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Merge branch 'master' of https://git.irsamc.ups-tlse.fr/loos/srDFT_Ex

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Pierre-Francois Loos 2019-06-11 11:11:26 +02:00
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Manuscript/Ex-srDFT.tex
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@ -255,7 +255,7 @@ In other words, the correction vanishes in the CBS limit, hence guaranteeing an
\label{sec:rs} \label{sec:rs}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can efficiently approximate $\bE{}{\Bas}[\n{}{}]$ by short-range correlation functionals with multi-determinantal (ECMD) reference, \cite{TouGorSav-TCA-05} $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$. \manu{As it was initially proposed in Ref~\onlinecite{GinPraFerAssSavTou-JCP-18} and further developed in} Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can efficiently approximate $\bE{}{\Bas}[\n{}{}]$ by short-range correlation functionals with multi-determinantal (ECMD) reference, \cite{TouGorSav-TCA-05} $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$.
The ECMD functionals admit, for any $\n{}{}$, the following two limits The ECMD functionals admit, for any $\n{}{}$, the following two limits
\begin{align} \begin{align}
\label{eq:large_mu_ecmd} \label{eq:large_mu_ecmd}
@ -289,7 +289,7 @@ and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{
\f{}{\Bas}(\br{1},\br{2}) \f{}{\Bas}(\br{1},\br{2})
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation} \end{equation}
and $\V{pq}{rs}= \braket{pq}{rs}$ are two-electron Coulomb integrals. and $\V{pq}{rs}= \braket{pq}{rs}$ are two-electron Coulomb integrals. \manu{An important consequence of the definition of $\W{}{\Bas}(\br{1},\br{2})$ is that, when $\Bas$ is complete, the effective interaction tends to the regular $1/r_{12}$ which ensures that the present approximations $\bE{}{\Bas}[\n{}{}]$ vanishes when $\Bas$. }
We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JPCL-19} for additional details. We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JPCL-19} for additional details.
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@ -332,7 +332,7 @@ This computationally-lighter functional will be referred to as PBE.
\label{sec:compdetails} \label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
In the present study, we compute the ground- and excited-state energies, one-electron and on-top densities with a selected configuration interaction (sCI) method known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15} In the present study, we compute the ground- and excited-state energies, one-electron and on-top densities with a selected configuration interaction (sCI) method known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
The total energy of each state is obtained via an efficient extrapolation procedure of the sCI energies designed to reach near-FCI accuracy. \cite{QP2} The total energy of each state is obtained via an efficient extrapolation procedure of the sCI energies designed to reach near-FCI accuracy. \cite{QP2}(\manu{We should add the reference to the original paper by cyrus})
These energies will be labeled exFCI in the following. These energies will be labeled exFCI in the following.
Using near-FCI excitation energies (within a given basis set) has the indisputable advantage to remove the error inherent to the WFT method. Using near-FCI excitation energies (within a given basis set) has the indisputable advantage to remove the error inherent to the WFT method.
Indeed, in the present case, the only source of error on the excitation energies is due to basis set incompleteness. Indeed, in the present case, the only source of error on the excitation energies is due to basis set incompleteness.
@ -507,7 +507,7 @@ These results are also depicted in Figs.~\ref{fig:H2O} and \ref{fig:NH3} for \ce
One would have noticed that the basis set effects are particularly strong for the third singlet excited state of water and the third and fourth singlet excited states of ammonia where this effect is even magnified. One would have noticed that the basis set effects are particularly strong for the third singlet excited state of water and the third and fourth singlet excited states of ammonia where this effect is even magnified.
In these cases, one really needs doubly-augmented basis sets to reach radial completeness. In these cases, one really needs doubly-augmented basis sets to reach radial completeness.
The first observation worth reporting is that all three RS-DFT correlation functionals have very similar behaviors and they significantly reduce the error on the excitation energies for most of the states. The first observation worth reporting is that all three RS-DFT correlation functionals have very similar behaviors and they significantly reduce the error on the excitation energies for most of the states.
However, these results also clearly evidence that special care has to be taken for very diffuse excited states where the present correction might not be enough to catch the radial incompleteness of the one-electron basis set. However, these results also clearly evidence that special care has to be taken for very diffuse excited states where the present correction might not be enough to catch the radial incompleteness of the one-electron basis set.\manu{It could be interesting to point out that such effects are far from being cusp like effects}
%%% TABLE 2 %%% %%% TABLE 2 %%%
@ -678,7 +678,7 @@ However, these results also clearly evidence that special care has to be taken f
It is interesting to study the behavior of $\rsmu{}{\Bas}(\br{})$ for different states as the basis set incompleteness error is obviously state specific. It is interesting to study the behavior of $\rsmu{}{\Bas}(\br{})$ for different states as the basis set incompleteness error is obviously state specific.
To do so, we consider the ground state (${}^{1}\Sigma^+$) of carbon monoxide as well as its lowest singlet excited state (${}^{1}\Pi$). To do so, we consider the ground state (${}^{1}\Sigma^+$) of carbon monoxide as well as its lowest singlet excited state (${}^{1}\Pi$).
The values of the vertical excitation energies obtained for various methods and basis sets are reported in Table \ref{tab:Mol}. The values of the vertical excitation energies obtained for various methods and basis sets are reported in Table \ref{tab:Mol}.
Figure \ref{fig:CO} represents $\rsmu{}{}(z)$ along the nuclear axis ($z$) for these two electronic states computed with the AVDZ, AVTZ and AVQZ basis sets. Figure \ref{fig:CO} represents $\rsmu{}{}(z)$ along the nuclear axis ($z$) for these two electronic states computed with the AVDZ, AVTZ and AVQZ basis sets. \manu{These figures illustrate several important things: i) the maximal values of $\rsmu{}{\Bas}(\br{})$ are systematically close to the nuclei, a signature of the atom-centered basis set, ii) the overall values of $\rsmu{}{\Bas}(\br{})$ increase with the basis set, which reflects the improvement of the description of the correlation effects when enlarging the basis set, iii) the value of $\rsmu{}{\Bas}(\br{})$ are slightly larger near the oxygen atom, which traduces the fact that the inter-electronic distance is higher than close to the carbon atom due to a higher nuclear charge. }
\titou{Add description about the graphs here.} \titou{Add description about the graphs here.}
%%% FIG 4 %%% %%% FIG 4 %%%