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Pierre-Francois Loos
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% https://bibdesk.sourceforge.io/
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%% Created for Pierre-Francois Loos at 2022-03-24 14:53:30 +0100
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%% Created for Pierre-Francois Loos at 2022-03-24 22:36:43 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@article{Sarkar_2022,
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author = {R. Sarkar and P. F. Loos and M. Boggio-Pasqua and D. Jacquemin.},
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date-added = {2022-03-24 22:00:41 +0100},
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date-modified = {2022-03-24 22:00:41 +0100},
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journal = {J. Chem. Theory Comput.},
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pages = {in press},
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title = {Assessing the performances of CASPT2 and NEVPT2 for vertical excitation energies,},
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year = {2022}}
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@article{Schapiro_2013,
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author = {Schapiro, Igor and Sivalingam, Kantharuban and Neese, Frank},
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date-added = {2022-03-24 22:00:34 +0100},
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date-modified = {2022-03-24 22:00:34 +0100},
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doi = {10.1021/ct400136y},
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journal = {J. Chem. Theory Comput.},
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number = {8},
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pages = {3567--3580},
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title = {Assessment of $n$-Electron Valence State Perturbation Theory for Vertical Excitation Energies},
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volume = {9},
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year = {2013},
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bdsk-url-1 = {https://doi.org/10.1021/ct400136y}}
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@article{Zobel_2017,
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abstract = {Multi-configurational second order perturbation theory (CASPT2) has become a very popular method for describing excited-state properties since its development in 1990. To account for systematic errors found in the calculation of dissociation energies{,} an empirical correction applied to the zeroth-order Hamiltonian{,} called the IPEA shift{,} was introduced in 2004. The errors were attributed to an unbalanced description of open-shell versus closed-shell electronic states and is believed to also lead to an underestimation of excitation energies. Here we show that the use of the IPEA shift is not justified and the IPEA should not be used to calculate excited states{,} at least for organic chromophores. This conclusion is the result of three extensive analyses. Firstly{,} we survey the literature for excitation energies of organic molecules that have been calculated with the unmodified CASPT2 method. We find that the excitation energies of 356 reference values are negligibly underestimated by 0.02 eV. This value is an order of magnitude smaller than the expected error based on the calculation of dissociation energies. Secondly{,} we perform benchmark full configuration interaction calculations on 137 states of 13 di- and triatomic molecules and compare the results with CASPT2. Also in this case{,} the excited states are underestimated by only 0.05 eV. Finally{,} we perform CASPT2 calculations with different IPEA shift values on 309 excited states of 28 organic small and medium-sized organic chromophores. We demonstrate that the size of the IPEA correction scales with the amount of dynamical correlation energy (and thus with the size of the system){,} and gets immoderate already for the molecules considered here{,} leading to an overestimation of the excitation energies. It is also found that the IPEA correction strongly depends on the size of the basis set. The dependency on both the size of the system and of the basis set{,} contradicts the idea of a universal IPEA shift which is able to compensate for systematic CASPT2 errors in the calculation of excited states.},
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author = {Zobel, J. Patrick and Nogueira, Juan J. and Gonzalez, Leticia},
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date-added = {2022-03-24 21:59:07 +0100},
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date-modified = {2022-03-24 21:59:07 +0100},
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doi = {10.1039/C6SC03759C},
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issue = {2},
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journal = {Chem. Sci.},
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pages = {1482--1499},
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publisher = {The Royal Society of Chemistry},
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title = {The IPEA Dilemma in CASPT2},
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url = {http://dx.doi.org/10.1039/C6SC03759C},
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volume = {8},
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year = {2017},
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bdsk-url-1 = {http://dx.doi.org/10.1039/C6SC03759C}}
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@article{Ghigo_2004,
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abstract = {A new shifted zeroth-order Hamiltonian is presented, which will be used in second-order multiconfigurational perturbation theory (CASPT2). The new approximation corrects for the systematic error of the original formulation, which led to an relative overestimate of the correlation energy for open shell system, resulting in too small dissociation and excitation energies. Errors in the De values for 49 diatomic molecules have been reduced with more than 50%. Calculations on excited states of the N2 and benzene molecules give a similar improvement.},
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author = {Giovanni Ghigo and Bj{\"o}rn O. Roos and Per-{\AA}ke Malmqvist},
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date-added = {2022-03-24 21:58:52 +0100},
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date-modified = {2022-03-24 21:58:52 +0100},
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doi = {https://doi.org/10.1016/j.cplett.2004.08.032},
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issn = {0009-2614},
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journal = {Chem. Phys. Lett.},
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number = {1},
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pages = {142--149},
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title = {A Modified Definition of the Zeroth-Order Hamiltonian in Multiconfigurational Perturbation Theory (CASPT2)},
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url = {http://www.sciencedirect.com/science/article/pii/S0009261404012242},
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volume = {396},
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year = {2004},
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bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261404012242},
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bdsk-url-2 = {https://doi.org/10.1016/j.cplett.2004.08.032}}
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@article{Hirata_1999,
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abstract = {A computationally simple method for molecular excited states, namely, the Tamm--Dancoff approximation to time-dependent density functional theory, is proposed and implemented. This method yields excitation energies for several closed- and open-shell molecules that are essentially of the same quality as those obtained from time-dependent density functional theory itself, when the same exchange-correlation functional is used.},
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author = {So Hirata and Martin Head-Gordon},
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@ -342,12 +342,15 @@ The CC2, \cite{Christiansen_1995,Hattig_2000} CC3 \cite{Christiansen_1995,Koch_1
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\label{sec:Multi}
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State-averaged CASSCF (SA-CASSCF) calculations are performed with MOLPRO. \cite{Werner_2020}
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For each excited state, a set of state-averaged orbitals is computed by taking into account the the excited state of interest as well as the ground state (even if it has a different symmetry).
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On top of these, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both partially contracted (PC) and strongly contracted (SC) scheme. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
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PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility.
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Two active spaces have been considered: (i) a minimal (4e,4o) active space including valence $\pi$ orbitals, and (ii) an extended (12e,12o) active space where we have additionally included the $\sigma_\text{CC}$ and $\sigma_\text{CC}^*$ orbitals.
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For ionic states, like the {\sBoneg} state of CBD, it is particularly important to take into account the $\sigma$-$\pi$ coupling.
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On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both partially contracted (PC) and strongly contracted (SC) scheme. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
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In order to avoid the intruders state problem, a real-valued level shift of \SI{0.3}{\hartree} is set in CASPT2, \cite{Roos_1995b,Roos_1996} with an additional ionization-potential-electron-affinity (IPEA) shift of \SI{0.3}{\hartree} to avoid systematic overestimation of the vertical excitation energies. \cite{Ghigo_2004,Schapiro_2013,Zobel_2017,Sarkar_2022}
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Note that PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility.
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For the sake of comparison, in some cases, we have also performed multi-reference CI (MRCI) calculations.
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All these calculations are also carried out with MOLPRO. \cite{Werner_2020}
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%and extended multistate (XMS) CASPT2 calculations are also performed in the cas of strong mixing between states with same spin and spatial symmetries.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -370,30 +373,38 @@ Note that all SF-TD-DFT calculations are done within the Tamm-Dancoff approximat
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Theoretical best estimates}
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When technically possible, each level of theory is tested with four Gaussian basis sets, namely, the 6-31+G(d) basis and the aug-cc-pVXZ with X $=$ D, T, Q. \cite{Dunning_1989}
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For each studied quantity, i.e., the autoisomerisation barrier and the vertical excitations, we provide a theoretical best estimate (TBE).
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These TBEs are provided using extrapolated CCSDTQ/aug-cc-pVTZ values when possible and using NEVPT2(12,12) otherwise.
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\alert{CIPSI calculations are a safety net to check the convergence of CCSDTQ.}
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The extrapolation of the CCSDTQ/aug-cc-pVTZ values is done using two schemes.
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The first one uses CC4 values for the extrapolation and proceed as follows
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When technically possible, each level of theory is tested with four Gaussian basis sets, namely, 6-31+G(d) and aug-cc-pVXZ with X $=$ D, T, and Q. \cite{Dunning_1989}
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This helps us to assess the convergence of each property with respect to the size of the basis set.
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More importantly, for each studied quantity (i.e., the autoisomerisation barrier and the vertical excitation energies), we provide a theoretical best estimate (TBE) established in the aug-cc-pVTZ basis.
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In most cases, these TBEs are defined using extrapolated CCSDTQ/aug-cc-pVTZ values or, in a single occasion, NEVPT2(12,12).
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The extrapolation of the CCSDTQ/aug-cc-pVTZ values is done via a ``pyramidal'' scheme, where we employ systematically the most accurate level of theory and the largest basis set available.
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For example, when CC4/aug-cc-pVTZ and CCSDTQ/aug-cc-pVDZ data are available, we proceed via the following basis set extrapolation:
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\begin{equation}
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\label{eq:aug-cc-pVTZ}
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\Delta E^{\text{CCSDTQ}}_{\text{aug-cc-pVTZ}} = \Delta E^{\text{CCSDTQ}}_{\text{aug-cc-pVDZ}} + \left[ \Delta E^{\text{CC4}}_{\text{aug-cc-pVTZ}} - \Delta E^{\text{CC4}}_{\text{aug-cc-pVDZ}} \right]
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\label{eq:aug-cc-pVTZ}
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\Delta \Tilde{E}^{\text{CCSDTQ}}_{\text{aug-cc-pVTZ}}
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= \Delta E^{\text{CCSDTQ}}_{\text{aug-cc-pVDZ}}
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+ \qty[ \Delta E^{\text{CC4}}_{\text{aug-cc-pVTZ}} - \Delta E^{\text{CC4}}_{\text{aug-cc-pVDZ}} ],
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\end{equation}
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and we evaluate the CCSDTQ/aug-cc-pVDZ values as
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while, when only CCSDTQ/6-31G+(d) values are available, we further extrapolate the CCSDTQ/aug-cc-pVDZ value as follows:
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\begin{equation}
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\label{eq:aug-cc-pVDZ}
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\Delta E^{\text{CCSDTQ}}_{\text{aug-cc-pVDZ}} = \Delta E^{\text{CCSDTQ}}_{6-31\text{G}+\text{(d)}} + \left[ \Delta E^{\text{CC4}}_{\text{aug-cc-pVDZ}} - \Delta E^{\text{CC4}}_{6-31\text{G}+\text{(d)}} \right]
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\label{eq:aug-cc-pVDZ}
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\Delta \Tilde{E}^{\text{CCSDTQ}}_{\text{aug-cc-pVDZ}}
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= \Delta E^{\text{CCSDTQ}}_{6-31\text{G}+\text{(d)}}
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+ \qty[ \Delta E^{\text{CC4}}_{\text{aug-cc-pVDZ}} - \Delta E^{\text{CC4}}_{6-31\text{G}+\text{(d)}} ].
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\end{equation}
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when CC4/aug-cc-pVTZ values have been obtained.
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If it is not the case we extrapolate CC4/aug-cc-pVTZ values using the CCSDT ones as follows
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If we lack the CC4 data, we can follow the same philosophy and rely on CCSDT.
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For example,
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\begin{equation}
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\label{eq:CC4_aug-cc-pVTZ}
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\Delta E^{\text{CC4}}_{\text{aug-cc-pVTZ}} = \Delta E^{\text{CC4}}_{\text{aug-cc-pVDZ}} + \left[ \Delta E^{\text{CCSDT}}_{\text{aug-cc-pVTZ}} - \Delta E^{\text{CCSDT}}_{\text{aug-cc-pVDZ}} \right]
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\label{eq:CC4_aug-cc-pVTZ}
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\Delta \Tilde{E}^{\text{CC4}}_{\text{aug-cc-pVTZ}}
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= \Delta E^{\text{CC4}}_{\text{aug-cc-pVDZ}}
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+ \qty[ \Delta E^{\text{CCSDT}}_{\text{aug-cc-pVTZ}} - \Delta E^{\text{CCSDT}}_{\text{aug-cc-pVDZ}} ],
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\end{equation}
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Then if the CC4 values have not been obtained then we use the second scheme which is the same as the first one but instead of the CC4 values we use CCSDT to extrapolate CCSDTQ.
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If none of the two schemes is possible then we use the NEVPT2(12,12) values.
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Note that a NEVPT2(12,12) value is used only once for one vertical excitation of the \Dfour structure.
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and so on.
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If neither CC4, nor CCSDT are feasible, then we rely on NEVPT2(12,12).
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The procedure for each extrapolated value is explicitly mentioned as a footnote.
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Note that, due to error bar inherently linked to the CIPSI calculations (see Subsection \ref{sec:SCI}), these are mostly used as a safety net to further check the convergence of the CCSDTQ values.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -404,7 +415,12 @@ Note that a NEVPT2(12,12) value is used only once for one vertical excitation of
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%================================================
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\subsection{Geometries}
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\label{sec:geometries}
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Two different sets of geometries obtained with different level of theory are considered for the ground state property and for the excited states of the CBD molecule. First, for the autoisomerization barrier because we consider an energy difference between two geometries it is needed to obtain these geometries at the same level of theory. Due to the fact that the square CBD is an open-shell molecule it is difficult to optimize the geometry so the most accurate method that we can use for both structures is the CASPT2(12,12) with the aug-cc-pVTZ basis without frozen core. Then, for the excited states because we look at vertical energy transitions in one particular geometry we can use different methods for the different structures and use the most accurate method for each geometry. So in the case of the excited states of the CBD molecule we use CC3 without frozen core with the aug-cc-pVTZ basis for the rectangular ({\Dtwo}) geometry and we use RO-CCSD(T) with the aug-cc-pVTZ basis again without frozen core for the square ({\Dfour}) geometry. Table \ref{tab:geometries} shows the results on the geometry parameters obtained with the different methods.
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Two different sets of geometries obtained with different levels of theory are considered for the ground state property and for the excited states of the CBD molecule.
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First, for the autoisomerization barrier because we consider an energy difference between two geometries it is paramount to obtain these geometries at the same level of theory.
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Due to the fact that the square CBD is an open-shell molecule, it is difficult to optimize the geometry so the most accurate method that we can use for both structures is CASPT2(12,12)/aug-cc-pVTZ.
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Then, for the excited states because we look at vertical energy transitions in one particular geometry we can use different methods for the different structures and use the most accurate method for each geometry.
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So in the case of the excited states of the CBD molecule we use CC3/aug-cc-pVTZ for the rectangular ({\Dtwo}) geometry and we use RO-CCSD(T)/aug-cc-pVTZ for the square ({\Dfour}) geometry.
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Table \ref{tab:geometries} shows the results on the geometry parameters obtained with the different methods.
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%%% TABLE I %%%
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\begin{squeezetable}
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@ -846,7 +862,7 @@ Figure \ref{fig:D4h} shows the vertical energies of the studied excited states d
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\end{figure*}
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%%% %%% %%% %%%
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\subsubsection{Theoretical Best Estimates}
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\subsubsection{Theoretical best estimates}
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\label{sec:TBE}
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Table \ref{tab:TBE} shows the energy differences, for the autoisomerization barrier (AB) and the different states considered, between the various methods treated and the Theoretical Best Estimate (TBE) at the aug-cc-pVTZ level for the AB and the states. The percentage \% T1 shown in parentheses for the excited states of the {\Dtwo} geometry is a metric that gives the percentage of single excitation calculated at the CC3/aug-cc-pVTZ level and it allows us to characterize the transition. First, we look at the AB energy difference. SF-TD-DFT shows large variations of the energy with errors of \SIrange{1.42}{10.81}{\kcalmol} compared to the TBE value. SF-ADC schemes provide smaller errors with \SIrange{0.30}{1.44}{\kcalmol} where we have that the SF-ADC(2)-x gives a worse error than the SF-ADC(2)-s method. CC methods also give small energy differences with \SIrange{0.11}{1.05}{\kcalmol} and where the CC4 provides an energy very close to the TBE one.
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