Instantaneous eigenstate coupling ($A_{mn}=⟨m(t)|{\partial }_{t}n(t)⟩$)

Expanding a quantum state in instantaneous energy eigenstates separates dynamical phases from the coupling $A_{mn}$. orthonormality makes $A$ anti-Hermitian. For distinct eigenvalues, differentiating the eigenvalue equation gives $A_{mn}=⟨m|\dot{H}|n⟩/(E_n-E_m)$. Its off-diagonal entries drive nonadiabatic transitions; diagonal entries describe phase changes of the basis and generate the Berry phase.