Manu: started polishing the exact theory part

Manuscript/eDFT.tex

@@ -94,7 +94,7 @@
 \newcommand{\beq}{\begin{eqnarray}}
 \newcommand{\eeq}{\nonumber\end{eqnarray}}
 \newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector
-\newcommand{\bmg}{\bm{\gamma}} % orbital rotation vector
+\newcommand{\bmg}{\bm{\Gamma}} % orbital rotation vector
 \newcommand{\bfx}{\bf{x}}
 \newcommand{\bfr}{\bf{r}}
 %%%% 
@@ -173,28 +173,82 @@ Atomic units are used throughout.
 \label{sec:geKS}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\alert{Manu, you might want to add general details about the eDFT here.}
+Since Hartree and exchange energy contributions cannot be separated in
-\manu{Yes. Copy paste from the SI. Will polish the all thing.}
+the one-dimensional case, we introduce in the following an alternative
+formulation of KS-eDFT where, in complete analogy with the generalized
+KS scheme, a HF-like Hartree-exchange energy is employed. This
+formulation is in principle exact and applicable to higher dimensions.
+Let us start from the analog for ensembles of Levy's universal
+functional,   
+\beq
+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
+$\hat{\gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
+is performed under the following density constraint:
+\beq
+{\rm
+Tr}\left[\hat{\gamma}^{{\bw}}\hat{n}(\br)\right]=\sum^M_{K=0}w^{(K)}n_{\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
+$w^{(0)}=1-\sum^M_{K>0}w^{(K)}\geq 0$. 
+
+\beq
+F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min}
+\bra{\Psi}\hat{T}+\hat{W}_{\rm
+ee}\ket{\Psi}
+\eeq
+
+\beq
+F[n]&=&
+\underset{\Phi\rightarrow n}{\rm min}
+\bra{\Phi}\hat{T}+\hat{W}_{\rm
+ee}\ket{\Phi}+\overline{E}_{\rm c}[n]
+\nonumber\\
+&=&
+\underset{\Phi\rightarrow n}{\rm min}
+\left\{{\rm
+Tr}\left[\bmg^\Phi{\bm t}\right]+W_{\rm
+HF}\left[{\bmg}^{\Phi}\right]\right\}+
+\overline{E}_{\rm c}[n]
+\eeq
+where ${\bm t}$ is the matrix representation of the one-electron kinetic
+energy operator, $\bmg^\Phi$ is the one-electron reduced density
+matrix (just referred to as density matrix in the following) of $\Phi$,
+and     
+\beq
+W_{\rm
+HF}\left[{\bmg}\right]\equiv\frac{1}{2} \Tr(\bmg \, \bG \, \bmg) 
+\eeq
+is the conventional density-matrix functional HF Hartree-exchange
+energy. By analogy with Eq.~(\ref{}),   
 
 \beq
 F^{\bw}_{\rm HF}[n]&=&
-\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
+\underset{\hat{\Gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
-Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\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
+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
+$\hat{\Gamma}^{{\bw}}=\sum^M_{K=0}w^{(K)}\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum^M_{K=0}w^{(K)}\hat{\Gamma}^{(K)}$ is an ensemble density matrix operator constructed
-from Slater determinants, the ensemble 1RDM elements are $\gamma_{pq}^{\bw}={\rm
+from Slater determinants, the ensemble 1RDM elements are $\Gamma_{pq}^{\bw}={\rm
-Tr}\left[\hat{\gamma}^{{\bw}}\hat{a}^\dagger_p\hat{a}_q\right]$,
+Tr}\left[\hat{\Gamma}^{{\bw}}\hat{a}^\dagger_p\hat{a}_q\right]$,
 and $W_{\rm
 HF}\left[{\bmg}\right]=\frac{1}{2}\sum_{pqrs}\langle \varphi_p\varphi_q\vert\vert
 \varphi_r\varphi_s\rangle
 %\times
-\gamma_{pr}\gamma_{qs}$.\\
+\Gamma_{pr}\Gamma_{qs}$.\\
 
 In-principle-exact decomposition:
 
@@ -231,17 +285,17 @@ 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\{
+E^{{\bw}}=\underset{\hat{\Gamma}^{{\bw}}}{\rm min}\Big\{
 &&{\rm
-Tr}\left[\hat{\gamma}^{{\bw}}\hat{T}\right]+W_{\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]
+Hxc}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
-%+E^{{\bw}}_{\rm c}\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})
++\int d{\br}\;v_{\rm ext}({\bfr})n_{\hat{\Gamma}^{{\bw}}}({\bfr})
 \Big\}
 \eeq
  
@@ -258,7 +312,7 @@ n({\br})}\left.\dfrac{\partial n^{{\bw}}({\bmk},{\br})}{\partial \kappa_{lm}}
 \eeq
 where   
 \beq
-n^{{\bw}}({\bmk},{\br})=\sum_\sigma\sum_{pq}\varphi_p({\bmk},{\bfx})\varphi_q({\bmk},{\bfx})\gamma_{pq}^{\bw}
+n^{{\bw}}({\bmk},{\br})=\sum_\sigma\sum_{pq}\varphi_p({\bmk},{\bfx})\varphi_q({\bmk},{\bfx})\Gamma_{pq}^{\bw}
 \eeq
 thus leading to
 \beq
@@ -266,7 +320,7 @@ thus leading to
 \overline{E}^{{\bw}}_{\rm Hxc}\left[n^{{\bw}}({\bmk})\right]
 }{\partial \kappa_{lm}}
 \right|_{{\bmk}=0}=
-\sum_{pq}\gamma_{pq}^{\bw}
+\sum_{pq}\Gamma_{pq}^{\bw}
 \nonumber\\
 &&\times\left.\dfrac{\partial} 
 {\partial \kappa_{lm}}
@@ -281,7 +335,7 @@ 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]
+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
@@ -290,8 +344,8 @@ n({\br})}\right)\varphi^{{\bw}}_p({\bfx})
 \eeq
 
 
-Since $\partial \gamma_{pq}^{\bw}/\partial
+Since $\partial \Gamma_{pq}^{\bw}/\partial
-w^{(I)}=\gamma_{pq}^{(I)}-\gamma_{pq}^{(0)}$, it comes
+w^{(I)}=\Gamma_{pq}^{(I)}-\Gamma_{pq}^{(0)}$, it comes
 
 \manu{just for me ...
 \beq
@@ -299,25 +353,25 @@ w^{(I)}=\gamma_{pq}^{(I)}-\gamma_{pq}^{(0)}$, it comes
 \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}
+\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)
+ \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}
+\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)
+\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)
+HF}\left[\Gamma^{\bw}\right]\right]_{pp}\left(\nu_p^{(I)}-\nu_p^{(0)}\right)
 \eeq
 }