The Basic Equations of Vibrational Spectroscopy

The potential energy function V(x) of a molecule with K atoms describes the increase in energy upon a displacement of the atomic nuclei from their equilibrium positions by a Cartesian displacement vector x = (xi yi, Z]. xk, yK/ zk) . Expanding the potential energy in a Taylor series and neglecting all higher order terms one obtains for V(x) expression (1) [1-6]  

Inserting (5) and (6) into (4) leads to the basic equation of vibrational spectroscopy [1-6]  

Generally, internal and external coordinates couple in the kinetic energy term, however they can be decoupled by inserting (8) into (3) and using (14), which leads to 

Vector d represents the normal mode p in internal coordinate space. It can be transformed to Cartesian coordinate space according to Eq. (21)  

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An adiabatic mode analysis of measured vibrational spectra is possible with a simple perturbation theory approach that was already published in the sixties [44]. The basic equation of vibrational spectroscopy (compare with Eq. 19) can be written in matrix form according to Eq. (74)... 

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