equations of motion of the $c_i$ co-efficients proceeds as –   $$\begin{aligned}
  \imath \hbar 
  \frac{\partial}
       {\partial t}
  \sum_i c_i e^{-\imath W_i t}
    \left| i \right>
&=&
\left( H_0 + V \right) 
  \sum_j c_j e^{-\imath W_j t}
    \left| j \right>
\\
  \imath \hbar 
  \sum_i 
  \left\{
    -\imath W_i 
    c_i 
    e^{-\imath W_i t}
  +
    e^{-\imath W_i t}
    \frac{\partial c_i}
         {\partial t}
  \right\}
    \left| i \right>
&=&
\left( H_0 + V \right) 
  \sum_j c_j e^{-\imath W_j t}
    \left| j \right>
\\
  \imath \hbar 
\left< n \right|
  \sum_i 
  \left\{
    -\imath W_i 
    c_i 
    e^{-\imath W_i t}
  +
    e^{-\imath W_i t}
    \frac{\partial c_i}
         {\partial t}
  \right\}
    \left| i \right>
&=&
\left< n \right|
\left( H_0 + V \right) 
  \sum_j c_j e^{-\imath W_j t}
    \left| j \right>
\\
  \imath \hbar 
  \left\{
    -\imath W_n 
    c_n 
    e^{-\imath W_n t}
  +
    e^{-\imath W_n t}
    \frac{\partial c_n}
         {\partial t}
  \right\}
&=&
  \hbar
  c_n
  W_n
    e^{-\imath W_n t}
 +
  \sum_j 
    c_j 
    e^{-\imath W_j t}
    \left< n \right|
      -d.E
    \left| j \right>
\\
  \imath \hbar 
  \left\{
    -\imath W_n 
    c_n
  +
    \frac{\partial c_n}
         {\partial t}
  \right\}
&=&
  \hbar
  W_n
  c_n
 -
  \sum_j 
    c_j 
    e^{-\imath W_j t + \imath W_n t}
    \left< n \right|
      d.E
    \left| j \right>
\\
  \imath \hbar 
    \frac{\partial c_n}
         {\partial t}
&=&
 -
  