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I have been conducting lattice vibration calculations using the Density Functional Perturbation Theory (DFPT) method. While investigating the convergence behavior, I noticed that when I increase the number of KPOINTS, I start observing imaginary frequencies in the results. This behavior seems counterintuitive, as I expected more accurate results with higher k-point sampling.

Could someone help me understand the possible causes behind the appearance of these imaginary frequencies with increasing KPOINTS? Are there any specific considerations or potential issues I should be aware of when interpreting these results? Additionally, how can I determine the appropriate number of k-points for accurate lattice vibration calculations? Any insights or suggestions would be greatly appreciated.

I am using VASP/DFPT and here is the INCAR file:

EDIFF = 1E-8
ENCUT = 400
ALGO = Fast
IBRION = 8
ISMEAR = 0
SIGMA = 0.2
PREC = Accurate
LREAL = .F.
ADDGRID = .TRUE.
LASPH = .TRUE.
NWRITE   = 3
LEPSILON = True

The corresponding KPOINTS:

Monkhorst-Pack
   7   4   2
0.0  0.0  0.0

The first observed frequency is positive (real):

   normal_modes:
- band_indices: [ 1 ]
  frequency: 0.0009280870
  ir_label: B1

As I increase the number of KPOINTS:

Monkhorst-Pack
   8  14   4
0.0  0.0  0.0

negative (imaginary) frequency will appear:

normal_modes:
- band_indices: [ 1 ]
  frequency: -0.0009088724
  ir_label: A1

The corresponding POSCAR file for both cases:

        6.2820000648         0.0000000000         0.0000000000
        0.0000000000         3.4960000515         0.0000000000
        0.0000000000         0.0000000000        14.0699996948
    W   Te
    4    8
Selective dynamics
Direct
     0.541400000  0.000000000  0.985100000
     0.458600000  0.500000000  0.485100000
     0.900500000  0.500000000  0.000000000
     0.099500000  0.000000000  0.500000000
     0.294100000  0.500000000  0.096500000
     0.705900000  0.000000000  0.596500000
     0.355900000  0.000000000  0.344900000
     0.644100000  0.500000000  0.844900000
     0.851700000  0.500000000  0.389300000
     0.148300000  0.000000000  0.889300000
     0.800200000  0.000000000  0.140000000
     0.199800000  0.500000000  0.640000000

So as you can see it happens only with the first band and despite that it is a small frequency, yet it is very strange to observe such thing.

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    $\begingroup$ Are you re-optimising the geometry when you increase the k-point sampling? $\endgroup$ Commented Dec 2, 2023 at 2:54
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    $\begingroup$ @PhilHasnip Thanks for the reply, actually no I didn't re-optimize the geometry, and to be more specific I am just experiementing here and I used some automated way to generate the kpoints, the scenario was that I am directly using a POSCAR file that was reported before without doing any optimizations either on the cell or the ion positions, I have generated kpoints using vaspkit, the first case with an accuracy of 0.04 leading to the (7,4,2) kpoints which gave accurate frequencies, in the second case, I just increased the sampling of kpoints (accuracy of 0.02) that gave (8,14,4) kpoints. $\endgroup$ Commented Dec 2, 2023 at 5:19
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    $\begingroup$ Negative modes usually mean that the structure was not well optimised for the given cut-off and k-point set. Since you're explicitly changing the k-points, this behaviour is what would be expected unless the k-point set was very well converged. $\endgroup$ Commented Dec 4, 2023 at 14:35
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    $\begingroup$ You are right, I have re-optimized the system according to the new set of kpoints and the modes were all real. $\endgroup$ Commented Dec 5, 2023 at 0:34
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    $\begingroup$ Great! I'll write something as an answer so other people can find it better $\endgroup$ Commented Dec 6, 2023 at 0:21

1 Answer 1

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In the usual perturbation theory approach, we optimise a structure to minimise the forces (to zero, within some small tolerance) and obtain the ground state structure (or a local energy minimum). Perturbing the atomic positions in any way will move them away from this ground state, and so the energy will rise and a force will act to oppose the perturbation. Within the quasi-harmonic approximation, we treat this as a simple harmonic oscillator, and from the mass and force constant matrix (derivatives of the force with respect to atomic position) we can compute the frequency, $\omega$, of the resultant oscillation. For any perturbation, $\omega$ should be positive (semi-definite).

If a phonon calculation results in "negative frequencies" (by which we actually mean negative $\omega^2$, i.e. imaginary frequencies), it means that the energy actually decreases with respect to the perturbation. This cannot happen if the structure was in the ground state, or indeed in any local energy minimum, so a negative frequency means either:

  • the structure is mechanically unstable
  • the structure was not well optimised (i.e. the forces are not small enough)

Since you expect your particular system to be mechanically stable, and your negative frequencies are small, we strongly suspect that the structure was not sufficiently well optimised.

Why should these frequencies get worse with increasing k-point sampling? The issue is almost certainly that the lowest energy atomic configuration changes as you increase the k-point sampling -- probably not by very much, but enough that the forces increase as you increase the k-point sampling for fixed atomic positions. If you re-optimise your structure for the improved k-point sampling, you should find that your phonon frequencies are much better.

This same behaviour can happen whenever you change any parameter which affects the geometry of your system. For example, you might see it when using more (or fewer) k-points, or increasing (or decreasing) the plane-wave cut-off energy, or changing the resolution of the fine density grid, even switching pseudopotentials. The solution is always the same: re-optimise your structure.

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