Formulation

In this section we briefly explore the underlying theory and formulation made used in the workflows. For a more in depth explanation, please refer to the main paper of the package.

Considered good sources are:

• Theoretical backgroud books:

  • Max Born and Kun Huang, Dynamical Theory of Crystal Lattices (1954)

  • Peter Brueescvh, Phonons: Theory and Experiments II (1982)

• Ab-initio related articles:

  • Stefano Baroni et al., Phonons and related crystal properties from density-functional perturbation theory, Rev. Modern Phys., 73, 515 (2001)

  • Paolo Umari and Alfredo Pasquarello, Infrared and Raman spectra of disordered materials from first principles, Diamond and Rel. Mat., 14, 9 (2005)

In the code, all properties are computed within the Born-Oppenheimer and harmonic approximation. The vibrational spectra are computed in the first-order non-resonant regime: the infrared using the dipole-dipole approximation, and the Raman using the Placzek approximation.

Important

These are considered good approximations for insulators. Nevertheless, a frequency dependent solution form is usually used also for the resonant case and for metals. Nevertheless, one must be aware that in such cases (resonance, metals) these approximations might not hold, as multiphonon processes, non-adiabaticity, excitonic effects (i.e. electronic excitations), or even exciton-phonon interactions might be non negligible, thus comparison with experiments could result poor. If these effects are important for your case, you can refer to S. Reichardt and L. Wirtz, Science Advances, 7, 32 (2020).

Moreover, temperature effects can also play a crucial role, as anharmonic effects (of ions) should be incorporate to the phonons. A state-of-the-art approach, which differs from the classical molecular dynamics solutions, can be found using the time-dependent self-consistent harmonic approximation.

Thus, for insulators, one needs to evaluate the following (static) tensors:

	\(\partial/\partial \tau_{K,k}\)	\(\partial/\partial \mathcal{E}_i\)	\(\partial^2/\partial \mathcal{E}_i \partial \mathcal{E}_j\)
\(\partial/\partial \tau_{L,l}\)	\(\Phi_{KL,kl}\)	\(Z^*_{K,ki}\)	\(\partial \chi_{ij}/\partial \tau_{K,k}\)
\(\partial/\partial \mathcal{E}_i\)	\(Z^*_{K,ik}\)	\(\epsilon^{\infty}_{ij}\)	\(\chi^{(2)}_{ijk}\)
\(\partial^2/\partial \mathcal{E}_i \partial \mathcal{E}_j\)	\(\partial \chi_{ij}/\partial \tau_{K,k}\)	\(\chi^{(2)}_{ijk}\)	“-”