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    Harmonic function and vibrational energy states

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    Why can a harmonic function be used to compute vibrational energy states for a DIATOMIC molecule, and under what conditions are the computed values inaccurate?

    © BrainMass Inc. brainmass.com March 4, 2021, 6:07 pm ad1c9bdddf
    https://brainmass.com/physics/atoms-molecules/harmonic-function-vibrational-energy-states-30191

    Solution Preview

    First of all, every motion under a potential, with a local minima, can be described as a harmonic motion about this minima. We can see that if we expand the potential function about the minima coordinate x0 as a Taylor's series:
    U(x) = U(x0) = +dU/dx|x0 (x-x0) + 1/2(d^(2)U)/(dx^(2))| (x-x0)^(2) + ...+ 1/n!(d^(n)U)/(dx^(n))|x0 (x-x0)^(n)

    Now, since the minima point is x0 the by definition of a minima:
    dU/dx|x0 = 0

    Furthermore, since the motion is centered about this minima, we can say that (x-x0)<<1, so we can discard all the terms for n>2.

    So we are left with:
    U(x) = U(x0) + 1/2(d^(2)U)/(dx^(2))|x0 (x-x0)^(2) + O(3), where O(3) is a the error of ...

    Solution Summary

    The solution answers why a harmonic function can be used to compute vibrational energy states for a diatomic molecule, using minima and maxima points in a function to demonstrate this. However, if a harmonic function is found at a higher energy state, far away from the ground state, then its computed energy states may be inaccurate.

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