From d5a492cf7a2206fe20170271472502c263934f85 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 4 Jun 2020 15:31:32 +0200 Subject: [PATCH] revision done --- Manuscript/FarDFT.tex | 70 ++++++----------------------- Response_Letter/Response_Letter.tex | 46 +++++++++---------- 2 files changed, 33 insertions(+), 83 deletions(-) diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index ef8e83a..388f21e 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -188,7 +188,7 @@ In other words, memory effects are absent from the xc functional which is assume Third and more importantly in the present context, a major issue of TD-DFT actually originates directly from the choice of the (ground-state) xc functional, and more specifically, the possible (not to say likely) substantial variations in the quality of the excitation energies for two different choices of xc functionals. -Because of its popularity, approximate TD-DFT has been studied extensively by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies. +Because of its popularity, approximate TD-DFT has been studied \alert{extensively}, and some researchers have quickly unveiled various theoretical and practical deficiencies. For example, TD-DFT has problems with charge-transfer \cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Maitra_2017} and Rydberg \cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the semi-local xc functional. The development of range-separated hybrids provides an effective solution to this problem. \cite{Tawada_2004,Yanai_2004} From a practical point of view, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent. @@ -295,7 +295,7 @@ Their dependence on the density is determined from the ensemble density constrai \end{equation} Note that the original decomposition \cite{Gross_1988b} shown in Eq.~\eqref{eq:FGOK_decomp}, where the conventional (weight-independent) Hartree functional -\beq +\beq \label{eq:Hartree} \E{\Ha}{}[\n{}{}]=\frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}' \eeq is separated @@ -390,24 +390,24 @@ We will also adopt the usual decomposition, and write down the weight-dependent where $\e{\ex}{\bw{}}(\n{}{})$ and $\e{\co}{\bw{}}(\n{}{})$ are the weight-dependent density-functional exchange and correlation energies per particle, respectively. -\manu{As shown in Sec.~\ref{subsubsec:weight-dep_corr_func}, the weight +\alert{As shown in Sec.~\ref{subsubsec:weight-dep_corr_func}, the weight dependence of the correlation energy can be extracted from a FUEG model. In order to make the resulting weight-dependent -correlation functional truly universal, \ie~ +correlation functional truly universal, \ie, independent on the number of electrons in the FUEG, one could use the curvature of the Fermi hole~\cite{Loos_2017a} as an additional variable in the density-functional approximation. The development of such a -generalized correlation eLDA is left for future work. Even though a similar strategy could be applied +generalised correlation eLDA is left for future work. Even though a similar strategy could be applied to the weight-dependent exchange part, we explore in the present work a different path where the (system-dependent) exchange functional -parameterization relies on the ensemble energy linearity +parameterisation relies on the ensemble energy linearity constraint (see Sec.~\ref{subsubsec:weight-dep_x_fun}). Finally, let us stress that, in order to further improve the description of the ensemble correlation energy, a post-treatment of the recently revealed density-driven -correlations~\cite{Gould_2019,Gould_2019_insights,gould_2020,Fromager_2020} [which, by construction, are absent -from FUEGs] might be necessary. An orbital-dependent correction derived +correlations~\cite{Gould_2019,Gould_2019_insights,gould_2020,Fromager_2020} (which, by construction, are absent +from FUEGs) might be necessary. An orbital-dependent correction derived in Ref.~\onlinecite{Fromager_2020} might be used for that purpose. Work is currently in progress in this direction.\\ @@ -505,7 +505,7 @@ linear ensemble energy and, hence, the same value of the excitation energy indep \includegraphics[width=\linewidth]{fig1} \caption{ \ce{H2} at equilibrium bond length: deviation from linearity of the ensemble energy $\E{}{\ew{}}$ (in hartree) as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set. - See main text for the definition of the various functionals' acronyms. + See main text for the definition of the various \alert{functionals'} acronyms. \label{fig:Ew_H2} } \end{figure} @@ -516,7 +516,7 @@ linear ensemble energy and, hence, the same value of the excitation energy indep \includegraphics[width=\linewidth]{fig2} \caption{ \ce{H2} at equilibrium bond length: error (with respect to FCI) in the excitation energy $\Ex{}{(2)}$ (in eV) associated with the doubly-excited state as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set. - See main text for the definition of the various functionals' acronyms. + See main text for the definition of the various \alert{functionals'} acronyms. \label{fig:Om_H2} } \end{figure} @@ -553,24 +553,6 @@ makes the ensemble energy $\E{}{(0,\ew{2})}$ almost perfectly linear (by constru It also allows to ``flatten the curve'' making 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.~\eqref{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. @@ -591,7 +573,8 @@ We enforce this type of \textit{exact} constraint (to the maximum possible exten 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 $\Cx{\ew{2}}$ reduces to $\Cx{}$ in these two limits. 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 $\ew{2} = 0$, 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. +\titou{Nonetheless, the CC-S functional also includes quadratic terms in order to compensate the spurious curvature of the ensemble energy originating, mainly, from the Hartree term [see Eq.~\eqref{eq:Hartree}].} +%We shall come back to this point later on. %%% FIG 3 %%% \begin{figure} @@ -608,7 +591,7 @@ We shall come back to this point later on. \subsubsection{Weight-independent correlation functional} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -Third, we include correlation effects via the conventional VWN5 local correlation functional. \cite{Vosko_1980} +Third, we \titou{include} correlation effects via the conventional VWN5 local correlation functional. \cite{Vosko_1980} For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}. The combination of the (weight-independent) Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (green curve in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a smaller curvature and improved excitation energies (red curve in Figs.~\ref{fig:Ew_H2} and \ref{fig:Om_H2}), especially at small weights, where the CC-SVWN5 excitation energy is almost spot on. @@ -758,10 +741,6 @@ In particular, we report the excitation energies obtained with GOK-DFT in the zero-weight limit (\ie, $\ew{} = 0$) and for equi-weights (\ie, $\ew{} = 1/3$). These excitation energies are computed using Eq.~\eqref{eq:dEdw}. -%\manuf{OK but, again, how do you compute the exchange ensemble -%derivative for both excited states when it seems like the functional in -%Eqs.~\eqref{eq:ensemble_Slater_func} and \eqref{eq:Cxw} -%only depends on $\ew{}$ rather than $\ew{1}$ AND $\ew{2}$.} For comparison, we also report results obtained with the linear interpolation method (LIM). \cite{Senjean_2015,Senjean_2016} The latter simply consists in extracting the excitation energies (which are @@ -774,16 +753,6 @@ follows: \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{subequations} -%\manu{ -%$\frac{1}{2}\Ex{\LIM}{(1)}=\frac{1}{2}\left(E_1-E_0\right)$\\ -%$\E{}{\bw{}=(1/3,1/3)}=\frac{1}{3}\left(E_0+E_1+E_2\right)$\\ -%$\E{}{\bw{}=(1/2,0)}=\frac{1}{2}\left(E_0+E_1\right)$\\ -%$3 \qty[\E{}{\bw{}=(1/3,1/3)} - -%\E{}{\bw{}=(1/2,0)}]=-\frac{1}{2}\left(E_0+E_1\right)+E_2$ -%\\ -%$3 \qty[\E{}{\bw{}=(1/3,1/3)} - -%\E{}{\bw{}=(1/2,0)}]+\frac{1}{2} \Ex{\LIM}{(1)}=E_2-E_0$ -%}\\ For a general expression with multiple (and possibly degenerate) states, we refer the reader to Eq.~(106) of Ref.~\onlinecite{Senjean_2015}, where LIM is shown to interpolate linearly the ensemble energy between equi-ensembles. Note that two calculations are needed to get the first LIM excitation energy, with an additional equi-ensemble calculation for each higher excitation energy. @@ -896,10 +865,6 @@ We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}]. The weight dependence of $\Cx{\ew{2}}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve). -%\manuf{Again, it would be nice to say -%explicitly if you construct a functional, function of $\ew{1}$ and -%$\ew{2}$ (how then) or just $\ew{}$ (how to compute the separate -%derivatives then?)} One clearly sees that the correction brought by CC-S is much more gentle than at $\RHH = 1.4$ bohr, which means that the ensemble energy obtained with the LDA exchange functional is much more linear at $\RHH = 3.7$ bohr. %In other words, the curvature ``hole'' depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry. @@ -907,15 +872,6 @@ Note that this linearity at $\RHH = 3.7$ bohr was also observed using weight-ind Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers basis set converged results), the same set of calculations as in Table \ref{tab:BigTab_H2}. As a reference value, we computed a FCI/aug-cc-pV5Z excitation energy of $8.69$ eV, which compares well with previous studies. \cite{Senjean_2015} For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the closest match being reached with HF exchange and VWN5 correlation at equi-weights. -%\manu{We did not mention HF exchange neither in the theory section nor -%in the computational details. We should be clear about this. Is this an -%ad-hoc correction, like in our previous work on ringium? Is HF exchange -%used for the full ensemble energy (i.e. the HF interaction energy is -%computed with the ensemble density matrix and therefore with -%ghost-interaction errors) or for -%individual energies (that you state-average then), like in our previous -%work. I guess the latter option is what you did. We need to explain more -%what we do!!!} As expected from the linearity of the ensemble energy, the CC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits. As a direct consequence of this linearity, LIM and MOM do not provide any noticeable improvement on the excitation diff --git a/Response_Letter/Response_Letter.tex b/Response_Letter/Response_Letter.tex index 78e914d..2484df4 100644 --- a/Response_Letter/Response_Letter.tex +++ b/Response_Letter/Response_Letter.tex @@ -64,11 +64,10 @@ density-driven correlations [62,92-94] (which, by construction, are absent from \item {Not clear what density is used in equation 21: From the discussion that follows this appears to not be the ensemble density (which is weight dependent and I would expect it to be used here) but the density only for the ground state Slater determinant. The authors should explain why this is so. } \\ - \alert{The density $n$ used in Eq.~(21), (9), (10) can be any density and does not represent any specific density. - In the case of Eq.~(21), we simply present the well-known Dirac-exchange density functional and, by definition of a density functional, does not need to specify in its formulation to which density it is applied but only that it is a mathematical object applying to any density $n(r)$. - Of course, when we will use this functional or any other one in -our work we will surely apply it to the {\it minimizing} ensemble density -$n^w(r)$ [see Eqs. (5) and (11)] and the notation will be carefully modified accordingly.} + \alert{The density $n$ used in Eqs.~(21), (9), (10) can be any density, and, mathematically, there is no need to specify its origin. + In the case of Eq.~(21), we simply define the well-known Dirac exchange functional. + By definition of a density functional, we do not need to specify in its formulation to which density it is applied but only that it is a mathematical object applying to any density $n(r)$. + Of course, when using this functional (or any other ones) in our work we surely apply it to the {\it minimizing} ensemble density $n^w(r)$ [see Eqs. (5) and (11)] and the notation is carefully modified accordingly.} \item {Change "Third, we add up correlation effects" to "Third, we include correlation effects"} @@ -84,33 +83,28 @@ $n^w(r)$ [see Eqs. (5) and (11)] and the notation will be carefully modified acc \item {They need to be a bit more consistent with their notation as in equation 9 and elsewhere "$n(r)$" should be the ensemble (weight-dependent) density "$n^w(r)$". - I don?t believe they defined "$n(r)$" in the paper so I don?t know which density it represents. } + I don't believe they defined "$n(r)$" in the paper so I don?t know which density it represents. } \\ - \alert{See our response to 4.} + \alert{See our response to 3.} \item {Even if it sounds trivial, they should explain why the exact xc functional should have linear dependence in the excitation energies as a function of the weight value.} \\ - \alert{GOK variational principle states that the expectation value of the ensemble energy admits/possesses a lower bond which is linear with respect to each of the ensemble-weights $w_i$ and is the exact ensemble energy of the studied system (equation 1). - Moreover, by construction, one can easily see that the slope of the exact ensemble energy with respect to a specific weight $w_i$ corresponds to the excitation energy of the system defined between the ground state and the ith-excited state associated to this specific weight (equation 4). + \alert{As clearly stated in the original manuscript, GOK variational principle states that the expectation value of the ensemble energy admits a lower bond which is linear with respect to each of the ensemble weight $w_I$ and is the exact ensemble energy of the studied system [Eq.~(1)]. + Moreover, by construction, one can easily see that the slope of the exact ensemble energy with respect to a specific weight $w_I$ corresponds to the excitation energy of the system defined between the ground state and the $I$th excited state associated to this specific weight [Eq.~(4)]. It is important that the reader keeps in mind that the exact excitation energies are based on pure-state energies and, therefore, do not depend on the weights of the ensemble. - In practice, the ensemble energy is rarely w-linear (linear in w ?) because of the use of approximate xc-functionals. - Indeed, by inserting the ensemble density in the Hartree interaction functional (equation 9), it introduces spurious quadratic curvature with respect to the weight in the ensemble energy. - Some of those terms are responsible of the unphysical phenomenon called ghost-interaction errors. - Therefore, the ensemble-Khon-Sham gap obtained at the end of the ensemble-HF-calculation is, somehow, "weight-contaminated" and doesn't possess the right weight-dependence. - (two first terms of the right-hand side of equation16) - By taking its first derivative with regard to the weight, the xc-functional is expected to compensate those parasite-quadratic terms in order to retrieve the linear behavior of the exact ensemble energy and one can understand that only a weight-dependant xc-functional could do so. - At the best of my knowledge, I cannot see any reason why the xc-functional should be w-linear. - The important idea is that the linearity must be in the ensemble energy but the main constraint on the xc-functional should be that it is weight-dependant. - We emphasize that only the exact ensemble-xc-functional would have the ideal weight-dependency that would make the corresponding ensemble energy reproduce perfectly the linear behavior of the exact ensemble energy and lead to weight-independant excitation energies, that is exact excitation energies. - The use of an approximate weight-dependant xc-functional could reduce the ensemble energy curvature and give less weight-dependant excitation energies but it is reasonable to admit that it also could make things worse it the weight-dependency of the functional is poorly chosen. - That is why the construction of "good" weight-dependant -xc-functionals is a really challenging matter in eDFT.\\ -As a final comment, we insist after Eq. (28) on the fact that, in the -exact theory, the xc ensemble density functional has no reason to vary (for a fixed density $n$) -linearly with the ensemble weights. We refer to Ref. 92 where exact -expressions for the individual xc energies are derived.} - + In practice, the ensemble energy is rarely linear in $w$ because of the approximate nature of the xc functionals. + Indeed, by inserting the ensemble density in the Hartree interaction functional [Eq.~(9)], one introduces, in the ensemble energy, spurious quadratic curvature with respect to the weight. + Some of those terms are responsible of the unphysical phenomenon called ghost interaction, as explained in the manuscript. + Therefore, the ensemble Khon-Sham gap is, somehow, "weight contaminated" and does not possess the correct weight dependence [see first two terms of the right-hand side of Eq.~(16)]. + By taking its derivative with respect to the weight, the weight-dependent xc functional is expected to compensate those spurious quadratic terms in order to retrieve a linear behavior of the exact ensemble energy: only a weight-dependent xc functional can achieve such a feat. + In other words, the xc functional does not have to be linear with respect to $w$. + We have clarified this point in the revised manuscript. + We emphasize that the exact ensemble xc functional has the ideal weight dependency, and would make the corresponding ensemble energy perfectly linear, hence leading to weight-independent excitation energies. + As shown in the present manuscript, the use of an approximate weight-dependent xc functional reduces the ensemble energy curvature as well as the variation of the excitation energies with respect to $w$. + This illustrates why the construction of reliable weight-dependent xc functionals is a challenging task in eDFT. + As a final comment, we insist after Eq.~(28) on the fact that, in the exact theory, the xc ensemble density functional has no reason to vary (for a fixed density $n$) linearly with the ensemble weights. + We refer to Ref.~92 where exact expressions for the individual xc energies are derived.} \end{enumerate} \end{letter}