# Partition Function from Dispersion Relation in Molecular Dynamics

I’ve seen there are ways to compute the dispersion relation of a crystal from molecular dynamics. An example of how to do this is discussed in this question: Computing phonon dispersion from molecular dynamics simulations?.

Once we have the dispersion relation, my understanding is that we can obtain the phonon Density of States and from it we can compute the partition function and therefore thermodynamic quantities like free energy. However, I have not seen this method used in practice for computing thermodynamic quantities in Molecular Dynamics. Therefore my questions: Is this method actually used in practice? Or is it incorrect to obtain the partition function from dispersion relations computed from molecular dynamics?

There is nothing wrong with obtaining the partition function from the dispersion relations computed from molecular dynamics or from any other method, including the diagonalization of the dynamic matrix. However, whatever is the method used for obtaining the dispersion relations, this procedure is only able to provide the free energy of a solid within harmonic approximation.

However in some cases, for example for the determination of liquid/solid or solid/solid phase boundaries, the harmonic approximation is not accurate enough.

• Thank you very much for your answer. I have a follow-up question: If the potential used in the Molecular Dynamics simulation is anharmonic such that it shows thermal expansion for example, is the provided free energy still within the harmonic approximation?
– sp1n
Oct 14, 2021 at 7:15
• @sp1n If the free energy is obtained from the dispersion relations, the free energy is by construction, within harmonic approximation. To go beyond harmonic approximation, one needs different methods to evaluate the free energy., based on motion beyond the harmonic regime. Oct 14, 2021 at 21:25
• Thank you again, it was very helpful.
– sp1n
Oct 15, 2021 at 15:56
• Do you know any references where the harmonic or quasi-harmonic approximation errors on the free energy of a solid/solid phase boundary are measured or analyzed? Thanks!
– sp1n
Oct 12, 2022 at 16:17