Nuclei Move on PES

Since we may assume that is not very much changed for close values of Q, the first term on the second row, where the derivatives of with respect to the nuclear coordinates are taken may be neglected. Subsequently, we just divide through by in Equation 4.6. If is replaced by =, we obtain 

This is the general equation for the nuclei obtained by Bom and Oppenheimer. For a single (K = 1) nuclear coordinate R (for example, diatomic molecules), where R is the distance between the atoms, we have 

Quantization is obtained by using boundary conditions where the wave function exponentially tends to zero as R tends to infinity. E is an energy eigenvalue. Equations 4.7 and 4.8 show that the nuclei move with E (Q) as potential energy and with % as the eigenfunction. This may be referred to as the second part of the BO approximation. 

The vibrational motion of the nuclei thus results in stationary states with quantized vibrational energy levels via the time-independent SE and its boundary conditions. Transitions between the energy levels give rise to a spectrum in the infrared (IR) region. 

If the derivatives of with respect to Q cannot be neglected in Equation 4.6, they 

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