From 3729f786c45726eceec90a06046499fc67fa4154 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Wed, 11 Mar 2020 12:55:50 +0100 Subject: [PATCH] Manu: saving work in the discussion --- Manuscript/eDFT.tex | 94 +++++++++++++++++++++++++++++++-------------- 1 file changed, 65 insertions(+), 29 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 4c29881..5dd0d84 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -1295,18 +1295,21 @@ 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. -\titou{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 first-excited-state weight $\ew{1}$, thus showing that we -``deteriorate'' these states by optimizing the orbitals. The reverse actually occurs for the ground state in the triensemble -as $\ew{2}$ increases.} The variations in the ensemble +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 +first-excited-state weight $\ew{1}$, thus showing that, \manu{in this +case}, we +``deteriorate'' these states by optimizing the orbitals \manu{for the +ensemble, rather than for each state individually}. 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. They are induced by the eLDA functional, as readily seen from Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and \eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first -\titou{excited-state energy} is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble +excited-state energy is reduced as the correlation increases. 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 in this case, as $\ew{2}$ increases. -The reverse is observed for the second \titou{excited state}. +ground-state energy, as $\ew{2}$ increases. +The reverse is observed for the second excited state. %%% FIG 3 %%% \begin{figure} @@ -1325,7 +1328,7 @@ For small $L$, the single and double excitations can be labeled as ``pure'', as revealed by a thorough analysis of the FCI wavefunctions. In other words, each excitation is dominated by a sole, well-defined reference Slater determinant. However, when the box gets larger (\ie, as $L$ increases), there is a strong mixing between the different excitation degrees. -In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more \titou{disputable}. \cite{Loos_2019} +In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more disputable. \cite{Loos_2019} This can be clearly evidenced by the weights of the different configurations in the FCI wave function. % TITOU: shall we keep the paragraph below? %Therefore, it is paramount to construct a two-weight correlation functional @@ -1353,8 +1356,8 @@ When the box gets larger, they start to deviate. For the single excitation, TDLDA is extremely accurate up to $L = 2\pi$, but yields more significant errors at larger $L$ by underestimating the excitation energies. TDA-TDLDA slightly corrects this trend thanks to error compensation. Concerning the eLDA functional, our results clearly evidence that the equiweight [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$]. -This is especially true for the single excitation -which is significantly improved by using equal weights, \titou{especially in the strong correlation regime}. +This is especially true\manu{, in the strong correlation regime,} for the single excitation +which is significantly improved by using equal weights. The effect on the double excitation is less pronounced. Overall, one clearly sees that, with equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations. @@ -1424,7 +1427,7 @@ It is also interesting to investigate the influence of the correlation ensemble derivative contribution $\DD{c}{(I)}$ to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}]. In our case, both single ($I=1$) and double ($I=2$) excitations are considered. -To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, \titou{for $\nEl = 3$, $5$, and $7$}, the error percentage (with respect to FCI) as a function of the box length $L$ +To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, for $\nEl = 3$, $5$, and $7$, the error percentage (with respect to FCI) as a function of the box length $L$ on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}]. %\manu{Manu: there is something I do not understand. If you want to %evaluate the importance of the ensemble correlation derivatives you @@ -1436,21 +1439,37 @@ on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)} %\eeq %%rather than $E^{(I)}_{\rm HF}$ %} -\titou{We first stress that although for $\nEl=3$ both single and double excitation energies are +We first stress that although for $\nEl=3$ both single and double excitation energies are systematically improved, as the strength of electron correlation increases, when taking into account -the correlation ensemble derivative, this is not systematically the case for larger number of electrons.} -\trashPFL{This statement holds in both zero-weight and equal-weight limits.} +the correlation ensemble derivative, this is not +\trashEF{systematically} \manu{always} the case for larger numbers of electrons. The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime. -\titou{For 3-boxium, in the zero-weight limit, its contribution is also significantly larger in the case of the single -excitation; the reverse is observed in the equal-weight triensemble +For 3-boxium, in the zero-weight limit, its contribution is +\trashEF{also} significantly larger \manu{for the single +excitation as compared to the double excitation}; the reverse is observed in the equal-weight triensemble case. However, for 5- and 7-boxium, the correlation ensemble derivative hardly influences the double excitation (except when the correlation is strong), and slightly deteriorates the single excitation in the intermediate and strong correlation regimes. -This non-systematic behavior in terms of the number of electrons might be a consequence of how we constructed the weight-dependent functional. -Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of the eLDA functional is based on a two-electron finite uniform electron gas. -Therefore, it might be more appropriate to model the derivative discontinuity in few-electron systems.} +This non-systematic behavior in terms of the number of electrons might +be a consequence of how we constructed \trashEF{the weight-dependent +functional} \manu{eLDA}. +Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of +the eLDA functional is based on a \manu{\it two-electron} finite uniform electron gas. +\manu{Incorporating an $N$-dependence in the functional through the +curvature of the Fermi hole, in the spirit of Ref. \cite{Loos_2017a}, would be +valuable in this respect. This is left for future work.} +\trashEF{Therefore, it might be more appropriate to model the derivative +discontinuity in few-electron systems.}\\ +\\ +\manu{Manu: I am sorry to insist but I have a real problem with what follows. If +we look at the N=3 results, one has the impression that, indeed, for the +single excitation, a zero-weight calculation with the ensemble derivative +is almost equivalent to an equal-weight calculation without the +derivative. This is not the case for $N=5$ or 7, maybe because our +derivative is based on two electrons. }\\ +{\it Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as compared to the zero-weight calculations). @@ -1461,16 +1480,33 @@ This could explain why equiensemble calculations are clearly more accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative: for a given method, equiensemble orbitals partially remove the burden of modelling properly the ensemble correlation derivative. -%\manu{Manu: I -%do not like this statement. As I wrote above, the ensemble derivative is -%still substantial in the strongly correlated limit of the equi -%triensemble for the double -%excitation. -%} -Note also that, in our case, the second term in +}\\ +\manu{Manu: I propose to rephrase this part as follows:}\\ +\\ +\manu{ +Interestingly, for the single excitation in the 3-boxium, the magnitude of the correlation ensemble +derivative is substantially reduced when switching from a zero-weight to +an equal-weight calculation, while giving similar excitation energies, +even in the strongly correlated regime. A possible interpretation is +that, at least for the single excitation, equiensemble orbitals partially remove the burden +of modelling properly the correlation ensemble derivative. +This conclusion does not hold for larger +$N=5$ or $N=7$ numbers of +electrons, possibly because eLDA extracts density-functional correlation ensemble +derivatives from a two-electron gas, as mentioned previously. +For the +double excitation, the ensemble derivative remains important, even in +the equiensemble case. +To summarize, in all cases, the equiensemble calculation +is always more accurate than a zero-weight +(\ie, a conventional ground-state DFT) one, with or without including the ensemble +derivative correction. +} +\\ +Note that the second term in Eq.~\eqref{eq:Om-eLDA}, which involves the weight-dependent correlation potential and the density difference between ground and excited states, -has a negligible effect on the excitation energies \titou{(results not shown)}. +has a negligible effect on the excitation energies (results not shown). %\manu{Manu: Is this %something that you checked but did not show? It feels like we can see %this in the Figure but we cannot, right?} @@ -1494,8 +1530,8 @@ has a negligible effect on the excitation energies \titou{(results not shown)}. Finally, in Fig.~\ref{fig:EvsN_DD}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime). The difference between the solid and dashed curves undoubtedly show that, even in the strong correlation regime, the -ensemble correlation derivative has \titou{a rather significant impact} on the double -excitations \titou{(around $10\%$)} with a slight tendency of worsening the excitation energies +ensemble correlation derivative has a rather significant impact on the double +excitations (around $10\%$) with a slight tendency of worsening the excitation energies in the case of equal weights, as the number of electrons increases. It has a rather large influence on the single excitation energies obtained in the zero-weight limit, showing once