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Manuscript/CI-F12.aux
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Schematic representation of the various orbital spaces and their notation. The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket *{_{ij}^{\alpha \beta }}$ (green), the mixed doubles $\ket *{_{ij}^{a \beta }}$ (magenta) and the pure singles $\ket *{_{i}^{\alpha }}$ (orange).}}{2}{figure.1}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Schematic representation of the various orbital spaces and their notation. The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket *{_{ij}^{\alpha \beta }}$ (green), the mixed doubles $\ket *{_{ij}^{a \beta }}$ (magenta) and the pure singles $\ket *{_{i}^{\alpha }}$ (orange).}}{2}{figure.1}}
\newlabel{fig:CBS}{{1}{2}{Schematic representation of the various orbital spaces and their notation. The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket *{_{ij}^{\alpha \beta }}$ (green), the mixed doubles $\ket *{_{ij}^{a \beta }}$ (magenta) and the pure singles $\ket *{_{i}^{\alpha }}$ (orange)}{figure.1}{}} \newlabel{fig:CBS}{{1}{2}{Schematic representation of the various orbital spaces and their notation. The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket *{_{ij}^{\alpha \beta }}$ (green), the mixed doubles $\ket *{_{ij}^{a \beta }}$ (magenta) and the pure singles $\ket *{_{i}^{\alpha }}$ (orange)}{figure.1}{}}
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@ -74,7 +78,6 @@
\@writefile{toc}{\contentsline {section}{\numberline {V}Computational details}{3}{section*.7}} \@writefile{toc}{\contentsline {section}{\numberline {V}Computational details}{3}{section*.7}}
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\@writefile{toc}{\contentsline {section}{\numberline {VII}Conclusion}{3}{section*.9}} \@writefile{toc}{\contentsline {section}{\numberline {VII}Conclusion}{3}{section*.9}}
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\bibdata{CI-F12Notes,CI-F12,CI-F12-control} \bibdata{CI-F12Notes,CI-F12,CI-F12-control}
@ -95,57 +98,62 @@
\bibcite{Hattig12}{{15}{2012}{{Hattig\ \emph {et~al.}}}{{Hattig, Klopper, Kohn,\ and\ Tew}}} \bibcite{Hattig12}{{15}{2012}{{Hattig\ \emph {et~al.}}}{{Hattig, Klopper, Kohn,\ and\ Tew}}}
\bibcite{Kong12}{{16}{2012}{{Kong, Bischo,\ and\ Valeev}}{{}}} \bibcite{Kong12}{{16}{2012}{{Kong, Bischo,\ and\ Valeev}}{{}}}
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\bibcite{Huron73}{{18}{1973}{{Huron, Malrieu,\ and\ Rancurel}}{{}}} \bibcite{Malrieu85}{{18}{1985}{{Malrieu, Durand,\ and\ Daudey}}{{}}}
\bibcite{Evangelisti83}{{19}{1983}{{Evangelisti, Daudey,\ and\ Malrieu}}{{}}} \bibcite{Heully92}{{19}{1992}{{Heuilly\ and\ Malrieu}}{{}}}
\bibcite{Heully92}{{20}{1992}{{Heuilly\ and\ Malrieu}}{{}}} \bibcite{Garniron18}{{20}{2018}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Giner, Caffarel,\ and\ Loos}}}
\bibcite{Garniron18}{{21}{2018}{{Garniron\ \emph {et~al.}}}{{Garniron, Scemama, Giner, Caffarel,\ and\ Loos}}} \bibcite{Knowles84}{{21}{1984}{{Knowles\ and\ Handy}}{{}}}
\bibcite{Giner13}{{22}{2013}{{Giner, Scemama,\ and\ Caffarel}}{{}}} \bibcite{Knowles89}{{22}{1989}{{Knowles\ and\ Handy}}{{}}}
\bibcite{Scemama13a}{{23}{2013{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Caffarel, Oseret,\ and\ Jalby}}} \bibcite{Huron73}{{23}{1973}{{Huron, Malrieu,\ and\ Rancurel}}{{}}}
\bibcite{Scemama13b}{{24}{2013{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Caffarel, Oseret,\ and\ Jalby}}} \bibcite{Giner13}{{24}{2013}{{Giner, Scemama,\ and\ Caffarel}}{{}}}
\bibcite{Scemama14}{{25}{2014}{{Scemama\ \emph {et~al.}}}{{Scemama, Applencourt, Giner,\ and\ Caffarel}}} \bibcite{Giner15}{{25}{2015}{{Giner, Scemama,\ and\ Caffarel}}{{}}}
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\bibcite{Loos19}{{29}{2019}{{Loos\ \emph {et~al.}}}{{Loos, Boggio-Pasqua, Scemama, Caffarel,\ and\ Jacquemin}}} \bibcite{Loos18b}{{29}{2018}{{Loos\ \emph {et~al.}}}{{Loos, Scemama, Blondel, Garniron, Caffarel,\ and\ Jacquemin}}}
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\bibcite{Reine12}{{45}{2012}{{Reine, Helgaker,\ and\ Lind}}{{}}} \bibcite{Klopper04}{{45}{2004}{{Klopper}}{{}}}
\bibcite{GG16}{{46}{2016}{{Barca\ and\ Gill}}{{}}} \bibcite{Manby06}{{46}{2006}{{Manby\ \emph {et~al.}}}{{Manby, Werner, Adler,\ and\ May}}}
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\bibcite{Bubin11}{{57}{2011}{{Bubin\ and\ Adamowicz}}{{}}} \bibcite{IntF12}{{54}{2017}{{Barca\ and\ Loos}}{{}}}
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\bibcite{Cleland12}{{61}{2012}{{Cleland\ \emph {et~al.}}}{{Cleland, Booth, Overy,\ and\ Alavi}}} \bibcite{Nakashima07}{{58}{2007}{{Nakashima\ and\ Nakatsuji}}{{}}}
\bibcite{AlmoraDiaz14}{{62}{2014}{{Almora-Diaz}}{{}}} \bibcite{Puchalski09}{{59}{2009}{{Puchalski, Kedziera,\ and\ Pachucki}}{{}}}
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@ -6,7 +6,7 @@
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Phys.}\ }\textbf {\bibinfo {volume} {149}},\ \bibinfo {pages} {064103} Phys.}\ }\textbf {\bibinfo {volume} {149}},\ \bibinfo {pages} {064103}
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{Handy}},\ }\href {\doibase 10.1016/0010-4655(89)90033-7} {\bibfield
{journal} {\bibinfo {journal} {Comput. Phys. Commun.}\ }\textbf {\bibinfo
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{NoStop}%
\bibitem [{\citenamefont {Huron}, \citenamefont {Malrieu},\ and\ \citenamefont
{Rancurel}(1973)}]{Huron73}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
{Huron}}, \bibinfo {author} {\bibfnamefont {J.~P.}\ \bibnamefont {Malrieu}},
\ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Rancurel}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\
}\textbf {\bibinfo {volume} {58}},\ \bibinfo {pages} {5745} (\bibinfo {year}
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\bibitem [{\citenamefont {Giner}, \citenamefont {Scemama},\ and\ \citenamefont \bibitem [{\citenamefont {Giner}, \citenamefont {Scemama},\ and\ \citenamefont
{Caffarel}(2013)}]{Giner13}% {Caffarel}(2013)}]{Giner13}%
\BibitemOpen \BibitemOpen
@ -236,42 +253,6 @@
{\doibase 10.1139/cjc-2013-0017} {\bibfield {journal} {\bibinfo {journal} {\doibase 10.1139/cjc-2013-0017} {\bibfield {journal} {\bibinfo {journal}
{Can. J. Chem.}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {879} {Can. J. Chem.}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {879}
(\bibinfo {year} {2013})}\BibitemShut {NoStop}% (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Giner}, \citenamefont {Scemama},\ and\ \citenamefont \bibitem [{\citenamefont {Giner}, \citenamefont {Scemama},\ and\ \citenamefont
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@ -292,6 +273,29 @@
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\
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\bibitem [{\citenamefont {Scemama}\ \emph {et~al.}(2018)\citenamefont
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\bibitem [{\citenamefont {Loos}\ \emph {et~al.}(2018)\citenamefont {Loos}, \bibitem [{\citenamefont {Loos}\ \emph {et~al.}(2018)\citenamefont {Loos},
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\citenamefont {Caffarel},\ and\ \citenamefont {Jacquemin}}]{Loos18b}% \citenamefont {Caffarel},\ and\ \citenamefont {Jacquemin}}]{Loos18b}%
@ -354,6 +358,29 @@
{Kutzelnigg}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. {Kutzelnigg}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
Phys. Chem.}\ }\textbf {\bibinfo {volume} {97}},\ \bibinfo {pages} {2425} Phys. Chem.}\ }\textbf {\bibinfo {volume} {97}},\ \bibinfo {pages} {2425}
(\bibinfo {year} {1993})}\BibitemShut {NoStop}% (\bibinfo {year} {1993})}\BibitemShut {NoStop}%
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@ -1,26 +1,50 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-03-10 21:06:16 +0100 %% Created for Pierre-Francois Loos at 2019-03-23 10:31:56 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Garniron19, @article{Knowles84,
Author = {Y. Garniron and K. Gasperich and T. Applencourt 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{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama}, Author = {P. J. Knowles and N. C. Handy},
Date-Added = {2019-03-10 21:06:09 +0100}, Date-Added = {2019-03-23 10:31:23 +0100},
Date-Modified = {2019-03-10 21:06:15 +0100}, Date-Modified = {2019-03-23 10:31:32 +0100},
Journal = {J. Chem. Theory Comput.}, Journal = {Chem. Phys. Lett.},
Title = {Quantum Package 2.0: a open-source determinant-driven suite of programs}, Keywords = {correlation},
Volume = {in press}, Pages = {315--321},
Year = {2019}} Title = {A new determinant-based full configuration interaction method},
Volume = {111},
Year = {1984}}
@article{Knowles89,
Author = {P. J. Knowles and N. C. Handy},
Date-Added = {2019-03-23 10:31:23 +0100},
Date-Modified = {2019-03-23 10:31:28 +0100},
Doi = {10.1016/0010-4655(89)90033-7},
Journal = {Comput. Phys. Commun.},
Pages = {75--83},
Title = {A determinant based full configuration interaction program},
Volume = {54},
Year = {1989},
Bdsk-Url-1 = {https://doi.org/10.1016/0010-4655(89)90033-7}}
@article{eee15,
Author = {P. F. Loos and N. J. Bloomfield and P. M. W. Gill},
Date-Added = {2019-03-23 10:29:52 +0100},
Date-Modified = {2019-03-23 10:29:52 +0100},
Journal = {J. Chem. Phys.},
Pages = {181101},
Title = {Three-electron coalescence points in two and three dimensions},
Volume = {143},
Year = {2015}}
@article{Scemama18a, @article{Scemama18a,
Author = {A. Scemama and Y. Garniron and M. Caffarel and P. F. Loos}, Author = {A. Scemama and Y. Garniron and M. Caffarel and P. F. Loos},
Date-Added = {2019-03-10 21:04:59 +0100}, Date-Added = {2019-03-23 10:21:00 +0100},
Date-Modified = {2019-03-10 21:05:07 +0100}, Date-Modified = {2019-03-23 10:21:00 +0100},
Doi = {10.1021/acs.jctc.7b01250}, Doi = {10.1021/acs.jctc.7b01250},
Journal = {J. Chem. Theory Comput.}, Journal = {J. Chem. Theory Comput.},
Pages = {1395}, Pages = {1395},
@ -30,21 +54,60 @@
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b01250}} Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b01250}}
@article{Scemama18b, @article{Scemama18b,
Author = {Anthony Scemama and Anouar Benali and Denis Jacquemin and Michel Caffarel and Pierre-Fran{\c{c}}ois Loos}, Author = {A. Scemama and A. Benali and D. Jacquemin and M. Caffarel and P. F. Loos},
Date-Added = {2019-03-10 21:04:59 +0100}, Date-Added = {2019-03-23 10:21:00 +0100},
Date-Modified = {2019-03-10 21:05:12 +0100}, Date-Modified = {2019-03-23 10:21:00 +0100},
Doi = {10.1063/1.5041327}, Doi = {10.1063/1.5041327},
Journal = {J. Chem. Phys.}, Journal = {J. Chem. Phys.},
Month = {jul}, Pages = {xxxxxx},
Number = {3}, Title = {Excitation energies from diffusion Monte Carlo using selected Configuration Interaction nodes},
Pages = {034108},
Publisher = {{AIP} Publishing},
Title = {Excitation energies from diffusion Monte Carlo using selected configuration interaction nodes},
Url = {https://doi.org/10.1063%2F1.5041327},
Volume = {149}, Volume = {149},
Year = 2018, Year = {in press},
Bdsk-Url-1 = {https://doi.org/10.1063%2F1.5041327}, Bdsk-Url-1 = {https://arxiv.org/pdf/1805.09553.pdf}}
Bdsk-Url-2 = {https://doi.org/10.1063/1.5041327}}
@article{Heully96,
Author = {Heully, J. L. and Malrieu, J. P. and Zaitsevskii, A.},
Date-Added = {2019-03-23 10:17:49 +0100},
Date-Modified = {2019-03-23 10:18:00 +0100},
Doi = {10.1063/1.471982},
File = {/Users/loos/Zotero/storage/D3SYMXH6/Heully et al. - 1996 - On the origin of size inconsistency of the second‐.pdf},
Issn = {0021-9606, 1089-7690},
Journal = {J. Chem. Phys.},
Language = {en},
Month = oct,
Number = {16},
Pages = {6887--6891},
Title = {On the Origin of Size Inconsistency of the Second-order State-specific Effective {{Hamiltonian}} Method},
Volume = {105},
Year = {1996},
Bdsk-Url-1 = {https://doi.org/10.1063/1.471982}}
@article{Malrieu85,
Abstract = {The theory of effective Hamiltonians is well established. However, limitations appear in its applicability for many problems in molecular physics and quantum chemistry. The standard effective Hamiltonians may become strongly non-Hermitian when there is a large coupling between the model space, in which they are defined, and the outer space. Moreover, in the presence of intruder states, discontinuities appear in the matrix elements of these effective Hamiltonians as a function of the internuclear distances. To solve these difficulties, a new class of effective Hamiltonians (called intermediate Hamiltonians) is presented: only one part of their roots are exact eigen energies of the full Hamiltonian. The theory of these intermediate Hamiltonians is presented by means of a new waveoperator R which is the analogue of the wave-operator R in the theory of effective Hamiltonians. Solutions are obtained by a generalised degenerate perturbation theory (GDPT) and by iterative procedures. Two model systems are numerically solved which demonstrate the good convergence properties of GDFT with respect to standard degenerate perturbation theory (DPT). Continuity of the solutions is also checked in the presence of an intruder state.},
Author = {Malrieu, J P and Durand, P and Daudey, J P},
Date-Added = {2019-03-23 10:17:09 +0100},
Date-Modified = {2019-03-23 10:17:13 +0100},
Doi = {10.1088/0305-4470/18/5/014},
File = {/Users/loos/Zotero/storage/KUDRGJEN/Malrieu et al. - 1985 - Intermediate Hamiltonians as a new class of effect.pdf},
Issn = {0305-4470, 1361-6447},
Journal = {J. Phys. Math. Gen.},
Language = {en},
Month = apr,
Number = {5},
Pages = {809--826},
Title = {Intermediate {{Hamiltonians}} as a New Class of Effective {{Hamiltonians}}},
Volume = {18},
Year = {1985},
Bdsk-Url-1 = {https://doi.org/10.1088/0305-4470/18/5/014}}
@article{Garniron19,
Author = {Y. Garniron and K. Gasperich and T. Applencourt 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{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
Date-Added = {2019-03-10 21:06:09 +0100},
Date-Modified = {2019-03-10 21:06:15 +0100},
Journal = {J. Chem. Theory Comput.},
Title = {Quantum Package 2.0: a open-source determinant-driven suite of programs},
Volume = {in press},
Year = {2019}}
@article{Veril_2018, @article{Veril_2018,
Author = {M. Veril and P. Romaniello and J. A. Berger and P. F. Loos}, Author = {M. Veril and P. Romaniello and J. A. Berger and P. F. Loos},

2
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@ -1,7 +1,7 @@
\BOOKMARK [0][-]{section*.2}{Dressing the configuration interaction matrix with explicit correlation}{}% 2 \BOOKMARK [0][-]{section*.2}{Dressing the configuration interaction matrix with explicit correlation}{}% 2
\BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1 \BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1
\BOOKMARK [1][-]{section*.3}{Introduction}{section*.2}% 3 \BOOKMARK [1][-]{section*.3}{Introduction}{section*.2}% 3
\BOOKMARK [1][-]{section*.4}{Ansatz}{section*.2}% 4 \BOOKMARK [1][-]{section*.4}{Ans\344tz}{section*.2}% 4
\BOOKMARK [1][-]{section*.5}{Dressing}{section*.2}% 5 \BOOKMARK [1][-]{section*.5}{Dressing}{section*.2}% 5
\BOOKMARK [1][-]{section*.6}{Matrix elements}{section*.2}% 6 \BOOKMARK [1][-]{section*.6}{Matrix elements}{section*.2}% 6
\BOOKMARK [1][-]{section*.7}{Computational details}{section*.2}% 7 \BOOKMARK [1][-]{section*.7}{Computational details}{section*.2}% 7

56
Manuscript/CI-F12.tex
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@ -65,16 +65,16 @@
\affiliation{\LCPQ} \affiliation{\LCPQ}
\author{Michel Caffarel} \author{Michel Caffarel}
\affiliation{\LCPQ} \affiliation{\LCPQ}
\author{Pierre-Fran{\c c}ois Loos} \author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr} \email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ} \affiliation{\LCPQ}
\begin{abstract} \begin{abstract}
We present an explicitly-correlated version of the configuration interaction (CI) method. We present an explicitly-correlated version of the configuration interaction (CI) method.
An explicitly-correlated term is introduced via dressing of the CI matrix. An explicitly-correlated term is introduced via a dressing of the CI matrix.
The dressing is guided by electron-electron cusp conditions. The dressing is guided by electron-electron cusp conditions.
This greatly enhances the convergence with respect to the one-electron basis set compared to conventional CI methods. This greatly enhances the convergence with respect to the one-electron basis set compared to conventional CI methods.
The performance of the newly-designed explicitly-correlated dressing CI method is illustrated on atoms and molecules. The performance of the newly-designed explicitly-correlated CI-F12 method is illustrated on atoms and molecules.
\end{abstract} \end{abstract}
\keywords{configuration interaction; explicitly-correlated methods; effective Hamiltonian theory} \keywords{configuration interaction; explicitly-correlated methods; effective Hamiltonian theory}
@ -90,31 +90,36 @@ The performance of the newly-designed explicitly-correlated dressing CI method i
\\ \\
\textit{Theor.~Chim.~Acta} (1985) \textbf{68} 445--469. \textit{Theor.~Chim.~Acta} (1985) \textbf{68} 445--469.
\end{quotation} \end{quotation}
One of the most fundamental and severe error in electronic structure methods is the basis set incompleteness. One of the most fundamental and severe error in electronic structure methods is the (one-electron) basis set incompleteness.
In particular, conventional quantum chemistry wave function methods typically display a slow energy convergence with respect to the size of the one-electron basis set. In particular, conventional quantum chemistry wave function methods typically display a slow energy convergence with respect to the size of the one-electron basis set.
This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kutzelnigg85} This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kutzelnigg85}
He subsequently proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kutzelnigg91, Termath91, Klopper91a, Klopper91b, Noga94} To palliate this, he proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kutzelnigg85, Kutzelnigg91, Termath91, Klopper91a, Klopper91b, Noga94}
This yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis). This yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis).
This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$. \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05} This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$. \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}
The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Tenno12a, Tenno12b, Hattig12, Kong12} The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Tenno12a, Tenno12b, Hattig12, Kong12}
For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis. \cite{Tew07b} For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis. \cite{Tew07b}
In the present study, following Kutzelnigg's idea, we propose to introduce the explicit correlation between electrons within the configuration interaction (CI) method via a dressing of the CI matrix. \cite{Huron73, Evangelisti83} In the present study, following Kutzelnigg's idea, we propose to introduce the explicit correlation between electrons within the configuration interaction (CI) method via a dressing of the CI matrix.
This method, involving effective Hamiltonian theory, has been shown to be successful in other scenarios. \cite{Heully92, Garniron18} This method, involving effective Hamiltonian theory, \cite{Malrieu85} has been shown to be successful in other scenarios. \cite{Heully92, Garniron18}
Compared to other explicitly-correlated methods, this dressing strategy has the advantage of introducing the explicit correlation at a relatively low computational cost. Compared to other explicitly-correlated strategies, this dressing strategy has the advantage of introducing the explicit correlation at a relatively low computational cost.
The present explicitly-correlated dressed CI method is completely general and can be applied to any type of truncated, full, or even selected CI methods. \cite{Giner13, Scemama13a, Scemama13b, Scemama14, Giner15, Caffarel16, Loos18b, Loos19} The present explicitly-correlated dressed CI method is completely general and can be applied to any type of truncated, full, \cite{Knowles84, Knowles89} or even selected CI methods. \cite{Huron73, Giner13, Giner15, Caffarel16, Scemama18a, Scemama18b, Loos18b, Loos19}
However, for the sake of generality, we will discuss here the dressing of the full CI (FCI) matrix. However, for the sake of generality, we will discuss here the dressing of the full CI (FCI) matrix.
Atomic units are used throughout. Atomic units are used throughout.
%---------------------------------------------------------------- %----------------------------------------------------------------
\section{Ansatz} \section{Ans{\"a}tz}
%---------------------------------------------------------------- %----------------------------------------------------------------
Inspired by a number of previous research, \cite{Shiozaki11} our electronic wave function ansatz $\ket{\Psi} = \kD + \kF$ is simply written as the sum of a ``conventional'' part Inspired by a number of previous research (see Ref.~\onlinecite{Shiozaki11} and references therein), our electronic wave function \emph{ans{\"a}tz}
\begin{equation}
\label{eq:ansatz}
\ket{\Psi} = \kD + \kF
\end{equation}
is simply written as the sum of a \emph{``conventional''} part
\begin{equation} \begin{equation}
\label{eq:D} \label{eq:D}
\kD = \sum_{I} \cD{I} \kI \kD = \sum_{I} \cD{I} \kI
\end{equation} \end{equation}
composed by a linear combination of determinants $\kI$ of coefficients $\cD{I}$ and an ``explicitly-correlated'' part composed by a linear combination of determinants $\kI$ with CI coefficients $\cD{I}$ and an \emph{``explicitly-correlated''} part
\begin{equation} \begin{equation}
\label{eq:WF-F12-CIPSI} \label{eq:WF-F12-CIPSI}
\kF = \sum_{I} \cF{I} \hQ f \kI \kF = \sum_{I} \cF{I} \hQ f \kI
@ -140,7 +145,7 @@ is a correlation factor with
\end{equation} \end{equation}
\alert{The correlation factor \eqref{eq:Ja} is not size-consistent.} \alert{The correlation factor \eqref{eq:Ja} is not size-consistent.}
As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various authors \cite{Pack66, Morgan93}), for small $r_{12}$, the two-electron correlation factor $f_{12}$ in Eq.~\eqref{eq:Ja} must behave as As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various authors \cite{Pack66, Morgan93, Tew08, ExSpherium10, eee15}), for small $r_{12}$, the two-electron correlation factor $f_{12}$ in Eq.~\eqref{eq:Ja} must behave as
\begin{equation} \begin{equation}
f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}. f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}.
\end{equation} \end{equation}
@ -148,8 +153,8 @@ As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various
%---------------------------------------------------------------- %----------------------------------------------------------------
\section{Dressing} \section{Dressing}
%---------------------------------------------------------------- %----------------------------------------------------------------
Our primary goal is to introduce the explicit correlation between electrons at low computational cost. Our primary goal is to introduce the explicit correlation between electrons at relatively low computational cost.
Therefore, assuming that $\hH \ket{\Psi} = E \Psi$, one can write, by projection over $\bra{I}$, Therefore, assuming that $\hH \ket{\Psi} = E\,\Psi$, one can write, by projection over $\bra{I}$,
\begin{equation} \begin{equation}
\cD{I} \qty[ H_{II} + \cD{I}^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} \cD{J} H_{IJ} = 0. \cD{I} \qty[ H_{II} + \cD{I}^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} \cD{J} H_{IJ} = 0.
\end{equation} \end{equation}
@ -170,7 +175,8 @@ with
\mel{I}{\hH}{F} = \sum_J \cF{J} \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ], \mel{I}{\hH}{F} = \sum_J \cF{J} \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ],
\end{equation} \end{equation}
and $f_{IJ} = \mel{I}{f}{J}$. and $f_{IJ} = \mel{I}{f}{J}$.
It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F} = 0$, which demonstrates that our dressed CI method becomes exact in the limit of a complete one-electron basis. We refer to this strategy as diagonal dressing as only the diagonal of $\hH$ is modified in Eq.~\eqref{eq:DrH}.
It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F} = 0$, which demonstrates that the dressed term vanishes in the limit of a complete one-electron basis, as one would expect.
At this stage, two key comments are in order. At this stage, two key comments are in order.
First, as one may have realized, the coefficients $\cF{I}$ are unknown. First, as one may have realized, the coefficients $\cF{I}$ are unknown.
@ -186,9 +192,25 @@ which can be easily solved using standard linear algebra packages (where $\delta
Second, because Eq.~\eqref{eq:DrH} depends on the CI coefficient $\cD{I}$, one must iterate the diagonalization process self-consistently until convergence of the desired eigenvalues of the dressed Hamiltonian $\oH$. Second, because Eq.~\eqref{eq:DrH} depends on the CI coefficient $\cD{I}$, one must iterate the diagonalization process self-consistently until convergence of the desired eigenvalues of the dressed Hamiltonian $\oH$.
At each iteration, we solve Eq.~\eqref{eq:tI} to obtain the coefficients $\cF{I}$ and dress the Hamiltonian [see Eq.~\eqref{eq:DrH}]. At each iteration, we solve Eq.~\eqref{eq:tI} to obtain the coefficients $\cF{I}$ and dress the Hamiltonian [see Eq.~\eqref{eq:DrH}].
In practice, we initially start with a CI vector obtained by the diagonalization of the undressed Hamiltonian, and convergence is usually reached within few cycles. In practice, we initially start with a CI vector obtained by the diagonalization of the undressed Hamiltonian, and convergence is usually reached within few cycles.
%For pathological cases, a DIIS-like procedure may be employed. \cite{Pulay82}
We refer the interested reader to Ref.~\onlinecite{Garniron18} for additional details about our dressing scheme. We refer the interested reader to Ref.~\onlinecite{Garniron18} for additional details about our dressing scheme.
Note that the present formalism is state-specific and only focus on the ground state.
Multi-state can potentially developed following our work in Ref.~\onlinecite{Garniron18}.
In the state-specific case, it is possible to avoid the potentially troublesome division by $\cD{I}^{-1}$ shuffling around the dressing term.
Assuming, witout loss of generality that $\cD{0}$ is the largest coefficient $\cD{I}$, we have
\begin{equation}
\label{eq:DrH}
\oH_{IJ} =
\begin{cases}
H_{I0} + \frac{\mel*{I}{\hH}{F}}{\cD{0}}, & I > 0 \wedge J = 0
\\
H_{0J} + \frac{\mel*{J}{\hH}{F}}{\cD{0}}, & I = 0 \wedge J > 0
\\
H_{00} + \frac{2 \mel*{0}{\hH}{F}}{\cD{0}} - \sum_I \frac{\cD{I}}{\cD{0}} \frac{\mel*{I}{\hH}{F}}{\cD{0}}, & I = 0 \wedge J = 0
\\
H_{IJ}, & \text{otherwise}.
\end{cases}
\end{equation}
%%% FIG 1 %%% %%% FIG 1 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{fig1} \includegraphics[width=\linewidth]{fig1}