Manu: copy paste from the SI (to be polished)

Manuscript/eDFT.tex

@@ -93,6 +93,10 @@
 \newcommand{\manu}[1]{{\textcolor{blue}{ Manu: #1 }} }
 \newcommand{\beq}{\begin{eqnarray}}
 \newcommand{\eeq}{\nonumber\end{eqnarray}}
+\newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector
+\newcommand{\bmg}{\bm{\gamma}} % orbital rotation vector
+\newcommand{\bfx}{\bf{x}}
+\newcommand{\bfr}{\bf{r}}
 %%%% 
 
 \begin{document}	
@@ -170,6 +174,186 @@ Atomic units are used throughout.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \alert{Manu, you might want to add general details about the eDFT here.}
+\manu{Yes. Copy paste from the SI. Will polish the all thing.}
+
+\beq
+F^{\bw}_{\rm HF}[n]&=&
+\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm
+HF}\left[{\bmg}^{\bw}\right]\right\}
+\nonumber\\
+&=&{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}[n]\hat{T}\right]+W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right]
+\eeq
+where
+$\hat{\gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum^M_{K=0}w^{(K)}\hat{\gamma}^{(K)}$ is an ensemble density matrix operator constructed
+from Slater determinants, the ensemble 1RDM elements are $\gamma_{pq}^{\bw}={\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{a}^\dagger_p\hat{a}_q\right]$,
+and $W_{\rm
+HF}\left[{\bmg}\right]=\frac{1}{2}\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\gamma_{pr}\gamma_{qs}$.\\
+
+In-principle-exact decomposition:
+
+\beq
+F^{\bw}[n]= F^{\bw}_{\rm HF}[n]+\overline{E}^{{\bw}}_{\rm
+Hx}[n]+\overline{E}^{{\bw}}_{\rm c}[n]
+\eeq
+
+The complementary ensemble Hx energy removes the ghost-interaction
+errors introduced in $W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right]$:
+\beq
+\overline{E}^{{\bw}}_{\rm
+Hx}[n]=\sum^M_{K=0}w^{(K)}W_{\rm
+HF}\left[{\bmg}^{(K)}[n]\right]
+-W_{\rm
+HF}\left[{\bmg}^{\bw}[n]\right],
+\eeq
+which gives in the canonical orbital basis
+\beq
+&&\overline{E}^{{\bw}}_{\rm
+Hx}[n]=
+\dfrac{1}{2}\sum_{pq}
+\langle \varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\vert\vert
+\varphi^{{\bw}}_p[n]\varphi^{{\bw}}_q[n]\rangle
+\nonumber\\
+&&\times\left[\sum^M_{K=0}w^{(K)}\nu^{(K)}_p \left(\nu^{(K)}_q
+-\sum^M_{L=0}w^{(L)} \nu^{(L)}_q\right)\right]
+.\eeq
+\manu{I would guess that, in a uniform system, the GOK-DFT and our
+canonical orbitals are the same. This is nice since we can construct
+in a clean way density-functional approximations for both $\overline{E}^{{\bw}}_{\rm
+Hx}[n]$ and $E^{{\bw}}_{\rm c}[n]$ functionals. Am I right ?}
+
+Variational expression for the ensemble energy:
+\beq
+E^{{\bw}}=\underset{\hat{\gamma}^{{\bw}}}{\rm min}\Big\{
+&&{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\rm
+HF}\left[{\bmg}^{\bw}\right]
++
+\overline{E}^{{\bw}}_{\rm
+Hxc}\left[n_{\hat{\gamma}^{{\bw}}}\right]
+%+E^{{\bw}}_{\rm c}\left[n_{\hat{\gamma}^{{\bw}}}\right]
+\nonumber\\
+&&
++\int d{\br}\;v_{\rm ext}({\bfr})n_{\hat{\gamma}^{{\bw}}}({\bfr})
+\Big\}
+\eeq
+ 
+Note that, if we use orbital rotations, the gradient of the DFT energy
+contributions can be expressed as follows,
+\beq
+\left.\dfrac{\partial 
+\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
+}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}=\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n^{{\bw}}\right]}{\delta
+n({\br})}\left.\dfrac{\partial n^{{\bw}}({\bmk},{\br})}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}, 
+\eeq
+where   
+\beq
+n^{{\bw}}({\bmk},{\br})=\sum_\sigma\sum_{pq}\varphi_p({\bmk},{\bfx})\varphi_q({\bmk},{\bfx})\gamma_{pq}^{\bw}
+\eeq
+thus leading to
+\beq
+&&\left.\dfrac{\partial 
+\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
+}{\partial \kappa_{lm}}
+\right|_{{\bmk}=0}=
+\sum_{pq}\gamma_{pq}^{\bw}
+\nonumber\\
+&&\times\left.\dfrac{\partial} 
+{\partial \kappa_{lm}}
+\Big[\left\langle\varphi_p(\bmk)\middle\vert\hat{\overline{v}}^{{\bw}}_{\rm
+Hxc}
+\middle\vert \varphi_q(\bmk)\right\rangle
+\Big]
+\right|_{{\bmk}=0}.
+\eeq
+
+In conclusion, the minimizing canonical orbitals fulfill the following
+hybrid HF/GOK-DFT equation,
+\beq
+&&\left(-\frac{\nabla_{\bfr}^2}{2}+v_{\rm
+ext}({\bfr})+\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]
++\dfrac{\delta \overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]}{\delta
+n({\br})}\right)\varphi^{{\bw}}_p({\bfx})
+\nonumber
+\\
+&&=\varepsilon^{{\bw}}_p\varphi^{{\bw}}_p({\bfx}).
+\eeq
+
+
+Since $\partial \gamma_{pq}^{\bw}/\partial
+w^{(I)}=\gamma_{pq}^{(I)}-\gamma_{pq}^{(0)}$, it comes
+
+\manu{just for me ...
+\beq
+&&+\dfrac{1}{2}
+\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs}
+\nonumber\\
+&&+\dfrac{1}{2}\sum_{pqrs}\langle \varphi_q\varphi_p\vert\vert
+\varphi_s\varphi_r\rangle
+%\times
+ \gamma^{\bw}_{pr}\left(\gamma_{qs}^{(I)}-\gamma_{qs}^{(0)}\right)
+\nonumber\\
+&&=
+\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
+\varphi_r\varphi_s\rangle
+%\times
+\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)\gamma^{\bw}_{qs}
+\nonumber\\
+&&=
+\sum_{pr}\left[\hat{u}_{\rm HF}\left[\gamma^{\bw}\right]\right]_{pr}\left(\gamma_{pr}^{(I)}-\gamma_{pr}^{(0)}\right)
+\nonumber\\
+&&=
+\sum_p\left[\hat{u}_{\rm
+HF}\left[\gamma^{\bw}\right]\right]_{pp}\left(\nu_p^{(I)}-\nu_p^{(0)}\right)
+\eeq
+}
+
+\beq
+\dfrac{dE^{\bw}}{dw^{(I)}}=\sum_p\varepsilon^{{\bw}}_p\left(\nu_p^{(I)}-\nu_p^{(0)}\right)+\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}.
+\eeq
+
+LZ shift in this context: $\varepsilon^{{\bw}}_p\rightarrow
+\overline{\varepsilon}^{{\bw}}_p=\varepsilon^{{\bw}}_p+\overline{\Delta}_{\rm
+LZ}^{{\bw}}$ where
+
+\beq
+N\overline{\Delta}_{\rm
+LZ}^{{\bw}}&=&\overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}\right]
+-\int d{\br}\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n^{{\bw}}\right]}{\delta
+n({\br})}n^{{\bw}}({\bfr})
+\nonumber\\
+&&
+-W_{\rm
+HF}\left[{\bmg}^{\bw}\right]
+\eeq 
+
+such that
+\beq
+E^{{\bw}}=\sum^M_{K=0}w^{(K)}\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p.
+\eeq
+
+Thus we conclude that individual energies can be expressed in principle
+exactly as follows,
+
+\beq
+E^{(K)}=\sum_p\nu_p^{(K)}\overline{\varepsilon}^{{\bw}}_p+\sum^M_{I>0}\left(\delta_{IK}-w^{(I)}\right)\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n\right]}{\partial w^{(I)}}\right|_{n=n^{{\bw}}}.
+\eeq
 
 %In eDFT, the ensemble energy
 %\begin{equation}