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Which one is better in case of a phonon calculation and does DFPT also have small negative frequencies near Gamma due to the numerical noise? What are their advantages and disadvantages?

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From my understanding, the main difference is that DFPT is theoretically more sound where the electric field perturbations enter the hamiltonian and thus is calculated self consistently in the electronic and geometric optimization. This allows for several physical properties to be extracted (dielectric tensor, polarizabilities, Born charges, etc.) besides the phonon frequencies.

Computational limitations arise in DFPT to be restricted to small unit cells, while finite differences allow handling much larger unit cells.

If you are a VASP user, you may recognize the tags, IBRION=5/6 for FD and LEPSILON=True for DFPT, which may tell you some more technical differences.

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    $\begingroup$ The fact that for DFPT you only need to use a primitive cell is not a computational limitation, it is an advantage. Supercells are needed in finite differences because to describe a phonon of wave vector $\mathbf{q}$ you need a supercell commensurate with it. This somewhat limits finite differences because sometimes you cannot build a large enough supercell to do the calculation you want. In DFPT, you can have a phonon of any wave vector in the primitive cell, so if DFPT is available, it is computationally better, not worse. $\endgroup$
    – ProfM
    Aug 15 '20 at 18:36
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    $\begingroup$ DFPT is not theoretically sounder than finite differences, they are largely equivalent, and in fact most DFPT implementations are benchmarked against finite differences because the latter are conceptually simpler. You are mixing different things when you talk about electric fields. To give you an example, you can calculate the Born effective charge in a finite difference context by calculating the derivative of the polarization with respect to an atomic displacement. $\endgroup$
    – ProfM
    Aug 15 '20 at 18:39
  • $\begingroup$ Valid points and thanks for the corrections. I guess I jumped to premature conclusions. $\endgroup$
    – gogo
    Aug 16 '20 at 2:06

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