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An extension of the Hellmann-Feynman theorem to one employing dynamicalparameters that vary with time according to quantum dynamics is rigorouslyderived, avoiding any linear response or other approximations. The resultingtheorem for the dynamics of observables, valid to all orders in externalfields, is found to contain generalized Berry curvature type of quantities thatincorporate the dynamics through explicit and nontrivial time-dependence; theseBerry quantities resemble the so called anomalous terms (in semiclassicalequations of motion in solids) but are both of magnetic and electric type, theformer being associated with Quantum Hall Effect type of behaviors and thelatter with Polarization type of behaviors. By way of application of the newtheorem, the quantum equations of motion of a spinless and a spinfull electronin a solid are derived without any adiabatic or semiclassical approximation.The charge current formula for a many-body and interacting spinfull system isalso derived and is found to consist of a longitudinal and a transverse part;phenomenological interpretations with respect to polarization and magnetizationcurrents respectively then emerge in a natural way. In addition, a formula forthe topological magnetoelectric effect for an interacting spinfull electronsystem is also provided. By carefully defining singlevaluedness in parameterspace, in a nonstandard fashion and in higher rigor than usual, we are able todiscuss in clarity the issue of possible obstruction of this singlevaluedness,the associated creation of Berry monopoles in parameter space and thequantization of the flux of Berry curvature (but with nontrivial dynamicsincluded in its definition).

Dynamical extension of Hellmann-Feynman theorem and application to nonadiabatic quantum processes in Topological and Correlated Matter