The real wave packet approach applies either to the usual Schrödinger equation or to a modified Schrödinger equation, , where replaces and replaces [12]. Since only the real part is manipulated, computational requirements can be reduced relative to complex wave packet propagation. The modified Schrödinger equation, through judicious choice of f, results in greater savings because f can be chosen to simplify the propagation. Derivations of S matrix or reaction probability formulae are simpler if one refers to the usual Schrödinger equation. The results are then generalized to the modified Schrödinger equation without difficulty.
Consider the modified Schrödinger equation, and take , where with a_{s} and b_{s} chosen such that the maximum and minimum eigenvalues of lie between -1 and 1. Ref. [12] showed that the real part = can be propagated directly and satisfies
where is an arbitrary time step. We have also introduced an absorption operator to remove wave packet amplitude as it approaches grid edges. (Eq. (14) is also the damped Chebyshev iteration introduced in Ref. [9].) While this real wave packet approach focuses on the real part of a wave packet , it should be noted that the initial condition can be complex, as discussed in Ref. [12] and explicitly implemented in Ref. [16].