From b7ce410e5f68018b816aa9e19c5140a3f04c2add Mon Sep 17 00:00:00 2001 From: bsenjean Date: Fri, 3 Apr 2020 15:08:58 +0200 Subject: [PATCH] Corrected my comments. --- Manuscript/FarDFT.tex | 59 ++++++++++++++++++++++++++++++------------- 1 file changed, 41 insertions(+), 18 deletions(-) diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 781e4a2..8177f7b 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -193,36 +193,51 @@ Unless otherwise stated, atomic units are used throughout. %%%%%%%%%%%%%%%%%%%% \section{Theory} \label{sec:theo} -As mentioned above, eDFT for excited states is based on the GOK variational principle \cite{Gross_1988a} which states that a variational principle holds for the ensemble energy\bruno{I would write the variational principle equation here} + +Let us consider a GOK ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\nEns-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$. +The corresponding ensemble energy \begin{equation} \E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)} \end{equation} -built from an ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw = (\ew{0},\ldots,\ew{M-1})$, \ie, $\sum_{I=0}^{\nEns-1} \ew{I} = 1$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$. +fulfils the variational principle +as follows\cite{Gross_1988a} +\begin{eqnarray}\label{eq:ens_energy} +\E{}{\bw} = \min_{\hat{\Gamma}^w} {\rm Tr}\left[ \hat{\Gamma}^w \hat{H} \right], +\end{eqnarray} +where $\hat{H} = \hat{T} + \hat{W}_{\rm ee} + \hat{V}_{\rm ne}$ contains the kinetic, electron-electron and nuclei-electron interaction potential operators, respectively, +Tr denotes the trace and $\hat{\Gamma}^w$ +is a trial density matrix of the form +\begin{eqnarray} +\hat{\Gamma}^w = \sum_{I=0}^{\nEns - 1} +\ew{I} \dyad{\overline{\Psi}^{(I)}}, +\end{eqnarray} +where $\lbrace \overline{\Psi}^{(I)} \rbrace$ is a set of $M$ orthonormal trial wavefunctions. The lower bound of Eq.~(\ref{eq:ens_energy}) is reached when the set of wavefunctions correspond to the exact eigenstates of $\hat{H}$, \ie, $\lbrace \Psi^{(I)} \rbrace$. Multiplet degeneracies can be easily handled by assigning the same weight to the degenerate states. - -One of the key feature of GOK-DFT in the present context is that one can easily extract individual excitation energies from the ensemble energy via differentiation with respect to individual weights: +One of the key feature of the GOK ensemble is that individual excitation energies are extracted from the ensemble energy via differentiation with respect to individual weights: \begin{equation} \pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{}{(I)}. \end{equation} -\bruno{Turning to the DFT formulation... $\rightarrow$ I think you should not mention GOK-DFT prior to this part, as you simply described an ensemble without any DFT contributions.} -In GOK-DFT, one defines a universal (weight-dependent) ensemble functional $\F{}{\bw}[\n{}{}]$ such that +Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles allows to rewrite the exact variational expression for the ensemble energy as\cite{Gross_1988a} \begin{equation} \label{eq:Ew-GOK} \E{}{\bw} = \min_{\n{}{}} \qty{ \F{}{\bw}[\n{}{}] + \int \vext(\br{}) \n{}{}(\br{}) d\br{} }, \end{equation} -where $\vext(\br{})$ is the external potential. -In the KS formulation of GOK-DFT, the universal ensemble functional (the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles) is decomposed as +where $\vext(\br{})$ is the external potential +and $\F{}{\bw}[\n{}{}]$ is the universal ensemble functional +(the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles). +In the KS formulation, this functional is decomposed as \begin{equation} \F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}] - = \Tr[ \hGam{\bw} \hT ] + \Tr[ \hGam{\bw} \hWee ], + = \Tr[ \hat{\gamma}^w \hT ] + \Tr[ \hat{\gamma}^w \hWee ], \end{equation} -where $\hT$ and $\hWee$ are the kinetic and electron-electron interaction potential operators, respectively, $\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional, +where +$\Ts{\bw}[\n{}{}]$ is the noninteracting ensemble kinetic energy functional, \begin{equation} - \hGam{\bw} = \sum_{I=0}^{M-1} \ew{I} \dyad{\Det{I}{\bw}} + \hat{\gamma}^w = \sum_{I=0}^{M-1} \ew{I} \dyad{\Det{I}{\bw}} \end{equation} -is the density matrix operator, $\Det{I}{\bw}$ are single-determinant wave functions (or configuration state functions) built with KS orbitals $\MO{p}{\bw}(\br{})$, and +is the density matrix operator, $\lbrace \Det{I}{\bw} \rbrace_{0 \leq I \leq \nEns - 1}$ are single-determinant wave functions (or configuration state functions) built with KS orbitals $\lbrace \MO{p}{\bw}(\br{}) \rbrace$, and \begin{equation} \label{eq:exc_def} \begin{split} @@ -230,7 +245,7 @@ is the density matrix operator, $\Det{I}{\bw}$ are single-determinant wave funct & = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}] \\ & = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}' - + \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{}. + + \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{} \end{split} \end{equation} is the ensemble Hartree-exchange-correlation (Hxc) functional. @@ -256,7 +271,8 @@ are the ensemble and individual one-electron densities, respectively, \label{eq:KS-energy} \Eps{I}{\bw} = \sum_{p}^{\Norb} \ON{p}{(I)} \eps{p}{\bw} \end{equation} -is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ [$\ON{p}{(I)}$ being its occupancy for the state $I$]. The latters are determined by solving the ensemble KS equation +is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ [$\ON{p}{(I)}$ being its occupancy for the state $I$]. +The latters are determined by solving the ensemble KS equation \begin{equation} \label{eq:eKS} \qty( \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}), @@ -484,11 +500,11 @@ Equation \eqref{eq:becw} can be recast which nicely highlights the centrality of the LDA in the present eDFA. In particular, $\be{\xc}{(0)}(\n{}{}) = \e{\xc}{\LDA}(\n{}{})$. Consequently, in the following, we name this weight-dependent xc functional ``eLDA'' as it is a natural extension of the LDA for ensembles. -Also, we note that, by construction,\bruno{no need to specify $n = n^w$ for now right ?} +Also, we note that, by construction, \begin{equation} \label{eq:dexcdw} - \left. \pdv{\be{\xc}{\ew{}}[\n{}{}]}{\ew{}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} - = \be{\xc}{(1)}(\n{}{\ew{}}(\br)) - \be{\xc}{(0)}(\n{}{\ew{}}(\br)). + \pdv{\be{\xc}{\ew{}}(\n{}{})}{\ew{}} + = \be{\xc}{(1)}(n(\br)) - \be{\xc}{(0)}(n(\br)). \end{equation} This embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles (GACE) @@ -501,7 +517,14 @@ This embedding procedure can be theoretically justified by the generalised adiab (where $\E{\xc}{}[\n{}{}]$ is the usual ground-state xc functional) originally derived by Franck and Fromager. \cite{Franck_2014} Within this in-principle-exact formalism, the (weight-dependent) xc energy of the ensemble is constructed from the (weight-independent) ground-state functional. In the case of a homogeneous system (or equivalently within the LDA), substituting Eq.~\eqref{eq:dexcdw} into \eqref{eq:GACE} yields, in the case of a bi-ensemble, Eq.~\eqref{eq:eLDA}. \bruno{La formule me semble pas juste. Les exposants sont pas bons d'après moi. Pour le bi-ensemble, on devrait avoir -$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ?} +$\pdv{\E{\xc}{(1-\xi,\xi)}[\n{}{}]}{\xi}$ dans l'intégrale, non ? Can it be : +\begin{equation} +\label{eq:GACE2} + \E{\xc}{\bw}[\n{}{}] + = \E{\xc}{}[\n{}{}] + + \sum_{I=0}^{\nEns-1} \int_0^{\ew{I}} \pdv{\E{\xc}{(0,\ldots,0,\xi,\ew{I+1},\ldots,\ew{\nEns-1})}[\n{}{}]}{\xi} d\xi +\end{equation} +?} %%% TABLE I %%% \begin{table*}