From 6b49657428f04ae7bff517b6c86c901bd25257fe Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 6 Apr 2022 12:21:30 +0200 Subject: [PATCH] almost done with D4h --- Manuscript/CBD.tex | 55 ++++++++++++++++++++-------------------------- 1 file changed, 24 insertions(+), 31 deletions(-) diff --git a/Manuscript/CBD.tex b/Manuscript/CBD.tex index da06ecd..d3a3647 100644 --- a/Manuscript/CBD.tex +++ b/Manuscript/CBD.tex @@ -595,7 +595,7 @@ Considering the aug-cc-pVTZ basis, the evolution of the vertical excitation ener 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. -First, let us discuss basis set effects at the SF-TD-DFT level. +First, let us discuss basis set effects at the SF-TD-DFT level (Table \ref{tab:sf_D2h}). 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 fraction 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} @@ -615,7 +615,7 @@ Note that, as evidenced by the data reported in {\SupInf}, none of these states %The M11 vertical energies are close to the BH\&HLYP ones for the triplet and the first singlet excited states with \SIrange{0.01}{0.02}{\eV} and \SIrange{0.08}{0.10}{\eV} of energy difference, respectively. %For the {\twoAg} state the M11 energies are closer to the $\omega$B97X-V ones with \SIrange{0.05}{0.09}{\eV} of energy difference. -Second, we discuss the various SF-ADC schemes, \ie, SF-ADC(2)-s, SF-ADC(2)-x, and SF-ADC(3). +Second, we discuss the various SF-ADC schemes (Table \ref{tab:sf_D2h}), \ie, SF-ADC(2)-s, SF-ADC(2)-x, and SF-ADC(3). At the SF-ADC(2)-s level, going from the smallest 6-31+G(d) basis to the largest aug-cc-pVQZ basis, we see a small decrease in vertical excitation energies of about \SI{0.03}{\eV} for the {\tBoneg} state and around \SI{0.06}{\eV} for {\twoAg} state, while the transition energy of the {\sBoneg} state drops more significantly by about \SI{0.2}{\eV}. [The SF-ADC(2)-x and SF-ADC(3) calculations with aug-cc-pVQZ were not feasible with our computational resources.] These basis set effects are fairly transferable to the other wave function methods that we have considered here. @@ -631,7 +631,7 @@ This feature is characteristic of the inadequacy of the active space. For the two other states, {\tBoneg} and {\twoAg}, the errors at the CASPT2(4,4) and NEVPT2(4,4) levels are much smaller and typically below \SI{0.1}{\eV}. Using a larger active resolves most of these issues: CASSCF predicts the correct state ordering (though the ionic state is still badly described in term of energetics), CASPT2 and NEVPT2 excitation energies are much closer, and their accuracy is often improved (especially for the triplet state) although it is difficult to reach chemical accuracy (\ie, an error below \SI{0.043}{\eV}) on a systematic basis. -Finally, for the CC models, the two states with a large $\%T_1$ value, {\tBoneg} and {\sBoneg}, are already extremely accurate at the CC3 level, and systematically improved by CCSDT and CC4. +Finally, for the CC models (Table \ref{tab:D2h}), the two states with a large $\%T_1$ value, {\tBoneg} and {\sBoneg}, are already extremely accurate at the CC3 level, and systematically improved by CCSDT and CC4. For the doubly-excited state, {\twoAg}, the convergence of the CC expansion is much slower but it is worth pointing out that the inclusion of approximate quadruples via CC4 is particularly effective in the present case. The CCSDTQ excitation energies (which are used to define the TBEs) are systematically within the error bar of the CIPSI calculations, which confirms the outstanding performance of CC methods including quadruple excitations in the context of excited states. @@ -804,7 +804,7 @@ The vertical excitation energies computed at various levels of theory are depict Unfortunately, due to technical limitations, we could not compute $\%T_1$ values associated with the {\Atwog}, {\Aoneg}, and {\Btwog} excited states in the {\Dfour} symmetry. However, it is clear from the inspection of the wave function that, with respect to the {\sBoneg} ground state, {\Atwog} and {\Btwog} are dominated by single excitations, while {\Aoneg} is strongly dominated by double excitations. -As for the previous geometry we start by discussing the SF-TD-DFT results, and in particular the singlet-triplet gap, \ie, the energy difference between {\sBoneg} and {\Atwog}. +As for the previous geometry we start by discussing the SF-TD-DFT results (Table \ref{tab:sf_D4h}), and in particular the singlet-triplet gap, \ie, the energy difference between {\sBoneg} and {\Atwog}. For all functionals, this gap is small (basically below \SI{0.1}{\eV} while the TBE value is \SI{0.144}{\eV}) but it is worth mentioning that B3LYP and PBE0 predict a negative singlet-triplet gap (hence a triplet ground state). Increasing the exact exchange in hybrids or relying on RSHs (even with a small amount of short-range exact exchange) allows to recover a positive gap and a singlet ground state. At the SF-TD-DFT level, the energy gap between the two singlet excited states, {\Aoneg} and {\Btwog}, is particularly small and grows moderately with the amount of short-range exact exchange. @@ -813,12 +813,22 @@ As for the excitation energies computed at the {\Dtwo} equilibrium structure and Yet, the transition energy to {\Btwog} is off by more than half an \si{\eV} compared to the TBE, while the doubly-excited state is much closer to the reference value (errors of \SI{-0.251}{}, \SI{-0.097}{}, and \SI{-0.312}{\eV} for BH\&HLYP, M06-2X, and M11, respectively) Again, for all the excited states, the basis set effects are extremely small at the SF-TD-DFT level. -Then we discuss the various ADC scheme results, note that we were not able to obtain the vertical energies with the aug-cc-pVQZ basis due to computational resources. -For the ADC(2)-s scheme we can see that the energy difference for the triplet state are smaller than for the two other states, indeed we have an energy difference of \SI{0.09}{\eV} for the triplet state whereas we have \SI{0.15}{\eV} and \SI{0.25}{eV} for the {\Aoneg} and {\Btwog} states, again when considering all bases. -The energy difference for each state and through the bases are similar for the two other ADC schemes. -We can notice a large variation of the vertical energies for the {\Aoneg} state between ADC(2)-s and ADC(2)-x with around \SIrange{0.52}{0.58}{\eV} through all bases. -The ADC(3) vertical energies are very similar to the ADC(2) ones for the {\Btwog} state with an energy difference of \SIrange{0.01}{0.02}{\eV} for all bases, whereas we have an energy difference of \SIrange{0.04}{0.11}{\eV} and \SIrange{0.17}{0.22}{\eV} for the {\Aoneg} and {\Btwog} states, respectively. +Next, we discuss the various ADC schemes (Table \ref{tab:sf_D4h}) where we were not able to compute the vertical energies with the aug-cc-pVQZ basis due to our limited computational resources. +Overall, we observe similar trends than the ones mentioned in Sec.~\ref{sec:D2h}. +Concerning the singlet-triplet gap, each scheme predicts it to be positive. +Although it provides a decent singlet-triplet gap value, SF-ADC(2)-x seems to particularly struggle with the singlet excited states ({\Aoneg} and {\Btwog}), especially for the doubly-excited state {\Aoneg} where it underestimates the vertical excitation energy by \SI{0.4}{\eV}. +Again, averaging the SF-ADC(2)-s and SF-ADC(3) transition energies is beneficial in most cases at the exception of {\Aoneg}. +Although the basis set effect are larger than at the SF-TD-DFT level, they remain quite moderate at the SF-ADC level, and for any wave function method in general. +Then, we discuss the multi-reference results (Table \ref{tab:D4h}). +For both active spaces, expectedly, CASSCF does not provide a quantitive energetic description of the excited states, although it is worth mentioning that the right state ordering is preserved. +This is, of course, magnified with the (4e,4o) active space for which the second-order perturbative treatment is unable to provide a faithful description due to the restricted active space. +In particular SC-NEVPT2(4,4)/aug-cc-pVTZ and PC-NEVPT2(4,4)/aug-cc-pVTZ underestimate the singlet-triplet gap by \SI{0.072}{} and \SI{0.097}{\eV} and, more importantly, swap {\Aoneg} and {\Btwog}. +Although {\Aoneg} is not badly described, the excitation energy of the ionic state {\Btwog} is off by \SI{1}{\eV}. +Thanks to the IPEA shift in CASPT2(4,4), the singlet-triplet gap is accurate and the state ordering remains correct but the ionic state is still far from being accurate. +The (12e,12o) active space significantly damp these effects, and, as usually, the agreement between CASPT2 and NEVPT2 is very much improved for each state, though the accuracy of multi-configurational approaches remains questionable for the ionic state with an error up to \SI{-0.278}{\eV} at the PC-NEVPT2(12,12)/aug-cc-pVTZ level. + +Finally, let us consider the excitation energies computed with various CC models and gathered in Table \ref{tab:D4h}. Now we look at Table \ref{tab:D4h} for vertical energies obtained with standard methods, we have first the various CC methods, for all the CC methods we were only able to reach the aug-cc-pVTZ basis. %For the CCSD method we cannot obtain the vertical energy for the $1\,{}^1B_{2g}$ state, we can notice a small energy variation through the bases for the triplet state with around 0.06 eV. The energy variation is larger for the $2\,{}^1A_{1g}$ state with 0.20 eV. Again, we have computed the vertical energies at the CCSDT and CCSDTQ level as well as their approximation the CC3 and CC4 methods, respectively. @@ -828,28 +838,6 @@ Considering all bases for the {\Aoneg}and {\Btwog} states we have an energy dif The CCSDT energies are close to the CC3 ones for the {\Aoneg} state with an energy difference of around \SIrange{0.03}{0.06}{\eV} considering all bases. For the{\Btwog} state the energy difference between the CC3 and the CCSDT values is larger with \SIrange{0.18}{0.27}{\eV}. We can make a similar observation between the CC4 and the CCSDTQ values, for the {\Aoneg} state we have an energy difference of about \SI{0.01}{\eV} and this time we have smaller energy difference for the {\Btwog} with \SI{0.01}{\eV}. -Then we discuss the multi-reference results and this time we were able to reach the aug-cc-pVQZ basis. -Again we consider two different active spaces, four electrons in four orbitals and twelve electrons in twelve orbitals. -If we compare the CASSCF(4,4) and CASPT2(4,4) values we can see large energy differences for the {\Aoneg} and {\Btwog} states, for the {\Aoneg} state we have an energy difference of about \SIrange{0.67}{0.74}{\eV} and \SIrange{1.65}{1.81}{\eV} for the {\Btwog} state. -The energy difference is smaller for the triplet state with \SIrange{0.27}{0.31}{\eV}, we can notice that all the CASSCF(4,4) vertical energies are higher than the CASPT2(4,4) ones. -Then we have the NEVPT2(4,4) methods (SC-NEVPT2(4,4) and PC-NEVPT2(4,4)) for which the vertical energies are quite similar, however we can notice that we have a different ordering of the {\Aoneg} and {\Btwog} states with {\Btwog} higher in energy than {\Aoneg} for the two NEVPT2(4,4) methods. -Then we have the results for the same methods but with a larger active space. -For the CASSCF(12,12) we have similar values for the triplet state with energy difference of about \SI{0.06}{\eV} for all bases but larger energy difference for the {\Aoneg} state with around \SIrange{0.28}{0.29}{\eV} and \SIrange{0.79}{0.81}{\eV} for the {\Btwog} state. -Note that from CASSCF(4,4) to CASSCF(12,12) every vertical energy is smaller. -We can make the contrary observation for the CASPT2 method where from CASPT2(4,4) to CASPT2(12,12) every vertical energy is larger. -For the CASPT2(12,12) method we have similar values for the triplet state and for the {\Aoneg} state, considering all bases, with an energy difference of around \SIrange{0.05}{0.06}{\eV} and \SIrange{0.02}{0.05}{\eV} respectively. -The energy difference is larger for the {\Btwog} state with about \SIrange{0.27}{0.29}{\eV}. -Next we have the NEVPT2 methods, the first observation that we can make is that by increasing the size of the active space we retrieve the right ordering of the {\Aoneg} and {\Btwog} states. -Again, every vertical energy is higher in the NEVPT2(12,12) case than for the NEVPT2(4,4) one. - -Figure \ref{fig:D4h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_D4h} and \ref{tab:D4h}. -We see that all methods exhibit larger variations of the vertical energies due to the high symmetry of the {\Dfour} structure. -For the triplet state most of the methods used are able to give a vertical energy close to the TBE one, we can although see that CASSCF, with the two different active spaces, show larger energy error than most of the DFT functionals used. -Then for the, strongly multi-configurational character, {\Aoneg} state we have a good description by the CC and multi-reference methods with the largest active space, except for CASSCF. -The SF-ADC(2) and SF-ADC(3) schemes are also able to provide a good description of the {\Aoneg} state and even for the {\Btwog} we see that SF-ADC(2)-x give a worst result than SF-ADC(2). -Multi-configurational methods with the smallest active space do not demonstrate a good description of the two singlet excited states especially for the CASSCF method and for the the NEVPT2 methods that give the vertical energy of the {\Btwog} state below the {\Aoneg} one. -Note that CASPT2 improve a lot the description of all the states compared to CASSCF. -The various TD-DFT functionals are not able to describe correctly the two singlet excited states. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion} @@ -867,6 +855,11 @@ This has been shown to be beneficial for the automerization barrier and the vert \titou{At the SF-ADC level, we have found that the extended scheme SF-ADC(2)-x is not good, while SF-ADC(2)-s and SF-ADC(3) have opposite behavior which means that SF-ADC(2.5) is really good.} +\titou{For the multireference methods, we have found that NEVPT2 and CASPT2 can provide different results for the small active space, but they becomes very similar when the larger active space is considered. +Fro ma more general perspective, a singificant difference between NEVPT2 and CASPT2 can be then seen as a warning that the active space has been poorly chosen. +Also, the ionic state is usually significantly worse than the other state. +CASSCF cannot be advised for such a purpose.} + In order to provide a benchmark of the automerization barrier and vertical energies we used coupled-cluster (CC) methods with doubles (CCSD), with triples (CCSDT and CC3) and with quadruples (CCSTQ and CC4). Due to the presence of multi-configurational states we used multi-reference methods (CASSCF, CASPT2 and NEVPT2) with two active spaces ((4,4) and (12,12)). We also used spin-flip (SF-) within two frameworks, in TD-DFT with various global and range-separated hybrids functionals, and in ADC with the ADC(2)-s, ADC(2)-x and ADC(3) schemes.