## Normal frequencies of vibration of 1, 3-Butadiene

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https://hdl.handle.net/2144/8366##### Abstract

The study of infra-red and Raman spectra and their relationship to the vibrations and rotations of atoms in molecules has become more and more a matter of interest to the physicist and chemist within the past several decades. The study of the vibrations of the atoms which give rise to the broad band spectra in the near infra-red has led to a more thorough understanding of the forces in the molecules including a theoretical explanation of valence. The study of the rotation of groups of atoms within a molecule as well as the rotation of molecules as a whole which give rise to the band spectra in the far infra-red and to fine structure in the near infra-red has led to a rather accurate means for determining the structure of molecules.
This paper deals with the theoretical determination of the vibrational frequencies of 1, 3- butadiene. As is well known, the frequencies obtained from a quantum mechanical treatment of the problem are the same as those resulting from a classical determination; consequently, though the quantum mechanical methods are outlined, the actual calculations are carried out by the classical method of treating small oscillations. The normal frequencies of vibration determined for an oscillating system having particles of masses equal to those of the masses of the atoms with force constants of magnitude equal to the forces actually existing in a molecule will be the same as the infra-red and Raman frequencies.
In general the force constants are unknown so that the problem essentially has to be worked backward by carrying it out algebraically and then, by making use of observed spectra, determining the force constants. For a large molecule, such as butadiene, this is not easy nor even feasible. Consequently, force constants determined by others for smaller molecules where the structure of part of the molecule is similar to a part of butadiene are used for a determination of the normal frequencies. The force constants in this work were transferred from the work on propylene by Wilson and Wells^1 and from a paper on force constants in organic molecules by Crawford and Brinkley^2.
The cis-form was selected as the normal configuration for butadiene. The force field chosen was the valence type field. This means that the changes in bond length and changes in the bond angles from the equilibrium configuration were chosen as the coordinates for the potential function. The application of group theory to the symmetry in the molecule factored the secular equation obtained from the coefficients of the variables in the kinetic and potential energy expressions from a determinant of order twenty four to four determinants of order nine, eight, four and three. Since the latter two contained only out-of-plane vibrations and would thus give zero frequencies with the type of force field chosen, only the two equations of order nine and eight had to be solved. These are, however, in determinantal form, and the process of expanding secular equations of this size is, in general, very laborious and tedious.
In order to reduce the amount of labor involved, an approximation scheme was attempted. As a first approximation, the carbon-hydrogen bond force constants, both stretching and bending, were considered infinite. This served then to increase the mass of the carbon atoms slightly while eliminating the hydrogen frequencies. Two secular equations of order three and two had to be expanded, which yielded an approximate value for the carbon skeletal vibrational frequencies. Secondly, the carbon masses were considered infinite and then the motions of the hydrogen atoms with respect to the carbon atoms were determined. This required the expansion of two secular equations of order five and four.
In the process of making these approximations, three frequencies disappeared, which should have or would have been determined by an exact expansion of the original secular equations. A method was then used for an exact expansion of the secular equations which was due to J. G. Bryan^3 and was developed for expanding matrix characteristic equations and determining the latent vectors or characteristic vectors of a characteristic equation. The characteristic equation of a matrix is defined by
|A-λI| = 0
where A is the original matrix and I is the unit matrix. While the secular equation is not, in general, in the form defined above, it can be put into this form by adding and subtracting rows and columns or multiples of rows and columns until the variable terms appear only on the diagonal and then dividing each column by the coefficient of the variable in that column. Since the secular equation will then be in the form as defined above, it is possible to make use of Bryan's method for an exact expansion of the characteristic equation. The method is described completely and the expansion of the eighth order secular equation is carried out in detail as an illustration of the method. The actual expansion of the secular equations by this process requires less time and effort, in general, than the approximation scheme. This method has an additional advantage in that a continual check is provided on the numerical work as the expansion proceeds. Furthermore, the method is readily adaptable to I.B.M. automatic punch card computors which can result in an additional saving in time and effort while at the same time carrying out an exact expansion rather than an approximation with its attendant loss or shift of frequencies.
After the equations were expanded, they were solved by Horner's method yielding the seventeen required roots. These roots were proportional to the normal frequencies of vibration of the molecule and after solving for the frequencies in terms of wave numbers, they were compared with observed frequencies. The comparison showed a fairly good agreement between the observed and calculated frequencies. A comparison was also made between the frequencies obtained by the exact expansion and the approximation. This showed that the carbon vibrations in the approximation were all fairly uniformly displaced by about two hundred wave numbers toward lower frequencies as is to be expected from the general theory of small oscillations. The hydrogen frequencies were displaced by an even larger amount as also was to be expected but not in as regular a manner as the carbon oscillations.
The results of this paper again indicate that it is possible to transfer force constants from parts of one molecule to another where they exist in similar surroundings. The fairly good correlation between the observed and calculated frequencies indicates that the force constants as chosen are very nearly the correct values.
Probably of more significance in this paper is the application of Bryan's method to the exact expansion of the secular equation. This eliminates the necessity of introducing approximations, and the expansion can be carried out with less work and more of a check on the work than previously. This expansion as well as the solution of the expanded equation can be carried out on I.B.M. computors. Furthermore, the determination of the normal modes, which requires the solution of the ratios of the minors of any row of the secular equation with a given value for the frequency substituted, was perk formed by Crout's^4 method for the solution of determinants which is also readily adaptable to I.B.M. computors.
1. Wilson, E.B. Jr., and Wells, A.J., J. Chem. Phy. 9, 314 (1941)
2. Crawford, B.L. Jr., and Brinkley, S.R. Jr., J. Chem. Phy. 9, 69 (1941)
3. Bryan, J.G., A Method for the Exact Determination of the Characteristic Equation and Latent Vectors of a Matrix with Applications to the Discriminant Function for More than Two Groups. Harvard Graduate School of Education Dissertation (1950)
4. Crout, P.D., Trans. Am. Inst. Elec. Eng., LX (1941)

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This item was digitized by the Internet Archive. Thesis (Ph.D.)--Boston University