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Negative frequencies "in general" mean the direction of displacement is reverse to that in positive frequencies.

Does it mean two frequencies with opposite signs but the same magnitude give the same energy in the context of the energy-angular frequency relation?

\begin{equation} \omega = \frac{\epsilon}{\bar{h}} \end{equation}

What does this mean for two phonons with frequencies with the same magnitude but opposite signs?

(This question came to mind because the code phonopy outputs DoS for negative frequencies as well.)

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    $\begingroup$ +1. I just added the phonopy tag since this question came up due to phonopy outputting negative frequencies. $\endgroup$ – Nike Dattani Oct 29 at 22:11
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Phonons are a measure of the curvature of the potential energy surface about a stationary point. In particular, the matrix of force constants is calculated as:

$$ D_{i\alpha,i^{\prime}\alpha^{\prime}}(\mathbf{R}_p,\mathbf{R}_{p^{\prime}})=\frac{\partial^2 E}{\partial u_{p\alpha i}\partial u_{p^{\prime}\alpha^{\prime}i^{\prime}}}, $$

where $E$ is the potential energy surface in which the nuclei move, $u_{p\alpha i}$ is the displacement of atom $\alpha$ (of all atoms in the basis), in Cartesian direction $i$ ($x$, $y$, $z$), and located in the cell within the supercell at $\mathbf{R}_p$. This quantity is the second order derivative of the energy in all possible directions, so it measures the curvature about the reference point. To obtain phonons, one transforms the matrix of force constants into the dynamical matrix:

$$ D_{i\alpha;i^{\prime}\alpha^{\prime}}(\mathbf{q})=\frac{1}{N_p\sqrt{m_{\alpha}m_{\alpha^{\prime}}}}\sum_{\mathbf{R}_p,\mathbf{R}_{p^{\prime}}}D_{i\alpha;i^{\prime}\alpha^{\prime}}(\mathbf{R}_p,\mathbf{R}_{p^{\prime}})e^{i\mathbf{q}\cdot(\mathbf{R}_p-\mathbf{R}_{p^{\prime}})}, $$

where $N_p$ is the number of cells in the supercell over which periodic boundary conditions are applied, and $m_{\alpha}$ is the mass of atom $\alpha$. Using standard mathematical language, these two matrices are essentially Hessians. Diagonalizing the dynamical matrix gives eigenvalues $\omega^2_{\mathbf{q}\nu}$ and eigenvectors $v_{\mathbf{q}\nu;i\alpha}$. The key quantity for our discussion are the eigenvalues $\omega^2_{\mathbf{q}\nu}$ which can be:

  1. Positive. Positive eigenvalues indicate a positive curvature of the potential energy surface, so the energy increases quadratically if you displace the atoms in the directions given by the associated eigenvector, and the eigenvalue magnitude tells you how "fast" the energy increases.
  2. Negative. Negative eigenvalues indicate a negative curvature of the potential energy surface, so the energy decreases quadratically if you displace the atoms in the directions given by the associated eigenvector, and the eigenvalue magnitude tells you how "fast" the energy decreases.

If you are performing calculations for a structure at a (local) minimum of the potential energy surface, then all eigenvalues will be positive (positive-definite Hessian). If you are performing calculations for a structure at a saddle point of the potential energy surface, then most eigenvalues will be positive, but those associated with the directions which lower the energy will be negative.

Now we come to the key point: phonon frequencies are given by the square root of the eigenvalues of the dynamical matrix. As these eigenvalues are either positive or negative, then phonon frequencies are either positive real numbers or purely imaginary numbers. Phonon frequencies cannot be negative: they are either positive or imaginary. Many codes output imaginary frequencies as "negative" numbers, but this is a convention that in principle assumes that the user knows that "negative" frequencies are really imaginary, but which I think has traditionally led to big confusions, particularly for new people in the field.

Having clarified this, I will rephrase your question: what does it mean when two eigenvalues of the dynamical matrix, $\omega^2_{\mathbf{q}\nu}$, have the same magnitude but opposite sign? In terms of phonon frequencies the equivalent question would be: what does it mean that two phonon frequenies, $\omega_{\mathbf{q}\nu}$, have the same magnitude when one is real and one imaginary? In both cases, what that means is that the magnitude of the curvature of the potential energy surface is the same, but in one case the energy increases and in the other the energy decreases.

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    $\begingroup$ Thank you for the answer, ProfM. You are right when you say the negative notation for complex frequencies creates confusion. You've spoken of phonon calculations in terms of the PES before, but somehow that slips out of my mind. If I may, I have a suggestion/request for you. You could start a 2nd youtube channel about crystals and lattices, and I'm sure it'd become a sort of videographed textbook! $\endgroup$ – Hitanshu Sachania Oct 29 at 22:53
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    $\begingroup$ +1. How did I know ProfM would answer this? Our #1 user in the phonons tag: mattermodeling.stackexchange.com/tags/phonons/topusers $\endgroup$ – Nike Dattani Oct 29 at 23:00
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    $\begingroup$ @HitanshuSachania, thanks for the suggestion, I do have plans to start a new series on condensed matter once the quantum mechanics series is at a more advanced stage. $\endgroup$ – ProfM Oct 30 at 7:41

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