From 1a0fec1269a6e5941ef162acdd76cec15bfc83bc Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Wed, 11 Mar 2020 22:32:58 +0100 Subject: [PATCH] Manu: done with the discussion --- Manuscript/eDFT.tex | 35 +++++++++++++++++++---------------- 1 file changed, 19 insertions(+), 16 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 015e05b..f06e073 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -1349,7 +1349,8 @@ For small $L$, the single and double excitations can be labeled as 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 disputable. \cite{Loos_2019} -This can be clearly evidenced by the weights of the different configurations in the FCI wave function. +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 %(\ie, a triensemble functional, as we have done here) which @@ -1382,7 +1383,7 @@ 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. This conclusion is verified for smaller and larger numbers of electrons -(see {\SI}). +(see {\SI}).\\ %\\ %\manu{Manu: now comes the question that is, I believe, central in this %work. How important are the @@ -1465,11 +1466,11 @@ increases) when taking into account the correlation ensemble derivative, this is not always the case for larger numbers of electrons. -For 3-boxium, in the zero-weight limit, the ensemble derivative is +For 3-boxium, in the zero-weight limit, the correlation ensemble derivative is significantly larger 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 +However, for 5- and 7-boxium, it 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 eLDA. @@ -1499,7 +1500,6 @@ valuable in this respect. This is left for future work. %}\\ %\manu{Manu: I propose to rephrase this part as follows:}\\ %\\ -\titou{ Interestingly, for the single excitation in 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, @@ -1511,16 +1511,15 @@ numbers of electrons ($N=5$ or $7$), possibly because eLDA extracts density-func derivatives from a two-electron uniform 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 +To summarize, 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 +derivative correction. Note that the second term on the right-hand side +of 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 (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?} @@ -1543,13 +1542,17 @@ has a negligible effect on the excitation energies (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 a rather significant impact on the double -excitations (around $10\%$) with a slight tendency of worsening the excitation energies +undoubtedly show that \trashEF{, even in the strong correlation regime,} +\manu{Manu: in the light of what we already discussed, we expect the +derivative to be important in the strongly correlated regime so the +sentence "even in ..." is useless (that's why I would remove it)} the +correlation ensemble derivative has a rather significant impact on the double +excitation (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 +increases. It has a rather large influence \manu{(which decreases with the +number of electrons)} on the single excitation energies obtained in the zero-weight limit, showing once -again that the usage of equal weights has the benefit of significantly reducing the magnitude of the ensemble correlation derivative. +again that the usage of equal weights has the benefit of significantly reducing the magnitude of the correlation ensemble derivative. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Concluding remarks}