# Why phonon density of state depends on velocity autocorrelation?

We know that if we take the Fourier Transformation of velocity autocorrelation function, we will get the phonon density of state. But why phonon density of state depends on this? What is the physical explanation behind this? And if one has a velocity autocorrelation function of a nanomaterials obtained from molecular simulation, which factors may affect the accuracy of the calculation of phonon density of state?

• are you sure it is phonon density of states? It seems natural that the Fourier transform of velocity autocorrelation function is related to vibrational spectrum, but that is the spectrum of the currently occupied vibrational modes. If we talk about density of states, that includes both occupied and unoccupied modes. – wcc Mar 14 at 1:25

turash haque pial is correct that the density of states can be derived from the velocity autocorrelation function.

The motion of the system is a combination of it's normal modes, and if you look at the fourier transform of this "motion", you get the frequencies of the normal modes. In a finite system and perfect data on the motion, these normal modes will show up as delta functions in the fourier transform. If you look at the definition of the density of states (first equation here), you'll see that the density of states is really these delta functions (since $$E \propto \omega$$, where $$\omega$$ is the angular frequency).

In reality, you won't have perfect data, so the delta functions get smeared out into something that looks more like the density of states that we're used to.

You typically use the velocity autocorrelation function to get information about the "motion" of the system, but in principle, you could use other quantities to get at the motion.

I don't have experience doing this in an MD simulation, but I imagine things that could affect the accuracy include: accurately calculating the autocorrelation function (duh), running the simulation for long enough, and being sure to excite all the normal modes in the system (i.e. not exciting the system for only a narrow range of frequencies).

EDIT: Just to be clear, the relationship between the density of states and the velocity autocorrelation function is not universal. It should hold as long as your system is in equilibrium and is classical. If not, this technique probably won't work. So, I don't think it's right to say that the density of states depends on the velocity autocorrelation function. It's really the other way around. The density of states is the fundamental thing, and it's only in the right situation that it will reveal itself in the velocity autocorrelation function.

• if I may clarify my question to OP, my concern was that you will see thermal population of modes (basically, thermal occupancy convolved with density of states). Why is it not the case? To me, the answer will be density of states only in the limit of infinite temperature. – wcc Mar 15 at 0:23
• This technique for determining the density of states is usually used with equilibrium molecular dynamics simulations. These are classical simulations (in equilibrium), so the equipartition theorem applies, and the thermal occupancy problems basically come out in the wash; I think that you just need to divide the result by the system temperature (and some constants). Just to be clear, this is a well-established technique, and if you search around, you can find proofs for why it works. I'm just trying to give some intuition, not a proof. – lnmaurer Mar 15 at 4:32
• I should add that, yes, if you apply this technique to a non-equilibrium and/or non-classical system, you'll almost certainly get garbage out unless you are more careful about the thermal occupancy. (Maybe even then you'd get garbage out.) – lnmaurer Mar 15 at 4:37
• Thanks for pointing out the equipartition theorem. Now it makes sense to me in the very classical limit. But would you agree that at lower temperature, quantum effect begins to weigh in so that this is not accurate anymore? (analogous to how Dulong-Petit law for specific heat of solid is only valid at high temperature) -edit: just saw your EDIT in the post, where you state that system is considered classical. – wcc Mar 15 at 6:39