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An Analysis of the Vibrations of Certain Large Molecules

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An Analysis of the Vibrations of Certain Large Molecules AN ANALYSIS OF THE VIBRATIONS OF CERTAIN LARGE MOLECULES INCLUDING SPIROPENTANE, METHYLENECYCLOPROPANE} AND CYCLOPROPANE DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By WALTER BERNARD LOEWENSTEIN, B.S. The Ohio State University 195^ Approved by: Adviser ACKNOWLEDGMENT The author wishes to express his sincere grat itude to Professor W. H. Shaffer for his guidance, counsel and encouragement throughout the recent years. Pie would also like to express his thanks to the Texas Company for the award of a fellowship during the tenure of which this work was completed He is also grateful for the constant help and en­ couragement given by his parents. i TABLE OF CONTENTS >ag I INTRODUCTION 1 II THE SPIROPENTANE MOLECULE 5 Introduction 5 The Multiple Origin Method 5 The Structure of Spiropentane 7 Coordinates 9 III SYMMETRY PROPERTIES OF SPIROPENTANE 16 Group Theoretical Predictions 16 Covering Operations 19 Symmetry Coordinates 21 IV THE SECULAR DETERMINANT OF SPIROPENTANE 31 Introduction 31 The Kinetic Energy 31 The Potential Energy 3b The Secular Determinant k-5 V FREQUENCY ASSIGNMENTS AND CALCULATIONS 53 Possible Assignments 53 Calculations 59 More Definite Assignments 66 VI THE METHYLENECYC LOPROPANE MOLECULE 69 Introduction 69 The Geometry 69 Coordinate Systems Ik Symmetry Coordinates 78 Eckart Conditions 8 0 iii TABLE OF CONTENTS (Continued) Page CHAPTER VII THE SECULAR DETERMINANT OF METHYLENE- CYCLOPROPANE 85 A. Introduction 85 B. The Kinetic Energy 86 C. The Potential Energy 91 D. The Secular Determinant 95 CHAPTER VIII FREQUENCY ASSIGNMENTS AND CALCULATIONS 98 A. Assignments of Fundamental Vibrations 98 B. Further Assumptions 101 C. Revised Equations 102 D. Calculations 106 E. The Force Constants 111 CHAPTER IX THE VIBRATIONAL ANALYSIS OF PLANAR DISTRIBUTION OF IDENTICAL CHARACTERISTIC GROUPS 115 A. Introduction 115 B. The Boundary Conditions 117 C. Geometry 117 D. The Kinetic Energy 121 E. The Potential Energy 122 F. The Coordinate Axes 129 G. The Equations of Motion 132 H. The Secular Determinant lk-1 I. The Discrete Frequencies 1A5 CHAPTER X THE CYCLOPROPANE MOLECULE 150 A. Introduction 150 B. The Geometry of Cyclopropane 151 C . The Kinetic Energy 159 D. The Potential Energy 162 E. The Secular Determinant 163 F. The Fundamental Frequency Assignments 165 G. Calculation of Force Constants 167 SUMMARY AND CONCLUSIONS 171 BIBLIOGRAPHY 175 1 CHAPTER. I INTRODUCTION The proper interpretation of the infrared and Raman spectra obtained from a macroscopic collection of molecular systems is one of the most powerful methods of obtaining information concerning the inherent physical properties of such systems. Such information can include the structure of a molecule, the forces determining the structure, thermodynamic properties and even some of the proper­ ties of nuclei present in the molecule. It is usually true that the number of distinct observables is much smaller than the number of unknown parameters. To uniquely determine the unknown parameters with the observables it is necessary to introduce information from other studies and experiments to reduce the number of unknowns. This procedure leaves a problem with a reduced number of unknown parameters which still exceeds the number of distinct observables. At this point the particular type of analysis may suggest simplifi­ cations which lead to a successful completion. In a molecular vibration problem, the unknown parameters which may be immediately introduced are those relating to the geometrical structure of the molecule. These may be obtained from electron and x-ray diffraction data and to some extent from the rotational spec­ tra If they can be resolved for the molecules of Interest. The primary unknowns in a vibrational analysis are the parameters associated with the force field within a given molecule. The de­ termination of a potential function, from which a force field may 2 be obtained, for two related molecules Is the objective of this dissertation. There is a growing interest in determining the properties of what may be, at present, called large molecules. It is true that as the number of atoms is increased, the complexity of the analysis increases. It is therefore desirable to describe the system in a manner such that the number of unknown parameters is greatly de­ creased, thus simplifying the analysis. In i. vibration problem this is equivalent to describing the system as vibrating in or almost In a normal mode. It is found that the application of the rotating axes theorem of classical mechanics will tend to give such a description. A vibrational analysis utilizing the multiple origin method as developed by Deeds is the result. The details of the method are discussed In Chapter II. The choice of the two molecules was governed by the fact that there is a growing interest in investigating the properties of molecules containing supposedly strained rings. It is also true that, in general, many of the force constants associated with the multiple origin method cannot be freely transferred from molecule to molecule. However, the structure of these two molecules is such that it is geometrically permissible to attempt to transfer certain ones. It is of extreme interest to treat the two molecules inde­ pendently and compare such constants. Further, in the application of the multiple origin method to the study of a three member strained ring, portions of the analysis reduce to those arising in the study of the bent symmetrical XY2 molecule. Because of this reduction,, it should be possible to incorporate those well known treatments to simplify the analysis. The correct assignment of fundamental vibration frequencies is one of the major obstacles to be surmounted in the analysis. As molecules of interest become larger it can be expected that associated moments of inertia become larger. As a result, one finds that the differences in frequency between adjacent lines in the rotation-vibration spectra decrease. For large molecules this usually results in an essentially unresolved rotation-vibration band, so much so that quite frequently it is very difficult to differentiate between a perpendicular and a parallel band. There­ fore the frequency assignments must rely more heavily on the basis of calculated results. It is apparent that a rigorous analysis of an unresolved vibration-rotation band is extremely unlikely. In line with the low resolution approach of the analysis, the centers of the bands will be assumed to correspond to the vibration frequen­ cies. This is a practice frequently used by those studying large molecules. It can be expected that an-harmonicity will be quite pronounced in vibrations involving a strained ring. The description of a molecular system must be quantum mechanical in nature. However, such a description must be one covering all degrees of freedom, some having and others not having classical analogues. It Is well known that the total SchrtSdinger equation can, to a high degree of approximation, be separated into a number of separate equations. In a molecular vibrational analysis one deals with one of the separated Schro&inger equations obtained by the assumption of separation of variables. Then the diagonal- ization of the kinetic and potential energy expressions yields the transformation to normal modes. With a knowledge of the nor­ mal modes it then becomes possible to obtain information concerning energy levels, selection rules and intensities. 5 CHAPTER II THE SPIROPENTANE MOLECULE A . Introduction The infrared and Raman spectra of spiropentane (see Fig. l) were first obtained by Cleveland, Murray and Gallaway (l). Scott, Finite, Hubbard, McCullough, Gross, Williamson, Waddington and Huffman (2) carried out a vibrational analysis, assuming cyclopro­ pane force constants, in order to place the fundamental frequency assignments on a stronger basis. Some of these assignments agreed with the tentative assignments given by the former authors, others differed considerably. Blau (3) used the calculated force con­ stants of the latter authors in his study of methylenecyclopropane (C1).H^) but seriously questioned some of the assignments in spiro­ pentane . It will be seen that the method of analysis used here will considerably simplify the vibrational analysis and will make some of the assignments more definite. B. The Multiple Origin Method The vibrational analysis will incorporate the Multiple Origin Method, hereafter referred to as M.O.M. The M.O.M. has been developed in the course of a program to study the vibrations of large molecules carried on at this university under the direction of Professor W. H. Shaffer. The method was In­ troduced by H. S. Long (^l-), further developed by W. A. Pliskin (5) and R. E. Kidder (6 ). W. E. Deeds (7 ) improved the method and applied it to the paraffin chains. R. G. Breene(8 ) applied it to 6 dimethyl amine. B. Gurnutte (9 ) applied it to closed ring hydro­ carbons . The essence of the M.O.M. lies in the repeated application of the !Moving Axes " Theorem (10) of classical mechanics. This approach is ordinarily used to separate the free rotation of the molecule as a whole from the internal motion of the constituent particles, but here it will be used to separate further the various kinds of motion occurring under a given vibration. It is generally known that a C-H bond is much stronger than a C-C bond. To apply the M.O.M. we seek ensembles of atoms within the molecule which are more tightly bound to each other than to any neighboring atoms or groups of atoms. Such an ensemble of atoms will then be consid­ ered a characteristic group and the motion of its constituent par­ ticles under a given vibration of the molecule as a whole will be considered as consisting of the hindered translation of the charac­ teristic group as a whole In the molecular framework, the hindered rotation of the characteristic group about a principal axis through its center of mass and the motion of the particles constituting the characteristic group relative to its center of mass.
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