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@ -179,7 +179,7 @@
% \centering
% \includegraphics[width=\linewidth]{TOC}
%\end{wrapfigure}
Similarly to other electron correlation methods, many-body perturbation theory methods, such as the so-called GW approximation, suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis functions due to the lack of explicit electron-electron terms modeling the infamous electron-electron cusp.
Similarly to other electron correlation methods, many-body perturbation theory methods, such as the so-called GW approximation, suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set due to the lack of explicit electron-electron terms modeling the infamous electron-electron cusp.
Here, we propose a density-based basis set correction based on short-range correlation density functionals which significantly speed up the convergence of energetics towards the complete basis set limit.
\end{abstract}
@ -189,7 +189,10 @@ Here, we propose a density-based basis set correction based on short-range corre
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%
The so-called GW approximation has a long and successful history in the calculation of the electronic structure of solids \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} and is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2013, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b}.
The purpose of many-body perturbation theory (MBPT) is to solve the formidable many-body problem by adding the electron-electron Coulomb interaction perturbatively starting from an independent-particle model.
In MBPT, the ``screening'' of the Coulomb interaction plays a central role, and is responsible for a rich variety of phenomena that would be otherwise absent (such as quasiparticle satellites and lifetimes).
The so-called GW approximation is the workhorse of MBPT and has a long and successful history in the calculation of the electronic structure of solids \cite{Aryasetiawan_1998, Onida_2002, Reining_2017} and is getting increasingly popular in molecular systems \cite{Blase_2011, Faber_2011, Bruneval_2012, Bruneval_2013, Bruneval_2015, Bruneval_2016, Bruneval_2016a, Boulanger_2014, Blase_2016, Li_2017, Hung_2016, Hung_2017, vanSetten_2015, vanSetten_2018, Ou_2016, Ou_2018, Faber_2014} thanks to efficient implementation relying on local basis functions. \cite{Blase_2011, Blase_2018, Bruneval_2016, vanSetten_2013, Kaplan_2015, Kaplan_2016, Krause_2017, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b}
The GW approximation stems from the acclaimed Hedin's equations \cite{Hedin_1965}
\begin{subequations}
\begin{align}
@ -212,31 +215,43 @@ The GW approximation stems from the acclaimed Hedin's equations \cite{Hedin_1965
& \Sigma(12) = i \int G(13) W(14) \Gamma(324) d(34),
\end{align}
\end{subequations}
which connects the Green function $G$, its non-interacting version $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb interaction $W$
and the self-energy $\Sigma$, where $v$ is the bare Coulomb interaction, $\delta(12)$ is Dirac's delta function \cite{NISTbook} and $(1)$ is a composite coordinate gathering spin, space and time variables $(\sigma_1,\br{1},t_1)$
which connects the Green's function $G$, its non-interacting version $G_0$, the irreducible vertex function $\Gamma$, the irreducible polarizability $P$, the dynamically-screened Coulomb interaction $W$
and the self-energy $\Sigma$, where $v$ is the bare Coulomb interaction, $\delta(12)$ is Dirac's delta function \cite{NISTbook} and $(1)$ is a composite coordinate gathering spin, space and time variables $(\sigma_1,\br{1},t_1)$.
Within the GW approximation, one bypasses the calculation of the vertex corrections by setting \cite{Aryasetiawan_1998, Onida_2002, Reining_2017, Blase_2018}
\begin{equation}
\label{eq:GW}
\Gamma(123) = \delta(12) \delta(13).
\end{equation}
Depending on the degree of self-consistency one is willing to do, there exists several types of GW calculations.
The simplest and most popular variant is perturbative GW, or {\GOWO}, \cite{Hybertsen_1985a, Hybertsen_1986} which has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
Depending on the degree of self-consistency one is willing to perform, there exists several types of GW calculations. \cite{Loos_2018}
The simplest and most popular variant of GW is perturbative GW, or {\GOWO}, \cite{Hybertsen_1985a, Hybertsen_1986} which has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
For finite systems such as atoms and molecules, partially or fully self-consistent GW methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017}
Similarly to other electron correlation methods, MBPT methods suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set.
Pioneered by Hyllerras \cite{Hyl-ZP-29} in the 1930's and popularized in the 1990's by Kutzelnigg and coworkers, \cite{NogKut-JCP-94,KutMor-ZPD-96,Kut-TCA-85,KutKlo-JCP-91} (and subsequently others \cite{KonBisVal-CR-12, HatKloKohTew-CR-12, TenNog-WIREs-12, GruHirOhnTen-JCP-17}) this can be tracked down to the lack of explicit electron-electron terms modeling the infamous electron-electron Kato cusp. \cite{Kat-CPAM-57}
The basis-set correction presented here follow a different avenue, and relies on the range-separated density-functional theory (RS-DFT) formalism to capture, thanks to a short-range correlation functional, the missing part of the short-range correlation effects. \cite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19}
As we shall illustrate later on in this manuscript, it significantly speeds up the convergence of energetics towards the complete basis set (CBS) limit.
Explicitly correlated F12 correction schemes have been derived for second-order Green's function methods (GF2) \cite{SzaboBook, Casida_1989, Casida_1991, Stefanucci_2013, Ortiz_2013, Phillips_2014, Phillips_2015, Rusakov_2014, Rusakov_2016, Hirata_2015, Hirata_2017, Loos_2018} by Ten-no and coworkers \cite{Ohnishi_2016, Johnson_2018} and Valeev and coworkers. \cite{Pavosevic_2017, Teke_2019}
However, to the best of our knowledge, a F12-based correction for GW has not been designed yet.
In the present manuscript, we illustrate the performance of the density-based basis set correction on ionization potentials (IPs) obtained within {\GOWO}.
Note that the the present basis set correction can be straightforwardly applied to other properties (e.g., electron affinities and fundamental gap), as well as other flavours of GW or Green's function-based methods, such as GF2 (and its higher-order variants).
Moreover, we are currently investigating the performances of the present approach for linear response theory, in order to speed up the convergence of excitation energies obtained within the random-phase approximation (RPA) \cite{Dreuw_2005} and Bethe-Salpeter equation (BSE) formalism. \cite{Strinati_1988, Leng_2016, Blase_2018}
The paper is organised as follows.
%In Sec.~\ref{sec:paradigm}, we briefly review the GW equations for spherium.
%Section~\ref{sec:green} provides details about our perturbative and self-consistent GW implementations, and give the expression of the self-energy for GF2 and GW+SOSEX.
In Sec.~\ref{sec:theory}, we provide details about the theory behind the present basis set correction and its adaptation to GW methods.
Results are reported and discussed in Sec.~\ref{sec:results}.
Finally, we draw our conclusions in Sec.~\ref{sec:conclusion}.
Atomic units are used throughout.
Unless otherwise stated, atomic units are used throughout.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Many-body Green-function theory with DFT basis-set correction}
\subsection{Many-body Green's function theory with DFT basis-set correction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we start by defining, for a $N$-electron system with nuclei-electron potential $v_\text{ne}(\b{r})$, the approximate ground-state energy for one-electron densities $n$ which are ``representable'' in a finite basis set ${\cal B}$
@ -253,34 +268,34 @@ where $F^{\cal B}[n]$ is the Levy-Lieb density functional with wave functions $\
\begin{equation}
F^{\cal B}[n] = \min_{\Psi^{\cal B}\to n} \bra{\Psi^{\cal B}} \hat{T} + \hat{W}_\text{ee}\ket{\Psi^{\cal B}},
\end{equation}
and $\bar{E}^{\cal B}[n]$ is the complementary basis-correction density functional. In the present work, instead of using wave-function methods for calculating $F^{\cal B}[n]$, we reexpress it with a contrained search over $N$-representable one-electron Green functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$
and $\bar{E}^{\cal B}[n]$ is the complementary basis-correction density functional. In the present work, instead of using wave-function methods for calculating $F^{\cal B}[n]$, we reexpress it with a contrained search over $N$-representable one-electron Green's functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$
\begin{equation}
F^{\cal B}[n] = \min_{G^{\cal B}\to n} \Omega^{\cal B}[G^{\cal B}],
\label{FBn}
\end{equation}
where $\Omega^{\cal B}[G]$ is chosen to be a Klein-like energy functional of the Green function (see, e.g., Refs.~\onlinecite{SteLee-BOOK-13,MarReiCep-BOOK-16,DahLee-JCP-05,DahLeeBar-IJQC-05,DahLeeBar-PRA-06})
where $\Omega^{\cal B}[G]$ is chosen to be a Klein-like energy functional of the Green's function (see, e.g., Refs.~\onlinecite{SteLee-BOOK-13,MarReiCep-BOOK-16,DahLee-JCP-05,DahLeeBar-IJQC-05,DahLeeBar-PRA-06})
\begin{equation}
\Omega^{\cal B}[G] = \Tr \left[\ln ( - G ) \right] - \Tr \left[ (G_\text{s}^{\cal B})^{-1} G -1 \right] + \Phi_\text{Hxc}^{\cal B}[G],
\label{OmegaB}
\end{equation}
where $(G_\text{s}^{\cal B})^{-1}$ is the projection into ${\cal B}$ of the inverse free-particle Green function $(G_\text{s})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 )\delta(\b{r}-\b{r}')$ and we have used the notation $\Tr [A B] = 1/(2\pi i) \int_{-\infty}^{+\infty} \! \d \omega \, e^{i \omega 0^+} \! \iint \! \d \b{r} \d \b{r}' A(\b{r},\b{r}',\omega) B(\b{r}',\b{r},\omega)$. In Eq.~(\ref{OmegaB}), $\Phi_\text{Hxc}^{\cal B}[G]$ is a Hartree-exchange-correlation (Hxc) functional of the Green functional such as its functional derivatives yields the Hxc self-energy in the basis: $\delta \Phi_\text{Hxc}^{\cal B}[G]/\delta G(\b{r},\b{r}',\omega) = \Sigma_\text{Hxc}^{\cal B}[G](\b{r},\b{r}',\omega)$. Inserting Eqs.~(\ref{Fn}) and~(\ref{FBn}) into Eq.~(\ref{E0B}), we finally arrive at
where $(G_\text{s}^{\cal B})^{-1}$ is the projection into ${\cal B}$ of the inverse free-particle Green's function $(G_\text{s})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 )\delta(\b{r}-\b{r}')$ and we have used the notation $\Tr [A B] = 1/(2\pi i) \int_{-\infty}^{+\infty} \! \d \omega \, e^{i \omega 0^+} \! \iint \! \d \b{r} \d \b{r}' A(\b{r},\b{r}',\omega) B(\b{r}',\b{r},\omega)$. In Eq.~(\ref{OmegaB}), $\Phi_\text{Hxc}^{\cal B}[G]$ is a Hartree-exchange-correlation (Hxc) functional of the Green's functional such as its functional derivatives yields the Hxc self-energy in the basis: $\delta \Phi_\text{Hxc}^{\cal B}[G]/\delta G(\b{r},\b{r}',\omega) = \Sigma_\text{Hxc}^{\cal B}[G](\b{r},\b{r}',\omega)$. Inserting Eqs.~(\ref{Fn}) and~(\ref{FBn}) into Eq.~(\ref{E0B}), we finally arrive at
\begin{equation}
E_0^{\cal B} = \min_{G^{\cal B}} \left\{ \Omega^{\cal B}[G^{\cal B}] + \int v_\text{ne}(\b{r}) n_{G^{\cal B}}(\b{r}) \d\b{r} + \bar{E}^{\cal B}[n_{G^{\cal B}}] \right\},
\label{E0BGB}
\end{equation}
where the minimization is over $N$-representable one-electron Green functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$.
where the minimization is over $N$-representable one-electron Green's functions $G^{\cal B}(\b{r},\b{r}',\omega)$ representable in the basis set ${\cal B}$.
The stationary condition from Eq.~(\ref{E0BGB}) gives the following Dyson equation
\begin{equation}
(G^{\cal B})^{-1} = (G_\text{0}^{\cal B})^{-1}- \Sigma_\text{Hxc}^{\cal B}[G^{\cal B}]- \bar{\Sigma}^{\cal B}[n_{G^{\cal B}}],
\label{Dyson}
\end{equation}
where $(G_\text{0}^{\cal B})^{-1}$ is the basis projection of the inverse non-interacting Green function with potential $v_\text{ne}(\b{r})$,
where $(G_\text{0}^{\cal B})^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $v_\text{ne}(\b{r})$,
$(G_\text{0})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 + v_\text{ne}(\b{r}) + \lambda)\delta(\b{r}-\b{r}')$ with the chemical potential $\lambda$, and $\bar{\Sigma}^{\cal B}$ is a frequency-independent local self-energy coming from functional derivative of the complementary basis-correction density functional
\begin{equation}
\bar{\Sigma}^{\cal B}[n](\b{r},\b{r}') = \bar{v}^{\cal B}[n](\b{r}) \delta(\b{r}-\b{r}'),
\end{equation}
with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green function $G^{\cal B}(\b{r},\b{r}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bar{\Sigma}^{\cal B}[n]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^{\cal B}$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{{\cal B}\to \text{CBS}}=0$, and the Green function becomes exact, $G^{{\cal B}\to \text{CBS}}=G$.
with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green's function $G^{\cal B}(\b{r},\b{r}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bar{\Sigma}^{\cal B}[n]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^{\cal B}$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{{\cal B}\to \text{CBS}}=0$, and the Green's function becomes exact, $G^{{\cal B}\to \text{CBS}}=G$.
%From Julien:
%\begin{equation}
@ -446,7 +461,7 @@ Moreover, the infinitesimal $\eta$ has been set to zero.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and Discussion}
\label{sec:res}
\label{sec:results}
%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we study a subset of atoms and molecules from the GW100 test set.
@ -544,7 +559,8 @@ IPs (in eV) of the 20 smallest molecule of the GW100 set computed at the {\GOWO}
\begin{table*}
\caption{
IPs (in eV) of the five canonical nucleobases computed at the {\GOWO}@PBE level of theory for various basis sets.
The deviation with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values are reported in square brackets.
The deviation with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values are reported in square brackets.
The CCSD(T)/def2-TZVPP and experimental results are reported for comparison purposes.
\label{tab:DNA}
}
\begin{ruledtabular}
@ -601,12 +617,13 @@ The deviation with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values are
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supporting Information Available}
\section*{Supporting Information Available}
%%%%%%%%%%%%%%%%%%%%%%%%
Additional graphs reporting the convergence of the ionization potentials of the 20 smallest molecules of the GW100 set.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
@ -617,6 +634,6 @@ This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the
\end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{GW,GW-srDFT,GW-srDFT-control,biblio}
\bibliography{GW-srDFT,GW-srDFT-control,biblio}
\end{document}

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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-10-04 21:21:08 +0200
%% Created for Pierre-Francois Loos at 2019-10-08 22:10:37 +0200
%% Saved with string encoding Unicode (UTF-8)
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Bdsk-Url-1 = {https://doi.org/10.1063/1.2161183%F4%8F%B0%83}}
@misc{Dal-PROG-11,
Author = {C. Angeli and K. L. Bak and V. Bakken and O. Christiansen and R. Cimiraglia and S. Coriani and P. Dahle and E. K. Dalskov and T. Enevoldsen and B. Fernandez and L. Ferrighi and L. Frediani and C. H\"{a}ttig and K. Hald and A. Halkier and H. Heiberg and T. Helgaker and H. Hettema and B. Jansik and H. J. Aa. Jensen and D. Jonsson and P. J\{\o}rgensen and S. Kirpekar and W. Klopper and S. Knecht and R. Kobayashi and J. Kongsted and H. Koch and A. Ligabue and O. B. Lutn{\ae}s and K. V. Mikkelsen and C. B. Nielsen and P. Norman and J. Olsen and A. Osted and M. J. Packer and T. B. Pedersen and Z. Rinkevicius and E. Rudberg and T. A. Ruden and K. Ruud and P. Salek and C. C. M. Samson and A. Sanchez de Meras and T. Saue and S. P. A. Sauer and B. Schimmelpfennig and A. H. Steindal and K. O. Sylvester-Hvid and P. R. Taylor and O. Vahtras and D. J. Wilson and H.{\AA}gren},
Date-Added = {2019-10-08 21:49:49 +0200},
Date-Modified = {2019-10-08 21:49:49 +0200},
Title = {DALTON, a molecular electronic structure program, Release DALTON2011 (2011), see http://daltonprogram.org}}
@article{NeiHatKlo-JCP-06,
Author = {C. Neiss and C. Hattig and W. Klopper},
Date-Added = {2019-10-08 21:49:49 +0200},
Date-Modified = {2019-10-08 21:49:49 +0200},
Doi = {10.1063/1.2335443},
Journal = {J. Chem. Phys.},
Pages = {064111},
Title = {Extensions of r12 corrections to CC2-R12 for excited states},
Volume = {125},
Year = {2006},
Bdsk-Url-1 = {https://doi.org/10.1063/1.2335443%F4%8F%B0%83}}
@article{HauKlo-JCP-12,
Author = {Haunschild, Robin and Klopper, Wim},
Date-Added = {2019-10-08 21:49:49 +0200},
Date-Modified = {2019-10-08 21:49:49 +0200},
Journal = {J. Chem. Phys.},
Number = {16},
Pages = {164102},
Title = {New accurate reference energies for the G2/97 test set},
Volume = {136},
Year = {2012}}
@article{TewKloNeiHat-PCCP-07,
Author = {D. P. Tew and W. Klopper and C. Neiss and C. Hattig},
Date-Added = {2019-10-08 21:49:49 +0200},
Date-Modified = {2019-10-08 21:49:49 +0200},
Doi = {10.1039/b617230j},
Journal = {Phys. Chem. Chem. Phys.},
Pages = {1921},
Title = {Quintuple-{{$\zeta$}} Quality Coupled-Cluster Correlation Energies With Triple-{{$\zeta$}} Basis Sets},
Volume = {9},
Year = {2007},
Bdsk-Url-1 = {https://doi.org/10.1039/b617230j}}
@article{HofSchKloKoh-JCP-19,
Author = {S. Hofener and N. Schieschke and W. Klopper and A. Kohn},
Date-Added = {2019-10-08 21:49:49 +0200},
Date-Modified = {2019-10-08 21:49:49 +0200},
Doi = {10.1063/1.5094434},
Journal = {J. Chem. Phys.},
Pages = {184110},
Title = {The extended explicitly-correlated second- order approximate coupled-cluster singles and doubles ansatz suitable for response theory},
Volume = {150},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5094434}}
@article{KutKlo-JCP-91,
Author = {W. Kutzelnigg and W. Klopper},
Date-Added = {2019-10-08 21:49:49 +0200},
Date-Modified = {2019-10-08 21:49:49 +0200},
Doi = {10.1063/1.459921},
Journal = {J. Chem. Phys.},
Pages = {1985},
Title = {Wave Functions With Terms Linear In The Interelectronic Coordinates To Take Care Of The Correlation Cusp. I. General Theory},
Volume = {94},
Year = {1991},
Bdsk-Url-1 = {https://doi.org/10.1063/1.459921}}
@article{NogKut-JCP-94,
Author = {J. Noga and W. Kutzelnigg},
Date-Added = {2019-10-08 21:49:33 +0200},
Date-Modified = {2019-10-08 21:49:33 +0200},
Doi = {10.1063/1.468266},
Journal = {J. Chem. Phys.},
Pages = {7738},
Title = {Coupled Cluster Theory That Takes Care Of The Correlation Cusp By Inclusion Of Linear Terms In The Interelectronic Coordinates},
Volume = {101},
Year = {1994},
Bdsk-Url-1 = {https://doi.org/10.1063/1.468266}}
@article{KutMor-ZPD-96,
Author = {W. Kutzelnigg and J. D. {Morgan III}},
Date-Added = {2019-10-08 21:49:33 +0200},
Date-Modified = {2019-10-08 21:49:33 +0200},
Doi = {10.1007/BF01426405},
Journal = {Z. Phys. D},
Pages = {197},
Title = {Hund's rules},
Volume = {36},
Year = {1996},
Bdsk-Url-1 = {https://doi.org/10.1007/BF01426405}}
@article{MorKut-JPC-93,
Author = {J. D. {Morgan III} and W. Kutzelnigg},
Date-Added = {2019-10-08 21:49:33 +0200},
Date-Modified = {2019-10-08 21:49:33 +0200},
Doi = {10.1021/j100112a051},
Journal = {J. Phys. Chem.},
Pages = {2425},
Title = {Hund's rules, the alternating rule, and symmetry holes},
Volume = {97},
Year = {1993},
Bdsk-Url-1 = {https://doi.org/10.1021/j100112a051}}
@article{Kut-TCA-85,
Author = {W. Kutzelnigg},
Date-Added = {2019-10-08 21:49:33 +0200},
Date-Modified = {2019-10-08 21:49:33 +0200},
Doi = {10.1007/BF00527669},
Journal = {Theor. Chim. Acta},
Pages = {445},
Title = {R12-Dependent Terms In The Wave Function As Closed Sums Of Partial Wave Amplitudes For Large L},
Volume = {68},
Year = {1985},
Bdsk-Url-1 = {https://doi.org/10.1007/BF00527669}}
@article{KloKut-JCP-91,
Author = {W. Klopper and W. Kutzelnigg},
Date-Added = {2019-10-08 21:49:33 +0200},
Date-Modified = {2019-10-08 21:49:33 +0200},
Journal = {J. Chem. Phys.},
Pages = {2020},
Volume = {94},
Year = {1991}}
@article{KloRohKut-CPL-91,
Author = {W. Klopper and R. Rohse and W. Kutzelnigg},
Date-Added = {2019-10-08 21:49:33 +0200},
Date-Modified = {2019-10-08 21:49:33 +0200},
Journal = {Chem. Phys. Lett.},
Pages = {455},
Volume = {178},
Year = {1991}}
@article{KutMor-JCP-92,
Author = {{W. Kutzelnigg and J. D. Morgan III}},
Date-Added = {2019-10-08 21:49:33 +0200},
Date-Modified = {2019-10-08 21:49:33 +0200},
Journal = {J. Chem. Phys.},
Pages = {4484},
Volume = {96},
Year = {1992}}
@article{TerKloKut-JCP-91,
Author = {V. Termath and W. Klopper and W. Kutzelnigg},
Date-Added = {2019-10-08 21:49:33 +0200},
Date-Modified = {2019-10-08 21:49:33 +0200},
Journal = {J. Chem. Phys.},
Pages = {2002},
Volume = {94},
Year = {1991}}
@book{AbrSte-BOOK-72,
Address = {New York},
Author = {M. Abramowitz and I. A. Stegun},
@ -71,20 +383,6 @@
Volume = {128},
Year = {2008}}
@article{AkiTen-CPL-08,
Author = {Y. Akinaga and S. Ten-no},
Journal = {Chem. Phys. Lett.},
Pages = {348},
Volume = {462},
Year = {2008}}
@article{AkiTen-IJQC-09,
Author = {Y. Akinaga and S. Ten-no},
Journal = {Int. J. Quantum Chem.},
Pages = {1905},
Volume = {109},
Year = {2009}}
@article{AlaDeuKneFro-JCP-17,
Author = {M. M. Alam and K. Deur and S. Knecht and E. Fromager},
Journal = {J. Chem. Phys.},
@ -162,9 +460,6 @@
@misc{BraTouCafUmr-JJJ-XX-note3,
Note = {Bouab\c{c}a {\it et al.}~\cite{BouBraCaf-JCP-10} have introduced a wave function with several Jastrow factors attached to individual molecular orbitals. With such a wave function, the correlation effects can be treated differently in atomic and binding regions. In the case of the FH molecule, no atomic core Jastrow was used and two different valence Jastrow factors (one for the lone pairs paying a role in the bond and the other one for the $\sigma$-bond) were introduced. The resulting binding energy was essentially exact within error bars. Combining the various VB wave functions discussed in this work with this multi-Jastrow approach is presently under investigation.}}
@misc{BraTouCafUmr-JJJ-XX-note,
Note = {In the condensed-matter community, ``stricly localized orbitals'' often refers to orbitals that vanish exactly at some finite distance. In the present work, ``stricly localized orbitals'' is employed in the sense usually used in the quantum chemistry community, i.e. orbitals expanded on Gaussian or Slater basis functions centered on a single atom. Thus, these orbitals vanish exactly only at infinite distance.}}
@article{CocAssLupTou-JCP-17,
Author = {E. Coccia and R. Assaraf and E. Luppi and J. Toulouse},
Doi = {10.1063/1.4991563},
@ -296,16 +591,6 @@
Volume = {382-383},
Year = {2014}}
@article{LooPraSceTouGin-JPCL-19,
Author = {P.-F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner},
Doi = {10.1021/acs.jpclett.9b01176},
Journal = {J. Phys. Chem. Lett.},
Pages = {2931},
Title = {A density-based basis-set correction for wave function theory},
Volume = {10},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.9b01176}}
@incollection{MusCocAssOttUmrTou-AQC-18,
Author = {B. Mussard and E. Coccia and R. Assaraf and M. Otten and C. J. Umrigar and J. Toulouse},
Booktitle = {Novel Electronic Structure Theory: General Innovations and Strongly Correlated Systems},
@ -387,9 +672,6 @@
Year = {2016},
Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4943003}}
@misc{RebTou-JJJ-XX-note1,
Note = {The last two terms of the kernel in Eq. (31) of Ref.~\onlinecite{ZhaSteYan-JCP-13} contain non-antisymmetrized two-electron integrals. However, these terms can also be written with a factor of $1/2$ and antisymmetrized two-electron integrals, leading to our Eq.~(\ref{eq:eff kernel iajb}).}}
@misc{RebTouSav-JJJ-XX-note,
Note = {With the notations used here, the Hubbard model is obtained for $\Delta\varepsilon = 2 t$ and $J_{11}=J_{22}=J_{12}=J_{12}=K_{12}=U/2$ where $t$ is the hopping parameter and $U$ is the on-site Coulomb interaction.}}
@ -2702,10 +2984,6 @@
Author = {T. Helgaker and H. J. Aa. Jensen and P. Jorgensen and J. Olsen and K. Ruud and H. Agren and A. A. Auer and K. L. Bak and V. Bakken and O. Christiansen and S. Coriani and P. Dahle and E. K. Dalskov and T. Enevoldsen and B. Fernandez and C. H\"{a}ttig and K. Hald and A. Halkier and H. Heiberg and H. Hettema and D. Jonsson and S. Kirpekar and R. Kobayashi and H. Koch and K.V. Mikkelsen and P. Norman and M.J. Packer and T. B. Pedersen and T. A. Ruden and A. Sanchez and T. Saue and S. P. A. Sauer and B. Schimmelpfennig and K. O. Sylvester-Hvid and P. R. Taylor and O. Vahtras},
Title = {DALTON, a molecular electronic structure program, Release 1.2 (2001)}}
@misc{Dal-PROG-11,
Author = {C. Angeli and K. L. Bak and V. Bakken and O. Christiansen and R. Cimiraglia and S. Coriani and P. Dahle and E. K. Dalskov and T. Enevoldsen and B. Fernandez and L. Ferrighi and L. Frediani and C. H\"{a}ttig and K. Hald and A. Halkier and H. Heiberg and T. Helgaker and H. Hettema and B. Jansik and H. J. Aa. Jensen and D. Jonsson and P. J\{\o}rgensen and S. Kirpekar and W. Klopper and S. Knecht and R. Kobayashi and J. Kongsted and H. Koch and A. Ligabue and O. B. Lutn{\ae}s and K. V. Mikkelsen and C. B. Nielsen and P. Norman and J. Olsen and A. Osted and M. J. Packer and T. B. Pedersen and Z. Rinkevicius and E. Rudberg and T. A. Ruden and K. Ruud and P. Salek and C. C. M. Samson and A. Sanchez de Meras and T. Saue and S. P. A. Sauer and B. Schimmelpfennig and A. H. Steindal and K. O. Sylvester-Hvid and P. R. Taylor and O. Vahtras and D. J. Wilson and H.{\AA}gren},
Title = {DALTON, a molecular electronic structure program, Release DALTON2011 (2011), see http://daltonprogram.org}}
@misc{Dalshort-PROG-11,
Title = {DALTON, a molecular electronic structure program, Release Dalton2011 (2011), see \url{http://daltonprogram.org}}}
@ -5010,15 +5288,6 @@
Volume = {119},
Year = {2008}}
@article{HauKlo-JCP-12,
Author = {Haunschild, Robin and Klopper, Wim},
Journal = {J. Chem. Phys.},
Number = {16},
Pages = {164102},
Title = {New accurate reference energies for the G2/97 test set},
Volume = {136},
Year = {2012}}
@article{HauKlo-TCA-12,
Author = {Haunschild, Robin and Klopper, Wim},
Journal = {Theor. Chem. Acc.},
@ -6465,20 +6734,6 @@
Volume = {59},
Year = {1999}}
@article{KutKlo-JCP-91,
Author = {W. Kutzelnigg and W. Klopper},
Journal = {J. Chem. Phys.},
Pages = {1985},
Volume = {94},
Year = {1991}}
@article{KutMor-JCP-92,
Author = {{W. Kutzelnigg and J. D. Morgan III}},
Journal = {J. Chem. Phys.},
Pages = {4484},
Volume = {96},
Year = {1992}}
@article{Kva-JCP-12,
Author = {S. Kvaal},
Journal = {J. Chem. Phys.},
@ -11381,13 +11636,6 @@
Year = {1991},
Bdsk-Url-1 = {http://link.aip.org/link/?JCP/94/8054/1}}
@article{YamKocTen-JCP-07,
Author = {D. Yamaki and H. Koch and S. Ten-no},
Journal = {J. Chem. Phys.},
Pages = {144104},
Volume = {127},
Year = {2007}}
@article{YamNakUkaTakYam-IJQC-06,
Author = {S. Yamanaka and K. Nakata and T. Ukai and T. Takada and K. Yamaguchi},
Journal = {Int. J. Quantum Chem.},