diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex
index 1caf281..fdacc2d 100644
--- a/Manuscript/eDFT.tex
+++ b/Manuscript/eDFT.tex
@@ -414,9 +414,28 @@ integrals.
 %%% Manu's derivation
 \blue{
 \beq
-\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^\sigma q^\tau  r^\sigma
-s^\tau\in(K)}\langle p^\sigma q^\tau\vert r^\sigma 
-s^\tau\rangle -\langle p^\sigma q^\tau \vert s^\sigma r^\tau\rangle
+&&\dfrac{1}{2}\sum_{PQRS}\langle PQ\vert\vert
+RS\rangle\Gamma_{PR}\Gamma_{QS}
+\nonumber\\
+&&
+=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}RS}
+\nonumber\\
+&&\Big(\langle p^\sigma\sigma q^\tau\tau\vert RS\rangle -\langle
+p^\sigma\sigma q^\tau\tau
+\vert SR\rangle
+\Big)\Gamma_{p^\sigma\sigma,R}\Gamma_{q^\tau\tau, S}
+\nonumber\\
+&&
+=\dfrac{1}{2}\sum_{\sigma,\tau}\sum_{p^{\sigma} q^{\tau}}
+\nonumber\\
+&&\Big(\sum_{r^\sigma s^\tau}\langle p^\sigma q^\tau\vert r^\sigma s^\tau\rangle
+\Gamma^\sigma_{p^\sigma r^\sigma}\Gamma^\tau_{q^\tau s^\tau}
+\nonumber\\
+&& -\sum_{s^\sigma r^\tau}\langle
+p^\sigma q^\tau
+\vert s^\sigma r^\tau\rangle
+\delta_{\sigma\tau}\Gamma^\sigma_{p^\sigma
+r^\sigma}\Gamma^\sigma_{q^\sigma s^\sigma}\Big)
 \eeq
 }
 %%%%