Manu: started polishing the theory section

Manuscript/eDFT.tex

@@ -98,6 +98,7 @@
 \newcommand{\bfx}{\bf{x}}
 \newcommand{\bfr}{\bf{r}}
 \DeclareMathOperator*{\argmin}{arg\,min}
+\newcommand{\blue}[1]{{\textcolor{blue}{#1}}}
 %%%% 
 
 \begin{document}	
@@ -169,33 +170,9 @@ Atomic units are used throughout.
 \label{sec:eDFT}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\subsection{Kohn--Sham formulation of GOK-DFT}
+\subsection{GOK-DFT}
 
-Let us start from the analog for ensembles of Levy's universal
-functional,   
-\beq\label{eq:ens_LL_func}
-F^{\bw}[n]&=&
-\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
-Tr}\left[\hat{\gamma}^{{\bw}}\left(\hat{T}+\hat{W}_{\rm
-ee}\right)\right]\right\}
-\eeq
-where ${\rm
-Tr}$ denotes the trace. The minimization over trial ensemble density matrix operators
-$\hat{\gamma}^{{\bw}}=\sum_{K\geq0}w_K\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
-is performed under the following density constraint:
-\beq
-{\rm
-Tr}\left[\hat{\gamma}^{{\bw}}\hat{n}(\br)\right]=\sum_{K\geq0}w_Kn_{\Psi^{(K)}}(\br)=n(\br),
-\eeq
-where $\hat{n}(\br)$ is the density operator, $n_{\Psi}$ denotes the
-density of wavefunction $\Psi$, and
-$\bw\equiv\left(w^{(1)},w^{(2)},\ldots\right)$ is the collection of
-(decreasing) ensemble weights assigned to the excited states. Note that
-$w_0=1-\sum_{K>0}w_K\geq 0$.
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Hybrid GOK-DFT}
-\label{sec:geKS}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 Since Hartree and exchange energy contributions cannot be separated in
 the one-dimensional case, we introduce in the following an alternative
@@ -268,28 +245,8 @@ determinants $\Phi^{(K)}$. Note that the density matrices
 ${\bmg}^{(K)}={\bmg}^{\Phi^{(K)}}$ are idempotent and diagonal in the
 same spin-orbital basis). On the other hand, the ensemble
 density matrix ${\bmg}^{{\bw}}=\sum_{K\geq 0}w_K{\bmg}^{(K)}$, which is a convex combination of the ${\bmg}^{(K)}$
-matrices, is {\it not} idempotent, unless ${\bw}=0$. Indeed,
+matrices, is {\it not} idempotent, unless ${\bw}=0$.
-\beq
+Using an ensemble is, in this context,
-\left[{\bmg}^{{\bw}}\right]^2&=&\sum_{K,L\geq
-0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
-\nonumber\\
-&=&\sum_{K\geq
-0}\left(w_K\right)^2{\bmg}^{(K)}+\sum_{K\neq L\geq
-0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
-\nonumber\\
-&=&
-{\bmg}^{{\bw}}+\sum_{K,L\geq
-0}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
-\nonumber\\
-&=&{\bmg}^{{\bw}}+w_0{\bmg}^{(0)}\times\sum_{K>0}w_K\left(2{\bmg}^{(K)}-1\right)
-\nonumber\\
-&&+\sum_{K, L >0
-}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
-\nonumber\\
-&\neq&{\bmg}^{{\bw}}
-.
-\eeq
-This is of course expected since using an ensemble is, in this context,
 analogous to assigning
 fractional occupation numbers (which are determined from the ensemble
 weights) to the KS orbitals.\\ 
@@ -532,6 +489,120 @@ Hxc}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
 \eeq
 }
 
+\subsection{Approximations}
+
+In order to compute (approximate) energy levels within generalized
+GOK-DFT we use a two-step procedure. The first step consists in
+optimizing variationally the ensemble density matrix according to
+Eq.~(\ref{eq:var_princ_Gamma_ens}) with an approximate Hxc ensemble
+functional where (i) the ghost-interaction correction functional $\overline{E}^{{\bw}}_{\rm
+Hx}[n]$ in
+Eq.~(\ref{eq:exact_GIC}) is
+neglected, for simplicity, and (ii) the weight-dependent correlation
+energy is described at the local density level of approximation. 
+At this
+level of approximation, the two correlation functionals $\overline{E}^{{\bw}}_{\rm
+c}[n]$ and ${E}^{{\bw}}_{\rm
+c}[n]$ are actually identical and can be expressed as
+\beq\label{eq:eLDA_corr_fun}
+{E}^{{\bw}}_{\rm
+c}[n]=\int d\br\;n(\br)\epsilon_{c}^{\bw}(n(\br)). 
+\eeq
+More
+details about the construction of such a functional will be given in the
+following. In order to remove ghost interactions from the variational energy
+expression used in the first step, we then employ the (in-principle-exact)
+expression in Eq.~(\ref{eq:exact_ind_ener_OEP-like}). In this second
+step, the response of the individual density matrices to weight
+variations (last term on the right-hand side of
+Eq.~(\ref{eq:exact_ind_ener_OEP-like})) is neglected. The complete GIC
+procedure can be summarized as follows,    
+\beq
+{\bmg}^{\bw}\approx\argmin_{{\bm\gamma}^{\bw}}
+\Big\{
+{\rm
+Tr}\left[{\bm \gamma}^{{\bw}}{\bm h}\right]+W_{\rm
+HF}\left[{\bm\gamma}^{\bw}\right]
++
+{E}^{{\bw}}_{\rm
+c}\left[n_{\bm\gamma^{\bw}}\right]
+%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
+\Big\},
+\nonumber\\
+\eeq
+and
+\beq
+E^{(I)}&&\approx{\rm
+Tr}\left[{\bmg}^{(I)}{\bm h}\right]
++\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
+\bmg^{(I)})
+\nonumber\\
+&&+{E}^{{\bw}}_{\rm
+c}\left[n_{\bmg^{\bw}}\right]
++\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
+c}\left[n_{\bmg^{\bw}}\right]}{\delta
+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 {E}^{{\bw}}_{\rm
+c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
+,
+\eeq
+thus leading to the final implementable expression [see Eq.~(\ref{eq:eLDA_corr_fun})]
+\beq
+E^{(I)}&&\approx{\rm
+Tr}\left[{\bmg}^{(I)}{\bm h}\right]
++\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
+\bmg^{(I)})
+\nonumber\\
+&&+\int d\br\,
+{\epsilon}^{{\bw}}_{\rm
+c}(n_{\bmg^{\bw}}(\br))\,n_{\bmg^{(I)}}(\br)
+\nonumber\\
+&&
++\int d\br\,\left.\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
+c}(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.
+\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
+c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
+\eeq
+
+\blue{$================================$}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Theory (old)}
+\label{sec:eDFT_old}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\subsection{Kohn--Sham formulation of GOK-DFT}
+
+Let us start from the analog for ensembles of Levy's universal
+functional,   
+\beq\label{eq:ens_LL_func}
+F^{\bw}[n]&=&
+\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\left(\hat{T}+\hat{W}_{\rm
+ee}\right)\right]\right\}
+\eeq
+where ${\rm
+Tr}$ denotes the trace. The minimization over trial ensemble density matrix operators
+$\hat{\gamma}^{{\bw}}=\sum_{K\geq0}w_K\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
+is performed under the following density constraint:
+\beq
+{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{n}(\br)\right]=\sum_{K\geq0}w_Kn_{\Psi^{(K)}}(\br)=n(\br),
+\eeq
+where $\hat{n}(\br)$ is the density operator, $n_{\Psi}$ denotes the
+density of wavefunction $\Psi$, and
+$\bw\equiv\left(w^{(1)},w^{(2)},\ldots\right)$ is the collection of
+(decreasing) ensemble weights assigned to the excited states. Note that
+$w_0=1-\sum_{K>0}w_K\geq 0$.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Hybrid GOK-DFT}
+\label{sec:geKS}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
 \subsection{Exact ensemble exchange in hybrid GOK-DFT}
 
 
@@ -635,82 +706,6 @@ Tr}\left[{\bm F}^{(L)}\frac{\partial \bmg^{(L)}}{\partial w_K}\right]
 correlation LDA and ghost interaction correction 
 }
 
-In order to compute (approximate) energy levels within generalized
-GOK-DFT we use a two-step procedure. The first step consists in
-optimizing variationally the ensemble density matrix according to
-Eq.~(\ref{eq:var_princ_Gamma_ens}) with an approximate Hxc ensemble
-functional where (i) the ghost-interaction correction functional $\overline{E}^{{\bw}}_{\rm
-Hx}[n]$ in
-Eq.~(\ref{eq:exact_GIC}) is
-neglected, for simplicity, and (ii) the weight-dependent correlation
-energy is described at the local density level of approximation. 
-At this
-level of approximation, the two correlation functionals $\overline{E}^{{\bw}}_{\rm
-c}[n]$ and ${E}^{{\bw}}_{\rm
-c}[n]$ are actually identical and can be expressed as
-\beq\label{eq:eLDA_corr_fun}
-{E}^{{\bw}}_{\rm
-c}[n]=\int d\br\;n(\br)\epsilon_{c}^{\bw}(n(\br)). 
-\eeq
-More
-details about the construction of such a functional will be given in the
-following. In order to remove ghost interactions from the variational energy
-expression used in the first step, we then employ the (in-principle-exact)
-expression in Eq.~(\ref{eq:exact_ind_ener_OEP-like}). In this second
-step, the response of the individual density matrices to weight
-variations (last term on the right-hand side of
-Eq.~(\ref{eq:exact_ind_ener_OEP-like})) is neglected. The complete GIC
-procedure can be summarized as follows,    
-\beq
-{\bmg}^{\bw}\approx\argmin_{{\bm\gamma}^{\bw}}
-\Big\{
-{\rm
-Tr}\left[{\bm \gamma}^{{\bw}}{\bm h}\right]+W_{\rm
-HF}\left[{\bm\gamma}^{\bw}\right]
-+
-{E}^{{\bw}}_{\rm
-c}\left[n_{\bm\gamma^{\bw}}\right]
-%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
-\Big\},
-\nonumber\\
-\eeq
-and
-\beq
-E^{(I)}&&\approx{\rm
-Tr}\left[{\bmg}^{(I)}{\bm h}\right]
-+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
-\bmg^{(I)})
-\nonumber\\
-&&+{E}^{{\bw}}_{\rm
-c}\left[n_{\bmg^{\bw}}\right]
-+\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
-c}\left[n_{\bmg^{\bw}}\right]}{\delta
-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 {E}^{{\bw}}_{\rm
-c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
-,
-\eeq
-thus leading to the final implementable expression [see Eq.~(\ref{eq:eLDA_corr_fun})]
-\beq
-E^{(I)}&&\approx{\rm
-Tr}\left[{\bmg}^{(I)}{\bm h}\right]
-+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
-\bmg^{(I)})
-\nonumber\\
-&&+\int d\br\,
-{\epsilon}^{{\bw}}_{\rm
-c}(n_{\bmg^{\bw}}(\br))\,n_{\bmg^{(I)}}(\br)
-\nonumber\\
-&&
-+\int d\br\,\left.\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
-c}(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.
-\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
-c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
-\eeq
 
 \alert{Secs. \ref{sec:KS-eDFT}-\ref{sec:E_I} should maybe be moved to an appendix or merged
 with the theory section (?)}
@@ -1062,6 +1057,34 @@ E.~F.~thanks the \textit{Agence Nationale de la Recherche} (MCFUNEX project, Gra
 \end{acknowledgements}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+%%%% REMOVED FROM THE MAIN TEXT by Manu %%%%%%%%%%%%
+\iffalse%%%%
+Indeed,
+\beq
+\left[{\bmg}^{{\bw}}\right]^2&=&\sum_{K,L\geq
+0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
+\nonumber\\
+&=&\sum_{K\geq
+0}\left(w_K\right)^2{\bmg}^{(K)}+\sum_{K\neq L\geq
+0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
+\nonumber\\
+&=&
+{\bmg}^{{\bw}}+\sum_{K,L\geq
+0}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
+\nonumber\\
+&=&{\bmg}^{{\bw}}+w_0{\bmg}^{(0)}\times\sum_{K>0}w_K\left(2{\bmg}^{(K)}-1\right)
+\nonumber\\
+&&+\sum_{K, L >0
+}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
+\nonumber\\
+&\neq&{\bmg}^{{\bw}}
+.
+\eeq
+%%%% End -- REMOVED FROM THE MAIN TEXT by Manu %%%%%%%%%%%%
+\fi%%%
+
+
+
 \bibliography{eDFT}
 
 \end{document}