diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 9b62104..3d5d6ed 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -458,10 +458,34 @@ Hxc}(n^{\bw}(\br))\,n^{(I)}(\br) Hxc}(n)}{\partial n}\right|_{n=n^{\bw}(\br)}n^{\bw}(\br)\left(n^{(I)}(\br)-n^{\bw}(\br)\right) \nonumber\\ && -+\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm -Hxc}\left[n\right]}{\partial w^{(K)}}\right|_{n=n^{{\bw}}}. ++\int d\br\,\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)n^{{\bw}}(\br)\left. +\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm +Hxc}(n)}{\partial w^{(K)}}\right|_{n=n^{{\bw}}(\br)}. \eeq - +\alert{ +or, equivalently, +\beq +&&E^{(I)}\rightarrow +{\rm +Tr}\left[{\bmg}^{(I)}{\bm h}\right]+ +\Tr\left[\left(\bmg^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right) \, \bG \, \bmg^{\bw}\right] +\nonumber\\ +&& ++\int d\br\, +\dfrac{\delta \overline{E}^{{\bw}}_{\rm +Hxc}\left[n^{\bw}\right]}{\delta +n({\br})}\,n^{(I)}(\br) +\nonumber\\ +&& +-\int d\br\,\left.\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm +Hxc}(n)}{\partial n}\right|_{n=n^{\bw}(\br)}\Big(n^{\bw}(\br)\Big)^2 +\nonumber\\ +&& ++\int d\br\,\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)n^{{\bw}}(\br)\left. +\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm +Hxc}(n)}{\partial w^{(K)}}\right|_{n=n^{{\bw}}(\br)}. +\eeq +} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{KS-eDFT for excited states}