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Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view. Note that, although we have dropped the weight-dependency in the individual densities $\n{}{(I)}(\br{})$ defined in Eq.~\eqref{eq:nI}, these do not match the \textit{exact} individual-state densities as the non-interacting KS ensemble is expected to reproduce the true interacting ensemble density $\n{}{\bw}(\br{})$ defined in Eq.~\eqref{eq:nw}, and not each individual density. +In the following, we adopt the usual decomposition, and write down the weight-dependent xc functional as +\begin{equation} + \e{\xc}{\ew{}}(\n{}{}) = \e{\ex}{\ew{}}(\n{}{}) + \e{\co}{\ew{}}(\n{}{}), +\end{equation} +where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-dependent exchange and correlation functionals, respectively. + %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% COMPUTATIONAL DETAILS %%% @@ -298,6 +304,46 @@ The self-consistent GOK-DFT calculations have been performed with the \texttt{Qu For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found. For all calculations, we use the aug-cc-pVXZ (X = D, T, and Q) Dunning's family of atomic basis sets. Numerical quadratures are performed with the \texttt{numgrid} library using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988,Lindh_2001} +This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities). +Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$) and the first doubly-excited state ($I=1$) are considered. +Although we will sometimes ``violate'' this variational constraint, we should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle. + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Results} +\label{sec:res} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Weight-independent local exchange-correlation functionals} +\label{sec:S51} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +First, we compute the ensemble energy of the \ce{H2} molecule using the weight-independent Slater's local exchange functional, \cite{Dirac_1930} +which is explicitly given by +\begin{align} + \e{\ex}{\text{S51}}(\n{}{}) & = \Cx{} \n{}{1/3}, + & + \Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}. +\end{align} +The ensemble energy $\E{}{w}$ is depicted in Fig.~\ref{fig:Ew-H2} as a function of the weight $0 \le \ew{} \le 1$. +Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}]. +As anticipated, $\E{}{w}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies greatly with the weight (see Fig.~\ref{fig:Om-H2}). +Note that the exact xc correlation ensemble functional should yield a perfectly linear energy and the same excitation energy independently of $\ew{}$. + + +As a first example, we compute the ensemble energy of the \ce{H2} molecule as a function of the weight $\ew{}$ using the SVWN5 local functional which corresponds to the combination of Slater's local exchange functional \cite{Dirac_1930} and the VNW5 local correlation functional. \cite{Vosko_1980} +The SVWN5 xc functional is explicitly given by +\begin{equation} + \e{\xc}{\text{SVWN5}}(\n{}{}) = \e{\ex}{\text{S51}}(\n{}{}) + \e{\co}{\text{VWN5}}(\n{}{}), +\end{equation} +with +\begin{align} + \e{\ex}{\text{S51}}(\n{}{}) & = \Cx{} \n{}{1/3}, + & + \Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}. +\end{align} +For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}. + %%%%%%%%%%%%%%%%%% @@ -306,17 +352,9 @@ Numerical quadratures are performed with the \texttt{numgrid} library using 194 \section{Functional} \label{sec:func} The present work deals with the explicit construction of the (reduced) LDA xc functional $\e{\xc}{\bw}[\n{}{}]$ defined in Eq.~\eqref{eq:exc_def}. -Here, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$) and the first doubly-excited state ($I=1$) are considered. -Thus, we have $0 \le \ew{} \le 1/2$. The generalisation to a larger number of states (in particular the inclusion of the first singly-excited state) is trivial and left for future work. -We adopt the usual decomposition, and write down the weight-dependent xc functional as -\begin{equation} - \e{\xc}{\ew{}}(\n{}{}) = \e{\ex}{\ew{}}(\n{}{}) + \e{\co}{\ew{}}(\n{}{}), -\end{equation} -where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-dependent exchange and correlation functionals, respectively. The construction of these two functionals is described below. -Here, we restrict our study to spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities). Extension to spin-polarised systems will be reported in future work. To build our weight-dependent xc functional, we propose to consider the singlet ground state and the first singlet doubly-excited state of a two-electron FUEG which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e} @@ -585,26 +623,6 @@ Ensemble energies (in hartree) of \ce{H2} with $\RHH = 1.4$ bohr as a function o \end{figure} %%% %%% %%% %%% -%%%%%%%%%%%%%%% -%%% RESULTS %%% -%%%%%%%%%%%%%%% -\section{Results and discussion} -\label{sec:resdis} -Here, we consider as testing ground the \ce{H2} molecule, and study the behaviour of the total energy of \ce{H2} as a function of the internuclear distance $\RHH$ (in bohr). - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Results} -\label{sec:res} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Discussion} -\label{sec:dis} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -Numerical results are reported in Table \ref{tab:Energies}. - %%%%%%%%%%%%%%%%%% %%% CONCLUSION %%%