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first complete version of Green-function theory

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Manuscript/GW-srDFT.tex
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@ -194,62 +194,74 @@
\subsection{Many-body Green-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 $v_{ne}(\b{r})$, the approximate ground-state energy for one-electron densities $n$ representable in a finite basis set ${\cal B}$
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 ${\cal B}$
\begin{equation}
E_0^{\cal B} = \min_{n \in {\cal D}^{\cal B}} \left\{ F[n] + \int v_{ne}(\b{r}) n(\b{r}) \d\b{r}\right\},
E_0^{\cal B} = \min_{n \in {\cal D}^{\cal B}} \left\{ F[n] + \int v_\text{ne}(\b{r}) n(\b{r}) \d\b{r}\right\},
\label{E0B}
\end{equation}
where ${\cal D}^{\cal B}$ is the set of $N$-representable densities which can be extracted from a wave function $\Psi^{\cal B}$ expandable in the Hilbert space generated by ${\cal B}$. In this expression, $F[n]=\min_{\Psi\to n} \bra{\Psi} \hat{T} + \hat{W}_\text{ee}\ket{\Psi}$ is the exact Levy-Lieb universal density functional, where $\hat{T}$ and $\hat{W}_\text{ee}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
\begin{equation}
F[n] = F^{\cal B}[n] + \bar{E}^{\cal B}[n],
\label{Fn}
\end{equation}
where $F^{\cal B}[n]$ is the Levy-Lieb density functional with wave functions $\Psi^{\cal B}$ expandable in the Hilbert space generated by ${\cal B}$
\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 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 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[G^{\cal B}],
F^{\cal B}[n] = \min_{G^{\cal B}\to n} \Omega^{\cal B}[G^{\cal B}],
\label{FBn}
\end{equation}
where $\Omega[G]$ is a universal Luttinger-Ward-like functional of the Green function
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})
\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
\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 ${\cal B}$.
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 $v_\text{ne}(\b{r})$,
$(G_\text{0})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 + v_\text{ne}(\b{r}) + \lambda)\delta(\b{r}-\b{r}')$ 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 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, $G^{{\cal B}\to \text{CBS}}=G$.
%From Julien:
%\begin{equation}
%\Omega[G] = - \Tr \left[\ln ( - G^{-1} ) \right] - \Tr \left[ G_\text{s}^{-1} G -1 \right] + \Phi[G]
%\fdv{E[n_G]}{G(r,r',\omega)} = \int \fdv{E[n_G]}{n(r'')}] \fdv{n_G(r'')}{G(r,r',w)} dr''
%\end{equation}
%
%\begin{equation}
%n_G(r'') = i \int G(r'',r'',w) d\omega
%\end{equation}
%
%
%\begin{equation}
%\fdv{n_G(r'')}{G(r,r',w)} = \delta(r -r') \delta (r'-r'')
%\end{equation}
%
%
%\begin{equation}
%\begin{split}
% \fdv{E[n_G]}{G(r,r',w)}
% & = \int \fdv{E[n_G]}{n(r'')} \delta(r -r') \delta (r'-r'') dr''
% \\
% & = \fdv{E[n_G]}{n(r)} \delta(r -r')
% \\
% & = v[n_G](r) \delta(r -r')
%\end{split}
%\end{equation}
%$\Tr [A B] = 1/(2\pi i) \int \! \d \omega \, e^{i \omega 0^+} \! \iint \! \d \b{r} \d \b{r}' A(\b{r},\b{r}',\omega) B(\b{r}',\b{r},\omega)$
\begin{equation}
E_0^{\cal B} = \min_{G^{\cal B}} \left\{ \Omega[G^{\cal B}] + \int v_{ne}(\b{r}) n_{G^{\cal B}}(\b{r}) \d\b{r} + \bar{E}^{\cal B}[n_{G^{\cal B}}] \right\}
\end{equation}
ddd
\begin{equation}
G^{\cal B}(\omega)^{-1} = G_\text{0}^{-1}(\omega) - \Sigma_\text{Hxc}(\omega) - \bar{v}_\text{c}^{\cal B}[n_{G^{\cal B}}]
\end{equation}
From Julien:
\begin{equation}
\fdv{E[n_G]}{G(r,r',\omega)} = \int \fdv{E[n_G]}{n(r'')}] \fdv{n_G(r'')}{G(r,r',w)} dr''
\end{equation}
\begin{equation}
n_G(r'') = i \int G(r'',r'',w) d\omega
\end{equation}
\begin{equation}
\fdv{n_G(r'')}{G(r,r',w)} = \delta(r -r') \delta (r'-r'')
\end{equation}
\begin{equation}
\begin{split}
\fdv{E[n_G]}{G(r,r',w)}
& = \int \fdv{E[n_G]}{n(r'')} \delta(r -r') \delta (r'-r'') dr''
\\
& = \fdv{E[n_G]}{n(r)} \delta(r -r')
\\
& = v[n_G](r) \delta(r -r')
\end{split}
\end{equation}
\subsection{The GW Approximation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

943
Manuscript/biblio.bib
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@ -157,6 +157,917 @@ year = {2017}
year = {2002}
}
%Letter RSH+MP2
@misc{AngGerSavTou-JJJ-XXa,
author = {J. G. \'Angy\'an and I. Gerber and A. Savin and J. Toulouse},
title = {},
journal = {},
year = {},
pages = {},
volume = {{}},
note = {in preparation}
}
%ACDF+DFT
@misc{AngGerSavTou-JJJ-XX,
author = {J. G. \'Angy\'an and I. Gerber and A. Savin and J. Toulouse},
title = {},
journal = {},
year = {},
pages = {},
volume = {{}},
note = {in preparation}
}
%RSH+MP2
@article{AngGerSavTou-PRA-05,
author = {J. G. \'Angy\'an and I. C. Gerber and A. Savin and J. Toulouse},
title = {van der Waals forces in density functional theory: perturbational long-range electron interaction corrections},
journal = {Phys. Rev. A},
volume = {72},
pages = {012510},
year = {2005}
}
%
@article{AngLiuTouJan-JCTC-11,
author = {J. G. \'Angy\'an and R.-F. Liu and J. Toulouse and G. Jansen},
title = {Correlation Energy Expressions from the Adiabatic-Connection Fluctuation-Dissipation Theorem Approach},
journal = {J. Chem. Theory Comput.},
volume = {7},
pages = {3116},
year = {2011}
}
%
@misc{AngLiuTouJan-JJJ-XX,
author = {J. G. \'Angyan and R.-F. Liu and J. Toulouse and G. Jansen},
note = {J. Chem. Theory Comput., in press.}
}
@article{BraTouCafUmr-JCP-11,
author = {B. Bra\"ida and J. Toulouse and M. Caffarel and C. J. Umrigar},
title = {Quantum Monte Carlo with Jastrow-valence-bond wave functions},
journal = {J. Chem. Phys.},
volume = {134},
pages = {084108},
year = {2011}
}
@misc{BraTouCafUmr-JJJ-XX-note2,
note = {In our BOVB wave functions, we use different sets of orbital coefficients, but a single set of basis exponents shared by all orbital sets.}
}
@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},
title = {Ab initio lifetime correction to scattering states for time-dependent electronic-structure calculations with incomplete basis sets},
journal = {J. Chem. Phys.},
volume = {147},
pages = {014106},
year = {2017},
doi = {10.1063/1.4991563}
}
@article{CocMusLabCaiTaiTouLup-IJQC-16,
author = {E. Coccia and B. Mussard and M. Labeye and J. Caillat and R. Ta\"{\i}eb and J. Toulouse and E. Luppi},
title = {Gaussian continuum basis functions for calculating high-harmonic generation spectra},
journal = {Int. J. Quantum Chem.},
volume = {116},
pages = {1120-1131},
year = {2016},
doi = {10.1002/qua.25146}
}
@article{FerGinTou-JCP-19,
author = {Anthony Fert\'e and Emmanuel Giner and Julien Toulouse},
title = {Range-separated multideterminant density-functional theory with a short-range correlation functional of the on-top pair density},
journal = {J. Chem. Phys.},
volume = {150},
pages = {084103},
year = {2019},
doi = {10.1063/1.5082638}
}
@misc{FilTouUmr-JJJ-XX,
author = {Claudia Filippi and Julien Toulouse and C. J. Umrigar},
title = {in preparation}
}
@misc{FraLupTou-JJJ-XX,
author = {O. Franck and E. Luppi and J. Toulouse},
title = {},
journal = {},
year = {},
pages = {},
volume = {},
note = {unpublished}
}
@article{FraMusLupTou-JCP-15,
author = {O. Franck and B. Mussard and E. Luppi and J. Toulouse},
title = {Basis convergence of range-separated density-functional theory},
journal = {J. Chem. Phys.},
volume = {142},
pages = {074107},
year = {2015}
}
%doi ={http://dx.doi.org/10.1063/1.4907920}
@misc{FraMusLupTou-JJJ-XX,
author = {O. Franck and B. Mussard and E. Luppi and J. Toulouse},
note = {Basis convergence of range-separated density-functional theory, arXiv:1412.2613, to appear in J. Chem. Phys. }
}
@misc{FraMusLupTou-JJJ-XX-note1,
note = {Note that the partial-wave expansion of any {\it odd} power of $r_{12}$ contains an infinite number of terms. In contrast, the partial-wave expansion of any {\it even} power of $r_{12}$ terminates at a finite $\ell$.}
}
@misc{FraMusLupTou-JJJ-XX-note2,
note = {For a discussion of angular and radial correlation and the shape of the correlation hole in the He atom with the long-range interaction, see Ref.~\onlinecite{PolColLeiStoWerSav-IJQC-03}.}
}
@misc{FraMusLupTou-JJJ-XX-note3,
note = {For $\mu=1$ the difference between the long-range MP2 correlation energies for $X=5$ and $X=6$ is still as small as about 0.2 mhartree.}
}
@misc{FraMusLupTou-JJJ-XX-note4,
note = {For $r_1=r_2$, $f\lr_\ell >0$ for $\ell=0$ and $f\lr_\ell <0$ for $\ell \geq 1$.}
}
% MCSCF+DFT
@article{FroTouJen-JCP-07,
author = {E. Fromager and J. Toulouse and H. J. Aa. Jensen},
title = {On the universality of the long-/short-range separation in multiconfigurational density-functional theory},
journal = {J. Chem. Phys.},
volume = {126},
pages = {074111},
year = {2007}
}
%
@article{GarAppGasBenFerPaqPraAssReiTouBarRenDavMalVerCafLooGinSce-JCTC-19,
author = "{Y. Garniron, T. Applencourt, K. Gasperich, A. Benali, A. Fert\'e, J. Paquier, B. Pradines, R. Assaraf, P. Reinhardt, J. Toulouse, P. Barbaresco, N. Renon, G. David, J.-P. Malrieu, M. V\'eril, M. Caffarel, P.-F. Loos, E. Giner, and A. Scemama}",
title = {Quantum Package 2.0: An open-source determinant-driven suite of programs},
journal = {J. Chem. Theory Comput.},
volume = {15},
pages = {3591},
year = {2019},
doi = {10.1021/acs.jctc.9b00176}
}
@misc{GarAppGasBenFerPaqPraAssReiTouBarRenDavMalVerCafLooGinSce-JJJ-XX,
author = {Y. Garniron and T. Applencourt and K. Gasperich and A. Benali and A. Fert\'e and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J.-P. Malrieu and M. V\'eril and M. Caffarel and P.-F. Loos and E. Giner and A. Scemama},
note = {Quantum Package 2.0: An open-source determinant-driven suite of programs, {\it submitted to J. Chem. Theory Comput.}, \href{https://arxiv.org/abs/1902.08154}{arXiv:1902.08154}}
}
%MP2+DFT
@misc{GerAngSavTou-JJJ-XX,
author = {I. Gerber and J. G. \'Angy\'an and A. Savin and J. Toulouse},
title = {},
journal = {},
year = {},
pages = {},
volume = {{}},
note = {in preparation}
}
@article{GinAssTou-MP-16,
author = {E. Giner and R. Assaraf and J. Toulouse},
title = {Quantum Monte Carlo with reoptimized perturbatively selected configuration-interaction wave functions},
journal = {Mol. Phys.},
volume = {114},
pages = {910-920},
year = {2016},
doi = {http://dx.doi.org/./..}
}
@article{GinPraFerAssSavTou-JCP-18,
author = {Emmanuel Giner and Barth\'elemy Pradines and Anthony Fert\'e and Roland Assaraf and Andreas Savin and Julien Toulouse},
title = {Curing basis-set convergence of wave-function theory using
density-functional theory: A systematically improvable approach},
journal = {J. Chem. Phys.},
volume = {149},
pages = {194301},
year = {2018}
}
@article{GorHelScuSilTou-MP-16,
author = {Paola Gori-Giorgi and Trygve Helgaker and Gustavo Scuseria and Bernard Silvi and Julien Toulouse},
title = {Foreword for special issue of Molecular Physics in honour of Andreas Savin},
journal = {Molecular Physics},
volume = {114},
pages = {909},
year = {2016},
doi = {http://dx.doi.org/./..}
}
%
@article{GouTouAngDob-JCTC-17,
author = {Tim Gould and Julien Toulouse and J\'anos G. \'Angy\'an and John F. Dobson},
title = {Casimir--Polder Size Consistency: A Constraint Violated by Some Dispersion Theories},
journal = {J. Chem. Theory Comput.},
volume = {13},
pages = {5829},
year = {2017},
doi = {10.1021/acs.jctc.7b00996},
}
@article{GouTou-PRA-14,
author = {T. Gould and J. Toulouse},
title = "{Kohn-Sham potentials in exact density-functional theory at noninteger electron numbers}",
journal = {Phys. Rev. A},
volume = {90},
pages = {050502(R)},
year = {2014},
doi ={http://dx.doi.org/10.1103/PhysRevA.90.050502}
}
%
@misc{HedTouJen-ARX-17,
author = {E. D. Hedeg{\aa}rd and J. Toulouse and H. J. Aa. Jensen},
title = {Multiconfigurational short-range density-functional theory for open-shell systems},
note={arXiv:1711.03882}
}
@article{HedTouJen-JCP-18,
author = {E. D. Hedeg{\aa}rd and J. Toulouse and H. J. Aa. Jensen},
title = {Multiconfigurational short-range density-functional theory for open-shell systems},
journal = {J. Chem. Phys.},
volume = {148},
pages = {214103},
year = {2018},
doi = {10.1063/1.5013306}
}
@misc{KalMusTou-JJJ-19,
author = {C. Kalai and B. Mussard and J. Toulouse},
note = {Range-separated double-hybrid density-functional theory with coupled-cluster and random-phase approximations \href{https://arxiv.org/abs/1905.01014}{arXiv:1905.01014}}
}
@article{KalTou-JCP-18,
author = {C. Kalai and J. Toulouse},
title = {A general range-separated double-hybrid density-functional theory},
journal = {J. Chem. Phys.},
volume = {148},
pages = {164105},
year = {2018},
doi = {10.1063/1.5025561}
}
@misc{KalTou-JJJ-XX-note1,
note = {The geometries are available in the Minnesota Database at http://comp.chem.umn.edu/db/.}
}
@misc{KalTou-JJJ-XX-note2,
note = {The term ``range-separated double hybrid (RSDH)'' has already been used in Ref.~\onlinecite{SanCivUsvTouShaMas-JCP-15} to designate the RSH+MP2 method and in Refs.~\onlinecite{CorStoJenFro-PRA-13,CorFro-IJQC-14} to generically designate several kinds of range-separated MP2/DFT hybrids. In the present work, we use RSDH to designate the specific scheme corresponding to Eq.~(\ref{EmulRSDH}).}
}
@misc{KalTou-JJJ-XX-note3,
note = {We use the AE49 set of Fast {\it et al.}~\cite{FasCorSanTru-JPCA-99} to facilitate comparisons with previous related works~\cite{ShaTouSav-JCP-11,SouShaTou-JCP-14,MusReiAngTou-JCP-15} where this set was used. We do not expect any special difficulties with the six molecules that have been dropped from the G2-1 set.}
}
%
@article{LabZapCocVenTouCaiTaiLup-JCTC-18,
author = {Marie Labeye and Felipe Zapata and Emanuele Coccia and Val\'erie V\'eniard and Julien Toulouse and J\'er\'emie Caillat and Richard Ta\"{\i}eb and Eleonora Luppi},
title = {Optimal Basis Set for Electron Dynamics in Strong Laser Fields: The case of Molecular Ion H$_2$^+},
journal = {J. Chem. Theory Comput.},
volume = {14},
pages = {5846},
year = {2018},
doi = {10.1021/acs.jctc.8b00656}
}
@article{LawBauTouFilUmr-CPL-08,
author = "{J. W. Lawson, C. W. Bauschlicher Jr, J. Toulouse, C. Filippi, C. J. Umrigar}",
title = "{Quantum Monte Carlo study of the cooperative binding of NO2 to fragment models of carbon nanotubes}",
journal = {Chem. Phys. Lett.},
volume = {466},
pages = {170},
year = {2008}
}
@article{LeiTou-AC-14,
author = {T. Leininger and J. Toulouse},
title = "{Relever le d\'efi de la r\'esolution de l'\'equation de Schr\"odinger}",
journal = {L'Actualit\'e Chimique},
volume = {382-383},
pages = {13-21},
year = {2014}
}
%
@article{LooPraSceTouGin-JPCL-19,
author = {P.-F. Loos and B. Pradines and A. Scemama and J. Toulouse and E. Giner},
title = {A density-based basis-set correction for wave function theory},
journal = {J. Phys. Chem. Lett.},
volume = {10},
pages = {2931},
year = {2019},
doi ={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},
title = {Time-dependent linear-response variational Monte Carlo},
booktitle = {Novel Electronic Structure Theory: General Innovations and Strongly Correlated Systems},
series = {Advances in Quantum Chemistry Vol. 76},
editor = {P. E. Hoggan},
publisher = {Academic Press},
pages = {255-270},
year = {2018},
doi = {10.1016/bs.aiq.2017.05.005}
}
@article{MusReiAngTou-JCP-15,
author = {B. Mussard and P. Reinhardt and J. G. \'Angy\'an and J. Toulouse},
title = {Spin-unrestricted random-phase approximation with range separation: Benchmark on atomization energies and reaction barrier heights},
journal = {J. Chem. Phys.},
volume = {142},
pages = {154123},
year = {2015},
note = {Erratum: J. Chem. Phys. {\bf 142}, 219901 (2015)}
}
@misc{MusReiAngTou-JJJ-XX-note,
note = {See the supplementary material for the detailed results of the calculations on the AE49 and DBH24/08 datasets.}
}
@misc{MusTou-JJJ-XX,
author = {B. Mussard and J. Toulouse},
title = {},
journal = {},
year = {},
pages = {},
volume = {},
note = {Fractional-charge and fractional-spin errors in range-separated density-functional theory, Mol. Phys., to appear (2016); preprint arXiv:1607.03621}
}
@article{MusTou-MP-17,
author = {Bastien Mussard and Julien Toulouse},
title = {Fractional-charge and fractional-spin errors in range-separated density-functional theory},
journal = {Mol. Phys.},
volume = {115},
pages = {161-173},
year = {2017}
}
@article{OleTouSch-JCP-19,
author = {Valerio Olevano and Julien Toulouse and Peter Schuck},
title = {A formally exact one-frequency-only Bethe-Salpeter-like equation. Similarities and differences between GW+BSE and self-consistent RPA},
journal = {J. Chem. Phys.},
volume = {150},
pages = {084112},
year = {2019},
doi = {10.1063/1.5080330}
}
@article{PaqTou-JCP-18,
author = {Julien Paquier and Julien Toulouse},
title = {Four-component relativistic range-separated density-functional
theory: Short-range exchange local-density approximation},
journal = {J. Chem. Phys.},
volume = {149},
pages = {174110},
year = {2018},
doi = {10.1063/1.5049773}
}
@misc{PaqTou-JJJ-XX-note,
note = {See Supplementary Information for details on the calculation of the sums over spins in Eqs.~(\ref{epsxC}) and (\ref{epsxB}), details on the calculation of the large-$\ct$ expansions in Eqs.~(\ref{epsCsrasymp}) and~(\ref{epsBsrasymp}), and for a Mathematica notebook containing the complete expressions of the large-$\ct$ expansions in Eqs.~(\ref{epsCsrasymp}) and~(\ref{epsBsrasymp}) and of the Pad\'e approximants in Eqs.~(\ref{PadeC}) and~(\ref{PadeB}).}
}
@article{PauPilTouEll-JCP-10,
author = {Fran\c{c}oise Pauzat and Julien Pilm\'e and Julien Toulouse and Yves Ellinger},
title = {About the collapse of the 3.3 µm CH stretching band with ionization in polycyclic aromatic hydrocarbons: Configuration interaction and quantum Monte Carlo studies of the CH fragment},
journal = {J. Chem. Phys.},
volume = {133},
pages = {054301},
year = {2010}
}
@article{PetTouUmr-JCP-11,
author = {F. R. Petruzielo and J. Toulouse and C. J. Umrigar},
title = {Basis set construction for molecular electronic structure theory: Natural orbital and Gauss–Slater basis for smooth pseudopotentials},
journal = {J. Chem. Phys.},
volume = {134},
pages = {064104},
year = {2011}
}
@article{PetTouUmr-JCP-12,
author = {F. R. Petruzielo and J. Toulouse and C. J. Umrigar},
title = "{Approaching chemical accuracy with quantum Monte Carlo}",
journal = {J. Chem. Phys.},
volume = {136},
pages = {124116},
year = {2012}
}
% range-separated TDDFT
@misc{RebSavTou-JJJ-XX,
author = {E. Rebolini and A. Savin and J. Toulouse},
title = {},
journal = {},
year = {},
pages = {},
volume = {},
note = {unpublished}
}
%
@article{RebSavTou-MP-13,
author = {E. Rebolini and A. Savin and J. Toulouse},
title = {Electronic excitations from a linear-response range-separated hybrid scheme},
journal = {Mol. Phys.},
volume = {111},
pages = {1219},
year = {2013}
}
@article{RebTeaHelSavTou-MP-18,
author = {E. Rebolini and A. M. Teale and T. Helgaker and A. Savin and J. Toulouse},
title = {Excitation energies from G\"orling--Levy perturbation theory along the
range-separated adiabatic connection},
journal = {Mol. Phys.},
volume = {116},
pages = {1443-1451},
year = {2018},
doi ={doi.org/10.1080/00268976.2017.1422811}
}
@article{RebTou-JCP-16,
author = {E. Rebolini and J. Toulouse},
title = {Range-separated time-dependent density-functional theory with a frequency-dependent second-order Bethe-Salpeter correlation kernel},
journal = {J. Chem. Phys.},
volume = {144},
pages = {094107},
year = {2016},
doi ={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}).}
}
%
@incollection{RebTouSav-INC-13,
author = {E. Rebolini and J. Toulouse and A. Savin},
title = {Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H$_2$ molecule},
booktitle = {Electronic Structure and Reactivity},
series = {Concepts and Methods in Modern Theoretical Chemistry Vol. 1},
editor = {S. K. Ghosh and P. K. Chattaraj},
publisher = {CRC Press},
pages = {367-390},
year = {2013},
note = {preprint at http://arxiv.org/abs/1304.1314}
}
@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.}
}
@article{RebTouTeaHelSav-JCP-14,
author = {E. Rebolini and J. Toulouse and A. M. Teale and T. Helgaker and A. Savin},
title = {Excitation energies along a range-separated adiabatic connection},
journal = {Journal of Chemical Physics},
volume = {141},
pages = {044123},
year = {2014},
doi ={http://dx.doi.org/10.1063/1.4890652}
}
@misc{RebTouTeaHelSav-JJJ-XX-sup,
note = {See supplementary material at http://xxxxxxxx for the fits of the total and excitation energies.}
}
@article{RebTouTeaHelSav-MP-15,
author = {E. Rebolini and J. Toulouse and A. M. Teale and T. Helgaker and A. Savin},
title = {Excited states from range-separated density-functional perturbation theory},
journal = {Mol. Phys.},
volume = {113},
pages = {1740},
year = {2015},
doi ={http://dx.doi.org/10.1080/00268976.2015.1011248}
}
@article{RebTouTeaHelSav-PRA-15,
author = {E. Rebolini and J. Toulouse and A. M. Teale and T. Helgaker and A. Savin},
title = {Calculating excitation energies by extrapolation along adiabatic connections},
journal = {Phys. Rev. A},
volume = {91},
pages = {032519},
year = {2015},
doi ={http://dx.doi.org/10.1103/PhysRevA.91.032519}
}
% Be2, Be3
@misc{ReiTouAngSav-JJJ-XX,
author = {P. Reinhardt and J. Toulouse and J. G. \'Angy\'an and A. Savin},
title = {},
journal = {},
year = {},
pages = {},
volume = {},
note = {unpublished}
}
%
@incollection{ReiTouAssUmrHog-INC-12,
author = {P. Reinhardt and J. Toulouse and R. Assaraf and C. J. Umrigar and P. E. Hoggan},
title = {Quantum Monte Carlo Facing the Hartree-Fock Symmetry Dilemma: The Case of Hydrogen Rings},
booktitle = {Advances in Quantum Monte Carlo},
series = {ACS Symposium Series Vol. 1094},
editor = "{S. Tanaka, S. M. Rothstein, and W. A. Lester, Jr.}",
publisher = {American Chemical Society},
address = {Washington, DC},
pages = {53-63},
year = {2012}
}
@article{ReiTouSav-TCA-18,
author = {Peter Reinhardt and Julien Toulouse and Andreas Savin},
title = {Range-separated density-functional theory applied to the beryllium dimer and trimer},
journal = {Theor. Chem. Acc.},
volume = {137},
pages = {168},
year = {2018},
doi = {10.1007/s00214-018-2370-5}
}
@article{RohTouPer-PRA-10,
author = {D. R. Rohr and J. Toulouse and K. Pernal},
title = {Combining density-functional theory and density-matrix-functional theory},
journal = {Phys. Rev. A},
volume = {82},
pages = {052502},
year = {2010}
}
@article{SanCivUsvTouShaMas-JCP-15,
author = {G. Sansone and B. Civalleri and D. Usvyat and J. Toulouse and K. Sharkas and L. Maschio},
title = {Range-separated double-hybrid density-functional theory applied to periodic systems},
journal = {J. Chem. Phys.},
volume= {143},
pages= {102811},
year = {2015},
doi = {http://dx.doi.org/10.1063/1.4922996}
}
@article{ShaSavJenTou-JCP-12,
author = {K. Sharkas and A. Savin and H. J. Aa. Jensen and J. Toulouse},
title = {A multiconfigurational hybrid density-functional theory},
journal = {J. Chem. Phys.},
volume = {137},
pages = {044104},
year = {2012}
}
@article{ShaTouMasCiv-JCP-14,
author = {K. Sharkas and J. Toulouse and L. Maschio and B. Civalleri},
title = {Double-hybrid density-functional theory applied to molecular crystals},
journal = {J. Chem. Phys.},
volume = {141},
pages = {044105},
year = {2014}
}
%doi ={http://dx.doi.org/10.1063/1.4890439}
@article{ShaTouSav-JCP-11,
author = {K. Sharkas and J. Toulouse and A. Savin},
title = {Double-hybrid density-functional theory made rigorous},
journal = {J. Chem. Phys.},
volume = {134},
pages = {064113},
year = {2011}
}
@article{SmiFraMusBukGraLupTou-JCP-16,
author = {S. \'Smiga and O. Franck and B. Mussard and A. Buksztel and I. Grabowski and E. Luppi and J. Toulouse},
title = {Self-consistent double-hybrid density-functional theory using the optimized-effective-potential method},
journal = {J. Chem. Phys.},
volume = {145},
pages = {144102},
year = {2016},
doi = {http://dx.doi.org/10.1063/1.4964319}
}
@article{SouShaTou-JCP-14,
author = {Sidi M. O. Souvi and Kamal Sharkas and Julien Toulouse},
title = {Double-hybrid density-functional theory with meta-generalized-gradient approximations},
journal = {J. Chem. Phys.},
volume = {140},
pages = {084107},
year = {2014}
}
@misc{SouShaTou-JJJ-XX-sup,
note = {See supplementary material.}
}
@article{StoTeaTouHelFro-JCP-13,
author = {Alexandrina Stoyanova and Andrew M. Teale and Julien Toulouse and Trygve Helgaker and Emmanuel Fromager},
title = {Alternative separation of exchange and correlation energies in multi-configuration range-separated density-functional theory},
journal = {J. Chem. Phys.},
volume = {139},
pages = {134113},
year = {2013}
}
@article{TayAngGalZhaGygHirSonRahLilPodBulHenScuTouPevTruSza-JCP-16,
author = "Taylor, D. E. and \'Angy\'an, J. G. and Galli, G. and Zhang, C. and Gygi, F. and Hirao, K. and Song, J. W. and Rahul, K. and von Lilienfeld, O. A. and Podeszwa, R. and Bulik, I. W. and Henderson, T. M. and Scuseria, G. E. and Toulouse, J. and Peverati, R. and Truhlar, D. G. and Szalewicz, K.",
title = "Blind test of density-functional-based methods on intermolecular interaction energies",
journal = "J. Chem. Phys.",
year = "2016",
volume = "145",
pages = "124105",
url = "http://scitation.aip.org/content/aip/journal/jcp/145/12/10.1063/1.4961095",
doi = "http://dx.doi.org/10.1063/1.4961095"
}
%
@incollection{TouAssUmr-INC-16,
author = {J. Toulouse and R. Assaraf and C. J. Umrigar},
title = {Introduction to the Variational and Diffusion Monte Carlo Methods},
booktitle = {Electron Correlation in Molecules - ab initio Beyond Gaussian Quantum Chemistry},
series = {Advances in Quantum Chemistry Vol. 73},
editor = {P. E. Hoggan and T. Ozdogan},
publisher = {Academic Press},
pages = {285-314},
year = {2016},
doi = {10.1016/bs.aiq.2015.07.003}
}
% QMC calculations of intracule densities
@article{TouAssUmr-JCP-07,
author = {J. Toulouse and R. Assaraf and C. J. Umrigar},
title = "{Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density}",
journal = {J. Chem. Phys.},
volume = {126},
pages = {244112},
year = {2007}
}
%
@incollection{TouCafReiHogUmr-INC-12,
author = {J. Toulouse and M. Caffarel and P. Reinhardt and P. E. Hoggan and C. J. Umrigar},
title = {Quantum Monte Carlo calculations of electronic excitation energies: The case of the singlet $n\to\pi^*$ (CO) transition in acrolein},
booktitle = {Advances in the Theory of Quantum Systems in Chemistry and Physics},
series = {Progress in Theoretical Chemistry and Physics Vol. 22},
editor = {P. E. Hoggan and J. Maruani and P. Piecuch and G. Delgado-Barrio and E. J. Br\"andas},
publisher = {Springer},
address = {Dordrecht Heidelberg London New York},
pages = {345-353},
year = {2012}
}
%Beyond LDA for short-range functionals
@article{TouColSav-JCP-05,
author = {J. Toulouse and F. Colonna and A. Savin},
title = {Short-range exchange and correlation energy density functionals: Beyond the local-density approximation},
journal = {J. Chem. Phys.},
volume = {122},
pages = {014110},
year = {2005},
note = {}
}
%exact data, epsilons
@misc{TouColSav-JJJ-XXa,
author = {J. Toulouse and F. Colonna and A. Savin},
title = {in preparation}
}
% potentials and epsilons
@article{TouColSav-MP-05,
author = {Julien Toulouse and Francois Colonna and Andreas Savin},
title = {Exchange-correlation potentials and local energies per particle along non-linear adiabatic connections},
journal = {Mol. Phys.},
volume = {103},
pages = {2725},
year = {2005}
}
% erfgau, expansions
@article{TouColSav-PRA-04,
author = {J. Toulouse and F. Colonna and A. Savin},
title = {Long-range--short-range separation of the electron-electron interaction in density-functional theory},
publisher = {APS},
year = {2004},
journal = {Phys. Rev. A},
volume = {70},
number = {6},
pages = {062505},
keywords = {density functional theory; wave functions; exchange interactions (electron); electron correlations},
}
@misc{TouGerJanSavAng-JJJ-XX-note,
note = {For Ne$_{2}$ at equilibrium distance with aug-cc-pVQZ basis, the counterpoise correction on the interaction energy is 16 $\mu$H for RSH+RPAx and 62 $\mu$H for standard MP2.}
}
% ACFDT+DFT (RSH+RPAx)
@article{TouGerJanSavAng-PRL-09,
author = {J. Toulouse and I. C. Gerber and G. Jansen and A. Savin and J. G. \'Angy\'an},
title = {Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation},
journal = {Phys. Rev. Lett.},
volume = {102},
pages = {096404},
year = {2009}
}
% scaling relation, virial theorem, energy densities
@article{TouGorSav-IJQC-06,
author = {J. Toulouse and P. Gori-Giorgi and A. Savin},
title = {Scaling relations, virial theorem and energy densities for long-range and short-range density functionals},
journal = {Int. J. Quantum Chem.},
volume = {106},
pages = {2026},
year = {2006},
note = {}
}
@article{TouGorSav-TCA-05,
author = {J. Toulouse and P. Gori-Giorgi and A. Savin},
journal = {Theor. Chem. Acc.},
volume = {114},
pages = {305},
year = {2005}
}
% cours de QMC
@misc{Tou-JJJ-XX,
author = {S. Dubillard and E. Eliav and L. Visscher and R. Bast and T. Saue},
note = {unpublished}
}
% lr/sr fxc for uniform electron gas
@article{Tou-PRB-05,
author = {Julien Toulouse},
title = {Simple model of the static exchange-correlation kernel of a uniform electron gas with long-range electron-electron interaction},
journal = {Phys. Rev. B},
volume = {72},
pages = {035117},
year = {2005}
}
@article{TouRebGouDobSeaAng-JCP-13,
author = {Julien Toulouse and Elisa Rebolini and Tim Gould and John F. Dobson and Prasenjit Seal and J\'anos G. \'Angy\'an},
title = {Assessment of range-separated time-dependent density-functional theory for calculating $C_6$ dispersion coefficients},
journal = {J. Chem. Phys.},
volume = {138},
pages = {194106},
year = {2013}
}
%Ueg erf and erfgau
@article{TouSavFla-IJQC-04,
author = {J. Toulouse and A. Savin and H.-J. Flad},
title = {Short-range exchange-correlation energy of a uniform electron gas with modified electron-electron interaction},
journal = {Int. J. Quantum Chem.},
volume = {100},
pages = {1047},
year = {2004},
note = {}
}
%scaling relations and virial theorem
@misc{TouSav-JJJ-XX,
author = {J. Toulouse and A. Savin},
title = {in preparation}
}
%
@article{TouSav-JMS-06,
author = {Julien Toulouse and Andreas Savin},
title = {Local density approximation for long-range or for short-range energy functionals?},
journal = {J. Mol. Struct. (Theochem)},
volume = {762},
pages = {147},
year = {2006}
}
@article{TouShaBreAda-JCP-11,
author = {J. Toulouse and K. Sharkas and E. Br\'emond and C. Adamo},
title = {Rationale for a new class of double-hybrid approximations in density-functional theory},
journal = {J. Chem. Phys.},
volume = {135},
pages = {101102},
year = {2011}
}
%these
@phdthesis{Tou-THESIS-05,
author = {J. Toulouse},
title = {},
school = {Universit\'e Pierre et Marie Curie (Paris 6)},
year = {2005},
note = {tel.archives-ouvertes.fr/tel-00550772}
}
% Wave function optimization
@article{TouUmr-JCP-07,
author = {J. Toulouse and C. J. Umrigar},
title = "{Optimization of quantum Monte Carlo wave functions by energy minimization}",
journal = {J. Chem. Phys.},
volume = {126},
pages = {084102},
year = {2007}
}
% Wave function optimization with applications to diatomics
@article{TouUmr-JCP-08,
author = {J. Toulouse and C. J. Umrigar},
title = "{Full optimization of Jastrow-Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules}",
journal = {J. Chem. Phys.},
volume = {128},
pages = {174101},
year = {2008}
}
% energy optimization in QMC
@misc{TouUmr-JJJ-XX,
author = {J. Toulouse and C. J. Umrigar},
title = {},
journal = {},
year = {},
pages = {},
volume = {},
note = {unpublished}
}
@misc{TouZhuAngSav-JJJ-XX-note2,
note = {In the context of density-functional theory RPA is usually derived from the Kohn-Sham reference, while in the context of many-body perturbation theory (see appendices) RPA is usually derived from the HF reference. Therefore, both PBE+RPA and HF+RPA are theoretically justified.}
}
@misc{TouZhuAngSav-JJJ-XX-note3,
note = {The inverse of a 4-point function $\chi (1,2;1',2')$ is defined according to $\int d1'd2' \chi (1,2;1',2') \chi^{-1} (2',1';4,3) = \delta(1,3) \delta (2,4)$.}
}
@misc{TouZhuAngSav-JJJ-XX-note,
note = {The short-range self-energy correction $\Delta \Sigma^\sr_\l$ is wrongly missing in Eq.~(11) of Ref.~\onlinecite{TouGerJanSavAng-PRL-09}. However, in practice, this term vanishes in the RPA or RPAx approximation so that the results of Ref.~\onlinecite{TouGerJanSavAng-PRL-09} are correct.}
}
@article{TouZhuAngSav-PRA-10,
author = {Julien Toulouse and Wuming Zhu and J\'anos G. \'Angy\'an and Andreas Savin},
title = {Range-separated density-functional theory with the random-phase approximation: Detailed formalism and illustrative applications},
journal = {Phys. Rev. A},
volume = {82},
pages = {032502},
year = {2010}
}
@article{TouZhuSavJanAng-JCP-11,
author = {J. Toulouse and W. Zhu and A. Savin and G. Jansen and J. G. \'Angy\'an},
title = {Closed-shell ring coupled cluster doubles theory with range separation
applied on weak intermolecular interactions},
journal = {J. Chem. Phys.},
volume = {135},
pages = {084119},
year = {2011}
}
%
@misc{TouZhuSavJanAng-JJJ-XX,
author = {J. Toulouse and W. Zhu and A. Savin and G. Jansen and J. G. \'Angyan},
note = {unpublished}
}
@misc{TouZhuSavJanAng-JJJ-XX-note,
note = {In Ref.~\onlinecite{Hes-JCP-11}, AC-RPA, NRPA1, NRPA3, and NRPA4 refer to what we call here RPAx-I, RPAx-II, RPA-SO2, and RPA-SO1, respectively. In addition, NRPA2 corresponds to the variant of Fukuda {\it et al.}, i.e. $2E_{c,\RPAxII}-E_{c,\text{MP2}}$.}
}
% PRL, linear method + systematic elimination of fixed-node error
@article{UmrTouFilSorHen-PRL-07,
author = {C. J. Umrigar and J. Toulouse and C. Filippi and S. Sorella and R. G. Hennig},
title = {Alleviation of the Fermion-sign problem by optimization of many-body wave functions},
journal = {Phys. Rev. Lett.},
volume = {98},
pages = {110201},
year = {2007}
}
@article{ZapLupTou-JCP-19,
author = {F. Zapata and E. Luppi and J. Toulouse},
title = {Linear-response range-separated density-functional theory for atomic photoexcitation and photoionization spectra},
journal = {J. Chem. Phys.},
volume = {150},
pages = {234104},
year = {2019},
doi = {10.1063/1.5096037}
}
@misc{ZapLupTou-JJJ-XX-note,
note = {Contrary to our Fig. \ref{Hespectra}a, the TDLDA spectrum of the He atom shown in Fig. 6 of Ref.~\onlinecite{WasMaiBur-PRL-03} has a larger maximum than the LDA spectrum. This discrepancy is due to the fact that the TDLDA spectrum shown in Ref.~\onlinecite{WasMaiBur-PRL-03} comes in fact from Ref.~\onlinecite{SteDecGor-JCP-01} where it was calculated by replacing the LDA 1s orbital energy by the opposite of the exact ionization energy. We have checked that this results not only in an energy shift of the spectrum but also to larger oscillator strengths. The true TDLDA spectrum of the He atom is thus the one shown in the present Fig. \ref{Hespectra}a.}
}
@article{ZhuTouSavAng-JCP-10,
author = {Wuming Zhu and Julien Toulouse and Andreas Savin and J\'anos G. \'Angy\'an},
title = {Range-separated density-functional theory with random phase approximation applied to noncovalent intermolecular interactions},
journal = {J. Chem. Phys.},
volume = {132},
pages = {244108},
year = {2010}
}
% RSH+RPAx on S22
@misc{ZhuTouSavAng-JJJ-XX,
author = {W. Zhu and J. Toulouse and A. Savin and J. G. \'Angy\'an},
title = {},
journal = {},
year = {},
pages = {},
volume = {},
note = {unpublished}
}
@article{ZimTouZhaMusUmr-JCP-09,
author = {P. M. Zimmerman and J. Toulouse and Z. Zhang and C. B. Musgrave and C. J. Umrigar},
title = {Excited states of methylene from quantum Monte Carlo},
journal = {J. Chem. Phys.},
volume = {131},
pages = {124103},
year = {2009}
}
% asymptotic properties of density, orbitals
@article{AlmBar-PRB-85,
author = {C.-O. Almbladh and U. von Barth},
@ -2728,23 +3639,31 @@ year = {2013}
}
@ARTICLE{DahLeeBar-IJQC-05,
author = {Nils Erik Dahlen and Robert Van Leeuwen and Ulf Von Barth},
author = "{N. E. Dahlen, R. van Leeuwen, and U. Von Barth}",
title = {Variational energy functionals of the Green function tested on molecules},
journal = {Int. J. Quantum Chem.},
year = {2005},
volume = {101},
pages = {512-519},
doi = {10.1002/qua.20306}
pages = {512-519}
}
@ARTICLE{DahLeeBar-PRA-06,
author = {Nils Erik Dahlen and Robert van Leeuwen and Ulf von Barth},
author = "{N. E. Dahlen, R. van Leeuwen, and U. von Barth}",
journal = {Phys. Rev. A},
year = {2006},
volume = {73},
pages = {012511}
}
%
@article{DahLee-JCP-05,
author = "{N. E. Dahlen and R. van Leeuwen}",
journal = {J. Chem. Phys.},
volume = {122},
pages = {164102},
year = {2005}
}
%
@article{DahLee-PRL-07,
author = "{N. E. Dahlen and R. van Leeuwen}",
@ -8773,6 +9692,14 @@ year = {1978}
year = {1958},
}
%
@book{MarReiCep-BOOK-16,
author = {Richard M. Martin and Lucia Reining and David M. Ceperley},
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year = {2016},
publisher = {Cambridge University Press}
}
%
@article{MarWer-JPCA-09,
author = {O. Marchetti and H.-J. Werner},
@ -12751,6 +13678,14 @@ year = {2009}
pages = {244119}
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%
@book{SteLee-BOOK-13,
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year = {2013},
publisher = {Cambridge University Press}
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