From cba8d0360a01765bafb7ad4f896bfb49092ea228 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Wed, 18 Sep 2019 17:15:01 +0200 Subject: [PATCH] Manu: saving work --- Manuscript/eDFT.tex | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 347563d..7d212a2 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%