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); 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.