From 2684d6cbded10b8167030a20811709ac7bc985b4 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Sun, 10 May 2020 11:17:49 +0200 Subject: [PATCH] Manu: IV A 2 --- Manuscript/FarDFT.tex | 47 ++++++++++++++++++++++++++++++++++++++----- 1 file changed, 42 insertions(+), 5 deletions(-) diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 333f42b..b22afe8 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -242,7 +242,7 @@ Let us consider a GOK ensemble of $\nEns$ electronic states with individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and (normalised) monotonically decreasing weights $\bw \equiv (\ew{1},\ldots,\ew{M-1})$, \ie, $\ew{0}=1-\sum_{I=1}^{\nEns-1} \ew{I}$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$. The corresponding ensemble energy -\begin{equation} +\begin{equation}\label{eq:exp_ens_ener} \E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)} \end{equation} can be obtained from the GOK variational principle @@ -576,7 +576,8 @@ the excitation energy associated with the doubly-excited state obtained via the derivative of the ensemble energy \manu{with respect to $\ew{2}$ (and taken at $\ew{2}=\ew{}=\ew{1}$)} revaries 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 $1/3$. -Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of the ensemble weights. +Note that the exact xc ensemble functional would yield a perfectly +linear \manu{ensemble} energy and, hence, the same value of the excitation energy independently of the ensemble weights. %%% FIG 1 %%% \begin{figure} @@ -605,7 +606,7 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Second, in order to remove some of this spurious curvature of the ensemble -energy (which is mostly due to the ghost-interaction error, \cite{Loos_2020} but not only), +energy (which is mostly due to the ghost-interaction error \cite{Gidopoulos_2002}, but not only \cite{Loos_2020}), one can easily reverse-engineer (for this particular system, geometry, basis set, and excitation) a local exchange functional to make $\E{}{(0,\ew{2})}$ as linear as possible for $0 \le \ew{2} \le 1$ assuming a perfect linearity between the pure-state limits $ \ew{1} = \ew{2} = 0$ (ground state) and $\ew{1} = 0 \land \ew{2} = 1$ (doubly-excited state). Doing so, we have found that the following weight-dependent exchange functional (denoted as CC-S for ``curvature-corrected'' Slater functional) \begin{equation} @@ -627,13 +628,49 @@ and \end{align} \end{subequations} makes the ensemble energy $\E{}{(0,\ew{2})}$ almost perfectly linear (by construction), and removes some of the curvature of $\E{}{\ew{}}$ (see yellow curve in Fig.~\ref{fig:Ew_H2}). -It also makes the excitation energy much more stable (with respect to $\ew{}$), and closer to the FCI reference (see yellow curve in Fig.~\ref{fig:Om_H2}). +It also makes the excitation energy much more stable (with respect to +$\ew{}$), and closer to the FCI reference (see yellow curve in +Fig.~\ref{fig:Om_H2}).\\ + +\manuf{One point is not clear to me at all. If I understood correctly, +the optimization of $\alpha$, $\beta$, and $\gamma$ is done for +$\ew{1}=0$. So, once the optimisation is done, we have a coefficient +$\Cx{\ew{2}}$ that is a function of $\ew{2}$. Then, how do you obtain +a coefficient $\Cx{\ew{}}$ that is supposed to describe a {\it +different} ensemble defined as $\ew{1}=\ew{2}=\ew{}$ (it says in the +computational details that, ultimately, this is what we are looking at)? Did you just +replace $\ew{2}$ by $\ew{}$? This should be clarified. Another point: in +order to apply Eq.~(\ref{eq:dEdw}) for computing excitation energies, +you need $\ew{1}$ and $\ew{2}$ to be independent variables before +differentiating (and taking the value of the derivatives at +$\ew{1}=\ew{2}=\ew{}$). I do not see how you can do this (and generate +the results in Fig.~\ref{fig:Om_H2}) if the only ensemble functional you +have depends on $\ew{}$ and not on both $\ew{1}$ and $\ew{2}$. Regarding +Fig.~\ref{fig:Om_H2}, I would suspect +that you took $\ew{1}=0$, which is questionable and not clear at all from +the text.} + The parameters $\alpha$, $\beta$, and $\gamma$ entering Eq.~\eqref{eq:Cxw} have been obtained via a least-square fit of the non-linear component of the ensemble energy computed between $\ew{2} = 0$ and $\ew{2} = 1$ by steps of $0.025$. Although this range of weights is inconsistent with GOK theory, we have found that it is important, from a practical point of view, to ensure a correct behaviour in the whole range of weights in order to obtain accurate excitation energies. The present procedure can be related to optimally-tuned range-separated hybrid functionals, \cite{Stein_2009} where the range-separation parameters (which control the amount of short- and long-range exact exchange) are determined individually for each system by iteratively tuning them in order to enforce non-empirical conditions related to frontier orbitals (\eg, ionisation potential, electron affinity, etc) or, more importantly here, the piecewise linearity of the ensemble energy for ensemble states described by a fractional number of electrons. \cite{Stein_2009,Stein_2010,Stein_2012,Refaely-Abramson_2012} - +\manu{In this context, the analog of the ``IP theorem'' for the first +(neutral) +excitation, for example, would read as follows [see +Eqs.~(\ref{eq:exp_ens_ener}), (\ref{eq:diff_Ew}), and (\ref{eq:dEdw})]: +\beq +2\left(E^{\ew{1}=1/2}-E^{\ew{1}=0}\right)&\overset{0\leq \ew{1}\leq 1/2}{=}&\Eps{1}{\ew{1}} - \Eps{0}{\ew{1}} + \left. +\pdv{\E{\xc}{\ew{1}}[\n{}{}]}{\ew{1}} \right|_{\n{}{} = +\n{}{\ew{1}}}.%,\hspace{0.2cm}0\leq \ew{1}\leq 1/2. +\eeq +We enforce this type of {\it exact} constraint (to the +maximum possible extent) when optimizing the parameters in +Eq.~(\ref{eq:Cxw}) in order to minimize the curvature of the ensemble energy.} As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw} (red curve), the weight-dependent correction does not affect the two ghost-interaction-free limits at $\ew{1} = \ew{2} = 0$ and $\ew{1} = 0 \land \ew{2} = 1$ (\ie, the pure-state limits), as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits. +\manuf{again, when reading the text and looking at the figure, I feel +like $\ew{}$ on the $x$ axis is in fact $\ew{2}$, and $\ew{1}$ is set to +zero. Nothing to do with the $\ew{1}=\ew{2}=\ew{}$ case expected from +the computational details. This is very confusing.} Indeed, it is important to ensure that the weight-dependent functional does not alter these pure-state limits, which are genuine saddle points of the KS equations, as mentioned above. Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear: this is the main feature that one needs to catch in order to get accurate excitation energies in the zero-weight limit. We shall come back to this point later on.