The properties of molecules and solids derive from the quantum mechanics of the electrons and nuclei they are made of. As a full quantum many-body treatment is impractical, virtually all theories begin from the Born-Oppenheimer approximation, where the many-electron problem is solved under the assumption that the usually sluggish nuclei are fixed in space. Solving the many-electron problem again and again for different fixed nuclear positions maps out an electronic wave function Phi_R, a conditional probability amplitude, depending parametrically on the set of nuclear coordinates R. To complete the approximation, one solves for the nuclear wave function chi, a marginal probability amplitude, and constructs the full wave function as the product Phi_R * chi. A recent breakthrough known as the exact factorization method  showed that this type of factorization into conditional and marginal amplitudes can be made exact, establishing a fundamentally new way of tackling nonadiabatic (beyond Born-Oppenheimer) effects. In this talk, I will introduce the basics of the exact factorization method and survey a variety of developments carried out in my group over the past several years. The foundation of a nonadiabatic density functional theory , seamlessly encompassing quantum nuclei, gives us a computationally tractable approach for timely problems in condensed matter physics, materials science and chemistry. I will describe applications to Jahn-Teller-active defects in crystals  and the renormalization of electronic band structure due to electron-phonon interactions .
 A. Abedi, N. T. Maitra and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)  R. Requist and E. K. U. Gross, Phys. Rev. Lett. 117, 193001 (2016)  R. Requist, F. Tandetzky and E. K. U. Gross, Phys. Rev. A 93, 042108 (2016)  R. Requist, C. R. Proetto and E. K. U. Gross, Phys. Rev. B 99, 165136 (2019)