 other one composed of cross terms. The first one is given by $$\begin{aligned}
&& \left| S^{(2)}_{\alpha \beta} \right|^2_{\text{1st}} = 
\sum_{k, K} \sum_{l, L} 
\frac{ 1 }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) } 
\nonumber \\ 
&\times&
\biggl[
x^2 e^{- i ( h_{k} - h_{l} ) x} 
- (ix) \frac{e^{- i ( \Delta_{K} - h_{l} ) x} - e^{- i ( h_{k} - h_{l} ) x} }{ ( \Delta_{K} - h_{k} )  } 
+ (ix) \frac{e^{- i ( h_{k} - \Delta_{L} ) x} - e^{- i ( h_{k} - h_{l} ) x} }{ ( \Delta_{L} - h_{l} )  } 
\nonumber \\ 
&+& 
\frac{ 1 }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) } 
\biggl\{
e^{- i ( \Delta_{K} - \Delta_{L} ) x} + e^{- i ( h_{k} - h_{l} ) x} 
- e^{- i ( \Delta_{K} - h_{l} ) x} - e^{- i ( h_{k} - \Delta_{L} ) x} 
\biggr\}
\biggr] 
\nonumber \\ 
&\times&
(UX)_{\alpha k} (UX)^*_{\beta k} 
\left\{ (UX)^{\dagger} A W \right\}_{k K} 
\left\{ W ^{\dagger} A (UX) \right\}_{K k} 
\nonumber \\
&\times& 
(UX)_{\alpha l}^* (UX)_{\beta l} 
\left\{ (UX)^{\dagger} A W \right\}_{l L} 
\left\{ W^{\dagger} A (UX) \right\}_{L l} 
\nonumber \\ 
&+& 
\sum_{k \neq m} \sum_{K}  \sum_{l \neq n} \sum_{L}  
\frac{ 1 }{ ( h_{m}  - h_{k} ) (\Delta_{K} - h_{k}) (\Delta_{K} - h_{m}) } 
\frac{ 1 }{ ( h_{n}  - h_{l} ) (\Delta_{L} - h_{l}) (\Delta_{L} - h_{n}) } 
\nonumber \\
&\times& 
\biggl[
\left( \Delta_{K} - h_{k} \right) e^{- i h_{m} x} 
- \left( \Delta_{K} - h_{m} \right) e^{- i h_{k} x} 
- ( h_{m}  - h_{k} )  e^{- i \Delta_{K} x} 
\biggr]
\nonumber \\
&\times& 
\biggl[
\left( \Delta_{L} - h_{l} \right) e^{+ i h_{n} x} 
- \left( \Delta_{L} - h_{n} \right) e^{+ i h_{l} x} 
- ( h_{n}  - h_{l} )  e^{+ i \Delta_{L} x} 
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} (UX)^*_{\beta m} 
\left\{ (UX)^{\dagger} A W \right\}_{k K} 
\left\{ W ^{\dagger} A (UX) \right\}_{K m} 
\nonumber \\
&\times& 
(UX)_{\alpha l}^* (UX)_{\beta n} 
\left\{ (UX)^{\dagger} A W \right\}_{n L} 
\left\{ W^{\dagger} A (UX) \right\}_{L l} 
\nonumber \\
&+& 
\sum_{k, K} \sum_{l, L} 
\frac{1}{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{l} ) } 
\left( e^{- i \Delta_{K} x}