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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 ...

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The phonon dispersion relates the phonon frequencies $\omega_{\mathbf{q}\nu}$ for each branch $\nu$ with the phonon wave vector $\mathbf{q}$, typically along a path in the Brillouin zone joining high-symmetry points. The phonon density of states compresses this information by integrating over $\mathbf{q}$ and summing over $\nu$: $$\tag{1} g(\omega)=\sum_{\... 5 There are two possible things tripping you up here: Phonons are collective oscillations: they involve the motion of all the atoms together. Therefore it only makes sense to talk about the phonons of the whole system, not any individual atom. The density of states only makes sense as you take N\to \infty. For finite N, there is a finite/discrete number ... 4 Background. The phonon density of states g is given by:$$ \tag{1} g(\omega)=\sum_{\nu}\int\frac{d\mathbf{q}}{(2\pi)^3}\delta(\omega-\omega_{\mathbf{q}\nu})\approx\frac{1}{N_{\mathbf{q}}}\sum_{\nu}\sum_{\mathbf{q}}\Delta(\omega-\omega_{\mathbf{q}\nu}), $$where \omega is the energy and \omega_{\mathbf{q}\nu} the energy of a phonon of wave vector \... 4 Short answer. Yes, it is possible to calculate the phonon density of states using density functional theory. You can calculate the phonon frequencies on an arbitrarily large \mathbf{q}-point grid to construct the density of states, and most DFT codes will have the functionality to do this. Longer answer. The density of states is given by:$$ \tag{1} g(\...

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As known , the eigenvalues of the dynamical Hessian matrix represent the phonon frequencies, whereas the eigenvectors represent the particular atomic displacement patterns contributing to the vibrations. Therefore you might be interested in analyzing the eigenvectors and building the atomic visualizations based on them. In many cases (especially in the ...

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