MEAN AMPLITUDES OF VIBRATION: BENT XYZ MOLECULES

Transcription

1 MEAN AMPLITUDES OF VIBRATION: BENT XYZ MOLECULES BY DR. K. VENKATESWARLU, F.A.Sc. AND DR. S. MARIAM (Department of Physics, Kerala University, C/o U.C. College, Alwaye, S. India) Rr August 22, 1964 ABSTRACT The theory of mean-square amplitude matrices is applied to the bent XYZ molecules. The principal mean amplitudes of vibration for the molecules investigated here ate reported for 300 ~ K. INTRODUCTION THE study of mean amplitudes of vibration is of great interest for electron diffraction investigations. Ir provides an additional set of parameters required in defining the nonrigid model of the molecules. Ir serves as an additional information to normal frequencies in the determination of a complete harmonic force field. The theory of computing the mean amplitudes of vibration from spectroscopic data is developed by Morino and Hirota. 1 A slightly modified method is suggested by Cyvin, 2 who gave a new secular equation, relating the mean-square amplitudes of vibration to the normal frequencies. In this investigation, Cyvin's method has been applied to the evaluation of mean amplitudes of ten bent XYZ moiecules at 300 ~ K. using their vibrational frequencies. All the bonded and nonbonded mean square amplitude quantities and the corresponding mean amplitude quantities of these molecules are reported here for the time. 2. SYMMETRY CO-ORDINATES The bent XYZ molecules (Fig. 1) belong to the point group Cs and possess three A' normal modes of vibration. The following ate the symmetry r formed out of the valenee force co-ordinates ~.D, AR and Aa, where D, R and ~ denote the equilibrium values. 260

2 Mean Amplitudes of Vibration : Bem XYZ~ Molecules 261 S, = AD $2 = AR S,~ = (DR)~ A~. Th-. angle displacement co-ordinate has been multiplied by (DR)~ in order to have the same dimension to all the symmetry co-ordinates. Y Fin. 1. Configuration of bent XYZ molecules. 3. SYMMETRIZED MEAN-SQUARE AMPLlTUDE MATRIX The harmonic potential energy function for XYZ molecules is given below in terms of the valence force co-ordinates. 2V fd (AD) ~ +fr (AR) ~ (Aa) 2 + 2fD~ (AD) (AR) + 2fD~ (AD) (Da) 2fR~ (AR) (Aa). Where fd and fr are the XY and YZ stretching constants respectively, fa the bending constant and fd~, fd~, fr~ are the interaction constants concerning the respective parameters. Using the above potential function, the symmetrized mean-square amplitude matrix 27 is formed and the elements of Z' matrix are given below in terms of the symmetry co-ordinates. S~~ = (S#> = ~D S~.~ = (S~ ~) = ~,~ S18 = <SlS3) = "Da

3 262 K. VENKATESWARLU AND S. MAmAM 4. ADDITIONAL MEAN-SQUARE AMPLITUDE Considering the nonbonded distance deviation Ar*, the fol'.ow'ng additional mean-square amplitude quantity is obtained in terms of the symmetr zed mean-square amplitude matrix elements.,~~. = ((Ar*) s) 1 = ~ [ASeo + Bg"CrR ")- C~cra "1-2AB~DR + 2BCcrR~ + 2AC~Da]. Where r* is the interatomie displacement between nonbonded atoms, A = (D--R cos -), B = (R -- D cos a), C = (DR) t sin a and r* = (D z + R DR cos a)89 5. G MATR:X Using Wilson's method the following G matrix elements are obtained: Gxx =/~x q- t~y COS a'~ s +(1 c~~)=+2(1 CD,~ )(1 CR~ )cos~] Glz = q G~~ =- COS a (DR)~ ~n~ ~~ [L(RI G33 = --(DR)89 s~~/zv [(-ID cosa) ] D cos ~. 127G-1-- EA I = 0 6. RESULTS AND D SCUSSlON The elements of the mean-square amplitude matrix 27 were obtained by solving the secular equation where G -1 is the inverse of the kinetie energy matlix and E is the un't Matrix. The values of A are connected w~th the normal frequencies vk by the relation

4 Mean Amplitudes of V, bration : Bem XYZ~. Molecules 263 /Xk = ~ coth \ 2,:~T) where h is the P;anck's constant, k is the Boltzmann's constant and T absolute temperature. The molecular parameters and the observed fundamental frequencies of the molecules studied in this investigation are given in Table I. The mean-square amplitudes of vibration evaluated for the bonded and nonbonded distances respectively are listed in Table II. The mean amplitude quantides, gd ~, (rg89 and ~rr*89 are given in Table III. It can be seen from the results presented in Table III that the mean amplitudes of vibration for some bonded interatomic distances are found to possess characteristic values, namely, nearly for OH bond in HDO and HOC1 molecules and for N = O bond in the nitrosyl compounds. The mean amplitude values for nitrogen halogen bond in nitrosyl compounds increase from lower to higher members of the series and the same trend is noted for the oxygen,halogen nonbonded distances also but the increase is more rapid in the case of nonbonded mean amplitudes. TABLE l Molecular parameters and observed frequencies (cm. -~) of bent XYZ molecule~~ Bond length Observed frequencies Molecules Bond angle ~ References D A R A v I v 2 v a DOH DSH DSeH O1GSOlS HOCI DOCI ONF ONI~C1 ON14CI ONBr 104 ~ 31' , 4 92 ~ 1, , 6 91 ~ " , ~ , ~ , ~ , ~ 1" , ~ 1" , ~ , ~ " ,16

5 264 K. VENKAIESWARLU AND S. MARIAM TaBLE 11 Mean-square amplitudes (,~~) of bent XYZ molecules at 300 ~ K. Molecules Mean square amplitudes (7 D 0"1~ 0'(1 O'r~ DOH DSH DSeH 016S01~ HOCI DOCI ONF 0N15C1 ONtaC1 ONBr \ TABLE 11I Mean amplitudes (d) ofbentxyz mo&eu&s at 300 ~ K. "\ Distance (Y -- Z) (X...Z) "k\ Molecule\\\ DOH.. O DSH DSeH., Ot6SO l~ HOC DOC ONF ONI~CI ONlaCI ONBr " " " " " " " " " " " " " " " " "

6 Mean Amplitudes oj Vibration" Bent XYZ,,. Molecules 265 ACKNOWLEDGMENT One of the authors (S.M.) is thankful to the Council of Scientific and Industrial Research, Government of India, New Delhi, for the award of a Senior Research Fellowship. 1. Morino, Y. and Hirota, I6. 2. Cyvin, S.J Delmison, D. M. 4. Benedict, W. S., Gailar, N. and Plyler, E. K. 5. Sponer, H. 6. Landolt-B6rnstcin 7. Palik, E.D Cameron, D. M., Sears, W. C. and Nielson, H. H. 9. Cross, P. C. and Brockway, L.O. 10. Polo, S. R. and Wilson, M.K. t 1. Hedberg, K. and Badger, R.M. 12. Magnuson, D.W Woltz, P. J. H., Jones, E. A. and Nielsen, A. H. 14. Ketelaar, J. A. A. and Palmer, K. J. 15. Landau, L Burns, W. G. and Bernstein, H. J. REFERENCES J. Chenl. Phys., 1955, 23, 737. Acta Polytechnica Scand., PH. 6, 279/1960. Revs. ~~/Iod. Phy., 1940, 12, 175. J. Chem. Phys., 1956, 24, Mo&ktdspektren, Springer, Atom and Mo&cular Physik, Spfinger Verlag, Berlin, 195l. J. Ch~n. Ph)~., 1955, 23, 280. lbid., 1939, 7, 994. Ibid., 1935, 3, 82. lbkl, 1954, 22, 900. lbid., 1951, 19, 508. lbm., 1951, 19, Ibid., t952, 20, 378..LAmer. Chem. Soc., t937, 59, J. 5IoL Spectroscopy, 1960, 4, 276. J. Chem. Phys., 1950, 18, 1669.