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valence only complementary functional

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Emmanuel Giner 2019-03-24 13:13:11 +01:00
parent 883ead2ba1
commit 5876f02a34
2 changed files with 46 additions and 12 deletions
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Manuscript/G2-srDFT.tex
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@ -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}