diff --git a/Manuscript/Be.pdf b/Manuscript/Be.pdf index 98e9b94..04e1e76 100644 Binary files a/Manuscript/Be.pdf and b/Manuscript/Be.pdf differ diff --git a/Manuscript/sfBSE.bib b/Manuscript/sfBSE.bib index 6fc2d7b..86990cd 100644 --- a/Manuscript/sfBSE.bib +++ b/Manuscript/sfBSE.bib @@ -1,13 +1,37 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2021-01-10 22:26:03 +0100 +%% Created for Pierre-Francois Loos at 2021-01-11 09:47:39 +0100 %% Saved with string encoding Unicode (UTF-8) +@article{Koch_1994, + author = {Koch,Henrik and Kobayashi,Rika and Sanchez de Mer{\'a}s,Alfredo and Jorgensen, Poul}, + date-added = {2021-01-11 09:32:50 +0100}, + date-modified = {2021-01-11 09:32:50 +0100}, + doi = {10.1063/1.466321}, + journal = {J. Chem. Phys.}, + pages = {4393-4400}, + title = {Calculation of size‐intensive transition moments from the coupled cluster singles and doubles linear response function}, + volume = {100}, + year = {1994}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.466321}} + +@article{Stanton_1993, + author = {Stanton,John F. and Bartlett,Rodney J.}, + date-added = {2021-01-11 09:32:39 +0100}, + date-modified = {2021-01-11 09:32:39 +0100}, + doi = {10.1063/1.464746}, + journal = {J. Chem. Phys.}, + pages = {7029-7039}, + title = {The equation of motion coupled‐cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties}, + volume = {98}, + year = {1993}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.464746}} + @article{Sarkar_2021, author = {R. Sarkar and M. Boggio-Pasqua and P. F. Loos and D. Jacquemin}, date-added = {2021-01-10 18:27:30 +0100}, @@ -245,10 +269,10 @@ year = {2019}, Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00289}} -@article{Chrayteh_2020, +@article{Chrayteh_2021, author = {Chrayteh, Amara and Blondel, Aymeric and Loos, Pierre-Fran{\c c}ois and Jacquemin, Denis}, date-added = {2020-12-09 09:59:26 +0100}, - date-modified = {2020-12-09 09:59:26 +0100}, + date-modified = {2021-01-11 09:19:56 +0100}, doi = {10.1021/acs.jctc.0c01111}, journal = {Journal of Chemical Theory and Computation}, number = {0}, @@ -3487,44 +3511,25 @@ @article{Holzer_2018a, author = {Holzer,Christof and Klopper,Wim}, date-added = {2020-05-18 21:40:28 +0200}, - date-modified = {2020-05-18 21:40:28 +0200}, + date-modified = {2021-01-11 09:18:51 +0100}, doi = {10.1063/1.5051028}, - eprint = {https://doi.org/10.1063/1.5051028}, journal = {J. Chem. Phys.}, number = {10}, pages = {101101}, title = {Communication: A hybrid Bethe--Salpeter/time-dependent density-functional-theory approach for excitation energies}, - url = {https://doi.org/10.1063/1.5051028}, volume = {149}, year = {2018}, Bdsk-Url-1 = {https://doi.org/10.1063/1.5051028}} -@article{Holzer_2018b, - author = {Holzer, Christof and Gui, Xin and Harding, Michael E. and Kresse, Georg and Helgaker, Trygve and Klopper, Wim}, - date-added = {2020-05-18 21:40:28 +0200}, - date-modified = {2020-05-18 21:40:28 +0200}, - doi = {10.1063/1.5047030}, - eprint = {https://doi.org/10.1063/1.5047030}, - journal = {J. Chem. Phys.}, - number = {14}, - pages = {144106}, - title = {Bethe-Salpeter correlation energies of atoms and molecules}, - url = {https://doi.org/10.1063/1.5047030}, - volume = {149}, - year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.5047030}} - @article{Holzer_2019, author = {Holzer,Christof and Teale,Andrew M. and Hampe,Florian and Stopkowicz,Stella and Helgaker,Trygve and Klopper,Wim}, date-added = {2020-05-18 21:40:28 +0200}, - date-modified = {2020-05-18 21:40:28 +0200}, + date-modified = {2021-01-11 09:18:12 +0100}, doi = {10.1063/1.5093396}, - eprint = {https://doi.org/10.1063/1.5093396}, journal = {J. Chem. Phys.}, number = {21}, pages = {214112}, title = {GW Quasiparticle Energies of Atoms in Strong Magnetic Fields}, - url = {https://doi.org/10.1063/1.5093396}, volume = {150}, year = {2019}, Bdsk-Url-1 = {https://doi.org/10.1063/1.5093396}} @@ -4538,15 +4543,12 @@ @article{Stein_2009, author = {Stein, Tamar and Kronik, Leeor and Baer, Roi}, date-added = {2020-05-18 21:40:28 +0200}, - date-modified = {2020-05-18 21:40:28 +0200}, + date-modified = {2021-01-11 09:20:50 +0100}, doi = {10.1021/ja8087482}, - eprint = {https://doi.org/10.1021/ja8087482}, journal = {J. Am. Chem. Soc.}, - note = {PMID: 19239266}, number = {8}, pages = {2818-2820}, title = {Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory}, - url = {https://doi.org/10.1021/ja8087482}, volume = {131}, year = {2009}, Bdsk-Url-1 = {https://doi.org/10.1021/ja8087482}} @@ -4996,10 +4998,10 @@ year = {2012}, Bdsk-Url-1 = {https://doi.org/10.1002/cphc.201100200}} -@article{Holzer_2018, +@article{Holzer_2018b, author = {Christof Holzer and Xin Gui and Michael E. Harding and Georg Kresse and Trygve Helgaker and Wim Klopper}, date-added = {2020-01-04 20:49:55 +0100}, - date-modified = {2020-02-05 20:58:26 +0100}, + date-modified = {2021-01-11 09:18:02 +0100}, doi = {10.1063/1.5047030}, journal = {J. Chem. Phys.}, pages = {144106}, @@ -8699,12 +8701,9 @@ @article{Koch_1990, author = {Koch, Henrik and Jensen, Hans Jo/rgen Aa. and Jo/rgensen, Poul and Helgaker, Trygve}, date-added = {2020-01-01 21:36:51 +0100}, - date-modified = {2020-01-01 21:36:52 +0100}, + date-modified = {2021-01-11 09:32:58 +0100}, doi = {10.1063/1.458815}, - issn = {0021-9606, 1089-7690}, journal = {J. Chem. Phys.}, - language = {en}, - month = sep, number = {5}, pages = {3345-3350}, title = {Excitation Energies from the Coupled Cluster Singles and Doubles Linear Response Function ({{CCSDLR}}). {{Applications}} to {{Be}}, {{CH}} {\textsuperscript{+}} , {{CO}}, and {{H}} {\textsubscript{2}} {{O}}}, @@ -9006,19 +9005,6 @@ volume = {478}, year = {2009}} -@article{Levine_2006a, - author = {Levine, Benjamin G. and Ko, Chaehyuk and Quenneville, Jason and Mart\'Inez, Todd J.}, - date-added = {2020-01-01 21:36:51 +0100}, - date-modified = {2020-01-01 21:36:52 +0100}, - doi = {10.1080/00268970500417762}, - journal = {Mol. Phys.}, - month = mar, - pages = {1039-1051}, - title = {Conical Intersections and Double Excitations in Time-Dependent Density Functional Theory}, - volume = {104}, - year = {2006}, - Bdsk-Url-1 = {https://doi.org/10.1080/00268970500417762}} - @article{Levy_1995, author = {Levy, Mel}, date-added = {2020-01-01 21:36:51 +0100}, @@ -12828,14 +12814,11 @@ Bdsk-Url-1 = {https://doi.org/10.1063/1.1633756}} @article{Levine_2006, - author = {Levine, Benjamin G. and Ko, Chaehyuk and Quenneville, Jason and Mart\'Inez, Todd J.}, + author = {Levine, Benjamin G. and Ko, Chaehyuk and Quenneville, Jason and Martinez, Todd J.}, date-added = {2020-01-01 21:36:32 +0100}, - date-modified = {2020-01-01 21:36:32 +0100}, + date-modified = {2021-01-11 09:19:29 +0100}, doi = {10.1080/00268970500417762}, - issn = {0026-8976, 1362-3028}, journal = {Mol. Phys.}, - language = {en}, - month = mar, number = {5-7}, pages = {1039-1051}, title = {Conical Intersections and Double Excitations in Time-Dependent Density Functional Theory}, diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index 53707b5..e437489 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -577,10 +577,12 @@ In the following, all linear response calculations are performed within the TDA %\titou{As one-electron basis sets, we employ Pople's 6-31G basis or the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.} Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations. -All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}. -The SF-ADC, EOM-SF-CC and SF-TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and the EOM-CCSD calculation with Gaussian 09. \cite{g09} -As a consistency check, we systematically perform the SF-CIS calculations with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies. -Throughout this work, all spin-flip calculations have been performed with a UHF reference. +All the static and dynamic BSE calculations (labeled in the following as SF-BSE and SF-dBSE respectively) are performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}. +The standard and extended spin-flip ADC(2) calculations [SF-ADC(2)-s and SF-ADC(2)-x, respectively] as well as the SF-ADC(3) \cite{Lefrancois_2015} are performed with Q-CHEM 5.2.1. \cite{qchem4} +Spin-flip TD-DFT calculations \cite{Shao_2003} considering the BLYP, B3LYP, and BH\&HLYP functionals with contains $0\%$, $20\%$, and $50\%$ of exact exchange are labeled as SF-TD-BLYP, SF-TD-B3LYP, and SF-TD-BH\&HLYP, respectively, and are also performed with Q-CHEM 5.2.1. +EOM-CCSD excitation energies \cite{Koch_1990,Stanton_1993,Koch_1994} are computed with Gaussian 09. \cite{g09} +As a consistency check, we systematically perform the SF-CIS calculations \cite{Krylov_2001a} with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies. +Throughout this work, all spin-flip calculations are performed with a UHF reference. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results} @@ -606,7 +608,7 @@ The excitation energies corresponding to the first singlet and triplet single ex \begin{tabular}{lcccc} & \mc{4}{c}{Excitation energies (eV)} \\ \cline{2-5} - Method & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$ + Method & $^3P(1s^22s^12p^1)$ & $^1P(1s^22s^12p^1)$ & $^3P(1s^22p^2)$ & $^1D(1s^22p^2)$ \\ \hline SF-TD-BLYP\fnm[1] & 3.210 & 3.210 & 6.691 & 7.598 \\ @@ -625,9 +627,9 @@ The excitation energies corresponding to the first singlet and triplet single ex FCI\fnm[3] & 2.862 & 6.577 & 7.669 & 8.624 \\ \end{tabular} \end{ruledtabular} - \fnt[1]{Excitation energies extracted from Ref.~\onlinecite{Casanova_2020}.} + \fnt[1]{Values from Ref.~\onlinecite{Casanova_2020}.} \fnt[2]{This work.} - \fnt[3]{Excitation energies taken from Ref.~\onlinecite{Krylov_2001a}.} + \fnt[3]{Values from Ref.~\onlinecite{Krylov_2001a}.} \end{table} \end{squeezetable} %%% %%% %%% %%% @@ -637,7 +639,7 @@ The excitation energies corresponding to the first singlet and triplet single ex \includegraphics[width=\linewidth]{Be} \caption{ Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained with the 6-31G basis for various levels of theory: - SD-TD-DFT \cite{Casanova_2020} (red), SF-BSE (blue), SF-CIS \cite{Krylov_2001a} and SF-ADC (orange), and FCI \cite{Krylov_2001a} (black). + SF-TD-DFT \cite{Casanova_2020} (red), SF-BSE (blue), SF-CIS \cite{Krylov_2001a} and SF-ADC (orange), and FCI \cite{Krylov_2001a} (black). All the spin-flip calculations have been performed with a UHF reference. \label{fig:Be}} \end{figure} @@ -751,7 +753,15 @@ The excitation energies corresponding to the first singlet and triplet single ex \subsection{Hydrogen molecule} \label{sec:H2} %=============================== +<<<<<<< HEAD The second system of interest is the \ce{H2} molecule where we streched the bond. The ground state of the \ce{H2} molecule is a singlet with $(1\sigma_g)^2$ configuration. Three excited states were investigated during the streching: the singly excited state B${}^1 \Sigma_u^+$ with $(1\sigma_g )~ (1\sigma_u)$ configuration, the singly excited state E${}^1 \Sigma_g^+$ with $(1\sigma_g )~ (2\sigma_g)$ configuration and the doubly excited state F${}^1 \Sigma_g^+$ with $(1\sigma_u )~ (1\sigma_u)$ configuration. +======= +Our second example deals with the dissociation of the \ce{H2} molecule. +The $\text{X}\,{}^1 \Sigma_g^+$ ground state of \ce{H2} has an electronic configuration $1\sigma_g^2$, and the lowest singlet excited state is of $\text{B}\,{}^1 \Sigma_u^+$ symmetry with a configuration $1\sigma_g 1\sigma_u$. +Of particular interest here are the two lowest singlet excited states of $\Sigma_g^+$ spatial symmetry: the singly excited $\text{E}\,{}^1 \Sigma_g^+$ with configuration $1\sigma_g 2\sigma_g$, and the doubly excited $\text{F}\,{}^1 \Sigma_g^+$ of configuration $1\sigma_u^2$ . +Because these two excited states interact strongly and form an avoided crossing around $R_{\ce{H-H}} = 1.4$ \AA, they are usually labeled as the $\text{EF}\,{}^1 \Sigma_g^+$ state. + +>>>>>>> ec35e13a169105bfc5b5f7b6435e4aab238b63e6 \begin{figure} \includegraphics[width=\linewidth]{H2_BSE_RHF.png} @@ -795,7 +805,8 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations %%% TABLE ?? %%% \begin{table} \caption{ - Vertical excitation energies (with respect to the singlet $X\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $X\,{}^1 A_{g}$ singlet ground state. + Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ singlet ground state. + All the spin-flip calculations have been performed with a UHF reference. \label{tab:CBD_D2h}} \begin{ruledtabular} \begin{tabular}{lccc} @@ -810,12 +821,17 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations SF-ADC(2)-s\fnm[2] & & & \\ SF-ADC(2)-x\fnm[2] & & & \\ SF-ADC(3)\fnm[2] & & & \\ +<<<<<<< HEAD SF-BSE@{\GOWO}@UHF\fnm[3] & 1.438 & 2.704 &4.540 \\ SF-dBSE@{\GOWO}@UHF\fnm[3] & 1.403 &2.883 &4.621 \\ +======= + SF-BSE@{\GOWO}\fnm[3] & & & \\ + SF-dBSE@{\GOWO}\fnm[3] & & & \\ +>>>>>>> ec35e13a169105bfc5b5f7b6435e4aab238b63e6 \end{tabular} \end{ruledtabular} - \fnt[1]{Value from Ref.~\onlinecite{Manohar_2008} using a UHF reference.} - \fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015} using a UHF reference.} + \fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.} + \fnt[2]{Values from Ref.~\onlinecite{Lefrancois_2015}.} \fnt[3]{This work.} \end{table} %%% %%% %%% %%% @@ -823,8 +839,9 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations %%% TABLE ?? %%% \begin{table} \caption{ - Vertical excitation energies (with respect to the singlet $X\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $X\,{}^1B_{1g}$ singlet ground state. - \label{tab:CBD_D2h}} + Vertical excitation energies (with respect to the singlet $X\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $\text{X}\,{}^1B_{1g}$ singlet ground state. + All the spin-flip calculations have been performed with a UHF reference. + \label{tab:CBD_D4h}} \begin{ruledtabular} \begin{tabular}{lccc} & \mc{3}{c}{Excitation energies (eV)} \\ @@ -838,12 +855,17 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations SF-ADC(2)-s\fnm[2] & & & \\ SF-ADC(2)-x\fnm[2] & & & \\ SF-ADC(3)\fnm[2] & & & \\ +<<<<<<< HEAD SF-BSE@{\GOWO}@UHF\fnm[3] & -0.049 & 1.189 & 1.480 \\ SF-dBSE@{\GOWO}@UHF\fnm[3] & 0.012 & 1.507 & 1.841 \\ +======= + SF-BSE@{\GOWO}\fnm[3] & & & \\ + SF-dBSE@{\GOWO}\fnm[3] & & & \\ +>>>>>>> ec35e13a169105bfc5b5f7b6435e4aab238b63e6 \end{tabular} \end{ruledtabular} - \fnt[1]{Value from Ref.~\onlinecite{Manohar_2008} using a UHF reference.} - \fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015} using a UHF reference.} + \fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.} + \fnt[2]{Values from Ref.~\onlinecite{Lefrancois_2015}.} \fnt[3]{This work.} \end{table} %%% %%% %%% %%% diff --git a/Notebooks/sf-BSE.nb b/Notebooks/sf-BSE.nb index 6766cd1..51e6c61 100644 --- a/Notebooks/sf-BSE.nb +++ b/Notebooks/sf-BSE.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 268110, 4885] -NotebookOptionsPosition[ 265513, 4828] -NotebookOutlinePosition[ 265907, 4844] -CellTagsIndexPosition[ 265864, 4841] +NotebookDataLength[ 261405, 4767] +NotebookOptionsPosition[ 258809, 4710] 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