Manu: saving work

Manuscript/eDFT.tex

@@ -336,171 +336,132 @@ ee}\right)\right]
 ee}\ket{\Psi^{(K)}[n]}
 \nonumber\\
 &&-\bra{\Phi^{(K)}[n]}\hat{T}+\hat{W}_{\rm
-ee}\ket{\Phi^{(K)}[n]}\Bigg)
+ee}\ket{\Phi^{(K)}[n]}\Bigg),
 \eeq
 where $\hat{\gamma}^{{\bw}}[n]$ and $\hat{\Gamma}^{{\bw}}[n]$ are the minimizing density matrix
-operators in Eqs.~(\ref{eq:ens_LL_func}) and (\ref{eq:generalized_KS-DFT_decomp}), respectively.
-Variational expression of the ensemble energy:
+operators in Eqs.~(\ref{eq:ens_LL_func}) and
+(\ref{eq:generalized_KS-DFT_decomp}), respectively.\\
+
+
+In eDFT, the ensemble energy $E^{{\bw}}=\sum_{K\geq
+0}w^{(K)}E^{(K)}$ is obtained variationally as follows:
 \beq
-E^{{\bw}}=\underset{{\bmg}^{{\bw}}}{\rm min}\Big\{
+E^{{\bw}}=\underset{n}{\rm min}\Big\{
+F^{\bw}[n]+\int d\br\,v_{\rm ext}(\br)n(\br)
+\Big\}.
+\eeq
+Combining the latter expression with the decomposition in
+Eq.~(\ref{eq:generalized_KS-DFT_decomp}) leads to 
+\beq
+E^{{\bw}}=
+\underset{n}{\rm min}\Bigg\{
+\underset{{\bmg}^{{\bw}}\rightarrow n}{\rm min}\Big\{
 {\rm
 Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+W_{\rm
 HF}\left[{\bmg}^{\bw}\right]
 +
 \overline{E}^{{\bw}}_{\rm
-Hxc}\left[n_{{\bmg}^{{\bw}}}\right]
+Hxc}\left[n^{{\bw}}\right]
 %+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
 \Big\}
+\Bigg\}
+\nonumber\\
 \eeq
-
-For $K>0$
-
-\alert{
+or, equivalently,
+\beq\label{eq:var_princ_Gamma_ens}
+E^{{\bw}}=
+\underset{{\bmg}^{{\bw}}}{\rm min}\Big\{
+{\rm
+Tr}\left[{\bmg}^{{\bw}}{\bm h}\right]+W_{\rm
+HF}\left[{\bmg}^{\bw}\right]
++
+\overline{E}^{{\bw}}_{\rm
+Hxc}\left[n^{{\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
+${\bmg}^{\bw}$ 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
+ground- and excited-state energies as follows:\cite{Deur_2018b} 
+\beq\label{eq:indiv_ener_from_ens}
+E^{(I)}&=&E^{{\bw}}+\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)\dfrac{\partial
+E^{{\bw}}}{\partial w^{(K)}}.
+\eeq
+Since, according to the Hellmann--Feynman theorem, the ensemble energy
+derivative reads 
 \beq
 \dfrac{\partial E^{{\bw}}}{\partial w^{(K)}}&=&{\rm
-Tr}\left[{\bmg}^{(K)}{\bm h}\right]-{\rm
-Tr}\left[{\bmg}^{(0)}{\bm h}\right]
+Tr}\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right){\bm h}\right]
 \nonumber\\
-&&+\Tr(\bmg^{(K)} \, \bG \, \bmg^{\bw})
--\Tr(\bmg^{(0)} \, \bG \, \bmg^{\bw})
-+...
+&&+\Tr\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right) \, \bG \, \bmg^{\bw}\right]
+\nonumber\\
+&&+
+\int d\br\,\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n^{{\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}}},
 \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: 
 \beq
-E^{(I)}&=&E^{{\bw}}+\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)\dfrac{\partial E^{{\bw}}}{\partial w^{(K)}}
-\nonumber\\
-&=&
-...+\Tr(\bmg^{(I)} \, \bG \, \bmg^{\bw})
--\dfrac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})+...
-\nonumber\\
-&=&...+
+&&E^{(I)}=
+{\rm
+Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
 \Tr\left[\left(\bmg^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right) \, \bG \, \bmg^{\bw}\right]
-+...
-\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
+\nonumber\\
+&&+\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})}\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})
+n({\br})}\left(n^{(I)}(\br)-n^{\bw}(\br)\right)
 \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.
++\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}}}.
 \eeq
+%+\Tr(\bmg^{(I)} \, \bG \, \bmg^{\bw})
+%-\dfrac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})+...
 
-Thus we conclude that individual energies can be expressed in principle
-exactly as follows,
+At the eLDA level:
 
 \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}}}.
+\overline{E}^{{\bw}}_{\rm
+Hxc}\left[n\right]\rightarrow\int d\br\,n(\br)\overline{\epsilon}^{{\bw}}_{\rm
+Hxc}(n(\br))
+\eeq
+\beq
+\dfrac{\delta \overline{E}^{{\bw}}_{\rm
+Hxc}\left[n\right]}{\delta
+n({\br})}\rightarrow \overline{\epsilon}^{{\bw}}_{\rm
+Hxc}(n(\br))+n(\br)\left.\dfrac{\partial \overline{\epsilon}^{{\bw}}_{\rm
+Hxc}(n)}{\partial n}\right|_{n=n(\br)}
+\eeq
+
+\beq
+&&E^{(I)}\rightarrow
+{\rm
+Tr}\left[{\bmg}^{(I)}{\bm h}\right]+
+\Tr\left[\left(\bmg^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right) \, \bG \, \bmg^{\bw}\right]
+\nonumber\\
+&&
++\int d\br\,
+\overline{\epsilon}^{{\bw}}_{\rm
+Hxc}(n^{\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)
+\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}}}.
 \eeq
 
-%In eDFT, the ensemble energy
-%\begin{equation}
-%	\E{}{\bw} = \qty(1-\sum_{I>0}\ew{I})\E{}{(0)}+\sum_{I>0} \ew{I} \E{}{(I)}
-%\end{equation}
-%is obtained variationally as follows,
-%\begin{equation}
-%	\E{}{\bw} = \qty(1-\sum_{I>0}\ew{I})\E{}{(0)}+\sum_{I>0} \ew{I} \E{}{(I)}
-%\end{equation}
-%
-%In analogy with ground-state generalized KS-DFT, we consider the following partitioning of the ensemble Levy-Lieb functional 
-%\begin{equation}
-%	F^{\bw}[n]=\underset{\hat{\Gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\Gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
-%\end{equation}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{KS-eDFT for excited states}