Anharmonicity in Lattice Dynamics and Vibrational Spectroscopies: Development of Algorithms in the Crystal Code

The description of vibrational properties in solids is a major challenge in theoretical chemistry and condensed-matter physics. Vibrational spectra, such as IR and Raman, are highly sensitive to structural and dynamical features and are essential for validating first-principles simulations. Proper interpretation of experimental spectra requires a theoretical framework linking structural details to observable quantities. The harmonic approximation is the standard starting point for lattice dynamics, treating nuclear motion as independent oscillators around equilibrium. While successful for many thermodynamic and vibrational properties, it neglects higher-order contributions of the potential energy surface and therefore exhibits several limitations. For predictive modeling, accurate inclusion of anharmonicity is essential. A variety of methods exists, ranging from perturbative schemes to fully variational approaches accounting for mode couplings beyond the harmonic level. Their application to periodic systems, however, remains computationally demanding and is still an active research area. This thesis was developed within the Theoretical Chemistry group at the University of Torino, which develops advanced electronic structure and vibrational methods in the Crystal software suite. Crystal, based on localized Gaussian-type orbitals, provides extensive electronic and vibrational capabilities for molecular and periodic systems. The aim of this work is the development and implementation of new theoretical tools enabling a quantitative anharmonic description of lattice dynamics in solids. Building upon the existing VSCF and VCI infrastructure in Crystal, the work extends the code to compute IR and Raman intensities within different anharmonic frameworks, ensuring a balanced compromise between accuracy and computational feasibility. Within Crystal, the variational VSCF and VCI methods already offer a route to include anharmonicity through mean-field treatments and correlated vibrational wavefunctions. A subspace iterative algorithm was implemented to improve the efficiency of VCI calculations in collaboration with Prof. Jean- Pierre Flament (Universit´e de Lille). In parallel, perturbative techniques provide an alternative description of anharmonic effects. Vibrational perturbation theory to second order (VPT2) allows the evaluation of frequency shifts and intensities at low cost. Building on this formalism, this thesis introduces vibrational quasi-degenerate perturbation 1 theory to second order (VQDPT2), improving the treatment of resonant systems and mitigating limitations of conventional schemes. All these anharmonic approaches were integrated into Crystal with particular attention to IR and Raman intensity evaluation, enabling direct comparison with experiments. To validate the implementations, applications were performed on two strongly anharmonic molecular crystals: dry ice (solid CO2) and thiourea. Dry ice presents quasi-degeneracy between its symmetric stretching and bending modes, making it a challenging benchmark. The study, conducted with Prof. Mino (University of Torino), provided new insight into observed resonances and predicted an additional, previously unreported one. Thiourea displays highly anharmonic low-frequency terahertz modes. Collaboration with Prof. Michael T. Ruggiero (University of Rochester) enabled comparison with terahertz timedomain spectroscopy, confirming the key role of anharmonicity for accurate modeling of lattice dynamics in molecular crystals