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Pierre-Francois Loos 2022-03-21 22:29:11 +01:00
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@ -266,15 +266,26 @@ Are represented {\Ag} and {\tBoneg} states for the rectangular geometry and {\sB
Due to energy scaling doubly excited state {\sBoneg} and {\Aoneg} for the {\Dtwo} and {\Dfour} structures, respectively, are not drawn.
Doubly excited states are known to be challenging to represent for adiabatic time-dependent density functional theory \cite{casida_1995} (TD-DFT) and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{christiansen_1995,koch_1997} or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT). \cite{kucharski_1991,kallay_2004,hirata_2000,hirata_2004}
In order to tackle the problems of multi-configurational character and double excitations several ways are explored. The most evident way that one can think about to describe multiconfigurational and double excitations are multiconfigurational methods. Among these methods, one can find complete active space self-consistent field (CASSCF) \cite{roos_1996}, the second perturbation-corrected variant (CASPT2) \cite{andersson_1990} and the second-order $n$-electron valence state perturbation theory (NEVPT2). \cite{angeli_2001,angeli_2001a,angeli_2002}The exponential scaling of these methods with the size of the active space is the limitation to the application of these ones to big molecules.
In order to tackle the problems of multi-configurational character and double excitations several ways are explored. The most evident way that one can think about to describe multiconfigurational and double excitations are multiconfigurational methods.
Among these methods, one can find complete active space self-consistent field (CASSCF) \cite{roos_1996}, the second perturbation-corrected variant (CASPT2) \cite{andersson_1990} and the second-order $n$-electron valence state perturbation theory (NEVPT2). \cite{angeli_2001,angeli_2001a,angeli_2002}
The exponential scaling of these methods with the size of the active space is the limitation to the application of these ones to large molecules.
Another way to deal with double excitations is to use high level truncation of the equation-of-motion (EOM) formalism of coupled-cluster (CC) theory. However, to provide a correct description of doubly excited states one have to take into account contributions from the triple excitations in the CC expansion. Again, due to the scaling of CC methods with the number of basis functions the applicability of these methods is limited to small molecules.
Another way to deal with double excitations is to use high level truncation of the equation-of-motion (EOM) formalism of coupled-cluster (CC) theory.
However, to provide a correct description of doubly excited states one have to take into account contributions from the triple excitations in the CC expansion.
Again, due to the scaling of CC methods with the number of basis functions the applicability of these methods is limited to small molecules.
An alternative to multiconfigurational and CC methods is the use of selected CI (SCI) methods for the computation of transition energies for singly and doubly excited states that are known to reach near full CI energies for small molecules. These methods allow to avoid an exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, using a second-order energetic criterion to select perturbatively determinants in the FCI space.
An alternative to multiconfigurational and CC methods is the use of selected CI (SCI) methods for the computation of transition energies for singly and doubly excited states that are known to reach near full CI energies for small molecules.
These methods allow to avoid an exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, using a second-order energetic criterion to select perturbatively determinants in the FCI space.
Finally, to describe doubly excited states, one can think of spin-flip formalism established by Krylov in 2001.\cite{casanova_2020} To briefly introduce the spin-flip idea we can present it like: instead of taking the singlet ground state as reference, the reference configuration is taken as the lowest triplet state. So one can access the singlet ground state and the singlet doubly-excited state via a spin-flip deexcitation and excitation (respectively), the difference of these two excitation energies providing an estimate of the double excitation. Obviously spin-flip methods have their own flaws, especially the spin contamination \cite{casanova_2020} (i.e., an artificial mixing of electronic states of different spin multiplicities) due to spin incompleteness of the spin-flip expansion and by spin contamination of the reference configuration. One can address part of this problem by expansion of the excitation order but with an increase of the computational cost or by complementing the spin-incomplete configuration set with the missing configurations.
Finally, to describe doubly excited states, one can think of spin-flip formalism established by Krylov in 2001.\cite{casanova_2020}
To briefly introduce the spin-flip idea we can present it like: instead of taking the singlet ground state as reference, the reference configuration is taken as the lowest triplet state.
So one can access the singlet ground state and the singlet doubly-excited state via a spin-flip deexcitation and excitation (respectively), the difference of these two excitation energies providing an estimate of the double excitation.
Obviously spin-flip methods have their own flaws, especially the spin contamination \cite{casanova_2020} (\ie, an artificial mixing of electronic states of different spin multiplicities) due to spin incompleteness of the spin-flip expansion and by spin contamination of the reference configuration.
One can address part of this problem by expansion of the excitation order but with an increase of the computational cost or by complementing the spin-incomplete configuration set with the missing configurations.
In the present work we investigate $1{}^1A_{g}$, {\tBoneg}, {\sBoneg}, {\Ag} and {\sBoneg}, {\Atwog}, {\Aoneg},{\Btwog} excited states for the {\Dtwo} and {\Dfour} geometries, respectively. Computational details are reported in Section \ref{sec:compmet} for SCI (Subsection \ref{sec:SCI}), EOM-CC (Subsection \ref{sec:CC}), multiconfigurational (Subsection \ref{sec:Multi}) and spin-flip (Subsection \ref{sec:sf}) methods. Section \ref{sec:res} is devoted to the discussion of our results, first we consider the ground state property studied which is the AB (Subsection \ref{sec:auto}) and then we study the excited states (Subsection \ref{sec:states}) of the CBD molecule for both geometries.
In the present work we investigate $1{}^1A_{g}$, {\tBoneg}, {\sBoneg}, {\Ag} and {\sBoneg}, {\Atwog}, {\Aoneg},{\Btwog} excited states for the {\Dtwo} and {\Dfour} geometries, respectively.
Computational details are reported in Section \ref{sec:compmet} for SCI (Subsection \ref{sec:SCI}), EOM-CC (Subsection \ref{sec:CC}), multiconfigurational (Subsection \ref{sec:Multi}) and spin-flip (Subsection \ref{sec:sf}) methods.
Section \ref{sec:res} is devoted to the discussion of our results, first we consider the ground state property studied which is the AB (Subsection \ref{sec:auto}) and then we study the excited states (Subsection \ref{sec:states}) of the CBD molecule for both geometries.
\begin{figure}
\includegraphics[width=0.6\linewidth]{figure2.png}