## ANTONIO SICILIANO

Dottore di ricerca**ciclo**: XXXVI

**supervisore**: Giancarlo Ruocco

**relatore**: Francesco Mauri

**Titolo della tesi:**Quantum Nuclear Dynamics: Non-Gaussian Fluctuations and Multi-Phonon Scattering

The atomic motion controls essential properties of materials, such as thermal transport, phase transitions, and vibrational spectra. However, simulating the ionic dynamics can be exceptionally challenging when quantum fluctuations are relevant (e.g., at low temperatures or with light atoms) and the energy landscape is anharmonic. The most straightforward solution to this problem is to constrain the quantum-thermal fluctuations of atoms to be Gaussian, as done in the Self-Consistent Harmonic Approximation (SCHA). Unfortunately, this approximation fails for rotational modes, diffusive states, liquids, mixed order-disorder transitions, and tunneling effects, where the nuclei density matrix exhibits strong non-Gaussian fluctuations.
In the first part of the thesis, I bridge this methodological gap by introducing the nonlinear SCHA (NLSCHA) method. With a nonlinear transformation, I map Cartesian coordinates into an auxiliary manifold. So I describe non-Gaussian nuclear fluctuations using a Gaussian density matrix in the nonlinear coordinate system. The nonlinearity of the mapping ensures that NLSCHA goes beyond SCHA, and its invertibility conserves information, so I can evaluate the entropy in the auxiliary space where it has a simple analytical form. Consequently, there is no need for thermodynamic integration or heavy diagonalizations to include finite-temperature effects. Additionally, I demonstrate the method's efficacy for rotational degrees of freedom, outperforming Gaussian-based approaches.
In the second part of the thesis, I investigate the response of quantum nuclei to external time-dependent potentials. The Time-Dependent SCHA (TDSCHA) provides an efficient numerical solution for finite-temperature nuclear dynamics with quantum and anharmonic fluctuations beyond perturbative approaches. TDSCHA approximates the nuclear density matrix with the most general Gaussian in Cartesian coordinates. However, the original formulation of TDSCHA is exceptionally complex, hindering clear physical interpretations. So, I employ Wigner's formalism to rewrite the equations of motion in the quantum phase space. Besides the improved numerical efficiency, the Wigner formalism unveils the classical limit of TDSCHA and provides a link with the many-body perturbation theory of Feynman diagrams. I further extend the method to account for the nonlinear couplings between phonons and photons, responsible, e.g., for a nonvanishing Raman signal in high-symmetry Raman inactive crystals, first discussed by Rasetti and Fermi. Finally, I benchmark the method simulating vibrational spectra of high-pressure hydrogen phase III.