  An extension of the Hellmann-Feynman theorem to one employing dynamical
parameters that vary with time according to quantum dynamics is rigorously
derived, avoiding any linear response or other approximations. The resulting
theorem for the dynamics of observables, valid to all orders in external
fields, is found to contain generalized Berry curvature type of quantities that
incorporate the dynamics through explicit and nontrivial time-dependence; these
Berry quantities resemble the so called anomalous terms (in semiclassical
equations of motion in solids) but are both of magnetic and electric type, the
former being associated with Quantum Hall Effect type of behaviors and the
latter with Polarization type of behaviors. By way of application of the new
theorem, the quantum equations of motion of a spinless and a spinfull electron
in a solid are derived without any adiabatic or semiclassical approximation.
The charge current formula for a many-body and interacting spinfull system is
also derived and is found to consist of a longitudinal and a transverse part;
phenomenological interpretations with respect to polarization and magnetization
currents respectively then emerge in a natural way. In addition, a formula for
the topological magnetoelectric effect for an interacting spinfull electron
system is also provided. By carefully defining singlevaluedness in parameter
space, in a nonstandard fashion and in higher rigor than usual, we are able to
discuss in clarity the issue of possible obstruction of this singlevaluedness,
the associated creation of Berry monopoles in parameter space and the
quantization of the flux of Berry curvature (but with nontrivial dynamics
included in its definition).
