+
    \frac{1}
         {W_{ij}+\omega_0}
  \right\}
;
~~ \textrm{by discarding the $2 \omega_0$ terms;}
\\
&=&
-\frac{1}{ \hbar}
\sum_i
    \sum_j
      c_j
      d_{ni} 
      d_{ij} 
      A A^*
          \exp \left[ -\imath W_{jn} t \right]
    \frac{W_{ij}+\omega_0 ~~ + ~~ {W_{ij}-\omega_0}}
         { \left( W_{ij}-\omega_0 \right) \left( W_{ij}+\omega_0 \right) }
\\
&=&
-\frac{1}{ \hbar}
\sum_i
    \sum_j
      c_j
      d_{ni} 
      d_{ij} 
      A A^*
          \exp \left[ -\imath W_{jn} t \right]
    \frac{2 W_{ij} }
         {W_{ij}^2-\omega_0^2}
\\
&=&
-    
  A A^*
\sum_j
  c_j
  \sum_i
      \frac{1}{ \hbar}
      d_{ni} 
      d_{ij} 
          \exp \left[ -\imath W_{jn} t \right]
    \frac{2 W_{ij} }
         {W_{ij}^2-\omega_0^2}
\\
&=&
-
\sum_j
  c_j
  \alpha_{nj}'
   ~~ . 2 \left| A \right|^2
\label{eqn-single-A-DcnDt}\end{aligned}$$   where   $$\begin{aligned}
\alpha_{nj}'
&=&
      \frac{1}{ \hbar}
          \exp \left[ -\imath W_{jn} t \right]
  \sum_i
      d_{ni} 
      d_{ij} 
    \frac{W_{ij} }
         {W_{ij}^2-\omega_0^2}
\label{eqn-single-A-alpha}
~~~~ ~~~~ = ~~
      \frac{1}{ \hbar}
          \exp \left[ +\imath W_{nj} t \right]
  \sum_i
      d_{ni} 
      d_{ij} 
    \frac{W_{ij} }
         {W_{ij}^2-\omega_0^2}\end{aligned}$$

These redefined $\alpha_{nj}'$ parameters still oscillate, as they must because unlike in the HPB derivation, there is no frequency difference between field components to cancel with the Raman transition frequency. The coupling also varies with frequency, which is discussed next.

Raman coupling parameters: approximations
-----------------------------------------

I now discuss two particular (and vital) approximations applied to the Raman coupling parameters.

First, note that (as in HPB), 