# How are Fermi resonances between normal modes possible?

As I understand, Fermi resonances occur when two (nearly) resonant vibrational states—most commonly a fundamental and overtone or combination mode—that are of the same symmetry interact, which, in the usual way, causes mixing between the two states and an increase in the energy difference between the two (a Rabi splitting, essentially).

But it puzzles me that this is even possible if I assume that both of the two states that comprise the resonance are actually normal modes. The resonance/interaction relies on the matrix element between the two states, $$\langle \psi_1 | H | \psi_2 \rangle$$, being nonzero. But by the definition of normal modes, these states should be eigenstates of $$H$$ and orthogonal, shouldn't they? Then the matrix element is necessarily zero, so is the resonance only possible in the presence of some additional perturbation or did I misunderstand something?

• I think you have it backwards: the matrix element in this case is zero because you are imposing the constraint via normal modes. Commented Sep 8, 2023 at 17:24
• @ToddMinehardt After reading and thinking a bit more, I think my reasoning is correct but irrelevant, since the concept of Fermi resonances is (apparently?) only applied to local modes, which are generally not the normal modes. And then the matrix element is non-zero and everything works out. Do you agree? Commented Sep 8, 2023 at 17:34
• Your reasoning is correct - local modes are not the normal modes unless an approximation has been made. I was glues to a copy of Wilson, Decius, and Cross for 4 years. R = LQ = B$\delta$, where local modes R are related to normal modes Q by the L-matrix, which is sparse but not just on the diagonal. And the expectation value of the transition dipole moment operator - sometimes taken as the vector r - is then calculated with the force field parameters which include but are not limited to Morse functions plus assorted higher-order terms. Commented Sep 8, 2023 at 20:15