Quantum geometry and adiabaticity in molecules and in condensed matter

The adiabatic theorem states that when the time evolution of the Hamiltonian is "infinitely slow," a system, when started in the ground state, remains in the instantaneous ground state at all times. This, however, does not mean that the adiabatic evolution of a generic observable obtains simply as its expectation value over the instantaneous eigenstate. As a general principle, there is an additional adiabatic term, of quantum-geometrical nature, which is the relevant one for several static or adiabatic observables. This is shown explicitly for the cases of polarizability and infrared tensors (in molecules and condensed matter) and rotational g factor and magnetizability (in molecules only). Quantum geometry allows for a transparent derivation and a compact expression for these observables, alternative to the well-known sum-over-states Kubo formulas.