diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 33ebf6b..a569f08 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -178,7 +178,7 @@ The present eLDA functional is specifically designed to compute double excitatio The paper is organised as follows. In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented. Section \ref{sec:func} provides details about the construction of the weight-dependent xc LDA functional. -The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:res}. +The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:resdis}. Finally, we draw our conclusions in Sec.~\ref{sec:ccl}. Unless otherwise stated, atomic units are used throughout. @@ -418,13 +418,13 @@ Combining these, we build a two-state weight-dependent correlation functional: Parameters of the correlation functionals for each individual state defined in Eq.~\eqref{eq:ec}. The values of $a_1$ are obtained to reproduce the exact high density correlation energy of each individual state, while $a_2$ and $a_3$ are fitted on the numerical values reported in Table \ref{tab:Ref}.} \begin{ruledtabular} - \begin{tabular}{ldd} - & \tabc{Ground state} & \tabc{Doubly-excited state} \\ - & \tabc{$I=0$} & \tabc{$I=1$} \\ + \begin{tabular}{lll} + & \tabc{Ground state} & \tabc{Doubly-excited state} \\ + & \tabc{$I=0$} & \tabc{$I=1$} \\ \hline - $a_1$ & -0.023\,818\,4 & -0.014\,463\,3 \\ - $a_2$ & +0.005\,409\,94 & -0.050\,601\,9 \\ - $a_3$ & +0.083\,076\,6 & +0.033\,141\,7 \\ + $a_1$ & $-0.023\,818\,4$ & $-0.014\,463\,3$ \\ + $a_2$ & $+0.005\,409\,94$ & $-0.050\,601\,9$ \\ + $a_3$ & $+0.083\,076\,6$ & $+0.033\,141\,7$ \\ \end{tabular} \end{ruledtabular} \end{table} @@ -552,8 +552,8 @@ Ensemble energies (in hartree) of \ce{H2} with $\RHH = 1.4$ bohr as a function o %%%%%%%%%%%%%%% %%% RESULTS %%% %%%%%%%%%%%%%%% -\section{Results} -\label{sec:res} +\section{Results and discussion} +\label{sec:resdis} Here, we consider as testing ground the minimal-basis \ce{H2} molecule. We select STO-3G as minimal basis, and study the behaviour of the total energy of \ce{H2} as a function of the internuclear distance $\RHH$ (in bohr). This minimal-basis example is quite pedagogical as the molecular orbitals are fixed by symmetry. @@ -561,6 +561,11 @@ We have then access to the individual densities of the ground and doubly-excited Moreover, thanks to the spatial symmetry and the minimal basis, the individual densities extracted from the ensemble density are equal to the \textit{exact} individual densities. In other words, there is no density-driven error and the only error that we are going to observe is the functional-driven error (and this is what we want to study). +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Results} +\label{sec:res} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + The bonding and antibonding orbitals of the \ce{H2} molecule are given by \begin{subequations} \begin{align} @@ -703,7 +708,6 @@ Alternatively to Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA}, o \end{equation} (This is what one would do in practice, \ie, by performing a KS ensemble calculation.) We will label these energies as $\tE{}{\ew{}}$ to avoid confusion with the expressions reported in Eqs.~\eqref{eq:EwHF}, \eqref{eq:EwLDA} and \eqref{eq:EweLDA}. -\begin{widetext} For HF, we have \begin{equation} \label{eq:bEwHF} @@ -711,10 +715,12 @@ For HF, we have \tE{\HF}{\ew{}} & = \Ts{\ew{}}[\n{}{\ew{}}(\br{})] + \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{} - + \frac{1}{2} \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' \\ - & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} - + (1-\ew{})^2 (2\eJ{11}- \eK{11}) + \ew{}^2 (2\eJ{22}- \eK{22}) + 2 (1-\ew{})\ew{} (2 \eJ{12} - \eK{12}), + & + \frac{1}{2} \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' + \\ + & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} + (1-\ew{})^2 (2\eJ{11}- \eK{11}) + \\ + & + \ew{}^2 (2\eJ{22}- \eK{22}) + 2 (1-\ew{})\ew{} (2 \eJ{12} - \eK{12}), \end{split} \end{equation} which is clearly quadratic with respect to $\ew{}$ due to the ghost interaction error in the Hartree term. @@ -725,14 +731,17 @@ In the case of the LDA, it reads \tE{\LDA}{\ew{}} & = \Ts{\ew{}}[\n{}{\ew{}}(\br{})] + \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{} - + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' + \\ + & + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' + \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{} \\ & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} - + 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12} + \\ + & + 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12} \\ & + (1-\ew{}) \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} - + \ew{} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}, + \\ + & + \ew{} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}, \end{split} \end{equation} which is also clearly quadratic with respect to $\ew{}$ because the (weight-independent) LDA functional cannot compensate the ``quadraticity'' of the Hartree term. @@ -743,25 +752,33 @@ For eLDA, the ensemble energy can be decomposed as \tE{\eLDA}{\ew{}} & = \Ts{\ew{}}[\n{}{\ew{}}(\br{})] + \int \vext(\br{}) \n{}{\ew{}}(\br{}) d\br{} - + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' + \\ + & + \iint \frac{\n{}{\ew{}}(\br{})\n{}{\ew{}}(\br{}')}{\abs{\br{} - \br{}'}} d\br{} d\br{}' + \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{} \\ & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} - + 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12} \\ - & + (1-\ew{})^2 \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} - + \ew{}^2 \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} + & + 2(1-\ew{})^2 \eJ{11} + 2\ew{}^2 \eJ{22} + 4 (1-\ew{})\ew{} \eJ{12} \\ - & + (1-\ew{})\ew{} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} - + \ew{}(1-\ew{}) \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} - \\ - & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} - + (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ] - + \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} ] - \\ - & + 2 (1-\ew{})\ew{} \qty[ 2\eJ{12} - + \frac{1}{2} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} - + \frac{1}{2} \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ], + & + \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{\ew{}}(\br{}) d\br{} +% & + (1-\ew{})^2 \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} +% \\ +% & + \ew{}^2 \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} +% \\ +% & + (1-\ew{})\ew{} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} +% \\ +% & + \ew{}(1-\ew{}) \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} +% \\ +% & = 2 (1-\ew{}) \eHc{1} + 2 \ew{} \eHc{2} +% \\ +% & +% + (1-\ew{})^2 \qty[ 2\eJ{11} + \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ] +% \\ +% & + \ew{}^2 \qty[ 2\eJ{22} + \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} ] +% \\ +% & + 2 (1-\ew{})\ew{} \qty[ 2\eJ{12} +% + \frac{1}{2} \int \be{\xc}{(0)}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{} +% & + \frac{1}{2} \int \be{\xc}{(1)}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{} ], \end{split} \end{equation} which \textit{could} be linear with respect to $\ew{}$ if the weight-dependent xc functional compensates exactly the quadratic terms in the Hartree term. @@ -782,36 +799,55 @@ The two first terms are simply $\Eps{0}{\ew{}} = 2 \eps{1}{\ew{}}$, $\Eps{1}{\ew \begin{subequations} \begin{align} +\begin{split} \eps{1}{\ew{},\LDA} & = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12} - + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) - + \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) } \n{}{(0)}(\br{}) d\br{}, \\ + & + \frac{1}{2} \int \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(0)}(\br{}) d\br{}, + \\ + & + \frac{1}{2} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}, +\end{split} + \\ +\begin{split} \eps{2}{\ew{},\LDA} & = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2 \ew{} \eJ{22} - + \frac{1}{2} \int \qty{ \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) - + \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) } \n{}{(1)}(\br{}) d\br{}, + \\ + & + \frac{1}{2} \int \left. \pdv{\e{\xc}{\LDA}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(1)}(\br{}) d\br{}, + \\ + & + \frac{1}{2} \int \e{\xc}{\LDA}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}, +\end{split} \end{align} \end{subequations} \begin{subequations} \begin{align} +\begin{split} \eps{1}{\ew{},\eLDA} & = \eHc{1} + 2(1-\ew{}) \eJ{11} + 2\ew{} \eJ{12} - + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) - + \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) } \n{}{(0)}(\br{}) d\br{}, \\ + & + \frac{1}{2} \int \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(0)}(\br{}) d\br{}, + \\ + & + \frac{1}{2} \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{(0)}(\br{}) d\br{}, +\end{split} + \\ +\begin{split} \eps{2}{\ew{},\eLDA} & = \eHc{2} + 2(1-\ew{}) \eJ{12} + 2\ew{} \eJ{22} - + \frac{1}{2} \int \qty{ \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) - + \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) } \n{}{(1)}(\br{}) d\br{}. + \\ + & + \frac{1}{2} \int \left. \pdv{\be{\xc}{\ew{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{\ew{}}(\br{})} \n{}{\ew{}}(\br{}) \n{}{(1)}(\br{}) d\br{}. + \\ + & + \frac{1}{2} \int \be{\xc}{\ew{}}(\n{}{\ew{}}(\br{})) \n{}{(1)}(\br{}) d\br{}. +\end{split} \end{align} \end{subequations} -\end{widetext} - The derivative discontinuity is modelled by the last term of the right-hand-side of Eq.~\eqref{eq:dEdw}. Note that this contribution is only non-zero in the case of an explicitly weight-dependent functional [see Eq.~\eqref{eq:dexcdw}]. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Discussion} +\label{sec:dis} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + Numerical results are reported in Table \ref{tab:Energies}.