diff --git a/Manuscript/ufGW.tex b/Manuscript/ufGW.tex index 25997d4..268e5e3 100644 --- a/Manuscript/ufGW.tex +++ b/Manuscript/ufGW.tex @@ -353,7 +353,7 @@ Inspection of their corresponding eigenvectors reveals that the $(\Ne+1)$-electr By construction, the quasiparticle solution diabatically follows the reference determinant $\ket*{1\Bar{1}3}$ through the avoided crossing (thick lines in Fig.~\ref{fig:H2_zoom}) which is precisely the origin of the energetic discontinuity. A similar scenario is at play in the region around $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ (right panel of Fig.~\ref{fig:H2_zoom}) but it now involves three solutions ($s = 5$, $s = 6$, and $s = 7$). -The electronic configurations of the Slater determinant involved are the $\ket*{1\Bar{1}4}$ reference determinant as well as two external determinants of configuration \titou{$\ket*{1\Bar{?}?}$} and \titou{$\ket*{1\Bar{?}?}$}. +The electronic configurations of the Slater determinant involved are the $\ket*{1\Bar{1}4}$ reference determinant as well as two external determinants of configuration $\ket*{1\Bar{2}3}$ and $\ket*{12\Bar{3}}$. These forms two avoided crossings in rapid successions, which create two discontinuities in the energy surface (see Fig.~\ref{fig:H2}). In this region, although the ground-state wave function is well described by the $\Ne$-electron HF determinant, a situation that can be safely labeled as single-reference, one can see that the $(\Ne+1)$-electron state involves three Slater determinants and can then be labeled as a multi-reference (or strongly-correlated) situation with near-degenerate electronic configurations. Therefore, one can conclude that this downfall of $GW$ is a key signature of strong correlation in the $(\Ne\pm1)$-electron states that yields a significant redistribution of weights amongst electronic configurations. @@ -380,7 +380,6 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str % FIGURE 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure} -% \includegraphics[width=\linewidth]{fig4a} \includegraphics[width=\linewidth]{fig4} \caption{ \label{fig:H2reg} @@ -438,16 +437,17 @@ Figure \ref{fig:H2reg_zoom} compares the non-regularized and regularized quasipa It clearly shows how the regularization of the $GW$ self-energy diabatically linked the two solutions to get rid of the discontinuities. However, this diabatization is more or less accurate depending on the value of $\eta$. For $\eta = 10$, the value is clearly too large inducing a large difference between the two sets of quasiparticle energies (purple curves). -For $\eta = 0.1$, we have the opposite scenario where the value is too small and some irregularities remain (green curves). +For $\eta = 0.1$, we have the opposite scenario where $\eta$ is too small and some irregularities remain (green curves). We have found that $\eta = 1$ is a good compromise that does not alter significantly the quasiparticle energies while providing a smooth transition between the two solutions. -\titou{This values could be further refined for specfici applications.} +This value can be certainly refined for specific applications. -To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized and non-regularized quasiparticle energies as functions of $\RHH$. +To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized and non-regularized quasiparticle energies as functions of $\RHH$ for each orbital. +The principal observation is that, in the absence of intruder states, the regularization induces an error below \SI{10}{\milli\eV} ($p = 1$ and $p = 2$), which is practically variable . +Of course, in the troublesome regions ($p = 3$ and $p = 4$), the correction brought by the regularization procedure is larger but it has the undeniable advantage to provide smooth curves. As a final example, we report in Fig.~\ref{fig:F2} the ground-state potential energy surface of the \ce{F2} molecule obtained at various levels of theory with the cc-pVDZ basis. In particular, we compute, with and without regularization, the total energy at the Bethe-Salpeter equation (BSE) level \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} within the adiabatic connection fluctuation dissipation formalism \cite{Maggio_2016,Holzer_2018b,Loos_2020e} following the same protocol detailed in Ref.~\onlinecite{Loos_2020e}. These results are compared to high-level coupled-cluster calculations \cite{Purvis_1982,Christiansen_1995b} extracted from the same work. - As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' and it is clear that the regularization scheme (black curve) allows to smooth it out without significantly altering the overall accuracy. Morevoer, while it is extremely challenging to perform self-consistent $GW$ calculations without regularization, it is now straightforward to compute the BSE@ev$GW$@HF potential energy surface. diff --git a/Notebooks/GW_conundrum.nb b/Notebooks/GW_conundrum.nb index 12b68cf..f361678 100644 --- a/Notebooks/GW_conundrum.nb +++ b/Notebooks/GW_conundrum.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 205048, 4688] -NotebookOptionsPosition[ 197356, 4566] -NotebookOutlinePosition[ 197837, 4585] -CellTagsIndexPosition[ 197794, 4582] +NotebookDataLength[ 205237, 4691] +NotebookOptionsPosition[ 197451, 4568] +NotebookOutlinePosition[ 197932, 4587] +CellTagsIndexPosition[ 197889, 4584] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -2585,6 +2585,8 @@ Cell[CellGroupData[{ Cell["PES", "Section",ExpressionUUID->"8a3d16ca-0779-480a-b23d-aa8d77e59793"], +Cell[BoxData["AspectRatio"], "Input",ExpressionUUID->"bcdffa1d-f22d-40ee-a1b9-2dc03ba0be9c"], + Cell[BoxData[{ RowBox[{ RowBox[{"SizeTitle", "=", "20"}], ";"}], "\[IndentingNewLine]", @@ -4673,19 +4675,20 @@ Cell[88169, 2530, 1629, 50, 64, "Input",ExpressionUUID->"920b9a14-7358-437b-8b2d }, Closed]], Cell[CellGroupData[{ Cell[89835, 2585, 77, 0, 65, "Section",ExpressionUUID->"8a3d16ca-0779-480a-b23d-aa8d77e59793"], -Cell[89915, 2587, 1894, 46, 448, "Input",ExpressionUUID->"d754cecd-0dc1-44bc-aadc-26d9c91dd570", +Cell[89915, 2587, 92, 0, 37, "Input",ExpressionUUID->"bcdffa1d-f22d-40ee-a1b9-2dc03ba0be9c"], +Cell[90010, 2589, 1894, 46, 448, "Input",ExpressionUUID->"d754cecd-0dc1-44bc-aadc-26d9c91dd570", InitializationCell->True], Cell[CellGroupData[{ -Cell[91834, 2637, 84, 0, 67, "Subsection",ExpressionUUID->"14db7a36-714c-4139-a9a1-b9adf8150988"], +Cell[91929, 2639, 84, 0, 67, "Subsection",ExpressionUUID->"14db7a36-714c-4139-a9a1-b9adf8150988"], Cell[CellGroupData[{ -Cell[91943, 2641, 165, 3, 56, "Subsubsection",ExpressionUUID->"6995328e-6a17-410f-8738-8a1d6809e67a"], -Cell[92111, 2646, 1148, 32, 168, "Input",ExpressionUUID->"61300761-bb45-4b64-b105-cf2d5871979f"] +Cell[92038, 2643, 165, 3, 56, "Subsubsection",ExpressionUUID->"6995328e-6a17-410f-8738-8a1d6809e67a"], +Cell[92206, 2648, 1148, 32, 168, "Input",ExpressionUUID->"61300761-bb45-4b64-b105-cf2d5871979f"] }, Open ]], Cell[CellGroupData[{ -Cell[93296, 2683, 314, 5, 56, "Subsubsection",ExpressionUUID->"dcd678bb-6dc4-4a3c-9206-61c1204c9bdb"], +Cell[93391, 2685, 314, 5, 56, "Subsubsection",ExpressionUUID->"dcd678bb-6dc4-4a3c-9206-61c1204c9bdb"], Cell[CellGroupData[{ -Cell[93635, 2692, 2548, 64, 428, "Input",ExpressionUUID->"cd7ca353-7a15-4a03-a45c-503c6b68d417"], -Cell[96186, 2758, 101118, 1802, 617, "Output",ExpressionUUID->"ede19c11-5d07-4864-bb51-b02c94eff86a"] +Cell[93730, 2694, 2548, 64, 428, "Input",ExpressionUUID->"cd7ca353-7a15-4a03-a45c-503c6b68d417"], +Cell[96281, 2760, 101118, 1802, 603, "Output",ExpressionUUID->"ede19c11-5d07-4864-bb51-b02c94eff86a"] }, Open ]] }, Open ]] }, Open ]]