bib
This commit is contained in:
Pierre-Francois Loos
parent
ed0798260f
commit
4cd336196d
16266
Manuscript/ufGW.bib
Normal file
16266
Manuscript/ufGW.bib
Normal file
File diff suppressed because it is too large
Load Diff
@ -1,5 +1,5 @@
|
||||
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
|
||||
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
|
||||
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,siunitx}
|
||||
\usepackage[version=4]{mhchem}
|
||||
|
||||
\usepackage[utf8]{inputenc}
|
||||
@ -25,7 +25,6 @@
|
||||
\newcommand{\fnm}{\footnotemark}
|
||||
\newcommand{\fnt}{\footnotetext}
|
||||
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
|
||||
\newcommand{\SI}{\textcolor{blue}{supporting information}}
|
||||
\newcommand{\QP}{\textsc{quantum package}}
|
||||
\newcommand{\T}[1]{#1^{\intercal}}
|
||||
|
||||
@ -112,25 +111,15 @@
|
||||
\newcommand{\EgOpt}{\Eg^\text{opt}}
|
||||
\newcommand{\EB}{E_B}
|
||||
|
||||
\newcommand{\sig}{\sigma}
|
||||
\newcommand{\bsig}{{\Bar{\sigma}}}
|
||||
\newcommand{\sigp}{{\sigma'}}
|
||||
\newcommand{\bsigp}{{\Bar{\sigma}'}}
|
||||
\newcommand{\taup}{{\tau'}}
|
||||
\newcommand{\RHH}{R_{\ce{H-H}}}
|
||||
|
||||
\newcommand{\up}{\uparrow}
|
||||
\newcommand{\dw}{\downarrow}
|
||||
\newcommand{\upup}{\uparrow\uparrow}
|
||||
\newcommand{\updw}{\uparrow\downarrow}
|
||||
\newcommand{\dwup}{\downarrow\uparrow}
|
||||
\newcommand{\dwdw}{\downarrow\downarrow}
|
||||
|
||||
% addresses
|
||||
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\title{The $GW$ conundrum}
|
||||
\title{Unphysical Discontinuities in $GW$ Methods and the Role of Intruder States}
|
||||
|
||||
\author{Enzo \surname{Monino}}
|
||||
\affiliation{\LCPQ}
|
||||
@ -157,11 +146,11 @@ We consider {\GOWO} for the sake of simplicity but the same analysis can be perf
|
||||
\section{Downfold: The non-linear $GW$ problem}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Here, for the sake of simplicity, we consider a Hartree-Fock (HF) starting point.
|
||||
Here, for the sake of simplicity, we consider a restricted Hartree-Fock (HF) starting point but the present analysis can be straightforwardly extended to a Kohn-Sham (KS) starting point.
|
||||
Within the {\GOWO} approximation, in order to obtain the quasiparticle energies and the corresponding satellites, one solve, for each spatial orbital $p$, the following (non-linear) quasiparticle equation
|
||||
\begin{equation}
|
||||
\label{eq:qp_eq}
|
||||
\omega = \eps{p}{\HF} + \SigC{p}(\omega)
|
||||
\eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0
|
||||
\end{equation}
|
||||
where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of the {\GOWO} self-energy reads
|
||||
\begin{equation}
|
||||
@ -197,7 +186,8 @@ Note that we have the following important conservation rules
|
||||
\section{Upfolding: the linear $GW$ problem}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
The non-linear quasiparticle equation \eqref{eq:qp_eq} can be transformed into a larger linear problem via an upfolding process where the 2h1p and 2p1h sectors
|
||||
is downfolded on the 1h and 1p sectors via their interaction with the 2h1p and 2p1h sectors:
|
||||
are upfolded from the 1h and 1p sectors.
|
||||
For each orbital $p$, this yields a linear eigenvalue problem of the form
|
||||
\begin{equation}
|
||||
\bH^{(p)} \bc{}{(p,s)} = \eps{p,s}{} \bc{}{(p,s)}
|
||||
\end{equation}
|
||||
@ -225,15 +215,54 @@ and the corresponding coupling blocks read
|
||||
&
|
||||
V^\text{2p1h}_{p,cld} & = \sqrt{2} \ERI{pd}{kc}
|
||||
\end{align}
|
||||
The size of this eigenvalue problem is $N = 1 + N^\text{2h1p} + N^\text{2p1h} = 1 + O^2 V + O V^2$.
|
||||
The size of this eigenvalue problem is $N = 1 + N^\text{2h1p} + N^\text{2p1h} = 1 + O^2 V + O V^2$, and this eigenvalue problem has to be solved for each orbital that one wants to correct.
|
||||
Because the renormalization factor corresponds to the projection of the vector $\bc{}{(p,s)}$ onto the reference space, the weight of a solution $(p,s)$ is given by the the first coefficient of their corresponding eigenvector $\bc{}{(p,s)}$, \ie,
|
||||
\begin{equation}
|
||||
Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
|
||||
\end{equation}
|
||||
|
||||
It is important to understand that diagonalizing $\bH^{(p)}$ in Eq.~\eqref{eq:Hp} is completely equivalent to solving the quasiparticle equation \eqref{eq:qp_eq}.
|
||||
This can be further illustrate by expanding the secular equation associated with Eq.~\eqref{eq:Hp}
|
||||
\begin{equation}
|
||||
\det[ \bH^{(p)} - \omega \bI ] = 0
|
||||
\end{equation}
|
||||
and comparing it with Eq.~\eqref{eq:qp_eq} by setting
|
||||
\begin{multline}
|
||||
\SigC{p}(\omega)
|
||||
= \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
|
||||
\\
|
||||
+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
|
||||
\end{multline}
|
||||
The main difference between the two approaches is that, by diagonalizing Eq.~\eqref{eq:Hp}, one has directly access to the eigenvectors associated with each quasiparticle and satellites.
|
||||
|
||||
One can see this downfolding process as the construction of an frequency-dependent effective Hamiltonian where the reference (zeroth-order) space is composed by a single determinant of the 1h or 1p sector and the external (first-order) space by all the singly-excited states built from to this reference electronic configuration.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{An illustrative example}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
Multiple solution issues in $GW$ appears all the time, especially for orbitals that are far in energy from the Fermi level.
|
||||
Therefore, such issues are ubiquitous when one wants to compute core ionized states for example.
|
||||
|
||||
In order to illustrate the appearance and the origin of these multiple solutions, we consider the hydrogen molecule in the 6-31G basis set which corresponds to a system with 2 electrons and 4 spatial orbitals (one occupied and three virtual).
|
||||
This example was already considered in our previous work but here we provide further insights on the origin of the appearances of these multiple solutions.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% FIGURE 1
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{figure*}
|
||||
% \includegraphics[width=\linewidth]{fig1a}
|
||||
% \includegraphics[width=\linewidth]{fig1b}
|
||||
\caption{
|
||||
\label{fig:H2}
|
||||
Quasiparticle energies (left), correlation part of the self-energy (center) and renormalization factor (right) as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) for various orbitals of \ce{H2} at the {\GOWO}@HF/6-31G level.
|
||||
}
|
||||
\end{figure*}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Figure \ref{fig:H2} shows the evolution of the quasiparticle energies at the {\GOWO}@HF/6-31G level as a function on the internuclear distance $\RHH$.
|
||||
As one can see there are two problematic regions showing obvious discontinuities around $\RHH = \SI{1}{\AA}$ and $\RHH = \SI{1}{\AA}$
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Introducing regularized $GW$ methods}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
@ -248,5 +277,8 @@ This helps greatly convergence for (partially) self-consistent $GW$ methods.
|
||||
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\bibliography{ufGW}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\end{document}
|
||||
|
||||
@ -10,10 +10,10 @@
|
||||
NotebookFileLineBreakTest
|
||||
NotebookFileLineBreakTest
|
||||
NotebookDataPosition[ 158, 7]
|
||||
NotebookDataLength[ 42511, 1034]
|
||||
NotebookOptionsPosition[ 40318, 991]
|
||||
NotebookOutlinePosition[ 40770, 1009]
|
||||
CellTagsIndexPosition[ 40727, 1006]
|
||||
NotebookDataLength[ 48297, 1218]
|
||||
NotebookOptionsPosition[ 45590, 1169]
|
||||
NotebookOutlinePosition[ 46070, 1188]
|
||||
CellTagsIndexPosition[ 46027, 1185]
|
||||
WindowFrame->Normal*)
|
||||
|
||||
(* Beginning of Notebook Content *)
|
||||
@ -987,12 +987,191 @@ Cell[BoxData[
|
||||
CellChangeTimes->{{3.85122542357023*^9, 3.851225434992806*^9}, {
|
||||
3.851225486314489*^9, 3.851225493333332*^9}},
|
||||
CellLabel->"Out[34]=",ExpressionUUID->"2a2efd1e-215b-4367-831c-d3eaaa63ec02"]
|
||||
}, Open ]]
|
||||
}, Open ]],
|
||||
|
||||
Cell[BoxData[
|
||||
RowBox[{
|
||||
SuperscriptBox["H",
|
||||
RowBox[{"(", "p", ")"}]], "=",
|
||||
RowBox[{"(", "\[NoBreak]", GridBox[{
|
||||
{
|
||||
SubsuperscriptBox["\[Epsilon]", "p", "HF"],
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "h1p"}]],
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "p1h"}]]},
|
||||
{
|
||||
RowBox[{
|
||||
RowBox[{"(",
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "h1p"}]], ")"}], "\[Transpose]"}],
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "h1p"}]], "0"},
|
||||
{
|
||||
RowBox[{
|
||||
RowBox[{"(",
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "p1h"}]], ")"}], "\[Transpose]"}], "0",
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "p1h"}]]}
|
||||
}], "\[NoBreak]", ")"}]}]], "Input",
|
||||
CellChangeTimes->{{3.853860765989546*^9,
|
||||
3.853860843155305*^9}},ExpressionUUID->"6feb960f-bb32-4591-80f4-\
|
||||
4f095d8668c2"],
|
||||
|
||||
Cell[BoxData[
|
||||
RowBox[{
|
||||
RowBox[{"Det", "[",
|
||||
RowBox[{
|
||||
SuperscriptBox["H",
|
||||
RowBox[{"(", "p", ")"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}], "]"}], "==", "0"}]], "Input",
|
||||
CellChangeTimes->{{3.853860845482641*^9,
|
||||
3.8538608764071417`*^9}},ExpressionUUID->"21c50754-febc-41cd-a91a-\
|
||||
d483a7bc8140"],
|
||||
|
||||
Cell[BoxData[
|
||||
RowBox[{
|
||||
RowBox[{"Det", "[",
|
||||
RowBox[{"(", "\[NoBreak]", GridBox[{
|
||||
{
|
||||
RowBox[{
|
||||
SubsuperscriptBox["\[Epsilon]", "p", "HF"], "-", "\[Omega]"}],
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "h1p"}]],
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "p1h"}]]},
|
||||
{
|
||||
RowBox[{
|
||||
RowBox[{"(",
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "h1p"}]], ")"}], "\[Transpose]"}],
|
||||
RowBox[{
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "h1p"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}], "0"},
|
||||
{
|
||||
RowBox[{
|
||||
RowBox[{"(",
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "p1h"}]], ")"}], "\[Transpose]"}], "0",
|
||||
RowBox[{
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "p1h"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}]}
|
||||
}], "\[NoBreak]", ")"}], "]"}], "==", "0"}]], "Input",
|
||||
CellChangeTimes->{{3.853860882936471*^9,
|
||||
3.853860914930173*^9}},ExpressionUUID->"4dfd87b2-28e1-4b90-98bd-\
|
||||
12ef5c33e7cd"],
|
||||
|
||||
Cell[BoxData[
|
||||
RowBox[{
|
||||
RowBox[{
|
||||
RowBox[{
|
||||
RowBox[{"(",
|
||||
RowBox[{
|
||||
SubsuperscriptBox["\[Epsilon]", "p", "HF"], "-", "\[Omega]"}], ")"}],
|
||||
RowBox[{"Det", "[",
|
||||
RowBox[{"(", "\[NoBreak]", GridBox[{
|
||||
{
|
||||
RowBox[{
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "h1p"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}], "0"},
|
||||
{"0",
|
||||
RowBox[{
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "p1h"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}]}
|
||||
}], "\[NoBreak]", ")"}], "]"}]}], "-",
|
||||
RowBox[{
|
||||
RowBox[{
|
||||
RowBox[{"(",
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "h1p"}]], ")"}], "\[Transpose]"}],
|
||||
RowBox[{"(", "\[NoBreak]", GridBox[{
|
||||
{
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "h1p"}]],
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "p1h"}]]},
|
||||
{"0",
|
||||
RowBox[{
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "p1h"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}]}
|
||||
}], "\[NoBreak]", ")"}]}], "+",
|
||||
RowBox[{
|
||||
RowBox[{
|
||||
RowBox[{"(",
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "p1h"}]], ")"}], "\[Transpose]"}],
|
||||
RowBox[{"(", "\[NoBreak]", GridBox[{
|
||||
{
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "h1p"}]],
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "p1h"}]]},
|
||||
{
|
||||
RowBox[{
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "h1p"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}], "0"}
|
||||
}], "\[NoBreak]", ")"}]}]}], "=", "0"}]], "Input",
|
||||
CellChangeTimes->{{3.853860944854035*^9,
|
||||
3.853860994667194*^9}},ExpressionUUID->"f6448161-1952-4500-b5f5-\
|
||||
566946f9b807"],
|
||||
|
||||
Cell[BoxData[
|
||||
RowBox[{
|
||||
RowBox[{
|
||||
RowBox[{
|
||||
RowBox[{"(",
|
||||
RowBox[{
|
||||
SubsuperscriptBox["\[Epsilon]", "p", "HF"], "-", "\[Omega]"}], ")"}],
|
||||
RowBox[{"(",
|
||||
RowBox[{
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "h1p"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}], ")"}],
|
||||
RowBox[{"(",
|
||||
RowBox[{
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "p1h"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}], ")"}]}], "-",
|
||||
RowBox[{
|
||||
RowBox[{
|
||||
RowBox[{"(",
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "h1p"}]], ")"}], "\[Transpose]"}],
|
||||
RowBox[{"(",
|
||||
RowBox[{
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "p1h"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}], ")"}],
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "h1p"}]]}], "-",
|
||||
RowBox[{
|
||||
RowBox[{
|
||||
RowBox[{"(",
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "p1h"}]], ")"}], "\[Transpose]"}],
|
||||
RowBox[{"(",
|
||||
RowBox[{
|
||||
SuperscriptBox["C",
|
||||
RowBox[{"2", "h1p"}]], "-",
|
||||
RowBox[{"\[Omega]", " ", "1"}]}], ")"}],
|
||||
SubsuperscriptBox["V", "p",
|
||||
RowBox[{"2", "p1h"}]]}]}], "=", "0"}]], "Input",
|
||||
CellChangeTimes->{{3.853861004235111*^9,
|
||||
3.8538610705718946`*^9}},ExpressionUUID->"260633fc-4386-4e7b-a747-\
|
||||
77a336ee6b4d"]
|
||||
},
|
||||
WindowSize->{808, 911},
|
||||
WindowMargins->{{Automatic, 710}, {Automatic, 181}},
|
||||
WindowSize->{1440, 847},
|
||||
WindowMargins->{{0, Automatic}, {Automatic, 0}},
|
||||
PrintingCopies->1,
|
||||
PrintingPageRange->{1, Automatic},
|
||||
Magnification:>1.25 Inherited,
|
||||
FrontEndVersion->"13.0 for Mac OS X ARM (64-bit) (December 2, 2021)",
|
||||
StyleDefinitions->"Default.nb",
|
||||
ExpressionUUID->"c947be63-9aef-4192-bf26-b635965432d0"
|
||||
@ -1008,34 +1187,39 @@ CellTagsIndex->{}
|
||||
*)
|
||||
(*NotebookFileOutline
|
||||
Notebook[{
|
||||
Cell[558, 20, 1276, 35, 94, "Input",ExpressionUUID->"682f75fb-1edd-4aa6-a4ad-b597c5f869c9"],
|
||||
Cell[1837, 57, 1001, 27, 73, "Input",ExpressionUUID->"35c12977-2872-4eab-be4f-329f455172b3"],
|
||||
Cell[558, 20, 1276, 35, 117, "Input",ExpressionUUID->"682f75fb-1edd-4aa6-a4ad-b597c5f869c9"],
|
||||
Cell[1837, 57, 1001, 27, 91, "Input",ExpressionUUID->"35c12977-2872-4eab-be4f-329f455172b3"],
|
||||
Cell[CellGroupData[{
|
||||
Cell[2863, 88, 328, 7, 30, "Input",ExpressionUUID->"59810817-5144-4818-800a-a2ceb8fcd500"],
|
||||
Cell[3194, 97, 4095, 96, 247, "Output",ExpressionUUID->"814e3769-9eaf-4e1a-8956-617c8e61d77e"]
|
||||
Cell[2863, 88, 328, 7, 37, "Input",ExpressionUUID->"59810817-5144-4818-800a-a2ceb8fcd500"],
|
||||
Cell[3194, 97, 4095, 96, 308, "Output",ExpressionUUID->"814e3769-9eaf-4e1a-8956-617c8e61d77e"]
|
||||
}, Open ]],
|
||||
Cell[7304, 196, 1274, 35, 94, "Input",ExpressionUUID->"5f735bd6-ba1a-4987-b9f7-8ff4ac674345"],
|
||||
Cell[8581, 233, 981, 27, 73, "Input",ExpressionUUID->"5928cf25-152d-4cb2-9ce3-3f99ce9557d3"],
|
||||
Cell[7304, 196, 1274, 35, 117, "Input",ExpressionUUID->"5f735bd6-ba1a-4987-b9f7-8ff4ac674345"],
|
||||
Cell[8581, 233, 981, 27, 91, "Input",ExpressionUUID->"5928cf25-152d-4cb2-9ce3-3f99ce9557d3"],
|
||||
Cell[CellGroupData[{
|
||||
Cell[9587, 264, 310, 7, 30, "Input",ExpressionUUID->"33a86cd9-4a3d-4b09-91de-2cbb886a6250"],
|
||||
Cell[9900, 273, 5200, 125, 248, "Output",ExpressionUUID->"d32fbdb3-4ad8-4d05-aa55-4c32ff841fbc"]
|
||||
Cell[9587, 264, 310, 7, 37, "Input",ExpressionUUID->"33a86cd9-4a3d-4b09-91de-2cbb886a6250"],
|
||||
Cell[9900, 273, 5200, 125, 310, "Output",ExpressionUUID->"d32fbdb3-4ad8-4d05-aa55-4c32ff841fbc"]
|
||||
}, Open ]],
|
||||
Cell[15115, 401, 1275, 35, 94, "Input",ExpressionUUID->"051138e6-33b2-49e1-b570-b5e33fd33d7f"],
|
||||
Cell[16393, 438, 982, 27, 73, "Input",ExpressionUUID->"37bd2bc4-e119-4f55-88d4-845984e63c22"],
|
||||
Cell[17378, 467, 230, 6, 30, "Input",ExpressionUUID->"c2496421-c71c-4de1-a28d-dd9067951fda"],
|
||||
Cell[15115, 401, 1275, 35, 117, "Input",ExpressionUUID->"051138e6-33b2-49e1-b570-b5e33fd33d7f"],
|
||||
Cell[16393, 438, 982, 27, 91, "Input",ExpressionUUID->"37bd2bc4-e119-4f55-88d4-845984e63c22"],
|
||||
Cell[17378, 467, 230, 6, 37, "Input",ExpressionUUID->"c2496421-c71c-4de1-a28d-dd9067951fda"],
|
||||
Cell[CellGroupData[{
|
||||
Cell[17633, 477, 312, 7, 30, "Input",ExpressionUUID->"bbbc4763-dda3-4da9-8ff0-9b7dc9932b23"],
|
||||
Cell[17948, 486, 4933, 120, 246, "Output",ExpressionUUID->"cad1c286-e9fa-457d-9d0f-e5060da9432b"]
|
||||
Cell[17633, 477, 312, 7, 37, "Input",ExpressionUUID->"bbbc4763-dda3-4da9-8ff0-9b7dc9932b23"],
|
||||
Cell[17948, 486, 4933, 120, 307, "Output",ExpressionUUID->"cad1c286-e9fa-457d-9d0f-e5060da9432b"]
|
||||
}, Open ]],
|
||||
Cell[22896, 609, 513, 11, 52, "Input",ExpressionUUID->"c351c0eb-ffe9-41ea-b9a4-ee9c001a879d"],
|
||||
Cell[22896, 609, 513, 11, 65, "Input",ExpressionUUID->"c351c0eb-ffe9-41ea-b9a4-ee9c001a879d"],
|
||||
Cell[CellGroupData[{
|
||||
Cell[23434, 624, 1050, 23, 52, "Input",ExpressionUUID->"3dc691e1-f05a-4c4c-8978-c337f01f76cf"],
|
||||
Cell[24487, 649, 8346, 163, 468, "Output",ExpressionUUID->"dad67ae9-473d-49b3-a4ed-6cc27aca3cc2"]
|
||||
Cell[23434, 624, 1050, 23, 65, "Input",ExpressionUUID->"3dc691e1-f05a-4c4c-8978-c337f01f76cf"],
|
||||
Cell[24487, 649, 8346, 163, 583, "Output",ExpressionUUID->"dad67ae9-473d-49b3-a4ed-6cc27aca3cc2"]
|
||||
}, Open ]],
|
||||
Cell[CellGroupData[{
|
||||
Cell[32870, 817, 859, 20, 52, "Input",ExpressionUUID->"e9d7c473-f53c-4be4-9780-1cba856893ff"],
|
||||
Cell[33732, 839, 6570, 149, 248, "Output",ExpressionUUID->"2a2efd1e-215b-4367-831c-d3eaaa63ec02"]
|
||||
}, Open ]]
|
||||
Cell[32870, 817, 859, 20, 37, "Input",ExpressionUUID->"e9d7c473-f53c-4be4-9780-1cba856893ff"],
|
||||
Cell[33732, 839, 6570, 149, 308, "Output",ExpressionUUID->"2a2efd1e-215b-4367-831c-d3eaaa63ec02"]
|
||||
}, Open ]],
|
||||
Cell[40317, 991, 878, 28, 116, "Input",ExpressionUUID->"6feb960f-bb32-4591-80f4-4f095d8668c2"],
|
||||
Cell[41198, 1021, 316, 9, 40, "Input",ExpressionUUID->"21c50754-febc-41cd-a91a-d483a7bc8140"],
|
||||
Cell[41517, 1032, 1051, 32, 116, "Input",ExpressionUUID->"4dfd87b2-28e1-4b90-98bd-12ef5c33e7cd"],
|
||||
Cell[42571, 1066, 1724, 56, 68, "Input",ExpressionUUID->"f6448161-1952-4500-b5f5-566946f9b807"],
|
||||
Cell[44298, 1124, 1288, 43, 42, "Input",ExpressionUUID->"260633fc-4386-4e7b-a747-77a336ee6b4d"]
|
||||
}
|
||||
]
|
||||
*)
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user