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Manuscript/FCI2.tex
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@ -19,9 +19,9 @@
% energies
\newcommand{\EFCI}{E_\text{FCI}}
\newcommand{\EexCI}{E_\text{exCI}}
\newcommand{\EsCI}{E_\text{sCI}}
\newcommand{\ESCI}{E_\text{SCI}}
\newcommand{\EPT}{E_\text{PT2}}
\newcommand{\PsisCI}{\Psi_\text{sCI}}
\newcommand{\PsiSCI}{\Psi_\text{SCI}}
\newcommand{\Ndet}{N_\text{det}}
\newcommand{\ex}[6]{$^{#1}#2_{#3}^{#4}(#5 \ra #6)$}
@ -44,7 +44,7 @@
\newcommand{\CCSDTQ}{CCSDTQ}
\newcommand{\CCSDTQP}{CCSDTQP}
\newcommand{\CI}{CI}
\newcommand{\sCI}{sCI}
\newcommand{\SCI}{SCI}
\newcommand{\exCI}{exCI}
\newcommand{\FCI}{FCI}
@ -104,7 +104,7 @@
Following our previous work focussing on compounds containing up to 3 non-hydrogen atoms [\emph{J. Chem. Theory Comput.} {\bfseries 14} (2018) 4360--4379], we present here highly-accurate vertical transition energies
obtained for 27 molecules encompassing 4, 5, and 6 non-hydrogen atoms: acetone, acrolein, benzene, butadiene, cyanoacetylene, cyanoformaldehyde, cyanogen, cyclopentadiene, cyclopropenone, cyclopropenethione,
diacetylene, furan, glyoxal, imidazole, isobutene, methylenecyclopropene, propynal, pyrazine, pyridazine, pyridine, pyrimidine, pyrrole, tetrazine, thioacetone, thiophene, thiopropynal, and triazine.
To obtain these energies, we use equation-of-motion coupled cluster theory up to the highest technically possible excitation order for these systems ({\CCT}, {\CCSDT}, and {\CCSDTQ}), selected configuration interaction ({\sCI}) calculations (with tens of millions of determinants in the reference space),
To obtain these energies, we use equation-of-motion coupled cluster theory up to the highest technically possible excitation order for these systems ({\CCT}, {\CCSDT}, and {\CCSDTQ}), selected configuration interaction ({\SCI}) calculations (with tens of millions of determinants in the reference space),
as well as the multiconfigurational $n$-electron valence state perturbation theory (NEVPT2) method.
All these approaches are applied in combination with diffuse-containing atomic basis sets. For all transitions, we report at least {\CCT}/{\AVQZ} vertical excitation
energies as well as {\CCT}/{\AVTZ} oscillator strengths for each dipole-allowed transition. We show that {\CCT} almost systematically delivers transition energies in agreement with higher-level theoretical methods with a typically deviation of $\pm 0.04$ eV, except for transitions with a dominant double excitation character where the error is much larger.
@ -125,7 +125,9 @@ one is typically limited to the use of time-dependent density-functional theory
choose an ``appropriate'' exchange-correlation functional, which is difficult yet primordial as the impact of the exchange-correlation functional is exacerbated within TD-DFT as compared to DFT. \cite{Lau13} Such selection can, of course, rely
on the intrinsic features of the various exchange-correlation functional families, \eg, it is well-known that range-separated hybrids provide a more physically-sound description of long-range charge-transfer transitions than semi-local exchange-correlation functionals. \cite{Dre04,Pea08}
However, to obtain a quantitative assessment of the accuracy that can be expected from TD-DFT calculations, benchmarks cannot be avoided. This is why so many assessments of TD-DFT performances for various properties are
available. \cite{Lau13} While several of these benchmarks rely on experimental data as reference (typically band shapes \cite{Die04,Die04b,Avi13,Cha13,Lat15b,Mun15,Vaz15,San16b} or 0-0 energies
available. \cite{Lau13}
While several of these benchmarks rely on experimental data as reference (typically band shapes \cite{Die04,Die04b,Avi13,Cha13,Lat15b,Mun15,Vaz15,San16b} or 0-0 energies
\cite{Die04b,Goe10a,Jac12d,Chi13b,Win13,Fan14b,Jac14a,Jac15b,Loo19b}), reference from theoretical best estimates (TBE) based on state-of-the-art computational methods \cite{Sch08,Sau09,Sil10b,Sil10c,Sch17,Loo18a}
is advantageous as it allows comparisons on a perfectly equal footing (same geometry, vertical transitions, no environmental effects, etc). In such a case, the challenge is in fact to obtain accurate TBE, as these top-notch theoretical models
generally come with a dreadful scaling with system size and, in addition, typically require large atomic basis sets to deliver transition energies close to the complete basis set (CBS) limit.
@ -138,22 +140,28 @@ However, beyond comparisons with experiments, which are always challenging when
Nowadays, it is common knowledge that CASPT2 has the tendency of underestimating vertical excitation energies in organic molecules.
A decade ago, Thiel and coworkers defined TBE for 104 singlet and 63 triplet valence ES in 28 small and medium conjugated CNOH organic molecules. \cite{Sch08,Sil10b,Sil10c} These TBE
were computed on MP2/6-31G(d) structures with several levels of theories, notably {\CASPT} and various coupled cluster (CC) variants ({\CCD}, {\CCSD}, and {\CCT}). Interestingly, while the default reference approach used by Thiel and coworkers to define his
first series of TBE was {\CASPT}, \cite{Sch08} the majority of the most recent TBE (so-called ``TBE-2'' in Ref.~\citenum{Sil10c}) were determined at the {\CCT}/{\AVTZ} level, often using a basis set extrapolation technique.
In more details, CC3/TZVP values were typically corrected for basis set effects by the difference between {\CCD}/{\AVTZ} and {\CCD}/TZVP results. \cite{Sil10b,Sil10c} Many works used Thiel's TBE for
assessing low-order methods, \cite{Sil08,Goe09,Jac09c,Roh09,Sau09,Jac10c,Jac10g,Sil10,Mar11,Jac11a,Hui11,Del11,Tra11,Pev12,Dom13,Dem13,Sch13b,Voi14,Har14,Yan14b,Sau15,Pie15,Taj16,Mai16,Ris17,Dut18,Hel19} highlighting
their value for the community. In contrast, the number of extensions of this original set remains quite limited, \ie, {\CCSDT}/TZVP reference energies computed for 17 singlet states of six molecules appeared in 2014, \cite{Kan14}
and 46 {\CCSDT}/{\AVTZ} transition energies in small compounds containing two or three non-hydrogen atoms (ethylene, acetylene, formaldehyde, formaldimine, and formamide) have been described in 2017. \cite{Kan17}
This has motivated us to propose two years ago a set of 106 transition energies for which it was technically possible to reach the Full Configuration Interaction (FCI) limit by performing high-order CC calculations (up to {\CCSDTQP}) and selected
CI (sCI) transition energy calculations on {\CCT} GS structures. \cite{Loo18a} We used these TBE to benchmark many ES theories. \cite{Loo18a} Amongst our conclusions, we found that {\CCSDTQ} was on the {\FCI} spot, whereas we could not
detect significant differences of accuracies between {\CCT} and {\CCSDT}, both being very accurate with mean absolute errors (MAE) as small as 0.03 eV compared to {\FCI} for ES with a single excitation character. These conclusions
agree well with earlier studies. \cite{Wat13,Kan14,Kan17} We also recently proposed a set of 20 TBE for transitions presenting a large double excitation character. \cite{Loo19c} For such transitions, one can distinguish the ES
as a function of $\%T_1$, the percentage of single excitation calculated at the {\CCT} level. For ES with a significant yet not dominant double excitation character, such as the famous $A_g$ ES of butadiene ($\%T_1 = 75\%$),
CC methods including triples in the GS treatment deliver rather accurate estimates (MAE of $0.11$ eV with {\CCT} and $0.06$ eV with {\CCSDT}), surprisingly outperforming second-order multi-reference schemes such as {\CASPT} or
the generally robust $N$-Electron Valence State Perturbation Theory ({\NEV}). In contrast, for ES with a dominant double excitation character, \eg, the low-lying $n,n \ra \pis,\pis$ excitation in nitrosomethane ($\%T_1$=2\%),
single-reference methods are not suited (MAE of $0.86$ eV with {\CCT} and $0.42$ eV with {\CCSDT}) and multi-reference methods are, in practice, required to obtain accurate results. \cite{Loo19c} Obviously, a clear limit of our 2018 work\cite{Loo18a}
is that we treated only compounds containing 1--3 non-hydrogen atoms, hence introducing a significant chemical bias. Therefore we have decided to go for larger molecules and we consider in the present contribution organic
compounds encompassing 4, 5, and 6 non-hydrogen atoms. For such systems, performing {\CCSDTQ} calculations with large bases remains a dream, and, the convergence of {\sCI} with the number of determinants is slower
were computed on MP2/6-31G(d) structures with several levels of theories, notably {\CASPT} and various coupled cluster (CC) variants ({\CCD}, {\CCSD}, and {\CCT}). Interestingly, while the default theoretical protocol used by Thiel and coworkers to define their
first series of TBE used {\CASPT}, \cite{Sch08} the vast majority of their most recent TBE (the so-called ``TBE-2'' in Ref.~\citenum{Sil10c}) were determined at the {\CCT}/{\AVTZ} level of theory, often using a basis set extrapolation technique.
More specifically, CC3/TZVP values were corrected for basis set incompleteness errors by the difference between {\CCD}/{\AVTZ} and {\CCD}/TZVP results. \cite{Sil10b,Sil10c} Many works exploited Thiel's TBE for
assessing low-order methods, \cite{Sil08,Goe09,Jac09c,Roh09,Sau09,Jac10c,Jac10g,Sil10,Mar11,Jac11a,Hui11,Del11,Tra11,Pev12,Dom13,Dem13,Sch13b,Voi14,Har14,Yan14b,Sau15,Pie15,Taj16,Mai16,Ris17,Dut18,Hel19} highlighting further
their value for the electronic structure community. In contrast, the number of extensions/improvements of this original set remains quite limited.
For example, K\'ann\'ar and Szalay computed, in 2014, {\CCSDT}/TZVP reference energies for 17 singlet states of six molecules. \cite{Kan14}
Three years later, the same authors reported 46 {\CCSDT}/{\AVTZ} transition energies in small compounds containing two or three non-hydrogen atoms (ethylene, acetylene, formaldehyde, formaldimine, and formamide). \cite{Kan17}
Following the same philosophy, two years ago, we reported a set of 106 transition energies for which it was technically possible to reach the full configuration interaction (FCI) limit by performing high-order CC (up to {\CCSDTQP}) and selected
CI (SCI) calculations on {\CCT}/{\AVTZ} GS structures. \cite{Loo18a} We exploited these TBE to benchmark many ES theories. \cite{Loo18a} Amongst our conclusions, we found that {\CCSDTQ} yields near-{\FCI} quality excitation energies, whereas we could not
detect any significant differences between {\CCT} and {\CCSDT} transition energies, both being very accurate with mean absolute errors (MAE) as small as $0.03$ eV compared to {\FCI}.
Although these conclusions agree well with earlier studies, \cite{Wat13,Kan14,Kan17} they obviously only hold for single excitations, \ie, transitions with $\%T_1$ in the range $90$--$100\%$. Therefore, we also recently proposed a set of 20 TBE for transitions exhibiting a significant double-excitation character (\ie, with $\%T_1$ typically below $80\%$). \cite{Loo19c}
Unsurprisingly, our results clearly evidenced that the error in CC methods is intimately related to the $\%T_1$ value.
For example, ES with a significant yet \titou{not dominant} double excitation character, such as the infamous $A_g$ ES of butadiene ($\%T_1 = 75\%$),
CC methods including triples deliver rather accurate estimates (MAE of $0.11$ eV with {\CCT} and $0.06$ eV with {\CCSDT}), surprisingly outperforming second-order multi-reference schemes such as {\CASPT} or
the generally robust $n$-electron valence state perturbation theory ({\NEV}). In contrast, for ES with a dominant double excitation character, \eg, the low-lying $(n,n) \ra (\pis,\pis)$ excitation in nitrosomethane ($\%T_1 = 2\%$),
single-reference methods have been found to be unsuitable with MAEs of $0.86$ and $0.42$ eV for {\CCT} and {\CCSDT}, respectively.
In this case, multireference methods are required to obtain accurate results. \cite{Loo19c}
Obviously, a clear limit of our 2018 work\cite{Loo18a} is that we treated only compounds containing 1--3 non-hydrogen atoms, hence introducing a significant chemical bias. Therefore we have decided to go for larger molecules and we consider in the present contribution organic
compounds encompassing 4, 5, and 6 non-hydrogen atoms. For such systems, performing {\CCSDTQ} calculations with large bases remains a dream, and, the convergence of {\SCI} with the number of determinants is slower
as well, so that extrapolating to the {\FCI} limit with a ca.~$0.01$ eV error bar is rarely doable in practice. Consequently, the ``brute-force'' determination of {\FCI}/CBS estimates, as in our earlier work,\cite{Loo18a} is beyond reach.
Anticipating this problem, we have previously demonstrated that one can very accurately estimate such limit by correcting values obtained with a high-level of theory and a double-$\zeta$ basis set by {\CCT} results obtained with a
larger basis, \cite{Loo18a} and we glibally follow such strategy here. In addition, we also performed {\NEV} calculations in an effort to check the consistency of our estimates, especially for ES with intermediate $\%T_1$ values.
@ -180,21 +188,21 @@ Several structures have been extracted from previous contributions, \cite{Bud17,
\subsection{Selected Configuration Interaction methods}
All the sCI calculations have been performed in the frozen-core approximation with the latest version of QUANTUM PACKAGE \cite{Gar19} using the Configuration Interaction using a Perturbative Selection made Iteratively (CIPSI) algorithm to select the
All the SCI calculations have been performed in the frozen-core approximation with the latest version of QUANTUM PACKAGE \cite{Gar19} using the Configuration Interaction using a Perturbative Selection made Iteratively (CIPSI) algorithm to select the
most important determinants in the FCI space. Instead of generating all possible excited determinants like a conventional CI calculation, the iterative CIPSI algorithm performs a sparse exploration of the FCI space via a selection of the most relevant
determinants using a second-order perturbative criterion. At each iteration, the variational (or reference) space is enlarged with new determinants. CIPSI can be seen as a deterministic version of the FCIQMC algorithm developed by Alavi and
coworkers. \cite{Boo09} We refer the interested reader to Ref.~\citenum{Gar19} where our implementation of the CIPSI algorithm is detailed.
Excited-state calculations are performed within a state-averaged formalism which means that the CIPSI algorithm select determinants simultaneously for the GS and ES. Therefore, all electronic states share the same set of determinants with different CI coefficients.
Our implementation of the CIPSI algorithm for ES is detailed in Ref.~\citenum{Sce19}. For each system, a preliminary sCI calculation is performed using Hartree-Fock orbitals in order to generate sCI wavefunctions with at least 5,000,000 determinants.
State-averaged natural orbitals are then computed based on this wavefunction, and a new, larger sCI calculation is performed with this new set of orbitals. This has the advantage to produce a smoother and faster convergence of the sCI energy to the FCI limit.
Our implementation of the CIPSI algorithm for ES is detailed in Ref.~\citenum{Sce19}. For each system, a preliminary SCI calculation is performed using Hartree-Fock orbitals in order to generate SCI wavefunctions with at least 5,000,000 determinants.
State-averaged natural orbitals are then computed based on this wavefunction, and a new, larger SCI calculation is performed with this new set of orbitals. This has the advantage to produce a smoother and faster convergence of the SCI energy to the FCI limit.
For the largest systems, an additional iteration is sometimes required in order to obtain better quality natural orbitals and hence well-converged calculations.
The total sCI energy is defined as the sum of the (zeroth-order) variational energy (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction which takes into account the external determinants, \ie,
The total SCI energy is defined as the sum of the (zeroth-order) variational energy (computed via diagonalization of the CI matrix in the reference space) and a second-order perturbative correction which takes into account the external determinants, \ie,
the determinants which do not belong to the variational space but are linked to the reference space via a non-zero matrix element. The magnitude of this second-order correction, $E^{(2)}$, provides a qualitative idea of the ``distance" to the FCI limit.
For maximum efficiency, the total sCI energy is linearly extrapolated to $E^{(2)} = 0$ (which effectively corresponds to the FCI limit) using the two largest sCI wavefunctions. These extrapolated total energies simply (labeled as FCI in the remaining of the paper)
are then used to computed vertical excitation energies. Although it is not possible to provide a theoretically-sound error bar, we estimate the extrapolation error by the difference in excitation energy between the largest sCI wavefunction and its corresponding
extrapolated value. We believe that it provides a very safe estimate of the extrapolation error. Additional information about the sCI wavefunctions and excitation energies as well as their extrapolated values can be found in the SI.
For maximum efficiency, the total SCI energy is linearly extrapolated to $E^{(2)} = 0$ (which effectively corresponds to the FCI limit) using the two largest SCI wavefunctions. These extrapolated total energies simply (labeled as FCI in the remaining of the paper)
are then used to computed vertical excitation energies. Although it is not possible to provide a theoretically-sound error bar, we estimate the extrapolation error by the difference in excitation energy between the largest SCI wavefunction and its corresponding
extrapolated value. We believe that it provides a very safe estimate of the extrapolation error. Additional information about the SCI wavefunctions and excitation energies as well as their extrapolated values can be found in the SI.
\subsection{NEVPT2}
@ -227,7 +235,7 @@ Default program settings were applied. We note that for {\STEOM} we report only
\label{sec-res}
In the following, we present the results obtained for molecules containing four, five, and six (non-hydrogen) atoms. In all cases, we test several atomic basis sets and push the CC excitation order as high as technically possible.
Given that the {\sCI} results converges rather slowly for these larger systems, we provide an estimated error bar for these extrapolated {\FCI} values (\emph{vide supra}). In most cases, these extrapolated FCI reference data are used as a safety net to demonstrate the
Given that the {\SCI} results converges rather slowly for these larger systems, we provide an estimated error bar for these extrapolated {\FCI} values (\emph{vide supra}). In most cases, these extrapolated FCI reference data are used as a safety net to demonstrate the
consistency of the approaches rather than as definitive TBE (see next Section). We also show the results of {\NEV}/{\AVTZ} calculations for all relevant states to have a further consistency check. We underline that, except
when specifically discussed, all ES present a dominant single-excitation character (see also next Section), so that we do not expect serious CC breakdowns. This is especially true for the triplet ES that are known to show
very large \%$T_1$ for the vast majority of states, \cite{Sch08} and we consequently put our maximal computational effort on determining accurate transition energies for singlet states. To assign the different ES, we use literature data, as well as
@ -1490,13 +1498,13 @@ in the $0.12$--$0.23$ eV range.
Paraphrasing Thiel and coworkers, \cite{Sch08} we hope that this new set of vertical transition energies, combined or not with the ones described in our previous works, \cite{Loo18a,Loo19c} will be useful for the community,
will stimulate further developments and analyses in the field, and will provide new grounds for appraising the \emph{pros} and \emph{cons} of ES models already available or currently under development. We can
crystal-ball that the emergence of new {\sCI} algorithms optimized for modern computer architectures will likely lead to the revision of some the present TBE, allowing to climb even higher on the accuracy ladder.
crystal-ball that the emergence of new {\SCI} algorithms optimized for modern computer architectures will likely lead to the revision of some the present TBE, allowing to climb even higher on the accuracy ladder.
\begin{suppinfo}
Geometries.
Basis set and frozen core effects.
Definition of the active spaces for the multi-reference calculations.
Additional details about the {\sCI} calculations and their extrapolation.
Additional details about the {\SCI} calculations and their extrapolation.
Benchmark data and further statistical analysis.
\end{suppinfo}