W_i - W_j$,   $$\begin{aligned}
\imath \hbar \frac{d c_n}{dt}
&=&
-
\sum_i
  \left[
    \frac{1}{ \hbar}
    \sum_j
      c_j
      d_{ij} E
    \frac{
          \exp \left[ -\imath W_{ji} t \right]
         }
         {W_i-W_j}
  \right]
  d_{ni} E 
  \exp \left[ -\imath W_{in} t \right]
\\
&=&
-\frac{1}{ \hbar}
\sum_i
    \sum_j
      c_j
      d_{ij} E
    \frac{
          \exp \left[ -\imath W_{ji} t \right]
         }
         {W_{ij} }
  d_{ni} E 
  \exp \left[ -\imath W_{in} t \right]
\\
&=&
-
      E^2
  \sum_j 
    \frac{1}{\hbar}
    \sum_i
      d_{ni} d_{ij} c_j
    \frac{
          \exp \left[ -\imath W_{jn} t \right]
         }
         {W_i-W_j}
\\
&=&
-
  \sum_j 
    c_j
    \alpha_{nj} E^2
,
\label{eqn-single-E-DcnDt}\end{aligned}$$   where   $$\begin{aligned}
\alpha_{nj}
&=&
\frac{1}{\hbar}
\exp \left[ -\imath W_{jn} t \right]
    \sum_i
    \frac{ d_{ni} d_{ij} }
         {W_i-W_j}
\label{eqn-single-E-alpha}
\\
\textrm{UNITS:} ~~~~
\left[ \alpha_{nj} \right]
&\equiv& J^{-1} s^{-1} . Cm . Cm . \left( s^{-1} \right)^{-1}
~~~~ = C^2 m^2 J^{-1}
~~~~ = \left( J V^{-1} \right)^2 m^2 J^{-1}
~~~~ = J m^2 V^{-2}
\\
\textrm{UNITS:} ~~~~
\left[ \frac{\alpha_{nj}}{\hbar} E^2\right]
&\equiv&
J m^2 V^{-2} . J^{-1} s^{-1} . \left( V m^{-1} \right)^2
~~~~ = J . J^{-1}  \times V^{-2} . V^2 \times  m^2 . m^{-2} \times s^{-1}
~~~~ = s^{-1}\end{aligned}$$

Since we are only interested in the end states $j=1, 2$, between which the Raman transition occurs, we only need 