FarDFT/FarDFT.nb

2711 lines
104 KiB
Mathematica

(* Content-type: application/vnd.wolfram.mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
(* CreatedBy='Mathematica 11.3' *)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 158, 7]
NotebookDataLength[ 106462, 2702]
NotebookOptionsPosition[ 95830, 2533]
NotebookOutlinePosition[ 96167, 2548]
CellTagsIndexPosition[ 96124, 2545]
WindowFrame->Normal*)
(* Beginning of Notebook Content *)
Notebook[{
Cell[CellGroupData[{
Cell["Initialization", "Title",
CellChangeTimes->{{3.782624936755896*^9,
3.78262494500487*^9}},ExpressionUUID->"9a643c6d-1044-4612-8f7a-\
377e028cd777"],
Cell[BoxData[{
RowBox[{"Needs", "[", "\"\<MaTeX`\>\"", "]"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"SetOptions", "[",
RowBox[{"MaTeX", ",",
RowBox[{"\"\<Preamble\>\"", "\[Rule]",
RowBox[{
"{", "\"\<\\\\usepackage{amssymb,amsmath,latexsym,amsfonts,amsthm,\
mathpazo,xcolor,bm,mhchem}\>\"", "}"}]}]}], "]"}], ";"}]}], "Input",
InitializationCell->True,
CellChangeTimes->{{3.7288240181604652`*^9, 3.728824027007351*^9}, {
3.733131339213026*^9, 3.733131352923026*^9}},
CellLabel->"In[1]:=",ExpressionUUID->"8b7c0bbd-9f99-42dc-b8a5-a4dc3cb8cc89"],
Cell[BoxData[
RowBox[{
RowBox[{"HaToeV", "=", "27.21138602"}], ";"}]], "Input",
InitializationCell->True,
CellChangeTimes->{{3.7208031947801647`*^9, 3.7208032000677156`*^9}, {
3.7208034541742477`*^9, 3.720803455246439*^9}},
CellLabel->"In[3]:=",ExpressionUUID->"b24e8243-3c38-4b03-af3a-708cb1afa004"]
}, Closed]],
Cell[CellGroupData[{
Cell["DFAs", "Title",
CellChangeTimes->{{3.7826249735359907`*^9,
3.7826249744308434`*^9}},ExpressionUUID->"cdfcc3a6-7299-4a1b-a9d0-\
f29078225f1e"],
Cell[CellGroupData[{
Cell["LDA exchange", "Section",
CellChangeTimes->{{3.7826249982517223`*^9,
3.782625006542748*^9}},ExpressionUUID->"23d29e55-92a0-4dbd-8800-\
76494f7702a3"],
Cell[CellGroupData[{
Cell["Dirac", "Subsection",
CellChangeTimes->{{3.725782094153554*^9, 3.7257820943572903`*^9}, {
3.754455221403211*^9, 3.754455222014764*^9}, {3.782624874655992*^9,
3.782624879193289*^9}},ExpressionUUID->"5917ba4c-2cd7-4975-9b3b-\
073e72bd0e40"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"{",
RowBox[{
RowBox[{"\[Alpha]", "=",
RowBox[{
RowBox[{"-",
FractionBox["3", "2"]}],
SuperscriptBox[
RowBox[{"(",
FractionBox["3",
RowBox[{"4", "\[Pi]"}]], ")"}],
RowBox[{"1", "/", "3"}]]}]}], ",",
RowBox[{
RowBox[{
RowBox[{"-",
FractionBox["3", "2"]}],
SuperscriptBox[
RowBox[{"(",
FractionBox["3",
RowBox[{"4", "\[Pi]"}]], ")"}],
RowBox[{"1", "/", "3"}]]}], "//", "N"}]}], "}"}]], "Input",
CellChangeTimes->{{3.72598648647053*^9, 3.725986487865756*^9}, {
3.754455176229362*^9, 3.754455189038566*^9}},
CellLabel->"In[44]:=",ExpressionUUID->"c8ade4e2-951f-4fce-ae16-fdd4ea72db6f"],
Cell[BoxData[
RowBox[{"{",
RowBox[{
RowBox[{"-",
FractionBox[
RowBox[{"3", " ",
SuperscriptBox[
RowBox[{"(",
FractionBox["3", "\[Pi]"], ")"}],
RowBox[{"1", "/", "3"}]]}],
RowBox[{"2", " ",
SuperscriptBox["2",
RowBox[{"2", "/", "3"}]]}]]}], ",",
RowBox[{"-", "0.9305257363491`"}]}], "}"}]], "Output",
CellChangeTimes->{{3.754455185027546*^9, 3.754455189495618*^9}},
CellLabel->"Out[44]=",ExpressionUUID->"e2c1e6f8-35e1-43ee-b47e-a64bb64f8585"]
}, Open ]],
Cell[BoxData[
RowBox[{
RowBox[{
RowBox[{"\[Epsilon]", "[", "\[Rho]_", "]"}], "=",
RowBox[{"\[Alpha]", " ",
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}]}], ";", "\t",
RowBox[{
RowBox[{"v", "[", "\[Rho]_", "]"}], "=",
RowBox[{"\[Alpha]", " ",
SuperscriptBox["\[Rho]",
RowBox[{"4", "/", "3"}]]}]}], ";"}]], "Input",
CellChangeTimes->{{3.725782108438846*^9, 3.725782132849185*^9}, {
3.725986489670062*^9, 3.725986491162725*^9}, {3.754455193988475*^9,
3.754455223428444*^9}, {3.7544552613310947`*^9, 3.75445526326921*^9}},
CellLabel->"In[48]:=",ExpressionUUID->"02e08ec6-7c15-4841-9609-90296edfe494"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Simplify", "[",
RowBox[{
RowBox[{
SubscriptBox["\[PartialD]", "\[Rho]"],
RowBox[{"v", "[", "\[Rho]", "]"}]}], "\[Equal]",
RowBox[{
FractionBox["4", "3"], " ", "\[Alpha]", " ",
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}]}], "]"}]], "Input",
CellChangeTimes->{{3.725782136304735*^9, 3.725782163374041*^9}, {
3.725986494110373*^9, 3.725986498834753*^9}, {3.7544552437159224`*^9,
3.754455264690394*^9}},
CellLabel->"In[49]:=",ExpressionUUID->"150e916f-8806-4d8e-b5d0-c0bc713b1f4d"],
Cell[BoxData["True"], "Output",
CellChangeTimes->{{3.7257821552371197`*^9, 3.725782163950963*^9}, {
3.725986496082922*^9, 3.725986499250463*^9}, {3.754455244234871*^9,
3.754455267609223*^9}},
CellLabel->"Out[49]=",ExpressionUUID->"fb35d58c-512f-4f74-aa01-67c7b7b81a77"]
}, Open ]]
}, Closed]]
}, Open ]],
Cell[CellGroupData[{
Cell["LDA correlation", "Section",
CellChangeTimes->{{3.7826249982517223`*^9, 3.782625006542748*^9}, {
3.7826250653667927`*^9,
3.7826250672414913`*^9}},ExpressionUUID->"8fd0ca40-ccd3-410e-8538-\
798441b5cf4b"],
Cell[CellGroupData[{
Cell["VWN5", "Subsection",
CellChangeTimes->{{3.754982974070456*^9, 3.754982978077882*^9},
3.75506102068367*^9},ExpressionUUID->"10e37c4d-fcc0-4a25-949c-\
6383c67ff9eb"],
Cell[CellGroupData[{
Cell[BoxData[{
RowBox[{
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "x"], "[", "\[Zeta]_", "]"}], "=",
FractionBox[
RowBox[{
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "+", "\[Zeta]"}], ")"}],
RowBox[{"4", "/", "3"}]], "+",
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "-", "\[Zeta]"}], ")"}],
RowBox[{"4", "/", "3"}]]}], "2"]}], ";", "\t",
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "c"], "[", "\[Zeta]_", "]"}], "=",
FractionBox[
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "x"], "[", "\[Zeta]", "]"}], "-",
"1"}],
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "x"], "[", "1", "]"}], "-", "1"}]]}],
";", "\t",
RowBox[{
RowBox[{
FractionBox[
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "x"], "[", "\[Zeta]", "]"}], "-",
"1"}],
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "x"], "[", "1", "]"}], "-", "1"}]],
"\[Equal]",
RowBox[{"(",
RowBox[{
FractionBox["1", "2"],
FractionBox[
RowBox[{
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "+", "\[Zeta]"}], ")"}],
RowBox[{"4", "/", "3"}]], "+",
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "-", "\[Zeta]"}], ")"}],
RowBox[{"4", "/", "3"}]], "-", "2"}],
RowBox[{
SuperscriptBox["2",
RowBox[{"1", "/", "3"}]], "-", "1"}]]}], ")"}]}], "//",
"Simplify"}]}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c", "VWN"], "[",
RowBox[{
SubscriptBox["r", "s"], ",", "\[Zeta]"}], "]"}], "=",
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c", "VWN"], "[",
RowBox[{
SubscriptBox["r", "s"], ",", "0"}], "]"}], "+",
RowBox[{
RowBox[{
SubscriptBox["e", "a"], "[",
SubscriptBox["r", "s"], "]"}],
FractionBox[
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "c"], "[", "\[Zeta]", "]"}],
RowBox[{
SubsuperscriptBox["\[CapitalUpsilon]", "c", "\[DoublePrime]"], "[",
"0", "]"}]],
RowBox[{"(",
RowBox[{"1", "-",
SuperscriptBox["\[Zeta]", "4"]}], ")"}]}], "+",
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c", "VWN"], "[",
RowBox[{
SubscriptBox["r", "s"], ",", "1"}], "]"}], "-",
RowBox[{
SubsuperscriptBox["e", "c", "VWN"], "[",
RowBox[{
SubscriptBox["r", "s"], ",", "0"}], "]"}]}], ")"}],
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "c"], "[", "\[Zeta]", "]"}],
SuperscriptBox["\[Zeta]", "4"]}]}]}], ";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{
SubscriptBox["\[PartialD]",
RowBox[{"\[Zeta]", ",", "\[Zeta]"}]],
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "c"], "[", "\[Zeta]", "]"}]}],
"\[Equal]",
FractionBox["4",
RowBox[{"9", " ",
RowBox[{"(",
RowBox[{
SuperscriptBox["2",
RowBox[{"1", "/", "3"}]], "-", "1"}], ")"}]}]]}], "/.",
RowBox[{"\[Zeta]", "\[Rule]", "0"}]}]}], "Input",
CellChangeTimes->{{3.64953708589915*^9, 3.6495371040691566`*^9}, {
3.649537598690679*^9, 3.6495376202405977`*^9}, {3.6495378262544413`*^9,
3.649537829400469*^9}, {3.6495379773256607`*^9, 3.649537998778248*^9}, {
3.6495515782672*^9, 3.649551590769746*^9}, 3.649551667423394*^9, {
3.649573415465891*^9, 3.649573460769969*^9}, {3.6495735032778273`*^9,
3.649573558905994*^9}, {3.6502588937824087`*^9, 3.650258894980505*^9}, {
3.6711711046859093`*^9, 3.671171136347082*^9}, {3.6711711756949053`*^9,
3.671171216091024*^9}, {3.671171535080681*^9, 3.67117153659056*^9},
3.67117161343187*^9, 3.671218234223638*^9, {3.671218313087159*^9,
3.671218320467162*^9}, {3.6712183625084667`*^9, 3.671218440631269*^9}, {
3.6712185343897257`*^9, 3.671218555241634*^9}, {3.67121872124428*^9,
3.6712187515692987`*^9}, {3.75506070706028*^9, 3.755060820580678*^9}, {
3.755060993278182*^9, 3.755061012195376*^9}, {3.755062368242033*^9,
3.7550623825126963`*^9}, {3.755062641574308*^9, 3.755062643019628*^9}},
CellLabel->"In[86]:=",ExpressionUUID->"b7e33c7b-f239-4612-b339-911d66bee804"],
Cell[BoxData["True"], "Output",
CellChangeTimes->{3.755061014203738*^9, 3.755062643688386*^9,
3.755077988753642*^9},
CellLabel->"Out[86]=",ExpressionUUID->"3acd8e38-829c-4dd5-ac53-54c357a30852"],
Cell[BoxData["True"], "Output",
CellChangeTimes->{3.755061014203738*^9, 3.755062643688386*^9,
3.755077988820902*^9},
CellLabel->"Out[88]=",ExpressionUUID->"0a957f43-bd43-468d-a336-addbe42a4e30"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{
RowBox[{"D", "[",
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "c"], "[", "\[Zeta]", "]"}], ",",
"\[Zeta]"}], "]"}], "\[Equal]",
RowBox[{
FractionBox["4", "3"],
FractionBox[
RowBox[{
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "+", "\[Zeta]"}], ")"}],
RowBox[{"1", "/", "3"}]], "-",
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "-", "\[Zeta]"}], ")"}],
RowBox[{"1", "/", "3"}]]}],
RowBox[{"2",
RowBox[{"(",
RowBox[{
SuperscriptBox["2",
RowBox[{"1", "/", "3"}]], "-", "1"}], ")"}]}]]}]}], "//",
"Simplify"}]], "Input",
CellChangeTimes->{{3.755077992746286*^9, 3.7550780506722403`*^9}},
CellLabel->"In[93]:=",ExpressionUUID->"5bc098a3-a8b3-48df-b81b-4249253ebc03"],
Cell[BoxData["True"], "Output",
CellChangeTimes->{{3.755078029958152*^9, 3.755078051141384*^9}},
CellLabel->"Out[93]=",ExpressionUUID->"3597039b-63a4-48ef-b436-f5b1fb49d7fd"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[{
RowBox[{
RowBox[{
RowBox[{"D", "[",
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "c"], "[", "\[Zeta]", "]"}], ",",
RowBox[{"{",
RowBox[{"\[Zeta]", ",", "2"}], "}"}]}], "]"}], "\[Equal]",
RowBox[{
FractionBox["1",
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "x"], "[", "1", "]"}], "-", "1"}]],
RowBox[{"D", "[",
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "x"], "[", "\[Zeta]", "]"}], ",",
RowBox[{"{",
RowBox[{"\[Zeta]", ",", "2"}], "}"}]}], "]"}]}]}], "//",
"Simplify"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{"D", "[",
RowBox[{
RowBox[{
SubscriptBox["\[CapitalUpsilon]", "x"], "[", "\[Zeta]", "]"}], ",",
RowBox[{"{",
RowBox[{"\[Zeta]", ",", "2"}], "}"}]}], "]"}], "\[Equal]",
RowBox[{
FractionBox["4", "9"],
FractionBox[
RowBox[{
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "+", "\[Zeta]"}], ")"}],
RowBox[{
RowBox[{"-", "2"}], "/", "3"}]], "+",
SuperscriptBox[
RowBox[{"(",
RowBox[{"1", "-", "\[Zeta]"}], ")"}],
RowBox[{
RowBox[{"-", "2"}], "/", "3"}]]}], "2"]}]}], "//",
"Simplify"}]}], "Input",
CellChangeTimes->{{3.7550616534653797`*^9, 3.755061765205967*^9}},
CellLabel->"In[71]:=",ExpressionUUID->"7707f078-0791-4cee-b4ea-44b37115dc20"],
Cell[BoxData["True"], "Output",
CellChangeTimes->{3.7550617026823*^9, 3.755061765661969*^9,
3.755062645801415*^9},
CellLabel->"Out[71]=",ExpressionUUID->"d4f8c9be-6e4c-40eb-ba9a-d4053731284e"],
Cell[BoxData["True"], "Output",
CellChangeTimes->{3.7550617026823*^9, 3.755061765661969*^9,
3.7550626458247223`*^9},
CellLabel->"Out[72]=",ExpressionUUID->"2e03225f-8957-4361-919c-4f6a7c36dc6e"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{
RowBox[{"D", "[",
RowBox[{
RowBox[{
SubscriptBox["e", "a"], "[", "x", "]"}], ",", "x"}], "]"}], "\[Equal]",
RowBox[{"A", " ",
RowBox[{"(",
RowBox[{
RowBox[{"+", " ",
FractionBox["2", "x"]}], "-",
FractionBox[
RowBox[{"4", " ", "b"}],
RowBox[{" ",
RowBox[{
SuperscriptBox[
RowBox[{"(",
RowBox[{"b", "+",
RowBox[{"2", " ", "x"}]}], ")"}], "2"], "+",
SuperscriptBox[
RowBox[{"Q", "[",
RowBox[{"b", ",", "c"}], "]"}], "2"]}]}]], "-",
FractionBox[
RowBox[{" ",
RowBox[{
SuperscriptBox["X",
TagBox[
RowBox[{"(",
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None], "[",
RowBox[{"x", ",", "b", ",", "c"}], "]"}]}],
RowBox[{"X", "[",
RowBox[{"x", ",", "b", ",", "c"}], "]"}]], "-",
RowBox[{
FractionBox[
RowBox[{"b", " ", "x0"}],
RowBox[{"X", "[",
RowBox[{"x0", ",", "b", ",", "c"}], "]"}]], " ",
RowBox[{"(", " ",
RowBox[{
FractionBox["2",
RowBox[{"x", "-", "x0"}]], "-",
FractionBox[
RowBox[{"4", " ",
RowBox[{"(",
RowBox[{"b", "+",
RowBox[{"2", " ", "x0"}]}], ")"}]}],
RowBox[{" ",
RowBox[{
SuperscriptBox[
RowBox[{"(",
RowBox[{"b", "+",
RowBox[{"2", " ", "x"}]}], ")"}], "2"], "+",
SuperscriptBox[
RowBox[{"Q", "[",
RowBox[{"b", ",", "c"}], "]"}], "2"]}]}]], " ", "-",
FractionBox[
RowBox[{" ",
RowBox[{
SuperscriptBox["X",
TagBox[
RowBox[{"(",
RowBox[{"1", ",", "0", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None], "[",
RowBox[{"x", ",", "b", ",", "c"}], "]"}]}],
RowBox[{"X", "[",
RowBox[{"x", ",", "b", ",", "c"}], "]"}]]}], ")"}]}]}], ")"}]}]}],
"//", "Simplify"}]], "Input",
CellChangeTimes->{{3.755078452482924*^9, 3.755078456060067*^9}, {
3.755078514862273*^9, 3.75507859827311*^9}, {3.755078633288357*^9,
3.755078701168396*^9}, {3.7550788358255777`*^9, 3.755078836868124*^9}, {
3.7550789360932302`*^9, 3.755078974047309*^9}, {3.755093941986959*^9,
3.755093942181332*^9}, {3.755102551539661*^9, 3.7551025658665733`*^9}},
CellLabel->"In[64]:=",ExpressionUUID->"731b5390-31ed-4914-91db-765023c263dd"],
Cell[BoxData["True"], "Output",
CellChangeTimes->{
3.755078456572246*^9, {3.755078490433147*^9, 3.755078517939692*^9}, {
3.755078564736414*^9, 3.7550785986865063`*^9}, {3.755078638048624*^9,
3.7550786600597563`*^9}, {3.755078693091856*^9, 3.755078701737822*^9},
3.7550788375806637`*^9, {3.7550789379495983`*^9, 3.755078945513707*^9},
3.755078976190452*^9, 3.755093942709054*^9, 3.755102712713393*^9},
CellLabel->"Out[64]=",ExpressionUUID->"ed71661b-3a8d-4eee-a4d0-69211937142a"]
}, Open ]],
Cell[BoxData[
RowBox[{
RowBox[{
RowBox[{
SubscriptBox["e", "a"], "[", "x_", "]"}], "=",
RowBox[{"A", " ",
RowBox[{"(",
RowBox[{
RowBox[{"Log", "[",
FractionBox[
SuperscriptBox["x", "2"],
RowBox[{"X", "[",
RowBox[{"x", ",", "b", ",", "c"}], "]"}]], "]"}], "+",
RowBox[{
FractionBox[
RowBox[{"2", "b"}],
RowBox[{"Q", "[",
RowBox[{"b", ",", "c"}], "]"}]], " ",
RowBox[{"ArcTan", "[",
FractionBox[
RowBox[{"Q", "[",
RowBox[{"b", ",", "c"}], "]"}],
RowBox[{
RowBox[{"2", "x"}], "+", "b"}]], "]"}]}], "-",
RowBox[{
FractionBox[
RowBox[{"b", " ", "x0"}],
RowBox[{"X", "[",
RowBox[{"x0", ",", "b", ",", "c"}], "]"}]],
RowBox[{"(",
RowBox[{
RowBox[{"Log", "[",
FractionBox[
SuperscriptBox[
RowBox[{"(",
RowBox[{"x", "-", "x0"}], ")"}], "2"],
RowBox[{"X", "[",
RowBox[{"x", ",", "b", ",", "c"}], "]"}]], "]"}], "+",
RowBox[{"2",
FractionBox[
RowBox[{"(",
RowBox[{"b", "+",
RowBox[{"2", "x0"}]}], ")"}],
RowBox[{"Q", "[",
RowBox[{"b", ",", "c"}], "]"}]],
RowBox[{"ArcTan", "[",
FractionBox[
RowBox[{"Q", "[",
RowBox[{"b", ",", "c"}], "]"}],
RowBox[{
RowBox[{"2", "x"}], "+", "b"}]], "]"}]}]}], ")"}]}]}], ")"}]}]}],
";"}]], "Input",
CellChangeTimes->{{3.671218757744111*^9, 3.6712189001826477`*^9}, {
3.671262493487727*^9, 3.671262494966117*^9}, {3.755060508105689*^9,
3.755060539068595*^9}, {3.755060622387308*^9, 3.755060636469385*^9}, {
3.755060995209078*^9, 3.755060995660227*^9}, {3.7550611599982033`*^9,
3.755061166338585*^9}, {3.7550612340212917`*^9, 3.755061251761766*^9}, {
3.755062649580003*^9, 3.755062650928399*^9}, {3.7550711084786367`*^9,
3.7550711099313087`*^9}, 3.7550784458832827`*^9, {3.7550784813506203`*^9,
3.755078486479793*^9}, {3.782625033589868*^9,
3.7826250338691883`*^9}},ExpressionUUID->"1f5927dc-663a-4f11-a2d1-\
7bc195f79332"]
}, Closed]]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["eDFAs", "Title",
CellChangeTimes->{{3.7826249522813463`*^9,
3.782624958928115*^9}},ExpressionUUID->"e756f09e-4f67-4fb6-a4c1-\
1a4fcf53e013"],
Cell[CellGroupData[{
Cell["eLDA exchange", "Section",
CellChangeTimes->{{3.7826249982517223`*^9, 3.782625006542748*^9}, {
3.7826250653667927`*^9, 3.7826250672414913`*^9}, {3.782625110014524*^9,
3.782625116494841*^9}},ExpressionUUID->"81f07a4a-071b-4e38-8018-\
d116e770b20b"],
Cell[CellGroupData[{
Cell["Ground state exchange functional for glomium", "Subsection",
CellChangeTimes->{{3.782625187684318*^9, 3.782625205187807*^9}, {
3.7826253413716393`*^9, 3.7826253532482758`*^9}, {3.782625846786179*^9,
3.782625848068905*^9}},ExpressionUUID->"7471e1f3-36cd-497f-a3e0-\
06aa7035995d"],
Cell["\<\
The ground state two-electron glomium system has an uniform electron density \
\
\>", "Text",
CellChangeTimes->{{3.78262523411093*^9,
3.782625281581828*^9}},ExpressionUUID->"c0165a48-1e75-4237-86f9-\
15c7a0fc2c6e"],
Cell[BoxData[
RowBox[{"\[Rho]", "\[Equal]",
RowBox[{
FractionBox["2",
RowBox[{"2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]], "\t",
SubscriptBox["\[Rho]", "\[Alpha]"]}], "\[Equal]",
SubscriptBox["\[Rho]", "\[Beta]"], "\[Equal]",
FractionBox["1",
RowBox[{"2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]]}]], "Input",
CellChangeTimes->{{3.7826252862043962`*^9, 3.7826253053578*^9}, {
3.782625777108284*^9,
3.7826257868938847`*^9}},ExpressionUUID->"eeea1fbc-15c7-4b79-b5b2-\
0d2e682c7cb1"],
Cell["The reduced HF energy of glomium is", "Text",
CellChangeTimes->{{3.782625207745901*^9, 3.7826252161060266`*^9}, {
3.7826254294319897`*^9,
3.782625432722817*^9}},ExpressionUUID->"6509cf0c-3cb5-4484-a79b-\
9a4e9a4c867b"],
Cell[BoxData[
RowBox[{
SubsuperscriptBox["e", "HF",
RowBox[{"(", "0", ")"}]], "\[Equal]",
FractionBox["4",
RowBox[{"3", " ", "\[Pi]", " ", "R"}]]}]], "Input",
CellChangeTimes->{{3.7826252222025003`*^9, 3.782625227795396*^9}, {
3.782625336400528*^9, 3.7826253365779533`*^9}, 3.782625434707806*^9,
3.782625956802226*^9, {3.7826261353898563`*^9,
3.782626136687096*^9}},ExpressionUUID->"ea3020ca-486a-47b1-95e0-\
aa46f448646d"],
Cell["This energy can be decomposed as", "Text",
CellChangeTimes->{{3.782625326608221*^9,
3.7826253316050653`*^9}},ExpressionUUID->"dc6b5921-169b-4f2d-9120-\
4c7ca89c05f3"],
Cell[BoxData[
RowBox[{
SubsuperscriptBox["t", "s",
RowBox[{"(", "0", ")"}]], "\[Equal]",
RowBox[{"0", "\t",
SubsuperscriptBox["e", "H",
RowBox[{"(", "0", ")"}]]}], "\[Equal]",
RowBox[{
FractionBox["8",
RowBox[{"3", " ", "\[Pi]", " ", "R"}]], "\t",
SubsuperscriptBox["e", "x",
RowBox[{"(", "0", ")"}]]}], "\[Equal]",
RowBox[{"-",
FractionBox["4",
RowBox[{"3", " ", "\[Pi]", " ", "R"}]]}]}]], "Input",
CellChangeTimes->{{3.78262533887177*^9, 3.782625382698719*^9}, {
3.78262543658897*^9, 3.78262544697281*^9}, {3.7826261396725388`*^9,
3.782626144976527*^9}},ExpressionUUID->"12ac5e19-b8fd-477e-a85d-\
a74e029b6d4c"],
Cell["Knowing that the exchange functional has the following form", "Text",
CellChangeTimes->{{3.782625462881948*^9,
3.782625490540832*^9}},ExpressionUUID->"21ac5bf4-2ae7-4b41-8495-\
ff3b6d5d2017"],
Cell[BoxData[
RowBox[{
SubsuperscriptBox["\[CapitalEpsilon]",
RowBox[{"x", ",", "\[Alpha]"}],
RowBox[{"(", "0", ")"}]], "\[Equal]",
RowBox[{"\[Integral]",
RowBox[{
SubsuperscriptBox["e", "x",
RowBox[{"(", "0", ")"}]],
SubscriptBox["\[Rho]", "\[Alpha]"], " ",
RowBox[{"\[DifferentialD]", "r"}]}]}], "\[Equal]",
RowBox[{
SubscriptBox["C", "x"],
RowBox[{"\[Integral]",
RowBox[{
SubsuperscriptBox["\[Rho]", "\[Alpha]",
RowBox[{"4", "/", "3"}]], " ",
RowBox[{"\[DifferentialD]", "r"}]}]}]}]}]], "Input",
CellChangeTimes->{{3.782625494452404*^9, 3.782625542139269*^9}, {
3.7826257924766693`*^9, 3.782625804666955*^9}, {3.7826261512295113`*^9,
3.782626165142663*^9}},ExpressionUUID->"c8865179-994b-4123-9657-\
507eae2fd52f"],
Cell["This yields ", "Text",
CellChangeTimes->{{3.78262554551197*^9,
3.782625550141124*^9}},ExpressionUUID->"d352d13e-d96f-4f21-9cc5-\
32da8a2e376d"],
Cell[BoxData[
RowBox[{
RowBox[{
SubsuperscriptBox["\[CapitalEpsilon]", "x",
RowBox[{"(", "0", ")"}]], "\[Equal]",
RowBox[{"-",
FractionBox["8",
RowBox[{"3", "\[Pi]", " ", "R"}]]}], "\[Equal]",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]],
SuperscriptBox[
RowBox[{"(",
FractionBox["1",
RowBox[{"2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]], ")"}],
RowBox[{"4", "/", "3"}]], "2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]}], "\t", "\[Implies]", "\t",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]], "\[Equal]",
RowBox[{
RowBox[{"-",
FractionBox["4", "3"]}], " ",
SuperscriptBox[
RowBox[{"(",
FractionBox["2", "\[Pi]"], ")"}],
RowBox[{"1", "/", "3"}]]}]}]}]], "Input",
CellChangeTimes->{{3.782625551516776*^9, 3.782625586658499*^9},
3.7826256377657146`*^9, {3.782625680007085*^9, 3.782625692521583*^9}, {
3.782625807297289*^9, 3.782625835810197*^9}, {3.7826261728530617`*^9,
3.7826261808273573`*^9}},ExpressionUUID->"e2a90d7c-9765-4224-b6c1-\
0cb3ba56719e"],
Cell["The ground-state exchange functional for glomium is", "Text",
CellChangeTimes->{{3.782626083447263*^9,
3.782626096765339*^9}},ExpressionUUID->"4256ed0c-1e05-4553-a95a-\
370962e94038"],
Cell[BoxData[
FrameBox[
RowBox[{
RowBox[{
SubsuperscriptBox["e", "x",
RowBox[{"(", "0", ")"}]], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]], "\t",
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]]}], "\[Equal]",
RowBox[{
RowBox[{"-",
FractionBox["4", "3"]}], " ",
SuperscriptBox[
RowBox[{"(",
FractionBox["2", "\[Pi]"], ")"}],
RowBox[{"1", "/", "3"}]]}]}]]], "Input",
CellChangeTimes->{{3.782626098078829*^9, 3.782626129767681*^9},
3.7826261943234*^9},ExpressionUUID->"d7507220-76aa-479e-af51-0886ad0e4f5c"]
}, Open ]],
Cell[CellGroupData[{
Cell["Doubly-excited state exchange functional for glomium", "Subsection",
CellChangeTimes->{{3.782625187684318*^9, 3.782625205187807*^9}, {
3.7826253413716393`*^9, 3.7826253532482758`*^9}, {3.782625846786179*^9,
3.7826258696716547`*^9}},ExpressionUUID->"77af7129-5b6f-46d1-972e-\
088c04a397e1"],
Cell["\<\
Any doubly-excited state two-electron glomium system has the same uniform \
electron density as the ground state\
\>", "Text",
CellChangeTimes->{{3.78262523411093*^9, 3.782625281581828*^9}, {
3.782625873933075*^9,
3.782625893319673*^9}},ExpressionUUID->"137a2815-3e41-4c54-bb77-\
2baab3309f67"],
Cell[BoxData[
RowBox[{"\[Rho]", "\[Equal]",
RowBox[{
FractionBox["2",
RowBox[{"2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]], "\t",
SubscriptBox["\[Rho]", "\[Alpha]"]}], "\[Equal]",
SubscriptBox["\[Rho]", "\[Beta]"], "\[Equal]",
FractionBox["1",
RowBox[{"2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]]}]], "Input",
CellChangeTimes->{{3.7826252862043962`*^9, 3.7826253053578*^9}, {
3.782625777108284*^9,
3.7826257868938847`*^9}},ExpressionUUID->"ee1a49a3-d78d-48fd-b18b-\
b45c556268d6"],
Cell["\<\
The reduced HF energy of the first doubly-excited state of glomium is\
\>", "Text",
CellChangeTimes->{{3.782625207745901*^9, 3.7826252161060266`*^9}, {
3.7826254294319897`*^9, 3.782625432722817*^9}, {3.782625903968584*^9,
3.782625912039195*^9}},ExpressionUUID->"7f6139ea-87a7-485f-96a7-\
072785ed5a50"],
Cell[BoxData[
RowBox[{
SubsuperscriptBox["e", "HF",
RowBox[{"(", "1", ")"}]], "\[Equal]",
RowBox[{
FractionBox["3",
RowBox[{"2",
SuperscriptBox["R", "2"]}]], "+",
FractionBox["176",
RowBox[{"105", " ", "\[Pi]", " ", "R"}]]}]}]], "Input",
CellChangeTimes->{{3.7826252222025003`*^9, 3.782625227795396*^9}, {
3.782625336400528*^9, 3.7826253365779533`*^9}, 3.782625434707806*^9, {
3.782625929973528*^9, 3.782625972713728*^9}, {3.7826263054795427`*^9,
3.782626306892585*^9}},ExpressionUUID->"e2512605-e53f-4895-bc29-\
b005c70f53b5"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{
FractionBox["3",
RowBox[{"2",
SuperscriptBox["R", "2"]}]], "+",
FractionBox["176",
RowBox[{"105", " ", "\[Pi]", " ", "R"}]]}], "/.",
RowBox[{"R", "\[Rule]",
FractionBox["1",
RowBox[{
SuperscriptBox["\[Pi]",
RowBox[{"2", "/", "3"}]], " ",
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}]]}]}]], "Input",
CellChangeTimes->{{3.782672032733528*^9, 3.782672036312338*^9},
3.782672105472003*^9},
CellLabel->"In[6]:=",ExpressionUUID->"2caf6c5b-6ae4-41ea-a5bc-6e2553b311b0"],
Cell[BoxData[
RowBox[{
FractionBox[
RowBox[{"176", " ",
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}],
RowBox[{"105", " ",
SuperscriptBox["\[Pi]",
RowBox[{"1", "/", "3"}]]}]], "+",
RowBox[{
FractionBox["3", "2"], " ",
SuperscriptBox["\[Pi]",
RowBox[{"4", "/", "3"}]], " ",
SuperscriptBox["\[Rho]",
RowBox[{"2", "/", "3"}]]}]}]], "Output",
CellChangeTimes->{3.782672036741086*^9, 3.7826721062008343`*^9},
CellLabel->"Out[6]=",ExpressionUUID->"71a67885-8142-46b8-9292-410654424b59"]
}, Open ]],
Cell["This energy can be decomposed as", "Text",
CellChangeTimes->{{3.782625326608221*^9,
3.7826253316050653`*^9}},ExpressionUUID->"32015b28-f5cc-4eff-8d8d-\
b8691b888e8f"],
Cell[BoxData[
RowBox[{
SubsuperscriptBox["t", "s",
RowBox[{"(", "1", ")"}]], "\[Equal]",
RowBox[{
FractionBox["3",
RowBox[{"2",
SuperscriptBox["R", "2"]}]], "\t",
SubsuperscriptBox["e", "H",
RowBox[{"(", "1", ")"}]]}], "\[Equal]",
RowBox[{
FractionBox["352",
RowBox[{"105", " ", "\[Pi]", " ", "R"}]], "\t",
SubsuperscriptBox["e", "x",
RowBox[{"(", "1", ")"}]]}], "\[Equal]",
RowBox[{"-",
FractionBox["176",
RowBox[{"105", " ", "\[Pi]", " ", "R"}]]}]}]], "Input",
CellChangeTimes->{{3.78262533887177*^9, 3.782625382698719*^9}, {
3.78262543658897*^9, 3.78262544697281*^9}, {3.782625933482102*^9,
3.782625946143937*^9}, {3.7826259789921513`*^9, 3.782626004728611*^9}, {
3.782626310216545*^9,
3.7826263163180437`*^9}},ExpressionUUID->"1fcbd290-f3ea-43bd-b2ef-\
f31601137679"],
Cell["Knowing that the exchange functional has the following form", "Text",
CellChangeTimes->{{3.782625462881948*^9,
3.782625490540832*^9}},ExpressionUUID->"9b1e04fb-55cb-4e81-8daf-\
ba707c78dfce"],
Cell[BoxData[
RowBox[{
SubsuperscriptBox["\[CapitalEpsilon]",
RowBox[{"x", ",", "\[Alpha]"}],
RowBox[{"(", "1", ")"}]], "\[Equal]",
RowBox[{"\[Integral]",
RowBox[{
SubsuperscriptBox["e", "x",
RowBox[{"(", "1", ")"}]],
SubscriptBox["\[Rho]", "\[Alpha]"], " ",
RowBox[{"\[DifferentialD]", "r"}]}]}], "\[Equal]",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]],
RowBox[{"\[Integral]",
RowBox[{
SubsuperscriptBox["\[Rho]", "\[Alpha]",
RowBox[{"4", "/", "3"}]], " ",
RowBox[{"\[DifferentialD]", "r"}]}]}]}]}]], "Input",
CellChangeTimes->{{3.782625494452404*^9, 3.782625542139269*^9}, {
3.7826257924766693`*^9, 3.782625804666955*^9}, {3.782626419125482*^9,
3.782626430524787*^9}},ExpressionUUID->"c51ccbbc-5ee1-4cca-ade7-\
af230c7f6ba6"],
Cell["This yields ", "Text",
CellChangeTimes->{{3.78262554551197*^9,
3.782625550141124*^9}},ExpressionUUID->"8bfb35af-a4e4-4813-ae15-\
3703f4aeab96"],
Cell[BoxData[
RowBox[{
RowBox[{
SubsuperscriptBox["\[CapitalEpsilon]", "x",
RowBox[{"(", "1", ")"}]], "\[Equal]",
RowBox[{"-",
FractionBox["176",
RowBox[{"105", " ", "\[Pi]", " ", "R"}]]}], "\[Equal]",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]],
SuperscriptBox[
RowBox[{"(",
FractionBox["1",
RowBox[{"2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]], ")"}],
RowBox[{"4", "/", "3"}]], "2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]}], "\t", "\[Implies]", "\t",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]], "\[Equal]",
RowBox[{
RowBox[{"-",
FractionBox["176", "105"]}], " ",
SuperscriptBox[
RowBox[{"(",
FractionBox["2", "\[Pi]"], ")"}],
RowBox[{"1", "/", "3"}]]}]}]}]], "Input",
CellChangeTimes->{{3.782625551516776*^9, 3.782625586658499*^9},
3.7826256377657146`*^9, {3.782625680007085*^9, 3.782625692521583*^9}, {
3.782625807297289*^9, 3.782625835810197*^9}, {3.7826260201249437`*^9,
3.782626047402648*^9}, {3.7826264334320097`*^9,
3.782626440114307*^9}},ExpressionUUID->"60e340d1-6bd1-4928-9814-\
51c59353edd1"],
Cell["The double-excited state exchange functional for glomium is", "Text",
CellChangeTimes->{{3.782626083447263*^9, 3.782626096765339*^9}, {
3.782626446351823*^9,
3.782626452345066*^9}},ExpressionUUID->"f5f45693-a80c-4a46-ad73-\
07811620c0d9"],
Cell[BoxData[
FrameBox[
RowBox[{
RowBox[{
SubsuperscriptBox["e", "x",
RowBox[{"(", "1", ")"}]], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]], "\t",
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]]}], "\[Equal]",
RowBox[{
RowBox[{"-",
FractionBox["176", "105"]}], " ",
SuperscriptBox[
RowBox[{"(",
FractionBox["2", "\[Pi]"], ")"}],
RowBox[{"1", "/", "3"}]]}]}]]], "Input",
CellChangeTimes->{{3.782626098078829*^9, 3.782626129767681*^9},
3.7826261943234*^9, {3.782626456225254*^9, 3.782626460826289*^9},
3.7826266660833883`*^9},ExpressionUUID->"f99857fd-0430-402d-a733-\
929b6b359e23"]
}, Open ]],
Cell[CellGroupData[{
Cell["Weight-dependent exchange functional for glomium", "Subsection",
CellChangeTimes->{{3.782625187684318*^9, 3.782625205187807*^9}, {
3.7826253413716393`*^9, 3.7826253532482758`*^9}, {3.782625846786179*^9,
3.7826258696716547`*^9}, {3.782626481389626*^9,
3.7826264948791018`*^9}},ExpressionUUID->"ef62b877-dbc0-41f4-bb50-\
f7bab5069a50"],
Cell["\<\
We can now combine these two exchange functionals to create a \
weight-dependent exchange functional\
\>", "Text",
CellChangeTimes->{{3.7826264968248787`*^9,
3.782626540001565*^9}},ExpressionUUID->"5b10be0f-a732-422f-9446-\
681c78cbd103"],
Cell[BoxData[
RowBox[{
RowBox[{
SubsuperscriptBox["e", "x", "w"], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"1", "-", "w"}], ")"}],
RowBox[{
SubsuperscriptBox["e", "x",
RowBox[{"(", "0", ")"}]], "[", "\[Rho]", "]"}]}], "+",
RowBox[{"w", " ",
RowBox[{
SubsuperscriptBox["e", "x",
RowBox[{"(", "1", ")"}]], "[", "\[Rho]", "]"}]}]}]}]], "Input",
CellChangeTimes->{{3.782626542606078*^9,
3.7826265610373163`*^9}},ExpressionUUID->"009be4dc-79a6-499d-b9af-\
ddbb9078f85a"],
Cell[BoxData[{
RowBox[{
RowBox[{
SubsuperscriptBox["e", "x", "w"], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"1", "-", "w"}], ")"}],
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}], "+",
RowBox[{"w", " ",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]], "[", "\[Rho]", "]"}],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}]}]}], "\[IndentingNewLine]",
RowBox[{" ",
RowBox[{"\[Equal]", " ",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "w", ")"}]],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}]}]}]}], "Input",
CellChangeTimes->{{3.782626542606078*^9, 3.782626608609075*^9}, {
3.782627105138702*^9,
3.782627108847715*^9}},ExpressionUUID->"afd8bd15-999f-411c-a673-\
53d5b75e5201"],
Cell["The weight dependent exchange functional for glomium is then", "Text",
CellChangeTimes->{{3.782626083447263*^9, 3.782626096765339*^9}, {
3.782626446351823*^9, 3.782626452345066*^9}, {3.782626617430833*^9,
3.782626634809102*^9}},ExpressionUUID->"1fe4835f-3b82-458a-9e2a-\
9272922b081a"],
Cell[BoxData[
FrameBox[
RowBox[{
RowBox[{
SubsuperscriptBox["e", "x", "w"], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
SubsuperscriptBox["C", "x", "w"],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]], "\t",
SubsuperscriptBox["C", "x", "w"]}], "\[Equal]",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"1", "-", "w"}], ")"}],
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]]}], "+",
RowBox[{"w", " ",
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]], "\t",
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]]}]}], "\[Equal]",
RowBox[{
RowBox[{"-",
FractionBox["4", "3"]}], " ",
SuperscriptBox[
RowBox[{"(",
FractionBox["2", "\[Pi]"], ")"}],
RowBox[{"1", "/", "3"}]], "\t",
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]]}], "\[Equal]",
RowBox[{
RowBox[{"-",
FractionBox["176", "105"]}], " ",
SuperscriptBox[
RowBox[{"(",
FractionBox["2", "\[Pi]"], ")"}],
RowBox[{"1", "/", "3"}]]}]}]]], "Input",
CellChangeTimes->{{3.782626098078829*^9, 3.782626129767681*^9},
3.7826261943234*^9, {3.782626456225254*^9, 3.782626460826289*^9}, {
3.782626637071652*^9,
3.782626680771*^9}},ExpressionUUID->"b5e6e435-424a-4f3e-8416-709ae4356a73"],
Cell[TextData[{
StyleBox["Amazingly, the weight dependence of the exchange functional can be \
transfered to the ",
FontColor->RGBColor[1, 0, 0]],
Cell[BoxData[
FormBox[
SubscriptBox["C", "x"], TraditionalForm]],
FontColor->RGBColor[1, 0, 0],ExpressionUUID->
"fa14d797-4f81-4166-97c2-b14e12078e3f"],
StyleBox[" coefficient.\nThis is obvious but kind of nice.",
FontColor->RGBColor[1, 0, 0]]
}], "Text",
CellChangeTimes->{{3.7826266896125298`*^9,
3.782626751796768*^9}},ExpressionUUID->"d05917ae-a767-4a78-834a-\
c4b18c1c0b13"]
}, Open ]],
Cell[CellGroupData[{
Cell["\<\
LDA-centered weight-dependent exchange functional for molecules\
\>", "Subsection",
CellChangeTimes->{{3.782625187684318*^9, 3.782625205187807*^9}, {
3.7826253413716393`*^9, 3.7826253532482758`*^9}, {3.782625846786179*^9,
3.7826258696716547`*^9}, {3.782626481389626*^9, 3.7826264948791018`*^9}, {
3.782626761697323*^9,
3.78262677063722*^9}},ExpressionUUID->"bcd158fd-9aa7-4c7b-837e-\
afd37c54f55a"],
Cell["\<\
In order to create a more \[OpenCurlyDoubleQuote]universal\
\[CloseCurlyDoubleQuote] functional (that does not depend on the number of \
electrons in particular), we can \[OpenCurlyDoubleQuote]shift\
\[CloseCurlyDoubleQuote] the weight-dependent exchange functional designed \
for glomium to make it LDA centered.
The corresponding definition of the so-called LDA-centered weight-dependent \
exchange functional is then\
\>", "Text",
CellChangeTimes->{{3.782626772664956*^9,
3.7826268526294622`*^9}},ExpressionUUID->"a0112f10-d580-4370-a6ed-\
bff98b24a4a6"],
Cell[BoxData[{
RowBox[{
RowBox[{
SubsuperscriptBox[
OverscriptBox["e", "_"], "x", "w"], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"1", "-", "w"}], ")"}],
RowBox[{"(",
RowBox[{
RowBox[{
SubsuperscriptBox["e", "x",
RowBox[{"(", "0", ")"}]], "[", "\[Rho]", "]"}], "+",
RowBox[{"(",
RowBox[{
RowBox[{
SubsuperscriptBox["e", "x", "LDA"], "[", "\[Rho]", "]"}], "-",
RowBox[{
SubsuperscriptBox["e", "x",
RowBox[{"(", "0", ")"}]], "[", "\[Rho]", "]"}]}], ")"}]}], ")"}]}],
"+",
RowBox[{"w", " ",
RowBox[{"(",
RowBox[{
RowBox[{
SubsuperscriptBox["e", "x",
RowBox[{"(", "1", ")"}]], "[", "\[Rho]", "]"}], "+",
RowBox[{"(",
RowBox[{
RowBox[{
SubsuperscriptBox["e", "x", "LDA"], "[", "\[Rho]", "]"}], "-",
RowBox[{
SubsuperscriptBox["e", "x",
RowBox[{"(", "0", ")"}]], "[", "\[Rho]", "]"}]}], ")"}]}],
")"}]}]}]}], "\[IndentingNewLine]",
RowBox[{"\t ",
RowBox[{"\[Equal]",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"1", "-", "w"}], ")"}],
RowBox[{"(",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]], "+",
SubsuperscriptBox["C", "x", "LDA"], "-",
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]]}], ")"}],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}], "+",
RowBox[{"w", " ",
RowBox[{"(",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]], "+",
SubsuperscriptBox["C", "x", "LDA"], "-",
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]]}], ")"}],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}]}]}]}], "\[IndentingNewLine]",
RowBox[{"\t ",
RowBox[{"\[Equal]",
RowBox[{
RowBox[{"(",
RowBox[{
SubsuperscriptBox["C", "x", "LDA"], "+",
RowBox[{"w", " ",
RowBox[{"(",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]], "-",
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]]}], ")"}]}]}], ")"}],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}]}]}], "\[IndentingNewLine]",
RowBox[{"\t ",
RowBox[{"\[Equal]",
RowBox[{
SubsuperscriptBox[
OverscriptBox["C", "_"], "x", "w"],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}]}]}]}], "Input",
CellChangeTimes->{{3.782627011728672*^9, 3.782627035640417*^9}, {
3.7826270657535143`*^9, 3.782627143829431*^9}, {3.782627423090889*^9,
3.782627582604439*^9}},ExpressionUUID->"aabf1284-1a62-408f-8d1c-\
50c99b5acc35"],
Cell["\<\
where we recall that the usual LDA exchange functional (D30) is given by \
\>", "Text",
CellChangeTimes->{{3.782627269045208*^9, 3.7826272742216253`*^9}, {
3.7826273182152567`*^9,
3.7826273287284517`*^9}},ExpressionUUID->"88a4bd50-0ba7-4f7c-9fc0-\
e0feb01831fc"],
Cell[BoxData[
RowBox[{
RowBox[{
SubsuperscriptBox["e", "x", "LDA"], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
SubsuperscriptBox["C", "x", "LDA"],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]], "\t",
SubsuperscriptBox["C", "x", "LDA"]}], "\[Equal]",
RowBox[{
RowBox[{"-",
FractionBox["3", "2"]}],
SuperscriptBox[
RowBox[{"(",
FractionBox["3",
RowBox[{"4", "\[Pi]"}]], ")"}],
RowBox[{"1", "/", "3"}]]}]}]], "Input",
CellChangeTimes->{{3.782627279816565*^9,
3.782627314689026*^9}},ExpressionUUID->"a889f096-9a50-426c-9cf3-\
81c363ec2eaf"],
Cell["\<\
The LDA-centered weight dependent exchange functional is then\
\>", "Text",
CellChangeTimes->{{3.782626083447263*^9, 3.782626096765339*^9}, {
3.782626446351823*^9, 3.782626452345066*^9}, {3.782626617430833*^9,
3.782626634809102*^9}, {3.782627156867894*^9,
3.782627163608507*^9}},ExpressionUUID->"d3c69495-05ef-49fe-8484-\
fa7fd62cc408"],
Cell[BoxData[
FrameBox[
RowBox[{
RowBox[{
SubsuperscriptBox[
OverscriptBox["e", "_"], "x", "w"], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
SubsuperscriptBox[
OverscriptBox["C", "_"], "x", "w"],
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]], "\t",
SubsuperscriptBox[
OverscriptBox["C", "_"], "x", "w"]}], "\[Equal]",
RowBox[{
SubsuperscriptBox["C", "x", "LDA"], "+",
RowBox[{"w", " ",
RowBox[{"(",
RowBox[{
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]], "-",
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]]}], ")"}], "\t",
SubsuperscriptBox["C", "x",
RowBox[{"(", "0", ")"}]]}]}], "\[Equal]",
RowBox[{
RowBox[{"-",
FractionBox["4", "3"]}], " ",
SuperscriptBox[
RowBox[{"(",
FractionBox["2", "\[Pi]"], ")"}],
RowBox[{"1", "/", "3"}]], "\t",
SubsuperscriptBox["C", "x",
RowBox[{"(", "1", ")"}]]}], "\[Equal]",
RowBox[{
RowBox[{"-",
FractionBox["176", "105"]}], " ",
SuperscriptBox[
RowBox[{"(",
FractionBox["2", "\[Pi]"], ")"}],
RowBox[{"1", "/", "3"}]], "\t",
SubsuperscriptBox["C", "x", "LDA"]}], "\[Equal]",
RowBox[{
RowBox[{"-",
FractionBox["3", "2"]}],
SuperscriptBox[
RowBox[{"(",
FractionBox["3",
RowBox[{"4", "\[Pi]"}]], ")"}],
RowBox[{"1", "/", "3"}]]}]}]]], "Input",
CellChangeTimes->{{3.782626098078829*^9, 3.782626129767681*^9},
3.7826261943234*^9, {3.782626456225254*^9, 3.782626460826289*^9}, {
3.782626637071652*^9, 3.782626680771*^9}, {3.782627174977717*^9,
3.7826271819954042`*^9}, {3.782627246703974*^9, 3.7826272510149527`*^9}, {
3.782627360793895*^9, 3.78262736318638*^9}, {3.782627591509573*^9,
3.7826276031945353`*^9}},ExpressionUUID->"0138b572-760b-4f82-a618-\
73197e776fa2"],
Cell[TextData[StyleBox["This is the final form of our functional. The Cx \
coefficient is weight dependent and it appears as a correction over Dirac\
\[CloseCurlyQuote]s Cx coefficient.",
FontColor->RGBColor[1, 0, 0]]], "Text",
CellChangeTimes->{{3.7826266896125298`*^9, 3.782626751796768*^9}, {
3.782627617737678*^9, 3.7826277263447857`*^9}, {3.782627817686021*^9,
3.78262782048987*^9}},ExpressionUUID->"b83db319-04e5-42b8-ae4c-\
57f93abe09fd"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["eLDA correlation", "Section",
CellChangeTimes->{{3.7826249982517223`*^9, 3.782625006542748*^9}, {
3.7826250653667927`*^9, 3.7826250672414913`*^9},
3.782625112301036*^9},ExpressionUUID->"258ca12d-c8c5-4bc9-84ec-\
1711f7860786"],
Cell[CellGroupData[{
Cell["Ground state correlation functional for glomium", "Subsection",
CellChangeTimes->{{3.782625187684318*^9, 3.782625205187807*^9}, {
3.7826253413716393`*^9, 3.7826253532482758`*^9}, {3.782625846786179*^9,
3.782625848068905*^9}, {3.7826293000392017`*^9,
3.7826293054927187`*^9}},ExpressionUUID->"ee1991f0-6627-4afa-871c-\
73710071f6fd"],
Cell["\<\
The ground-state correlation energy of glomium can be very accurately \
computed with Hylleraas-type calculations.
We use a Pade-type fit to obtain the functional which reads\
\>", "Text",
CellChangeTimes->{{3.782629312827221*^9,
3.782629358270515*^9}},ExpressionUUID->"a6c10556-4f53-4da2-8cf0-\
79bbf3ec1242"],
Cell[BoxData[
FrameBox[
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c",
RowBox[{"(", "0", ")"}]], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
FractionBox[
SuperscriptBox["a",
RowBox[{"(", "0", ")"}]],
RowBox[{"1", "+",
RowBox[{
SuperscriptBox["b",
RowBox[{"(", "0", ")"}]],
SuperscriptBox["\[Rho]",
RowBox[{
RowBox[{"-", "1"}], "/", "6"}]]}], "+",
RowBox[{
SuperscriptBox["c",
RowBox[{"(", "0", ")"}]],
SuperscriptBox["\[Rho]",
RowBox[{
RowBox[{"-", "1"}], "/", "3"}]]}]}]], "\t",
SuperscriptBox["a",
RowBox[{"(", "0", ")"}]]}], "\[Equal]",
RowBox[{
RowBox[{"-", "0.0238184"}], "\t",
SuperscriptBox["b",
RowBox[{"(", "0", ")"}]]}], "\[Equal]",
RowBox[{"0.00575719", "\t",
SuperscriptBox["c",
RowBox[{"(", "0", ")"}]]}], "\[Equal]", "0.0830576"}]]], "Input",
CellChangeTimes->{{3.782626098078829*^9, 3.782626129767681*^9},
3.7826261943234*^9, {3.782626456225254*^9, 3.782626460826289*^9},
3.7826266660833883`*^9, {3.782629374026113*^9,
3.78262943004636*^9}},ExpressionUUID->"1885d9ca-6bb6-490d-a0bb-\
a3b96576a614"],
Cell["\<\
The first coefficient is actually not a fitting coefficient but is obtained \
via the exact value of the correlation energy is the high-density limit.\
\>", "Text",
CellChangeTimes->{{3.782629456743466*^9,
3.782629484288484*^9}},ExpressionUUID->"ead26072-79d4-4549-aab6-\
46fe8cf9b6db"]
}, Open ]],
Cell[CellGroupData[{
Cell["Doubly-excited state correlation functional for glomium", "Subsection",
CellChangeTimes->{{3.782625187684318*^9, 3.782625205187807*^9}, {
3.7826253413716393`*^9, 3.7826253532482758`*^9}, {3.782625846786179*^9,
3.7826258696716547`*^9}, {3.782629497369335*^9,
3.782629498706218*^9}},ExpressionUUID->"ac951d60-fbce-44ba-8f12-\
9e5545df5beb"],
Cell["\<\
Similarly, the correlation functional for the first doubly-excited states is\
\>", "Text",
CellChangeTimes->{{3.7826295051913967`*^9,
3.782629522030589*^9}},ExpressionUUID->"dca5790f-2358-43e0-b359-\
677842457a7d"],
Cell[BoxData[
FrameBox[
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c",
RowBox[{"(", "1", ")"}]], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
FractionBox[
SuperscriptBox["a",
RowBox[{"(", "1", ")"}]],
RowBox[{"1", "+",
RowBox[{
SuperscriptBox["b",
RowBox[{"(", "1", ")"}]],
SuperscriptBox["\[Rho]",
RowBox[{
RowBox[{"-", "1"}], "/", "6"}]]}], "+",
RowBox[{
SuperscriptBox["c",
RowBox[{"(", "1", ")"}]],
SuperscriptBox["\[Rho]",
RowBox[{
RowBox[{"-", "1"}], "/", "3"}]]}]}]], "\t",
SuperscriptBox["a",
RowBox[{"(", "1", ")"}]]}], "\[Equal]",
RowBox[{
RowBox[{"-", "0.0144633"}], "\t",
SuperscriptBox["b",
RowBox[{"(", "1", ")"}]]}], "\[Equal]",
RowBox[{
RowBox[{"-", "0.0504501"}], "\t",
SuperscriptBox["c",
RowBox[{"(", "1", ")"}]]}], "\[Equal]", "0.0331287"}]]], "Input",
CellChangeTimes->{{3.782626098078829*^9, 3.782626129767681*^9},
3.7826261943234*^9, {3.782626456225254*^9, 3.782626460826289*^9},
3.7826266660833883`*^9, {3.782629374026113*^9, 3.782629446040477*^9}, {
3.782629575500882*^9,
3.782629578454645*^9}},ExpressionUUID->"cce86f90-52f7-41f4-a63d-\
73a6c518b4c3"]
}, Open ]],
Cell[CellGroupData[{
Cell["Weight-dependent exchange functional for glomium", "Subsection",
CellChangeTimes->{{3.782625187684318*^9, 3.782625205187807*^9}, {
3.7826253413716393`*^9, 3.7826253532482758`*^9}, {3.782625846786179*^9,
3.7826258696716547`*^9}, {3.782626481389626*^9,
3.7826264948791018`*^9}},ExpressionUUID->"9f2ce783-2740-403c-974c-\
46a7ef9321fe"],
Cell["\<\
We can now combine these two correlations functionals to create a \
weight-dependent correlation functional\
\>", "Text",
CellChangeTimes->{{3.7826264968248787`*^9, 3.782626540001565*^9}, {
3.782629540323526*^9,
3.7826295457859783`*^9}},ExpressionUUID->"6c7b6dc7-06c2-4f95-a398-\
fe8f7adea7db"],
Cell[BoxData[
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c", "w"], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"1", "-", "w"}], ")"}],
RowBox[{
SubsuperscriptBox["e", "c",
RowBox[{"(", "0", ")"}]], "[", "\[Rho]", "]"}]}], "+",
RowBox[{"w", " ",
RowBox[{
SubsuperscriptBox["e", "c",
RowBox[{"(", "1", ")"}]], "[", "\[Rho]", "]"}]}]}]}]], "Input",
CellChangeTimes->{{3.782626542606078*^9, 3.7826265610373163`*^9}, {
3.782629548814664*^9,
3.782629551048087*^9}},ExpressionUUID->"c5451bea-41ca-4be3-9429-\
a6d8ddd06c4a"]
}, Open ]],
Cell[CellGroupData[{
Cell["\<\
LDA-centered weight-dependent correlation functional for molecules\
\>", "Subsection",
CellChangeTimes->{{3.782625187684318*^9, 3.782625205187807*^9}, {
3.7826253413716393`*^9, 3.7826253532482758`*^9}, {3.782625846786179*^9,
3.7826258696716547`*^9}, {3.782626481389626*^9, 3.7826264948791018`*^9}, {
3.782626761697323*^9, 3.78262677063722*^9}, {3.782629584583153*^9,
3.782629586157028*^9}},ExpressionUUID->"ab8ea4d9-6951-44dc-96fc-\
10c74ccd1a4f"],
Cell["\<\
The corresponding LDA-centered weight-dependent correlation functional is then\
\>", "Text",
CellChangeTimes->{{3.782626772664956*^9, 3.7826268526294622`*^9}, {
3.782629595887004*^9,
3.782629603557575*^9}},ExpressionUUID->"f0dac186-7114-4a5e-b53c-\
f938d8c4fd37"],
Cell[BoxData[
RowBox[{
RowBox[{
SubsuperscriptBox[
OverscriptBox["e", "_"], "c", "w"], "[", "\[Rho]", "]"}], "\[Equal]",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{"1", "-", "w"}], ")"}],
RowBox[{"(",
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c",
RowBox[{"(", "0", ")"}]], "[", "\[Rho]", "]"}], "+",
RowBox[{"(",
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c", "LDA"], "[", "\[Rho]", "]"}], "-",
RowBox[{
SubsuperscriptBox["e", "c",
RowBox[{"(", "0", ")"}]], "[", "\[Rho]", "]"}]}], ")"}]}], ")"}]}],
"+",
RowBox[{"w", " ",
RowBox[{"(",
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c",
RowBox[{"(", "1", ")"}]], "[", "\[Rho]", "]"}], "+",
RowBox[{"(",
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c", "LDA"], "[", "\[Rho]", "]"}], "-",
RowBox[{
SubsuperscriptBox["e", "c",
RowBox[{"(", "0", ")"}]], "[", "\[Rho]", "]"}]}], ")"}]}],
")"}]}]}]}]], "Input",
CellChangeTimes->{{3.782627011728672*^9, 3.782627035640417*^9}, {
3.7826270657535143`*^9, 3.782627143829431*^9}, {3.782627423090889*^9,
3.782627582604439*^9}, {3.782629605968178*^9,
3.782629619974502*^9}},ExpressionUUID->"2ca8802e-361f-4487-a3f0-\
67d18fc8fe5f"],
Cell["\<\
where we use the VWN5 functional as the LDA correlation functional (cf DFAs \
section)\
\>", "Text",
CellChangeTimes->{{3.782627269045208*^9, 3.7826272742216253`*^9}, {
3.7826273182152567`*^9, 3.7826273287284517`*^9}, {3.782629622259294*^9,
3.782629656207426*^9}, {3.782629733745845*^9,
3.782629760414081*^9}},ExpressionUUID->"00fbe382-6c25-4893-a74d-\
546919a831b8"],
Cell[BoxData[
RowBox[{
RowBox[{
SubsuperscriptBox["e", "c", "LDA"], "[", "\[Rho]", "]"}], "\[Congruent]",
RowBox[{
SubsuperscriptBox["e", "c", "VWN5"], "[", "\[Rho]", "]"}]}]], "Input",
CellChangeTimes->{{3.782627279816565*^9, 3.782627314689026*^9}, {
3.782629630477713*^9,
3.7826296413735*^9}},ExpressionUUID->"b7e98868-257f-43e4-b057-ff1cbb083046"],
Cell[TextData[StyleBox["Note that this functional is only valid for \
closed-shell system.\nIf one wants to extend this to open-shell system, one \
needs to take into account spin polarization.\nThere are several ways of \
doing this...\nThe extension to GGAs is also more tricky as the GGA is not \
usual a multiplicative factor compared to the LDA functional.",
FontColor->RGBColor[1, 0, 0]]], "Text",
CellChangeTimes->{{3.7826266896125298`*^9, 3.782626751796768*^9}, {
3.782627617737678*^9, 3.7826277263447857`*^9}, {3.782627817686021*^9,
3.78262782048987*^9}, {3.782629782084593*^9, 3.782629838089308*^9}, {
3.782629871341579*^9,
3.7826299046888933`*^9}},ExpressionUUID->"336e8167-7e94-4334-9956-\
c0652866375e"],
Cell[BoxData[
RowBox[{
RowBox[{"With", "[",
RowBox[{
RowBox[{"{",
RowBox[{"M", "=", "20"}], "}"}], ",", "\[IndentingNewLine]",
RowBox[{
RowBox[{
RowBox[{
SubscriptBox["\[Psi]", "n_"], "[", "u_", "]"}], "=",
SuperscriptBox["u",
RowBox[{"n", "-", "1"}]]}], ";", "\[IndentingNewLine]",
RowBox[{"H", "=",
RowBox[{"Table", "[",
RowBox[{
RowBox[{
SubsuperscriptBox["\[Integral]", "0",
RowBox[{"2", "R"}]],
RowBox[{
RowBox[{
SubscriptBox["\[Psi]", "i"], "[", "u", "]"}],
RowBox[{"(",
RowBox[{
RowBox[{
RowBox[{"(",
RowBox[{
FractionBox[
SuperscriptBox["u", "2"],
RowBox[{"4",
SuperscriptBox["R", "2"]}]], "-", "1"}], ")"}],
RowBox[{
SubsuperscriptBox["\[Psi]", "j", "\[DoublePrime]"], "[", "u",
"]"}]}], "+",
RowBox[{
RowBox[{"(",
RowBox[{
FractionBox[
RowBox[{"5", "u"}],
RowBox[{"4",
SuperscriptBox["R", "2"]}]], "-",
FractionBox["2", "u"]}], ")"}], " ",
RowBox[{
SubsuperscriptBox["\[Psi]", "j", "\[Prime]"], "[", "u", "]"}]}],
"+",
RowBox[{
FractionBox["1", "u"],
RowBox[{
SubscriptBox["\[Psi]", "j"], "[", "u", "]"}]}]}], ")"}],
SuperscriptBox["u", "2"], " ",
SqrtBox[
RowBox[{"1", "-",
FractionBox[
SuperscriptBox["u", "2"],
RowBox[{"4", " ",
SuperscriptBox["R", "2"]}]]}]],
RowBox[{"\[DifferentialD]", "u"}]}]}], ",",
RowBox[{"{",
RowBox[{"i", ",", "M"}], "}"}], ",",
RowBox[{"{",
RowBox[{"j", ",", "M"}], "}"}]}], "]"}]}], ";",
"\[IndentingNewLine]",
RowBox[{"S", "=",
RowBox[{"Table", "[",
RowBox[{
RowBox[{
SubsuperscriptBox["\[Integral]", "0",
RowBox[{"2", "R"}]],
RowBox[{
RowBox[{
SubscriptBox["\[Psi]", "i"], "[", "u", "]"}],
RowBox[{
SubscriptBox["\[Psi]", "j"], "[", "u", "]"}],
SuperscriptBox["u", "2"], " ",
SqrtBox[
RowBox[{"1", "-",
FractionBox[
SuperscriptBox["u", "2"],
RowBox[{"4", " ",
SuperscriptBox["R", "2"]}]]}]],
RowBox[{"\[DifferentialD]", "u"}]}]}], ",",
RowBox[{"{",
RowBox[{"i", ",", "M"}], "}"}], ",",
RowBox[{"{",
RowBox[{"j", ",", "M"}], "}"}]}], "]"}]}], ";"}]}],
"\[IndentingNewLine]", "]"}], ";"}]], "Input",
CellChangeTimes->{{3.742918604620111*^9, 3.7429186131351423`*^9},
3.742918719955948*^9, {3.74291885295679*^9, 3.742918854648775*^9},
3.742922362967875*^9, 3.742980439148241*^9, 3.742980540306929*^9, {
3.742994989188118*^9, 3.742995016518692*^9}, {3.742995150772037*^9,
3.7429951513978567`*^9}, 3.7429952985264072`*^9, {3.74299538074587*^9,
3.7429953808123302`*^9}, {3.742996132486848*^9, 3.7429961349036427`*^9},
3.7429963078815823`*^9, {3.742996660743409*^9, 3.742996660843338*^9}, {
3.742996945034573*^9, 3.742996945161457*^9}, 3.7430081967857656`*^9,
3.743009059579871*^9, 3.743009234820531*^9, 3.743009382026333*^9, {
3.7430094920285807`*^9, 3.74300951455619*^9}, 3.74300961041049*^9,
3.743009666438663*^9, {3.7430098539824047`*^9, 3.743009854170718*^9},
3.743009922023739*^9, 3.743134546286873*^9, 3.743134666275275*^9, {
3.74322049794233*^9, 3.7432204980603647`*^9}, 3.743249005112269*^9, {
3.743249116887512*^9, 3.743249129269992*^9}, {3.743249272993807*^9,
3.743249285356189*^9}, {3.743250236465829*^9, 3.743250264555167*^9},
3.743252170343741*^9, {3.748320687040152*^9, 3.7483207268379183`*^9},
3.748320783735428*^9, {3.748361427987043*^9, 3.748361428073436*^9}, {
3.753773332985786*^9, 3.753773347232463*^9}, {3.7827277659801283`*^9,
3.7827277718121433`*^9}},
CellLabel->"In[4]:=",ExpressionUUID->"8d489c13-936e-4272-b8f6-29963888bb9f"],
Cell[BoxData[{
RowBox[{
RowBox[{"Eig0", "=",
RowBox[{"Table", "[",
RowBox[{
RowBox[{"{",
RowBox[{"r", ",",
RowBox[{
RowBox[{
FractionBox["1", "2"],
RowBox[{
RowBox[{"Sort", "[",
RowBox[{"Eigenvalues", "[",
RowBox[{"{",
RowBox[{
RowBox[{"N", "[",
RowBox[{
RowBox[{"H", "/.",
RowBox[{"R", "\[Rule]", "r"}]}], ",", "100"}], "]"}], ",",
RowBox[{"N", "[",
RowBox[{
RowBox[{"S", "/.",
RowBox[{"R", "\[Rule]", "r"}]}], ",", "100"}], "]"}]}],
"}"}], "]"}], "]"}], "\[LeftDoubleBracket]", "1",
"\[RightDoubleBracket]"}]}], "-",
RowBox[{"(",
FractionBox["4",
RowBox[{"3", " ", "\[Pi]", " ", "r"}]], ")"}]}]}], "}"}], ",",
RowBox[{"{",
RowBox[{"r", ",",
RowBox[{"{",
RowBox[{
FractionBox["1", "10"], ",",
FractionBox["1", "5"], ",",
FractionBox["1", "2"], ",", "1", ",", "2", ",", "5", ",", "10", ",",
"20", ",", "50", ",", "100"}], "}"}]}], "}"}]}], "]"}]}],
";"}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"Eig1", "=",
RowBox[{"Table", "[",
RowBox[{
RowBox[{"{",
RowBox[{"r", ",",
RowBox[{
RowBox[{
FractionBox["1", "2"],
RowBox[{
RowBox[{"Sort", "[",
RowBox[{"Eigenvalues", "[",
RowBox[{"{",
RowBox[{
RowBox[{"N", "[",
RowBox[{
RowBox[{"H", "/.",
RowBox[{"R", "\[Rule]", "r"}]}], ",", "100"}], "]"}], ",",
RowBox[{"N", "[",
RowBox[{
RowBox[{"S", "/.",
RowBox[{"R", "\[Rule]", "r"}]}], ",", "100"}], "]"}]}],
"}"}], "]"}], "]"}], "\[LeftDoubleBracket]", "2",
"\[RightDoubleBracket]"}]}], "-",
RowBox[{"(",
RowBox[{
FractionBox["3",
RowBox[{"2",
SuperscriptBox["r", "2"]}]], "+",
FractionBox["176",
RowBox[{"105", " ", "\[Pi]", " ", "r"}]]}], ")"}]}]}], "}"}], ",",
RowBox[{"{",
RowBox[{"r", ",",
RowBox[{"{",
RowBox[{
FractionBox["1", "10"], ",",
FractionBox["1", "5"], ",",
FractionBox["1", "2"], ",", "1", ",", "2", ",", "5", ",", "10", ",",
"20", ",", "50", ",", "1000"}], "}"}]}], "}"}]}], "]"}]}],
";"}]}], "Input",
CellChangeTimes->{{3.78271083027982*^9, 3.782710830463963*^9}, {
3.782727157320765*^9, 3.782727160418681*^9}, {3.782727197346158*^9,
3.7827272500340633`*^9}, {3.782727665220646*^9, 3.782727667067819*^9}, {
3.782729309834597*^9, 3.782729356516032*^9}},
CellLabel->"In[11]:=",ExpressionUUID->"af4fe8f8-c3eb-4495-a149-b4bd1830693b"],
Cell[CellGroupData[{
Cell[BoxData[{"Eig0", "\[IndentingNewLine]", "Eig1"}], "Input",
CellChangeTimes->{{3.782727737034244*^9, 3.782727738289781*^9}, {
3.782729308468261*^9, 3.782729308769576*^9}, {3.782729365139875*^9,
3.7827293677139893`*^9}},
CellLabel->"In[14]:=",ExpressionUUID->"53e9768f-d789-47f5-820c-8dfbf0b14efb"],
Cell[BoxData[
RowBox[{"{",
RowBox[{
RowBox[{"{",
RowBox[{
FractionBox["1", "10"], ",",
RowBox[{
"-", "0.02339191336882963545044265285591406983139444634921910976840013882\
56953922110387472065630809604443573392133004949331`97.74367715673696"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{
FractionBox["1", "5"], ",",
RowBox[{
"-", "0.02297901655935039101295412026620162141571705296745076066046658132\
48834205779522345892528398130248617853218753737683`98.03930102160574"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{
FractionBox["1", "2"], ",",
RowBox[{
"-", "0.02181711610532083888602293483539682997722072405179322009717415052\
95474616275729043980295275056673766654385130233253`98.42128697728288"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"1", ",",
RowBox[{
"-", "0.02010876852270148367649542200116664623743451705584995206693144096\
23494269685316612029992332471221233565142795299272`98.69667699310243"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"2", ",",
RowBox[{
"-", "0.01737071560943161152923210852432991464124470120775155316932578502\
95565706528155610500399662073146644638727472446797`98.9501487834867"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"5", ",",
RowBox[{
"-", "0.01235932606980560593688171522824266216069711458773985091120220853\
41912627539958127666653152453350733142842635147608`99.23151717139389"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"10", ",",
RowBox[{
"-", "0.00843639446517865578251244610350507660125512863737716451821926116\
47283509834858938014719351849733383919632313474685`99.3946150734676"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"20", ",",
RowBox[{
"-", "0.00525691078329769832390278959039984853783898074754459164844639134\
46698230623061254357883113338275339961099123057183`99.51759573499423"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"50", ",",
RowBox[{
"-", "0.00254625823060283805659756104579829286188066051627600051142029563\
28260250669615751370338402696173858195663426415303`99.63196940361895"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"100", ",",
RowBox[{
"-", "0.00139903975020139787216990557168825202873552852242121741214248794\
00931596129251375255449261709204554250231778176676`99.6917337295701"}]}],
"}"}]}], "}"}]], "Output",
CellChangeTimes->{
3.782727738627494*^9, 3.782729311780086*^9, {3.782729342449399*^9,
3.782729368120493*^9}},
CellLabel->"Out[14]=",ExpressionUUID->"d7da7e11-bc55-4150-b2ad-bdfec7391bc8"],
Cell[BoxData[
RowBox[{"{",
RowBox[{
RowBox[{"{",
RowBox[{
FractionBox["1", "10"], ",",
RowBox[{
"-", "0.01449679709086608514885966546115856610835800505524967911686214153\
70925790367467138497709277177312108747520673941785`95.97004192918902"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{
FractionBox["1", "5"], ",",
RowBox[{
"-", "0.01452285969558371227310294103546671866855094230499695708731820401\
88774637173093250626437056295437267737977042644123`96.5583317943252"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{
FractionBox["1", "2"], ",",
RowBox[{
"-", "0.01456053237881486462592686542925494685845554337884698850405543602\
62618659037474735897929005798696043555314247418171`97.3148319755496"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"1", ",",
RowBox[{
"-", "0.01451224002718319908519441647116840901653124450002940637934925370\
07071415258546576398618079805790387597621381958802`97.8565904419334"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"2", ",",
RowBox[{
"-", "0.01414160309608582221327798600484863153716522304924298349554903042\
61730258569113406769911005526748892089401905765969`98.35279329134964"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"5", ",",
RowBox[{
"-", "0.01233394674740096943216208541906921913044600623658599159057541682\
43617997893859181243105537784990542708632904360789`98.90252326081136"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"10", ",",
RowBox[{
"-", "0.00971554642045995125387693282917957280007847039576225895067189461\
23692495757191107039481821707179867559999827009389`99.2192787978546"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"20", ",",
RowBox[{
"-", "0.00674380747270659875675558067307724842021673106928613331172989226\
38623759705256609013168765533861636407714355420818`99.45445758300228"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"50", ",",
RowBox[{
"-", "0.00358448914768323622319381160565649631089857683641653137135953111\
38424207594564910893071945445871024503002354543464`99.66870028556174"}]}],
"}"}], ",",
RowBox[{"{",
RowBox[{"1000", ",",
RowBox[{
"-", "0.00025886562211731695773198754396153978115276030728213275369586926\
70402117576797890862832806057190488584126947703637`99.97187841505077"}]}],
"}"}]}], "}"}]], "Output",
CellChangeTimes->{
3.782727738627494*^9, 3.782729311780086*^9, {3.782729342449399*^9,
3.78272936812368*^9}},
CellLabel->"Out[15]=",ExpressionUUID->"b74c742a-f437-4d6e-a206-6a508ca5a0dd"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[{
RowBox[{"FindFit", "[",
RowBox[{"Eig0", ",",
RowBox[{
FractionBox["a",
RowBox[{"1", "+",
RowBox[{"b", " ",
SuperscriptBox["R",
RowBox[{"1", "/", "2"}]]}], "+",
RowBox[{"c", " ", "R"}]}]], "/.",
RowBox[{"a", "->",
RowBox[{"-", "0.0238184"}]}]}], ",",
RowBox[{"{",
RowBox[{"b", ",", "c"}], "}"}], ",", "R", ",",
RowBox[{"MaxIterations", "\[Rule]", "10000"}]}],
"]"}], "\[IndentingNewLine]",
RowBox[{"FindFit", "[",
RowBox[{"Eig1", ",",
RowBox[{
FractionBox["a",
RowBox[{"1", "+",
RowBox[{"b", " ",
SuperscriptBox["R",
RowBox[{"1", "/", "2"}]]}], "+",
RowBox[{"c", " ", "R"}]}]], "/.",
RowBox[{"a", "->",
RowBox[{"-", "0.0144633"}]}]}], ",",
RowBox[{"{",
RowBox[{"b", ",", "c"}], "}"}], ",", "R", ",",
RowBox[{"MaxIterations", "\[Rule]", "10000"}]}], "]"}]}], "Input",
CellChangeTimes->{{3.74490442616041*^9, 3.744904462164751*^9}, {
3.744904493398246*^9, 3.7449044986891108`*^9}, {3.7449046086546717`*^9,
3.744904615683036*^9}, {3.744904661010707*^9, 3.744904730630151*^9}, {
3.7449048318432503`*^9, 3.7449048371708803`*^9}, {3.7483209427957153`*^9,
3.748320947610834*^9}, 3.7483614480757113`*^9, {3.748365049730114*^9,
3.74836505539084*^9}, {3.748365092652234*^9, 3.748365121022852*^9}, {
3.748365229503859*^9, 3.748365229661584*^9}, {3.7537740097668867`*^9,
3.753774017885911*^9}, {3.753774087759344*^9, 3.753774102439625*^9},
3.782710634125111*^9, {3.78271069785186*^9, 3.782710724897832*^9}, {
3.7827271850674067`*^9, 3.7827271853642693`*^9}, {3.782727488018824*^9,
3.782727520526437*^9}},
CellLabel->"In[36]:=",ExpressionUUID->"2f2c8334-f314-4d97-9abf-fd397dbace2b"],
Cell[BoxData[
RowBox[{"{",
RowBox[{
RowBox[{"b", "\[Rule]", "0.007319975674478506`"}], ",",
RowBox[{"c", "\[Rule]", "0.17864659588226584`"}]}], "}"}]], "Output",
CellChangeTimes->{{3.744904447194687*^9, 3.744904462614192*^9},
3.7449044992640533`*^9, {3.744904616038113*^9, 3.7449047310161533`*^9},
3.744904838229162*^9, 3.744910399143386*^9, 3.744957251784482*^9,
3.7449642181369953`*^9, 3.744972167413603*^9, 3.74508253355338*^9,
3.745118966686862*^9, 3.7451441000025997`*^9, 3.7479804201223307`*^9,
3.748320948669895*^9, 3.748361529488284*^9, 3.748361579108712*^9, {
3.748361677134914*^9, 3.748361685861054*^9}, 3.748364775740555*^9,
3.7483650558506002`*^9, {3.748365101887542*^9, 3.748365121567154*^9}, {
3.7483652043125143`*^9, 3.748365230113633*^9}, 3.753774018593093*^9,
3.753774104519877*^9, {3.782710627979114*^9, 3.782710634588338*^9}, {
3.7827106837527514`*^9, 3.782710725915807*^9}, {3.782727482716896*^9,
3.782727521432979*^9}, 3.7827276688072357`*^9},
CellLabel->"Out[36]=",ExpressionUUID->"f93ad9a7-a92c-48af-8e73-e28770d152a4"],
Cell[BoxData[
RowBox[{"{",
RowBox[{
RowBox[{"b", "\[Rule]",
RowBox[{"-", "0.07631897424525488`"}]}], ",",
RowBox[{"c", "\[Rule]", "0.07211543198556991`"}]}], "}"}]], "Output",
CellChangeTimes->{{3.744904447194687*^9, 3.744904462614192*^9},
3.7449044992640533`*^9, {3.744904616038113*^9, 3.7449047310161533`*^9},
3.744904838229162*^9, 3.744910399143386*^9, 3.744957251784482*^9,
3.7449642181369953`*^9, 3.744972167413603*^9, 3.74508253355338*^9,
3.745118966686862*^9, 3.7451441000025997`*^9, 3.7479804201223307`*^9,
3.748320948669895*^9, 3.748361529488284*^9, 3.748361579108712*^9, {
3.748361677134914*^9, 3.748361685861054*^9}, 3.748364775740555*^9,
3.7483650558506002`*^9, {3.748365101887542*^9, 3.748365121567154*^9}, {
3.7483652043125143`*^9, 3.748365230113633*^9}, 3.753774018593093*^9,
3.753774104519877*^9, {3.782710627979114*^9, 3.782710634588338*^9}, {
3.7827106837527514`*^9, 3.782710725915807*^9}, {3.782727482716896*^9,
3.782727521432979*^9}, 3.7827276688088408`*^9},
CellLabel->"Out[37]=",ExpressionUUID->"db4cf8bb-6a9b-4089-bef1-9bddb143e41d"]
}, Open ]],
Cell[BoxData[{
RowBox[{
RowBox[{"\[Epsilon]c0", "[", "\[Rho]_", "]"}], ":=",
RowBox[{"Module", "[",
RowBox[{
RowBox[{"{",
RowBox[{
RowBox[{"R", "=",
FractionBox["1",
RowBox[{
SuperscriptBox["\[Pi]",
RowBox[{"2", "/", "3"}]], " ",
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}]]}], ",",
RowBox[{"a", "=",
RowBox[{"-", "0.0238184"}]}], ",",
RowBox[{"b", "=",
RowBox[{"-", "0.007319975674478506`"}]}], ",",
RowBox[{"c", "=", "0.17864659588226584`"}]}], "}"}], ",",
FractionBox["a",
RowBox[{"1", "+",
RowBox[{"b", " ",
SuperscriptBox["R",
RowBox[{"1", "/", "2"}]]}], "+",
RowBox[{"c", " ", "R"}]}]]}], "]"}]}], "\[IndentingNewLine]",
RowBox[{
RowBox[{"\[Epsilon]c1", "[", "\[Rho]_", "]"}], ":=",
RowBox[{"Module", "[",
RowBox[{
RowBox[{"{",
RowBox[{
RowBox[{"R", "=",
FractionBox["1",
RowBox[{
SuperscriptBox["\[Pi]",
RowBox[{"2", "/", "3"}]], " ",
SuperscriptBox["\[Rho]",
RowBox[{"1", "/", "3"}]]}]]}], ",",
RowBox[{"a", "=",
RowBox[{"-", "0.01446325"}]}], ",",
RowBox[{"b", "=",
RowBox[{"-", "0.07631897424525488`"}]}], ",",
RowBox[{"c", "=", "0.07211543198556991`"}]}], "}"}], ",",
FractionBox["a",
RowBox[{"1", "+",
RowBox[{"b", " ",
SuperscriptBox["R",
RowBox[{"1", "/", "2"}]]}], "+",
RowBox[{"c", " ", "R"}]}]]}], "]"}]}]}], "Input",
CellChangeTimes->{{3.753773568857679*^9, 3.7537735989190903`*^9}, {
3.753774000549728*^9, 3.7537740603360977`*^9}, {3.753774109143517*^9,
3.753774138102693*^9}, {3.753774206440414*^9, 3.7537742282053556`*^9}, {
3.75396524390508*^9, 3.753965286454199*^9}, {3.7827108471854877`*^9,
3.782710858528974*^9}, {3.7827275396833487`*^9, 3.7827275687159843`*^9}, {
3.7827276811573753`*^9, 3.782727703059353*^9}},
CellLabel->"In[42]:=",ExpressionUUID->"6a07f7bc-64fd-4691-a797-042e1daf8a9c"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Show", "[",
RowBox[{"{",
RowBox[{
RowBox[{"ListPlot", "[",
RowBox[{"{",
RowBox[{"Eig0", ",", "Eig1"}], "}"}], "]"}], ",",
RowBox[{"Plot", "[",
RowBox[{
RowBox[{"{",
RowBox[{
RowBox[{"\[Epsilon]c0", "[",
FractionBox["2",
RowBox[{"2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]], "]"}], ",",
RowBox[{"\[Epsilon]c1", "[",
FractionBox["2",
RowBox[{"2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]], "]"}]}], "}"}], ",",
RowBox[{"{",
RowBox[{"R", ",", "0", ",", "1"}], "}"}], ",",
RowBox[{"PlotRange", "\[Rule]", "All"}]}], "]"}]}], "}"}],
"]"}]], "Input",
CellLabel->"In[44]:=",ExpressionUUID->"81ad7be8-1895-4df7-8778-5cf91f63b3cb"],
Cell[BoxData[
GraphicsBox[{{{}, {{
{RGBColor[0.368417, 0.506779, 0.709798], PointSize[
0.012833333333333334`], AbsoluteThickness[1.6],
PointBox[{{0.1, -0.023391913368829433`}, {0.2, -0.02297901655935025}, {
0.5, -0.021817116105320795`}, {1., -0.02010876852270148}, {
2., -0.01737071560943161}, {5., -0.012359326069805606`}, {
10., -0.008436394465178656}, {20., -0.0052569107832976985`}, {
50., -0.002546258230602838}, {100., -0.001399039749842982}}]},
{RGBColor[0.880722, 0.611041, 0.142051], PointSize[
0.012833333333333334`], AbsoluteThickness[1.6],
PointBox[{{0.1, -0.014496797090865628`}, {0.2, -0.01452285969558331}, {
0.5, -0.014560532378814613`}, {1., -0.01451224002718311}, {
2., -0.01414160309608582}, {5., -0.01233394674740097}, {
10., -0.009715546420459952}, {20., -0.006743807472706599}, {
50., -0.0035844891474714016`}, {1000., -0.0002542398931663114}}]}}, {
{RGBColor[0.368417, 0.506779, 0.709798], PointSize[
0.012833333333333334`], AbsoluteThickness[1.6]},
{RGBColor[0.880722, 0.611041, 0.142051], PointSize[
0.012833333333333334`], AbsoluteThickness[1.6]}, {}, {}, {}}, {
{RGBColor[0.368417, 0.506779, 0.709798], PointSize[
0.012833333333333334`], AbsoluteThickness[1.6]},
{RGBColor[0.880722, 0.611041, 0.142051], PointSize[
0.012833333333333334`], AbsoluteThickness[
1.6]}, {}, {}, {}}}, {{}, {}}}, {{{}, {},
TagBox[
{RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6],
Opacity[1.], LineBox[CompressedData["
1:eJwV0H081AccB3BxK6tFelidki53QtaDoVoP39q1tEt0utBIktSupkSeqqE5
M120+j0ZPau1lJOcGtrXuEkmxEke4roukYdfvSRJsd/++L4+r/c/n8/r9RUE
7ffaZWxkZBTE3f8p2dX5sKRryyqlFRnaspNBntjyQDvfGcZOrLD0DGbwy3NM
5i3+Wjj0wj7EnbNhljAhytYDRLWXX7tw3vCgMe8MXwYRecvXTuMsVFpdHrXe
CgMW9836uL4Pkp30DtsAgM39LqWcfYIH757iB8O/TKo6kLNGbHm7LDMEpox/
t8GR8yKblblvrL+H6pGa4YEgBk31xy5624aCaGx4QSLnJwt/Fp/gR0BALfoS
Oxgkzs0oEfkdAl1c8nhvzhLzayv+yoyEBt0PxtM5F/RVubDWMdAv9TCiAxlM
vTHZTmr7I8iSaqNObmfw/DTPpBR+IoSnhFgc2Mbg4Qe0xrswEbzEP2U4cPZW
6Ixt/BQQ2DtHb/Bn8LOBsLiizCRQi+pzfTlHa4noHutfwPlIeeZKPwalp5vl
HrapIJi3NGbEl8EE01vzfuET4LEgrGaXjEEHt1+T8w4QsN4ubcY0znWK/V0t
9wi42p22TrOZ+5+JY/aCSBIO2U6XzeVc/jFrgbaOgnnnS9yeSLn9fsrFWpkO
6qMLJ3t6Msg8iRWrR86C4Eq7VCRh0HfrBdWr11kgjv+m8DowOGE0wd9ttgpG
iYIq7XwGRdK+zrLj+XBwyNzGeQaDUdXLgHnzJ9DXdx9pNGZwer5F25gxCIeC
vx153kdjgV5cVbOiFHiicIG0icZS+ZmZW3s0ELKp6uxCDY0DtivNb/xwD/Zm
idNTcmk0HsqeNFNUCRi/X+OVQaPraGDB5pwqiHxeMJz+M41Vic7V5Q41MO1V
1e/fRdBofbJ76I++WqhRRoad2k5j92B7W8OiOqi4Jg53c6cx62VRpVlqPYSu
W/M+ZgmNZla/swaDFtJnTxx1sKFR+b065LzwEexLnYqySTS+XZ1t2BPXCLv/
qT5t8oHCCCdN/udlj8GnLHifXReF7Tma4w8FzRBr6jfYqKVwWK7NsI9tgf29
2zS8vymcv2bcSEhRK7Q4Nx28kEOh20XB7pvT22B1T8fpmt8ojLLaPvepfzv4
r7GSyJMpdOzZMqFxow4C1/+qOhxJ4RTzV6aa9zpQN0RphTsp5GGGyibzKXiy
bpbOUgr3Ohc1fbpBD7ejyUW6lRRmfxt0/q+3emhYtZjkzadwoNtyUgf1DN45
BcmT+RSqN7uZEGsNkMTzEqnGUhi3XnEuv8MA+quZjxYPkCh4Inl8SfEcYrc0
60UGEr9akkjwnTrA37SJKX1Iov/smqEx9R3QaG5RkV9C4kDlCf2BhBdQUDBu
iyyHRIYXEOEl7AR7dYnhmzMkTmlTXzTWdEJ5WLLqspLET44myJaFdYFH32eB
d2JJNPojQKWf/BIMR+/mfC4n8eBvqhOTS1+CiYlQ7uhL4t35yaYWod1QUVeW
cHMdiRaSg47HzHpgQr2u4rAria1f1NbHFPdA3vK/nxuLSMw9C0JDQC8IPB49
/XcqidtUrhZWI72wtHpZ+kceifGKCwmt1/pAYnDa+LqfwBudVhlm7izEDiur
5xgITD6jCGzzYOHjjchdzc8IDPbqFeVIWQhtth85zXlWcXHuRh8W7gf3OI/j
nJLmV64MYoFldLd7nxK4xzX99fgYFiTu0VOK2gmcmzh1/bgrLEg3Dc/xaSFw
ZOmRiY1XWdhkOvhqEuem3md1V7JZKPYpLK1sJvCkz61t626yoIyn967iPOrg
Fa4oZsFBdq1O1ERg68O0syb1LMRrhW/fPCLwTtLgzvoGFoT9p/Q5nInl2+0v
PWbB7s692j2cN1xekP91GwsWQ4l5rQ0EFkY/uJ/QxUJhrvtxjZZA6guXNGkP
C7n+fYofOYfpM2UCloU0V+9jSznbue9rL3nDQtWS6Pjr9QTyxmizTg6yYL7G
6VgIZ516uTzwPQvGK/IUczgXyy8tXPSRBZr/LqW5jkDaesLA6CgLpZUmpwjO
/wG03zZv
"]]},
Annotation[#, "Charting`Private`Tag$211022#1"]& ],
TagBox[
{RGBColor[0.880722, 0.611041, 0.142051], AbsoluteThickness[1.6],
Opacity[1.], LineBox[CompressedData["
1:eJwV13c8ld8fAHBS8lUhJLIzUoTKSuMjNMjKCA3JHhXRjyKJKMVzKVKKIiNk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"]]},
Annotation[#, "Charting`Private`Tag$211022#2"]& ]}, {}}},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->{True, True},
AxesLabel->{None, None},
AxesOrigin->{0, 0},
DisplayFunction->Identity,
Frame->{{False, False}, {False, False}},
FrameLabel->{{None, None}, {None, None}},
FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
GridLines->{None, None},
GridLinesStyle->Directive[
GrayLevel[0.5, 0.4]],
Method->{
"OptimizePlotMarkers" -> True,
"CoordinatesToolOptions" -> {"DisplayFunction" -> ({
(Identity[#]& )[
Part[#, 1]],
(Identity[#]& )[
Part[#, 2]]}& ), "CopiedValueFunction" -> ({
(Identity[#]& )[
Part[#, 1]],
(Identity[#]& )[
Part[#, 2]]}& )}},
PlotRange->{{0, 124.69999999999999`}, {-0.023391913368829433`, 0}},
PlotRangeClipping->True,
PlotRangePadding->{{
Scaled[0.02],
Scaled[0.02]}, {
Scaled[0.05],
Scaled[0.02]}},
Ticks->{Automatic, Automatic}]], "Output",
CellChangeTimes->{{3.782727692531313*^9, 3.7827277043231907`*^9}},
CellLabel->"Out[44]=",ExpressionUUID->"4dcb86c2-50f4-49ce-b4f7-f37d9b77c839"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Show", "[",
RowBox[{"{",
RowBox[{"Plot", "[",
RowBox[{
RowBox[{"{",
RowBox[{"\[Epsilon]c1", "[",
FractionBox["2",
RowBox[{"2",
SuperscriptBox["\[Pi]", "2"],
SuperscriptBox["R", "3"]}]], "]"}], "}"}], ",",
RowBox[{"{",
RowBox[{"R", ",", "0", ",", "1"}], "}"}], ",",
RowBox[{"PlotRange", "\[Rule]", "All"}]}], "]"}], "}"}], "]"}]], "Input",\
CellChangeTimes->{{3.782727636731057*^9, 3.782727656428298*^9}, {
3.782727708316208*^9, 3.782727722763756*^9}},
CellLabel->"In[48]:=",ExpressionUUID->"ec27db6b-31e1-48cc-802b-692296080f56"],
Cell[BoxData[
GraphicsBox[{{{}, {},
TagBox[
{RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6], Opacity[
1.], LineBox[CompressedData["
1:eJwV13c8ld8fAHBS8lUhJLIzUoTKSuMjNMjKCA3JHhXRjyKJKMVzKVKKIiNk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"]]},
Annotation[#, "Charting`Private`Tag$212177#1"]& ]}, {}},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->{True, True},
AxesLabel->{None, None},
AxesOrigin->{Automatic, Automatic},
DisplayFunction->Identity,
Frame->{{False, False}, {False, False}},
FrameLabel->{{None, None}, {None, None}},
FrameTicks->{{Automatic,
Charting`ScaledFrameTicks[{Identity, Identity}]}, {Automatic,
Charting`ScaledFrameTicks[{Identity, Identity}]}},
GridLines->{None, None},
GridLinesStyle->Directive[
GrayLevel[0.5, 0.4]],
ImagePadding->All,
ImageSize->{1085.640625, Automatic},
Method->{
"DefaultBoundaryStyle" -> Automatic,
"DefaultGraphicsInteraction" -> {
"Version" -> 1.2, "TrackMousePosition" -> {True, False},
"Effects" -> {
"Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2},
"Droplines" -> {
"freeformCursorMode" -> True,
"placement" -> {"x" -> "All", "y" -> "None"}}}}, "DefaultMeshStyle" ->
AbsolutePointSize[6], "ScalingFunctions" -> None,
"CoordinatesToolOptions" -> {"DisplayFunction" -> ({
(Identity[#]& )[
Part[#, 1]],
(Identity[#]& )[
Part[#, 2]]}& ), "CopiedValueFunction" -> ({
(Identity[#]& )[
Part[#, 1]],
(Identity[#]& )[
Part[#, 2]]}& )}},
PlotRange->{All, All},
PlotRangeClipping->True,
PlotRangePadding->{{
Scaled[0.02],
Scaled[0.02]}, {
Scaled[0.05],
Scaled[0.05]}},
Ticks->{Automatic, Automatic}]], "Output",
CellChangeTimes->{{3.78272763715982*^9, 3.782727656952072*^9}, {
3.782727693660577*^9, 3.782727723885977*^9}},
CellLabel->"Out[48]=",ExpressionUUID->"442fc301-280b-415d-90fd-eaaaa4ce9269"]
}, Open ]]
}, Open ]]
}, Open ]]
}, Open ]]
},
WindowSize->{1811, 1395},
WindowMargins->{{Automatic, 0}, {Automatic, 0}},
FrontEndVersion->"12.0 for Mac OS X x86 (64-bit) (April 8, 2019)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)
(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[580, 22, 156, 3, 98, "Title",ExpressionUUID->"9a643c6d-1044-4612-8f7a-377e028cd777"],
Cell[739, 27, 574, 12, 131, "Input",ExpressionUUID->"8b7c0bbd-9f99-42dc-b8a5-a4dc3cb8cc89",
InitializationCell->True],
Cell[1316, 41, 308, 6, 46, "Input",ExpressionUUID->"b24e8243-3c38-4b03-af3a-708cb1afa004",
InitializationCell->True]
}, Closed]],
Cell[CellGroupData[{
Cell[1661, 52, 151, 3, 72, "Title",ExpressionUUID->"cdfcc3a6-7299-4a1b-a9d0-f29078225f1e"],
Cell[CellGroupData[{
Cell[1837, 59, 159, 3, 67, "Section",ExpressionUUID->"23d29e55-92a0-4dbd-8800-76494f7702a3"],
Cell[CellGroupData[{
Cell[2021, 66, 250, 4, 54, "Subsection",ExpressionUUID->"5917ba4c-2cd7-4975-9b3b-073e72bd0e40"],
Cell[CellGroupData[{
Cell[2296, 74, 723, 23, 48, "Input",ExpressionUUID->"c8ade4e2-951f-4fce-ae16-fdd4ea72db6f"],
Cell[3022, 99, 517, 15, 62, "Output",ExpressionUUID->"e2c1e6f8-35e1-43ee-b47e-a64bb64f8585"]
}, Open ]],
Cell[3554, 117, 651, 15, 30, "Input",ExpressionUUID->"02e08ec6-7c15-4841-9609-90296edfe494"],
Cell[CellGroupData[{
Cell[4230, 136, 550, 13, 48, "Input",ExpressionUUID->"150e916f-8806-4d8e-b5d0-c0bc713b1f4d"],
Cell[4783, 151, 276, 4, 34, "Output",ExpressionUUID->"fb35d58c-512f-4f74-aa01-67c7b7b81a77"]
}, Open ]]
}, Closed]]
}, Open ]],
Cell[CellGroupData[{
Cell[5120, 162, 215, 4, 67, "Section",ExpressionUUID->"8fd0ca40-ccd3-410e-8538-798441b5cf4b"],
Cell[CellGroupData[{
Cell[5360, 170, 174, 3, 54, "Subsection",ExpressionUUID->"10e37c4d-fcc0-4a25-949c-6383c67ff9eb"],
Cell[CellGroupData[{
Cell[5559, 177, 4321, 122, 191, "Input",ExpressionUUID->"b7e33c7b-f239-4612-b339-911d66bee804"],
Cell[9883, 301, 199, 3, 34, "Output",ExpressionUUID->"3acd8e38-829c-4dd5-ac53-54c357a30852"],
Cell[10085, 306, 199, 3, 34, "Output",ExpressionUUID->"0a957f43-bd43-468d-a336-addbe42a4e30"]
}, Open ]],
Cell[CellGroupData[{
Cell[10321, 314, 847, 27, 57, "Input",ExpressionUUID->"5bc098a3-a8b3-48df-b81b-4249253ebc03"],
Cell[11171, 343, 176, 2, 34, "Output",ExpressionUUID->"3597039b-63a4-48ef-b436-f5b1fb49d7fd"]
}, Open ]],
Cell[CellGroupData[{
Cell[11384, 350, 1432, 45, 94, "Input",ExpressionUUID->"7707f078-0791-4cee-b4ea-44b37115dc20"],
Cell[12819, 397, 197, 3, 34, "Output",ExpressionUUID->"d4f8c9be-6e4c-40eb-ba9a-d4053731284e"],
Cell[13019, 402, 199, 3, 34, "Output",ExpressionUUID->"2e03225f-8957-4361-919c-4f6a7c36dc6e"]
}, Open ]],
Cell[CellGroupData[{
Cell[13255, 410, 2671, 77, 124, "Input",ExpressionUUID->"731b5390-31ed-4914-91db-765023c263dd"],
Cell[15929, 489, 497, 7, 34, "Output",ExpressionUUID->"ed71661b-3a8d-4eee-a4d0-69211937142a"]
}, Open ]],
Cell[16441, 499, 2240, 61, 123, "Input",ExpressionUUID->"1f5927dc-663a-4f11-a2d1-7bc195f79332"]
}, Closed]]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell[18742, 567, 150, 3, 98, "Title",ExpressionUUID->"e756f09e-4f67-4fb6-a4c1-1a4fcf53e013"],
Cell[CellGroupData[{
Cell[18917, 574, 259, 4, 67, "Section",ExpressionUUID->"81f07a4a-071b-4e38-8018-d116e770b20b"],
Cell[CellGroupData[{
Cell[19201, 582, 291, 4, 54, "Subsection",ExpressionUUID->"7471e1f3-36cd-497f-a3e0-06aa7035995d"],
Cell[19495, 588, 228, 6, 35, "Text",ExpressionUUID->"c0165a48-1e75-4237-86f9-15c7a0fc2c6e"],
Cell[19726, 596, 570, 16, 49, "Input",ExpressionUUID->"eeea1fbc-15c7-4b79-b5b2-0d2e682c7cb1"],
Cell[20299, 614, 230, 4, 35, "Text",ExpressionUUID->"6509cf0c-3cb5-4484-a79b-9a4e9a4c867b"],
Cell[20532, 620, 449, 10, 48, "Input",ExpressionUUID->"ea3020ca-486a-47b1-95e0-aa46f448646d"],
Cell[20984, 632, 176, 3, 35, "Text",ExpressionUUID->"dc6b5921-169b-4f2d-9120-4c7ca89c05f3"],
Cell[21163, 637, 670, 18, 48, "Input",ExpressionUUID->"12ac5e19-b8fd-477e-a85d-a74e029b6d4c"],
Cell[21836, 657, 201, 3, 35, "Text",ExpressionUUID->"21ac5bf4-2ae7-4b41-8495-ff3b6d5d2017"],
Cell[22040, 662, 794, 21, 45, "Input",ExpressionUUID->"c8865179-994b-4123-9657-507eae2fd52f"],
Cell[22837, 685, 153, 3, 35, "Text",ExpressionUUID->"d352d13e-d96f-4f21-9cc5-32da8a2e376d"],
Cell[22993, 690, 1177, 34, 50, "Input",ExpressionUUID->"e2a90d7c-9765-4224-b6c1-0cb3ba56719e"],
Cell[24173, 726, 193, 3, 35, "Text",ExpressionUUID->"4256ed0c-1e05-4553-a95a-370962e94038"],
Cell[24369, 731, 707, 21, 58, "Input",ExpressionUUID->"d7507220-76aa-479e-af51-0886ad0e4f5c"]
}, Open ]],
Cell[CellGroupData[{
Cell[25113, 757, 301, 4, 54, "Subsection",ExpressionUUID->"77af7129-5b6f-46d1-972e-088c04a397e1"],
Cell[25417, 763, 310, 7, 35, "Text",ExpressionUUID->"137a2815-3e41-4c54-bb77-2baab3309f67"],
Cell[25730, 772, 570, 16, 49, "Input",ExpressionUUID->"ee1a49a3-d78d-48fd-b18b-b45c556268d6"],
Cell[26303, 790, 318, 6, 35, "Text",ExpressionUUID->"7f6139ea-87a7-485f-96a7-072785ed5a50"],
Cell[26624, 798, 571, 14, 48, "Input",ExpressionUUID->"e2512605-e53f-4895-bc29-b005c70f53b5"],
Cell[CellGroupData[{
Cell[27220, 816, 570, 17, 50, "Input",ExpressionUUID->"2caf6c5b-6ae4-41ea-a5bc-6e2553b311b0"],
Cell[27793, 835, 544, 16, 55, "Output",ExpressionUUID->"71a67885-8142-46b8-9292-410654424b59"]
}, Open ]],
Cell[28352, 854, 176, 3, 35, "Text",ExpressionUUID->"32015b28-f5cc-4eff-8d8d-b8691b888e8f"],
Cell[28531, 859, 847, 23, 48, "Input",ExpressionUUID->"1fcbd290-f3ea-43bd-b2ef-f31601137679"],
Cell[29381, 884, 201, 3, 35, "Text",ExpressionUUID->"9b1e04fb-55cb-4e81-8daf-ba707c78dfce"],
Cell[29585, 889, 827, 22, 45, "Input",ExpressionUUID->"c51ccbbc-5ee1-4cca-ade7-af230c7f6ba6"],
Cell[30415, 913, 153, 3, 35, "Text",ExpressionUUID->"8bfb35af-a4e4-4813-ae15-3703f4aeab96"],
Cell[30571, 918, 1240, 35, 50, "Input",ExpressionUUID->"60e340d1-6bd1-4928-9814-51c59353edd1"],
Cell[31814, 955, 250, 4, 35, "Text",ExpressionUUID->"f5f45693-a80c-4a46-ad73-07811620c0d9"],
Cell[32067, 961, 787, 23, 58, "Input",ExpressionUUID->"f99857fd-0430-402d-a733-929b6b359e23"]
}, Open ]],
Cell[CellGroupData[{
Cell[32891, 989, 348, 5, 54, "Subsection",ExpressionUUID->"ef62b877-dbc0-41f4-bb50-f7bab5069a50"],
Cell[33242, 996, 252, 6, 35, "Text",ExpressionUUID->"5b10be0f-a732-422f-9446-681c78cbd103"],
Cell[33497, 1004, 561, 17, 32, "Input",ExpressionUUID->"009be4dc-79a6-499d-b9af-ddbb9078f85a"],
Cell[34061, 1023, 918, 28, 53, "Input",ExpressionUUID->"afd8bd15-999f-411c-a673-53d5b75e5201"],
Cell[34982, 1053, 297, 4, 35, "Text",ExpressionUUID->"1fe4835f-3b82-458a-9e2a-9272922b081a"],
Cell[35282, 1059, 1329, 40, 58, "Input",ExpressionUUID->"b5e6e435-424a-4f3e-8416-709ae4356a73"],
Cell[36614, 1101, 546, 14, 58, "Text",ExpressionUUID->"d05917ae-a767-4a78-834a-c4b18c1c0b13"]
}, Open ]],
Cell[CellGroupData[{
Cell[37197, 1120, 419, 8, 54, "Subsection",ExpressionUUID->"bcd158fd-9aa7-4c7b-837e-afd37c54f55a"],
Cell[37619, 1130, 571, 11, 58, "Text",ExpressionUUID->"a0112f10-d580-4370-a6ed-bff98b24a4a6"],
Cell[38193, 1143, 2757, 86, 101, "Input",ExpressionUUID->"aabf1284-1a62-408f-8d1c-50c99b5acc35"],
Cell[40953, 1231, 278, 6, 35, "Text",ExpressionUUID->"88a4bd50-0ba7-4f7c-9fc0-e0feb01831fc"],
Cell[41234, 1239, 606, 19, 48, "Input",ExpressionUUID->"a889f096-9a50-426c-9cf3-81c363ec2eaf"],
Cell[41843, 1260, 355, 7, 35, "Text",ExpressionUUID->"d3c69495-05ef-49fe-8484-fa7fd62cc408"],
Cell[42201, 1269, 1887, 55, 58, "Input",ExpressionUUID->"0138b572-760b-4f82-a618-73197e776fa2"],
Cell[44091, 1326, 452, 7, 35, "Text",ExpressionUUID->"b83db319-04e5-42b8-ae4c-57f93abe09fd"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell[44592, 1339, 240, 4, 67, "Section",ExpressionUUID->"258ca12d-c8c5-4bc9-84ec-1711f7860786"],
Cell[CellGroupData[{
Cell[44857, 1347, 347, 5, 54, "Subsection",ExpressionUUID->"ee1991f0-6627-4afa-871c-73710071f6fd"],
Cell[45207, 1354, 324, 7, 58, "Text",ExpressionUUID->"a6c10556-4f53-4da2-8cf0-79bbf3ec1242"],
Cell[45534, 1363, 1209, 36, 64, "Input",ExpressionUUID->"1885d9ca-6bb6-490d-a0bb-a3b96576a614"],
Cell[46746, 1401, 300, 6, 35, "Text",ExpressionUUID->"ead26072-79d4-4549-aab6-46fe8cf9b6db"]
}, Open ]],
Cell[CellGroupData[{
Cell[47083, 1412, 353, 5, 54, "Subsection",ExpressionUUID->"ac951d60-fbce-44ba-8f12-9e5545df5beb"],
Cell[47439, 1419, 228, 5, 35, "Text",ExpressionUUID->"dca5790f-2358-43e0-b359-677842457a7d"],
Cell[47670, 1426, 1279, 38, 64, "Input",ExpressionUUID->"cce86f90-52f7-41f4-a63d-73a6c518b4c3"]
}, Open ]],
Cell[CellGroupData[{
Cell[48986, 1469, 348, 5, 54, "Subsection",ExpressionUUID->"9f2ce783-2740-403c-974c-46a7ef9321fe"],
Cell[49337, 1476, 310, 7, 35, "Text",ExpressionUUID->"6c7b6dc7-06c2-4f95-a398-fe8f7adea7db"],
Cell[49650, 1485, 610, 18, 32, "Input",ExpressionUUID->"c5451bea-41ca-4be3-9429-a6d8ddd06c4a"]
}, Open ]],
Cell[CellGroupData[{
Cell[50297, 1508, 468, 8, 54, "Subsection",ExpressionUUID->"ab8ea4d9-6951-44dc-96fc-10c74ccd1a4f"],
Cell[50768, 1518, 279, 6, 35, "Text",ExpressionUUID->"f0dac186-7114-4a5e-b53c-f938d8c4fd37"],
Cell[51050, 1526, 1342, 40, 33, "Input",ExpressionUUID->"2ca8802e-361f-4487-a3f0-67d18fc8fe5f"],
Cell[52395, 1568, 386, 8, 35, "Text",ExpressionUUID->"00fbe382-6c25-4893-a74d-546919a831b8"],
Cell[52784, 1578, 368, 8, 32, "Input",ExpressionUUID->"b7e98868-257f-43e4-b057-ff1cbb083046"],
Cell[53155, 1588, 728, 11, 104, "Text",ExpressionUUID->"336e8167-7e94-4334-9956-c0652866375e"],
Cell[53886, 1601, 4267, 104, 217, "Input",ExpressionUUID->"8d489c13-936e-4272-b8f6-29963888bb9f"],
Cell[58156, 1707, 2931, 82, 88, "Input",ExpressionUUID->"af4fe8f8-c3eb-4495-a149-b4bd1830693b"],
Cell[CellGroupData[{
Cell[61112, 1793, 308, 4, 52, "Input",ExpressionUUID->"53e9768f-d789-47f5-820c-8dfbf0b14efb"],
Cell[61423, 1799, 2594, 69, 282, "Output",ExpressionUUID->"d7da7e11-bc55-4150-b2ad-bdfec7391bc8"],
Cell[64020, 1870, 2593, 69, 282, "Output",ExpressionUUID->"b74c742a-f437-4d6e-a206-6a508ca5a0dd"]
}, Open ]],
Cell[CellGroupData[{
Cell[66650, 1944, 1772, 41, 87, "Input",ExpressionUUID->"2f2c8334-f314-4d97-9abf-fd397dbace2b"],
Cell[68425, 1987, 1096, 17, 34, "Output",ExpressionUUID->"f93ad9a7-a92c-48af-8e73-e28770d152a4"],
Cell[69524, 2006, 1115, 18, 34, "Output",ExpressionUUID->"db4cf8bb-6a9b-4089-bef1-9bddb143e41d"]
}, Open ]],
Cell[70654, 2027, 2064, 55, 93, "Input",ExpressionUUID->"6a07f7bc-64fd-4691-a797-042e1daf8a9c"],
Cell[CellGroupData[{
Cell[72743, 2086, 862, 25, 49, "Input",ExpressionUUID->"81ad7be8-1895-4df7-8778-5cf91f63b3cb"],
Cell[73608, 2113, 12337, 220, 226, "Output",ExpressionUUID->"4dcb86c2-50f4-49ce-b4f7-f37d9b77c839"]
}, Open ]],
Cell[CellGroupData[{
Cell[85982, 2338, 638, 17, 49, "Input",ExpressionUUID->"ec27db6b-31e1-48cc-802b-692296080f56"],
Cell[86623, 2357, 9155, 170, 674, "Output",ExpressionUUID->"442fc301-280b-415d-90fd-eaaaa4ce9269"]
}, Open ]]
}, Open ]]
}, Open ]]
}, Open ]]
}
]
*)