An empirical rule relating fundamental to harmonic frequencies

Abstract

An empirical approach to estimate harmonic frequencies from experimental fundamental vibrational frequencies is presented. It is based on a partitioning of the eigenvectors of each normal mode, s = 1, 2, 3,…, 3 N − 6, into internal motion components involving bond stretching, B s , angle bending, A s , torsional motion, T s , and out-of-plane motion, O s . This is called the BATO method. An empirical function, f s = (1250B s + 2000A s + 1300T s + 1600O s) A ̊ , is developed to assign relative anharmonic contributions to each kind of internal motion by relating harmonic frequencies, ω s , to the anharmonic or fundamental frequencies, ν s , with ν s = ω s + 2 χ s = ω s (1 − f s ω s ). The anharmonic states are assumed to satisfy a Morse expression, ν sn = ω s(n s + 1 2 ) + χ s(n s + 1 2 ) 2 , where n s = 0, 1, 2, ⋯ denote the vibrational states of mode s. The coefficients of f s are calibrated with the available experimental data for ν s exp and ω s exp with isotopes of 19 molecules which contain bonds and groups required for biochemical applications. The objective is to use the BATO method to estimate empirical harmonic frequencies, ω s emp, from experimental fundamental frequencies, ν s exp, in a self-consistent procedure in cases where ω s emp are not known. Of the 394 states in 19 molecules studied, the method yields results within 10 cm −1 for 232 states. 11–20 cm −1 for 86 states, 21–30 cm −1 for 45 states, 31–40 cm −1 for 19 states, 41–60 cm −1 for 9 states, and 61–81 cm −1 for 3 states tested. Detailed results are presented for the molecule containing the most common isotope. The maximum errors occur in C 2H 6 and C 3H 8, for which the method tends to overestimate the CH stretch in several cases where the comparison is against Dennison's rule. All molecules studied satisfy the product and summation rules well.