Ratios of Isotopic Frequencies Using HLFS Method

Abstract

Simple expressions involving elements of the kinetic energy m atrix are obtained for the ratios of isotopic frequencies in the case of vibrational species, of order two and three, associated w ith small molecules by employing the high and low frequency separation m ethod and the product rule. The applicability of the method is indicated.The high and low frequencj^ separation (HLFS) method developed by Wilson and others [1, 2] enables factoring out of the high and low fre quencies associated with any particular vibrational species in a molecule. The applicability of the method in the analysis of normal coordinates and in the evaluation of force constants has been recently illustrated by Müller et al. [3][4][5][6]. The present discussion is intended to throw light on the appli cability of the method in the evaluation of ratios of isotopic frequencies in the analysis of molecular vibrations.In a case where the high frequency is to be separated, we use the secular equation \ F -G -' A \ = 0(1) after dropping the row and column of G~l, F and A matrices corresponding to the high frequency. In a case where the low frequency is to be separated, we use the secular equationafter dropping the row and column of G, F and A matrices corresponding to the low frequency. As usual, (x-1 and F are the kinetic and potential energy matrices respectively and A is a diagonal matrix with its elements Ai proportional to the square of vibrational frequencies eo*. E represents a unit matrix. The separation of any frequency in this manner means decoupling of the normal mode represented by that particular frequency and holds better correspondence to the true situation, the more the frequency concerned is spaced from the other ones.R eprint requests to Dr. T. R. A nanthakrishnan, D ep art m ent of Physics, St. P a u l's College, Kalamassery-683104, India. In the second order vibrational problems, the separation of the high frequency would thus yield F2.2 -G%2 Azand the separation of the low frequency would yield £ 1 1 F u = A i . (4) Referring by asterisks to the case after isotopic substitution and emplojäng the condition of invariance of Fy elements under isotopic sub stitution, these two equations lead to m /m * = (Gu /G*)i/2 (5) and C02/C02* = ( G t -'I G^) '/ * . (6) Equations (5) and (6) are the same as the ones obtained earlier [7 ] by splitting the normal coordi nate transformation matrix L into two parts, one, a lower triangular matrix Lq obeying the constraint LqLq = G and the other, an orthogonal matrix C made up of a free parameter c such that L = LqC and then setting the condition c = 0. Since the results thus obtained [7] correspond to separation of the high and low frequencies, as shown here, we may conclude that the matrix C takes charge of the mixing between the two normal modes in the actual case. (Some properties of the matrix C relevant to this context have been given earlier [8, 9].) The validity and usefulness of Eqs. (5) and (6) have already been established [7] by considering several examples.In the third order vibrati...