d_{1i} 
      d_{i1} 
    \frac{ W_{i1} }
         {W_{i1}^2-\omega_0^2}
~~~~
-
      \frac{1}{ \hbar}
          \exp \left[ +\imath W_{22} t \right]
  \sum_i
      d_{2i}^* 
      d_{i2}^* 
    \frac{ W_{i2} }
         {W_{i2}^2-\omega_0^2}
\\
&=&
      \frac{1}{ \hbar}
  \sum_i
  \left[
      d_{i1}^*d_{i1} 
    \frac{ W_{i1} }
         {W_{i1}^2-\omega_0^2}
 - 
      d_{2i}^* d_{2i}
    \frac{ W_{i2} }
         {W_{i2}^2-\omega_0^2}
  \right]
\label{eqn-alpha-stark}\end{aligned}$$

Simplify assuming $W_{21} \ll W_{i1}, W_{i2}$, while assuming $W_{i1}/\left(W_{i1}^2-\omega_0^2 \right) \sim 1$; or, more accurately, that   $$\begin{aligned}
1 
&\gg&
      \frac{W_{21}^n}{ W_{i1}^n}
    \frac{ W_{i1}^2 }
         {W_{i1}^2-\omega_0^2}
~~~~ ~~~~ \textrm{for } ~~~~ n ~ = ~~1, ~2
,\end{aligned}$$

and keeping only the terms first order in $W_{21}/W_{i1}, W_{21}/W_{i2}$, etc   $$\begin{aligned}
\alpha_{11}' - \alpha_{22}'^*
&=&
      \frac{1}{ \hbar}
  \sum_i
  \left\{
      \left| d_{1i} \right|^2
    \frac{ W_{i1} }
         {W_{i1}^2-\omega_0^2}
 - 
      \left| d_{2i} \right|^2
    \frac{ W_{i1} + W_{21} }
         {W_{i1}^2+ 2W_{i1} W_{21} + W_{21}^2 -\omega_0^2}
  \right\}
\\
&=&
      \frac{1}{ \hbar}
  \sum_i
  \left\{
      \left| d_{1i} \right|^2
    \frac{ W_{i1} }
         {W_{i1}^2-\omega_0^2}
 - 
      \left| d_{2i} \right|^2
    \frac{ W_{i1} + W_{21} }
         {
           \left( W_{i1}^2 -\omega_0^2 \right) 
         }
    \frac{ 1}
         {
           \left[ 
                  1 
                + \left( 2W_{21} / W_{i1} + W_{21}^2/ W_{i1}^2 \right)
                   W_{i1}^2 / \left( W_{i1}^2 -\omega_0^2 \right)
           \right]
         }
  \right\}
\\
&=&
      \frac{1}{ \hbar}
  \sum_i
  \left\{
      \left| d_{1i} \right|^2
    \frac{ W_{i1} }
         {W_{i1}^2-\omega_0^2}
 - 
      \left| d_{2i} \right|^2
    \frac{ W_{i1} + W_{21} }
         { W_{i1}^2 -\omega_0^2 }
           \left[ 
                  