Manu: started polishing the theory section
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Emmanuel Fromager
parent
90a16fc541
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@ -98,6 +98,7 @@
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\newcommand{\bfx}{\bf{x}}
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\newcommand{\bfr}{\bf{r}}
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\DeclareMathOperator*{\argmin}{arg\,min}
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\newcommand{\blue}[1]{{\textcolor{blue}{#1}}}
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%%%%
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\begin{document}
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@ -169,33 +170,9 @@ Atomic units are used throughout.
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\label{sec:eDFT}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Kohn--Sham formulation of GOK-DFT}
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\subsection{GOK-DFT}
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Let us start from the analog for ensembles of Levy's universal
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functional,
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\beq\label{eq:ens_LL_func}
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F^{\bw}[n]&=&
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\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}\left(\hat{T}+\hat{W}_{\rm
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ee}\right)\right]\right\}
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\eeq
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where ${\rm
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Tr}$ denotes the trace. The minimization over trial ensemble density matrix operators
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$\hat{\gamma}^{{\bw}}=\sum_{K\geq0}w_K\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
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is performed under the following density constraint:
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\beq
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{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}\hat{n}(\br)\right]=\sum_{K\geq0}w_Kn_{\Psi^{(K)}}(\br)=n(\br),
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\eeq
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where $\hat{n}(\br)$ is the density operator, $n_{\Psi}$ denotes the
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density of wavefunction $\Psi$, and
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$\bw\equiv\left(w^{(1)},w^{(2)},\ldots\right)$ is the collection of
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(decreasing) ensemble weights assigned to the excited states. Note that
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$w_0=1-\sum_{K>0}w_K\geq 0$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Hybrid GOK-DFT}
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\label{sec:geKS}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Since Hartree and exchange energy contributions cannot be separated in
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the one-dimensional case, we introduce in the following an alternative
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@ -268,28 +245,8 @@ determinants $\Phi^{(K)}$. Note that the density matrices
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${\bmg}^{(K)}={\bmg}^{\Phi^{(K)}}$ are idempotent and diagonal in the
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same spin-orbital basis). On the other hand, the ensemble
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density matrix ${\bmg}^{{\bw}}=\sum_{K\geq 0}w_K{\bmg}^{(K)}$, which is a convex combination of the ${\bmg}^{(K)}$
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matrices, is {\it not} idempotent, unless ${\bw}=0$. Indeed,
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\beq
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\left[{\bmg}^{{\bw}}\right]^2&=&\sum_{K,L\geq
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0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&\sum_{K\geq
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0}\left(w_K\right)^2{\bmg}^{(K)}+\sum_{K\neq L\geq
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0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&
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{\bmg}^{{\bw}}+\sum_{K,L\geq
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0}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&{\bmg}^{{\bw}}+w_0{\bmg}^{(0)}\times\sum_{K>0}w_K\left(2{\bmg}^{(K)}-1\right)
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\nonumber\\
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&&+\sum_{K, L >0
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}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&\neq&{\bmg}^{{\bw}}
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.
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\eeq
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This is of course expected since using an ensemble is, in this context,
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matrices, is {\it not} idempotent, unless ${\bw}=0$.
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Using an ensemble is, in this context,
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analogous to assigning
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fractional occupation numbers (which are determined from the ensemble
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weights) to the KS orbitals.\\
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@ -532,6 +489,120 @@ Hxc}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
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\eeq
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}
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\subsection{Approximations}
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In order to compute (approximate) energy levels within generalized
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GOK-DFT we use a two-step procedure. The first step consists in
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optimizing variationally the ensemble density matrix according to
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Eq.~(\ref{eq:var_princ_Gamma_ens}) with an approximate Hxc ensemble
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functional where (i) the ghost-interaction correction functional $\overline{E}^{{\bw}}_{\rm
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Hx}[n]$ in
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Eq.~(\ref{eq:exact_GIC}) is
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neglected, for simplicity, and (ii) the weight-dependent correlation
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energy is described at the local density level of approximation.
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At this
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level of approximation, the two correlation functionals $\overline{E}^{{\bw}}_{\rm
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c}[n]$ and ${E}^{{\bw}}_{\rm
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c}[n]$ are actually identical and can be expressed as
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\beq\label{eq:eLDA_corr_fun}
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{E}^{{\bw}}_{\rm
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c}[n]=\int d\br\;n(\br)\epsilon_{c}^{\bw}(n(\br)).
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\eeq
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More
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details about the construction of such a functional will be given in the
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following. In order to remove ghost interactions from the variational energy
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expression used in the first step, we then employ the (in-principle-exact)
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expression in Eq.~(\ref{eq:exact_ind_ener_OEP-like}). In this second
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step, the response of the individual density matrices to weight
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variations (last term on the right-hand side of
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Eq.~(\ref{eq:exact_ind_ener_OEP-like})) is neglected. The complete GIC
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procedure can be summarized as follows,
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\beq
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{\bmg}^{\bw}\approx\argmin_{{\bm\gamma}^{\bw}}
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\Big\{
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{\rm
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Tr}\left[{\bm \gamma}^{{\bw}}{\bm h}\right]+W_{\rm
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HF}\left[{\bm\gamma}^{\bw}\right]
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+
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{E}^{{\bw}}_{\rm
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c}\left[n_{\bm\gamma^{\bw}}\right]
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%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
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\Big\},
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\nonumber\\
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\eeq
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and
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\beq
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E^{(I)}&&\approx{\rm
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Tr}\left[{\bmg}^{(I)}{\bm h}\right]
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+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
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\bmg^{(I)})
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\nonumber\\
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&&+{E}^{{\bw}}_{\rm
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c}\left[n_{\bmg^{\bw}}\right]
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+\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
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c}\left[n_{\bmg^{\bw}}\right]}{\delta
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n({\br})}\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
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\nonumber\\
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&&+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left. \dfrac{\partial {E}^{{\bw}}_{\rm
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c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
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,
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\eeq
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thus leading to the final implementable expression [see Eq.~(\ref{eq:eLDA_corr_fun})]
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\beq
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E^{(I)}&&\approx{\rm
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Tr}\left[{\bmg}^{(I)}{\bm h}\right]
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+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
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\bmg^{(I)})
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\nonumber\\
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&&+\int d\br\,
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{\epsilon}^{{\bw}}_{\rm
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c}(n_{\bmg^{\bw}}(\br))\,n_{\bmg^{(I)}}(\br)
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\nonumber\\
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&&
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+\int d\br\,\left.\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
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c}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}n_{\bmg^{\bw}}(\br)\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
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\nonumber\\
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&&
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+\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br)\left.
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\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
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c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
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\eeq
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\blue{$================================$}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory (old)}
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\label{sec:eDFT_old}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Kohn--Sham formulation of GOK-DFT}
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Let us start from the analog for ensembles of Levy's universal
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functional,
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\beq\label{eq:ens_LL_func}
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F^{\bw}[n]&=&
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\underset{\hat{\gamma}^{{\bw}}\rightarrow n}{\rm min}\left\{{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}\left(\hat{T}+\hat{W}_{\rm
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ee}\right)\right]\right\}
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\eeq
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where ${\rm
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Tr}$ denotes the trace. The minimization over trial ensemble density matrix operators
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$\hat{\gamma}^{{\bw}}=\sum_{K\geq0}w_K\vert\Psi^{(K)}\rangle\langle\Psi^{(K)}\vert$
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is performed under the following density constraint:
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\beq
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{\rm
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Tr}\left[\hat{\gamma}^{{\bw}}\hat{n}(\br)\right]=\sum_{K\geq0}w_Kn_{\Psi^{(K)}}(\br)=n(\br),
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\eeq
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where $\hat{n}(\br)$ is the density operator, $n_{\Psi}$ denotes the
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density of wavefunction $\Psi$, and
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$\bw\equiv\left(w^{(1)},w^{(2)},\ldots\right)$ is the collection of
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(decreasing) ensemble weights assigned to the excited states. Note that
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$w_0=1-\sum_{K>0}w_K\geq 0$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Hybrid GOK-DFT}
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\label{sec:geKS}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Exact ensemble exchange in hybrid GOK-DFT}
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@ -635,82 +706,6 @@ Tr}\left[{\bm F}^{(L)}\frac{\partial \bmg^{(L)}}{\partial w_K}\right]
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correlation LDA and ghost interaction correction
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}
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In order to compute (approximate) energy levels within generalized
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GOK-DFT we use a two-step procedure. The first step consists in
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optimizing variationally the ensemble density matrix according to
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Eq.~(\ref{eq:var_princ_Gamma_ens}) with an approximate Hxc ensemble
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functional where (i) the ghost-interaction correction functional $\overline{E}^{{\bw}}_{\rm
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Hx}[n]$ in
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Eq.~(\ref{eq:exact_GIC}) is
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neglected, for simplicity, and (ii) the weight-dependent correlation
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energy is described at the local density level of approximation.
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At this
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level of approximation, the two correlation functionals $\overline{E}^{{\bw}}_{\rm
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c}[n]$ and ${E}^{{\bw}}_{\rm
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c}[n]$ are actually identical and can be expressed as
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\beq\label{eq:eLDA_corr_fun}
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{E}^{{\bw}}_{\rm
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c}[n]=\int d\br\;n(\br)\epsilon_{c}^{\bw}(n(\br)).
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\eeq
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More
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details about the construction of such a functional will be given in the
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following. In order to remove ghost interactions from the variational energy
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expression used in the first step, we then employ the (in-principle-exact)
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expression in Eq.~(\ref{eq:exact_ind_ener_OEP-like}). In this second
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step, the response of the individual density matrices to weight
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variations (last term on the right-hand side of
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Eq.~(\ref{eq:exact_ind_ener_OEP-like})) is neglected. The complete GIC
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procedure can be summarized as follows,
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\beq
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{\bmg}^{\bw}\approx\argmin_{{\bm\gamma}^{\bw}}
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\Big\{
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{\rm
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Tr}\left[{\bm \gamma}^{{\bw}}{\bm h}\right]+W_{\rm
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HF}\left[{\bm\gamma}^{\bw}\right]
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+
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{E}^{{\bw}}_{\rm
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c}\left[n_{\bm\gamma^{\bw}}\right]
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%+E^{{\bw}}_{\rm c}\left[n_{\hat{\Gamma}^{{\bw}}}\right]
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\Big\},
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\nonumber\\
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\eeq
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and
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\beq
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E^{(I)}&&\approx{\rm
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Tr}\left[{\bmg}^{(I)}{\bm h}\right]
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+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
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\bmg^{(I)})
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\nonumber\\
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&&+{E}^{{\bw}}_{\rm
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c}\left[n_{\bmg^{\bw}}\right]
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+\int d\br\,\dfrac{\delta {E}^{{\bw}}_{\rm
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c}\left[n_{\bmg^{\bw}}\right]}{\delta
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n({\br})}\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
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\nonumber\\
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&&+\sum_{K>0}\left(\delta_{IK}-w_K\right)\left. \dfrac{\partial {E}^{{\bw}}_{\rm
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c}\left[n\right]}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}}
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,
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\eeq
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thus leading to the final implementable expression [see Eq.~(\ref{eq:eLDA_corr_fun})]
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\beq
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E^{(I)}&&\approx{\rm
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Tr}\left[{\bmg}^{(I)}{\bm h}\right]
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+\frac{1}{2} \Tr(\bmg^{(I)} \, \bG \,
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\bmg^{(I)})
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\nonumber\\
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&&+\int d\br\,
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{\epsilon}^{{\bw}}_{\rm
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c}(n_{\bmg^{\bw}}(\br))\,n_{\bmg^{(I)}}(\br)
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\nonumber\\
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&&
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+\int d\br\,\left.\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
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c}(n)}{\partial n}\right|_{n=n_{\bmg^{\bw}}(\br)}n_{\bmg^{\bw}}(\br)\left(n_{\bmg^{(I)}}(\br)-n_{\bmg^{\bw}}(\br)\right)
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\nonumber\\
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&&
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+\int d\br\,\sum_{K>0}\left(\delta_{IK}-w_K\right)n_{\bmg^{\bw}}(\br)\left.
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\dfrac{\partial {\epsilon}^{{\bw}}_{\rm
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c}(n)}{\partial w_K}\right|_{n=n_{\bmg^{\bw}}(\br)}.
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\eeq
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\alert{Secs. \ref{sec:KS-eDFT}-\ref{sec:E_I} should maybe be moved to an appendix or merged
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with the theory section (?)}
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@ -1062,6 +1057,34 @@ E.~F.~thanks the \textit{Agence Nationale de la Recherche} (MCFUNEX project, Gra
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\end{acknowledgements}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%% REMOVED FROM THE MAIN TEXT by Manu %%%%%%%%%%%%
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\iffalse%%%%
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Indeed,
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\beq
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\left[{\bmg}^{{\bw}}\right]^2&=&\sum_{K,L\geq
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0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&\sum_{K\geq
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0}\left(w_K\right)^2{\bmg}^{(K)}+\sum_{K\neq L\geq
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0}w_Kw_L{\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&
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{\bmg}^{{\bw}}+\sum_{K,L\geq
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0}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&=&{\bmg}^{{\bw}}+w_0{\bmg}^{(0)}\times\sum_{K>0}w_K\left(2{\bmg}^{(K)}-1\right)
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\nonumber\\
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&&+\sum_{K, L >0
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}w_K\left(w_L-\delta_{KL}\right){\bmg}^{(K)}{\bmg}^{(L)}
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\nonumber\\
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&\neq&{\bmg}^{{\bw}}
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.
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\eeq
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%%%% End -- REMOVED FROM THE MAIN TEXT by Manu %%%%%%%%%%%%
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\fi%%%
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\bibliography{eDFT}
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\end{document}
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