diff --git a/Manuscript/G2-srDFT.pdf b/Manuscript/G2-srDFT.pdf index af204c8..216937a 100644 Binary files a/Manuscript/G2-srDFT.pdf and b/Manuscript/G2-srDFT.pdf differ diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 23eb14c..57b09f7 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -63,7 +63,11 @@ \newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]} \newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]} \newcommand{\efuncbasislda}[0]{\bar{E}_{\text{LDA}}^{\basis,\psibasis}[\den]} +\newcommand{\efuncbasisldaval}[0]{\bar{E}_{\text{LDA, val}}^{\basis,\psibasis}[\den]} +\newcommand{\efuncbasispbe}[0]{\bar{E}_{\text{PBE}}^{\basis,\psibasis}[\den]} +\newcommand{\efuncbasispbeval}[0]{\bar{E}_{\text{PBE, val}}^{\basis,\psibasis}[\den]} \newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\psibasis)\right)} +\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\psibasis)\right)} @@ -120,7 +124,9 @@ \newcommand{\denrfci}[0]{\denr_{\psifci}} \newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\basis({\bf r})} \newcommand{\den}[0]{{n}} +\newcommand{\denval}[0]{{n}^{\text{val}}} \newcommand{\denr}[0]{{n}({\bf r})} +\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}} % wave functions \newcommand{\psifci}[0]{\Psi^{\basis}_{\text{FCI}}} @@ -415,6 +421,34 @@ and \beta(n,\nabla n;\,\mu) = \frac{3 e_c^{PBE}(n,\nabla n)}{2\sqrt{\pi}\left(1 - \sqrt{2}\right)n^{(2)}_{\text{UEG}}(n_{\uparrow} \, n_{\downarrow})}. \end{equation} +Therefore, we propose this approximation for the complementary functional $\efuncbasisfci$: +\begin{equation} + \label{eq:def_lda_tot} + \efuncbasispbe = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf r}),\nabla n({\bf r});\,\mur) +\end{equation} + +\subsection{Valence-only approximation for the complementary functional} +We now introduce a valence-only approximation for the complementary functional, which, as we shall see, performs much better than the usual approximations in the context of atomization energies. +Defining the valence one-body spin density matrix as +\begin{equation} + \begin{aligned} + \onedmval[\psibasis] & = \elemm{\psibasis}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\psibasis} \qquad \text{if }(i,j)\in \basisval \\ + & = 0 \qquad \text{in other cases} + \end{aligned} +\end{equation} +then one can define the valence density as: +\begin{equation} + \denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\psibasis] \phi_i({\bf r}) \phi_j({\bf r}) +\end{equation} +Therefore, we propose the following valence-only approximations for the complementary functional +\begin{equation} + \label{eq:def_lda_tot} + \efuncbasisldaval = \int \, \text{d}{\bf r} \,\, \denval({\bf r}) \,\, \emuldaval\,, +\end{equation} +\begin{equation} + \label{eq:def_lda_tot} + \efuncbasispbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval) +\end{equation} %%%%%%%%%%%%%%%%%%%%%%%% \section{Results} %%%%%%%%%%%%%%%%%%%%%%%% @@ -458,46 +492,46 @@ V5Z & 38.0 & 38.7 & 38.8 & exFCI & 132.0 & 140.3 & 143.6 & 144.3 & \\ \hline & exFCI+LDA & 141.9 & 142.8 & 145.8 & 146.2 & \\ - & exFCI+LDA(FC) & 143.0 & 145.4 & 146.4 & 146.0 & \\ + & exFCI+LDA-val & 143.0 & 145.4 & 146.4 & 146.0 & \\ \hline & exFCI+PBE & 146.1 & 143.9 & 145.9 & 145.12 & \\ - & exFCI+PBE (FC) & 147.4 & 146.1 & 146.4 & 145.9 & \\ + & exFCI+PBE -val & 147.4 & 146.1 & 146.4 & 145.9 & \\ \hline & exFCI+PBE-on-top& 142.7 & 142.7 & 145.3 & 144.9 & \\ - & exFCI+PBE-on-top(FC) & 143.3 & 144.7 & 145.7 & 145.6 & \\ + & exFCI+PBE-on-top-val & 143.3 & 144.7 & 145.7 & 145.6 & \\ \\ \ce{N2} & exFCI & 200.9 & 217.1 & 223.5 & 225.7 & 228.5\fnm[2] \\ \hline & exFCI+LDA & 216.3 & 223.1 & 227.9 & 227.9 & \\ - & exFCI+LDA(FC) & 217.8 & 225.9 & 228.1 & 228.5 & \\ + & exFCI+LDA-val & 217.8 & 225.9 & 228.1 & 228.5 & \\ \hline & exFCI+PBE & 225.3 & 225.6 & 228.2 & 227.9 & \\ - & exFCI+PBE (FC) & 227.6 & 227.8 & 228.4 & 228.5 & \\ + & exFCI+PBE -val & 227.6 & 227.8 & 228.4 & 228.5 & \\ \hline & exFCI+PBE-on-top& 222.3 & 224.6 & 227.7 & 227.7 & \\ - & exFCI+PBE-on-top(FC) & 224.8 & 226.7 & 228.3 & 228.3 & \\ + & exFCI+PBE-on-top-val & 224.8 & 226.7 & 228.3 & 228.3 & \\ \\ \ce{O2} & exFCI & 105.3 & 114.6 & 118.0 &119.1 & 120.2\fnm[2] \\ \hline & exFCI+LDA & 111.8 & 117.2 & 120.0 &119.9 & \\ - & exFCI+LDA(FC) & 112.4 & 118.5 & 120.2 & 120.3 & \\ + & exFCI+LDA-val & 112.4 & 118.5 & 120.2 & 120.3 & \\ \hline & exFCI+PBE & 115.9 & 118.4 & 120.1 &119.9 & \\ - & exFCI+PBE (FC) & 117.2 & 119.4 & 120.4 &120.3 & \\ + & exFCI+PBE -val & 117.2 & 119.4 & 120.4 &120.3 & \\ \hline & exFCI+PBE-on-top& 115.0 & 118.4 & 120.2 & & \\ - & exFCI+PBE-on-top(FC) & 116.1 & 119.4 & 120.5 & & \\ + & exFCI+PBE-on-top-val & 116.1 & 119.4 & 120.5 & & \\ \\ \ce{F2} & exFCI & 27.5 & 35.4 & 37.5 & 38.0 & 38.2\fnm[2] \\ \hline & exFCI+LDA & 30.8 & 37.0 & 38.7 & 38.7 & \\ - & exFCI+LDA(FC) & 31.1 & 37.5 & 38.8 & 38.8 & \\ + & exFCI+LDA-val & 31.1 & 37.5 & 38.8 & 38.8 & \\ \hline & exFCI+PBE & 33.3 & 37.8 & 38.8 & 38.7 & \\ - & exFCI+PBE (FC) & 33.7 & 38.2 & 39.0 & 38.8 & \\ + & exFCI+PBE -val & 33.7 & 38.2 & 39.0 & 38.8 & \\ \hline & exFCI+PBE-on-top& 32.1 & 37.5 & 38.7 & 38.7 & \\ - & exFCI+PBE-on-top(FC) & 32.4 & 37.8 & 38.8 & 38.8 & \\ + & exFCI+PBE-on-top-val & 32.4 & 37.8 & 38.8 & 38.8 & \\ \end{tabular} \end{ruledtabular}