Merge branch 'master' of https://git.irsamc.ups-tlse.fr/loos/eDFT_FUEG

Manuscript/eDFT.tex

@@ -188,17 +188,17 @@ ee}\right)\right]\right\}
 \eeq
 where ${\rm
 Tr}$ denotes the trace. The minimization over trial ensemble density matrix operators
-$\hat{\gamma}^{{\bw}}=\sum_{K\geq0}w^{(K)}\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
+$\hat{\gamma}^{{\bw}}=\sum_{K\geq0}w_K\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
 is performed under the following density constraint:
 \beq
 {\rm
-Tr}\left[\hat{\gamma}^{{\bw}}\hat{n}(\br)\right]=\sum_{K\geq0}w^{(K)}n_{\Psi^{(K)}}(\br)=n(\br),
+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
-$w^{(0)}=1-\sum_{K>0}w^{(K)}\geq 0$. When $\bw=0$, the
+$w_0=1-\sum_{K>0}w_K\geq 0$. When $\bw=0$, the
 conventional ground-state universal functional is recovered,
 \beq
 F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min}
@@ -256,30 +256,30 @@ Hxc}[n]
 where the minimization in Eq.~(\ref{eq:ens_LL_func}) has been restricted
 to density matrix operators 
 \beq
-\hat{\Gamma}^{{\bw}}=\sum_{K\geq 0}w^{(K)}\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum_{K\geq 0}w^{(K)}\hat{\Gamma}^{(K)}
+\hat{\Gamma}^{{\bw}}=\sum_{K\geq 0}w_K\vert\Phi^{(K)}\rangle\langle\Phi^{(K)}\vert=\sum_{K\geq 0}w_K\hat{\Gamma}^{(K)}
 \eeq 
 that are constructed from single Slater
 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 ${\bmg}^{(K)}$
+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,
 \beq
 \left[{\bmg}^{{\bw}}\right]^2&=&\sum_{K,L\geq
-0}w^{(K)}w^{(L)}{\bmg}^{(K)}{\bmg}^{(L)}
+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^{(K)}w^{(L)}{\bmg}^{(K)}{\bmg}^{(L)}
+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)}
+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)
+&=&{\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)}
+}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
 \nonumber\\
 &\neq&{\bmg}^{{\bw}}
 .
@@ -306,7 +306,7 @@ Tr}\left[\hat{\Gamma}^{{\bw}}[n]\hat{W}_{\rm ee}\right]-W_{\rm
 HF}\left[{\bmg}^{\bw}[n]\right]
 \nonumber\\
 &=&
-\sum_{K\geq0}w^{(K)}W_{\rm
+\sum_{K\geq0}w_KW_{\rm
 HF}\left[{\bmg}^{(K)}[n]\right]
 -W_{\rm
 HF}\left[{\bmg}^{\bw}[n]\right],
@@ -332,7 +332,7 @@ Tr}\left[\hat{\Gamma}^{{\bw}}[n]\left(\hat{T}+\hat{W}_{\rm
 ee}\right)\right]
 \nonumber\\
 &=&
-\sum_{K\geq 0}w^{(K)}\Bigg(\bra{\Psi^{(K)}[n]}\hat{T}+\hat{W}_{\rm
+\sum_{K\geq 0}w_K\Bigg(\bra{\Psi^{(K)}[n]}\hat{T}+\hat{W}_{\rm
 ee}\ket{\Psi^{(K)}[n]}
 \nonumber\\
 &&-\bra{\Phi^{(K)}[n]}\hat{T}+\hat{W}_{\rm
@@ -344,7 +344,7 @@ operators in Eqs.~(\ref{eq:ens_LL_func}) and
 
 
 In eDFT, the ensemble energy $E^{{\bw}}=\sum_{K\geq
-0}w^{(K)}E^{(K)}$ is obtained variationally as follows:
+0}w_KE^{(K)}$ is obtained variationally as follows:
 \beq
 E^{{\bw}}=\underset{n}{\rm min}\Big\{
 F^{\bw}[n]+\int d\br\,v_{\rm ext}(\br)n(\br)
@@ -387,13 +387,13 @@ 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)}}.
+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
+\dfrac{\partial E^{{\bw}}}{\partial w_K}&=&{\rm
 Tr}\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right){\bm h}\right]
 \nonumber\\
 &&+\Tr\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right) \, \bG \, \bmg^{\bw}\right]
@@ -404,7 +404,7 @@ 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}}},
+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: 
@@ -421,12 +421,50 @@ Hxc}\left[n^{{\bw}}\right]}{\delta
 n({\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}}}.
++\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})+...
-
+\alert{
+Note that
+\beq
+\overline{E}^{{\bw}}_{\rm
+Hx}\left[n^{{\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\\
+\eeq
+and
+\beq
+\left.\dfrac{\partial \overline{E}^{{\bw}}_{\rm
+Hx}[n]}{\partial w_K}\right|_{n=n^{\bw}}&=&
+\frac{1}{2} \Tr(\bmg^{(K)} \, \bG \, \bmg^{(K)})-\frac{1}{2}
+\Tr(\bmg^{(0)} \, \bG \, \bmg^{(0)})
+\nonumber\\
+&&-\Tr\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right) \, \bG \, \bmg^{\bw}\right] 
++\ldots
+\eeq
+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\right]}{\partial w_K}\right|_{n=n^{{\bw}}}
+\nonumber\\
+&&=
+\overline{E}^{{\bw}}_{\rm
+Hx}\left[n^{{\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]
++\ldots
+\nonumber\\
+&&=\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
+\bmg^{(I)})-\Tr\left[\left({\bmg}^{(I)}-\dfrac{1}{2}\bmg^{\bw}\right)
+\, \bG \, \bmg^{\bw}\right]
++\ldots
+\eeq
+}
 At the eLDA level:
 
 \beq
@@ -458,9 +496,9 @@ Hxc}(n^{\bw}(\br))\,n^{(I)}(\br)
 Hxc}(n)}{\partial n}\right|_{n=n^{\bw}(\br)}n^{\bw}(\br)\left(n^{(I)}(\br)-n^{\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^{{\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^{{\bw}}(\br)}.
 \eeq
 \alert{
 or, equivalently,
@@ -481,9 +519,9 @@ n({\br})}\,n^{(I)}(\br)
 Hxc}(n)}{\partial n}\right|_{n=n^{\bw}(\br)}\Big(n^{\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^{{\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^{{\bw}}(\br)}.
 \eeq
 }