diff --git a/FarDFT.nb b/FarDFT.nb index c203951..9de6a6c 100644 --- a/FarDFT.nb +++ b/FarDFT.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 16909420, 393790] -NotebookOptionsPosition[ 16852687, 392938] -NotebookOutlinePosition[ 16853080, 392954] -CellTagsIndexPosition[ 16853037, 392951] +NotebookDataLength[ 16925857, 394270] +NotebookOptionsPosition[ 16866329, 393390] +NotebookOutlinePosition[ 16866722, 393406] +CellTagsIndexPosition[ 16866679, 393403] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -363765,6 +363765,568 @@ Cell[BoxData["$Aborted"], "Output", Cell[CellGroupData[{ +Cell["GOK-DFT made simple\t", "Title", + CellChangeTimes->{{3.795577514535224*^9, 3.795577518626693*^9}, + 3.7955779011312513`*^9},ExpressionUUID->"241eb9ca-79cc-4b1d-9088-\ +fb01f2c65187"], + +Cell["\<\ +With an exchange-only functional (i.e., without correlation), the ensemble \ +energy is\ +\>", "Text", + CellChangeTimes->{{3.7955781824525433`*^9, 3.7955782235749826`*^9}, { + 3.795578809335644*^9, + 3.795578819926897*^9}},ExpressionUUID->"a34330ac-5af3-4a38-a37c-\ +0498abda882c"], + +Cell[BoxData[ + RowBox[{ + SuperscriptBox["E", "w"], "=", + RowBox[{ + SubsuperscriptBox["T", "s", "w"], "+", + SubsuperscriptBox["E", "H", "w"], "+", + SubsuperscriptBox["E", "x", "w"]}]}]], "Input", + CellChangeTimes->{{3.795577539185522*^9, 3.7955775818996143`*^9}, { + 3.7955779124805813`*^9, 3.795577934329053*^9}, {3.7955779737292747`*^9, + 3.795578036672875*^9}, {3.7955781030709267`*^9, + 3.7955781057653027`*^9}},ExpressionUUID->"8ad34083-207a-475d-9246-\ +e578c2e73d39"], + +Cell["\<\ +where, in particular, the Hartree and LDA exchange energy reads\ +\>", "Text", + CellChangeTimes->{{3.795578192946072*^9, 3.795578197347353*^9}, { + 3.7955782280169*^9, + 3.7955782401288977`*^9}},ExpressionUUID->"e4dc23bf-395f-4a29-b9de-\ +6640b6baca1c"], + +Cell[BoxData[ + RowBox[{ + SubsuperscriptBox["E", "H", "w"], "=", + RowBox[{ + RowBox[{ + FractionBox["1", "2"], + RowBox[{"\[Integral]", + RowBox[{"\[Integral]", + RowBox[{ + FractionBox[ + RowBox[{ + RowBox[{ + SuperscriptBox["n", "w"], "[", "r", "]"}], + RowBox[{ + SuperscriptBox["n", "w"], "[", + RowBox[{"r", "'"}], "]"}]}], + RowBox[{"\[LeftBracketingBar]", + RowBox[{"r", "-", + RowBox[{"r", "'"}]}], "\[RightBracketingBar]"}]], + RowBox[{"\[DifferentialD]", "r"}], + RowBox[{"\[DifferentialD]", + RowBox[{"r", "'"}]}], "\t", + SubsuperscriptBox["E", "x", "w"]}]}]}]}], "=", + RowBox[{ + SubscriptBox["C", "x"], + RowBox[{"\[Integral]", + RowBox[{ + SuperscriptBox[ + RowBox[{"(", + RowBox[{ + SuperscriptBox["n", "w"], "[", "r", "]"}], ")"}], + RowBox[{"4", "/", "3"}]], + RowBox[{"\[DifferentialD]", "r"}]}]}]}]}]}]], "Input", + CellChangeTimes->{{3.795577822370946*^9, 3.795577850755807*^9}, { + 3.7955778877909307`*^9, 3.7955779326016073`*^9}, {3.79557810944242*^9, + 3.7955781454801903`*^9}},ExpressionUUID->"7d144384-4fef-4ccc-87c0-\ +df2676f9f904"], + +Cell[TextData[{ + "and ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SuperscriptBox["n", "w"], "[", "r", "]"}], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "12b0869f-64f2-4549-b99f-06ea125bd8cd"], + " is the ensemble density. 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Evaluating the latest expression at ", + Cell[BoxData[ + FormBox[ + RowBox[{"w", "=", "\[Epsilon]"}], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "b6a3c5d7-f350-4a35-992f-303ee61e416e"], + ", we get" +}], "Text", + CellChangeTimes->{{3.795578345755261*^9, 3.795578373504217*^9}, { + 3.79558342932677*^9, 3.795583438677984*^9}, 3.795583796485182*^9, { + 3.795584068562727*^9, + 3.7955840745069227`*^9}},ExpressionUUID->"1e019b3f-0ec6-4996-b433-\ +436bfa68249a"], + +Cell[BoxData[ + FrameBox[ + RowBox[{"\[Alpha]", "=", + RowBox[{ + FractionBox[ + RowBox[{ + SuperscriptBox[ + OverscriptBox["E", "~"], + RowBox[{"w", "=", "\[Epsilon]"}]], "-", + SuperscriptBox["E", + RowBox[{"w", "=", "\[Epsilon]"}]]}], + RowBox[{" ", + SubsuperscriptBox["E", "x", + RowBox[{"w", "=", "\[Epsilon]"}]]}]], "=", + FractionBox[ + RowBox[{" ", + RowBox[{ + SuperscriptBox["E", + RowBox[{"w", "=", "1"}]], "-", + RowBox[{ + RowBox[{"(", + RowBox[{"1", "-", + FractionBox["1", "\[Epsilon]"]}], ")"}], + SuperscriptBox["E", + RowBox[{"w", "=", "0"}]]}], "-", + RowBox[{ + FractionBox["1", "\[Epsilon]"], + SuperscriptBox["E", + RowBox[{"w", "=", "\[Epsilon]"}]]}]}]}], + RowBox[{" ", + SubsuperscriptBox["E", "x", + RowBox[{"w", "=", "\[Epsilon]"}]]}]]}]}]]], "Input", + CellChangeTimes->{{3.795581646812381*^9, 3.7955817523669024`*^9}, { + 3.795582574191654*^9, 3.795582628310651*^9}, {3.795582844800824*^9, + 3.795582923398822*^9}, + 3.795583251664819*^9},ExpressionUUID->"2e469831-9930-4ad2-83c0-\ +4e30beb7bfe5"], + +Cell["This is our final expression.", "Text", + CellChangeTimes->{{3.7955841073088713`*^9, + 3.795584112964857*^9}},ExpressionUUID->"bb9376c5-3710-4f38-a0cd-\ +cd2c215d3da3"], + +Cell["Because", "Text", + CellChangeTimes->{{3.7955834753531322`*^9, + 3.795583476193413*^9}},ExpressionUUID->"c64cae63-68fa-4104-844c-\ +5db4044e531d"], + +Cell[BoxData[ + RowBox[{ + FractionBox[ + RowBox[{"\[PartialD]", + SubsuperscriptBox["E", "x", "w"]}], + RowBox[{"\[PartialD]", "w"}]], "=", + RowBox[{"\[Alpha]", "=", + FractionBox[ + RowBox[{" ", + RowBox[{ + SuperscriptBox["E", + RowBox[{"w", "=", "1"}]], "-", + RowBox[{ + RowBox[{"(", + RowBox[{"1", "-", + FractionBox["1", "\[Epsilon]"]}], ")"}], + SuperscriptBox["E", + RowBox[{"w", "=", "0"}]]}], "-", + RowBox[{ + FractionBox["1", "\[Epsilon]"], + SuperscriptBox["E", + RowBox[{"w", "=", "\[Epsilon]"}]]}]}]}], + RowBox[{" ", + SubsuperscriptBox["E", "x", + RowBox[{"w", "=", "\[Epsilon]"}]]}]]}]}]], "Input", + CellChangeTimes->{{3.795583465613015*^9, + 3.795583505865527*^9}},ExpressionUUID->"efa8df3f-4ea7-4372-9650-\ +2183b89661e5"], + +Cell[TextData[{ + "the previous expression is an estimate of the ensemble derivative at the \ +vicinity of ", + Cell[BoxData[ + FormBox[ + RowBox[{"w", "=", "0."}], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "241fb185-40ea-4bdb-8b9e-65ab5cd1683d"] +}], "Text", + CellChangeTimes->{{3.79558349098402*^9, 3.795583528656736*^9}, { + 3.795584122322145*^9, + 3.795584123312717*^9}},ExpressionUUID->"472707cc-c67f-4150-bffb-\ +346deebfe911"], + +Cell["Summarizing, in practice, one needs to perform:", "Text", + CellChangeTimes->{{3.7955835324931717`*^9, 3.795583554573839*^9}, { + 3.795584131821254*^9, + 3.795584135355208*^9}},ExpressionUUID->"33b16062-89ca-4816-bce3-\ +188c08586462"], + +Cell[CellGroupData[{ + +Cell[TextData[{ + "a zero-weight (i.e., ground state) calculation to get ", + Cell[BoxData[ + FormBox[ + SuperscriptBox["E", + RowBox[{"w", "=", "0"}]], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "fa19a9a4-3fb5-4bce-b930-c94f51a58314"], + ". Easy." +}], "Item", + CellChangeTimes->{{3.7955835646032867`*^9, 3.795583632216599*^9}, { + 3.795583676432597*^9, 3.795583681226728*^9}, {3.795583759742305*^9, + 3.795583761044952*^9}, {3.7955841440139723`*^9, + 3.795584145624457*^9}},ExpressionUUID->"af84314f-3568-4b2a-86ae-\ +79f56f67a074"], + +Cell[TextData[{ + "a MOM-style calculation at ", + Cell[BoxData[ + FormBox[ + RowBox[{"w", "=", "1"}], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "ee22702f-91b4-436d-91e6-74d7ec2b66b3"], + " to get ", + Cell[BoxData[ + FormBox[ + SuperscriptBox["E", + RowBox[{"w", "=", "1"}]], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "2fa89da7-986c-488f-9250-02cb37d26e5e"], + ". Less easy but doable with the new algorithms developed in Gill\ +\[CloseCurlyQuote]s group and Head-Gordon\[CloseCurlyQuote]s group (MOM, SGM, \ +etc)." +}], "Item", + CellChangeTimes->{{3.7955835646032867`*^9, 3.7955836633523397`*^9}, { + 3.795583762585924*^9, 3.7955837819945183`*^9}, {3.795584157782454*^9, + 3.795584176310916*^9}},ExpressionUUID->"8a822d39-2341-4596-ada7-\ +e75d9e6690e9"], + +Cell[TextData[{ + "a GOK-DFT calculation at ", + Cell[BoxData[ + FormBox[ + RowBox[{"w", "=", "\[Epsilon]"}], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "9999f8ce-914f-402d-9fd6-55f4678521c9"], + " to get ", + Cell[BoxData[ + FormBox[ + SuperscriptBox["E", + RowBox[{"w", "=", "\[Epsilon]"}]], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "f72c2cb9-9c03-48f4-85f7-b28b6d915ec5"], + " and ", + Cell[BoxData[ + FormBox[ + SubsuperscriptBox["E", "x", + RowBox[{"w", "=", "\[Epsilon]"}]], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "93920de6-9927-4bf9-a7af-530961d43aa6"], + ". Because ", + Cell[BoxData[ + FormBox["\[Epsilon]", TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "f0dad6c5-1ee2-4393-941b-dadaa8701da1"], + " is small, in practice, the calculation is going to converge easily, which \ +is a very important point, especially if one starts with the converged \ +ground-state density matrix." +}], "Item", + CellChangeTimes->{{3.7955835646032867`*^9, 3.795583705466729*^9}, { + 3.795583798079362*^9, + 3.7955838291067142`*^9}},ExpressionUUID->"6fce318d-b27e-4732-8652-\ +d9af60c9d126"] +}, Open ]], + +Cell[TextData[{ + "In such a way, one can approximate efficiently the ensemble derivative at \ +", + Cell[BoxData[ + FormBox[ + RowBox[{"w", "=", "0"}], TraditionalForm]], + FormatType->"TraditionalForm",ExpressionUUID-> + "39cdb037-a197-48f8-805e-9092aacf0854"], + ", and obtains excitation energies." +}], "Text", + CellChangeTimes->{{3.795583713553678*^9, + 3.795583746528973*^9}},ExpressionUUID->"2122a45a-faf8-43b9-ad55-\ +a691c6aeb453"], 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Boelelaan 1083, 1081 HV Amsterdam, The Netherlands} -\newcommand{\bruno}[1]{\textcolor{blue}{Bruno: #1}} + \begin{document} \title{Weight Dependence of Local Exchange-Correlation Functionals: Double Excitations in Two-Electron Systems} @@ -639,16 +640,16 @@ The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (gree One clearly sees that the correction brought by GIC-S is much more gentle than at $\RHH = 1.4$ bohr, which means that the ensemble energy obtained with the LDA exchange functional is much more linear at $\RHH = 3.7$ bohr. In other words, the ghost-interaction ``hole'' depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry. -Note that this linearity at $\RHH = 3.7$ bohr was -also observed using weight-independent xc-functionals in Ref.~\cite{Senjean_2015}. +Note that this linearity at $\RHH = 3.7$ bohr was also observed using weight-independent xc functionals in Ref.~\onlinecite{Senjean_2015}. Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers basis set converged results), the same set of calculations as in Table \ref{tab:BigTab_H2}. As a reference value, we computed a FCI/aug-cc-pV5Z excitation energy of $8.69$ eV, which compares well with previous studies. \cite{Senjean_2015} -For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the best match being reached with HF exchange \bruno{? I don't see it, for me HF is really bad here, especially due to its very hight dependence on the weight ! Maybe you are just referring to MOM ?}. +For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the closest match being reached with HF exchange and eVWN5 correlation at equi-ensemble. +%\bruno{? I don't see it, for me HF is really bad here, especially due to its very hight dependence on the weight ! Maybe you are just referring to MOM ?}. As expected from the linearity of the ensemble energy, the GIC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits. -Nonetheless, the excitation energy is still off by 3 eV. +Nonetheless, the excitation energy is still off by $3$ eV. The fundamental theoretical reason of such a poor agreement is not clear. The fact that HF exchange yields better excitation energies hints at the effect of self-interaction error. -\bruno{I'm a bit surprise that the ensemble correction to the correlation functional does not improve things at all... Is the derivative discontinuity, computed with this functional, almost 0 here ?} +%\bruno{I'm a bit surprise that the ensemble correction to the correlation functional does not improve things at all... Is the derivative discontinuity, computed with this functional, almost 0 here ?} %%% TABLE IV %%% \begin{table} @@ -700,11 +701,10 @@ The parameters of the GIC-S weight-dependent exchange functional (computed with In other words, the ghost-interaction hole is deeper. The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange. -The GIC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the GIC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree (or $0.22$ eV) off the reference value.\bruno{But also with GIC-SVWN5, as in the rest of this article, so one could wonder about the usefulness of the eVWN5 functional...} -As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were -made in Ref.~\cite{Loos_2020} thus strengthening the importance of -developing weight-dependent functionals that gives linear ensemble -energies, \ie, to get rid of the weight-dependency of the excitation energy. +The GIC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the GIC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree (or $0.22$ eV) off the reference value. +%\bruno{But also with GIC-SVWN5, as in the rest of this article, so one could wonder about the usefulness of the eVWN5 functional...} +As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were made in Ref.~\onlinecite{Loos_2020}. +This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy. As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the GIC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.