Calculation of vibrational frequencies and displacements

In order to find the equation of motion of the atoms in a molecule we need to express the kinetic and potential energies as a function of the atomic coordinates. The coordinates that we shall use describe the displacements of the atoms from their equilibrium positions, . Here ui), (ui)y, (ui) 2 are the magnitudes of displacements of the atom I in a molecule from its equilibrium position, referred to the Cartesian frame. Using the time derivatives of these coordinates, we can write the kinetic energy of a molecule, E]f° containing Adatom atoms with masses mi. 

We replace the (u/)x terms with mass-weighted coordinates, q, where, for example  

Double indices, like (/, x) are thus replaced by the expanded single index, i, the summation of Eq. (4.4) now runs up to STVatom, but is simplified to  

With the exception of diatomic molecules, the potential energy surface that each atom experiences is very complicated. We approximate 

since we are considering infinitesimal vibrational amplitudes, the terms higher than quadratic can be neglected because qt qj qt qj qt-(This approximation will be inadequate when strong anharmonicities are present.) If we choose the minimum of energy, Fo °, as the arbitrary zero of our energy scale, the potential energy, within our approximation is  

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