diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 7213200..faae180 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -724,7 +724,7 @@ However, an obvious drawback of using FUEGs is that the resulting eDFA will inex Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems by combining these FUEGs with the usual IUEG to construct a weigh-dependent LDA functional for ensembles (eLDA). As a FUEG, we consider the ringium model in which electrons move on a perfect ring (\ie, a circle) but interact \textit{through} the ring. \cite{Loos_2012, Loos_2013a, Loos_2014b} -The most appealing feature of ringium (regarding the development of functionals in the context of eDFT) is the fact that both ground- and excited-state densities are uniform. +The most appealing feature of ringium regarding the development of functionals in the context of eDFT is the fact that both ground- and excited-state densities are uniform. As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary. This is a necessary condition for being able to model derivative discontinuities. Moreover, it has been shown that, in the thermodynamic limit, the ringium model is equivalent to the ubiquitous IUEG paradigm. \cite{Loos_2013,Loos_2013a} @@ -824,21 +824,19 @@ In particular, $\be{c}{(0,0)}(\n{}{}) = \e{c}{\text{LDA}}(\n{}{})$. Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles. Finally, we note that, by construction, \begin{equation} - \left. \pdv{\be{c}{\bw}(\n{}{})}{\ew{J}}\right|_{\n{}{} = \n{\bGam{\bw}}{}(\br{})} = \be{c}{(J)}(\n{\bGam{\bw}}{}(\br{})) - \be{c}{(0)}(\n{\bGam{\bw}}{}(\br{})). + \pdv{\be{c}{\bw}(\n{}{})}{\ew{J}} = \be{c}{(J)}(\n{}{}(\br{})) - \be{c}{(0)}(\n{}{}(\br{})). \end{equation} -%\titou{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details} \label{sec:comp_details} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation. -Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\nEl$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\nEl$-boxium. +Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\nEl$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\nEl$-boxium in the following. In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \nEl \le 7$. -%\titou{Comment on the quality of these density: density- and functional-driven errors?} - These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length. For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime. \cite{Rogers_2017,Rogers_2016} + We use as basis functions the (orthonormal) orbitals of the one-electron system, \ie, \begin{equation} \AO{\mu}(x) = @@ -849,13 +847,13 @@ We use as basis functions the (orthonormal) orbitals of the one-electron system, \end{cases} \end{equation} with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations. -For the self-consistent calculations (such as HF, KS-DFT or KS-eDFT), the convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw} \bS - \bS \bGam{\bw} \bF{\bw}}}$ been set to $10^{-5}$. -For KS-DFT, KS-eDFT and TDDFT calculations, a Gauss-Legendre quadrature is employed to compute the various integrals that cannot be performed in closed form. +For the self-consistent calculations (such as HF, KS-DFT or KS-eDFT), the convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw} \bS - \bS \bGam{\bw} \bF{\bw}}}$ is set to $10^{-5}$. +For KS-DFT, KS-eDFT and TDDFT calculations, a 51-point Gauss-Legendre quadrature is employed to compute the various integrals that cannot be performed in closed form. -In order to test the present eLDA functional we have performed various sets of calculations. -To get reference excitation energies for both the single and double excitations, we have performed full configuration interaction (FCI) calculations with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}. -For the single excitations, we have also performed time-dependent LDA (TDLDA) calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}], and the effect of the Tamm-Dancoff approximation (TDA) has been also investigated. \cite{Dreuw_2005} -Concerning the KS-eDFT and eHF calculations, two sets of weight have been tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$. +In order to test the present eLDA functional we perform various sets of calculations. +To get reference excitation energies for both the single and double excitations, we compute full configuration interaction (FCI) energies with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}. +For the single excitations, we also perform time-dependent LDA (TDLDA) calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}], and the effect of the Tamm-Dancoff approximation (TDA) has been also investigated. \cite{Dreuw_2005} +Concerning the KS-eDFT and eHF calculations, two sets of weight are tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results and discussion} @@ -930,7 +928,7 @@ As a rule of thumb, in the weak and intermediate correlation regimes, we see tha Moreover, we note that, in the strong correlation regime (left graph of Fig.~\ref{fig:EvsN}), the single excitation energies obtained at the state-averaged KS-eLDA level remain in good agreement with FCI and are much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$. This also applies to double excitations, the discrepancy between FCI and KS-eLDA remaining of the order of a few percents in the strong correlation regime. These observations nicely illustrate the robustness of the present state-averaged GOK-DFT scheme in any correlation regime for both single and double excitations. -This is definitely a very pleasing outcome. +This is definitely a very pleasing outcome, which additionally shows that, even though we have designed the eLDA functional based on a two-electron model system, the present methodology is applicable to any 1D electronic system. %%% FIG 4 %%% \begin{figure} diff --git a/Notebooks/eDFT_FUEG.nb b/Notebooks/eDFT_FUEG.nb index 1c53692..193b933 100644 --- a/Notebooks/eDFT_FUEG.nb +++ b/Notebooks/eDFT_FUEG.nb @@ -42917,7 +42917,7 @@ Cell[BoxData[ 9fca6882a7bd"] }, Open ]] }, Closed]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ @@ -199697,7 +199697,7 @@ b345a0305ece"] }, Open ]] }, Closed]] }, Closed]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ @@ -216714,7 +216714,7 @@ Cell[BoxData[ }, Closed]] }, Closed]] }, Closed]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ @@ -221270,7 +221270,7 @@ Cell[1688128, 39639, 10865, 191, 165, "Input",ExpressionUUID->"86327531-5150-4c8 }, Closed]], Cell[CellGroupData[{ Cell[1699030, 39835, 253, 4, 38, "Subsection",ExpressionUUID->"f36904a3-cd7b-4efb-8113-4cdee9045345"], -Cell[1699286, 39841, 63415, 1300, 2853, "Input",ExpressionUUID->"61323d25-e201-440f-8dae-ca2381d8a855", +Cell[1699286, 39841, 63415, 1300, 2874, "Input",ExpressionUUID->"61323d25-e201-440f-8dae-ca2381d8a855", InitializationCell->True], Cell[1762704, 41143, 12032, 211, 137, "Input",ExpressionUUID->"5b2641d7-6ead-4ede-9d94-99584c7a1d04", InitializationCell->True] @@ -221289,9 +221289,9 @@ Cell[1834618, 42669, 545, 12, 30, "Input",ExpressionUUID->"3612c783-cf2d-4c9e-a9 Cell[1835166, 42683, 7972, 233, 100, "Output",ExpressionUUID->"7273514a-05e5-4bd6-b38e-9fca6882a7bd"] }, Open ]] }, Closed]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ -Cell[1843199, 42923, 152, 3, 98, "Title",ExpressionUUID->"aefd3997-e92c-4a00-bbab-c1e0d31c8049"], +Cell[1843199, 42923, 152, 3, 72, "Title",ExpressionUUID->"aefd3997-e92c-4a00-bbab-c1e0d31c8049"], Cell[CellGroupData[{ Cell[1843376, 42930, 160, 3, 67, "Section",ExpressionUUID->"aa75faa4-f188-4f1f-b111-6efc6d390d21"], Cell[CellGroupData[{ @@ -222069,9 +222069,9 @@ Cell[9340288, 199599, 4054, 96, 128, "Output",ExpressionUUID->"70ff7d28-c092-492 }, Open ]] }, Closed]] }, Closed]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ -Cell[9344415, 199703, 160, 3, 67, "Section",ExpressionUUID->"0cf5029a-c295-4f81-81d9-c59ab38a6cd6"], +Cell[9344415, 199703, 160, 3, 53, "Section",ExpressionUUID->"0cf5029a-c295-4f81-81d9-c59ab38a6cd6"], Cell[CellGroupData[{ Cell[9344600, 199710, 155, 3, 54, "Subsection",ExpressionUUID->"399c4cbb-a491-485e-a0af-c29d98a817c1"], Cell[CellGroupData[{ @@ -222283,9 +222283,9 @@ Cell[9977334, 216619, 4219, 92, 170, "Output",ExpressionUUID->"2c367f82-1c84-41b }, Closed]] }, Closed]] }, Closed]] -}, Open ]], +}, Closed]], Cell[CellGroupData[{ -Cell[9981638, 216720, 159, 3, 98, "Title",ExpressionUUID->"6491b1b0-852f-4bdd-8e0c-baaf27bf3326"], +Cell[9981638, 216720, 159, 3, 72, "Title",ExpressionUUID->"6491b1b0-852f-4bdd-8e0c-baaf27bf3326"], Cell[CellGroupData[{ Cell[9981822, 216727, 153, 3, 67, "Section",ExpressionUUID->"5127f77e-fe2f-4dda-bd7d-3f930b336d92"], Cell[CellGroupData[{