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Added some comments on LIM. I'm waiting for a confirmation of the expressions to go deeper in the discussion.

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Bruno Senjean 2020-05-05 18:34:49 +02:00
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Manuscript/FarDFT.tex
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@ -479,8 +479,10 @@ For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family
Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-7}$. \cite{Becke_1988b,Lindh_2001} Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-7}$. \cite{Becke_1988b,Lindh_2001}
This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities). This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), a singly-excited state ($I=1$ with weight $\ew{1}$), as well as the lowest doubly-excited state ($I=2$ with weight $\ew{2}$) are considered. Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), a singly-excited state ($I=1$ with weight $\ew{1}$), as well as the lowest doubly-excited state ($I=2$ with weight $\ew{2}$) are considered.
Assuming that the singly-excited state is lower in energy than the doubly-excited state (which is not always the case as one would notice later), one should have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$ to ensure the GOK variational principle. Assuming that the singly-excited state is lower in energy than the doubly-excited state, one should have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$ to ensure the GOK variational principle.
If the doubly-excited state is lower in energy than the singly-excited one (which can be the case as one would notice later), then one has to swap $w_1$ and $w_2$ in the
above inequalities.
Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{} \equiv \ew{1} = \ew{2}$). Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{} \equiv \ew{1} = \ew{2}$).
In this case, the ensemble energy will be written as a single-weight quantity, $\E{}{\ew{}}$. In this case, the ensemble energy will be written as a single-weight quantity, $\E{}{\ew{}}$.
The zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiweight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$) are considered in the following. The zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiweight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$) are considered in the following.
@ -517,7 +519,7 @@ First, we compute the ensemble energy of the \ce{H2} molecule at equilibrium bon
\end{align} \end{align}
In the case of \ce{H2}, the ensemble is composed by the $\Sigma_g^+$ ground state of electronic configuration $1\sigma_g^2$, the lowest singly-excited state of the same symmetry as the ground state with configuration $1\sigma_g 2\sigma_g$, and the lowest doubly-excited state of configuration $1\sigma_u^2$ (which is also of symmetry $\Sigma_g^+$, and has an autoionising resonance nature \cite{Bottcher_1974}). In the case of \ce{H2}, the ensemble is composed by the $\Sigma_g^+$ ground state of electronic configuration $1\sigma_g^2$, the lowest singly-excited state of the same symmetry as the ground state with configuration $1\sigma_g 2\sigma_g$, and the lowest doubly-excited state of configuration $1\sigma_u^2$ (which is also of symmetry $\Sigma_g^+$, and has an autoionising resonance nature \cite{Bottcher_1974}).
The ensemble energy $\E{}{\ew{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of weight $0 \le \ew{} \le 1/3$ (blue curve). The deviation from linearity of the ensemble energy $\E{}{\ew{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of weight $0 \le \ew{} \le 1/3$ (blue curve).
Because the Slater exchange functional defined in Eq.~\eqref{eq:Slater} does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}]. Because the Slater exchange functional defined in Eq.~\eqref{eq:Slater} does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy associated with the doubly-excited state obtained via the derivative of the ensemble energy varies significantly with $\ew{}$ (see blue curve in Fig.~\ref{fig:Om_H2}). As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy associated with the doubly-excited state obtained via the derivative of the ensemble energy varies significantly with $\ew{}$ (see blue curve in Fig.~\ref{fig:Om_H2}).
Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\ew{} = 0$ to $1/3$. Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\ew{} = 0$ to $1/3$.
@ -743,25 +745,38 @@ For comparison purposes, we also report the linear interpolation method (LIM) ex
a pragmatic way of getting weight-independent excitation energies, defined as a pragmatic way of getting weight-independent excitation energies, defined as
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\Ex{\LIM}{(1)} & = 2 \qty[\E{}{\bw{}=(1/2,0)} - \E{}{\bw{}=(0,0)}], \Ex{\LIM}{(1)} & = 2 \qty[\E{}{\bw{}=(1/2,0)} - \E{}{\bw{}=(0,0)}], \label{eq:LIM1}
\\ \\
\Ex{\LIM}{(2)} & = 3 \qty[\E{}{\bw{}=(1/3,1/3)} - \E{}{\bw{}=(1/2,0)}] + \frac{1}{2} \Ex{\LIM}{(1)}, \Ex{\LIM}{(2)} & = 3 \qty[\E{}{\bw{}=(1/3,1/3)} - \E{}{\bw{}=(1/2,0)}] + \frac{1}{2} \Ex{\LIM}{(1)}, \label{eq:LIM2}
\end{align} \end{align}
\end{subequations} \end{subequations}
which require three independent calculations, as well as the MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b} which require three independent calculations, as well as the MOM excitation energies \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\Ex{\MOM}{(1)} & = \E{}{\bw{}=(1,0)} - \E{}{\bw{}=(0,0)}, \Ex{\MOM}{(1)} & = \E{}{\bw{}=(1,0)} - \E{}{\bw{}=(0,0)}, \label{eq:MOM1}
\\ \\
\Ex{\MOM}{(2)} & = \E{}{\bw{}=(0,1)} - \E{}{\bw{}=(0,0)}. \Ex{\MOM}{(2)} & = \E{}{\bw{}=(0,1)} - \E{}{\bw{}=(0,0)}. \label{eq:MOM2}
\end{align} \end{align}
\end{subequations} \end{subequations}
which also require three separate calculations at a different set of ensemble weights. which also require three separate calculations at a different set of ensemble weights.
As readily seen in Eqs.~(\ref{eq:LIM1}) and (\ref{eq:LIM2}), LIM is a recursive strategy where the first excitation energy has to be determined
in order to compute the second one. In the above equations, we
assumed that the singly-excited state (with weight $w_1$) was lower
in energy compared to the doubly-excited one (with weight $w_2$).
If the ordering changes, then one should read $\E{}{\bw{}=(0,1/2)}$
instead of $\E{}{\bw{}=(1/2,0)}$ in Eqs.~(\ref{eq:LIM1}) and (\ref{eq:LIM2}) which then correspond to the excitation energies of the
doubly-excited state and the singly-excited one, respectively.
The same hold for the MOM excitation energies in
Eqs.~\ref{eq:MOM1} and \ref{MOM2}.
For a general expression with multiple (and possibly degenerate)
states, we
refer the reader to Eq.~106 of Ref.~\cite{Senjean_2015} (note
however that another convention were used to define the ensemble).
\bruno{Note that by construction, for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper), LIM and MOM can be reduced to a single calculation at $\ew{} = 1/4$ and $\ew{} = 1/2$, respectively, instead of performing an interpolation between two different calculations.}
The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weights are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional. The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weights are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI. The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI.
It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes in this limit (\textit{vide supra}). It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes in this limit (\textit{vide supra}).
\bruno{Note that by construction, for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper), LIM and MOM can be reduced to a single calculation at $\ew{} = 1/4$ and $\ew{} = 1/2$, respectively, instead of performing an interpolation between two different calculations.}
Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between the ghost-interaction-free pure-state limits. Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between the ghost-interaction-free pure-state limits.
%%% TABLE III %%% %%% TABLE III %%%