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FarDFT.nb
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Manuscript/FarDFT.bib
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@ -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},

49
Manuscript/FarDFT.tex
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@ -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*}
%%% %%% %%% %%%

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