An online source stated that molecules have resonant frequencies and then vaguely said that university textbooks should be consulted for more information. Another source contradicted the first in regard to the relationship between microwave absorption and molecular resonance. However I am not interested so much in microwave absorption as in molecular resonance frequencies. What is a good textbook for understanding the resonant frequencies of molecules?
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Most any quantum mechanics textbook will discuss molecular resonances.
The classic book on molecular resonant frequencies is Spectra of diatomic molecules by Gerhard Herzberg. For larger molecules, you will want to consult Infrared and Raman spectra of polyatomic molecules, also by G. Herzberg.
As for the microwave issue, the second link that you cite, cyberphysics, confusingly writes:
Some science books say that the frequency of microwaves used in a microwave oven is the 'natural frequecy' of water and resonance is the mechanism that causes the vibration. That is not the case - it is down to the pull of the electric and magnetic forces from the wave.
They are dealing with two halves of the same coin: microwaves do excite molecular resonances and they do that through "pull of the electric and magnetic forces" (with electric forces being the most important). So, both are true.
Separately, in the gas phase, molecular resonances are very sharp. In liquid phase, like water in a microwave, they are less sharp. As they become less and less sharp, some might choose to stop using the word resonance.