Explore advanced concepts in quantum mechanics through this lecture from the Advanced Quantum ESPRESSO school, focusing on beyond harmonic phonons, phase diagrams, and phase transitions. Delve into the intricacies of Hubbard and Koopmans functionals from linear response, gaining valuable insights into cutting-edge quantum physics research. Learn how these advanced techniques contribute to understanding complex material behaviors and their applications in materials science. Enhance your knowledge of computational methods used in quantum mechanics and their role in predicting and analyzing phase transitions in various systems.