Eberhard Gross - The Schrödinger equation of two-component systems and its density-functionalization

Описание: This lecture was part of the Symposium on "The World in One Line – Schrödinger’s Equation Turns 100" held at the ESI on January 22 & 23, 2026.

Quantum systems consisting of more than one species of particles are common in nature. If the masses of the particles are very different, as in the case of electrons and nuclei, this fact can be exploited to develop approximations. For example, the adiabatic approximation writes the full wave function of a molecule or solid as a single product of a Born-Oppenheimer state and a nuclear wave packet. This approximation is a corner stone of modern quantum chemistry and solid state physics. Yet, many fascinating phenomena appear outside the adiabatic regime such as the microscopic processes underlying the physiology of vision. Another example is the phenomenon of electronic decoherence, i.e. the loss of quantumness, infamous for its capacity of preventing genuine scalable quantum computing to this day. In this lecture a unified approach to non-adiabaticity, known as the exact factorization, will be presented. The approach starts from a formally exact representation of the full electron-nuclear wave function as a product of a purely nuclear part and a many-electron wave function which parametrically depends on the nuclear configuration and which has the meaning of a conditional probability amplitude. The equations of motion for these two factors provide an ideal starting point to develop efficient algorithms for the study of non-adiabatic phenomena. In particular, a time-dependent density functional theory will be developed for the complete system of electrons and nuclei. The resulting time-dependent Kohn-Sham equations have two interesting, and somewhat surprising, properties: (i) The time evolution is non-unitary. This feature is essential for the description of decoherence. (ii) Even if the initial state is a determinant, the time-propagation creates correlations among the electrons through non-adiabatic interactions with the nuclei. A similar effect is known in the Lindblad approach. The successful description of vibrational circlular dichroism, the ab-initio evaluation of decoherence, as well as calculations of the beyond-Born-Oppenheimer molecular Berry phase will demonstrate the power of the new method. The approach is applicable to arbitrary two-component systems.