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Manu: saving work

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This commit is contained in:
Emmanuel Fromager 2019-09-18 17:15:01 +02:00
parent 8a0b414989
commit cba8d0360a
1 changed files with 9 additions and 9 deletions
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18
Manuscript/eDFT.tex
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@ -403,7 +403,7 @@ Tr}\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right){\bm h}\right]
&&+
\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
Hxc}\left[n_{\bmg^{\bw}}\right]}{\delta
n({\br})}\left(n^{(K)}(\br)-n^{(0)}(\br)\right)
n({\br})}\left(n_{\bmg^{(K)}}(\br)-n_{\bmg^{(0)}}(\br)\right)
\nonumber\\
&&
+\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm
@ -422,7 +422,7 @@ Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
Hxc}\left[n_{\bmg^{\bw}}\right]
+\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
Hxc}\left[n_{\bmg^{\bw}}\right]}{\delta
n({\br})}\left(n^{(I)}(\br)-n_{\bmg^{\bw}}(\br)\right)
n({\br})}\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
\nonumber\\
&&
+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm
@ -493,11 +493,11 @@ Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
&&
+\int d\br\,
\overline{\epsilon}^{{\bw}}_{\rm
Hxc}(n_{\bmg^{\bw}}(\br))\,n^{(I)}(\br)
Hxc}(n_{\bmg^{\bw}}(\br))\,n_{\bmg^{(I)}}(\br)
\nonumber\\
&&
+\int d\br\,\left.\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
Hxc}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}n_{\bmg^{\bw}}(\br)\left(n^{(I)}(\br)-n_{\bmg^{\bw}}(\br)\right)
Hxc}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}n_{\bmg^{\bw}}(\br)\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
\nonumber\\
&&
+\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br)\left.
@ -516,7 +516,7 @@ Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
+\int d\br\,
\dfrac{\delta \overline{E}^{{\bw}}_{\rm
Hxc}\left[n_{\bmg^{\bw}}\right]}{\delta
n({\br})}\,n^{(I)}(\br)
n({\br})}\,n_{\bmg^{(I)}}(\br)
\nonumber\\
&&
-\int d\br\,\left.\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
@ -571,14 +571,14 @@ For $K>0$
&&\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)})
&&+\frac{1}{2}\Tr(\bmg^{(K)} \, \bG \, \bmg^{(K)})
-\frac{1}{2}\Tr(\bmg^{(0)} \, \bG \, \bmg^{(0)})
\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)
n({\br})}\left(n_{\bmg^{(K)}}(\br)-n_{\bmg^{(0)}}(\br)\right)
+\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm
c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
\nonumber\\
@ -592,7 +592,7 @@ Tr}\left[\dfrac{\partial\bmg^{(L)}}{\partial w_K}{\bm h}\right]
&&
+\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)
n({\br})}n_{\frac{\partial \bmg^{(L)}}{\partial w_K}}(\br).
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%