diff --git a/Notes/PerturbativeAnalysis.tex b/Notes/PerturbativeAnalysis.tex
index bcadbd0..751c9d2 100644
--- a/Notes/PerturbativeAnalysis.tex
+++ b/Notes/PerturbativeAnalysis.tex
@@ -500,10 +500,12 @@ Recalling that $\bHod{0} = \bO$ and $\bHd{1} = \bO$, we derive
         \dv{\bV{}{(2),\dagger}}{s} & \dv{\bC{}{(2)}}{s} 
       \end{pmatrix} \\
   \dv{\bF^{(2)}}{s}  &= \bF^{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF^{(0)} - 2 \bV{}{(1)}\bC{\text{d}}{(0)}\bV{}{(1),\dagger}\\
-  \dv{\bC{}{(2)}}{s}  &=  \\
+  \dv{\bC{}{(2)}}{s}  &= 2 \bC{\text{d}}{(0)}\bC{\text{od}}{(2)}\bC{\text{d}}{(0)}- (\bC{\text{d}}{(0)})^2\bC{\text{od}}{(2)} - \bC{\text{od}}{(2)}(\bC{\text{d}}{(0)})^2  \\
+  &-2 \bC{\text{d}}{(1)}\bC{\text{od}}{(0)}\bC{\text{d}}{(1)}- (\bC{\text{d}}{(1)})^2\bC{\text{od}}{(0)} - \bC{\text{od}}{(0)}(\bC{\text{d}}{(1)})^2 \notag \\
   \dv{\bV{}{(2)}}{s} &= 2 \bF^{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF^{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\
-  &- 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)}  + \bF^{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)}  \\
-  \dv{\bV{}{(2),\dagger}}{s} &=
+  &- 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)}  + \bF^{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \notag \\
+  \dv{\bV{}{(2),\dagger}}{s} &= 2 \bC{\text{d}}{(0)}\bV{}{(2),\dagger}\bF^{(0)} - \bV{}{(2),\dagger}(\bF^{(0)})^2 - (\bC{\text{d}}{(0)})^2\bV{}{(2),\dagger} \\
+  &- 2 \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \bV{}{(1),\dagger} +  \bC{\text{od}}{(1)} \bV{}{(1),\dagger}  \bF^{(0)} + \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} \bV{}{(1),\dagger} \notag
 \end{align}
 
 \begin{align}
@@ -516,6 +518,13 @@ Recalling that $\bHod{0} = \bO$ and $\bHd{1} = \bO$, we derive
  &\color{red}{\boxed{\color{black}{- \sum_R \frac{\eps^{(0)}_{p}  + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R}{(\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2}(1 - e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]})}}} \notag
 \end{align}
 
+\begin{align}
+  (\dv{\bV{}{(2)}}{s})_{pQ} &=  (2 \bF^{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF^{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\
+  & - 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)}  + \bF^{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)})_{pQ} \notag \\
+  v^{(2)}_{pQ}(s) &= v^{(2)}_{pQ}(0) e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2} + \text{Non-homogeneous solution} \notag \\
+  v^{(2)}_{pQ}(s) &= \text{Non-homogeneous solution}
+\end{align}
+
 %%%%%%%%%%%%%%%%%%%%%%
 \subsection{Downfolding the SRG-transformed matrix}
 %%%%%%%%%%%%%%%%%%%%%%