remove GIC print
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FarDFT.nb
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FarDFT.nb
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@ -1,7 +1,7 @@
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2020-04-09 14:58:35 +0200
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%% Created for Pierre-Francois Loos at 2020-04-09 21:27:50 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@ -17,7 +17,8 @@
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Pages = {3163--3183},
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Title = {Highly doubly excited S states of the helium atom},
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Volume = {28},
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Year = {1995}}
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Year = {1995},
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Bdsk-Url-1 = {https://doi.org/10.1088/0953-4075/28/15/010}}
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@article{Woon_1994,
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Author = {Woon, D. and Dunning, T. H.},
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@ -56,18 +57,15 @@
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Bdsk-Url-1 = {https://doi.org/10.1063/1.456153}}
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@misc{numgrid,
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Author = {R. Bast},
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Date-Added = {2020-04-09 09:23:10 +0200},
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Date-Modified = {2020-04-09 09:23:10 +0200},
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Doi = {10.5281/zenodo.2659208},
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Month = {May},
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Note = {\url{https://github.com/dftlibs/numgrid}},
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Author = {Bast, Radovan},
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Doi = {10.5281/zenodo.3746461},
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Month = {4},
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Publisher = {Zenodo},
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Title = {numgrid: numerical integration grid for molecules},
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Url = {https://github.com/dftlibs/numgrid},
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Year = {2019},
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Bdsk-Url-1 = {https://github.com/dftlibs/numgrid},
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Bdsk-Url-2 = {https://doi.org/10.5281/zenodo.2659208}}
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Title = {Numgrid: Numerical integration grid for molecules},
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Url = {https://doi.org/10.5281/zenodo.3746461},
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Version = {v1.1.1},
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Year = {2020},
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Bdsk-Url-1 = {https://doi.org/10.5281/zenodo.3746461}}
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@misc{QuAcK,
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Author = {P. F. Loos},
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@ -187,7 +187,7 @@ In particular, we combine these FUEGs with the usual infinite UEG (IUEG) to cons
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The paper is organised as follows.
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In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented.
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Section \ref{sec:compdet} provides the computational details.
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The results of our calculations for two-electron systems are reported and discussed in Secs.~\ref{sec:res_H2} and \ref{sec:res_He}.
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The results of our calculations for two-electron systems are reported and discussed in Sec.~\ref{sec:res}.
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Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
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Unless otherwise stated, atomic units are used throughout.
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@ -297,9 +297,7 @@ is the Hxc potential, with
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\end{subequations}
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Equation \eqref{eq:dEdw} is our working equation for computing excitation energies from a practical point of view.
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Note that the individual densities $\n{\Det{I}{\bw}}{}(\br{})$ defined in Eq.~\eqref{eq:nI} 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.
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Nevertheless,
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these densities can still be extracted in principle exactly
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from the KS ensemble as shown by Fromager~\cite{Fromager_2020}.
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Nevertheless, these densities can still be extracted in principle exactly from the KS ensemble as shown by Fromager. \cite{Fromager_2020}
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In the following, we adopt the usual decomposition, and write down the weight-dependent xc functional as
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\begin{equation}
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@ -324,12 +322,17 @@ Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a g
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\titou{Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Hydrogen molecule}
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\section{Results}
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Hydrogen molecule at equilibrium}
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\label{sec:H2}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-independent exchange functional}
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\subsubsection{Weight-independent exchange functional}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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First, we compute the ensemble energy of the \ce{H2} molecule (at equilibrium bond length, \ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent Slater-Dirac local exchange functional, \cite{Dirac_1930, Slater_1951} which is explicitly given by
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@ -363,7 +366,7 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent exchange functional}
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\subsubsection{Weight-dependent exchange functional}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Second, in order to remove this spurious curvature of the ensemble energy (which is mostly due to the ghost-interaction error, but not only), one can easily reverse-engineer (for this particular system, geometry, and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
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@ -403,7 +406,7 @@ We shall come back to this point later on.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-independent correlation functional}
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\subsubsection{Weight-independent correlation functional}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Third, we add up correlation effects via the VWN5 local correlation functional. \cite{Vosko_1980}
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@ -423,7 +426,7 @@ The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent correlation functional}
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\subsubsection{Weight-dependent correlation functional}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Fourth, in the spirit of our recent work, \cite{Loos_2020} we have designed a weight-dependent correlation functional.
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@ -434,11 +437,6 @@ As mentioned above, we confine our attention to paramagnetic (or unpolarised) sy
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Note that the present paradigm is equivalent to the conventional IUEG model in the thermodynamic limit. \cite{Loos_2011b}
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We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{Weight-dependent exchange functional}
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%\label{sec:Ex}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states are
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\begin{subequations}
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\begin{align}
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@ -564,10 +562,6 @@ Combining these, we build a two-state weight-dependent correlation functional:
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\end{table}
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{LDA-centered functional}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Because our intent is to incorporate into standard functionals (which are ``universal'' in the sense that they do not depend on the number of electrons) information about excited states that will be extracted from finite systems (whose properties may depend on the number of electrons), we employ a simple embedding scheme where the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
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As explained further in Ref.~\onlinecite{Loos_2020}, this embedding procedure can be theoretically justified by the generalised adiabatic connection formalism for ensembles originally derived by Franck and Fromager. \cite{Franck_2014}
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The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in \eqref{eq:ecw}], while the remaining effects are produced by the bath (\ie, the usual LDA correlation functional).
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@ -631,7 +625,7 @@ The GIC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less
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Finally, note that the GIC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}).
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%%% TABLE I %%%
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\begin{table*}
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\begin{table}
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\caption{
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Excitation energies (in eV) associated with the lowest double excitation of \ce{H2} with $\RHH = 1.4$ bohr for various methods, combinations of xc functionals, and basis sets.
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\label{tab:BigTab_H2}
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@ -640,12 +634,20 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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\begin{tabular}{llccccc}
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\mc{2}{c}{xc functional} & & \mc{2}{c}{GOK} \\
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\cline{1-2} \cline{4-5}
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exchange & correlation & Basis & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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\tabc{x} & \tabc{c} & Basis & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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\hline
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HF & & aug-cc-pVDZ & 38.52 & 30.86 & 34.55 & 28.65 \\
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& & aug-cc-pVTZ & 38.58 & 35.82 & 35.68 & 28.65 \\
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& & aug-cc-pVQZ & 39.12 & 35.94 & 35.64 & 28.65 \\
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\\
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HF & VWN5 & aug-cc-pVDZ & 37.83 & 31.19 & 35.66 & 29.17 \\
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& & aug-cc-pVTZ & 37.35 & 36.98 & 37.23 & 29.17 \\
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& & aug-cc-pVQZ & 37.07 & 37.07 & 37.21 & 29.17 \\
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\\
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HF & eVWN5 & aug-cc-pVDZ & 38.09 & 31.34 & 35.74 & 29.34 \\
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& & aug-cc-pVTZ & 37.61 & 37.04 & 37.28 & 29.34 \\
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& & aug-cc-pVQZ & 37.32 & 37.14 & 37.27 & 29.34 \\
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\\
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S & & aug-cc-pVDZ & 38.40 & 27.35 & 23.54 & 26.60 \\
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& & aug-cc-pVTZ & 39.21 & 27.42 & 23.62 & 26.67 \\
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& & aug-cc-pVQZ & 39.78 & 27.42 & 23.62 & 26.67 \\
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@ -669,20 +671,23 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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GIC-S & eVWN5 & aug-cc-pVDZ & 28.78 & 27.10 & 27.56 & 27.27 \\
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& & aug-cc-pVTZ & 28.90 & 27.16 & 27.64 & 27.34 \\
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& & aug-cc-pVQZ & 28.89 & 27.16 & 27.65 & 27.34 \\
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\\
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\hline
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B & LYP & aug-mcc-pV8Z & & & & 28.42 \\
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B3 & LYP & aug-mcc-pV8Z & & & & 27.77 \\
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HF & LYP & aug-mcc-pV8Z & & & & 29.18 \\
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HF & & aug-mcc-pV8Z & & & & 28.65 \\
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\\
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\hline
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\mc{5}{l}{Accurate (FCI/aug-mcc-pV8Z)\fnm[1]} & 28.75 \\
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\mc{6}{l}{Accurate\fnm[1]} & 28.75 \\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Reference \onlinecite{Barca_2018a}.}
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\end{table*}
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\fnt[1]{FCI/aug-mcc-pV8Z calculation from Ref.~\onlinecite{Barca_2018a}.}
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\end{table}
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Hydrogen molecule at stretched geometry}
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\label{sec:H2st}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% TABLE I %%%
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\begin{table}
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@ -694,26 +699,33 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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\begin{tabular}{llcccc}
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\mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\
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\cline{1-2} \cline{3-4}
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exchange & correlation & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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\tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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\hline
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HF & & 19.09 & 6.59 & 12.92 & 6.52 \\
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HF & VWN5 & 19.40 & 6.54 & 13.02 & \\
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HF & eVWN5 & 19.59 & 6.72 & 13.11 & \\
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S & & 5.31 & 5.60 & 5.46 & 5.56 \\
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S & VWN5 & 5.34 & 5.57 & 5.46 & 5.52 \\
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S & eVWN5 & 5.53 & 5.76 & 5.56 & 5.72 \\
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GIC-S & & 5.55 & 5.56 & 5.56 & 5.56 \\
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GIC-S & VWN5 & 5.58 & 5.53 & 5.57 & 5.52 \\
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GIC-S & eVWN5 & 5.77 & 5.72 & 5.66 & 5.72 \\
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\hline
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B & LYP & & & & 5.28 \\
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B3 & LYP & & & & 5.55 \\
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HF & LYP & & & & 6.68 \\
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\hline
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\mc{5}{l}{Accurate (FCI/aug-cc-pV5Z)\fnm[1]} & 8.69 \\
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\mc{5}{l}{Accurate\fnm[1]} & 8.69 \\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{FCI calculations performed with QUANTUM PACKAGE. \cite{QP2}}
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\fnt[1]{FCI/aug-cc-pV5Z calculation performed with QUANTUM PACKAGE. \cite{QP2}}
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\end{table}
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Helium atom}
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\label{sec:He}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% TABLE I %%%
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\begin{table}
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@ -725,23 +737,26 @@ Excitation energies (in hartree) associated with the lowest double excitation of
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\begin{tabular}{llcccc}
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\mc{2}{c}{xc functional} & \mc{2}{c}{GOK} \\
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\cline{1-2} \cline{3-4}
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exchange & correlation & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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\tabc{x} & \tabc{c} & $\ew{} = 0$ & $\ew{} = 1/2$ & LIM & MOM \\
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\hline
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HF & & 1.874 & 2.212 & 2.080 & 2.142 \\
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HF & VWN5 & 1.988 & 2.260 & 2.153 & 2.193 \\
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HF & eVWN5 & 2.000 & 2.264 & 2.156 & 2.196 \\
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S & & 1.062 & 2.056 & 1.547 & 2.030 \\
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S & VWN5 & 1.163 & 2.104 & 1.612 & 2.079 \\
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S & eVWN5 & 1.174 & 2.108 & 1.615 & 2.083 \\
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GIC-S & & 1.996 & 2.044 & 1.988 & 2.030 \\
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GIC-S & VWN5 & 2.107 & 2.097 & 2.060 & 2.079 \\
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GIC-S & eVWN5 & 2.118 & 2.100 & 2.063 & 2.083 \\
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\hline
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B & LYP & & & & 2.147 \\
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B3 & LYP & & & & 2.150 \\
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HF & LYP & & & & 2.171 \\
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\hline
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\mc{5}{l}{Accurate (explicitly-correlated method)\fnm[1] } & 2.126 \\
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\mc{5}{l}{Accurate\fnm[1]} & 2.126 \\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Reference \onlinecite{Burges_1995}.}
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\fnt[1]{Explicitly-correlated calculation from Ref.~\onlinecite{Burges_1995}.}
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\end{table}
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%%%%%%%%%%%%%%%%%%
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