal ac and be and polycrystalline in KBr, 800-71)0 cm-1 9 Ba(CH02)2 aqueous, Pb(CH02)o 3n KBr, and NaCH0 2 in KBr, 3 2 0 0 - 2 6 0 0 c m-l 13i Table 2 lists the observed infrared absorption frequencies of crystalline barium formate, along with assignments and polarization properties. Table 3 lists previous data for barium formate, sodium formate, and observed frequencies for sodium formate and barium formate-d^. Figure 1: Infrared spectrum of polycrystalline barium formate. Discontinuity at 970 cm' is part of background. Figure h: Infrared spectrum of barium fornate single crystal (ac) 2600-1000 cm"1 Figure 5 : Polarized infrared spectrum of barium formate, 3300-2500 cm - 1. Figure 6 : Polarized infrared spectrum of barium formate, 2000-1000 cm"1. Figure f: Polarized infrared spectrum of barium formate, 8OO-7I4.O cm - 1. The plane of polarization of the infrared light is indicated as being par a l l e l to the a, b or axis of the crystal. 4 1 1 1 f , 1— 800 780 760 740 Frequency cm Figure 7 15 Figure 8 : Infrared spectrum of barium formate, 800-7l|0 cm"1. (a) ac crystal (b) be crystal (c) polycrystalline in KBr. An ac type crystal is one in which both the a and c axes of the crystal are perpendicular to the direction of propagation of light through the crystal. Figure 9: Infrared spectra, 3200-2600 cm-1 (a) an aqueous solution of barium formate (b) polycrystalline NaCH02 in KBr (c) polycrystalline Pb(CH02)2 in KBr. 16 TABLE 2 The Infrared Absorptions of Barium Formate Frequency cm Polarization Assignment 744 w c v 3 (C 1 3) 749.5 vw b v 3 (C 1 5) 752 vw c 758 m a V3 ( B 3 ) 761.5 m c,b v (B.B ) 766.5 vs* c v ( B j 769.5 vs* a,b v 3 (B 2B 3) 1000 vw VL 1066 m c,a V6 ( B 1 B 3 } 1073 m c,b V6 ( B1 B2 } 1110 vw V L 1165 vw V L 1230 m VL 1280 w 1334 w 1350 s* V l 1357 s* v l 1386 s* V r. 1398 s* jv5 1465 vw VL 1535 w 2v 3 or v 4 - v L 1570 vs* v4 1590 vs* v4 1630 m a,b V4 + VL ( B 2 B 3 } 1640 m a,b V4 + VL ( B 2 B 3 ) 1680 vw V4 + VL 1720 vw a,b V4 + VL ( B 2 B 3 } 1750 m c V4 + VL ( V 2110 vw v„ + V, 3 1 2150 vw V + V 3 5 17 TABLE 2 (con't) Frequency cm 1 Polarization Assignment 2525 vw V3 + V4 2420 vw V6 + v l 2645 w V6 + V4 2700 m a,b,c 2vx (B1B2B3) 2740 m a,b(vw) v x + v 5 (B )^ 2750 s e v 2 (B x) 2763 m b,a(vw) 2v5 (B 2) 2823 s c 2863 s a,b v, (B2B3) 2910 3 a,e(w) V l + V4 ( B 3 ) 2920 s b V l + V4 ( B 2 } 2935 b,m a,b V l + V4 ( B 2 B 3 ) 2975 s c,b v 4 + v 5 ( B l B 2 ) 2985 s c v 4 + v 5 (Bx) 3020 w c 3134 w b,c 2v4 ( B lB 2) 3200 w c * observed in KBr pellets of barium formate symbols: vs = very strong, s = strong, m = medium, w = weak, vw = very weak, 13 b = broad, v = lattice mode, C = carbon 13 formate ion. 18 TABLE 3 Summary of Previously Reported Raman Absorptions and of Observed Infrared  Absorptions of Sodium Formate and Barium Formate-d., Ba(CH02)2 R(18) Ba(CH02)2 R(l6) NaCH02 R(20) NaCH02 IR Ba(CH02)2 IR 735 790 1070 1290 2832 773 (v3) 1069 (vg) 755 (v 3 C 1 3 ) 772 (v3) 1062 (vg) 1353 1352 1355 (v1) 1386 (v5) 1385 (v5) 1520 1584 < V 1590 (v4) 1630 2126 ( 2 V 6 ) 2732 (2v5) 2717 2825 (v2) 2828 (v2) 2950 (2v4) 745 (v3) 755:(v3) 908 (vg) 912' (v6) 1025 1320 1330 1350 1390 1545 1570 2046 2054 2080 2116 2163 2645 2700 2745 2820 2862 2881 2900 2955 3370 3440 3479 V V V H) » 5 H) V V 2v5) 2vx) 2vx H) v 2 H) v 2 H) 2v2 H. v 2 + v 3 ) V l + V v, + v 3) v 2 + V V2 + v 2 + v x ) Vl> 13 symbols: H = hydrogen species of formate-d^. C = carbon 13 formate ion. R = Raman data. IR = Infrared data, this work. 19 CHAPTER 3: THEORY 3-1 General theory of vibrations in crystals In studying the dynamics of a system of N particles, i t is necessary to specify 3N coordinates to describe the configuration of the system. These 3N coordinates generally are the three cartesian coordinates of each particle, the coordinates of the ith particle being Xj,, y^, and z^. Such a system has 3N degrees of freedom; three translational, three rotational, and 3N-6 vibrational degrees of freedom. In a vibrational analysis, of a system of N particles, an expression for the kinetic and potential energy is derived and is substituted into Newton's equations of motion. The solution of the 3N, equations obtained, depends on a parameter related to a frequency of oscillation of the system, of which there are 3N-6 possible values. These values are the fundamental frequencies or NORMAL modes of vibration of a system of N particles. The analysis of the vibrations which can occur in a crystal would appear to be most difficult i f the above procedure were to be adopted. 23 An average crystal contains of the order of 10 atoms, which would 23 correspond to 3 x 10 fundamental modes of vibration. On this basis, one might expect the vibrational spectrum of a crystal to be totally absorbing, there being 102-* absorption bands. However, i t is observed that there are a finite number of bands in the infrared and Raman spectra of crystals; in fact there are often not too many more than observed in the spectrum of the vapor. It is then necessary to consider whether a l l of the 3N vibrations which a crystal should sustain are physically distinct. 20 Born.',s treatment of the dynamics of crystal structures (13) showed that a three dimensional crystal should be able to sustain an infinite number of vibrations, the wave motion being of the nature of a continuum. However, an infinite number of lines is not observed in the infrared or Baman spectra of a crystal. Furthermore, there is often a close correspondence in the spectrum of a melt and that of the crystal of many substances, a fact which suggests small units-of-. atoms or subunits of this are very important in determining the nature of the normal oscillations. As a start in determining the vibrations of crystals, earlier workers considered the unit cell, which is the smallest repeating unit of the crystal, (lit) The theory adopted for present day use will be explained briefly in the following paragraph. Situated on the Bravais lattices of a crystal are sets of equivalent atoms, ordered in such a way that each atom in a set (or lattice) is both geometrically and physically related to its environment in exactly the same way as every other atom of the same set. Hence, the corres-ponding displacements in: a given normal mode of oscillation of every pair of equivalent atoms contiguously situated along any one of the three principal axes of the Bravais lattice should be the same. Thus, a crystal may be considered as a structure or a set of interpenetrating lattices, and the unit cell of the smallest possible size (that is the primitive unit cell) will contain as many atoms in i t as there are interpenetrating lattices in the structure. Yith p atoms in the unit cel l , there will be 3p oscillations, and three of them will correspond to pure translations. In these modes, the lattices can be regarded as moving in phase with each other. 21 When treated in this manner, a suitable number of frequencies is usually accounted for, but this is not the most general treatment. One has not taken account of the fact that the Bravais lattices may also vibrate out of phase with respect to each other, and that one should in the most general treatment consider a block of eight unit cells. This is the result of the Born-Karman boundary conditions. The physical significance of this general treatment is that we have considered the effect of the environment and possible interactions from other unit cells surrounding the one we are considering. However, the justification in the simplified treatment lies in the fact that a suitable number of frequencies is accounted for, and there is no need to make the treatment unduly complicated. 3-2 Application of symmetry In carrying out a vibrational analysis of a dynamical system of N particles, a 3N by 3N secular determinant has to be solved in order to determine the normal frequencies. However, i f the system (or molecule, as is usually the case in infrared spectroscopy) posseses symmetry, the secular determinant can be broken down into the sum of several components. The principle of the application of symmetry to a vibrational analysis is that the potential energy of the equilibrium configuration of a dynamical system must be invarient under the symmetry operations of the system. A group of operations which carry a molecule into a configuration identical with the original configuration are called the symmetry operations of the molecule, and this collection of operations form a group, in the mathematical sense. Associated with a group is a set of irreducible representations, which may be visualized as non-equivalent examples of the 22 symmetry of the molecule. The number of irreducible representations is limited, and the essence of the application of group theory to a vibrational analysis is that the potential energy of the system may be broken down into a sum of independent contributions. Each contribution is associated with some one irreducible representation of the group describing the symmetry of the configuration. With the above definitions in mind, we will define what is meant by a representation and an irreducible representation of a group. As stated earlier, a system of N particles may be completely determined by a knowledge of the 3 K coordinates, usually the Cartesian coordinates. In a vibrational analysis, i t is convenient to use displacement coordinates, measuring the displacement from the equilibrium position of the ith particle x^, y^, z^, or to simplify the notation 1 > 11 * 1> 11 '+ 2) "there being 3N q's in a l l . If the original configuration of a molecule is disturbed, then any new coordinate q 1 is changed into a linear combination of the original coordinates. Such a disturbance may be a symmetry operation, in which case the new coordinates q* may be represented analytically as: q'j - J Ryqi 3-1 •J or Q' = RQ 3-2 where Q' is a column matrix of order 1x3N, Q is a row matrix of order 3Nxl, and R is the square transformation matrix of order 3Mx3N. The set of linear transformations defined above for the symmetry operations of a molecule has a l l the properties of the symmetry group (that is the transformations have the same multiplication properties as the group) belonging to the molecule, andis said to be a representation of the 23 group. The coordinates are said to form a basis of the representation. The basis coordinates chosen were arbitrary, and i f a different set of coordinates were used (say a different orientation of the Cartesian coordinates) the same results should be obtained* When two representations of a group differ only-in that the basis coordinates of one are linear combinations of the basis coordinates of the other, then the two representations obtained are said to be equivalent. The characters of equivalent representations are the same, the character being defined as the sum of the diagonal elements of the transformation matrix R? 3n X - £ R,. 3-3 R i In general, Eq.. (3-1) consists of 3K equations in 3M unknowns. However, i t is usually possible by choosing a suitable set of basis coordinates: (different from the Cartesian coordinates generally) where-by the number of terms in Eq. (3-1) may be greatly reduced. In fact, in a symmetrical molecule i t is possible to choose coordinates which may be separated into sets which do not mix with one another, and the representation for which these coordinates form the basis, is said to be reduced. When it is possible %X) do this, the original representation is said to be reducible,and when i t is not possible to form smaller non-mixing sets, the resulting representation is said to be completely reduced, Sinee the coordinates which form the basis of the reduced representation separate into sets which do not mix with one another, the equations involving any one set form a representation of the group. Such a representation is irreducible, when the maximum reduction of the reducible representation has been carried out. .2k A fundamental theory of the representation of finite groups states that for each point group there are only a definite small number of non-equivalent irreducible representations possible. A well known formula shows the number of times a given irreducible representation may appear in a reduced representation; where h is the order-of the group,X R is the character for the reducible representation, andX^ is the character for the R'th operation of the R i'th irreducible representation. The sum isover the R operations of the group. A l l the quantities on the right of Eq. (3-h) are easily deduced. Associated with any normal mode of vibration of a molecule is its normal coordinate Q which describes the displacement of the atoms of the molecule relative to its equilibrium configuration. Each normal coordinate transforms according to one of the irreducible representations of the g roup describing the symmetry of the molecule, and with the use of Eq. O - h ) , the number >of normal coordinates belonging to each irreducible representation may be deduced. The selection rules for transitions in molecules can also be derived with the aid of group theory (1). A transition from a state i to a state j is allowed or forbidden according as the integral n; 3-h M 3-5 is equal to zero or not. Since this is a definite integral over the whole configuration of space, it must be invariant to any symmetry operation of the system considered. For fundamentals (from the ground 25 state which is symmetrical) the integral w i l l be symmetric i f the dipole moment operator^ and ^  j belong to the same irreducible representation, for the product of a representation with i t s e l f is always symmetric. Now the wave function of the f i r s t excited state has the symmetry of the normal coordinate for the transition considered (1) and the dipole moment operator belongs to the same irreducible representation as does the translational coordinates T , T , and T_. Therefore, a transition from the x y z ground state to a f i r s t excited state is allowed i f the functionV. belongs *J to the same symmetry species asyu . Hence, with the aid of group theory, selection rules for vibrational transitions can be conveniently derived. 3-3 Crystal symmetry If a crystal is assumed to be in f i n i t e , the theory of space groups shows that a la t t i c e structure must have the symmetry of one of the 230 space groups. The space group i s that group of operationsTrhich carry an atom into an equivalent atom in the crystal. The symmetry operations include rotation axes, mirror planes, glide planes, screw axes, inversion centers, inversion-rotation axes, and simple translations:, i t is the translations which generate the l a t t i c e of the crystal, while the other mentioned operations are the familiar space group (in the crystallographer 1s definition) operations. The smallest unit in which no atoms are equivalent under simple translation is called the primitive unit c e l l , and the crystal is constructed by translations of this unit c e l l which when carried out an inf i n i t e number of times, builds the in f i n i t e crystal. In this definition the space group i s an i n f i n i t e group: however, i f we define translations which carry a point in a unit c e l l into an equivalent point i n another unit c e l l as the identity operation, we define a f i n i t e group, which i s a factor group of the space group. This 26 f a c t o r group describes the symmetry of the u n i t c e l l , and although not a p o i n t group, i t i s isomorphous w i t h one of the 32 c r y s t a l l o g r a p h i c p o i n t groups. I f we d e f i n e the operations of the t r a n s l a t i o n group, the f a c t o r group and the space group by T, F, and S, r e s p e c t i v e l y , we can w r i t e S = F x T 3 - 6 From the group theory i t can be shown that an i r r e d u c i b l e r e p r e s e n t a t i o n of the f a c t o r group F i s a l s o an i r r e d u c i b l e r e p r e s e n t a t i o n of the space group S. About any p o i n t i n a c r y s t a l there i s a l o c a l symmetry, which f o r most p o i n t s w i l l c o n s i s t of the i d e n t i t y operation E. However, i f the p o i n t i n question i s s i t u a t e d on one or more of the symmetry elements of the f a c t o r * group which do not i n v o l v e t r a n s l a t i o n s , t h a t p o i n t w i l l be i n v a r i e n t under c e r t a i n operations of t h i s group, and w i l l belong t o a group which w i l l have an order l e s s than t h a t of the f a c t o r group. Such a c r y s t a l l o g r a p h i c " s p e c i a l p o s i t i o n " has been designated by H a l f o r d as a s i t e . (9) A given u n i t c e l l w i l l i n g e n e r a l e x h i b i t s e v e r a l d i f f e r e n t kinds of s i t e s , and may even have s e v e r a l d i s t i n c t s e t s of s i t e s of the same symmetry. The symmetry elements which leave t h i s p o i n t I n v a r i e n t form a p o i n t group, which i s c a l l e d a s i t e group. As a consequence of i t s d e f i n i t i o n , the s i t e group i s a sub group of the f a c t o r group. 3-h A n a l y s i s of v i b r a t i o n s Bhagavantam and Venkatarayudu were the f i r s t workers t o adapt the u n i t c e l l method described p r e v i o u s l y t o the determination of the allowed frequencies of v i b r a t i o n i n a c r y s t a l (7 , l y ) . That i s , they considered The " f a c t o r group" w i l l be used t o denote the f a c t o r group of the t r a n s l a t i o n group i n the space group. 27 an isolated unit cell as their dynamical system, and using the factor group characteristic of a crystal, derived the selection rules. This is the method used in the treatment of the results for barium formate to be described. Halford (9) has developed another method for determining the number of modes allowed in a crystal. Instead of treating the motions of one unit cell, he chose to treat the motionscof one molecule moving in a potential field reflecting the symmetry of the surrounding crystal. This symmetry will be characterized by the symmetry of the site group. In this method, the frequencies of a l l equivalent molecules of a given set of sites have the same frequency, and the selection rules are derived by the standard methods, using the site group as a basis. The site group of a molecule in a crystal is a subgroup of the molecular group, and it is well known that symmetrical molecules are usually situated on sites other than the site of tr i v i a l symmetry (C]_), which is the most general position. The site group method is only an approximation when applied to the analysis of the vibrations in a crystal lattice, and complete information of the vibrational modes and their polarization properties can only be obtained from the more complete factor group analysis. However,- the factor group and site group approach are both useful in.a vibrational analysis. The relationship between the two can be discussed in terms of the equation for the potential energy of a crystal, which can be written (15) v . I y • v L * v L j 3-7 28 Vj i s the harmonic oscillator function the sum being over a l l molecules, £ £ V-y contains a l l cross terms between internal coordinates of different molecules, V^ contains terms involving displacements of the centers of gravity and change of orientation of molecules, and V^ -j contains cross terms between internal coordinates and la t t i c e coordinates. When only the f i r s t term is considered, we are dealing with the orientated gas model approximation, and Vj reflects the symmetry of the crystal as viewed from the site of the molecule in the crystal. The site symmetry is usually different from the symmetry of the free ion or molecule, and Vj w i l l be different for a molecule in a crystal than the free gas phase molecule. Tnis is the so called static f i e l d effect, and is a measure of the effect a surrounding lattice has on the equilibrium configuration of a molecule. If the site group is of lower symmetry than the molecular symmetry, a relaxing of selection rules and removal of degeneracies may result. The latter results i n "static f i e l d s p l i t t i n g " of degenerate modes. The "dynamic crystal effect" arises from the £ *jT V y term, and measures the effect of interactions of the internal vibrations .of other molecules In the unit c e l l on the molecule being considered. Depending on the number of molecules per unit c e l l , each "gas phase" internal mode w i l l be s p l i t in the crystal. Such splitting are generally referred to as factor group, or correlation f i e l d splittings. From the above arguments i t can be seen that bands which are not allowed in the infrared spectrum of a free molecule may become active in a crystal, depending on the mapping of the molecular group into the site group. The correlation mapping of the site group into the factor group then shows the polarization properties of the molecular modes. -£29 CHAPTER' hi DISCUSSION  Infrared Spectra According to Sugawara et. al. (11) barium formate crystals are orthorhombic, a = 6 .80 A, b = 8 .89 A, c = 7*66 A, belonging to factor group P2J2J2J- D^ , with four molecules per unit cell. An X-ray-analysis carried out as part of this investigation indicated that the factor group and unit ce l l dimensions are as stated. The formate ion has point symmetry C2y, with a C-0 bond length of 1.25 - 0 . 0 1 A, and an 0C0 angle of 127 degrees ( l i ) . The ion has six normal modes of vibration, a l l allowed in the infrared, and distributed among the irreducible representations of Cgy as follows; 3Ai + 1B^ + 2B2. An infrared spectrum of the "free" formate ion can only be observed with an aqueous solution of the salt, and the observed vibrational frequencies have been reported by several workers from the Raman effect ( 1 6 - 2 0 ) . Our infrared frequencies are in good agreement with the Raman data of Fonteyne (20) for sodium formate. As the forces between molecules or ions in a crystal are • weak, the molecular fundamentals in the crystal are close in frequency to the corresponding "free" molecular modes. Since there are no non t r i v i a l subgroups of the factor group P 2 ] 2 i 2 i , the formate ion in barium formate is in a general position of site symmetry C-^ . This lack of site symmetry has no effect on the crystal spectrum since a l l molecular modes are allowed under C2y selection rules, and there are no degenerate modes'to be split due to the static field effect. Hence, the factor group must be considered to interpret the r esults. The character table for the point group 222 isomorphous with the factor group P2]_2^2]iis shown in table 3 . By application of Figure 10b: Projection of the positions of the protons of barium formate on (OOl) 30 Eq. (3-il), the number of normal modes for the unit c e l l associated with each irreducible representation has been determined. X ^ is given in the l e f t section of the table. X R i * 1 equation 3-h may easily be derived (see (li*) page 138, or (1) page 1 0 5 ) , and "h", the order of the group is four for 222. The symmetry operations are the identity operation E, which leaves a molecule unchanged, and the two fold rotation axis C 2. TABLE ij Character Table and Factor Group Analysis of Barium Formate Under 222 222 E C 2(z) c 2(y) C 2(x) n i T T i R' A 1 1 1 1 27 0 9 6 12 B l 1 -1 -1 -1 27 1 8 6 12 B2 1 -1 1 -1 27 1 8 6 12 B3 1 -1 -1 1 27 1 8 6 12 n i is the total number of modes for the unit c e l l , T gives the symmetry of the translation coordinates, T 1 is the number of translational l a t t i c e modes, R1 is the number of rotational lattice modes, and n| is the number of internal (or molecular) modes for the unit c e l l . The barium formate crystal has 36 atoms per unit c e l l , corresponding to 3x36 = 108 degrees of freedom, of which 3 are for translation and 1 0 5 are for vibration. Only 36 of the I48 internal molecular vibrations ( £ n|) are allowed in the infrared. The factor group splitting under the point group 222 w i l l be explained in the following paragraphs. 31 Under the symmetry operations of the factor group (three mutually perpendicular two fold screw axes) a given formate ion is transformed into its three crystallographically equivalent neighbours in the unit cell. For any internal molecular mode of vibration, any two of these three equivalent ions may be out of phase (antisymmetric) with respect to the generating ion (that is, transform according to one of the three B symmetry species above), or a l l three may be in phase with the generating ion (A species). This gives rise to a possible four fold splitting of each molecular mode. But the other crystallographically non equivalent formate ion also has the same possible phase relation-ship for its three other equivalent ions in the unit cell. However, this second set of ions can be symmetric or antisymmetric with respect to the first set and will s t i l l have the 222 phase relations among the ions of this set. Thus, a further two fold splitting of each of the four modes described above is possible, giving a total of eight modes for each normal mode of a formate ion. Each molecular fundamental of the formate ion will be split into 2A + 2B-^  + 2B2 + 2B-j modes under the factor group, of which only the B modes are active in the infrared. For light polarized along one of the principal axes only two of the six componen ts will be allowed. Denote the two non equivalent formate ions by F and F', and their three equivalent ions in the unit c e l l by F z, F y, F x, Fy, and F^ respectively, the subscript indicating which screw axis is doing the generating. Consider the V -|_ normal mode of the formate ion, and let F ion have arbitrary phase (+). Any other ion will either be symmetric (+) or antisymmetric (-) relative to F. For the two B3 factor group components of V p *he other seven ions can have the following phase relationship relative to F: 32 F (+) F a(-) F y(-) Fx(+) F'(+) F'(-) F^OO F^(+) ) ) ) symmetric or F (+) F z(-) F y„r s) Fx(+) F'(-) F<(+) F'(+) F x(-) ) ) ) antisymmetric These are the two components of V ^ that would be allowed for light polarized along the a axis of the crystal. The same type of phase diagram can be drawn for any species of 222, and for any molecular fundamental. This is the basis of the factor group splitting of barium formate under factor group P2;j2]2-j_. The infrared spectrum of each molecular mode will be discussed in turn. The out of plane mode V ^  which occurs at IO69 cm"1 in sodium formate*- . is split into a doublet at 1073 and 1066 cm"1 in the barium formate crystal. The 1073 cm"1 component is active along axis c and b, while the 1066 cm"1 component is active along axis c and a (Fig. 6). The two B^^y) modes of the formate ion, V ^ at 1 5 8 U cm""1 and "V ^  at I386 cm"1 in sodium formate are both split into doublets in the infrared spectrum of barium formate, located at 1 5 9 0 , 1 5 7 0 cm"1 and 1398, 1386 cm"1 respectively (Fig. 1). This splitting is only observed for polycrystalline material in KBr pellets, since the single crystals used were thick and totally absorbed over the range of these doublets. As a result, no polarization data is available for these fundamentals.. The three totally symmetric fundamentals of the free formate ion, V V 2> and V . occur at 1 3 5 2 , 2 8 2 5 , and 7 7 3 cm' respectively in the * The Hainan frequencies of aqueous sodium formate provide the closest approximation to a "free" formate ion. 33 Raman spectrum of aqueous sodium formate. In barium formate, is split into a doublet at 1357, 1350 cm"'1' as can be seen in Fig. 1, but is not resolved in the single crystal spectrum. A weak but very sharp line appears at 133ij cm""1 in the spectrum of the crystal which has been assigned 13 to the C species of barium formate for this vibration. The basis for this assignment will be discussed later. The region of the V ^  fundamental (3200-2600 cm"1) is complicated due to the presence of at least 15 absorptions of which three only are due to Vg. This strong three fold splitting is readily observed in the spectrum of barium formate-d-j_ (Fig. 2), where three sharp linesodf equal intensity appear at 2163, 2116, and 20I46 cm"1. In the region of V ^ no less than seven distinct components are observed between 770-7Wi cm"-1 (Table 2). These two regionsof the spectrum are most interesting and will be discussed separately. Combinations and Overtones Of the kl observed frequencies in the spectrum of the single crystal of barium formate 13 are due to combinations andcovertones of molecular modes. These are listed in Table 2, along with their polarization properties when observed. Under 222 selection rules, no fi r s t overtones should be allowed. However, i t must be remembered that each fundamental has eight components under P2]2]2]_ factor group splitting, and that a fi r s t overtone of a fundamental, would in actual fact be just a combination of two different species of the same fundamental, and is therefore allowed under the selection rules. Then each first overtone theoretically consists of c| = 28 combination components, 27 being allowed. Direct overtones of the same species are not allowed. Splitting is also observed in some of the high frequency combinations. The V ), + V t combination at 2980 cm"1 has two components and the v l + V U c o mk Nation a t 2920 cm" has three (table 2 ) . This splitting is likely due to the factor group components-.of the fundamentals. This is substantiated by the 2 + V l c o m l 3 i n a - t : i- o n 1 1 1 barium formate-d-^ (Fig. 2) at 3k793 3kk0, and 3370 cm"1. The intensities are in the same ratio and the splittings between the components are nearly the same for the combination as for the "v^ fundamental. Lattice modes The six external degrees of freedom of the formate ion belong to the following irreducible representations of 02^: 1A]_ + I&2 + 2B-j_ + 2B2. These degrees of freedom become translatory and rotatory lattice modes when the ion is in a crystal. Table k lists the number of lattice modes of each type belonging to the various irreducible representations of 222. These modes usually occur in the frequency range between 20 to U00 cm"1, far below the region accessible on the spectrometer used for this study. However, lattice modes can be observed in the near infrared as combinations with molecular modes. Eleven frequencies are listed in Table 2 assigned to combinations with lattice modes. The most prominant of these occur at 1750, l6 i i0 , and 1630 cm"1, which have been assigned as sums of lattice modes trith V ^ at 1580 cm"1 (mean of the doublet). Another sharp peak at 1535 is possibly the difference mode corresponding to the sum mode at I63O cm-1, or may be the overtone 35 TABLE 5 Lattice Modes of Barium Flarmate Combination, cm Lattice frequency, cm" = 1750 170 + V L = 1720 l5o v u + *L = 161*0 60 = 1630 50 = 1535 1*5 The difference frequencies corresponding to these modes are listed in Table 5. Other weak absorptions occur at 11*65, 1165, 1110, 98O cm"1 with a fairly strong one at 1230 cm-1. These are also attributed to combinations of fundamentals with lattice modes, but i t is very difficult to ascertain which fundamental is responsible. No lattice frequencies are listed for these latter modes. "*> 2 Region, 3200-2600 cm"1 The region around the C-H stretching fundamental V 2 is extremely complicated, and at the same time interesting. It was only decided with the aid of deuterated barium formate that the fundamental was split into three components, and from frequency separations, which three these were. The fundamental appears at 2825 in the Raman spectrum of sodium formate (2828 cm"1 in the infrared spectrum reported in Table 3) and at 2820 cm"1 in the spectrum of aqueous barium formate (Fig. 9). In the spectrum of the single crystal, a and b, active components appear at 2863 cm"1, and c active components at 2823 and 2750 cm"1. No mention has been made up to this point concerning the infrared spectrum of lead formate or sodium formate. Crystalline lead formate has the same factor group symmetry (11) as does barium formate, P2^2^2p and since both salts have two formate ions per molecule, theyshould have a similar infrared spectrum. In a l l fundamentals except V 2 this similarity is observed. However, the 1) 2 region differs from that in barium formate, and is shown in Fig. 9. On the same diagram is shown the y 2 region of sodium formate, and of an aqueous solution of barium formate. A l l three spectra have the same general features: there is no evidence of factor group splitting in lead formate, or, as would be expected, in aqueous barium formate. (It is worth mentioning here that no splitting was observed in any of the other fundamentals of aqueous barium formate either.) The factor group analysis of sodium formate is presented in the appendix, where the theory predicts that each molecular fundamental should be split into two infrared active components in the crystal. This splitting is not observed here for sodium formate, nor is i t for any other fundamental. The remaining 12 of the 15 components in the barium formate spectrum between 3300-2600 cm""1" can be assigned to combinations and overtones, as indicated in Table 2 . It is then necessary to explain why these other lines do not appear in the spectrum of lead formate. This author believes the clue is furnished by the disappearance or decrease in intensity of these combinations and overtones in the solution spectrum of barium formate between 3300 and 2600 cm"1. Further, these same bands are extremely weak in the spectrum of crystalline lead formate in KBr, *jhich exhibits no factor group splitting, as does barium formate. The explanation of this intensity hehavior is probably due to Fermi resonance between the components of the fundamental 37 of barium formate, and the overtone and combination frequencies. Due to the factor group splitting, the fundamentals are now close (50 cm"1) in frequency to the overtones, which will produce an equalization of intensity due to a mixing of the wave functions of the two states. ^ 3 Region. 770-7UO cm"1 Another interesting region of the spectrum of barium formate lies between 770 and 72*0 cm"1 where seven absorption lines of varying intensity are observed (Fig. 7,8). This "band" is the V 3 fundamental, which occurs at 773 cm"1 in the Raman spectrum of sodium formate. The strong absorption between 770-765 cm"1 actually consists of two distinct lines at 769.5 cm"1, and 766.5 cm"1, the splitting being easily observed in KBr pellet of barium formate (Fig. 8c). The polarization properties and frequencies of these lines are shown in Table 2. The polarization properties of the components of the unresolved doublet were inferred from the width and frequency of the absorption maxima, when the polarization was changed. The three lines at 702, 7u9.5» and 7kk cm-1 are extremely weak relative to the higher frequency components, and have been assigned to C1-^ barium formate (1$ natural abundance). Factor group splitting is again observed in this fundamental, but further discussion of the polarization properties will be left until the isotope effect has been discussed. Isotopic Shifts The assignment of the three low frequency modes to is questionable, but there is much evidence in its favor. Frequency shifts were calculated for a C 1 ^ relative to a C 1 2 formate ion using a Urey-Bradley potential function and equations developed by Duchesne (21) for an XYZ2 "type molecule. The shifts for the three A^ (C2f) modes 38-are listed in Table 5. These shifts were based on the sodium formate model, since the original force constants used were derived from sodium formate data (20). TABLE 6 Calculated Frequency shifts of the Formate ion Mode Shift cm"1 ^x(1352) 27 M2825) 3 ^ 3(773) 6 Assuming the mean of the four components of is at about 765 cm"1, this would place the C1^ barium formate band at about 759 cm"1. There is ah absorption line here (758 cm"1) but this would not explain the component at 7i4.it cm"1. Now there is a sharp line at 133ll cm"1, which has been assigned as the Ci3 component of V ^  at 1353 cm"1 (mean of the doublet). This shift of 19 cm"1 is considerably less than the predicted 27 cm-1 for this mode. Hence, the splitting calculated for the other modes may be in error. A better approach is to use the product rule (1,6) which gives a mathematical relationship between the product of the frequencies of a given symmetry species for two isotopically substituted molecules. For the three A^ modes of the formate ion, this rule is given by the expression; 12 3 vl»2>3 = [m(C12) M' ra(Cl3) M 1 1 2 39 where the primed frequencies refer to the C X J species, m referes to the mass of the two atoms isotopically substituted, and M and M' refer to the total mass of the C 1 2 and C1^ i o n s respectively. Now, assuming a C1^ splitting of 19 era"1 for \>1 (which is observed), and assuming 3 cm"1 for V 2 (the product of the frequencies is relatively insensitive to a - 3 cm"1 change in this high frequency mode), the splitting for V ^  can be calculated. This turns out to be 12 cm"1, s t i l l less than the 17 cm"1 observed, but much closer than the previous calculated 6 cm-1. A further justification in the assignment of these low frequency modes is that the shiftsrof a l l three from their corresponding C 1 2 mode is constant (Table 7 ) . TABLE 7 Product Rule Frequency shifts of Formate Ion C12V , cm"1 C l 3 D ) c r a - l V l 2 > V 1 3 , cm"1 769.5 752 17.5 766.5 7u9.5 17.0 761 # 7UU 17.0 * weighted mean of 761.5 and 758 Confirmation of the magnitude of this splitting is also found in the infrared spectrum of the same fundamental of sodium formate. A single strong absorption appears at 772 cm"1 in the infrared, accompanied by a very weak shoulder at 755 cm"1, a separation of 17 cm"1, 13 which has also been assigned to the corresponding C J mode. The polarization assignments for the C components were very-difficult to make, especially for the two stronglines at 769.5 a n d ?66.5 cm"1. The former is eertainly a active, and the latter c active. However, the b axis spectrum was very broad between 770-7$6 cm"1 since the crystal used for this spectrum was slightly thicker than that for the c and a axis spectra. As a result, i t is difficult to ascertain i f both components are b active. The high frequency line at 769.5 cm"1, is definitely b active, and:it is felt from Intensity studies done by rotating the plane of polarization in small increments from c to b axis polarization, that the 766.5 is probably not b active. However, further study with thinner cyrstals might show this assignment to be incorrect. Although single crystals of lead formate were not prepared, this region in KBr pellets of lead formate showed five distinct components, whereas only four are detected for barium formate in KBr. (Fig. 8c). Wo assignments can be made for lead formate. It—2 N. M. R. Results It was pointed out in Chapter (1), that an n.m.r. analysis of a single crystal of barium formate failed to give the line shape predicted by the crystal structure of Sugawara et. al. ( l l ) . From this structure, the nearest proton-proton distance is 1.81 A, and the next nearest is It. 11 A. The predicted positions of the protons are shown in Fig. 10b, where a C-0 bond length of 1.25 A, a C-H bond length of 1.09 A, and an 0C0 angle of 127 degrees was assumed. The n.m.r. spectrum for proton resonance of polycrystalline barium formate should have a pronounced doublet splitting of 7.13 gauss, and a second moment of 12.3 gauss2. The experimental second moment was 1.9 - .3 gauss2, and showed no doublet splitting characteristic of two close protons. I hi A similar analysis for lead formate, with nearest proton-proton distance of 1.78 A, and next nearest at 3.3 A, also predicted a doublet splitting of about 7 gauss, and a second moment of 12.2 gauss . The experimental value was 1.6 ± 0.3 gauss . These results indicate that either both crystal structures are in error or that there is considerable rotation of the formate ion in the crystal. The latter suggestion can be discarded since in order to alter the position of the proton, a hindered rotation or rotational oscillation about an axis other than the symmetry axis (C2) of the ion would have to occur. It is believed that this type of motion is very unlikely to occur at room temperature. The f i r s t hypothesis is almost certainly correct in view of the fact that the sum of the van der Waals radii of two protons is 2.1). A, considerably greater than 1.8 A. Further, the Ba-H distance is only 1.3 A, whereas the sum of the radius of the barium ion and hydrogen is 3.2 A. Due to the large scattering power for X-rays of barium and lead relative to that of carbon and oxygen, i t is likely that the positions of the formate ions in the unit cell are 3n error. Assuming the protons are as close as possible in accordance with their van der Waals radii, say 2.li A apart, the second moment should be 1.8 gauss2 for two protons only. This is in excellent agreement with the experimental values. It is possible that the C and 0 positions are correct but that the C~H bond has been distorted from the line bisecting the 0C0 angle, to give this 2.1* A separation. However, the infrared evidence to be discussed below indicates that there is no distortion of the formate ion in the crystal. ' ii2 k-3 Crystal Structure of Barium Formate A careful study of the infrared spectra and polarization data shows that there is perhaps something unique about the c axis of the barium formate crystal. The c active absorptions are generally more intense than the others, and in the V ^ a n <* g fundamentals there are more apparent c axis absorptions than a or b. While this behavior is not inconsisteritwith the P2]2-]_2^  factor group predictions, i t was felt that i t could indicate a different factor group. As stated at the beginning of this chapter, an X-ray analysis carried out as part of this investigation indicated that the factor group of barium formate was in fact P2].2^ 2]_. Even with extremely long exposures, no evidence could be found to indicate otherwise. Therefore, i t appears that the barium, carbon and oxygen atoms are in a T2j2i2i arrangement in the lattice. But this symmetry may be destroyed when the hydrogen atoms are considered, i f there is a distortion of the C-H bond from a line bisecting the 0C0 angle. The effect of this distortion in destroying the P2]_2-L2]L symmetry would very likely not show up on an X-ray photograph when one considers the scattering power for X-rays of a barium atom relative to hydrogen. Thus, i f the factor group of the crystal is not P2i2]2x, it must be a subgroup of this, of which only P2^ and PI are possible. The latter factor group is the one of t r i v i a l symmetry and one would expect no polarization behavior under this group. But polarization properties are possible under P2j. A factor group analysis for barium formate under the point group 2 (isomorphous with P2j) is presented in Table 8. 143 TABLE 8 Factor Group Analysis of Barium Formate Under 2 2 E C 2(z) n i T T1 n' i A 1 1 5k 1 17 12 2U B 1 -1 5k 2 ' 16 12 2a If the C 2(z) screw axis is in the direction of the c axis in barium formate, 2ijc and 2liab active components are predicted for the six molecular fundamentals in the crystal. By similar reasoning as was applied to the P2-j2-j2-^  factor group analysis, each molecular mode should have kc + gab active components in the crystal. Now when an absorption has dual activity, say active for light polarized along the a and b axes, it is necessary to distinguish between an ab active absorption, which is a single normal mode active along the a and b crystal axes, and two accidently degenerate modes, one a active, and the other b active. For ?2-j_ symmetry, only c and ab type absorptions should appear. A great many observed absorptions have this property (cf. Table 2) which encourages an interpretation of the results in terms of ?2]_ factor group. However, there are some conspicuous exceptions to the expected P2^ behavior, namely the 313k cm-1 overtone which is b and c active only, and the V g fundamental which has a c and b active component at 1073 cm"1 and a c and a active component at 1066 cm"1. This violates the P2^ predictions where each a active absorption must also be b active and vice versa. However, i f we assume that the 1073 cm-1 _____ ^ absorption consists of two distinct modes accidentally degenerate, one hk mode c active and the other b active, there is no violation of V2J2J2^ predictions. The same consideration applied to other absorptions';of dual activity (i.e. that these absorptions are two distinct modes of different activity) is likewise amenable with a factor group of ?2]2]2\ for barium formate. The factor group splitting of the V 2 V^, and V5 fundamentals under ^J^J^X l s a s ^ o l l o w s : * n e v 2 fundamental has a and b active modes at 2863 cm-1, a c active mode at 2823 cm-1, and a c active mode at 2750 cm-1. The V ^  fundamental has four distinct components, two c active at 1073 and 1066 cm-1, onei-b active at 1073 cm"1, and one a active at 1066 cm"1, fhe fundamental has\the predicted six ^2j2-^2.-^ factor group components; a active lines at 769.5 and 758 cm"1, b active lines at 769.5 and 761.5 cm"1, and c active lines at 766.5 and 761.5 cm"1. The a, b or c activity of an absorption corresponds to a normal mode which belongs to the B-j, B 2 , or B]_ irreducible representation respectively of the factor group P2-j2-j_2p The symmetry species of the absorptions of barium formate are then shown in Table 2, when it was possible to determine such. k-h Conclusion Infrared spectroscopy has-proven to be a valuable tool in elucidating the crystal structure of a hydrogen containing compound. While X-ray diffraction studies can indicate the symmetry of heavy atoms in a crystal structure, infrared spectroscopy can show i f this symmetry is correct when the light atoms, and in particular hydrogen, are also considered. This study has shown that the symmetry of crystalline barium formate is in fact P 2]_2]2i, and that the error in Sugarawa's structure is probably due to an incorrect determination of the coordinates of the carbon and;oxygen atoms in the unit cell. Several weak infrared absorption lines in the spectrum of single crystal barium formate are s t i l l unassigned, and may be due to sums of lattice modes with molecular fundamentals. It would be very desirable to study the effect of temperature changes on the intensities of these suspected lattice modes. This would aid in assigning these modes since difference combinations appear less intense than corresponding sum modes as the temperature decreases. This effect is due to the decrease in population of the lattice vibrational states at lower temperatures as governed by the Boltzmann factor. U6 APPENDIX Factor Group Analysis of Sodium Formate The sodium formate crystal is monoclinic with factor group symmetry C2/c - Cg^, and has four molecules per unit cell (23). The unit cell dimensions are a = 6.19 A, b = 6.72 A, c = 6.1*9 A, and /3 = 121° 1*2'. A factor group analysis under point group 2/m (isomorphous with C2/c) is presented below. The symmetry elements are as follows; E is the identity operation, C2 is a two fold rotation axis, is a mirror plane, and i is a center of inversion. 2/m E c 2 i <T£ n.t T T 1 R 1 n! 1 x Ag Bg K Bu l l l l I - 1 1 - 1 I I - 1 - 1 1 - 1 - 1 1 12 0 1* 2 6 18 0 8 * 1* 6 12 1 3 2 6 18 2 6 1* 6 Sixty normal modes are possible for the unit cell ( T n ^ ) , of which twenty four are internal molecular modes ( E ' n p . T refers to the symmetry of the translation coordinates, T' indicates the number of translational lattice modes, and R' indicates the number of the rotational lattice modes. Only twelve of the txrenty four molecular modes are active in the infrared, (that is, belong to one of the translation coordinate symmetry species), and each fundamental of the formate ion should be split into two components in the crystal. U7 BIBLIOGRAPHY 1 . Wilson, Decius, Cross, "Molecular Vibrations", McGraw-Hill, 1955' 2. Halford, J. Chem. Phys. 35: 1097, and 1109 (1961) 3 . Halford, J. Chem. Phys. 3 6 : 2651* (1962) l i . Hornig, J. Chem. Phys. 3 6 : 263b (1962) 5. Hornig, J. Chem. Phys. 3k: 265 (1961) 6 . Herzberg, "Infrared and Raman Spectra", Van Nostrand, 19hh 7. Bhagavantam and Venkatarayudu, Proc. Ind. Acad. Sc. 9A: 22k (1939) 8. Bhagavantam and Venkatarayudu, Proc. Ind. Acad. Sc. 13A: 5h3 (I9I4I) 9. Halford, J. Chem. Phys. lU: 8 (I9I46) 10. Itoh, Kusaka, Kiriyama, Saito, Osaka University, Science and Industrial Research, Memoirs, Ik: 1 (1957) 1 1 . Sugawara, Kakuso, Saito, Nitta, X-Sen. (X-Rays), 6 : 85 (1951) 12. Charney, J. Opt. Soc. Am. k$: 980 (1955) 1 3 . Born and Huang, "Dynamical Theory of Crystal Lattices", Oxford, 1954 Hj. Bhagavantam and Venkatarayudu, "Theory of Groups", Andhra University, 1951 15. Hornig, Vedder, "Advances in Spectroscopy" Vol. II, Interscience, I96I 1 6 . Gupta, Indian. J . Phys. 1 0 : 117 (1936) 17. Lecomte, Comptes Rendus, 208: ll(01 (1939) 18. Lecomte, Cahiers Phys. 17: 1 (19U3) 19. Fonteyne, Naturwisse. 31: kkl (19li3) 20. Fonteyne, Natuurwetenschappelijk Tijdschrift, 25: 173 (1943) 21. Duchesne, Physica, 8: 5'25 (19iil) 22. Groth, "Chemische Krystallographie", Vol. 3 , Wilhelm Engelmann, 1910. 23. Zachariasen, J. Am. Chem. Soc. 62: 1011 (191)5) 


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