diff --git a/H8_cc-pvdz/pes_ooCIo2.dat b/H8_cc-pvdz/pes_ooCIo2.dat index c421f72..f9fa784 100644 --- a/H8_cc-pvdz/pes_ooCIo2.dat +++ b/H8_cc-pvdz/pes_ooCIo2.dat @@ -10,9 +10,9 @@ 1.8 -4.48292621 1.85 -4.48109504 1.9 -4.47708231 -1.95 -4.47024232 +1.95 -4.47065633 2.0 -4.46242175 -2.05 -4.45241887 +2.05 -4.45265761 2.1 -4.44160247 2.15 -4.42947635 2.2 -4.41646807 @@ -26,7 +26,7 @@ 2.9 -4.20336457 3.0 -4.17400938 3.1 -4.14572503 -3.2 -4.10746165 +3.2 -4.11861535 3.3 -4.09274385 3.4 -4.06815391 3.5 -4.04482786 @@ -57,7 +57,7 @@ 6.0 -3.74169432 6.1 -3.73609119 6.2 -3.73076556 -6.3 -3.72570301 +6.3 -3.72570304 6.4 -3.72089049 6.5 -3.71631549 6.6 -3.71196536 diff --git a/Manuscript/seniority.bib b/Manuscript/seniority.bib index d2f2a94..c6e08ba 100644 --- a/Manuscript/seniority.bib +++ b/Manuscript/seniority.bib @@ -687,3 +687,21 @@ volume = {7}, year = {1967}, bdsk-url-1 = {https://doi.org/10.1007/BF01151915}} + +@article{Loos_2018, +abstract = {Striving to define very accurate vertical transition energies, we perform both high-level coupled cluster (CC) calculations (up to CCSDTQP) and selected configuration interaction (sCI) calculations (up to several millions of determinants) for 18 small compounds (water, hydrogen sulfide, ammonia, hydrogen chloride, dinitrogen, carbon monoxide, acetylene, ethylene, formaldehyde, methanimine, thioformaldehyde, acetaldehyde, cyclopropene, diazomethane, formamide, ketene, nitrosomethane, and the smallest streptocyanine). By systematically increasing the order of the CC expansion, the number of determinants in the CI expansion as well as the size of the one-electron basis set, we have been able to reach near full CI (FCI) quality transition energies. These calculations are carried out on CC3/aug-cc-pVTZ geometries, using a series of increasingly large atomic basis sets systematically including diffuse functions. In this way, we define a list of 110 transition energies for states of various characters (valence, Rydberg, n → $\pi$∗, $\pi$ → $\pi$ ∗, singlet, triplet, etc.) to be used as references for further calculations. Benchmark transition energies are provided at the aug-cc-pVTZ level as well as with additional basis set corrections, in order to obtain results close to the complete basis set limit. These reference data are used to benchmark a series of 12 excited-state wave function methods accounting for double and triple contributions, namely ADC(2), ADC(3), CIS(D), CIS(D∞), CC2, STEOM-CCSD, CCSD, CCSDR(3), CCSDT-3, CC3, CCSDT., and CCSDTQ. It turns out that CCSDTQ yields a negligible difference with the extrapolated CI values with a mean absolute error as small as 0.01 eV, whereas the coupled cluster approaches including iterative triples are also very accurate (mean absolute error of 0.03 eV). Consequently, CCSDT-3 and CC3 can be used to define reliable benchmarks. This observation does not hold for ADC(3) that delivers quite large errors for this set of small compounds, with a clear tendency to overcorrect its second-order version, ADC(2). Finally, we discuss the possibility to use basis set extrapolation approaches so as to tackle more easily larger compounds.}, +archivePrefix = {arXiv}, +arxivId = {1807.02045}, +author = {Loos, Pierre Fran{\c{c}}ois and Scemama, Anthony and Blondel, Aymeric and Garniron, Yann and Caffarel, Michel and Jacquemin, Denis}, +doi = {10.1021/acs.jctc.8b00406}, +eprint = {1807.02045}, +issn = {15499626}, +journal = {Journal of Chemical Theory and Computation}, +number = {8}, +pages = {4360--4379}, +pmid = {29966098}, +title = {{A Mountaineering Strategy to Excited States: Highly Accurate Reference Energies and Benchmarks}}, +volume = {14}, +year = {2018} +} + diff --git a/Manuscript/seniority.tex b/Manuscript/seniority.tex index 25611a4..46eaf20 100644 --- a/Manuscript/seniority.tex +++ b/Manuscript/seniority.tex @@ -206,11 +206,12 @@ We have calculated the potential energy curves (PECs) for the dissociation of si which display a variable number of bond breaking. For the latter two molecules, we considered linearly arranged with equally spaced hydrogen atoms, and computed PECs along the symmetric dissociation coordinate. For ethylene, we considered the \ce{C=C} double bond breaking, while freezing the remaining internal coordinates. +Its equilibrium geometry was taken from Ref.~\cite{Loos_2018} and is reproduced in the \SupInf. Due to the (multiple) bond breaking, these are challenging systems for electronic structure methods, being often considered when assessing novel methodologies. We evaluated the convergence of four observables: the non-parallelity error (NPE), the distance error, the vibrational frequencies, and the equilibrium geometries. The NPE is defined as the maximum minus the minimum differences between the PECs obtained at given CI level and the exact FCI result. -We define the distance error as the maximum and the minimum differences between a given PEC and the FCI result. +We define the distance error as the maximum plus the minimum differences between a given PEC and the FCI result. Thus, while the NPE probes the similarity regarding the shape of the PECs, the distance error provides a measure of how their overall magnitudes compare. From the PECs, we have also extracted the vibrational frequencies and equilibrium geometries (details can be found in the \SupInf). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -227,8 +228,9 @@ which requires considerably fewer determinants than the formal number of determi Nevertheless, we decided to present the results as functions of the formal number of determinants, which are not related to the particular algorithmic choices of the CIPSI calculations. All CI calculations were performed for the cc-pVDZ basis set and with frozen core orbitals. -For \ce{HF} we have also tested basis set effects, by considered the cc-pVTZ and cc-pVQZ basis sets. +For the \ce{HF} molecule we have also tested basis set effects, by considered the cc-pVTZ and cc-pVQZ basis sets. \titou{Geometries? SI?} +\fk{Only the geometry of ethylene has to be given. I added it to the Sup. Info.} \titou{T2: I think it might be worth mentioning that the determinant-driven framework of {\QP} allows to include any arbitrary set of determinants. This would also justify why we are focusing on the number of determinants instead of the actual scaling of the method. I think this is a important point because the CISD Hilbert space has a size proportional to $N^4$ but the cost associated with solving the CISD equations scales as $N^6$... Actually, it follows the same rules as CC: CISD scales as $N^6$, CISDT as $N^8$, CISDTQ as $N^{10}$, etc. @@ -276,11 +278,7 @@ For \ce{H8}, hCI and excitation-based CI perform similarly. The convergence with respect to $\Ndet$ is slower in the latter, more challenging cases, irrespective of the class of CI methods, as would be expected. But more importantly, the superiority of the hCI methods appears to be highlighted in the multiple bond break systems (compare ethylene and \ce{N2} with \ce{HF} and \ce{F2} in Fig.~\ref{fig:plot_stat}). \titou{T2: Would it be a good idea to write the \ce{HF} molecule as \ce{FH}?} - -For \ce{HF} we have also evaluated how the convergence is affected by increasing the basis sets, going from cc-pVDZ to cc-pVTZ and cc-pVQZ (see Fig.~Sx in the \SupInf). -While a larger $\Ndet$ is required to achieve the same level of convergence, as expected, -the convergence profiles remain very similar for all basis sets. -We thus believe that the main findings discussed here for the other systems would be equally basis set independent. +\fk{Don't think so. I included ``molecule'' after HF whenever one could have understood Hartree-Fock instead.} %%% FIG 2 %%% \begin{figure}[h!] @@ -328,7 +326,6 @@ For both observables, hCI and excitation-based CI largely outperform seniority-b Similarly to what we observed for the NPEs, the convergence of hCI was also found to be non-monotonic in some cases. This oscillatory behavior is particularly evident for \ce{F2}, also noticeable for \ce{HF}, becoming less apparent for ethylene, virtually absent for \ce{N2}, and showing up again for \ce{H4} and \ce{H8}. -Results for \ce{HF} with larger basis sets (see Fig.Sx in the \SupInf) show very similar convergence behaviors, though with less oscillations for the hCI methods. Interestingly, equilibrium geometries and vibrational frequencies of \ce{HF} and \ce{F2} (single bond breaking), are rather accurate when evaluated at the hCI1.5 level, bearing in mind its relatively modest computational cost. @@ -350,6 +347,12 @@ are rather accurate when evaluated at the hCI1.5 level, bearing in mind its rela \end{figure} %%% %%% %%% +For the \ce{HF} molecule we have also evaluated how the convergence is affected by increasing the basis sets, going from cc-pVDZ to cc-pVTZ and cc-pVQZ (see Fig.~Sx and Fig.~Sy in the \SupInf). +While a larger $\Ndet$ is required to achieve the same level of convergence, as expected, +the convergence profiles remain very similar for all basis sets. +Vibrational frequency and equilibrium geometry present less oscillations for the hCI methods. +We thus believe that the main findings discussed here for the other systems would be equally basis set independent. + %\subsection{Orbital optimized configuration interaction} \titou{T2: Would it be a good idea to have mentioned that seniority-based schemes are not invariant with respect to orbital rotations?} diff --git a/Manuscript/sup.tex b/Manuscript/sup.tex index ec2db16..5ee5f3d 100644 --- a/Manuscript/sup.tex +++ b/Manuscript/sup.tex @@ -78,6 +78,22 @@ %x %\newpage +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\section{Equilibrium geometry of ethylene} +%\label{sec:ethylene} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Equilibrium geometry of ethylene, in atomic units. + +\begin{tabular}{ r r r r } + C & 0.00000000 & 1.26026583 & 0.00000000 \\ + C & 0.00000000 & -1.26026583 & 0.00000000 \\ + H & 0.00000000 & 2.32345976 & 1.74287672 \\ + H & 0.00000000 & -2.32345976 & 1.74287672 \\ + H & 0.00000000 & 2.32345976 & -1.74287672 \\ + H & 0.00000000 & -2.32345976 & -1.74287672 \\ +\end{tabular} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section{oo-CI} %\label{sec:oo-CI}