From 8a0b414989903898d7ffb38a5f041024e9305f73 Mon Sep 17 00:00:00 2001
From: Emmanuel Fromager <manu@Emmanuels-MacBook-Pro.local>
Date: Tue, 17 Sep 2019 18:18:35 +0200
Subject: [PATCH] Manu: saving work

---
 Manuscript/eDFT.tex | 26 +++++++++++++++++++++-----
 1 file changed, 21 insertions(+), 5 deletions(-)

diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex
index 510c2fe..347563d 100644
--- a/Manuscript/eDFT.tex
+++ b/Manuscript/eDFT.tex
@@ -405,8 +405,10 @@ Tr}\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right){\bm h}\right]
 Hxc}\left[n_{\bmg^{\bw}}\right]}{\delta
 n({\br})}\left(n^{(K)}(\br)-n^{(0)}(\br)\right)
 \nonumber\\
-&&+\left.       \dfrac{\partial \overline{E}^{{\bw}}_{\rm
-Hxc}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}},
+&&
++\left.       \dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
+,
 \eeq
 we finally obtain from Eqs.~(\ref{eq:var_princ_Gamma_ens}) and (\ref{eq:indiv_ener_from_ens}) the following in-principle-exact expressions for the
 energy levels within the ensemble: 
@@ -564,19 +566,33 @@ c}\left[n_{\bmg^{\bw}}\right]
 \Bigg\}
 \eeq
 
+For $K>0$
 \beq
-\dfrac{\partial E^{{\bw}}}{\partial w_K}&=&
+&&\dfrac{\partial E^{{\bw}}}{\partial w_K}=
 {\rm
 Tr}\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right){\bm h}\right]
 +\frac{1}{2}\Tr(\bmg^{(K)} \, \bG \, \bmg^{(K)})
 \nonumber\\
 &&-\frac{1}{2}\Tr(\bmg^{(0)} \, \bG \, \bmg^{(0)})
-+\sum_{L\geq0}w_L{\rm
-Tr}\left[\dfrac{\partial\bmg^{(L)}}{\partial w_K}{\bm h}\right]
 \nonumber\\
 &&
++\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+c}\left[n_{\bmg^{\bw}}\right]}{\delta
+n({\br})}\left(n^{(K)}(\br)-n^{(0)}(\br)\right)
++\left.       \dfrac{\partial \overline{E}^{{\bw}}_{\rm
+c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
+\nonumber\\
+&&+\sum_{L\geq0}w_L{\rm
+Tr}\left[\dfrac{\partial\bmg^{(L)}}{\partial w_K}{\bm h}\right]
+%\nonumber\\
+%&&
 +\sum_{L\geq0}w_L
 \Tr(\bmg^{(L)} \, \bG \, \dfrac{\partial\bmg^{(L)}}{\partial w_K})
+\nonumber\\
+&&
++\sum_{L\geq0}w_L\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+c}\left[n_{\bmg^{\bw}}\right]}{\delta
+n({\br})}n_{\frac{\partial \bmg^{(L)}}{\partial w_K}}(\br)
 \eeq
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%