Question 15.9: Find the vibration frequencies of a linear three-atom symmet...
Find the vibration frequencies of a linear three-atom symmetric molecule ABA. We assume that the potential energy of the molecule depends only on the distances AB and BA and the angle ABA. Write the Lagrangian of the molecule in appropriate coordinates (normal coordinates) where the Lagrangian has the form
L =\sum\limits_{α}{\frac{m_{α}}{2} ( \dot{Θ}^{2}_{α}−ω^{2}_{α} Θ^{2}_{α})}.The ω_{α} are the desired vibration frequencies of the normal modes. If one cannot find the normal coordinates of the system, one can proceed as follows: If a system has s degrees of freedom and does not vibrate, then the Lagrangian generally reads
L = \frac{1}{2} \sum\limits_{i,k}{(m_{ik} \dot{x}_{i} \dot{x}_{k} −k_{ik}x_{i}x_{k})}.The eigenfrequencies of the system are then determined by the so-called characteristic equation
det|k_{ik} −ω^{2}m_{ik}| = 0.The "Step-by-Step Explanation" refers to a detailed and sequential breakdown of the solution or reasoning behind the answer. This comprehensive explanation walks through each step of the answer, offering you clarity and understanding.
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