Manu: started writing about the OEP-like scheme

Manuscript/eDFT.tex

@@ -298,7 +298,7 @@ HF}\left[\bmg\right]$. This type of errors is specific to ensembles
 which explains why, in constrast to ground-state DFT [see
 Eq.~(\ref{eq:generalized_KS-DFT_decomp})], a complementary ensemble Hx
 energy is needed to recover a ghost-interaction-free energy: 
-\beq
+\beq\label{eq:exact_GIC}
 \overline{E}^{{\bw}}_{\rm
 Hx}[n]&=&
 {\rm
@@ -361,7 +361,7 @@ Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+W_{\rm
 HF}\left[{\bmg}^{\bw}\right]
 +
 \overline{E}^{{\bw}}_{\rm
-Hxc}\left[n^{{\bw}}\right]
+Hxc}\left[n_{\bmg^{\bw}}\right]
 %+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
 \Big\}
 \Bigg\}
@@ -376,12 +376,14 @@ Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+W_{\rm
 HF}\left[{\bmg}^{\bw}\right]
 +
 \overline{E}^{{\bw}}_{\rm
-Hxc}\left[n^{{\bw}}\right]
+Hxc}\left[n_{\bmg^{\bw}}\right]
 %+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
 \Big\}
 ,
 \eeq
-where $n^{\bw}$ is the density obtained from the density matrix
+where $n_{\bmg^{\bw}}$ 
+\manu{I am in favor of using $n_{{\bmg}^{{\bw}}}$ rather than $n_{\bmg^{\bw}}$,
+for clarity} is the density obtained from the density matrix
 ${\bmg}^{\bw}$ and ${\bm h}={\bm t}+{\bm v}_{\rm ext}$ is the total one-electron
 Hamiltonian matrix representation. When the minimum is reached, the
 ensemble energy and its derivatives can be used to extract individual
@@ -400,11 +402,11 @@ Tr}\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right){\bm h}\right]
 \nonumber\\
 &&+
 \int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
-Hxc}\left[n^{{\bw}}\right]}{\delta
+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^{{\bw}}},
+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: 
@@ -415,14 +417,14 @@ Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
 \Tr\left[\left(\bmg^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right) \, \bG \, \bmg^{\bw}\right]
 \nonumber\\
 &&+\overline{E}^{{\bw}}_{\rm
-Hxc}\left[n^{{\bw}}\right]
+Hxc}\left[n_{\bmg^{\bw}}\right]
 +\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
-Hxc}\left[n^{{\bw}}\right]}{\delta
+Hxc}\left[n_{\bmg^{\bw}}\right]}{\delta
-n({\br})}\left(n^{(I)}(\br)-n^{\bw}(\br)\right)
+n({\br})}\left(n^{(I)}(\br)-n_{\bmg^{\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}}}.
+Hxc}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}.
 \eeq
 %+\Tr(\bmg^{(I)} \, \bG \, \bmg^{\bw})
 %-\dfrac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})+...
@@ -430,7 +432,7 @@ Hxc}\left[n\right]}{\partial w_K}\right|_{n=n^{{\bw}}}.
 Note that
 \beq
 \overline{E}^{{\bw}}_{\rm
-Hx}\left[n^{{\bw}}\right]=
+Hx}\left[n_{\bmg^{\bw}}\right]=
 \frac{1}{2} \sum_{L\geq0}w_L \Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
 -\frac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})
 \nonumber\\
@@ -438,7 +440,7 @@ Hx}\left[n^{{\bw}}\right]=
 and
 \beq
 \left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
-Hx}[n]}{\partial w_K}\right|_{n=n^{\bw}}&=&
+Hx}[n]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}&=&
 \frac{1}{2} \Tr(\bmg^{(K)} \, \bG \, \bmg^{(K)})-\frac{1}{2}
 \Tr(\bmg^{(0)} \, \bG \, \bmg^{(0)})
 \nonumber\\
@@ -448,12 +450,12 @@ Hx}[n]}{\partial w_K}\right|_{n=n^{\bw}}&=&
 thus leading to
 \beq
 &&\overline{E}^{{\bw}}_{\rm
-Hx}\left[n^{{\bw}}\right]+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left.       \dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hx}\left[n_{\bmg^{\bw}}\right]+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left.       \dfrac{\partial \overline{E}^{{\bw}}_{\rm
-Hx}\left[n\right]}{\partial w_K}\right|_{n=n^{{\bw}}}
+Hx}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
 \nonumber\\
 &&=
 \overline{E}^{{\bw}}_{\rm
-Hx}\left[n^{{\bw}}\right]+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \, \bmg^{(I)})
+Hx}\left[n_{\bmg^{\bw}}\right]+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \, \bmg^{(I)})
 -\frac{1}{2} \sum_{L\geq0}w_L \Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
 \nonumber\\
 &&-\Tr\left[\left({\bmg}^{(I)}-{\bmg}^{\bw}\right) \, \bG \, \bmg^{\bw}\right]
@@ -489,16 +491,16 @@ Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
 &&
 +\int d\br\,
 \overline{\epsilon}^{{\bw}}_{\rm
-Hxc}(n^{\bw}(\br))\,n^{(I)}(\br)
+Hxc}(n_{\bmg^{\bw}}(\br))\,n^{(I)}(\br)
 \nonumber\\
 &&
 +\int d\br\,\left.\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
-Hxc}(n)}{\partial n}\right|_{n=n^{\bw}(\br)}n^{\bw}(\br)\left(n^{(I)}(\br)-n^{\bw}(\br)\right)
+Hxc}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}n_{\bmg^{\bw}}(\br)\left(n^{(I)}(\br)-n_{\bmg^{\bw}}(\br)\right)
 \nonumber\\
 &&
-+\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n^{{\bw}}(\br)\left.
++\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br)\left.
 \dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
-Hxc}(n)}{\partial w_K}\right|_{n=n^{{\bw}}(\br)}.
+Hxc}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
 \eeq
 \alert{
 or, equivalently,
@@ -511,20 +513,72 @@ Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
 &&
 +\int d\br\,
 \dfrac{\delta \overline{E}^{{\bw}}_{\rm
-Hxc}\left[n^{\bw}\right]}{\delta
+Hxc}\left[n_{\bmg^{\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
+Hxc}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}\Big(n_{\bmg^{\bw}}(\br)\Big)^2
 \nonumber\\
 &&
-+\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n^{{\bw}}(\br)\left.
++\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br)\left.
 \dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
-Hxc}(n)}{\partial w_K}\right|_{n=n^{{\bw}}(\br)}.
+Hxc}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
 \eeq
 }
 
+\subsection{OEP-like approach}
+
+
+In the exact theory, the minimizing density matrix in
+Eq.~(\ref{eq:var_princ_Gamma_ens}) is such that
+\beq
+{\bmg}^{(K)}[n_{{\bmg}^{{\bw}}}]={\bmg}^{(K)},\hspace{0.2cm}\forall
+K\geq0,
+\eeq
+and therefore
+\beq
+{\bmg}^{{\bw}}\left[n_{{\bmg}^{{\bw}}}\right]={\bmg}^{{\bw}}.
+\eeq
+Combining the latter Eqs. with
+Eqs. (\ref{eq:exact_GIC}), (\ref{eq:var_princ_Gamma_ens}) leads to
+the final ensemble energy expression
+\beq
+E^{{\bw}}={\rm
+Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+\frac{1}{2} \sum_{L\geq0}w_L
+\Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
++\overline{E}^{{\bw}}_{\rm
+c}\left[n_{\bmg^{\bw}}\right].
+\nonumber\\
+\eeq
+
+Note that
+\beq
+E^{{\bw}}&\neq& \underset{\left\{{\bmg}^{(L)}\right\}_{L\geq 0}}{\rm min}\Big\{
+{\rm
+Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+\frac{1}{2} \sum_{L\geq0}w_L
+\Tr(\bmg^{(L)} \, \bG \, \bmg^{(L)})
+\nonumber\\
+&&+\overline{E}^{{\bw}}_{\rm
+c}\left[n_{\bmg^{\bw}}\right]
+\Bigg\}
+\eeq
+
+\beq
+\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\\
+&&
++\sum_{L\geq0}w_L
+\Tr(\bmg^{(L)} \, \bG \, \dfrac{\partial\bmg^{(L)}}{\partial w_K})
+\eeq
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{KS-eDFT for excited states}
 \label{sec:KS-eDFT}