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\newlabel{fig:C2}{{4}{6}{Error in vertical excitation energies (in eV) for two doubly-excited states of the carbon dimer for various basis sets and methods. The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal } or 0.043 eV). See Table \ref {tab:Mol} for raw data}{figure.4}{}}
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\bibdata{Ex-srDFTNotes,Ex-srDFT}
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\bibcite{Kut-TCA-85}{{1}{1985}{{Kutzelnigg}}{{}}}
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\bibcite{KutKlo-JCP-91}{{2}{1991}{{Kutzelnigg\ and\ Klopper}}{{}}}
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\bibcite{NogKut-JCP-94}{{3}{1994}{{Noga\ and\ Kutzelnigg}}{{}}}
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||||
\bibcite{HatKloKohTew-CR-12}{{6}{2012}{{Hattig\ \emph {et~al.}}}{{Hattig, Klopper, Kohn,\ and\ Tew}}}
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||||
\bibcite{KonBisVal-CR-12}{{7}{2012}{{Kong\ \emph {et~al.}}}{{Kong, Bischo,\ and\ Valeev}}}
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||||
\bibcite{GruHirOhnTen-JCP-17}{{8}{2017}{{Gr\"uneis\ \emph {et~al.}}}{{Gr\"uneis, Hirata, Ohnishi,\ and\ Ten-no}}}
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||||
\bibcite{MaWer-WIREs-18}{{9}{2018}{{Ma\ and\ Werner}}{{}}}
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\bibcite{TewKloNeiHat-PCCP-07}{{10}{2007}{{Tew\ \emph {et~al.}}}{{Tew, Klopper, Neiss,\ and\ Hattig}}}
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||||
\bibcite{FliHatKlo-JCP-06}{{11}{2006}{{Fliegl\ \emph {et~al.}}}{{Fliegl, Hattig,\ and\ Klopper}}}
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||||
\bibcite{NeiHatKlo-JCP-06}{{12}{2006}{{Neiss\ \emph {et~al.}}}{{Neiss, Hattig,\ and\ Klopper}}}
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||||
\bibcite{HanKoh-JCP-09}{{13}{2009}{{Hanauer\ and\ Kohn}}{{}}}
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||||
\bibcite{Koh-JCP-09}{{14}{2009}{{Kohn}}{{}}}
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||||
\bibcite{ShiWer-JCP-10}{{15}{2010}{{Shiozaki\ and\ Werner}}{{}}}
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\bibcite{ShiKniWer-JCP-11}{{16}{2011}{{Shiozaki\ \emph {et~al.}}}{{Shiozaki, Knizia,\ and\ Werner}}}
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\bibcite{ShiWer-JCP-11}{{17}{2011}{{Shiozaki\ and\ Werner}}{{}}}
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||||
\bibcite{GinPraFerAssSavTou-JCP-18}{{20}{2018}{{Giner\ \emph {et~al.}}}{{Giner, Pradines, Fert\'e, Assaraf, Savin,\ and\ Toulouse}}}
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||||
\bibcite{LooPraSceTouGin-JPCL-19}{{21}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Pradines, Scemama, Toulouse,\ and\ Giner}}}
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||||
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces $\mu _{}^{\mathcal {B}}$ (left), $n_{}^{\mathcal {B}} \mathaccentV {bar}916{\varepsilon }_{\text {c,md}}^{\text {sr},\text {PBEot}}$ (center) and $n_{2}^{\mathcal {B}}$ (right) along the molecular axis ($z$) for the ground state (black curve) and second doubly-excited state (red curve) of \ce {C2} for various basis sets $\mathcal {B}$. The two electronic states are both of $\Sigma _g^+$ symmetry. The carbon nuclei are located at $z= \pm 1.180$ bohr and are represented by the thin black lines.}}{7}{figure.5}}
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\newlabel{fig:C2_mu}{{5}{7}{$\rsmu {}{\Bas }$ (left), $\n {}{\Bas } \be {\text {c,md}}{\sr ,\PBEot }$ (center) and $\n {2}{\Bas }$ (right) along the molecular axis ($z$) for the ground state (black curve) and second doubly-excited state (red curve) of \ce {C2} for various basis sets $\Bas $. The two electronic states are both of $\Sigma _g^+$ symmetry. The carbon nuclei are located at $z= \pm 1.180$ bohr and are represented by the thin black lines}{figure.5}{}}
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||||
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Error in vertical excitation energies (in eV) of ethylene for various basis sets and methods. The green region corresponds to chemical accuracy (i.e., error below 1 {kcal/mol} or 0.043 eV). See Table \ref {tab:Mol} for raw data.}}{7}{figure.6}}
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||||
\newlabel{fig:C2H4}{{6}{7}{Error in vertical excitation energies (in eV) of ethylene for various basis sets and methods. The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal } or 0.043 eV). See Table \ref {tab:Mol} for raw data}{figure.6}{}}
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||||
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\bibcite{Sav-INC-96}{{22}{1996}{{Savin}}{{}}}
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\bibcite{LeiStoWerSav-CPL-97}{{23}{1997}{{Leininger\ \emph {et~al.}}}{{Leininger, Stoll, Werner,\ and\ Savin}}}
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||||
\bibcite{TouColSav-PRA-04}{{24}{2004{}}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Colonna,\ and\ Savin}}}
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\bibcite{TouSavFla-IJQC-04}{{25}{2004{}}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Savin,\ and\ Flad}}}
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\bibcite{AngGerSavTou-PRA-05}{{26}{2005}{{\'Angy\'an\ \emph {et~al.}}}{{\'Angy\'an, Gerber, Savin,\ and\ Toulouse}}}
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\bibcite{GolWerSto-PCCP-05}{{27}{2005}{{Goll\ \emph {et~al.}}}{{Goll, Werner,\ and\ Stoll}}}
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\bibcite{PazMorGorBac-PRB-06}{{28}{2006}{{Paziani\ \emph {et~al.}}}{{Paziani, Moroni, Gori-Giorgi,\ and\ Bachelet}}}
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\bibcite{FroTouJen-JCP-07}{{29}{2007}{{Fromager\ \emph {et~al.}}}{{Fromager, Toulouse,\ and\ Jensen}}}
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||||
\bibcite{TouGerJanSavAng-PRL-09}{{30}{2009}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Gerber, Jansen, Savin,\ and\ \'Angy\'an}}}
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||||
\bibcite{JanHenScu-JCP-09}{{31}{2009}{{Janesko\ \emph {et~al.}}}{{Janesko, Henderson,\ and\ Scuseria}}}
|
||||
\bibcite{FroCimJen-PRA-10}{{32}{2010}{{Fromager\ \emph {et~al.}}}{{Fromager, Cimiraglia,\ and\ Jensen}}}
|
||||
\bibcite{TouZhuSavJanAng-JCP-11}{{33}{2011}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Zhu, Savin, Jansen,\ and\ \'Angy\'an}}}
|
||||
\bibcite{MusReiAngTou-JCP-15}{{34}{2015}{{Mussard\ \emph {et~al.}}}{{Mussard, Reinhardt, \'Angy\'an,\ and\ Toulouse}}}
|
||||
\bibcite{HedKneKieJenRei-JCP-15}{{35}{2015}{{Hedeg{\r a}rd\ \emph {et~al.}}}{{Hedeg{\r a}rd, Knecht, Kielberg, Jensen,\ and\ Reiher}}}
|
||||
\bibcite{HedTouJen-JCP-18}{{36}{2018}{{Hedeg{\r a}rd\ \emph {et~al.}}}{{Hedeg{\r a}rd, Toulouse,\ and\ Jensen}}}
|
||||
\bibcite{FerGinTou-JCP-19}{{37}{2019}{{Fert{\'e}\ \emph {et~al.}}}{{Fert{\'e}, Giner,\ and\ Toulouse}}}
|
||||
\bibcite{ParYan-BOOK-89}{{38}{1989}{{Parr\ and\ Yang}}{{}}}
|
||||
\bibcite{SzaOst-BOOK-96}{{39}{1996}{{Szabo\ and\ Ostlund}}{{}}}
|
||||
\bibcite{Kat-CPAM-57}{{40}{1957}{{Kato}}{{}}}
|
||||
\bibcite{FraMusLupTou-JCP-15}{{41}{2015}{{Franck\ \emph {et~al.}}}{{Franck, Mussard, Luppi,\ and\ Toulouse}}}
|
||||
\bibcite{MorKut-JPC-93}{{42}{1993}{{{Morgan III}\ and\ Kutzelnigg}}{{}}}
|
||||
\bibcite{KutMor-ZPD-96}{{43}{1996}{{Kutzelnigg\ and\ {Morgan III}}}{{}}}
|
||||
\bibcite{Tew-JCP-08}{{44}{2008}{{Tew}}{{}}}
|
||||
\bibcite{LooGil-MP-10}{{45}{2010}{{Loos\ and\ Gill}}{{}}}
|
||||
\bibcite{LooBloGil-JCP-2015}{{46}{2015}{{Loos\ \emph {et~al.}}}{{Loos, Bloomfield,\ and\ Gill}}}
|
||||
\bibcite{TouGorSav-TCA-05}{{47}{2005}{{Toulouse\ \emph {et~al.}}}{{Toulouse, Gori-Giorgi,\ and\ Savin}}}
|
||||
\bibcite{HohKoh-PR-64}{{48}{1964}{{Hohenberg\ and\ Kohn}}{{}}}
|
||||
\bibcite{KohSha-PR-65}{{49}{1965}{{Kohn\ and\ Sham}}{{}}}
|
||||
\bibcite{LooGil-WIRES-16}{{50}{2016}{{Loos\ and\ Gill}}{{}}}
|
||||
\bibcite{PerBurErn-PRL-96}{{51}{1996}{{Perdew\ \emph {et~al.}}}{{Perdew, Burke,\ and\ Ernzerhof}}}
|
||||
\bibcite{GorSav-PRA-06}{{52}{2006}{{Gori-Giorgi\ and\ Savin}}{{}}}
|
||||
\bibcite{HurMalRan-JCP-73}{{53}{1973}{{Huron\ \emph {et~al.}}}{{Huron, Malrieu,\ and\ Rancurel}}}
|
||||
\bibcite{GinSceCaf-CJC-13}{{54}{2013}{{Giner\ \emph {et~al.}}}{{Giner, Scemama,\ and\ Caffarel}}}
|
||||
\bibcite{GinSceCaf-JCP-15}{{55}{2015}{{Giner\ \emph {et~al.}}}{{Giner, Scemama,\ and\ Caffarel}}}
|
||||
\bibcite{SceCafBenJacLoo-RC-19}{{56}{2019}{{Scemama\ \emph {et~al.}}}{{Scemama, Caffarel, Benali, Jacquemin,\ and\ Loos}}}
|
||||
\bibcite{HolUmrSha-JCP-17}{{57}{2017}{{Holmes\ \emph {et~al.}}}{{Holmes, Umrigar,\ and\ Sharma}}}
|
||||
\bibcite{QP2}{{58}{2019}{{Garniron\ \emph {et~al.}}}{{Garniron, Gasperich, Applencourt, Benali, Fert{\'e}, Paquier, Pradines, Assaraf, Reinhardt, Toulouse, Barbaresco, Renon, David, Malrieu, V{\'e}ril, Caffarel, Loos, Giner,\ and\ Scemama}}}
|
||||
\bibcite{SceGarCafLoo-JCTC-18}{{59}{2018{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Garniron, Caffarel,\ and\ Loos}}}
|
||||
\bibcite{LooSceBloGarCafJac-JCTC-18}{{60}{2018{}}{{Loos\ \emph {et~al.}}}{{Loos, Scemama, Blondel, Garniron, Caffarel,\ and\ Jacquemin}}}
|
||||
\bibcite{SceBenJacCafLoo-JCP-18}{{61}{2018{}}{{Scemama\ \emph {et~al.}}}{{Scemama, Benali, Jacquemin, Caffarel,\ and\ Loos}}}
|
||||
\bibcite{LooBogSceCafJac-JCTC-19}{{62}{2019{}}{{Loos\ \emph {et~al.}}}{{Loos, Boggio-Pasqua, Scemama, Caffarel,\ and\ Jacquemin}}}
|
||||
\bibcite{DasHer-JCC-17}{{63}{2017}{{Dasgupta\ and\ Herbert}}{{}}}
|
||||
\bibcite{SheLeiVanSch-JCP-98}{{64}{1998}{{Sherrill\ \emph {et~al.}}}{{Sherrill, Leininger, Huis,\ and\ Schaefer}}}
|
||||
\bibcite{Sch-Science-86}{{65}{1986}{{{H. F. Schaeffer III}}}{{}}}
|
||||
\bibcite{BauTay-JCP-86}{{66}{1986}{{Bauschlicher\ and\ Taylor}}{{}}}
|
||||
\bibcite{JenBun-JCP-88}{{67}{1988}{{Jensen\ and\ Bunker}}{{}}}
|
||||
\bibcite{SheVanYamSch-JMS-97}{{68}{1997}{{Sherrill\ \emph {et~al.}}}{{Sherrill, {Van Huis}, Yamaguchi,\ and\ Schaefer}}}
|
||||
\bibcite{AbrShe-JCP-04}{{69}{2004}{{Abrams\ and\ Sherrill}}{{}}}
|
||||
\bibcite{AbrShe-CPL-05}{{70}{2005}{{Abrams\ and\ Sherrill}}{{}}}
|
||||
\bibcite{ZimTouZhaMusUmr-JCP-09}{{71}{2009}{{Zimmerman\ \emph {et~al.}}}{{Zimmerman, Toulouse, Zhang, Musgrave,\ and\ Umrigar}}}
|
||||
\bibcite{GouPieWlo-MP-10}{{72}{2010}{{Gour\ \emph {et~al.}}}{{Gour, Piecuch,\ and\ Wloch}}}
|
||||
\bibcite{ChiHolAdaOttUmrShaZim-JPCA-18}{{73}{2018}{{Chien\ \emph {et~al.}}}{{Chien, Holmes, Otten, Umrigar, Sharma,\ and\ Zimmerman}}}
|
||||
\bibcite{HelJorOls-BOOK-02}{{74}{2002}{{Helgaker\ \emph {et~al.}}}{{Helgaker, J{\o }rgensen,\ and\ Olsen}}}
|
||||
\bibcite{LooGalJac-JPCL-18}{{75}{2018{}}{{Loos\ \emph {et~al.}}}{{Loos, Galland,\ and\ Jacquemin}}}
|
||||
\bibcite{LooJac-JCTC-19}{{76}{2019}{{Loos\ and\ Jacquemin}}{{}}}
|
||||
\bibcite{CaiTozRei-JCP-00}{{77}{2000}{{Cai\ \emph {et~al.}}}{{Cai, Tozer,\ and\ Reimers}}}
|
||||
\bibcite{RubSerMer-JCP-08}{{78}{2008}{{Rubio\ \emph {et~al.}}}{{Rubio, Serrano-Andr{\'e}s,\ and\ Merch{\'a}n}}}
|
||||
\bibcite{LiPal-JCP-11}{{79}{2011}{{Li\ and\ Paldus}}{{}}}
|
||||
\bibcite{SchGoe-JCTC-17}{{80}{2017}{{Schwabe\ and\ Goerigk}}{{}}}
|
||||
\bibcite{BarDelPerMat-JMS-97}{{81}{1997}{{Bartlett\ \emph {et~al.}}}{{Bartlett, Bene, Perera,\ and\ Mattie}}}
|
||||
\bibcite{Var-JCP-08}{{82}{2008}{{Varandas}}{{}}}
|
||||
\bibcite{PurZhaKra-JCP-09}{{83}{2009}{{Purwanto\ \emph {et~al.}}}{{Purwanto, Zhang,\ and\ Krakauer}}}
|
||||
\bibcite{AngCimPas-MP-12}{{84}{2012}{{Angeli\ \emph {et~al.}}}{{Angeli, Cimiraglia,\ and\ Pastore}}}
|
||||
\bibcite{BooCleThoAla-JCP-11}{{85}{2011}{{Booth\ \emph {et~al.}}}{{Booth, Cleland, Thom,\ and\ Alavi}}}
|
||||
\bibcite{Sha-JCP-15}{{86}{2015}{{Sharma}}{{}}}
|
||||
\bibcite{SokCha-JCP-16}{{87}{2016}{{Sokolov\ and\ Chan}}{{}}}
|
||||
\bibcite{VarRoc-PTRSMPES-18}{{88}{2018}{{Varandas\ and\ Rocha}}{{}}}
|
||||
\bibcite{SerMarNebLinRoo-JCP-93}{{89}{1993}{{{Serrano-Andr\'es}\ \emph {et~al.}}}{{{Serrano-Andr\'es}, Merch\'an, Nebot-Gil, Lindh,\ and\ Roos}}}
|
||||
\bibcite{WatGwaBar-JCP-96}{{90}{1996}{{Watts\ \emph {et~al.}}}{{Watts, Gwaltney,\ and\ Bartlett}}}
|
||||
\bibcite{WibOliTru-JPCA-02}{{91}{2002}{{Wiberg\ \emph {et~al.}}}{{Wiberg, de~Oliveria,\ and\ Trucks}}}
|
||||
\bibcite{BarPaiLis-JCP-04}{{92}{0004}{{Barbatti\ \emph {et~al.}}}{{Barbatti, Paier,\ and\ Lischka}}}
|
||||
\bibcite{Ang-JCC-08}{{93}{2008}{{Angeli}}{{}}}
|
||||
\bibcite{SchSilSauThi-JCP-08}{{94}{2008}{{Schreiber\ \emph {et~al.}}}{{Schreiber, Silva-Junior, Sauer,\ and\ Thiel}}}
|
||||
\bibcite{SilSchSauThi-JCP-10}{{95}{2010{}}{{Silva-Junior\ \emph {et~al.}}}{{Silva-Junior, Schreiber, Sauer,\ and\ Thiel}}}
|
||||
\bibcite{SilSauSchThi-MP-10}{{96}{2010{}}{{Silva-Junior\ \emph {et~al.}}}{{Silva-Junior, Sauer, Schreiber,\ and\ Thiel}}}
|
||||
\bibcite{Ang-IJQC-10}{{97}{2010}{{Angeli}}{{}}}
|
||||
\bibcite{DadSmaBooAlaFil-JCTC-12}{{98}{2012}{{Daday\ \emph {et~al.}}}{{Daday, Smart, Booth, Alavi,\ and\ Filippi}}}
|
||||
\bibcite{FelPetDav-JCP-14}{{99}{2014}{{Feller\ \emph {et~al.}}}{{Feller, Peterson,\ and\ Davidson}}}
|
||||
\bibstyle{apsrev4-1}
|
||||
\newlabel{LastBibItem}{{99}{8}{}{section*.15}{}}
|
||||
\newlabel{LastPage}{{}{8}{}{}{}}
|
||||
1031
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||||
This is BibTeX, Version 0.99d (TeX Live 2017/Debian)
|
||||
Capacity: max_strings=100000, hash_size=100000, hash_prime=85009
|
||||
The top-level auxiliary file: Ex-srDFT.aux
|
||||
The style file: apsrev4-1.bst
|
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Reallocated singl_function (elt_size=4) to 100 items from 50.
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|
||||
Reallocated wiz_functions (elt_size=4) to 6000 items from 3000.
|
||||
Database file #1: Ex-srDFTNotes.bib
|
||||
Database file #2: Ex-srDFT.bib
|
||||
Reallocated singl_function (elt_size=4) to 100 items from 50.
|
||||
merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked
|
||||
Control: key (0)
|
||||
Control: author (72) initials jnrlst
|
||||
Control: editor formatted (1) identically to author
|
||||
Control: production of article title (-1) disabled
|
||||
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|
||||
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|
||||
Control: production of eprint (0) enabled
|
||||
You've used 99 entries,
|
||||
5847 wiz_defined-function locations,
|
||||
2322 strings with 35827 characters,
|
||||
and the built_in function-call counts, 99582 in all, are:
|
||||
= -- 6457
|
||||
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||||
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||||
+ -- 856
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:= -- 9771
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add.period$ -- 99
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call.type$ -- 99
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||||
change.case$ -- 397
|
||||
chr.to.int$ -- 94
|
||||
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|
||||
duplicate$ -- 9128
|
||||
empty$ -- 7400
|
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format.name$ -- 1572
|
||||
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|
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int.to.chr$ -- 6
|
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int.to.str$ -- 107
|
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missing$ -- 1193
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newline$ -- 349
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num.names$ -- 298
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pop$ -- 3708
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preamble$ -- 1
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quote$ -- 0
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skip$ -- 3669
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stack$ -- 0
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||||
\BOOKMARK [0][-]{section*.2}{Chemically accurate excitation energies with small basis sets}{}% 2
|
||||
\BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1
|
||||
\BOOKMARK [1][-]{section*.3}{Introduction}{section*.2}% 3
|
||||
\BOOKMARK [1][-]{section*.4}{Theory}{section*.2}% 4
|
||||
\BOOKMARK [2][-]{section*.5}{Range-separation function}{section*.4}% 5
|
||||
\BOOKMARK [2][-]{section*.6}{Short-range correlation functionals}{section*.4}% 6
|
||||
\BOOKMARK [1][-]{section*.7}{Computational details}{section*.2}% 7
|
||||
\BOOKMARK [1][-]{section*.8}{Results and Discussion}{section*.2}% 8
|
||||
\BOOKMARK [2][-]{section*.9}{Methylene}{section*.8}% 9
|
||||
\BOOKMARK [2][-]{section*.10}{Rydberg States of Water and Ammonia}{section*.8}% 10
|
||||
\BOOKMARK [2][-]{section*.11}{Doubly-Excited States of the Carbon Dimer}{section*.8}% 11
|
||||
\BOOKMARK [2][-]{section*.12}{Ethylene}{section*.8}% 12
|
||||
\BOOKMARK [1][-]{section*.13}{Conclusion}{section*.2}% 13
|
||||
\BOOKMARK [1][-]{section*.14}{Supplementary material}{section*.2}% 14
|
||||
\BOOKMARK [1][-]{section*.15}{Acknowledgments}{section*.2}% 15
|
||||
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|
||||
\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1}
|
||||
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,wrapfig}
|
||||
\usepackage{mathpazo,libertine}
|
||||
|
||||
\bibliographystyle{apsrev4-1}
|
||||
|
||||
\usepackage{hyperref}
|
||||
\hypersetup{
|
||||
colorlinks=true,
|
||||
linkcolor=blue,
|
||||
filecolor=blue,
|
||||
urlcolor=blue,
|
||||
citecolor=blue
|
||||
}
|
||||
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
||||
\definecolor{darkgreen}{HTML}{009900}
|
||||
\usepackage[normalem]{ulem}
|
||||
\newcommand{\titou}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\jt}[1]{\textcolor{purple}{#1}}
|
||||
\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}}
|
||||
\newcommand{\toto}[1]{\textcolor{brown}{#1}}
|
||||
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
|
||||
\newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}}
|
||||
\newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}}
|
||||
\newcommand{\trashAS}[1]{\textcolor{brown}{\sout{#1}}}
|
||||
\newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}}
|
||||
\newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}}
|
||||
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
|
||||
\newcommand{\AS}[1]{\toto{(\underline{\bf TOTO}: #1)}}
|
||||
|
||||
|
||||
\newcommand{\mc}{\multicolumn}
|
||||
\newcommand{\fnm}{\footnotemark}
|
||||
\newcommand{\fnt}{\footnotetext}
|
||||
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
|
||||
\newcommand{\SI}{\textcolor{blue}{supplementary material}}
|
||||
\newcommand{\QP}{\textsc{quantum package}}
|
||||
|
||||
% second quantized operators
|
||||
\newcommand{\ai}[1]{\hat{a}_{#1}}
|
||||
\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}}
|
||||
|
||||
% units
|
||||
\newcommand{\IneV}[1]{#1 eV}
|
||||
\newcommand{\InAU}[1]{#1 a.u.}
|
||||
\newcommand{\InAA}[1]{#1 \AA}
|
||||
\newcommand{\kcal}{kcal/mol}
|
||||
|
||||
% methods
|
||||
\newcommand{\D}{\text{D}}
|
||||
\newcommand{\T}{\text{T}}
|
||||
\newcommand{\Q}{\text{Q}}
|
||||
\newcommand{\X}{\text{X}}
|
||||
\newcommand{\UEG}{\text{UEG}}
|
||||
\newcommand{\HF}{\text{HF}}
|
||||
\newcommand{\ROHF}{\text{ROHF}}
|
||||
\newcommand{\LDA}{\text{LDA}}
|
||||
\newcommand{\PBE}{\text{PBE}}
|
||||
\newcommand{\PBEUEG}{\text{PBE-UEG}}
|
||||
\newcommand{\PBEot}{\text{PBEot}}
|
||||
\newcommand{\FCI}{\text{FCI}}
|
||||
\newcommand{\CBS}{\text{CBS}}
|
||||
\newcommand{\exFCI}{\text{exFCI}}
|
||||
\newcommand{\CCSDT}{\text{CCSD(T)}}
|
||||
\newcommand{\lr}{\text{lr}}
|
||||
\newcommand{\sr}{\text{sr}}
|
||||
|
||||
\newcommand{\Ne}{N}
|
||||
\newcommand{\NeUp}{\Ne^{\uparrow}}
|
||||
\newcommand{\NeDw}{\Ne^{\downarrow}}
|
||||
\newcommand{\Nb}{N_{\Bas}}
|
||||
\newcommand{\Ng}{N_\text{grid}}
|
||||
\newcommand{\nocca}{n_{\text{occ}^{\alpha}}}
|
||||
\newcommand{\noccb}{n_{\text{occ}^{\beta}}}
|
||||
|
||||
\newcommand{\n}[2]{n_{#1}^{#2}}
|
||||
\newcommand{\Ec}{E_\text{c}}
|
||||
\newcommand{\E}[2]{E_{#1}^{#2}}
|
||||
\newcommand{\DE}[2]{\Delta E_{#1}^{#2}}
|
||||
\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
|
||||
\newcommand{\DbE}[2]{\Delta \Bar{E}_{#1}^{#2}}
|
||||
\newcommand{\bEc}[1]{\Bar{E}_\text{c,md}^{#1}}
|
||||
\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
|
||||
\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
|
||||
\newcommand{\bec}[1]{\Bar{e}^{#1}}
|
||||
\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
|
||||
\newcommand{\W}[2]{W_{#1}^{#2}}
|
||||
\newcommand{\w}[2]{w_{#1}^{#2}}
|
||||
\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}}
|
||||
\newcommand{\rsmu}[2]{\mu_{#1}^{#2}}
|
||||
\newcommand{\V}[2]{V_{#1}^{#2}}
|
||||
\newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
|
||||
|
||||
\newcommand{\tX}{\text{X}}
|
||||
|
||||
\newcommand{\pbeotint}[0]{\be{\text{c,md}}{\sr,\PBEot}(\br{})\,\n{}{}(\br{})}
|
||||
\newcommand{\pbeint}[0]{\be{\text{c,md}}{\sr,\PBE}(\br{})\,\n{}{}(\br{})}
|
||||
|
||||
% basis sets
|
||||
\newcommand{\Bas}{\mathcal{B}}
|
||||
\newcommand{\BasFC}{\mathcal{A}}
|
||||
\newcommand{\FC}{\text{FC}}
|
||||
\newcommand{\occ}{\text{occ}}
|
||||
\newcommand{\virt}{\text{virt}}
|
||||
\newcommand{\val}{\text{val}}
|
||||
\newcommand{\Cor}{\mathcal{C}}
|
||||
|
||||
% operators
|
||||
\newcommand{\hT}{\Hat{T}}
|
||||
\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
|
||||
\newcommand{\updw}{\uparrow\downarrow}
|
||||
\newcommand{\f}[2]{f_{#1}^{#2}}
|
||||
\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
|
||||
|
||||
% coordinates
|
||||
\newcommand{\br}[1]{\mathbf{r}_{#1}}
|
||||
\newcommand{\dbr}[1]{d\br{#1}}
|
||||
|
||||
\newcommand{\ra}{\rightarrow}
|
||||
|
||||
% frozen core
|
||||
\newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}}
|
||||
\newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}}
|
||||
\newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}}
|
||||
\newcommand{\tn}[2]{\tilde{n}_{#1}^{#2}}
|
||||
\newcommand{\ttn}[2]{\mathring{n}_{#1}^{#2}}
|
||||
|
||||
|
||||
% energies
|
||||
\newcommand{\EHF}{E_\text{HF}}
|
||||
\newcommand{\EPT}{E_\text{PT2}}
|
||||
\newcommand{\EFCI}{E_\text{FCI}}
|
||||
\newcommand{\EsCI}{E_\text{sCI}}
|
||||
\newcommand{\EDMC}{E_\text{DMC}}
|
||||
\newcommand{\EexFCI}{E_\text{exFCI}}
|
||||
\newcommand{\EexDMC}{E_\text{exDMC}}
|
||||
\newcommand{\Ead}{\Delta E_\text{ad}}
|
||||
\newcommand{\Eabs}{\Delta E_\text{abs}}
|
||||
\newcommand{\Evert}{\Delta E_\text{vert}}
|
||||
|
||||
\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$}
|
||||
|
||||
\newcommand{\pis}{\pi^\star}
|
||||
\newcommand{\si}{\sigma}
|
||||
\newcommand{\sis}{\sigma^\star}
|
||||
|
||||
\newcommand{\extrfunc}[0]{\epsilon}
|
||||
|
||||
|
||||
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\title{\textcolor{blue}{Chemically accurate excitation energies with small basis sets}}
|
||||
|
||||
\author{Emmanuel Giner}
|
||||
\email[Corresponding author: ]{emmanuel.giner@lct.jussieu.fr}
|
||||
\affiliation{\LCT}
|
||||
\author{Anthony Scemama}
|
||||
\affiliation{\LCPQ}
|
||||
\author{Julien Toulouse}
|
||||
\affiliation{\LCT}
|
||||
\author{Pierre-Fran\c{c}ois Loos}
|
||||
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
|
||||
\affiliation{\LCPQ}
|
||||
|
||||
\begin{abstract}
|
||||
%\begin{wrapfigure}[13]{o}[-1.25cm]{0.5\linewidth}
|
||||
% \centering
|
||||
% \includegraphics[width=\linewidth]{TOC}
|
||||
%\end{wrapfigure}
|
||||
By combining extrapolated selected configuration interaction (sCI) energies obtained with the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm with the recently proposed short-range density-functional correction for basis-set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner \textit{et al.}, \textit{J.~Chem.~Phys.}~ \textbf{2018}, \textit{149}, 194301}], we show that one can get chemically accurate vertical and adiabatic excitation energies with, typically, augmented double-$\zeta$ basis sets.
|
||||
We illustrate the present approach on various types of excited states (valence, Rydberg, and double excitations) in several small organic molecules (methylene, water, ammonia, carbon dimer and ethylene).
|
||||
The present study clearly evidences that special care has to be taken with very diffuse excited states where the present correction does not catch the radial incompleteness of the one-electron basis set.
|
||||
\end{abstract}
|
||||
|
||||
\maketitle
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Introduction}
|
||||
\label{sec:intro}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
One of the most fundamental problems of conventional wave function electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set.
|
||||
The overall basis-set incompleteness error can be, qualitatively at least, split in two contributions stemming from the radial and angular incompleteness.
|
||||
Although for ground-state properties angular incompleteness is by far the main source of error, it is definitely not unusual to have a significant radial incompleteness in the case of excited states (especially for Rydberg states), which can be alleviated by using additional sets of diffuse basis functions (i.e.~augmented basis sets).
|
||||
|
||||
Explicitly-correlated F12 methods \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94} have been specifically designed to efficiently catch angular incompleteness. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18}
|
||||
Although they have been extremely successful to speed up convergence of ground-state energies and properties, such as correlation and atomization energies, \cite{TewKloNeiHat-PCCP-07} their performance for excited states \cite{FliHatKlo-JCP-06, NeiHatKlo-JCP-06, HanKoh-JCP-09, Koh-JCP-09, ShiWer-JCP-10, ShiKniWer-JCP-11, ShiWer-JCP-11, ShiWer-MP-13} has been much more conflicting. \cite{FliHatKlo-JCP-06, NeiHatKlo-JCP-06}
|
||||
However, very encouraging results have been reported recently using the extended explicitly-correlated second-order approximate coupled-cluster singles and doubles ansatz suitable for response theory on systems such as methylene, formaldehyde and imidazole. \cite{HofSchKloKoh-JCP-19}
|
||||
|
||||
Instead of F12 methods, here we propose to follow a different route and investigate the performance of the recently proposed density-based basis set
|
||||
incompleteness correction. \cite{GinPraFerAssSavTou-JCP-18}
|
||||
Contrary to our recent study on atomization and correlation energies, \cite{LooPraSceTouGin-JPCL-19} the present contribution focuses on vertical and adiabatic excitation energies in molecular systems which is a much tougher test for the reasons mentioned above.
|
||||
This density-based correction relies on short-range correlation density functionals (with multideterminant reference) from range-separated density-functional theory \cite{Sav-INC-96, LeiStoWerSav-CPL-97, TouColSav-PRA-04, TouSavFla-IJQC-04, AngGerSavTou-PRA-05, GolWerSto-PCCP-05, PazMorGorBac-PRB-06, FroTouJen-JCP-07, TouGerJanSavAng-PRL-09, JanHenScu-JCP-09, FroCimJen-PRA-10, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-19} (RS-DFT) to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set.
|
||||
Because RS-DFT combines rigorously density-functional theory (DFT) \cite{ParYan-BOOK-89} and wave function theory (WFT) \cite{SzaOst-BOOK-96} via a decomposition of the electron-electron interaction into a non-divergent long-range part and a (complementary) short-range part (treated with WFT and DFT, respectively), the WFT method is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points (the so-called electron-electron cusp). \cite{Kat-CPAM-57}
|
||||
Consequently, the energy convergence with respect to the size of the basis set is significantly improved, \cite{FraMusLupTou-JCP-15} and chemical accuracy can be obtained even with small basis sets.
|
||||
For example, in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can recover quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets for a much lower computational cost than F12 methods.
|
||||
|
||||
This work is organized as follows.
|
||||
In Sec.~\ref{sec:theory}, the main working equations of the density-based correction are reported and discussed.
|
||||
Computational details are given in Sec.~\ref{sec:compdetails}.
|
||||
In Sec.~\ref{sec:res}, we discuss our results for each system and draw our conclusions in Sec.~\ref{sec:ccl}.
|
||||
Unless otherwise stated, atomic units are used.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Theory}
|
||||
\label{sec:theory}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The present basis-set correction assumes that we have, in a given (finite) basis set $\Bas$, the ground-state and the $k$th excited-state energies, $\E{0}{\Bas}$ and $\E{k}{\Bas}$, their one-electron densities, $\n{k}{\Bas}(\br{})$ and $\n{0}{\Bas}(\br{})$, as well as their opposite-spin on-top pair densities, $\n{2,0}{\Bas}(\br{})$ and $\n{2,k}{\Bas}(\br{})$,
|
||||
Therefore, the complete-basis-set (CBS) energy of the ground and excited states may be approximated as \cite{GinPraFerAssSavTou-JCP-18}
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\label{eq:E0CBS}
|
||||
\E{0}{\CBS} & \approx \E{0}{\Bas} + \bE{}{\Bas}[\n{0}{\Bas}],
|
||||
\\
|
||||
\label{eq:EkCBS}
|
||||
\E{k}{\CBS} & \approx \E{k}{\Bas} + \bE{}{\Bas}[\n{k}{\Bas}],
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
where
|
||||
\begin{equation}
|
||||
\label{eq:E_funcbasis}
|
||||
\bE{}{\Bas}[\n{}{}]
|
||||
= \min_{\wf{}{} \rightsquigarrow \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
|
||||
- \min_{\wf{}{\Bas} \rightsquigarrow \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
|
||||
\end{equation}
|
||||
is the basis-dependent complementary density functional,
|
||||
\begin{align}
|
||||
\hT & = - \frac{1}{2} \sum_{i}^{\Ne} \nabla_i^2,
|
||||
&
|
||||
\hWee{} & = \sum_{i<j}^{\Ne} r_{ij}^{-1},
|
||||
\end{align}
|
||||
are the kinetic and electron-electron repulsion operators, respectively, and $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert spaces spanned by $\Bas$ and the complete basis, respectively.
|
||||
The notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E_funcbasis} states that $\wf{}{}$ yields the one-electron density $\n{}{}$.
|
||||
|
||||
Hence, the CBS excitation energy associated with the $k$th excited state reads
|
||||
\begin{equation}
|
||||
\begin{split}
|
||||
\DE{k}{\CBS}
|
||||
& = \E{k}{\CBS} - \E{0}{\CBS}
|
||||
\\
|
||||
& \approx \DE{k}{\Bas} + \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}],
|
||||
\end{split}
|
||||
\end{equation}
|
||||
where
|
||||
\begin{equation}
|
||||
\label{eq:DEB}
|
||||
\DE{k}{\Bas} = \E{k}{\Bas} - \E{0}{\Bas}
|
||||
\end{equation}
|
||||
is the excitation energy in $\Bas$ and
|
||||
\begin{equation}
|
||||
\label{eq:DbE}
|
||||
\DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}] = \bE{}{\Bas}[\n{k}{\Bas}] - \bE{}{\Bas}[\n{0}{\Bas}]
|
||||
\end{equation}
|
||||
its basis-set correction.
|
||||
An important property of the present correction is
|
||||
\begin{equation}
|
||||
\label{eq:limitfunc}
|
||||
\lim_{\Bas \to \CBS} \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}] = 0.
|
||||
\end{equation}
|
||||
In other words, the correction vanishes in the CBS limit, hence guaranteeing an unaltered limit. \cite{LooPraSceTouGin-JPCL-19}
|
||||
Note that in Eqs.~\eqref{eq:E0CBS} and \eqref{eq:EkCBS} we have assumed that the same density functional $\bE{}{\Bas}$ can be used for correcting all excited-state energies, which seems a reasonable approximation since the electron-electron cusp effects are largely universal. \cite{Kut-TCA-85, MorKut-JPC-93, KutMor-ZPD-96, Tew-JCP-08, LooGil-MP-10, LooBloGil-JCP-2015}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Range-separation function}
|
||||
\label{sec:rs}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
As initially proposed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further developed in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can efficiently approximate $\bE{}{\Bas}[\n{}{}]$ by short-range correlation functionals with multi-determinantal (ECMD) reference borrowed from RS-DFT. \cite{TouGorSav-TCA-05}
|
||||
The ECMD functional, $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$, is a function of the range-separation parameter $\mu$ and admits, for any $\n{}{}$, the following two limits
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\label{eq:large_mu_ecmd}
|
||||
\lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
|
||||
\\
|
||||
\label{eq:small_mu_ecmd}
|
||||
\lim_{\mu \to 0} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
which correspond to the WFT limit ($\mu \to \infty$) and the Kohn-Sham DFT (KS-DFT) limit ($\mu = 0$).
|
||||
In Eq.~\eqref{eq:small_mu_ecmd}, $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT. \cite{HohKoh-PR-64, KohSha-PR-65}
|
||||
|
||||
The key ingredient that allows us to exploit ECMD functionals for correcting the basis-set incompleteness error is the range-separated function
|
||||
\begin{equation}
|
||||
\label{eq:def_mu}
|
||||
\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}),
|
||||
\end{equation}
|
||||
which automatically adapts to the spatial non-homogeneity of the basis-set incompleteness error.
|
||||
It is defined such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides, at coalescence, with an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$. \cite{GinPraFerAssSavTou-JCP-18}
|
||||
The explicit expression of $\W{}{\Bas}(\br{1},\br{2})$ is given by
|
||||
\begin{equation}
|
||||
\label{eq:def_weebasis}
|
||||
\W{}{\Bas}(\br{1},\br{2}) =
|
||||
\begin{cases}
|
||||
\f{}{\Bas}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), & \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,}
|
||||
\\
|
||||
\infty, & \text{otherwise,}
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
where
|
||||
\begin{equation}
|
||||
\label{eq:n2basis}
|
||||
\n{2}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
|
||||
\end{equation}
|
||||
and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
|
||||
\begin{equation}
|
||||
\label{eq:fbasis}
|
||||
\f{}{\Bas}(\br{1},\br{2})
|
||||
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
|
||||
\end{equation}
|
||||
and $\V{pq}{rs}= \braket{pq}{rs}$ are two-electron Coulomb integrals.
|
||||
An important feature of $\W{}{\Bas}(\br{1},\br{2})$ is that it tends to the regular Coulomb operator $r_{12}^{-1}$ as $\Bas \to \CBS${, which implies that
|
||||
\begin{equation}
|
||||
\lim_{\Bas \rightarrow \CBS} \rsmu{}{\Bas}(\br{}) = \infty,
|
||||
\end{equation}
|
||||
ensuring that $\bE{}{\Bas}[\n{}{}]$ vanishes when $\Bas$ is complete.
|
||||
We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JPCL-19} for additional details.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Short-range correlation functionals}
|
||||
\label{sec:func}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The local-density approximation ($\LDA$) of the ECMD complementary functional is defined as
|
||||
\begin{equation}
|
||||
\label{eq:def_lda_tot}
|
||||
\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
|
||||
\end{equation}
|
||||
where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
|
||||
|
||||
The functional $\be{\text{c,md}}{\sr,\LDA}$ from Eq.~\eqref{eq:def_lda_tot} presents two main defects: i) at small $\mu$, it overestimates the correlation energy, and ii) UEG-based quantities are hardly transferable when the system becomes strongly correlated.
|
||||
An attempt to solve these problems was suggested by some of the authors in the context of RS-DFT. \cite{FerGinTou-JCP-19}
|
||||
They proposed to interpolate between the usual Perdew-Burke-Ernzerhof ($\PBE$) correlation functional \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ (where $s=\nabla n/n^{4/3}$ is the reduced density gradient) at $\mu = 0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
|
||||
In the context of RS-DFT, the large-$\mu$ behavior corresponds to an extremely short-range interaction in the short-range functional.
|
||||
In this regime, the ECMD energy
|
||||
\begin{align}
|
||||
\label{eq:exact_large_mu}
|
||||
\bE{\text{c,md}}{\sr} = \frac{2\sqrt{\pi} (1 - \sqrt{2})}{3\mu^3} \int \dbr{} \n{2}{}(\br{}) + \order*{\mu^{-4}}
|
||||
\end{align}
|
||||
only depends on the \textit{exact} on-top pair density $\n{2}{}(\br{}) \equiv \n{2}{}(\br{},\br{})$ which is obtained from the \textit{exact} ground-state wave function $\Psi$ belonging to the many-electron Hilbert space in the CBS limit.
|
||||
|
||||
Obviously, an exact quantity such as $\n{2}{}(\br{})$ is out of reach in practical calculations and must be approximated by a function referred here as $\tn{2}{}(\br{})$.
|
||||
For a given $\tn{2}{}(\br{})$, some of the authors proposed the following functional form in order to interpolate between $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ at $\mu = 0$ and Eq.~\eqref{eq:exact_large_mu} as $\mu \to \infty$: \cite{FerGinTou-JCP-19}
|
||||
\begin{subequations}
|
||||
\begin{gather}
|
||||
\label{eq:epsilon_cmdpbe}
|
||||
\be{\text{c,md}}{\sr,\PBE}(\n{}{},\tn{2}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBE(\n{}{},\tn{2}{},s,\zeta) \rsmu{}{3} },
|
||||
\\
|
||||
\label{eq:beta_cmdpbe}
|
||||
\beta^\PBE(\n{}{},\tn{2}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2})} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\tn{2}{}/\n{}{}}.
|
||||
\end{gather}
|
||||
\end{subequations}
|
||||
As illustrated in the context of RS-DFT, \cite{FerGinTou-JCP-19} such a functional form is able to treat both weakly and strongly correlated systems thanks to the explicit inclusion of $\e{\text{c}}{\PBE}$ and $\tn{2}{}$, respectively.
|
||||
Therefore, in the present context, we introduce the general form of the $\PBE$-based complementary functional within a given basis set $\Bas$
|
||||
\begin{multline}
|
||||
\bE{\PBE}{\Bas}[\n{}{},\tn{2}{},\rsmu{}{\Bas}] =
|
||||
\int \n{}{}(\br{})
|
||||
\\
|
||||
\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
|
||||
\end{multline}
|
||||
which has an explicit dependency on both the range-separation function $\rsmu{}{\Bas}(\br{})$ (instead of the range-separation parameter in RS-DFT) and the approximation level of $\tn{2}{}$.
|
||||
|
||||
In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the authors introduced a version of the $\PBE$-based functional, here-referred as $\PBEUEG$
|
||||
\begin{equation}
|
||||
\bE{\PBEUEG}{\Bas}
|
||||
\equiv
|
||||
\bE{\PBE}{\Bas}[\n{}{},\n{2}{\UEG},\rsmu{}{\Bas}],
|
||||
\end{equation}
|
||||
in which the on-top pair density was approximated by its UEG version, i.e., $\tn{2}{}(\br{}) = \n{2}{\UEG}(\br{})$, with
|
||||
\begin{equation}
|
||||
\label{eq:n2UEG}
|
||||
\n{2}{\UEG}(\br{}) \approx n(\br{})^2 [1-\zeta(\br{})^2] g_0(n(\br{})),
|
||||
\end{equation}
|
||||
and where $g_0(n)$ is the UEG on-top pair distribution function [see Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}].
|
||||
Note that in Eq.~\eqref{eq:n2UEG} the dependence on the spin polarization $\zeta$ is only approximate.
|
||||
As illustrated in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, the $\PBEUEG$ functional has clearly shown, for weakly correlated systems, to improve energetics over the pure UEG-based functional $\bE{\LDA}{\Bas}$ [see Eq.~\eqref{eq:def_lda_tot}] thanks to the leverage brought by the $\PBE$ functional in the small-$\mu$ regime.
|
||||
|
||||
However, the underlying UEG on-top pair density might not be suited for the treatment of excited states and/or strongly correlated systems.
|
||||
Besides, in the context of the present basis-set correction, $\n{2}{\Bas}(\br{})$, the on-top pair density in $\Bas$, must be computed anyway to obtain $\rsmu{}{\Bas}(\br{})$ [see Eqs.~\eqref{eq:def_mu} and \eqref{eq:def_weebasis}].
|
||||
Therefore, as in Ref.~\onlinecite{FerGinTou-JCP-19}, we define a better approximation of the exact on-top pair density as
|
||||
\begin{equation}
|
||||
\label{eq:ot-extrap}
|
||||
\ttn{2}{\Bas}(\br{}) = \n{2}{\Bas}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{\Bas}(\br{})})^{-1}
|
||||
\end{equation}
|
||||
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin \cite{GorSav-PRA-06} in the context of RS-DFT.
|
||||
Using this new ingredient, we propose here the ``$\PBE$-ontop'' (\PBEot) functional
|
||||
\begin{equation}
|
||||
\bE{\PBEot}{\Bas}
|
||||
\equiv
|
||||
\bE{\PBE}{\Bas}[\n{}{},\ttn{2}{\Bas},\rsmu{}{\Bas}].
|
||||
\end{equation}
|
||||
The sole distinction between $\PBEUEG$ and $\PBEot$ is the level of approximation of the exact on-top pair density.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Computational details}
|
||||
\label{sec:compdetails}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
In the present study, we compute the ground- and excited-state energies, one-electron densities and on-top pair densities with a selected configuration interaction (sCI) method known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
|
||||
Both the implementation of the CIPSI algorithm and the computational protocol for excited states is reported in Ref.~\onlinecite{SceCafBenJacLoo-RC-19}.
|
||||
The total energy of each state is obtained via an efficient extrapolation procedure of the sCI energies designed to reach near-FCI accuracy. \cite{HolUmrSha-JCP-17, QP2}
|
||||
These energies will be labeled exFCI in the following.
|
||||
Using near-FCI excitation energies (within a given basis set) has the indisputable advantage to remove the error inherent to the WFT method.
|
||||
Indeed, in the present case, the only source of error on the excitation energies is due to basis-set incompleteness.
|
||||
We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details.
|
||||
The one-electron densities and on-top pair densities are computed from a very large CIPSI expansion containing up to several million of Slater determinants.
|
||||
All the RS-DFT and exFCI calculations have been performed with {\QP}. \cite{QP2}
|
||||
For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
|
||||
Except for methylene for which FCI/TZVP geometries have been taken from Ref.~\onlinecite{SheLeiVanSch-JCP-98}, the other molecular geometries have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJac-JCTC-19} and have been obtained at the CC3/aug-cc-pVTZ level of theory.
|
||||
For the sake of completeness, all these geometries are reported in the {\SI}.
|
||||
Frozen-core calculations are systematically performed and defined as such: a \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
|
||||
The frozen-core density-based correction is used consistently with the frozen-core approximation in WFT methods.
|
||||
We refer the reader to Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} for an explicit derivation of the equations associated with the frozen-core version of the present density-based basis-set correction.
|
||||
Compared to the exFCI calculations performed to compute energies and densities, the basis-set correction represents, in any case, a marginal computational cost.
|
||||
In the following, we employ the AVXZ shorthand notations for Dunning's aug-cc-pVXZ basis sets.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Results and Discussion}
|
||||
\label{sec:res}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%=======================
|
||||
\subsection{Methylene}
|
||||
\label{sec:CH2}
|
||||
%=======================
|
||||
|
||||
Methylene is a paradigmatic system in electronic structure theory. \cite{Sch-Science-86}
|
||||
Due to its relative small size, its ground and excited states have been thoroughly studied with high-level ab initio methods. \cite{Sch-Science-86, BauTay-JCP-86, JenBun-JCP-88, SheVanYamSch-JMS-97, SheLeiVanSch-JCP-98, AbrShe-JCP-04, AbrShe-CPL-05, ZimTouZhaMusUmr-JCP-09, GouPieWlo-MP-10, ChiHolAdaOttUmrShaZim-JPCA-18}
|
||||
|
||||
As a first test of the present density-based basis-set correction, we consider the four lowest-lying states of methylene ($1\,^{3}B_1$, $1\,^{1}A_1$, $1\,^{1}B_1$ and $2\,^{1}A_1$) at their respective equilibrium geometry and compute the corresponding adiabatic transition energies for basis sets ranging from AVDZ to AVQZ.
|
||||
We have also computed total energies at the exFCI/AV5Z level and used these alongside the quadruple-$\zeta$ ones to extrapolate the total energies to the CBS limit with the usual extrapolation formula \cite{HelJorOls-BOOK-02}
|
||||
\begin{equation}
|
||||
\E{}{\text{AVXZ}}(\tX) = \E{}{\CBS} + \alpha \, \tX^{-3}.
|
||||
\end{equation}
|
||||
These results are illustrated in Fig.~\ref{fig:CH2} and reported in Table \ref{tab:CH2} alongside reference values from the literature obtained with various deterministic and stochastic approaches. \cite{ChiHolAdaOttUmrShaZim-JPCA-18, SheLeiVanSch-JCP-98, JenBun-JCP-88, SheLeiVanSch-JCP-98, ZimTouZhaMusUmr-JCP-09}
|
||||
Total energies for each state can be found in the {\SI}.
|
||||
The exFCI/CBS values are still off by a few tenths of a {\kcal} compared to the DMC results of Zimmerman \textit{et al.} \cite{ZimTouZhaMusUmr-JCP-09} which are extremely close from the experimentally-derived adiabatic energies.
|
||||
The reason of this discrepancy is probably due to the frozen-core approximation which has been applied in our case and has shown to significantly affect adiabatic energies. \cite{LooGalJac-JPCL-18, LooJac-JCTC-19}
|
||||
However, the exFCI/CBS energies are in perfect agreement with the semistochastic heat-bath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}, as expected.
|
||||
|
||||
Figure \ref{fig:CH2} clearly shows that, for the double-$\zeta$ basis, the exFCI adiabatic energies are far from being chemically accurate with errors as high as 0.15 eV.
|
||||
From the triple-$\zeta$ basis onward, the exFCI excitation energies are chemically accurate though (i.e. error below 1 {\kcal} or 0.043 eV), and converge steadily to the CBS limit when one increases the size of the basis set.
|
||||
Concerning the basis-set correction, already at the double-$\zeta$ level, the $\PBEot$ correction returns chemically accurate excitation energies.
|
||||
The performance of the $\PBEUEG$ and $\LDA$ functionals is less impressive.
|
||||
Yet, they still yield significant reductions of the basis-set incompleteness error, hence representing a good compromise between computational cost and accuracy.
|
||||
Note that the results for the $\PBEUEG$ functional are not represented in Fig.~\ref{fig:CH2} as they are very similar to the $\LDA$ ones (similar considerations apply to the other systems studied below).
|
||||
It is also quite evident that, the basis-set correction has the tendency of over-correcting the excitation energies via an over-stabilization of the excited states compared to the ground state.
|
||||
This trend is quite systematic as we shall see below.
|
||||
|
||||
%%% TABLE 1 %%%
|
||||
\begin{squeezetable}
|
||||
\begin{table}
|
||||
\caption{
|
||||
Adiabatic transition energies (in eV) of excited states of methylene for various methods and basis sets.
|
||||
The relative difference with respect to the exFCI/CBS result is reported in square brackets.
|
||||
See {\SI} for total energies.}
|
||||
\label{tab:CH2}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{lllll}
|
||||
& & \mc{3}{c}{Transitions} \\
|
||||
\cline{3-5}
|
||||
Method & Basis set & \tabc{$1\,^{3}B_1 \ra 1\,^{1}A_1$}
|
||||
& \tabc{$1\,^{3}B_1 \ra 1\,^{1}B_1$}
|
||||
& \tabc{$1\,^{3}B_1 \ra 2\,^{1}A_1$} \\
|
||||
\hline
|
||||
exFCI & AVDZ
|
||||
& $0.441$ [$+0.057$]
|
||||
& $1.536$ [$+0.152$]
|
||||
& $2.659$ [$+0.162$] \\
|
||||
& AVTZ
|
||||
& $0.408$ [$+0.024$]
|
||||
& $1.423$ [$+0.040$]
|
||||
& $2.546$ [$+0.049$] \\
|
||||
& AVQZ
|
||||
& $0.395$ [$+0.011$]
|
||||
& $1.399$ [$+0.016$]
|
||||
& $2.516$ [$+0.020$] \\
|
||||
& AV5Z
|
||||
& $0.390$ [$+0.006$]
|
||||
& $1.392$ [$+0.008$]
|
||||
& $2.507$ [$+0.010$] \\
|
||||
& CBS
|
||||
& $0.384$
|
||||
& $1.384$
|
||||
& $2.497$ \\
|
||||
\\
|
||||
exFCI+$\PBEot$ & AVDZ
|
||||
& $0.347$ [$-0.037$]
|
||||
& $1.401$ [$+0.017$]
|
||||
& $2.511$ [$+0.014$] \\
|
||||
& AVTZ
|
||||
& $0.374$ [$-0.010$]
|
||||
& $1.378$ [$-0.006$]
|
||||
& $2.491$ [$-0.006$] \\
|
||||
& AVQZ
|
||||
& $0.379$ [$-0.005$]
|
||||
& $1.378$ [$-0.006$]
|
||||
& $2.489$ [$-0.008$] \\
|
||||
\\
|
||||
exFCI+$\PBEUEG$ & AVDZ
|
||||
& $0.308$ [$-0.076$]
|
||||
& $1.388$ [$+0.004$]
|
||||
& $2.560$ [$+0.064$] \\
|
||||
& AVTZ
|
||||
& $0.356$ [$-0.028$]
|
||||
& $1.371$ [$-0.013$]
|
||||
& $2.510$ [$+0.013$] \\
|
||||
& AVQZ
|
||||
& $0.371$ [$-0.013$]
|
||||
& $1.375$ [$-0.009$]
|
||||
& $2.498$ [$+0.002$] \\
|
||||
\\
|
||||
exFCI+$\LDA$ & AVDZ
|
||||
& $0.337$ [$-0.047$]
|
||||
& $1.420$ [$+0.036$]
|
||||
& $2.586$ [$+0.089$] \\
|
||||
& AVTZ
|
||||
& $0.359$ [$-0.025$]
|
||||
& $1.374$ [$-0.010$]
|
||||
& $2.514$ [$+0.017$] \\
|
||||
& AVQZ
|
||||
& $0.370$ [$-0.014$]
|
||||
& $1.375$ [$-0.009$]
|
||||
& $2.499$ [$-0.002$] \\
|
||||
\\
|
||||
SHCI\fnm[1] & AVQZ
|
||||
& $0.393$
|
||||
& $1.398$
|
||||
& $2.516$ \\
|
||||
CR-EOMCC (2,3)D\fnm[2]& AV5Z
|
||||
& $0.430$
|
||||
& $1.464$
|
||||
& $2.633$ \\
|
||||
FCI\fnm[3] & TZ2P
|
||||
& $0.483$
|
||||
& $1.542$
|
||||
& $2.674$ \\
|
||||
DMC\fnm[4] &
|
||||
& $0.406$
|
||||
& $1.416$
|
||||
& $2.524$ \\
|
||||
Exp.\fnm[5] &
|
||||
& $0.406$
|
||||
& $1.415$
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\fnt[1]{Semistochastic heat-bath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}.}
|
||||
\fnt[2]{Completely-renormalized equation-of-motion coupled cluster (CR-EOMCC) calculations from Refs.~\onlinecite{GouPieWlo-MP-10}.}
|
||||
\fnt[3]{Reference \onlinecite{SheLeiVanSch-JCP-98}.}
|
||||
\fnt[4]{Diffusion Monte Carlo (DMC) calculations from Ref.~\onlinecite{ZimTouZhaMusUmr-JCP-09} obtained with a CAS(6,6) trial wave function.}
|
||||
\fnt[5]{Experimentally-derived values. See footnotes of Table II from Ref.~\onlinecite{GouPieWlo-MP-10} for additional details.}
|
||||
\end{table}
|
||||
\end{squeezetable}
|
||||
%%% %%% %%%
|
||||
|
||||
%%% FIG 1 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{fig1.pdf}
|
||||
\caption{Error in adiabatic excitation energies (in eV) of methylene for various basis sets and methods.
|
||||
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
|
||||
See Table \ref{tab:CH2} for raw data.}
|
||||
\label{fig:CH2}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
|
||||
%=======================
|
||||
\subsection{Rydberg States of Water and Ammonia}
|
||||
\label{sec:H2O-NH3}
|
||||
%=======================
|
||||
|
||||
For the second test, we consider the water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, SceCafBenJacLoo-RC-19} and ammonia \cite{SchGoe-JCTC-17, BarDelPerMat-JMS-97, LooSceBloGarCafJac-JCTC-18} molecules.
|
||||
They are both well studied and possess Rydberg excited states which are highly sensitive to the radial completeness of the one-electron basis set, as evidenced in Ref.~\onlinecite{LooSceBloGarCafJac-JCTC-18}.
|
||||
Table \ref{tab:Mol} reports vertical excitation energies for various singlet and triplet excited states of water and ammonia at various levels of theory (see the {\SI} for total energies).
|
||||
The basis-set corrected theoretical best estimates (TBEs) have been extracted from Ref.~\onlinecite{LooSceBloGarCafJac-JCTC-18} and have been obtained on the same geometries.
|
||||
These results are also depicted in Figs.~\ref{fig:H2O} and \ref{fig:NH3} for \ce{H2O} and \ce{NH3}, respectively.
|
||||
One would have noticed that the basis-set effects are particularly strong for the third singlet excited state of water and the third and fourth singlet excited states of ammonia where this effect is even magnified.
|
||||
In other words, substantial error remains in these cases even with the largest AVQZ basis set.
|
||||
In these cases, one really needs doubly augmented basis sets to reach radial completeness.
|
||||
The first observation worth reporting is that all three RS-DFT correlation functionals have very similar behaviors and they significantly reduce the error on the excitation energies for most of the states.
|
||||
However, these results also clearly evidence that special care has to be taken for very diffuse excited states where the present correction cannot catch the radial incompleteness of the one-electron basis set, a feature which is far from being a cusp-related effect.
|
||||
In other words, the DFT-based correction recovers dynamic correlation effects only and one must ensure that the basis set includes enough diffuse functions in order to describe Rydberg states.
|
||||
|
||||
%%% TABLE 2 %%%
|
||||
\begin{squeezetable}
|
||||
\begin{table*}
|
||||
\caption{
|
||||
Vertical excitation energies (in eV) of excited states of water, ammonia, carbon dimer and ethylene for various methods and basis sets.
|
||||
The TBEs have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJac-JCTC-19} on the same geometries.
|
||||
See the {\SI} for total energies.}
|
||||
\label{tab:Mol}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{lllddddddddddddd}
|
||||
& & & & \mc{12}{c}{Deviation with respect to TBE}
|
||||
\\
|
||||
\cline{5-16}
|
||||
& & & & \mc{3}{c}{exFCI}
|
||||
& \mc{3}{c}{exFCI+$\PBEot$}
|
||||
& \mc{3}{c}{exFCI+$\PBEUEG$}
|
||||
& \mc{3}{c}{exFCI+$\LDA$}
|
||||
\\
|
||||
\cline{5-7} \cline{8-10} \cline{11-13} \cline{14-16}
|
||||
Molecule & Transition & Nature & \tabc{TBE} & \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
|
||||
& \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
|
||||
& \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
|
||||
& \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVQZ}
|
||||
\\
|
||||
\hline
|
||||
Water & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & Ryd. & 7.70\fnm[1] & -0.17 & -0.07 & -0.02
|
||||
& 0.01 & 0.00 & 0.02
|
||||
& -0.02 & -0.01 & 0.00
|
||||
& -0.04 & -0.01 & 0.01
|
||||
\\
|
||||
& $1\,^{1}A_1 \ra 1\,^{1}A_2$ & Ryd. & 9.47\fnm[1] & -0.15 & -0.06 & -0.01
|
||||
& 0.03 & 0.01 & 0.03
|
||||
& 0.00 & 0.00 & 0.02
|
||||
& -0.03 & 0.00 & 0.00
|
||||
\\
|
||||
& $1\,^{1}A_1 \ra 2\,^{1}A_1$ & Ryd. & 9.97\fnm[1] & -0.03 & 0.02 & 0.06
|
||||
& 0.13 & 0.08 & 0.09
|
||||
& 0.10 & 0.07 & 0.08
|
||||
& 0.09 & 0.07 & 0.03
|
||||
\\
|
||||
& $1\,^{1}A_1 \ra 1\,^{3}B_1$ & Ryd. & 7.33\fnm[1] & -0.19 & -0.08 & -0.03
|
||||
& 0.02 & 0.00 & 0.02
|
||||
& 0.05 & 0.01 & 0.02
|
||||
& 0.00 & 0.00 & 0.04
|
||||
\\
|
||||
& $1\,^{1}A_1 \ra 1\,^{3}A_2$ & Ryd. & 9.30\fnm[1] & -0.16 & -0.06 & -0.01
|
||||
& 0.04 & 0.02 & 0.04
|
||||
& 0.07 & 0.03 & 0.04
|
||||
& 0.03 & 0.03 & 0.04
|
||||
\\
|
||||
& $1\,^{1}A_1 \ra 1\,^{3}A_1$ & Ryd. & 9.59\fnm[1] & -0.11 & -0.05 & -0.01
|
||||
& 0.07 & 0.02 & 0.03
|
||||
& 0.09 & 0.03 & 0.03
|
||||
& 0.06 & 0.03 & 0.04
|
||||
\\
|
||||
\\
|
||||
Ammonia & $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & Ryd. & 6.66\fnm[1] & -0.18 & -0.07 & -0.04
|
||||
& -0.04 & -0.02 & -0.01
|
||||
& -0.07 & -0.03 & -0.02
|
||||
& -0.07 & -0.03 & -0.02
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 1\,^{1}E$ & Ryd. & 8.21\fnm[1] & -0.13 & -0.05 & -0.02
|
||||
& 0.01 & 0.00 & 0.01
|
||||
& -0.03 & -0.01 & 0.00
|
||||
& -0.03 & 0.00 & 0.00
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & Ryd. & 8.65\fnm[1] & 1.03 & 0.68 & 0.47
|
||||
& 1.17 & 0.73 & 0.50
|
||||
& 1.12 & 0.72 & 0.49
|
||||
& 1.11 & 0.71 & 0.49
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 2\,^{1}A_{2}$ & Ryd. & 8.65\fnm[2] & 1.22 & 0.77 & 0.59
|
||||
& 1.36 & 0.83 & 0.62
|
||||
& 1.33 & 0.81 & 0.61
|
||||
& 1.32 & 0.81 & 0.61
|
||||
\\
|
||||
& $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & Ryd. & 9.19\fnm[1] & -0.18 & -0.06 & -0.03
|
||||
& -0.03 & 0.00 & -0.02
|
||||
& -0.07 & -0.02 & -0.03
|
||||
& -0.07 & -0.01 & -0.03
|
||||
\\
|
||||
\\
|
||||
Carbon dimer & $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ & Val. & 2.04\fnm[3] & 0.17 & 0.05 & 0.02
|
||||
& 0.04 & 0.00 & 0.00
|
||||
& 0.15 & 0.04 & 0.02
|
||||
& 0.17 & 0.05 & 0.02
|
||||
\\
|
||||
& $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ & Val. & 2.38\fnm[3] & 0.12 & 0.04 & 0.02
|
||||
& 0.00 & 0.00 & 0.00
|
||||
& 0.11 & 0.03 & 0.02
|
||||
& 0.13 & 0.04 & 0.02
|
||||
\\
|
||||
\\
|
||||
Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43\fnm[3] & -0.12 & -0.04 &
|
||||
& -0.05 & -0.01 &
|
||||
& -0.04 & -0.01 &
|
||||
& -0.02 & 0.00 &
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1u}$ & Val. & 7.92\fnm[3] & 0.01 & 0.01 &
|
||||
& 0.00 & 0.00 &
|
||||
& 0.06 & 0.03 &
|
||||
& 0.06 & 0.03 &
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{1}B_{1g}$ & Ryd. & 8.10\fnm[3] & -0.1 & -0.02 &
|
||||
& -0.03 & 0.00 &
|
||||
& -0.02 & 0.00 &
|
||||
& 0.00 & 0.01 &
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & Val. & 4.54\fnm[3] & 0.01 & 0.00 &
|
||||
& 0.05 & 0.03 &
|
||||
& 0.08 & 0.04 &
|
||||
& 0.07 & 0.04 &
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$ & Val. & 7.28\fnm[4] & -0.12 & -0.04 &
|
||||
& -0.04 & 0.00 &
|
||||
& 0.00 & 0.00 &
|
||||
& 0.00 & 0.02 &
|
||||
\\
|
||||
& $1\,^{1}A_{1g} \ra 1\,^{3}B_{1g}$ & Val. & 8.00\fnm[4] & -0.07 & -0.01 &
|
||||
& 0.00 & 0.03 &
|
||||
& 0.04 & 0.03 &
|
||||
& 0.05 & 0.04 &
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\fnt[1]{exFCI/AVQZ data corrected with the difference between CC3/d-AV5Z and exFCI/AVQZ values. \cite{LooSceBloGarCafJac-JCTC-18}
|
||||
d-AV5Z is the doubly augmented V5Z basis set.}
|
||||
\fnt[2]{exFCI/AVTZ data corrected with the difference between CC3/d-AV5Z and exFCI/AVTZ values. \cite{LooSceBloGarCafJac-JCTC-18}}
|
||||
\fnt[3]{exFCI/CBS obtained from the exFCI/AVTZ and exFCI/AVQZ data of Ref.~\onlinecite{LooBogSceCafJac-JCTC-19}.}
|
||||
\fnt[4]{exFCI/AVDZ data corrected with the difference between CC3/d-AV5Z and exFCI/AVDZ values. \cite{LooSceBloGarCafJac-JCTC-18}}
|
||||
\end{table*}
|
||||
\end{squeezetable}
|
||||
%%% %%% %%%
|
||||
|
||||
%%% FIG 2 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{fig2.pdf}
|
||||
\caption{Error in vertical excitation energies (in eV) of water for various basis sets and methods.
|
||||
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
|
||||
See Table \ref{tab:Mol} for raw data.}
|
||||
\label{fig:H2O}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
|
||||
%%% FIG 3 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{fig3.pdf}
|
||||
\caption{Error in vertical excitation energies (in eV) of ammonia for various basis sets and methods.
|
||||
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
|
||||
See Table \ref{tab:Mol} for raw data.}
|
||||
\label{fig:NH3}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
|
||||
%=======================
|
||||
\subsection{Doubly-Excited States of the Carbon Dimer}
|
||||
\label{sec:C2}
|
||||
%=======================
|
||||
In order to have a miscellaneous test set of excitations, in a third time, we propose to study some doubly-excited states of the carbon dimer \ce{C2}, a prototype system for strongly correlated and multireference systems. \cite{AbrShe-JCP-04, AbrShe-CPL-05, Var-JCP-08, PurZhaKra-JCP-09, AngCimPas-MP-12, BooCleThoAla-JCP-11, Sha-JCP-15, SokCha-JCP-16, HolUmrSha-JCP-17, VarRoc-PTRSMPES-18}
|
||||
These two valence excitations --- $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ and $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ --- are both of $(\pi,\pi) \ra (\si,\si)$ character.
|
||||
They have been recently studied with state-of-the-art methods, and have been shown to be ``pure'' doubly-excited states as they involve an insignificant amount of single excitations. \cite{LooBogSceCafJac-JCTC-19}
|
||||
The vertical excitation energies associated with these transitions are reported in Table \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}.
|
||||
An interesting point here is that one really needs to consider the $\PBEot$ functional to get chemically accurate excitation energies with the AVDZ atomic basis set.
|
||||
We believe that the present result is a direct consequence of the multireference character of the \ce{C2} molecule.
|
||||
In other words, the UEG on-top pair density used in the $\LDA$ and $\PBEUEG$ functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top pair density for the present system.
|
||||
|
||||
%%% FIG 5 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{fig4.pdf}
|
||||
\caption{Error in vertical excitation energies (in eV) for two doubly-excited states of the carbon dimer for various basis sets and methods.
|
||||
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
|
||||
See Table \ref{tab:Mol} for raw data.}
|
||||
\label{fig:C2}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
|
||||
It is interesting to study the behavior of the key quantities involved in the basis-set correction for different states as the basis-set incompleteness error is obviously state specific.
|
||||
In Fig.~\ref{fig:C2_mu}, we report $\rsmu{}{\Bas}(z)$, $\n{}{\Bas}(z) \be{\text{c,md}}{\sr,\PBEot}(z)$, and $\n{2}{\Bas}(z)$
|
||||
along the nuclear axis ($z$) for the two $^1 \Sigma_g^+$ electronic states of \ce{C2} computed with the AVDZ, AVTZ, and AVQZ basis sets.
|
||||
The graphs gathered in Fig.~\ref{fig:C2_mu} illustrate several general features regarding the present basis-set correction:
|
||||
\begin{itemize}
|
||||
\item the maximal values of $\rsmu{}{\Bas}(z)$ are systematically close to the nuclei, a signature of the atom-centered basis set;
|
||||
\item the overall magnitude of $\rsmu{}{\Bas}(z)$ increases with the basis set, which reflects the improvement of the description of the correlation effects when enlarging the basis set;
|
||||
\item the absolute value of the energetic correction decreases when the size of the basis set increases;
|
||||
\item there is a clear correspondence between the values of the energetic correction and the on-top pair density.
|
||||
\end{itemize}
|
||||
Regarding now the differential effect of the basis-set correction in the special case of the two $^1 \Sigma_g^+$ states studied here, we observe that:
|
||||
\begin{itemize}
|
||||
\item $\rsmu{}{\Bas}(z)$ has the same overall behavior for the two states, with slightly more fine structure in the case of the ground state.
|
||||
Such feature is consistent with the fact that the two states considered are both of $\Sigma_g^+$ symmetry and of valence character.
|
||||
\item $\n{2}{}(z)$ is overall larger in the excited state, specially in the bonding and outer regions.
|
||||
This is can be explained by the nature of the electronic transition which qualitatively corresponds to a double excitation from $\pi$ to $\sigma$ orbitals, therefore increasing the overall electronic population on the bond axis.
|
||||
\item The energetic correction clearly stabilizes preferentially the excited state rather than the ground state, illustrating that short-range correlation effects are more pronounced in the former than in the latter.
|
||||
This is linked to the larger values of the excited-state on-top pair density.
|
||||
\end{itemize}
|
||||
|
||||
%%% FIG 4 %%%
|
||||
\begin{figure*}
|
||||
\includegraphics[height=0.35\linewidth]{fig5a.pdf}
|
||||
\includegraphics[height=0.35\linewidth]{fig5b.pdf}
|
||||
\includegraphics[height=0.35\linewidth]{fig5c.pdf}
|
||||
\caption{$\rsmu{}{\Bas}$ (left), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBEot}$ (center) and $\n{2}{\Bas}$ (right) along the molecular axis ($z$) for the ground state (black curve) and second doubly-excited state (red curve) of \ce{C2} for various basis sets $\Bas$.
|
||||
The two electronic states are both of $\Sigma_g^+$ symmetry.
|
||||
The carbon nuclei are located at $z= \pm 1.180$ bohr and are represented by the thin black lines.}
|
||||
\label{fig:C2_mu}
|
||||
\end{figure*}
|
||||
%%% %%% %%%
|
||||
|
||||
%=======================
|
||||
\subsection{Ethylene}
|
||||
\label{sec:C2H4}
|
||||
%=======================
|
||||
|
||||
As a final example, we consider the ethylene molecule, yet another system which has been particularly scrutinized theoretically using high-level ab initio methods. \cite{SerMarNebLinRoo-JCP-93, WatGwaBar-JCP-96, WibOliTru-JPCA-02, BarPaiLis-JCP-04, Ang-JCC-08, SchSilSauThi-JCP-08, SilSchSauThi-JCP-10, SilSauSchThi-MP-10, Ang-IJQC-10, DadSmaBooAlaFil-JCTC-12, FelPetDav-JCP-14, ChiHolAdaOttUmrShaZim-JPCA-18}
|
||||
We refer the interested reader to the work of Feller \textit{et al.}\cite{FelPetDav-JCP-14} for an exhaustive investigation dedicated to the excited states of ethylene using state-of-the-art CI calculations.
|
||||
In the present context, ethylene is a particularly interesting system as it contains a mixture of valence and Rydberg excited states.
|
||||
Our basis-set corrected vertical excitation energies are gathered in Table \ref{tab:Mol} and depicted in Fig.~\ref{fig:C2H4}.
|
||||
Note that exFCI/AVQZ calculations are inaccessible for ethylene.
|
||||
The exFCI+$\PBEot$/AVDZ excitation energies are at near chemical accuracy and the errors drop further when one goes to the triple-$\zeta$ basis.
|
||||
Consistently with the previous examples, the $\LDA$ and $\PBEUEG$ functionals are slightly less accurate, although they still correct the excitation energies in the right direction.
|
||||
|
||||
%%% FIG 6 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{fig6.pdf}
|
||||
\caption{Error in vertical excitation energies (in eV) of ethylene for various basis sets and methods.
|
||||
The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
|
||||
See Table \ref{tab:Mol} for raw data.}
|
||||
\label{fig:C2H4}
|
||||
\end{figure}
|
||||
%%% %%% %%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Conclusion}
|
||||
\label{sec:ccl}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
We have shown that, by employing the recently proposed density-based basis-set correction developed by some of the authors, \cite{GinPraFerAssSavTou-JCP-18} one can obtain, using sCI methods, chemically accurate excitation energies with typically augmented double-$\zeta$ basis sets.4
|
||||
This nicely complements our recent investigation on ground-state properties, \cite{LooPraSceTouGin-JPCL-19} which has evidenced that one recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets.
|
||||
The present study clearly shows that, for very diffuse excited states, the present correction relying on short-range correlation functionals from RS-DFT might not be enough to catch the radial incompleteness of the one-electron basis set.
|
||||
Also, in the case of multireference systems, we have evidenced that the $\PBEot$ functional, which uses an accurate on-top pair density, is more appropriate than the $\LDA$ and $\PBEUEG$ functionals relying on the UEG on-top pair density.
|
||||
We are currently investigating the performance of the present basis-set correction for strongly correlated systems and we hope to report on this in the near future.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section*{Supplementary material}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
See {\SI} for geometries and additional information (including total energies and energetic correction of the various functionals).
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{acknowledgements}
|
||||
PFL would like to thank Denis Jacquemin for numerous discussions on excited states.
|
||||
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), CALMIP (Toulouse) under allocation 2019-18005 and the Jarvis-Alpha cluster from the \textit{Institut Parisien de Chimie Physique et Th\'eorique}.
|
||||
\end{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\bibliography{Ex-srDFT}
|
||||
|
||||
\end{document}
|
||||
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