diff --git a/Manuscript/Be.pdf b/Manuscript/Be.pdf new file mode 100644 index 0000000..98e9b94 Binary files /dev/null and b/Manuscript/Be.pdf differ diff --git a/Manuscript/sfBSE.bib b/Manuscript/sfBSE.bib index 35cd375..6fc2d7b 100644 --- a/Manuscript/sfBSE.bib +++ b/Manuscript/sfBSE.bib @@ -1,7 +1,7 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2021-01-10 18:27:42 +0100 +%% Created for Pierre-Francois Loos at 2021-01-10 22:26:03 +0100 %% Saved with string encoding Unicode (UTF-8) @@ -1492,23 +1492,6 @@ Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261401002871}, Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(01)00287-1}} -@article{Krylov_2001b, - abstract = {A new approach to the bond-breaking problem is proposed. Both closed and open shell singlet states are described within a single reference formalism as spin-flipping, e.g., α→β, excitations from a triplet (Ms=1) reference state for which both dynamical and non-dynamical correlation effects are much smaller than for the corresponding singlet state. Formally, the new theory can be viewed as an equation-of-motion (EOM) model where excited states are sought in the basis of determinants conserving the total number of electrons but changing the number of α and β electrons. The results for two simplest members of the proposed hierarchy of approximations are presented.}, - author = {Anna I. Krylov}, - date-added = {2020-12-06 14:36:46 +0100}, - date-modified = {2020-12-06 14:37:01 +0100}, - doi = {https://doi.org/10.1016/S0009-2614(01)00287-1}, - issn = {0009-2614}, - journal = {Chem. Phys. Lett.}, - number = {4}, - pages = {375 - 384}, - title = {Size-consistent wave functions for bond-breaking: the equation-of-motion spin-flip model}, - url = {http://www.sciencedirect.com/science/article/pii/S0009261401002871}, - volume = {338}, - year = {2001}, - Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261401002871}, - Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(01)00287-1}} - @article{Krylov_2002, author = {Krylov,Anna I. and Sherrill,C. David}, date-added = {2020-12-06 14:36:32 +0100}, @@ -4204,9 +4187,9 @@ @article{Refaely-Abramson_2012, author = {Sivan Refaely-Abramson and Sahar Sharifzadeh and Niranjan Govind and Jochen Autschbach and Jeffrey B. Neaton and Roi Baer and Leeor Kronik}, date-added = {2020-05-18 21:40:28 +0200}, - date-modified = {2020-05-18 21:40:28 +0200}, + date-modified = {2021-01-10 20:59:42 +0100}, doi = {10.1103/PhysRevLett.109.226405}, - journal = {Phys. Rev. X}, + journal = {Phys. Rev. Lett.}, pages = {226405}, title = {Quasiparticle Spectra from a Nonempirical Optimally Tuned Range-Separated Hybrid Density Functional}, volume = {109}, @@ -8832,10 +8815,10 @@ Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.075143}, Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.93.075143}} -@article{Krylov_2001c, +@article{Krylov_2001b, author = {Krylov, Anna I.}, date-added = {2020-01-01 21:36:51 +0100}, - date-modified = {2020-12-06 14:37:24 +0100}, + date-modified = {2021-01-10 22:26:03 +0100}, doi = {10.1016/S0009-2614(01)01316-1}, issn = {00092614}, journal = {Chem. Phys. Lett.}, diff --git a/Manuscript/sfBSE.tex b/Manuscript/sfBSE.tex index b7e4ed7..eaaed28 100644 --- a/Manuscript/sfBSE.tex +++ b/Manuscript/sfBSE.tex @@ -572,13 +572,15 @@ Note that, in any case, the entire set of orbitals and energies is corrected. Further details about our implementation of {\GOWO}, ev$GW$, and qs$GW$ can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h}. Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} and BSE@ev$GW$ excitation energies. This is left for future work. -However, it is worth mentioning that, for the present (small) molecular systems, HF is usually an excellent starting point. \cite{Loos_2020a,Loos_2020e,Loos_2020h} +However, it is worth mentioning that, for the present (small) molecular systems, HF is usually a good starting point, \cite{Loos_2020a,Loos_2020e,Loos_2020h} although improvements could certainly be obtained with starting orbitals and energies computed with, for example, optimally-tuned range-separated hybrid functionals. \cite{Stein_2009,Stein_2010,Refaely-Abramson_2012,Kronik_2012} In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results. -\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.} +%\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 TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and the EOM-CCSD calculation with Gaussian 09. \cite{g09} +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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results} @@ -589,40 +591,56 @@ The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and \subsection{Beryllium atom} \label{sec:Be} %=============================== +As a first example, we consider the simple case of the beryllium atom which was considered by Krylov in two of her very first papers on spin-flip methods \cite{Krylov_2001a,Krylov_2001b} and was also considered in later studies thanks to its pedagogical value. \cite{Sears_2003,Casanova_2020} +Beryllium has a $^1S$ ground state with $1s^2 2s^2$ configuration. +The excitation energies corresponding to the first singlet and triplet single excitations $2s \to 2p$ with $P$ spatial symmetry as well as the first singlet and triplet double excitations $2s^2 \to 2p^2$ with $P$ and $D$ spatial symmetries (respectively), obtained with the 6-31G basis set are reported in Table \ref{tab:Be} and depicted in Fig.~\ref{fig:Be}. %%% TABLE I %%% \begin{squeezetable} \begin{table} \caption{ Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained for various methods with the 6-31G basis. + All the spin-flip calculations have been performed with a UHF reference. \label{tab:Be}} \begin{ruledtabular} \begin{tabular}{lcccc} - & \mc{4}{c}{Excitation energies (eV)} \\ + & \mc{4}{c}{Excitation energies (eV)} \\ \cline{2-5} - Method & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$ - & $^3P(1s^22p^2)$ & $^1P(1s^22p^2)$ \\ + Method & $^3P(1s^22s2p)$ & $^1P(1s^22s2p)$ + & $^3P(1s^22p^2)$ & $^1D(1s^22p^2)$ \\ \hline - SF-TD-BLYP & 3.210 & 3.210 & 6.691 & 7.598 \\ - SF-TD-B3LYP & 3.332 & 4.275 & 6.864 & 7.762 \\ - SF-TD-BHHLYP & 2.874 & 4.922 & 7.112 & 8.188 \\ - SF-CIS & 2.111 & 6.036 & 7.480 & 8.945 \\ - SF-BSE@{\GOWO}@UHF & 2.399 & 6.191 & 7.792 & 9.373 \\ - SF-BSE@{\evGW}@UHF & 2.407 & 6.199 & 7.788 & 9.388 \\ - SF-BSE@{\qsGW}@UHF & 2.376 & 6.241 & 7.668 & 9.417 \\ - SF-dBSE@{\GOWO}@UHF & 2.363 & 6.263 & 7.824 & 9.424 \\ - SF-dBSE@{\evGW}@UHF & 2.369 & 6.273 & 7.820 & 9.441 \\ - SF-dBSE@{\qsGW}@UHF & 2.335 & 6.317 & 7.689 & 9.470 \\ - SF-ADC(2)-s & 2.433 & 6.255 & 7.745 & 9.047 \\ - SF-ADC(2)-x & 2.866 & 6.581 & 7.664 & 8.612 \\ - SF-ADC(3) & 2.863 & 6.579 & 7.658 & 8.618 \\ - FCI & 2.862 & 6.577 & 7.669 & 8.624 \\ + SF-TD-BLYP\fnm[1] & 3.210 & 3.210 & 6.691 & 7.598 \\ + SF-TD-B3LYP\fnm[1] & 3.332 & 4.275 & 6.864 & 7.762 \\ + SF-TD-BH\&HLYP\fnm[1] & 2.874 & 4.922 & 7.112 & 8.188 \\ + SF-BSE@{\GOWO}\fnm[2] & 2.399 & 6.191 & 7.792 & 9.373 \\ + SF-BSE@{\evGW}\fnm[2] & 2.407 & 6.199 & 7.788 & 9.388 \\ + SF-BSE@{\qsGW}\fnm[2] & 2.376 & 6.241 & 7.668 & 9.417 \\ + SF-dBSE@{\GOWO}\fnm[2] & 2.363 & 6.263 & 7.824 & 9.424 \\ + SF-dBSE@{\evGW}\fnm[2] & 2.369 & 6.273 & 7.820 & 9.441 \\ + SF-dBSE@{\qsGW}\fnm[2] & 2.335 & 6.317 & 7.689 & 9.470 \\ + SF-CIS\fnm[3] & 2.111 & 6.036 & 7.480 & 8.945 \\ + SF-ADC(2)-s\fnm[2] & 2.433 & 6.255 & 7.745 & 9.047 \\ + SF-ADC(2)-x\fnm[2] & 2.866 & 6.581 & 7.664 & 8.612 \\ + SF-ADC(3)\fnm[2] & 2.863 & 6.579 & 7.658 & 8.618 \\ + 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[2]{This work.} + \fnt[3]{Excitation energies taken from Ref.~\onlinecite{Krylov_2001a}.} \end{table} \end{squeezetable} %%% %%% %%% %%% +%%% FIG. 1 %%% +\begin{figure} + \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). + All the spin-flip calculations have been performed with a UHF reference. + \label{fig:Be}} +\end{figure} %%% TABLE II %%% %\begin{squeezetable} diff --git a/Notebooks/sf-BSE.nb b/Notebooks/sf-BSE.nb index 115e38d..6766cd1 100644 --- a/Notebooks/sf-BSE.nb +++ b/Notebooks/sf-BSE.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 245153, 4491] -NotebookOptionsPosition[ 242557, 4434] -NotebookOutlinePosition[ 242951, 4450] -CellTagsIndexPosition[ 242908, 4447] 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