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16
Manuscript/CBD.bib
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@ -1,13 +1,25 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2022-04-02 21:11:12 +0200
%% Created for Pierre-Francois Loos at 2022-04-04 11:49:40 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Giner_2019,
author = {E. Giner and A. Scemama and J. Toulouse and P. F. Loos},
date-added = {2022-04-04 11:49:35 +0200},
date-modified = {2022-04-04 11:49:35 +0200},
doi = {10.1063/1.5122976},
journal = {J. Chem. Phys.},
pages = {144118},
title = {Chemically accurate excitation energies with small basis sets},
volume = {151},
year = {2019},
bdsk-url-1 = {https://doi.org/10.1063/1.5052714}}
@article{Shao_2003,
author = {Shao,Yihan and Head-Gordon,Martin and Krylov,Anna I.},
date-added = {2022-04-02 21:11:09 +0200},

38
Manuscript/CBD.tex
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@ -220,7 +220,7 @@ The spin-flip version of our recently proposed composite approach, namely SF-ADC
We have also carried out spin-flip calculations within the TD-DFT framework (SF-TD-DFT), \cite{Shao_2003} and these are also performed with Q-CHEM 5.2.1. \cite{qchem}
The B3LYP, \cite{Becke_1988b,Lee_1988a,Becke_1993b} PBE0 \cite{Adamo_1999a,Ernzerhof_1999} and BH\&HLYP hybrid functionals are considered, which contain 20\%, 25\%, 50\% of exact exchange, respectively.
These calculations are labeled as SF-TD-BLYP, SF-TD-B3LYP, SF-TD-PBE0, and SF-TD-BH\&HLYP in the following.
Additionally, we have also computed SF-TD-DFT excitation energies using range-separated hybrid (RSH) functionals: CAM-B3LYP (19\% of short-range exact exchange and 65\% at long range), \cite{Yanai_2004a} LC-$\omega$PBE08 (16.7\% of short-range exact exchange and 100\% at long range), \cite{Weintraub_2009a} and $\omega$B97X-V (16.7\% of short-range exact exchange and 100\% at long range). \cite{Mardirossian_2014}
Additionally, we have also computed SF-TD-DFT excitation energies using range-separated hybrid (RSH) functionals: CAM-B3LYP (19\% of short-range exact exchange and 65\% at long range), \cite{Yanai_2004a} LC-$\omega$PBE08 (0\% of short-range exact exchange and 100\% at long range), \cite{Weintraub_2009a} and $\omega$B97X-V (16.7\% of short-range exact exchange and 100\% at long range). \cite{Mardirossian_2014}
Finally, the hybrid meta-GGA functional M06-2X (54\% of exact exchange) \cite{Zhao_2008} and the RSH meta-GGA functional M11 (42.8\% of short-range exact exchange and 100\% at long range) \cite{Peverati_2011} are also employed.
Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximation. \cite{Hirata_1999}
%EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
@ -274,11 +274,11 @@ Two different sets of geometries obtained with different levels of theory are co
First, because the autoisomerization barrier is obtained as a difference of energies computed at distinct geometries, it is paramount to obtain these at the same level of theory.
However, due to the fact that the ground state of the square arrangement is a transition state of singlet open-shell nature, it is technically difficult to optimize the geometry with high-order CC methods.
Therefore, we rely on CASPT2(12,12)/aug-cc-pVTZ for both the {\Dtwo} and {\Dfour} ground-state structures.
(Note that these optimizations are done with the IPEA shift.)
(Note that these optimizations are done without IPEA shift.)
Second, because the vertical transition energies are computed for a particular equilibrium geometry, we can afford to use different methods for the rectangular and square structures.
Hence, we rely on CC3/aug-cc-pVTZ to compute the equilibrium geometry of the {\oneAg} state in the rectangular ({\Dtwo}) arrangement and the restricted open-shell (RO) version of CCSD(T)/aug-cc-pVTZ to obtain the equilibrium geometry of the {\Atwog} state in the square ({\Dfour}) arrangement.
These two geometries are the lowest-energy equilibrium structure of their respective spin manifold (see Fig.~\ref{fig:CBD}).
The cartesian coordinates of all these geometries are provided in the {\SupInf}.
The cartesian coordinates of these geometries are provided in the {\SupInf}.
Table \ref{tab:geometries} reports the key geometrical parameters obtained at these levels of theory as well as previous geometries computed by Manohar and Krylov at the CCSD(T)/cc-pVTZ level.
%================================================
@ -335,8 +335,8 @@ SF-TD-BH\&HLYP & $11.90$ & $12.02$ & $12.72$ & $12.73$ \\
SF-TD-M06-2X & $9.32$ & $9.62$ & $10.35$ & $10.37$ \\
SF-TD-CAM-B3LYP& $18.05$ & $18.10$ & $18.83$ & $18.83$ \\
SF-TD-$\omega$B97X-V & $18.26$ & $18.24$ & $18.94$ & $18.92$ \\
SF-TD-M11 & $11.03$ & $10.25$ & $11.22$ & $11.12$ \\
SF-TD-LC-$\omega$PBE08 & $19.05$ & $18.98$ & $19.74$ & $19.71$ \\[0.1cm]
SF-TD-M11 & $11.03$ & $10.25$ & $11.22$ & $11.12$ \\
SF-ADC(2)-s & $6.69$ & $6.98$ & $8.63$ & \\
SF-ADC(2)-x & $8.63$ & $8.96$ &$10.37$ & \\
SF-ADC(2.5) & $7.36$ & $7.76$ & $9.11$ & \\
@ -377,7 +377,7 @@ CCSDTQ & $7.51$ & $[ 7.89]$\fnm[4]& $[ 8.93]$\fnm[5]& $[ 9.11]$\fnm[6]\\
The results concerning the autoisomerization barrier are reported in Table \ref{tab:auto_standard} for various basis sets and shown in Fig.~\ref{fig:AB} for the aug-cc-pVTZ basis.
First, one can see large variations of the energy barrier at the SF-TD-DFT level, with differences as large as \SI{10}{\kcalmol} for a given basis set between the different functionals.
First, one can see large variations of the energy barrier at the SF-TD-DFT level, with differences as large as \SI{10}{\kcalmol} between the different functionals for a given basis set.
Nonetheless, it is clear that the performance of a given functional is directly linked to the amount of exact exchange at short range.
Indeed, hybrid functionals with a large fraction of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) perform significantly better than the functionals having a small fraction of short-range exact exchange (\eg, B3LYP, PBE0, CAM-B3LYP, $\omega$B97X-V, and LC-$\omega$PBE08).
However, they are still off by \SIrange{1}{4}{\kcalmol} from the TBE reference value.
@ -385,21 +385,21 @@ For the RSH functionals, the autoisomerization barrier is much less sensitive to
Another important feature of SF-TD-DFT is the fast convergence of the energy barrier with the size of the basis set. \cite{Loos_2019d}
With the augmented double-$\zeta$ basis, the SF-TD-DFT results are basically converged to sub-\kcalmol accuracy, which is a drastic improvement compared to wave function approaches where this type of convergence is reached with the augmented triple-$\zeta$ basis.
For the SF-ADC family of methods, the energy differences are much smaller with a maximum deviation of \SI{2.0}{\kcalmol} between different versions.
For the SF-ADC family of methods, the energy differences are much smaller with a maximum deviation of \SI{2}{\kcalmol} between different versions.
In particular, we observe that SF-ADC(2)-s and SF-ADC(3), which scale as $\order*{N^5}$ and $\order*{N^6}$ respectively (where $N$ is the number of basis functions), under- and overestimate the autoisomerization barrier, making SF-ADC(2.5) a good compromise with an error of only \SI{0.18}{\kcalmol} with the aug-cc-pVTZ basis.
Nonetheless, at a $\order*{N^5}$ computational scaling, SF-ADC(2)-s is particularly accurate, even compared to high-order CC methods (see below).
We note that SF-ADC(2)-x is probably not worth its extra cost [as compared to SF-ADC(2)-s] as it overestimates the energy barrier even more than SF-ADC(3).
Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearly outperformed by the more expensive SF-ADC methods.
Concerning the multi-reference approaches with the smaller (4e,4o) active space, the TBEs are bracketed by the CASPT2 and NEVPT2 values that differ by approximately \SI{1.5}{\kcalmol} for all the bases.
In this case, the NEVPT2 values are fairly accurate with differences below half a \si{\kcalmol} with the TBEs.
Concerning the multi-reference approaches with the minimal (4e,4o) active space, the TBEs are bracketed by the CASPT2 and NEVPT2 values that differ by approximately \SI{1.5}{\kcalmol} for all bases.
In this case, the NEVPT2 values are fairly accurate with differences below half a \si{\kcalmol} compared to the TBEs.
The CASSCF results predict an even lower barrier than CASPT2 due to the well known lack of dynamical correlation at the CASSCF level.
For the larger (12e,12o) active space, we see larger differences of the order of \SI{3}{\kcalmol} through all the bases between CASSCF and the second-order variants (CASPT2 and NEVPT2).
The deviations between CASPT2(12,12) and NEVPT2(12,12) are much smaller with an energy difference of around \SIrange{0.1}{0.2}{\kcalmol} for all bases, CASPT2 being slightly more accurate than NEVPT2 in this case.
Thought, the deviations between CASPT2(12,12) and NEVPT2(12,12) are much smaller with an energy difference of around \SIrange{0.1}{0.2}{\kcalmol} for all bases, CASPT2 being slightly more accurate than NEVPT2 in this case.
For each basis set, both CASPT2(12,12) and NEVPT2(12,12) are less than a \si{\kcalmol} away from the TBEs.
For the two active spaces that we have considered here, the PC- and SC-NEVPT2 schemes provide nearly identical barriers independently of the size of the one-electron basis.
Finally, for the CC family of methods, we observe the usual systematic improvement following the series CCSD $<$ CC3 $<$ CCSDT $<$ CC4 $<$ CCSDTQ, which is also linked with the increase in computational cost of $\order*{N^6}$, $\order*{N^7}$, $\order*{N^8}$, $\order*{N^9}$, and $\order*{N^{10}}$, respectively.
Finally, for the CC family of methods, we observe the usual systematic improvement following the series CCSD $<$ CC3 $<$ CCSDT $<$ CC4 $<$ CCSDTQ, which is also linked to their increase in computational cost: $\order*{N^6}$, $\order*{N^7}$, $\order*{N^8}$, $\order*{N^9}$, and $\order*{N^{10}}$, respectively.
Note that the introduction of the triple excitations is clearly mandatory to have an accuracy beyond SF-TD-DFT, while it is also clear that the iterative triples and quadruples can be included approximately via the CC3 and CC4 methods.
%================================================
@ -580,17 +580,19 @@ CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
\end{figure*}
%%% %%% %%% %%%
Table \ref{tab:sf_tddft_D2h} reports, at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state, the singlet and triplet vertical transition energies associated with the {\tBoneg}, {\sBoneg}, and {\twoAg} states obtained using the spin-flip formalism, while Table \ref{tab:D2h} gathers the same quantities obtained with the multi-reference and CC methods.
Table \ref{tab:sf_tddft_D2h} reports, at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state, the singlet and triplet vertical transition energies associated with the {\tBoneg}, {\sBoneg}, and {\twoAg} states obtained using the spin-flip formalism, while Table \ref{tab:D2h} gathers the same quantities obtained with the multi-reference, CC, and CIPSI methods.
Considering the aug-cc-pVTZ basis, the evolution of the vertical excitation energies with respect to the level of theory is illustrated in Fig.~\ref{fig:D2h}.
At the CC3/aug-cc-pVTZ level, the percentage of single excitation involved in the {\tBoneg}, {\sBoneg}, and {\twoAg} are 99\%, 95\%, and 1\%, respectively.
Therefore, the two formers are dominated by single excitations, while the latter state is a genuine double excitation.
\alert{First, let us discuss the basis set effects at the SF-TD-DFT level.
These are small and the results are basically converged to the basis set limit with the triple-$\zeta$ basis.
Again, we clearly see that the functional with the largest amount of short-range exact exchange are the most accurate ones.
However, their accuracy remains average especially for the {\sBoneg} state (due to spin contamination?).
Mention that the amount of exact exchange is key for SF-TD-DFT as this is the only non-vanishing term in the spin-flip block. \cite{Shao_2003}}
First, let us discuss basis set effects at the SF-TD-DFT level.
As expected, these are found to be small and the results are basically converged to the basis set limit with the triple-$\zeta$ basis, which is definitely not the case for the wave function methods. \cite{Giner_2019}
Regarding now the accuracy of the vertical excitation energies, again, we clearly see that, for each transition, the functionals with the largest amount of short-range exact exchange (\eg, BH\&HLYP, M06-2X, and M11) are the most accurate.
Functionals with large amount of exact exchange are known to perform best in the SF-TD-DFT framework as the Hartree-Fock exchange term is the only non-vanishing term in the spin-flip block. \cite{Shao_2003}
However, their overall accuracy remains average especially for the singlet states, {\sBoneg} and {\twoAg}, with error of the order of \SIrange{0.2}{0.5}{\eV} compared to the TBEs for the best functionals.
The triplet state, {\tBoneg}, is much better described with errors below \SI{0.1}{\eV}.
Note that, as evidenced by the data reported in {\SupInf}, none of these states exhibit a strong spin contamination.
First, we discuss the SF-TD-DFT values with hybrid functionals.
For the B3LYP functional we can see that the energy differences for each state and throughout all bases are small with the largest one for the {\tBoneg} state with \SI{0.012}{\eV}.
@ -892,8 +894,8 @@ SF-TD-BHHLYP & $3.79$ & $0.078$ & $-0.393$ & $0.343$ & $-0.099$ & $-0.251$ & $-0
SF-TD-M06-2X & $1.42$ & $0.000$ & $-0.354$ & $0.208$ & $-0.066$ & $-0.097$ & $-0.432$ \\
SF-TD-CAM-B3LYP & $9.90$ & $0.280$ & $-0.807$ & $-0.011$ & $-0.134$ & $-0.920$ & $-1.370$ \\
SF-TD-$\omega $B97X-V & $10.01$ & $0.335$ & $-0.774$ & $0.064$ & $-0.118$ & $-0.928$ & $-1.372$ \\
SF-TD-M11 & $2.29$ & $0.097$ & $-0.474$ & $0.151$ & $-0.063$ & $-0.312$ & $-0.675$ \\
SF-TD-LC-$\omega $PBE08 & $10.81$ & $0.435$ & $-0.710$ & $0.199$ & $-0.086$ & $-0.939$ & $-1.376$ \\[0.1cm]
SF-TD-LC-$\omega $PBE08 & $10.81$ & $0.435$ & $-0.710$ & $0.199$ & $-0.086$ & $-0.939$ & $-1.376$ \\
SF-TD-M11 & $2.29$ & $0.097$ & $-0.474$ & $0.151$ & $-0.063$ & $-0.312$ & $-0.675$ \\[0.1cm]
SF-ADC(2)-s & $-0.30$ & $0.069$ & $-0.026$ & $-0.018$ & $0.112$ & $0.112$ & $-0.190$ \\
SF-ADC(2)-x & $1.44$ & $0.077$ & $-0.094$ & $-0.446$ & $0.068$ & $-0.409$ & $-0.303$ \\
SF-ADC(2.5) & $0.18$ & $0.013$ & $0.006$ & $0.029$ & $0.024$ & $0.094$ & $-0.185$ \\