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Computational calculations allow us to simulate the frequencies of molecules. They can even tell us if the optimized structure is a minimum, a saddle point or a maximum according to the number of imaginary frenquencies.

I was wondering how do quantum calculations could yield these frequencies. Does it have any link to the force constants of a molecule ?

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    $\begingroup$ Possible duplicate of How to calculate wavenumbers of normal modes from the eigenvalues of the Cartesian force constant matrix? $\endgroup$ – hBy2Py Aug 7 '18 at 16:50
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    $\begingroup$ Using a harmonic oscillator model, if you know the change in energy of displacement of atoms (the energy gradient) you can calculate force constants and from that and the mass of the fragments you can calculate frequencies. $\endgroup$ – DSVA Aug 7 '18 at 17:02
  • $\begingroup$ @hBy2Py I don't think this is a duplicate; maybe your answer there answers the question here, too, but from the question itself I cannot find any good reason to close it as a duplicate. $\endgroup$ – Martin - マーチン Aug 7 '18 at 17:40
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    $\begingroup$ @hBy2Py On the very contrary, this is a very low-level question (and the tl;dr is given in DSVA's comment). This question is basically How to calculate the Hessian in the DFT framework? The linked question already assumes to have that done. This is not a duplicate. $\endgroup$ – Martin - マーチン Aug 7 '18 at 18:08
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    $\begingroup$ @DSVA Is that worth expanding into an answer? $\endgroup$ – Zhe Aug 7 '18 at 18:30
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Within DFT the vicinity of the solution to the Kohn-Sham equations is explored treating small changes of nuclei positions as a perturbation. The restoring forces that act in the system upon such perturbations give us information about the force constant matrix, from which we can get the harmonic vibrational frequencies as explained elsewhere, e.g. in the book of Wilson and Decius. Basically, you need the second derivatives.

A key point is to do it efficiently. Doing it numerically is very costly. Since the basis functions are known and they have some nice properties one can do it analytically. Thus, researchers have devised a set of coupled equations to compute the effect of perturbations on the density matrix, which is the core of the calculation.

You will find a detailed explanation in this article by F. Neese.

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