W_{21} t \right]
  \sum_i
      d_{1i} 
      d_{i2} 
    \frac{ W_{i2} }
         {W_{i2}^2-\omega_0^2}
~~~~
+
      \frac{1}{ \hbar}
          \exp \left[ +\imath W_{12} t \right]
  \sum_i
      d_{2i}^* 
      d_{i1}^* 
    \frac{ W_{i1} }
         {W_{i2}^2-\omega_0^2}
\\
&=&
      \frac{1}{ \hbar}
  \sum_i
      d_{1i} 
      d_{i2} 
    \left[
      \frac{ W_{i2} }
           {W_{i2}^2-\omega_0^2} 
     +
      \frac{ W_{i1} }
           {W_{i2}^2-\omega_0^2}
    \right]
          \exp \left[ -\imath W_{21} t \right]
\\
&\approx&
\bar{\alpha}_{12}  
  e^{-\imath W_{21} t}
\label{eqn-alpha-bar}\end{aligned}$$

### Sum is cosine

I do not reduce the exponential sum to a (trig) cosine function, for potential use later when matching and/or differencing exponentials. However, this sum is does not occur, so the $\bar{a}_{12}$ is never used.   $$\begin{aligned}
\alpha_{12}' + \alpha_{21}'
&=&
      \frac{1}{ \hbar}
          \exp \left[ -\imath W_{21} t \right]
  \sum_i
      d_{1i} 
      d_{i2} 
    \frac{ W_{i2} }
         {W_{i2}^2-\omega_0^2}
~~~~
+
      \frac{1}{ \hbar}
          \exp \left[ -\imath W_{12} t \right]
  \sum_i
      d_{2i} 
      d_{i1} 
    \frac{ W_{i1} }
         {W_{i2}^2-\omega_0^2}
\\
&=&
      \frac{1}{ \hbar}
  \sum_i
      d_{1i} 
      d_{i2} 
    \left[
      \frac{ W_{i2} }
           {W_{i2}^2-\omega_0^2} 
          \exp \left[ -\imath W_{21} t \right]
     +
      \frac{ W_{i1} }
           {W_{i2}^2-\omega_0^2}
          \exp \left[ +\imath W_{21} t \right]
    \right]
\\
&\approx&
\bar{a}_{12}  \left[ e^{-\imath W_{21} t} + e^{+\imath W_{21} t} \right]
\label{eqn-alpha-bar-B}\end{aligned}$$

### Stark Shifts

Here I calculate the Stark shift term   $$\begin{aligned}
\alpha_{11}' - \alpha_{22}'^*
&=&
      \frac{1}{ \hbar}
          \exp \left[ -\imath W_{11} t \right]
  \sum_i
      