Manu: polished the theory and the approximations

Manuscript/eDFT.tex

@@ -333,7 +333,9 @@ c}\left[n\right]}{\partial w_K}
 \right|
 _{n=n_{\opGamma{\bw}}}.
 \eeq
-
+%%%%%%%%%%%%%%%%
+\subsection{One-electron reduced density matrix formulation}
+%%%%%%%%%%%%%%%%
 For implementation purposes, we will use in the rest of this work 
 (one-electron reduced) density matrices 
 as basic variables, rather than Slater determinants. If we expand the
@@ -362,7 +364,9 @@ n_{\bmg^{(K)}}(\br)=
 %\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\br,
 %\sigma})\AO{\nu}(\br,\sigma){\Gamma}^{(K)}_{\mu\nu}.
 \eeq 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % Manu's derivation %%%
+\iffalse%%
 \blue{
 \beq
 n_{\bmg^{(K)}}(\br)&=&\sum_\sigma\left\langle\hat{\Psi}^\dagger(\br\sigma)\hat{\Psi}(\br\sigma)\right\rangle^{(K)}
@@ -378,11 +382,13 @@ p}}c^\sigma_{{\nu p}}\AO{\mu}(\br)\AO{\nu}(\br)
 p}}c^\sigma_{{\nu p}}
 \eeq
 }
-%%%%
+\fi%%%
+%%%% end Manu
 We can then construct the ensemble density matrix
 and the ensemble density as follows:
 \beq
-{\bmg}^{{\bw}}=\sum_{K\geq 0}w_K{\bmg}^{(K)}
+{\bmg}^{{\bw}}=\sum_{K\geq 0}w_K{\bmg}^{(K)}\equiv
+\Gamma_{\mu\nu}^{\bw\sigma}=\sum_{K\geq 0}w_K \Gamma_{\mu\nu}^{(K)\sigma}
 \eeq
 and
 \beq
@@ -406,12 +412,43 @@ n({\br})}\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
 c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
 ,
 \eeq
-where ${\bm
+where 
-h}\equiv\left\{\langle\AO{\mu}\vert-\frac{1}{2}\nabla_{\br}^2+v_{\rm
+\beq
-ne}(\br)\vert\AO{\nu}\rangle\right\}_{\mu\nu}$ and ${\bm G}\equiv{\bm J}-{\bm K}$ denote
+{\bm
-the Coulomb-exchange
+h}\equiv h_{\mu\nu}=
-integrals. 
+%\langle\AO{\mu}\vert-\frac{1}{2}\nabla_{\br}^2+v_{\rm
+%ne}(\br)\vert\AO{\nu}\rangle
+\int d\br\;\AO{\mu}(\br)\left[-\frac{1}{2}\nabla_{\br}^2+v_{\rm
+ne}(\br)\right]\AO{\nu}(\br)
+\eeq
+denote the one-electron integrals matrix.
+The individual Hx energy is obtained from the following trace
+\beq
+\Tr(\bmg^{(K)} \, \bG \,
+\bmg^{(L)})=\sum_{\mu\nu\lambda\omega}\sum_{\sigma=\alpha,                                         \beta}\sum_{\tau=\alpha,\beta}G_{\mu\nu\lambda\omega}^{\sigma\tau}
+\Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
+\nonumber\\
+\eeq
+where the two-electron Coulomb-exchange integrals read
+\beq
+G_{\mu\nu\lambda\omega}^{\sigma\tau}=({\mu}{\nu}\vert{\lambda}{\omega})
+-\delta_{\sigma\tau}(\mu\omega\vert\lambda\nu),
+\eeq
+with
+\beq
+(\mu\nu|\la\omega) = \iint  \frac{\AO\mu(\br_1) \AO\nu(\br_1)
+\AO\la(\br_2)                          \AO\omega(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
+.
+\nonumber\\
+\eeq
+Note that, in Sec.~\ref{sec:results}, the theory is applied to (1D) spin
+polarized systems in which $\Gamma_{\mu\nu}^{(K)\beta}=0$ and
+$G_{\mu\nu\lambda\omega}^{\alpha\alpha}\equiv G_{\mu\nu\lambda\omega}=({\mu}{\nu}\vert{\lambda}{\omega})
+-(\mu\omega\vert\lambda\nu)$. 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%% Hx energy ...
 %%% Manu's derivation
+\iffalse%%%%
 \blue{
 \beq
 &&\dfrac{1}{2}\sum_{PQRS}\langle PQ\vert\vert
@@ -460,9 +497,10 @@ n_{p^\sigma}^{(K)\sigma}n_{q^\sigma}^{(L)\sigma}\right)
 \Gamma_{\mu\nu}^{(K)\sigma}\Gamma_{\lambda\omega}^{(L)\tau}
 \eeq
 }
+\fi%%%%%%%
 %%%%
 %%%%%%%%%%%%%%%%%%%%%
-%\iffalse%%%% Manu's derivation ...
+\iffalse%%%% Manu's derivation ...
 \blue{
 \beq
 n^{\bw}({\br})&=&\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
@@ -488,33 +526,8 @@ w}_K
 \nonumber\\
 &=&\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\bfx})\AO{\nu}({\bfx}){\Gamma}^{\bw}_{\mu\nu}
 \eeq
-With the notations T2 prefers ...
-\beq
-n^{\bw}({\br})&=&\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
-w}_Kn^{(K)}({\bfx})
-\nonumber\\
-&=&
-\sum_{K\geq 0}\sum_{\sigma=\alpha,\beta}{\tt
-w}_K\sum_{pq}\varphi_p({\br})\varphi_q({\br})\Gamma_{pq}^{(K)}
-\nonumber\\
-&=&
-\sum_{\sigma=\alpha,\beta}
-\sum_{K\geq 0}
-{\tt
-w}_K\sum_{p\in (K)}\varphi^2_p({\bfx})
-\nonumber\\
-&=&
-\sum_{\sigma=\alpha,\beta}
-\sum_{K\geq 0}
-{\tt
-w}_K
-\sum_{\mu\nu}
-\sum_{p\in (K)}c_{\mu p}c_{\nu p}\AO{\mu}({\bfx})\AO{\nu}({\bfx})
-\nonumber\\
-&=&\sum_{\sigma=\alpha,\beta}\sum_{\mu\nu}\AO{\mu}({\bfx})\AO{\nu}({\bfx}){\Gamma}^{\bw}_{\mu\nu}
-\eeq
 }
-%\fi%%%%%%%% end
+\fi%%%%%%%% end
 %%%%%%%%%%%%%%%
 %\subsection{Hybrid GOK-DFT}
 %%%%%%%%%%%%%%%
@@ -544,6 +557,29 @@ c}\left[n_{\bm\gamma^{\bw}}\right]
 \Big\}.
 \nonumber\\
 \eeq
+The minimizing ensemble density matrix fulfills the following
+stationarity condition
+\beq
+{\bm F}^{\bw\sigma}{\bm \Gamma}^{\bw\sigma}{\bm S}={\bm S}{\bm
+\Gamma}^{\bw\sigma}{\bm F}^{\bw\sigma},
+\eeq
+where ${\bm S}\equiv S_{\mu\nu}=\braket{\AO{\mu}}{\AO{\nu}}$ is the
+metric and the ensemble Fock-like matrix reads
+\beq
+F_{\mu\nu}^{\bw\sigma}=h^\bw_{\mu\nu}+\sum_{\lambda\omega}\sum_{\tau=\alpha,\beta}
+G_{\mu\nu\lambda\omega}^{\sigma\tau}\Gamma^{\bw\tau}_{\lambda\omega}
+\eeq 
+with
+\beq
+h^\bw_{\mu\nu}=h_{\mu\nu}+
+%\left\langle\AO{\mu}\middle\vert\dfrac{\delta E^\bw_{\rm
+%c}[n_{\bmg^\bw}]}{\delta n(\br)}\middle\vert\AO{\nu}\right\rangle
+\int d\br\;\AO{\mu}(\br)\dfrac{\delta E^\bw_{\rm
+c}[n_{\bmg^\bw}]}{\delta n(\br)}\AO{\nu}(\br).
+\eeq
+
+%%%%%%%%%%%%%%%
+\iffalse%%%%%%
 % Manu's derivation %%%%
 \color{blue}
 I am teaching myself ...\\
@@ -659,13 +695,11 @@ F_{\mu\nu}^\sigma=h_{\mu\nu}+\sum_{\lambda\omega}\sum_\tau
 G_{\mu\nu\lambda\omega}^{\sigma\tau}\Gamma^{\bw\tau}_{\lambda\omega}
 \eeq
 and
-\beq
-G_{\mu\nu\lambda\omega}^{\sigma\tau}=({\mu}{\nu}\vert{\lambda}{\omega})
--\delta_{\sigma\tau}(\mu\omega\vert\lambda\nu)
-\eeq
 \color{black}
 \\
-%%%%%
+\fi%%%%%%%%%%%
+%%%%% end Manu 
+%%%%%%%%%%%%%%%%%%%%
 Note that this approximation, where the ensemble density matrix is
 optimized from a non-local exchange potential [rather than a local one,
 as expected from Eq.~(\ref{eq:var_ener_gokdft})] is applicable to real
@@ -994,7 +1028,7 @@ For TDLDA, the validity of the Tamm-Dancoff approximation (TDA) has been also te
 Concerning the eKS calculations, two sets of weight have been tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Results and discussion}
+\section{Results and discussion}\label{sec:results}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 In Fig.~\ref{fig:EvsL}, we report the error (in \%) in excitation energies (compared to FCI) for various methods and box sizes in the case of 5-boxium (i.e., $\Nel = 5$).
 Similar graphs are obtained for the other $\Nel$ values and they can be found --- alongside the numerical data associated with each method --- in the {\SI}.