and write the time-dependent wave function by expanding it in terms of the eigenfunctions in the field-free (i.e. $E=0$) case. This means I can get the expansion coefficients by solving for an effective Schröedinger equation that contains a two-photon Rabi frequency created by means of an interaction term based on a field-dependent dipole moment. I assume a dispersionless medium and write all equations in terms of position $z$ and retarded times $t=t_{lab}-z/c$. Here I follow the method of HPB [@Hickman-PB-1986pra], but use only a single $E$ field rather than multiple components. Note that HPB use [*Gaussian*]{} units, so there can appear to be inconsistencies when comparing my formulae (in S.I.) to theirs.

I denote the known molecular eigenfunctions of the unperturbed Hamiltonian $H_0$ as $\left| n \right>$, and their corresponding energies $\hbar W_n$. I want to obtain the solution to   $$\begin{aligned}
\left( H_0 + V \right) \psi 
&=& 
  \imath \hbar 
  \frac{\partial \psi}
       {\partial t}
\label{eqn-hamltonian-def}
\\
\textrm{for} ~~~ ~~~
V &=& 
-d E 
\label{eqn-perturbation-def}
\\
\psi &=&
\sum_n
  c_n e^{-\imath W_n t} \left| n \right>
,
\label{eqn-psi-def}\end{aligned}$$   where $d$ is the electronic dipole moment operator and the $c_n$ are a set of complex probability amplitudes.

A standard derivation for the 