What would it mean to have "a formula for measuring the vibrational rate of an atom"? If you want to measure something, you use a measuring device. Formulas are for calculating the value you expect to find if you were to perform the measurement.
There are ways of calculating the expected vibration rate (which is properly called "frequency") of various oscillating systems, but you have to start with a fairly precise specification of the forces, masses, and energies involved. If you google for "Harmonic oscillator" that will get you started.