From 40f077a595904edd32231137c2df87d34fcdf445 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Thu, 7 May 2020 22:53:05 +0200 Subject: [PATCH] correction Fig 4 --- Notebooks/eDFT_FUEG.nb | 6867 ++++++++++++++------------- Response_Letter/Response_Letter.tex | 24 +- Revised_Manuscript/EIvsW_n5.pdf | Bin 106248 -> 0 bytes Revised_Manuscript/EvsL_5.pdf | Bin 73872 -> 0 bytes Revised_Manuscript/EvsL_DD.pdf | Bin 151952 -> 0 bytes Revised_Manuscript/EvsN.pdf | Bin 84032 -> 0 bytes Revised_Manuscript/EvsN_DD.pdf | Bin 92420 -> 0 bytes Revised_Manuscript/EvsW_n5.pdf | Bin 87240 -> 0 bytes Revised_Manuscript/eDFT.tex | 44 +- Revised_Manuscript/embedding.pdf | Bin 80714 -> 0 bytes Revised_Manuscript/rho.pdf | Bin 51929 -> 0 bytes 11 files changed, 3475 insertions(+), 3460 deletions(-) delete mode 100644 Revised_Manuscript/EIvsW_n5.pdf delete mode 100644 Revised_Manuscript/EvsL_5.pdf delete mode 100644 Revised_Manuscript/EvsL_DD.pdf delete mode 100644 Revised_Manuscript/EvsN.pdf delete mode 100644 Revised_Manuscript/EvsN_DD.pdf delete mode 100644 Revised_Manuscript/EvsW_n5.pdf delete mode 100644 Revised_Manuscript/embedding.pdf delete mode 100644 Revised_Manuscript/rho.pdf diff --git a/Notebooks/eDFT_FUEG.nb b/Notebooks/eDFT_FUEG.nb index 93a4b97..7709eeb 100644 --- a/Notebooks/eDFT_FUEG.nb +++ b/Notebooks/eDFT_FUEG.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 12318169, 261582] -NotebookOptionsPosition[ 12185160, 259415] -NotebookOutlinePosition[ 12185607, 259433] -CellTagsIndexPosition[ 12185564, 259430] +NotebookDataLength[ 12318270, 261587] +NotebookOptionsPosition[ 12185260, 259420] +NotebookOutlinePosition[ 12185707, 259438] +CellTagsIndexPosition[ 12185664, 259435] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -37,7 +37,7 @@ mathpazo,xcolor,bm,mhchem}\>\"", "}"}]}]}], "]"}], ";"}]}], "Input", InitializationCell->True, CellChangeTimes->{{3.7288240181604652`*^9, 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FrontEnd`AttachCell[Typeset`box$, - FrontEndResource["RGBColorValueSelector"], { - 0, {Left, Bottom}}, {Left, Top}, - "ClosingActions" -> { - "SelectionDeparture", "ParentChanged", - "EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator -> - Automatic, Method -> "Preemptive"], - RGBColor[0, 0, 1], Editable -> False, Selectable -> - False], ",", - RowBox[{"Thickness", "[", "0.01`", "]"}]}], "]"}], ",", RowBox[{"Directive", "[", RowBox[{ RowBox[{"PointSize", "[", "0.011000000000000001`", "]"}], @@ -69364,9 +69372,10 @@ AeRhYPycAYE34vD4q7n/45ZxtqwDevwCAObl4Mw= CellChangeTimes->{ 3.792064960680579*^9, 3.792065089886101*^9, {3.79206524763841*^9, 3.792065260353528*^9}, 3.792838396305697*^9, 3.792839482267396*^9, { - 3.792839520926496*^9, 3.792839572797037*^9}}, + 3.792839520926496*^9, 3.792839572797037*^9}, 3.797870490842554*^9, + 3.797870662524585*^9}, CellLabel-> - "Out[205]=",ExpressionUUID->"ea478f5f-0f87-4e63-9664-614026fb48e8"] + 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individual -correlation energies should vary with both the density {\it -and} the ensemble weights. There is in principle no reason to expect the -same variations for different ensembles and density regimes. The fact -that, for $L=8\pi$, electron correlation is strong and therefore the -density is more localized, is probably the reason for the disappearance -of the crossover point. We were not able to rationalize this observation -further but we still mention in the revised manuscript that it is an -illustration of the importance of both the density and the weights in -the evaluation of individual energies within an ensemble.} + \alert{The legend of Fig.~2 was incorrect (the curves were mislabeled), but this has now been corrected. + In the new Fig.~2 (which is now Fig.~4 in the revised manuscript), this crossover has disappeared and the discussion is much more fluid: + when the weight of a state increases, this state is stabilized while the two others increases in energy (as it should). + The discussion regarding this figure has been modified accordingly.} +% \alert{It is clear from our derivations that the individual +%correlation energies should vary with both the density {\it +%and} the ensemble weights. There is in principle no reason to expect the +%same variations for different ensembles and density regimes. The fact +%that, for $L=8\pi$, electron correlation is strong and therefore the +%density is more localized, is probably the reason for the disappearance +%of the crossover point. We were not able to rationalize this observation +%further but we still mention in the revised manuscript that it is an +%illustration of the importance of both the density and the weights in +%the evaluation of individual energies within an ensemble.} \item {Page 8: If the authors have evidence of behavior between $w=(0,0)$ and the equiensemble, instead of just these endpoints, that would be interesting to mention for the eDFT crowd. } diff --git a/Revised_Manuscript/EIvsW_n5.pdf b/Revised_Manuscript/EIvsW_n5.pdf deleted file mode 100644 index 73ebd43cbc5690a47f9c11f93f571e8fcd58bfe9..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 106248 zcmZ6ybCf2r^XR=}+qP|6JNAxk;~CqwZDU6}wz*^T8Qa#*o$n9leeeCFPfmBzl}dFb zm83tRP!gA9U}5BdrD(qV{SM1c%uMWHYz@oLPs}7|W^dtQNzC#6NClRdn3ze@%GSlq z`8%~WaxoJ(GjT9AgB28n1-Lky8QH;l57+r9vn-uyt%4AbV zC-C`1H2Kx=+ve=c?z4CEi|*^#;j8^)W+%=6!@%mz&wJZVEphtTwbsH;RsB>&-RED~ 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\includegraphics[width=0.7\linewidth]{fig1} \caption{ \label{fig:embedding} \titou{Schematic view of the ``embedding'' scheme: the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (the bath). @@ -952,7 +952,7 @@ For small $L$, the system is weakly correlated, while strong correlation effects %%% FIG 1 %%% \begin{figure} - \includegraphics[width=\linewidth]{rho} + \includegraphics[width=\linewidth]{fig2} \caption{ \titou{Ground-state one-electron density $\n{}{}(x)$ of 4-boxium (\ie, $N = 4$) for $L = \pi/32$ (left) and $L = 32\pi$ (right). In the weak correlation regime (small box length), the one-electron density is much more delocalized and uniform than in the strong correlation regime (large box length), where a Wigner crystal starts to appear. \cite{Rogers_2017,Rogers_2016}} @@ -996,7 +996,7 @@ equi-triensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$. %%% FIG 1 %%% \begin{figure*} - \includegraphics[width=\linewidth]{EvsW_n5} + \includegraphics[width=\linewidth]{fig3} \caption{ \label{fig:EvsW} Deviation from linearity of the weight-dependent KS-eLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost-interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right). @@ -1044,7 +1044,7 @@ Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:WHF}]. %%% FIG 2 %%% \begin{figure*} - \includegraphics[width=\linewidth]{EIvsW_n5} + \includegraphics[width=\linewidth]{fig4} \caption{ \label{fig:EIvsW} KS-eLDA individual energies, $\E{eLDA}{(0)}$ (black), $\E{eLDA}{(1)}$ (red), and $\E{eLDA}{(2)}$ (blue), as functions of the weights $\ew{1}$ (solid) and $\ew{2}$ (dashed) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).} @@ -1058,14 +1058,18 @@ energies, which is in agreement with the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the individual energies do not vary in the same way depending on the state considered and the value of the weights. -We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of -the ground and first excited-state increase with respect to the +\titou{On one hand,} we see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of +the ground and \titou{second} excited-state increase with respect to the first-excited-state weight $\ew{1}$, thus showing that, in this case, we ``deteriorate'' these states by optimizing the orbitals for the -ensemble, rather than for each state separately. The reverse actually occurs for the ground state in the triensemble -as $\ew{2}$ increases. The variations in the ensemble -weights are essentially linear or quadratic. +ensemble, rather than for each state separately. +\titou{The singly excited state is, on the other hand, stabilize in the biensemble, which is reasonable as the weight associated with this state increases. +For the triensemble, as $\ew{2}$ increases, the energy of the ground state increases, while the energy of the first excited state remains stable with a slight increase at large $L$. +The second excited state is obviously stabilized by the increase of its weight in the ensemble. +These are all very sensible observations.} + +The variations in the ensemble weights are essentially linear or quadratic. \manurev{This can be rationalized as follows. As readily seen from Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:ind_HF-like_ener}, the individual HF-like energies do not depend explicitly on the weights, which means @@ -1081,23 +1085,25 @@ weights $\bw$ [see Eqs.~\eqref{eq:ens1RDM}, \eqref{eq:ens_dens_from_ens_1RDM}, and \eqref{eq:decomp_ens_correner_per_part}], the latter contributions will contain both linear and quadratic terms in $\bw$, as evidenced by Eq.~\eqref{eq:Taylor_exp_DDisc_term} [see the second term on the right-hand -side].} In the biensemble, the weight dependence of the first -excitation energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble +side].} +!!! In the biensemble, the weight dependence of the first +excitation energy is \titou{increased?} as the correlation increases, \titou{while the weight dependence of the second excitation energy is reduced}. +On the other hand, switching from a bi- to a triensemble systematically enhances the weight dependence, due to the lowering of the -ground-state energy, as $\ew{2}$ increases. -The reverse is observed for the second excited state. -\manurev{Finally, we notice that the crossover point of the +ground-state energy, as $\ew{2}$ increases. +The reverse is observed for the second excited state. !!! \titou{THIS PARAGRAPH MIGHT NEED MODIFICATIONS.} +\trashPFL{Finally, we notice that the crossover point of the first-excited-state energies based on bi- and triensemble calculations, respectively, disappears in the strong correlation regime [see the right panel of Fig.~\ref{fig:EIvsW}], thus illustrating the importance of (individual and ensemble) densities, in addition to the weights, in the evaluation of individual energies within -an ensemble. +an ensemble. } %%% FIG 3 %%% \begin{figure} - \includegraphics[width=\linewidth]{EvsL_5} + \includegraphics[width=\linewidth]{fig5} \caption{ \label{fig:EvsL} Excitation energies (multiplied by $L^2$) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) of 5-boxium for various methods and box lengths $L$. @@ -1132,7 +1138,7 @@ This conclusion is verified for smaller and larger numbers of electrons %%% FIG 4 %%% \begin{figure*} - \includegraphics[width=\linewidth]{EvsN} + \includegraphics[width=\linewidth]{fig6} \caption{ \label{fig:EvsN} Error with respect to FCI in single and double excitation energies for $\nEl$-boxium for various methods and electron numbers $\nEl$ at $L=\pi/8$ (left), $L=\pi$ (center), and $L=8\pi$ (right). @@ -1170,7 +1176,7 @@ electrons. %%% FIG 5 %%% \begin{figure*} - \includegraphics[width=\linewidth]{EvsL_DD} + \includegraphics[width=\linewidth]{fig7} \caption{ \label{fig:EvsL_DD} Error with respect to FCI (in \%) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) as a function of the box length $L$ for 3-boxium (left), 5-boxium (center), and 7-boxium (right) at the KS-eLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$. @@ -1227,7 +1233,7 @@ shown). %%% FIG 6 %%% \begin{figure} - \includegraphics[width=\linewidth]{EvsN_DD} + \includegraphics[width=\linewidth]{fig8} \caption{ \label{fig:EvsN_DD} Error with respect to FCI in single and double excitation energies for $\nEl$-boxium (with a box length of $L=8\pi$) as a function of the number of electrons $\nEl$ at the KS-eLDA level with and without the contribution of the ensemble correlation derivative $\DD{c}{(I)}$. diff --git a/Revised_Manuscript/embedding.pdf b/Revised_Manuscript/embedding.pdf deleted file mode 100644 index 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