Numerical time-propagation describing photoionization

The integration over k in Eq. (3.57) may be approximated with a numerical quadrature over sampled kj up to a maximum photoelectron momentum Nk = max with quadrature weights Wj, 

Chemical Theory Beyond the Born-Oppenheimer Paradigm 

Now we see that the coupled equations of motion may be written in a compact matrix form, 

Y ) R) is the diagonal part of the potential energy matrix comprised of the neutral state potential energy function and the potential energy function of the ionized states. 

We may apply the split-operator method (Sec. 3.2.1) to the three matrices lijV) V/), and Vo for a short-time numerical propagation scheme. Ionization is now described by population of the neutral state Xn R,t) transferring to the ionized state partial-wave components Xc,kjix R,t) over time through the interaction represented by the matrix Vo(i ,f)- 

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