From 7b0b1b150eb844cd656821e292226c644a2a8540 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 9 Apr 2020 10:25:06 +0200 Subject: [PATCH] start clean up results --- FarDFT.nb | 8078 ++++++++++++++++++++++++++++++++++++++++- Manuscript/Cx_H2.pdf | Bin 15109 -> 15109 bytes Manuscript/Ew_H2.pdf | Bin 52732 -> 52732 bytes Manuscript/FarDFT.bib | 78 +- Manuscript/FarDFT.tex | 49 +- Manuscript/Om_H2.pdf | Bin 53890 -> 53890 bytes 6 files changed, 8081 insertions(+), 124 deletions(-) diff --git a/FarDFT.nb b/FarDFT.nb index 7cabac4..321ed4e 100644 --- a/FarDFT.nb +++ b/FarDFT.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 16166017, 379887] -NotebookOptionsPosition[ 16114493, 379116] -NotebookOutlinePosition[ 16114886, 379132] -CellTagsIndexPosition[ 16114843, 379129] +NotebookDataLength[ 16591353, 387787] +NotebookOptionsPosition[ 16536853, 386969] +NotebookOutlinePosition[ 16537246, 386985] 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zcmew}oB7Xd<_#^URV@sS4b2UVqBOY_6cqG*^HW?BOHvgyj0}tn3``85vYS_)4rT%X D&3X^} diff --git a/Manuscript/FarDFT.bib b/Manuscript/FarDFT.bib index 92d5bab..ccd21bc 100644 --- a/Manuscript/FarDFT.bib +++ b/Manuscript/FarDFT.bib @@ -1,13 +1,76 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-04-08 14:13:22 +0200 +%% Created for Pierre-Francois Loos at 2020-04-09 10:05:15 +0200 %% Saved with string encoding Unicode (UTF-8) +@article{Woon_1994, + Author = {Woon, D. and Dunning, T. H.}, + Date-Added = {2020-04-09 09:59:19 +0200}, + Date-Modified = {2020-04-09 10:00:56 +0200}, + Doi = {10.1063/1.466439}, + Journal = {J. Chem. Phys.}, + Pages = {2975--2988}, + Title = {Gaussian basis sets for use in correlated molecular calculations. IV. Calculation of static electrical response properties}, + Volume = {100}, + Year = {1994}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.466439}} + +@article{Kendall_1992, + Author = {Kendall, R. A. and Dunning, T. H. and Harisson, R. J.}, + Date-Added = {2020-04-09 09:58:17 +0200}, + Date-Modified = {2020-04-09 10:01:10 +0200}, + Doi = {10.1063/1.462569}, + Journal = {J. Chem. Phys.}, + Pages = {6796--6806}, + Title = {Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions}, + Volume = {96}, + Year = {1992}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.462569}} + +@article{Dunning_1989, + Author = {T. H. {Dunning, Jr.}}, + Date-Added = {2020-04-09 09:55:22 +0200}, + Date-Modified = {2020-04-09 09:55:22 +0200}, + Doi = {10.1063/1.456153}, + Journal = {J. Chem. Phys.}, + Pages = {1007}, + Title = {Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen}, + Volume = {90}, + Year = {1989}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.456153}} + +@misc{numgrid, + Author = {R. Bast}, + Date-Added = {2020-04-09 09:23:10 +0200}, + Date-Modified = {2020-04-09 09:23:10 +0200}, + Doi = {10.5281/zenodo.2659208}, + Month = {May}, + Note = {\url{https://github.com/dftlibs/numgrid}}, + Publisher = {Zenodo}, + Title = {numgrid: numerical integration grid for molecules}, + Url = {https://github.com/dftlibs/numgrid}, + Year = {2019}, + Bdsk-Url-1 = {https://github.com/LCPQ/quantum_package}, + Bdsk-Url-2 = {http://dx.doi.org/10.5281/zenodo.200970}} + +@misc{QuAcK, + Author = {P. F. Loos}, + Date-Added = {2020-04-09 09:19:41 +0200}, + Date-Modified = {2020-04-09 09:33:32 +0200}, + Doi = {10.5281/zenodo.3745928}, + Note = {\url{https://github.com/pfloos/QuAcK}}, + Publisher = {Zenodo}, + Title = {{{QuAcK: a software for emerging quantum electronic structure methods}}}, + Url = {https://github.com/pfloos/QuAcK}, + Year = {2019}, + Bdsk-Url-1 = {https://github.com/LCPQ/quantum_package}, + Bdsk-Url-2 = {http://dx.doi.org/10.5281/zenodo.200970}} + @article{Fromager_2020, Archiveprefix = {arXiv}, Author = {Emmanuel Fromager}, @@ -27,7 +90,8 @@ Pages = {L5}, Title = {Autoionizing States of the Hydrogen Molecule.}, Volume = {7}, - Year = {1974}} + Year = {1974}, + Bdsk-Url-1 = {https://doi.org/10.1088/0022-3700/7/1/002}} @article{Mielke_2005, Author = {S. L. Mielke and D. W. Schwenke and K. A. Peterson}, @@ -91,13 +155,6 @@ Pages = {arXiv:2003.05553}, Title = {A weight-dependent local correlation density-functional approximation for ensembles}, Year = {submitted}} - -@article{Fromager_2020, - title={Individual correlations in ensemble density-functional theory: State-driven/density-driven decomposition without additional Kohn-Sham systems}, - author={Fromager, Emmanuel}, - journal={arXiv:2001.08605}, - year={submitted}, - url={https://arxiv.org/abs/2001.08605}} @article{Lindh_2001, Author = {R. Lindh and P.-A. Malmqvist and L. Gagliardi}, @@ -376,7 +433,8 @@ @article{Perdew_1983, Author = {J. P. Perdew and M. Levy}, Date-Added = {2019-09-05 12:04:19 +0200}, - Date-Modified = {2019-09-05 12:13:34 +0200}, + Date-Modified = {2020-04-09 10:05:15 +0200}, + Doi = {10.1103/PhysRevLett.51.1884}, Journal = {Phys. Rev. Lett.}, Pages = {1884}, Title = {Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities}, diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 9e69b65..ca71a03 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -183,12 +183,12 @@ However, Loos and Gill have recently shown that there exists other UEGs which co Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b} In particular, we combine these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT, and automatically incorporates the infamous derivative discontinuity contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983} -%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:resdis}. -%Finally, we draw our conclusions in Sec.~\ref{sec:ccl}. -%Unless otherwise stated, atomic units are used throughout. +The paper is organised as follows. +In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented. +Section \ref{sec:compdet} provides the computational details. +The results of our calculations for two-electron systems are reported and discussed in Secs.~\ref{sec:res_H2} and \ref{sec:res_He}. +Finally, we draw our conclusions in Sec.~\ref{sec:ccl}. +Unless otherwise stated, atomic units are used throughout. %%%%%%%%%%%%%%%%%%%% %%% THEORY %%% @@ -248,7 +248,7 @@ is the density matrix operator, $\lbrace \Det{I}{\bw} \rbrace_{0 \le I \le \nEns \end{split} \end{equation} is the ensemble Hartree-exchange-correlation (Hxc) functional. -Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error (GIE) \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\E{\xc}{\bw}[\n{}{}]$. +Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\E{\xc}{\bw}[\n{}{}]$. From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain \cite{Gross_1988b,Deur_2019} \begin{equation} @@ -312,15 +312,15 @@ where $\e{\ex}{\ew{}}(\n{}{})$ and $\e{\co}{\ew{}}(\n{}{})$ are the weight-depen %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details} \label{sec:compdet} -The self-consistent GOK-DFT calculations have been performed with the \texttt{QuAcK} software, freely available on \texttt{github}, where the present functional has been implemented. +The self-consistent GOK-DFT calculations have been performed in a restricted formalism with the \texttt{QuAcK} software, \cite{QuAcK} which is freely available on \texttt{github}, and where the present weight-dependent functionals have been implemented. 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 a restricted formalism and 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} +For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family of atomic basis sets. \cite{Dunning_1989,Kendall_1992,Woon_1994} +Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} 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 ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered. -Although we should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint. +Although one should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint. Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b} -Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error. +\titou{Moreover, the limits $\ew{} = 0$ and $\ew{} = 1$ are the only two weights for which there is no ghost-interaction error.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Hydrogen molecule} @@ -340,9 +340,9 @@ First, we compute the ensemble energy of the \ce{H2} molecule (at equilibrium bo In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state and the lowest doubly-excited state of configuration $1\sigma_u^2$, which has an autoionising resonance nature. \cite{Bottcher_1974} The ensemble energy $\E{}{\ew{}}$ 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, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy obtained via the derivative of the local energy varies significantly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}). +As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy obtained via the derivative of the ensemble energy varies significantly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}). Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\ew{} = 0$ to $1/2$. -Note that the exact xc correlation ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of $\ew{}$. +Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of $\ew{}$. \begin{figure} \includegraphics[width=\linewidth]{Ew_H2} @@ -365,8 +365,8 @@ Note that the exact xc correlation ensemble functional would yield a perfectly l \subsection{Weight-dependent exchange functional} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -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 and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$. -Doing so, we have found that the present weight-dependent exchange functional (denoted as GIC-S in the following as its main purpose is to correct for the ghost-interaction error), represented in Fig.~\ref{fig:Cx_H2}, +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 $0 \le \ew{} \le 1$. +Doing so, we have found that the present weight-dependent exchange functional (denoted as GIC-S in the following as its main purpose is to correct for the ghost-interaction error) \begin{equation} \e{\ex}{\ew{},\text{GIC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3}, \end{equation} @@ -379,22 +379,23 @@ and \begin{subequations} \begin{align} \alpha & = + 0.575\,178, - \\ + & \beta & = - 0.021\,108, - \\ + & \gamma & = - 0.367\,189, \end{align} \end{subequations} makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}). -As readily seen from Eq.~\eqref{eq:Cxw}, $\Cx{\ew{}}$ reduces to $\Cx{}$ for $\ew{} = 0$. -Note that we are not only using data from $\ew{} = 0$ to $\ew{} = 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is strictly forbidden by the GOK variational principle. \cite{Gross_1988a} -However, it is important to ensure that the weight-dependent functional does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $\ew{} = 1$. -Therefore, by construction, the weight-dependent correction vanishes for these two limiting weight values (see Fig.~\ref{fig:Cx_H2}). +As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cx_H2}, the weight-dependent correction does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $\ew{} = 1$, as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits. +Note that we are not only using data from $0 \le \ew{} \le 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is strictly forbidden by the GOK variational principle. \cite{Gross_1988a} +However, it is important to ensure that the weight-dependent functional does not alter the $\ew{} = 1$, which corresponds to a genuine saddle point of the KS equations, as mentioned above. Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear. + \begin{figure} \includegraphics[width=0.8\linewidth]{Cx_H2} \caption{ - $\Cx{\ew{}}/\Cx{\ew{}=0}$ as a function of $\ew{}$ [see Eq.~\eqref{eq:Cxw}]. + $\Cx{\ew{}}/\Cx{\ew{}=0}$ as a function of $\ew{}$ [see Eq.~\eqref{eq:Cxw}] for the \ce{H2} molecule at equilibrium bond length and the aug-cc-pVTZ basis set. + \titou{T2: Add the same curve for He and stretch H2.} \label{fig:Cx_H2} } \end{figure} @@ -733,8 +734,6 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{ HF & FCI & aug-cc-pV5Z & & & & 8.69 \\ \end{tabular} \end{ruledtabular} -\fnt[1]{Reference \onlinecite{Mielke_2005}.} -\fnt[2]{Reference \onlinecite{Barca_2018a}.} \end{table*} %%% %%% %%% %%% diff --git a/Manuscript/Om_H2.pdf b/Manuscript/Om_H2.pdf index fda28a000016a6a2ec848bc95fc534c96657146f..6a3104f954da810b0a78dfaef9e78f3b38370ddb 100644 GIT binary patch delta 34 jcmZo#%G|V+c>~u4c1r^T6C*>j$$}TeVeHNN7lN4pzJv-( delta 34 jcmZo#%G|V+c>~u4b_+veLvv%($$}TeVeHNN7lN4pzc30{