326
WALTERF. EDGELL AND THOMAS R. RIETHOF
I
Vol. 56
substances shown in Table I ClOC-COCI provides different configurations with the change of stat'e.1p2 us with such an example. For this substance the For example, for ClH2C-CH2Cl the gauche molecule trans form is found to be the stable configuration, becomes more abundant in the liquid state than in wfiich would not be the case, if the steric repulsion the gaseous state a t the same temperature, while in is the predominant factor in determining the stable the solid state this molecule disappears almost configuration. The stability of the trans form in completely (see Table I). Such a change in equilithis case is due on one hand to the single bond- brium ratio must be explained in terms of the interdouble bond resonance of the structure of O=Cmolecular forces as was already discussed in our C=O and on the other hand to the fact that the previous papers. However, as we can see from steric repulsion between two C1-atoms is much Table I, the intermolecular force is not so strong greater than that between C1- and 0-atoms. as to change greatly the positions of maxima and So far we have mainly discussed the molecular minima of the hindering potential which is deterconfigurations and their energy differences in the mined mainly by steric repulsion in case there free state. We have, however, often reported the is no considerable single bond-double bond resoconsiderable change of the equilibrium rattio of nance.
THE CALCULATION OF VIBRATIONAL FREQUENCIES OF MOLECULES WITH MANY ATOMS ANR LITTLE SYMMETRY. I. SIMPLE DERIVATIVES OF SYMMETRICAL MOLECULES BY WALTERF. EDGELL AND THOMAS R. RIETHOF Department of Chenaistry, Purdue University, La.fayetle, Indiana Receiued December 86, 1961
The use of approximate normal coordinates is suggested as a basis for the calculation of vibrational frequencies in the more complicated molecules. Two basically similar methods have been proposed. The first, which takes advantage of the existence of group fre uencies, has not been tested sufficiently. The second method dealing with molecules which may be regarded as simple jerivatives of symmetrical molecules, is tested numerically with simple examples.
Introduction One of the most potent devices in the analysis of vibrational spectra is the calculation of the fundamental modes of vibration of a molecule based on a mechanical model. All of our progress in the correlation of observed frequencies with types of motion of the various component elements of structure of a molecule has its origin in such calculations. Fundamentally the use of infrared and Raman spect,ra iii qualitative structural analysis rests upon this base. Because of the high order of the matrices and determinants involved in the practical application of the methods so far developed, such calculations have been limited to molecules containing few atoms and of relatively high degree of symmetry. The extension of the methods of theoretical infrared and Raman spectroscopy t o complex molecules depends upon the penetration of this barrier. A tremendous wealth of information available in spectra is as yet untapped because we do not recognize its correlation with structure and environment. There is no reason why one should not apply the same methods to penicillin and cortisone as are nom applied to methane and ethyl alcohdl. Two fundamentally similar methods of extending vibrational calculations to complex molecules have been outlined recent1y.l They involve perturbation methods based upon the use of normal coordinates as a practical tool in numerical calculations. Such a use of normal coordinates has been (1) Walter F. Edgell, Paper No. 16,Sixty First Session of the Iowa Academy of Science, Inorganic and Physical Chemistry Section, April,
1949.
neglected except in the application to the isotope e f f e ~ t . ~The ? ~ first and simpler method applies to molecules which may be considered as a simple derivative of a molecule possessing greater symmetry. Examples would be cyclic C4F7C1 considered as a derivative of cyclic C4Fs or C2F6C1 and CFzClCFzCl as derivatives of C2Faand CsHbC1 as a derivative of CGHB. The second method regards a complex molecule as being built up of basic structural units such as methyl and ethyl groups, benzene rings, etc., joined together by coupling links. It puts the concept of group frequencies upon a quantitative basis. While it will be relatively simple in actual application, this method requires considerable ground work including the calculation of numerous tables for its practical use. The present communication deals with the numerical test of the first method. No loss in generality and considerable savings in labor are involved by restricting the test to simple molecules. Normal coordinates by definition yield a unit matrix for the kinetic energy matrix and a diagonal potential energy matrix whose elements are the square of the (circular) frequencies of vibration. Unfortunately frequencies of vibration must be calculated first before normal coordinates can be obtained. However it can ,be predicted that the normal coBrdinates for a molecule like cyclic C4F7C1 will be similar to those for the molecule cyclic C4Fs. The use of the normal coordinates of the latter or parent molecule in setting up the expressions for the former or derivative molecule will (2) Walter F. Edgell, J. Chem. P h ~ s . 13, , 306 (1925). (3) Ibid., 13, 539 (1945).
March, 1952
VIBRATIONAL FREQUENCIES OF SIMPLE DERIVATIVES OF SYMMETRICAL MOLECULES327
result in kinetic and potential energy matrices which are approximately diagonal. This permits the use of perturbation methods for obtaining the frequencies. Thus, for the example just cited, the necessity of expanding and finding the roots of a 16th order secular equation is replaced by the multiplication of matrices of the same size. This is a very much simpler (and possible) task and is readily carried out with ordinary calculating machines. Because of the symmetry of the parent molecule (cyclic C4Fs) the calculation of the basic normal coordinates is a relatively simple matter. The methods and hence the equations are very similar to those already applied to the isotope The main difference results from the fact that now the potential energy matrix will also contain relatively small off diagonal terms. Let the transformation to the normal coordinates of the parent molecule be given by the expression R
=
LQ
(1)
The inverse kinetic energy matrix of the derivative molecule based upon the approximate normal coordinates Q is L-IGZ-', where G is the corresponding matrix for the coordinates R. The potential energy matris for Q will be Z F L where F is the matrix for R. The square of the (circular) vibrational frequencies of the derivative molecule are the roots of the matris H = L-1 GFL
(3)
Numerical Test of the Method As pointed out in the introduction no loss in generality is involved by testing the method with simple molecules. Methyl chloride was taken as the parent molecule for the derivative molecules CH3Br and CHIF. Its normal coordinates were calculated and the H matrices for the derivative TABLEI: CHv3Br Frequency
\v, w 2 w
3
W 4 \v5
ivs
Ea. (3)
2953 1302 613 3074 1458 953
2953 1302 613 3074 1452 953
Diff.,
%
0 0 0 0 0 0
TABLE 11: CH:,F Frequency w 1
W? w.3
\v
4
w5
w
li
Exact, om-'
Ea. (3)
2860 1500 1037 2978 1484 1223
2860 1500 1037 2978 1487
1220
TABLE111: CHBn
Diff.,
%
0 0 0 0 0.20 0.25
Diff.,
Exact, cm.-1
Eq. (3)
%
iv6
3037 510 223 1135 645 154
3037 540 223 1132 645 153
0 0 0 0.26 0 0.65
Frequency
Exact, cm.-l
Frequency
WI
We W3 w 4 w 5
TABLEIV: CFa
w, TVZ wv3
004
Diff., Eq. (3)
%
904 437 126G 629
0 0 0 0.1G
w 4
437 1266 630
Frequenay
Exact, cm.?
Eq. (3)
%
269 123 672 183
269 123 672 183
0 0
(2)
Because the off diagonal terms of H are small the matrix can be diagonalized by perturbation methods to a sufficient degree of accuracy. Applying an appropriate transformation to H yields the following expression for the ith root of H , correct to the second order
Exact, cm. - 1
molecules were obtained from equation 2. The frequencies mere calculated by equation 3 and are compared with an exact calculation in Tables I and 11. Further tests are provided by considering bromoform as a derivative of chloroform and Cc14 as the parent molecule for CF4and CBr4. The results are found in Tables 111-V. It should be pointed out that equation 3 must yield exact results for WI and WZof the last two molecules. The force constants used in these calculations were taken from the work of Noether, Linnett, and Decius.4-6
'
T A B L E V : CBr4
w1 W2
Wa w 4
Diff.,
0
0
The results are uniformly excellent. It is clear that equation 3 yields the same results as the exact calculation for all practical purposes. This method is being applied to the interpretation of the Ramaii and infrared spectrum of cyclic C4F7C1and the results will be published in connection with that work. In some cases where the mass change is large, as for example CsH6F as a derivative of C~HF,, several terms in the inverse kinetic energy matrix, and hence in H , will be relatively large. A modification is required In such cases. The offending terms are collected into a diagonal submatrix by reordering the rows and columns. The roots of this submatrix (of H ) are obtained by solution of its secular equation and the results used to transform H to the new basis coordinates so determined. The roots, and hence the vibrational frequencies, are obtained by applying equation 3 to the transformed H matrix. This is analogous to the use of equation 23 of the paper on the isotope effect (3). The restriction of the method to simple derivatives ensures that the secular equation of the offending submatrix will never be of high order. Similar techniques will be required when several roots have nearly the same value. ( 4 ) H. Noether, J . Chem. Phys., 10, 867 (1942). ( 5 ) J. W. Linnett, {bid., 8, 91 (6) -1. C. Decius, ibid., 16, 214 (1948).
(1940).
