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Manu: now use the notation w_K

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Emmanuel Fromager 2019-09-11 19:37:09 +02:00
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Manuscript/eDFT.tex
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@ -188,17 +188,17 @@ ee}\right)\right]\right\}
\eeq \eeq
where ${\rm where ${\rm
Tr}$ denotes the trace. The minimization over trial ensemble density matrix operators 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: is performed under the following density constraint:
\beq \beq
{\rm {\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 \eeq
where $\hat{n}(\br)$ is the density operator, $n_{\Psi}$ denotes the where $\hat{n}(\br)$ is the density operator, $n_{\Psi}$ denotes the
density of wavefunction $\Psi$, and density of wavefunction $\Psi$, and
$\bw\equiv\left(w^{(1)},w^{(2)},\ldots\right)$ is the collection of $\bw\equiv\left(w^{(1)},w^{(2)},\ldots\right)$ is the collection of
(decreasing) ensemble weights assigned to the excited states. Note that (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, conventional ground-state universal functional is recovered,
\beq \beq
F^{\bw=0}[n]=F[n]=\underset{\Psi\rightarrow n}{\rm min} 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 where the minimization in Eq.~(\ref{eq:ens_LL_func}) has been restricted
to density matrix operators to density matrix operators
\beq \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 \eeq
that are constructed from single Slater that are constructed from single Slater
determinants $\Phi^{(K)}$. Note that the density matrices determinants $\Phi^{(K)}$. Note that the density matrices
${\bmg}^{(K)}={\bmg}^{\Phi^{(K)}}$ are idempotent and diagonal in the ${\bmg}^{(K)}={\bmg}^{\Phi^{(K)}}$ are idempotent and diagonal in the
same spin-orbital basis). On the other hand, the ensemble 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, matrices, is {\it not} idempotent, unless ${\bw}=0$. Indeed,
\beq \beq
\left[{\bmg}^{{\bw}}\right]^2&=&\sum_{K,L\geq \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\\ \nonumber\\
&=&\sum_{K\geq &=&\sum_{K\geq
0}\left(w^{(K)}\right)^2{\bmg}^{(K)}+\sum_{K\neq L\geq 0}\left(w_K\right)^2{\bmg}^{(K)}+\sum_{K\neq L\geq
0}w^{(K)}w^{(L)}{\bmg}^{(K)}{\bmg}^{(L)} 0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
\nonumber\\ \nonumber\\
&=& &=&
{\bmg}^{{\bw}}+\sum_{K,L\geq {\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\\ \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\\ \nonumber\\
&&+\sum_{K, L >0 &&+\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\\ \nonumber\\
&\neq&{\bmg}^{{\bw}} &\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] HF}\left[{\bmg}^{\bw}[n]\right]
\nonumber\\ \nonumber\\
&=& &=&
\sum_{K\geq0}w^{(K)}W_{\rm \sum_{K\geq0}w_KW_{\rm
HF}\left[{\bmg}^{(K)}[n]\right] HF}\left[{\bmg}^{(K)}[n]\right]
-W_{\rm -W_{\rm
HF}\left[{\bmg}^{\bw}[n]\right], 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] ee}\right)\right]
\nonumber\\ \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]} ee}\ket{\Psi^{(K)}[n]}
\nonumber\\ \nonumber\\
&&-\bra{\Phi^{(K)}[n]}\hat{T}+\hat{W}_{\rm &&-\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 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 \beq
E^{{\bw}}=\underset{n}{\rm min}\Big\{ E^{{\bw}}=\underset{n}{\rm min}\Big\{
F^{\bw}[n]+\int d\br\,v_{\rm ext}(\br)n(\br) 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 ensemble energy and its derivatives can be used to extract individual
ground- and excited-state energies as follows:\cite{Deur_2018b} ground- and excited-state energies as follows:\cite{Deur_2018b}
\beq\label{eq:indiv_ener_from_ens} \beq\label{eq:indiv_ener_from_ens}
E^{(I)}&=&E^{{\bw}}+\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)\dfrac{\partial E^{(I)}&=&E^{{\bw}}+\sum_{K>0}\left(\delta_{IK}-w_K\right)\dfrac{\partial
E^{{\bw}}}{\partial w^{(K)}}. E^{{\bw}}}{\partial w_K}.
\eeq \eeq
Since, according to the Hellmann--Feynman theorem, the ensemble energy Since, according to the Hellmann--Feynman theorem, the ensemble energy
derivative reads derivative reads
\beq \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] Tr}\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right){\bm h}\right]
\nonumber\\ \nonumber\\
&&+\Tr\left[\left({\bmg}^{(K)}-{\bmg}^{(0)}\right) \, \bG \, \bmg^{\bw}\right] &&+\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) n({\br})}\left(n^{(K)}(\br)-n^{(0)}(\br)\right)
\nonumber\\ \nonumber\\
&&+\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm &&+\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 \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 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: energy levels within the ensemble:
@ -421,8 +421,8 @@ Hxc}\left[n^{{\bw}}\right]}{\delta
n({\br})}\left(n^{(I)}(\br)-n^{\bw}(\br)\right) n({\br})}\left(n^{(I)}(\br)-n^{\bw}(\br)\right)
\nonumber\\ \nonumber\\
&& &&
+\sum_{K>0}\left(\delta_{IK}-w^{(K)}\right)\left. \dfrac{\partial \overline{E}^{{\bw}}_{\rm +\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}}}. Hxc}\left[n\right]}{\partial w_K}\right|_{n=n^{{\bw}}}.
\eeq \eeq
%+\Tr(\bmg^{(I)} \, \bG \, \bmg^{\bw}) %+\Tr(\bmg^{(I)} \, \bG \, \bmg^{\bw})
%-\dfrac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})+... %-\dfrac{1}{2}\Tr(\bmg^{\bw} \, \bG \, \bmg^{\bw})+...
@ -458,9 +458,9 @@ 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) Hxc}(n)}{\partial n}\right|_{n=n^{\bw}(\br)}n^{\bw}(\br)\left(n^{(I)}(\br)-n^{\bw}(\br)\right)
\nonumber\\ \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 \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 \eeq
\alert{ \alert{
or, equivalently, or, equivalently,
@ -481,9 +481,9 @@ n({\br})}\,n^{(I)}(\br)
Hxc}(n)}{\partial n}\right|_{n=n^{\bw}(\br)}\Big(n^{\bw}(\br)\Big)^2 Hxc}(n)}{\partial n}\right|_{n=n^{\bw}(\br)}\Big(n^{\bw}(\br)\Big)^2
\nonumber\\ \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 \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 \eeq
} }