Next, the electronic mapping variables are propagated using an linearly interpolated Hamiltonian between the time 0 and ∆t in the basis of adiabatic electronic states at time 0. The unitary rotation between the time ∆t and time 0 bases is exactly the time-overlap matrix S(0, ∆t). Explicitly, the diagonal energies at time ∆t, [ (∆t)], are rotated to the time 0 basis as H(∆t) = S(0, ∆t)×[ (∆t)]×S T (0, ∆t), which forms an off-diagonal Hamiltonian matrix. Here [ (∆t)] implies a matrix representation of the diagonal adiabatic energies µ (∆t). The linearly interpolated Hamiltonian then takes the form