From ec2df6a6802543387d4bb28a62eec1c9616501c8 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Wed, 11 Sep 2019 19:37:09 +0200 Subject: [PATCH] Manu: now use the notation w_K --- Manuscript/eDFT.tex | 48 ++++++++++++++++++++++----------------------- 1 file changed, 24 insertions(+), 24 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 3d5d6ed..5e695cd 100644 --- a/Manuscript/eDFT.tex +++ b/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: @@ -421,8 +421,8 @@ 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})+... @@ -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) \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 +481,9 @@ 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 }