diff --git a/Manuscript/EPAWTFT.aux b/Manuscript/EPAWTFT.aux index 1f5aa29..29f9a04 100644 --- a/Manuscript/EPAWTFT.aux +++ b/Manuscript/EPAWTFT.aux @@ -55,7 +55,7 @@ \citation{Moller_1934} \citation{SzaboBook} \citation{Gill_1994} -\citation{Gill_1986,Gill_1988,Handy_1985,Lepetit_1988} +\citation{Gill_1986,Gill_1988,Handy_1985,Lepetit_1988,Leininger_2000} \citation{JensenBook} \citation{Nesbet_1955,Epstein_1926} \citation{Seidl_2018} @@ -89,10 +89,10 @@ \citation{Sergeev_2005} \citation{Stillinger_2000} \citation{Olsen_2000} +\@writefile{toc}{\contentsline {subsection}{\numberline {C}The singularity structure}{5}{section*.13}\protected@file@percent } +\newlabel{eq:HamiltonianStillinger}{{23}{5}{}{equation.3.23}{}} \citation{Goodson_2004} \citation{Olsen_2000} -\@writefile{toc}{\contentsline {subsection}{\numberline {C}The singularity structure}{5}{section*.13}\protected@file@percent } -\newlabel{eq:HamiltonianStillinger}{{22}{5}{}{equation.3.22}{}} \citation{Heiss_1988,Heiss_2002,Cejnar_2005,Cejnar_2007,Cejnar_2009,Borisov_2015,Sindelka_2017} \citation{Cejnar_2009,Sachdev_2011,Cejnar_2015,Cejnar_2016,Caprio_2008,Macek_2019} \citation{Cejnar_2009,Sachdev_2011} @@ -148,34 +148,35 @@ \bibcite{Gill_1988}{{44}{1988}{{Gill\ \emph {et~al.}}}{{Gill, Pople, Radom,\ and\ Nobes}}} \bibcite{Handy_1985}{{45}{1985}{{Handy\ \emph {et~al.}}}{{Handy, Knowles,\ and\ Somasundram}}} \bibcite{Lepetit_1988}{{46}{1988}{{Lepetit\ \emph {et~al.}}}{{Lepetit, P{\'e}lissier,\ and\ Malrieu}}} -\bibcite{JensenBook}{{47}{2017}{{Jensen}}{{}}} -\bibcite{Nesbet_1955}{{48}{1955}{{Nesbet\ and\ Hartree}}{{}}} -\bibcite{Epstein_1926}{{49}{1926}{{Epstein}}{{}}} -\bibcite{Seidl_2018}{{50}{2018}{{Seidl\ \emph {et~al.}}}{{Seidl, Giarrusso, Vuckovic, Fabiano,\ and\ Gori-Giorgi}}} -\bibcite{Cremer_1996}{{51}{1996}{{Cremer\ and\ He}}{{}}} -\bibcite{Christiansen_1996}{{52}{1996}{{Christiansen\ \emph {et~al.}}}{{Christiansen, Olsen, J{\o }rgensen, Koch,\ and\ Malmqvist}}} -\bibcite{Sergeev_2005}{{53}{2005}{{Sergeev\ \emph {et~al.}}}{{Sergeev, Goodson, Wheeler,\ and\ Allen}}} -\bibcite{Sergeev_2006}{{54}{2006}{{Sergeev\ and\ Goodson}}{{}}} -\bibcite{Stillinger_2000}{{55}{2000}{{Stillinger}}{{}}} -\bibcite{Baker_1971}{{56}{1971}{{{Baker}}}{{}}} -\bibcite{Goodson_2004}{{57}{2004}{{Goodson\ and\ Sergeev}}{{}}} -\bibcite{Heiss_2002}{{58}{2002}{{Heiss\ and\ M{\"u}ller}}{{}}} +\bibcite{Leininger_2000}{{47}{2000}{{Leininger\ \emph {et~al.}}}{{Leininger, Allen, Schaefer,\ and\ Sherrill}}} +\bibcite{JensenBook}{{48}{2017}{{Jensen}}{{}}} +\bibcite{Nesbet_1955}{{49}{1955}{{Nesbet\ and\ Hartree}}{{}}} +\bibcite{Epstein_1926}{{50}{1926}{{Epstein}}{{}}} +\bibcite{Seidl_2018}{{51}{2018}{{Seidl\ \emph {et~al.}}}{{Seidl, Giarrusso, Vuckovic, Fabiano,\ and\ Gori-Giorgi}}} \@writefile{toc}{\contentsline {section}{\numberline {}Acknowledgments}{7}{section*.16}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {}References}{7}{section*.17}\protected@file@percent } -\bibcite{Cejnar_2005}{{59}{2005}{{Cejnar\ \emph {et~al.}}}{{Cejnar, Heinze,\ and\ Dobe{\v s}}}} -\bibcite{Cejnar_2007}{{60}{2007}{{Cejnar\ \emph {et~al.}}}{{Cejnar, Heinze,\ and\ Macek}}} -\bibcite{Cejnar_2009}{{61}{2009}{{Cejnar\ and\ Jolie}}{{}}} -\bibcite{Borisov_2015}{{62}{2015}{{Borisov\ \emph {et~al.}}}{{Borisov, Ru{\v z}i{\v c}ka,\ and\ Znojil}}} -\bibcite{Sindelka_2017}{{63}{2017}{{{\v S}indelka\ \emph {et~al.}}}{{{\v S}indelka, Santos,\ and\ Moiseyev}}} -\bibcite{Sachdev_2011}{{64}{2011}{{Sachdev}}{{}}} -\bibcite{Cejnar_2015}{{65}{2015}{{Cejnar\ \emph {et~al.}}}{{Cejnar, Str{\'a}nsk{\'y},\ and\ Kloc}}} -\bibcite{Cejnar_2016}{{66}{2016}{{Cejnar\ and\ Str{\'a}nsk{\'y}}}{{}}} -\bibcite{Caprio_2008}{{67}{2008}{{Caprio\ \emph {et~al.}}}{{Caprio, Cejnar,\ and\ Iachello}}} -\bibcite{Macek_2019}{{68}{2019}{{Macek\ \emph {et~al.}}}{{Macek, Str{\'a}nsk{\'y}, Leviatan,\ and\ Cejnar}}} -\bibcite{Stransky_2018}{{69}{2018}{{Str{\'a}nsk{\'y}\ \emph {et~al.}}}{{Str{\'a}nsk{\'y}, Dvo{\v r}{\'a}k,\ and\ Cejnar}}} -\bibcite{Lipkin_1965}{{70}{1965}{{Lipkin\ \emph {et~al.}}}{{Lipkin, Meshkov,\ and\ Glick}}} +\bibcite{Cremer_1996}{{52}{1996}{{Cremer\ and\ He}}{{}}} +\bibcite{Christiansen_1996}{{53}{1996}{{Christiansen\ \emph {et~al.}}}{{Christiansen, Olsen, J{\o }rgensen, Koch,\ and\ Malmqvist}}} +\bibcite{Sergeev_2005}{{54}{2005}{{Sergeev\ \emph {et~al.}}}{{Sergeev, Goodson, Wheeler,\ and\ Allen}}} +\bibcite{Sergeev_2006}{{55}{2006}{{Sergeev\ and\ Goodson}}{{}}} +\bibcite{Stillinger_2000}{{56}{2000}{{Stillinger}}{{}}} +\bibcite{Baker_1971}{{57}{1971}{{{Baker}}}{{}}} +\bibcite{Goodson_2004}{{58}{2004}{{Goodson\ and\ Sergeev}}{{}}} +\bibcite{Heiss_2002}{{59}{2002}{{Heiss\ and\ M{\"u}ller}}{{}}} +\bibcite{Cejnar_2005}{{60}{2005}{{Cejnar\ \emph {et~al.}}}{{Cejnar, Heinze,\ and\ Dobe{\v s}}}} +\bibcite{Cejnar_2007}{{61}{2007}{{Cejnar\ \emph {et~al.}}}{{Cejnar, Heinze,\ and\ Macek}}} +\bibcite{Cejnar_2009}{{62}{2009}{{Cejnar\ and\ Jolie}}{{}}} +\bibcite{Borisov_2015}{{63}{2015}{{Borisov\ \emph {et~al.}}}{{Borisov, Ru{\v z}i{\v c}ka,\ and\ Znojil}}} +\bibcite{Sindelka_2017}{{64}{2017}{{{\v S}indelka\ \emph {et~al.}}}{{{\v S}indelka, Santos,\ and\ Moiseyev}}} +\bibcite{Sachdev_2011}{{65}{2011}{{Sachdev}}{{}}} +\bibcite{Cejnar_2015}{{66}{2015}{{Cejnar\ \emph {et~al.}}}{{Cejnar, Str{\'a}nsk{\'y},\ and\ Kloc}}} +\bibcite{Cejnar_2016}{{67}{2016}{{Cejnar\ and\ Str{\'a}nsk{\'y}}}{{}}} +\bibcite{Caprio_2008}{{68}{2008}{{Caprio\ \emph {et~al.}}}{{Caprio, Cejnar,\ and\ Iachello}}} +\bibcite{Macek_2019}{{69}{2019}{{Macek\ \emph {et~al.}}}{{Macek, Str{\'a}nsk{\'y}, Leviatan,\ and\ Cejnar}}} +\bibcite{Stransky_2018}{{70}{2018}{{Str{\'a}nsk{\'y}\ \emph {et~al.}}}{{Str{\'a}nsk{\'y}, Dvo{\v r}{\'a}k,\ and\ Cejnar}}} +\bibcite{Lipkin_1965}{{71}{1965}{{Lipkin\ \emph {et~al.}}}{{Lipkin, Meshkov,\ and\ Glick}}} \bibstyle{apsrev4-1} \citation{REVTEX41Control} \citation{apsrev41Control} -\newlabel{LastBibItem}{{70}{8}{}{section*.17}{}} +\newlabel{LastBibItem}{{71}{8}{}{section*.17}{}} \newlabel{LastPage}{{}{8}{}{}{}} diff --git a/Manuscript/EPAWTFT.bbl b/Manuscript/EPAWTFT.bbl index ca104da..9adf5bc 100644 --- a/Manuscript/EPAWTFT.bbl +++ b/Manuscript/EPAWTFT.bbl @@ -6,7 +6,7 @@ %Control: page (0) single %Control: year (1) truncated %Control: production of eprint (0) enabled -\begin{thebibliography}{70}% +\begin{thebibliography}{71}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} @@ -521,6 +521,19 @@ {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {89}},\ \bibinfo {pages} {998} (\bibinfo {year} {1988})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Leininger}\ \emph {et~al.}(2000)\citenamefont + {Leininger}, \citenamefont {Allen}, \citenamefont {Schaefer},\ and\ + \citenamefont {Sherrill}}]{Leininger_2000}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~L.}\ \bibnamefont + {Leininger}}, \bibinfo {author} {\bibfnamefont {W.~D.}\ \bibnamefont + {Allen}}, \bibinfo {author} {\bibfnamefont {H.~F.}\ \bibnamefont {Schaefer}}, + \ and\ \bibinfo {author} {\bibfnamefont {C.~D.}\ \bibnamefont {Sherrill}},\ + }\href {\doibase 10.1063/1.481764} {\bibfield {journal} {\bibinfo {journal} + {The Journal of Chemical Physics}\ }\textbf {\bibinfo {volume} {112}},\ + \bibinfo {pages} {9213} (\bibinfo {year} {2000})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1063/1.481764} + {https://doi.org/10.1063/1.481764} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Jensen}(2017)}]{JensenBook}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont diff --git a/Manuscript/EPAWTFT.bib b/Manuscript/EPAWTFT.bib index f9cbd97..1cf6784 100644 --- a/Manuscript/EPAWTFT.bib +++ b/Manuscript/EPAWTFT.bib @@ -1,13 +1,28 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-11-12 10:02:40 +0100 +%% Created for Pierre-Francois Loos at 2020-11-12 14:54:07 +0100 %% Saved with string encoding Unicode (UTF-8) +@article{Leininger_2000, + author = {Leininger,Matthew L. and Allen,Wesley D. and Schaefer,Henry F. and Sherrill,C. David}, + date-added = {2020-11-12 14:50:57 +0100}, + date-modified = {2020-11-12 14:51:07 +0100}, + doi = {10.1063/1.481764}, + eprint = {https://doi.org/10.1063/1.481764}, + journal = {The Journal of Chemical Physics}, + number = {21}, + pages = {9213-9222}, + title = {Is Mo/ller--Plesset perturbation theory a convergent ab initio method?}, + url = {https://doi.org/10.1063/1.481764}, + volume = {112}, + year = {2000}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.481764}} + @article{Nesbet_1955, abstract = { A systematic method is developed for estimating or calculating corrections for configuration interaction in atomic, molecular and nuclear wave-function calculations. Solutions of the Hartree-Fock equations for a single Slater determinant or approximate Hartree-Fock solutions obtained by Roothaan's iterative procedure have special properties which are used to simplify the matrix of the many-particle Hamiltonian. A restricted self-consistent field method is proposed for treating states of low symmetry. This method avoids the off-diagonal Lagrange multipliers encountered in previous methods and is adapted to configuration interaction calculations. }, author = {Nesbet, R. K. and Hartree, Douglas Rayner}, @@ -240,17 +255,6 @@ year = {2019}, Bdsk-Url-1 = {https://doi.org/10.1063/1.5110554}} -@article{Leininger_2000, - author = {Leininger, Matthew L. and Allen, Wesley D. and Schaefer, Henry F. and Sherrill, C. David}, - date = {2000-05-17}, - doi = {10.1063/1.481764}, - journal = {J. Chem. Phys.}, - number = {21}, - pages = {9213--9222}, - title = {Is {M{\o}ller-Plesset} perturbation theory a convergent ab initio method?}, - volume = {112}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.481764}} - @article{Moller_1934, author = {M{\o}ller, Chr. and Plesset, M. S.}, date = {1934-10-01}, diff --git a/Manuscript/EPAWTFT.blg b/Manuscript/EPAWTFT.blg index 727fbc2..5362b8d 100644 --- a/Manuscript/EPAWTFT.blg +++ b/Manuscript/EPAWTFT.blg @@ -23,45 +23,45 @@ Control: production of article title (-1) disabled Control: page (0) single Control: year (1) truncated Control: production of eprint (0) enabled -You've used 72 entries, +You've used 73 entries, 5847 wiz_defined-function locations, - 2133 strings with 28024 characters, -and the built_in function-call counts, 70033 in all, are: -= -- 4473 -> -- 1850 -< -- 481 -+ -- 585 -- -- 453 -* -- 10828 -:= -- 6899 -add.period$ -- 70 -call.type$ -- 72 -change.case$ -- 282 -chr.to.int$ -- 78 -cite$ -- 72 -duplicate$ -- 6517 -empty$ -- 5216 -format.name$ -- 1078 -if$ -- 13975 + 2142 strings with 28296 characters, +and the built_in function-call counts, 71165 in all, are: += -- 4543 +> -- 1883 +< -- 488 ++ -- 595 +- -- 461 +* -- 11019 +:= -- 7005 +add.period$ -- 71 +call.type$ -- 73 +change.case$ -- 286 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Please write \usepackage[version=4]{mhchem} +(mhchem) in your preamble (or any lower number for +(mhchem) compatibility mode), because you might get slightly +(mhchem) different output with the same input in future versions +. + +) (/usr/local/texlive/2020/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2018/08/11 v1.3c Input encoding file +\inpenc@prehook=\toks28 +\inpenc@posthook=\toks29 +) +(/usr/local/texlive/2020/texmf-dist/tex/latex/base/fontenc.sty +Package: fontenc 2020/02/11 v2.0o Standard LaTeX package ) (/usr/local/texlive/2020/texmf-dist/tex/latex/txfonts/txfonts.sty Package: txfonts 2008/01/22 v3.2.1 @@ -319,10 +445,14 @@ LaTeX Font Info: Overwriting symbol font `operators' in version `bold' (Font) OT1/cmr/bx/n --> OT1/txr/m/n on input line 21. LaTeX Font Info: Overwriting symbol font `operators' in version `bold' (Font) OT1/txr/m/n --> OT1/txr/bx/n on input line 22. -\symitalic=\mathgroup9 +\symitalic=\mathgroup6 LaTeX Font Info: Overwriting symbol font `italic' in version `bold' (Font) OT1/txr/m/it --> OT1/txr/bx/it on input line 26. LaTeX Font Info: Redeclaring math alphabet \mathbf on input line 29. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `normal' +(Font) OT1/cmr/bx/n --> OT1/txr/bx/n on input line 29. +LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `bold' +(Font) OT1/cmr/bx/n --> OT1/txr/bx/n on input line 29. LaTeX Font Info: Redeclaring math alphabet \mathit on input line 30. LaTeX Font Info: Overwriting math alphabet `\mathit' in version `normal' (Font) OT1/cmr/m/it --> OT1/txr/m/it on input line 30. @@ -351,7 +481,7 @@ LaTeX Font Info: Overwriting symbol font `letters' in version `bold' (Font) OML/cmm/b/it --> OML/txmi/m/it on input line 58. LaTeX Font Info: Overwriting symbol font `letters' in version `bold' (Font) OML/txmi/m/it --> OML/txmi/bx/it on input line 59. -\symlettersA=\mathgroup10 +\symlettersA=\mathgroup7 LaTeX Font Info: Overwriting symbol font `lettersA' in version `bold' (Font) U/txmia/m/it --> U/txmia/bx/it on input line 67. LaTeX Font Info: Redeclaring math alphabet \mathfrak on input line 70. @@ -376,7 +506,7 @@ LaTeX Font Info: Overwriting symbol font `AMSb' in version `bold' (Font) U/msb/m/n --> U/txsyb/m/n on input line 102. LaTeX Font Info: Overwriting symbol font `AMSb' in version `bold' (Font) U/txsyb/m/n --> U/txsyb/bx/n on input line 103. -\symsymbolsC=\mathgroup11 +\symsymbolsC=\mathgroup8 LaTeX Font Info: Overwriting symbol font `symbolsC' in version `bold' (Font) U/txsyc/m/n --> U/txsyc/bx/n on input line 113. LaTeX Font Info: Redeclaring symbol font `largesymbols' on input line 120. @@ -386,85 +516,23 @@ LaTeX Font Info: Overwriting symbol font `largesymbols' in version `bold' (Font) OMX/cmex/m/n --> OMX/txex/m/n on input line 120. LaTeX Font Info: Overwriting symbol font `largesymbols' in version `bold' (Font) OMX/txex/m/n --> OMX/txex/bx/n on input line 121. -\symlargesymbolsA=\mathgroup12 +\symlargesymbolsA=\mathgroup9 LaTeX Font Info: Overwriting symbol font `largesymbolsA' in version `bold' (Font) U/txexa/m/n --> U/txexa/bx/n on input line 129. LaTeX Font Info: Redeclaring math symbol \mathsterling on input line 164. 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Package hyperref Info: Option `breaklinks' set `true' on input line 4338. Package hyperref Info: Hyper figures OFF on input line 4464. @@ -533,9 +601,9 @@ Package hyperref Info: Backreferencing OFF on input line 4484. Package hyperref Info: Implicit mode ON; LaTeX internals redefined. Package hyperref Info: Bookmarks ON on input line 4717. 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PDF statistics: - 457 PDF objects out of 1000 (max. 8388607) - 430 compressed objects within 5 object streams - 120 named destinations out of 1000 (max. 500000) - 47241 words of extra memory for PDF output out of 51595 (max. 10000000) + 462 PDF objects out of 1000 (max. 8388607) + 435 compressed objects within 5 object streams + 122 named destinations out of 1000 (max. 500000) + 52873 words of extra memory for PDF output out of 61914 (max. 10000000) diff --git a/Manuscript/EPAWTFT.pdf b/Manuscript/EPAWTFT.pdf index 4186eb2..fbc3c1c 100644 Binary files a/Manuscript/EPAWTFT.pdf and b/Manuscript/EPAWTFT.pdf differ diff --git a/Manuscript/EPAWTFT.synctex.gz b/Manuscript/EPAWTFT.synctex.gz index 1eca268..3a81c6b 100644 Binary files a/Manuscript/EPAWTFT.synctex.gz and b/Manuscript/EPAWTFT.synctex.gz differ diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index eb1b5f2..ccf5749 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -1,6 +1,5 @@ \documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1} -\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,txfonts,mhchem} - +\usepackage{graphicx,tabularx,array,booktabs,longtable,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,siunitx,mhchem} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{txfonts} @@ -27,7 +26,6 @@ \newcommand{\fnt}{\footnotetext} \newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\mr}{\multirow} -\newcommand{\SI}{\textcolor{blue}{supporting information}} % operators \newcommand{\bH}{\mathbf{H}} @@ -302,10 +300,10 @@ with $\mel{pq}{}{rs} = \braket{pq}{rs} - \braket{pq}{sr}$, and where \begin{equation} \braket{pq}{rs} = \iint \dd\vb{x}_1\dd\vb{x}_2\frac{\phi_p(\vb{x}_1)\phi_q(\vb{x}_2)\phi_r(\vb{x}_1)\phi_s(\vb{x}_2)}{\abs{\vb{r}_1 - \vb{r}_2}} \end{equation} -are two-electron integral in the spin-orbital basis. \cite{Gill_1994} +are two-electron integrals in the spin-orbital basis. \cite{Gill_1994} As mentioned earlier, there is, \textit{a priori}, no guarantee that the MP$m$ series converges to the exact energy when $m \to \infty$. -In fact, it is known that when the HF wave function is a poor approximation to the exact wave function, for example in multi-reference systems, the MP method yields deceptive results. \cite{Gill_1986, Gill_1988, Handy_1985, Lepetit_1988} +In fact, it is known that when the HF wave function is a poor approximation to the exact wave function, for example in multi-reference systems, the MP method yields deceptive results. \cite{Gill_1986,Gill_1988,Handy_1985,Lepetit_1988,Leininger_2000} A convenient way to investigate the convergence properties of the MP series is to analytically continue the coupling parameter $\lambda$ into the complex variable. By doing so, the Hamiltonian and the energy become complex-valued functions of $\lambda$, and the energy becomes a multivalued function on $K$ Riemann sheets (where $K$ is the number of basis functions). As mentioned above, by searching the singularities of the function $E(\lambda)$, one can get information on the convergence properties of the MP series. @@ -364,11 +362,16 @@ The discovery of this divergent behavior is worrying as in order to get meaningf A singularity in the unit circle is designated as an intruder state, more precisely as a front-door (respectively back-door) intruder state if the real part of the singularity is positive (respectively negative). Their method consists in performing a scan of the real axis to detect the avoided crossing responsible for the pair of dominant singularities. Then, by modeling this avoided crossing via a two-state Hamiltonian one can get an approximation of the dominant conjugate pair of singularities by finding the EPs of the following $2\times2$ matrix \begin{equation} -\underbrace{\mqty(\alpha & \delta \\ \delta & \beta)}_{\bH} = \underbrace{\mqty(\alpha + \alpha_s & 0 \\ 0 & \beta + \beta_s )}_{\bH^{(0)}} + \underbrace{\mqty(- \alpha_s & \delta \\ \delta & - \beta_s)}_{\bV}, + \underbrace{\mqty(\alpha & \delta \\ \delta & \beta)}_{\bH} = \underbrace{\mqty(\alpha & 0 \\ 0 & \beta + \gamma )}_{\bH^{(0)}} + \underbrace{\mqty( 0 & \delta \\ \delta & - \gamma)}_{\bV}, \end{equation} where the diagonal matrix is the unperturbed Hamiltonian matrix $\bH^{(0)}$ and the second matrix in the right-hand-side $\bV$ is the perturbation. -In Ref.~\onlinecite{Olsen_2019}, the simple $2 \times 2$ model proposed by Olsen \textit{et al.} is generalized to a non-symmetric (i.e, non-Hermitian) Hamiltonian and various choice of perturbation (not only the MP partioning). +In Ref.~\onlinecite{Olsen_2019}, the simple two-state model proposed by Olsen \textit{et al.} is generalized to a non-symmetric Hamiltonian +\begin{equation} + \underbrace{\mqty(\alpha & \delta_1 \\ \delta_2 & \beta)}_{\bH} = \underbrace{\mqty(\alpha & 0 \\ 0 & \beta + \gamma )}_{\bH^{(0)}} + \underbrace{\mqty( 0 & \delta_2 \\ \delta_1 & - \gamma)}_{\bV}. +\end{equation} +allowing an analysis of various choice of perturbation (not only the MP partioning) such as coupled cluster perturbation expansions and other non-Hermitian perturbation methods. +It is worth noting that only cases where $\text{sgn}(\delta_1) = - \text{sgn}(\delta_2)$ leads to new forms of perturbation expansions. Interestingly, they showed that the convergence pattern of a given perturbation method can be characterized by its archetype which defines the overall ``shape'' of the energy convergence and can be subdivided in five classes for Hermitian Hamiltonians (zigzag, interspersed zigzag, triadic, ripples, and geometric), while two additional archetypes (zigzag-geometric and convex-geometric) are observed in non-Hermitian Hamiltonians. Other features characterizing the convergence behavior of a perturbation method are its rate of convergence, its length of recurring period, and its sign pattern. diff --git a/References/Olsen_2019.pdf b/References/Olsen_2019.pdf index 5d85df3..82ee022 100644 Binary files a/References/Olsen_2019.pdf and b/References/Olsen_2019.pdf differ