Question calculating eigenfrequency triatomic molecule

Physics Q&A
1

Hi,

In exercise 1.5 of Bonds and spectra, they ask us to calculate the eigenfrequency for the antisymmetric mode. I was wondering if maybe there is an example somewhere on how they compose this, because I am having some trouble with creating the matrix as the masses are not the same…

Thanks in advance!
Liselotte

2

Indeed, you can write the -\omega^2 m_i u_i in matrix notation using a matrix with the masses on the diagonal:

-\omega^2 \begin{pmatrix} m_1 & 0 & 0 \\ 0 & m_2 & 0 \\ 0 & 0 & m_3 \end{pmatrix}\mathbf{u} = \kappa \begin{pmatrix} -1 &1 & 0 \\ 1 & -2 & 1 \\ 0 & 1 & -1 \end{pmatrix}\mathbf{u} .

You can then bring the mass matrix to the other side of the equation by multiplying with its inverse. This yields the eigenvalue problem in its usual form.

3

Thank you!