An Empirical Rule Relating Fundamental to Harmonic Frequencies
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Date
1990
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Ohio State University
Abstract
An empirical approach to estimate the harmonic from the experimental fundamental vibrational frequencies is presented. It is based on a partitioning of the eigenvectors of each normal mode into intenal motion components involving bond stretching, $B_{S}$, bond angle bending, $A_{S}$, torsional, $T_{S}$, and out-of-plane, $O_{s}$, motion. These normalized components. $B_{S}+A_{S}+T_{S}+O_{S}=1$, are used in an empirical formula, $f_{S} = (1200B_{S} + 2000A_{S} + 1300T_{S} + 1600O_{S})10^{-8} cm$, which relates fundamental to harmonic frequencies with $v_{S} =\omega_{S}(1 - 2x_{S})=\omega_{S}(1 - f_{S} \omega_{S})$ when the anharmonic states follow a Morse expression. $v_{sn}= \omega_{S} (n+1/2)- \omega_{S}x_{S}(n+1/2)^{2}$, where $n=0, 1.2\ldots$. denotes the vibrational states of mode s. This formula is tested with the available experimental data for $v_{s}\exp$ and $\omega_{s} \exp$. It allows one to estimate harmonic frequencies, $\omega_{s}$. from experimental fundamental frequencies, $v_{s}\exp$ In a self consistent procedure in cases where $\omega_{s}\exp$ are not known. Experimental data is available for approximately 20 molecules (and isotopes) including diatomics. $CO_{2}, HCN, O_{3},H_{2}O,H_{2}S, NH_{3}, CH_{2}O, C_{2}H_{2}, CH_{4}$ and $C_{3}H_{8}$. The method, calibrated with these molecules, yields results which suggest that harmonic frequencies can be predicted accurately from experimental anharmonic frequencies. Therefore the empirical harmonic frequencies obtained from $v_{s}\exp = \omega_{s}emp (-f_{s}\omega_{s}emp)$ can be used to adjust the force constants in Molecular Mechanics calculations until the theoretical eigenvalues, $\omega_{s}^{th}$, agree with the emprical harmonic frequencies, $\omega_{s}emp$.
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Author Institution: Rensselaer Polytechnic Institute