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D2h excited states update

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EnzoMonino 2021-12-24 15:43:09 +01:00
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@ -414,8 +414,10 @@ All the calculations are performed using four basis set, the 6-31+G(d) basis and
\subsubsection{D2h geometry}
Table \ref{tab:sf_tddft_D2h} shows the results obtained for the vertical transition energies using spin-flip methods and Table \ref{tab:D2h} shows the results obtained with the standard methods. We discuss first 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 $1\,{}^3B_{1g}$ state with 0.012 eV. The same observation can be done for the PBE0 and BH\&HLYP functionals, we can also oberve that adding exact exchange to the functional (20\% of exact exchange for B3LYP, 25\% for PBE0 and 50\% for BH\&HLYP increase the energy difference between the $1\,{}^1B_{1g}$ and the $1\,{}^3B_{1g}$ states. Put another way, increasing the amount of exact exchange in the functional reduces the energy difference between the $1\,{}^3B_{1g}$ and the $\text{X}\,{}^1A_{g}$ states. We can also notice that the vertical energies of the different states do not vary in the same way when adding exact exchange, for instance the energy variation for the $1\,{}^3B_{1g}$ state from PBE0 to BH\&HLYP is around 0.1 eV whereas for the $1\,{}^1B_{1g}$ and the $1\,{}^1A_{1g}$ states this energy variation is about 0.4-0.5 eV and 0.34 eV respectively. For the RSH functionals we can not make the same observation, indeed we can see that for the CAM-B3LYP functional we have that the energy difference between the $1\,{}^1B_{1g}$ and the $1\,{}^3B_{1g}$ states is larger than for the $\omega$B97X-V and LC-$\omega $PBE08 functionals despite the fact that the latter ones have a bigger amount of exact exchange. However we can observe that we have a small energy difference for each state throughout the bases and for each RSH functional. The M06-2X functional is an Hybrid meta-GGA functional and contains 54\% of Hartree-Fock exchange, we can compare the energies with the BH\&HLYP functional and we can see that the energy differences are small with around 0.03-0.08 eV. We can notice that the upper bound of 0.08 eV in the energy differences is due to the $1\,{}^3B_{1g}$ state.
Then we discuss the various ADC scheme (ADC(2)-s, ADC(2)-x and ADC(3)) results. For ADC(2) we have vertical energy differencies of about 0.03 eV for the $1\,{}^3B_{1g}$ state and around 0.06 eV for $1\,{}^1A_{1g}$ state throughout all bases. However for the $1\,{}^1B_{1g}$ state we have an energy difference of about 0.2 eV. For the ADC(2)-x and ADC(3) schemes the calculations with the AVQZ basis are not feasible with our resources. With ADC(2)-x we have similar vertical energies for the triplet and the $1\,{}^1B_{1g}$ states but for the $1\,{}^1A_{1g}$ state the energy difference between the ADC(2) and ADC(2)-x schemes is about 0.4-0.5 eV.
Then we discuss the various ADC scheme (ADC(2)-s, ADC(2)-x and ADC(3)) results. For ADC(2) we have vertical energy differencies of about 0.03 eV for the $1\,{}^3B_{1g}$ state and around 0.06 eV for $1\,{}^1A_{1g}$ state throughout all bases. However for the $1\,{}^1B_{1g}$ state we have an energy difference of about 0.2 eV. For the ADC(2)-x and ADC(3) schemes the calculations with the AVQZ basis are not feasible with our resources. With ADC(2)-x we have similar vertical energies for the triplet and the $1\,{}^1B_{1g}$ states but for the $1\,{}^1A_{1g}$ state the energy difference between the ADC(2) and ADC(2)-x schemes is about 0.4-0.5 eV. The ADC(3) values are closer from the ADC(2) than the ADC(2)-x except for the triplet state for which we have a energy difference of about 0.09-0.14 eV. Now, we look at Table \ref{tab:D2h} to discuss the results of standard methods. First we discuss the CC values, we have computed the vertical energies at the CCSDT and CCSDTQ level as well as their approximation the CC3 and CC4 methods, respectively. We can notice that for the $1\,{}^3B_{1g}$ and the $1\,{}^1B_{1g}$ states the CCSDT and the CC3 values are close with an energy difference of 0.009-0.02 eV for all bases. The energy difference is larger for the $1\,{}^1A_{1g}$ state with around 0.35-0.38 eV. The same observation can be done for CCSDTQ and CC4 with similar vertical energies for all bases. Note that the $1\,{}^1B_{1g}$ state can not be described with this methods.
Then we review the vertical energies obtained with multireference methods. The smallest active space considered is four electrons in four orbitals, for CASSCF(4,4) we have small energy variations throughout bases for the $1\,{}^3B_{1g}$ and the $1\,{}^1A_{1g}$ states but a larger variation for the $1\,{}^1B_{1g}$ state with around 0.1 eV. We can observe that we have the inversion of the states compared to all methods discussed so far between the $1\,{}^1A_{1g}$ and $1\,{}^1B_{1g}$ states with $1\,{}^1B_{1g}$ higher than $1\,{}^1A_{1g}$ due to the lack of dynamical correlation in the CASSCF methods. The $1\,{}^1B_{1g}$ state values in CASSCF(4,4) is much higher than for any of the other methods discussed so far. With CASPT2(4,4) we retrieve the right ordering between the states and we see large energy differences with the CASSCF values. Indeed, we have approximatively 0.22-0.25 eV of energy difference for the triplet state for all bases and 0.32-0.36 eV for the $1\,{}^1A_{1g}$ state, the largest energy difference is for the $1\,{}^1B_{1g}$ state with 1.5-1.6 eV.
For the XMS-CASPT2(4,4) only the $1\,{}^1A_{1g}$ state is described with values similar than for the CAPST2(4,4) method. For the NEVPT2(4,4) methods (SC-NEVPT2 and PC-NEVPT2) the vertical energies are similar for the $1\,{}^1B_{1g}$ and the $1\,{}^1A_{1g}$ states with approximatively 0.002-0.003 eV and 0.02-0.03 eV of energy difference for all bases,respectively. The energy difference for the $1\,{}^1B_{1g}$ state is slightly larger with 0.05 eV for all bases. Note that for this state the vertical energy varies of 0.23 eV from the 6-31+G(d) basis to the AVDZ one. Then we use a larger active space with twelve electrons in twelve orbitals, the CASSF(12,12) values are close to the the CASSCF(4,4) for the triplet state with 0.01-0.02 eV of energy differences. For the $1\,{}^1A_{1g}$ state we have an energy difference of about 0.2 eV between the CASSCF(4,4) and the CASSCF(12,12) values. We can notice that increasing the size of the active space gives the right ordering for the states and we have an energy difference of around 0.7 eV for the $1\,{}^1B_{1g}$ state between CASSCF(4,4) and the CASSCF(12,12) values. The CASPT2(12,12) method decreases the energy of all states compared to the CASSCF(12,12) method, again the decrease is not the same for all states. We have a diminution from CASSCF to CASPT2 of about 0.17 to 0.2 eV for the $1\,{}^3B_{1g}$ and the $1\,{}^1A_{1g}$ states and for the different bases. Again, the energy difference for the $1\,{}^1A_{1g}$ state is larger with 0.5-0.7 eV depending on the basis. In a similar way than with XMS-CASPT2(4,4), the XMS-CASPT(12,12) only describes the $1\,{}^1A_{1g}$ state and the vertical energies for this state are close to the CASPT(12,12) values. For the NEVPT2(12,12) schemes we see that for the triplet and the $1\,{}^1A_{1g}$ states the energies are similar with an energy difference between the SC-NEVPT2(12,12) and the PC-NEVPT2(12,12) values of about 0.03-0.04 eV and 0.02-0.03 eV respectively.
%%% TABLE II %%%
\begin{squeezetable}