Potential Energy Functions for Stretching, Bending, and Torsional Energy and Fundamental Vibrational Levels

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Ohio State University

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A combined classical and quantum mechanical method is presented to calculate fundamental vibrational frequencies with anharmonic potential energy functions. The potentials are expanded in internal coordinate displacements, $g = r-r_{eq}$ and $\alpha-\alpha_{eq}$ for stretching and bending and a cosine function of for torsional interactions. For asymmetric motion in one internal coordinate about its equilibrium position a modified Morse potential $kG^{2}/2$, is used where $G=[1-e^{-\beta g}]/\beta_{0}$ and $\beta=\beta_{0}[1+\beta_{1}]u+\beta_{2} u^{2}. \beta_{0}=[k/2D_{e}]^{1/2}$, with the force constant, k and dissociation energy, $D_{}e. u =g\gamma$. where $\gamma = \exp[-\alpha g^{2}/(2l+1)], \alpha=2\pi[\mu k]^{1/2}/h$ and l is an integer, damps the potential to the Morse asymptotes. For symmetric motion in one internal coordinate about its equilibrium position, such as linear and out of plane motion, a modified harmonic potential, $kG^{2}/2$, where $G = g[l+\beta_{1}g^{2}+\beta_{2}g^{4}]^{1/2}$ is used. For torsional motion, functions of the form $[l +\lambda/\lambda 1\cos \lambda\phi]^{n}, n >1$, are added for the anharmonic corrections. Standard correlation terms of second degree in the parameters G are introduced for stretch-stretch, bend-bend, stretch-bend, etc. At the minimum energy conformations. the second derivative energy matrix yields the normal coordinate directions, eigenvalues and force constants, k. A fit of an anharmonic potential with a fourth- and sixth-order polynomial along the normal modes permits adjustment of the parameters k, $D_{e}, \beta_{1}$ and $\beta_{2}$ to reproduce the experimental fundamental frequencies. Single term Hartree wavefunctions are obtained for the ground and excited states independently for each mode with all other modes in the ground state. The potential parameters are adjusted until the calculated and experimental fundamental frequencies are in good agreement. The product and sum rules are used to analyze and suggest corrections to the experimental harmonic frequencies.


Author Institution: Rensselaer Polytechnic Institute