5922 lines
215 KiB
Mathematica
5922 lines
215 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[ 219832, 5913]
|
|
NotebookOptionsPosition[ 200640, 5612]
|
|
NotebookOutlinePosition[ 200977, 5627]
|
|
CellTagsIndexPosition[ 200934, 5624]
|
|
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"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Delta]",
|
|
RowBox[{"a_", ",", "b_"}]], ":=",
|
|
RowBox[{"KroneckerDelta", "[",
|
|
RowBox[{"a", ",", "b"}], "]"}]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{3.5767843605069733`*^9},
|
|
CellLabel->"In[4]:=",ExpressionUUID->"5822ee69-a453-46a0-9a58-bd14b8a97301"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["F", "0"], "[", "t_", "]"}], ":=",
|
|
RowBox[{"If", "[",
|
|
RowBox[{
|
|
RowBox[{"t", "\[Equal]", "0"}], ",", "1", ",",
|
|
RowBox[{
|
|
FractionBox["1", "2"],
|
|
SqrtBox[
|
|
FractionBox["\[Pi]", "t"]],
|
|
RowBox[{"Erf", "[",
|
|
SqrtBox["t"], "]"}]}]}], "]"}]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.5764508510320177`*^9, 3.576450882688406*^9}, {
|
|
3.576460057492025*^9, 3.5764600828205853`*^9}, {3.5764671476781178`*^9,
|
|
3.5764671766921053`*^9}, {3.579916170955673*^9, 3.579916171041703*^9}, {
|
|
3.579916284781068*^9, 3.5799162848679523`*^9}, {3.579917335658855*^9,
|
|
3.5799173370087748`*^9}, 3.57991775752785*^9, {3.580013360449069*^9,
|
|
3.580013371032818*^9}},
|
|
CellLabel->"In[5]:=",ExpressionUUID->"43788edb-8cf4-404a-88d1-db5bf1515eb0"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"\[CapitalGamma]", "[", "x_", "]"}], ":=",
|
|
RowBox[{"Gamma", "[", "x", "]"}]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{
|
|
3.5767843605069733`*^9, {3.668756017902652*^9, 3.668756027660227*^9}},
|
|
CellLabel->"In[6]:=",ExpressionUUID->"058853fd-911e-44b3-8eac-1c3fbeb8c4a5"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"$Assumptions", "\[Rule]",
|
|
RowBox[{"R", ">", "0"}]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790612553197805*^9, 3.790612558626453*^9}},
|
|
CellLabel->"In[7]:=",ExpressionUUID->"5dbbd40c-d911-445f-b476-85cd91f8d014"]
|
|
}, Closed]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell["Matrix elements", "Title",
|
|
CellChangeTimes->{{3.779526821452816*^9,
|
|
3.77952682493222*^9}},ExpressionUUID->"98d632cb-7177-481c-9b9f-\
|
|
d6f48d10e284"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell["s-type ERIs", "Section",
|
|
CellChangeTimes->{{3.779526832772407*^9, 3.779526836020318*^9}, {
|
|
3.7796013836604347`*^9,
|
|
3.779601384213182*^9}},ExpressionUUID->"9458067c-0b7a-4f70-9a58-\
|
|
1f771c4b3842"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell["Explicit expressions", "Subsection",
|
|
CellChangeTimes->{{3.779526843620368*^9,
|
|
3.779526847828706*^9}},ExpressionUUID->"11bf4889-b9bd-4f4e-8dbb-\
|
|
a1e9e8a04d2a"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"\[ScriptCapitalS]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"\[Alpha]_", ",", "A_"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Beta]_", ",", "B_"}], "}"}]}], "]"}], ":=",
|
|
RowBox[{
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"2", "\[Alpha]"}], "\[Pi]"], ")"}],
|
|
RowBox[{"3", "/", "4"}]],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"2", "\[Beta]"}], "\[Pi]"], ")"}],
|
|
RowBox[{"3", "/", "4"}]],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["\[Pi]",
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}]], ")"}],
|
|
RowBox[{"3", "/", "2"}]],
|
|
RowBox[{"Exp", "[",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox[
|
|
RowBox[{"\[Alpha]", " ", "\[Beta]"}],
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}]]}], " ",
|
|
SuperscriptBox[
|
|
RowBox[{"Norm", "[",
|
|
RowBox[{"A", "-", "B"}], "]"}], "2"]}], "]"}]}]}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.576450521600048*^9, 3.576450601477757*^9}, {
|
|
3.576872585214479*^9, 3.576872668475618*^9}, {3.57687272885421*^9,
|
|
3.576872728944818*^9}, {3.579916092387648*^9, 3.579916142300425*^9}, {
|
|
3.6560325968246727`*^9, 3.656032627333782*^9}, {3.6560334237894363`*^9,
|
|
3.6560334543496313`*^9}, 3.656033501500393*^9, {3.656034168807974*^9,
|
|
3.656034169503957*^9}, 3.656034249838081*^9, 3.65603428628251*^9, {
|
|
3.657419237952764*^9, 3.657419243496664*^9}, {3.657432554859721*^9,
|
|
3.6574325625623617`*^9}, {3.7795268642448807`*^9, 3.779526892756715*^9}, {
|
|
3.790614097995901*^9, 3.790614134630106*^9}},
|
|
CellLabel->"In[8]:=",ExpressionUUID->"fd4d58e4-e6f8-4497-ac22-d536c7dc2652"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"\[ScriptCapitalT]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"\[Alpha]_", ",", "A_"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Beta]_", ",", "B_"}], "}"}]}], "]"}], ":=",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"\[Alpha]", " ", "\[Beta]"}],
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}]],
|
|
RowBox[{"(",
|
|
RowBox[{"3", "-",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"2", "\[Alpha]", " ", "\[Beta]"}],
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}]],
|
|
SuperscriptBox[
|
|
RowBox[{"Norm", "[",
|
|
RowBox[{"A", "-", "B"}], "]"}], "2"]}]}], ")"}],
|
|
RowBox[{"\[ScriptCapitalS]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"\[Alpha]", ",", "A"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Beta]", ",", "B"}], "}"}]}], "]"}]}]}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.5764506077735443`*^9, 3.576450636952054*^9}, {
|
|
3.579916143373057*^9, 3.579916148380529*^9}, {3.656032632015997*^9,
|
|
3.656032656876679*^9}, {3.6560341704633427`*^9, 3.656034171112013*^9},
|
|
3.6560342990101433`*^9, {3.656034587343053*^9, 3.656034588398306*^9}, {
|
|
3.657419246521282*^9, 3.657419251312986*^9}, {3.6574325567021017`*^9,
|
|
3.657432563402364*^9}, {3.779526874688856*^9, 3.779526896028626*^9},
|
|
3.790614198473736*^9, 3.79061641080725*^9},
|
|
CellLabel->"In[9]:=",ExpressionUUID->"f4215905-d329-4e5c-925a-feafdeee6fce"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"\[ScriptCapitalV]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"\[Alpha]_", ",", "A_"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Beta]_", ",", "B_"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"ZC_", ",", "C_"}], "}"}]}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox[
|
|
RowBox[{"2", "\[Pi]", " ", "ZC"}],
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}]]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"2", "\[Alpha]"}], "\[Pi]"], ")"}],
|
|
RowBox[{"3", "/", "4"}]],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"2", "\[Beta]"}], "\[Pi]"], ")"}],
|
|
RowBox[{"3", "/", "4"}]],
|
|
RowBox[{
|
|
SubscriptBox["F", "0"], "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}], ")"}],
|
|
SuperscriptBox[
|
|
RowBox[{"Norm", "[",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", " ", "A"}], "+",
|
|
RowBox[{"\[Beta]", " ", "B"}]}],
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}]], "-", "C"}], "]"}], "2"]}],
|
|
"]"}],
|
|
RowBox[{"Exp", "[",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox[
|
|
RowBox[{"\[Alpha]", " ", "\[Beta]"}],
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}]]}], " ",
|
|
SuperscriptBox[
|
|
RowBox[{"Norm", "[",
|
|
RowBox[{"A", "-", "B"}], "]"}], "2"]}], "]"}]}]}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{
|
|
3.5764506469916887`*^9, {3.576450689278904*^9, 3.576450845913303*^9}, {
|
|
3.576450891155459*^9, 3.5764508913303423`*^9}, {3.576459615586534*^9,
|
|
3.576459619742888*^9}, {3.5764599742451572`*^9, 3.576460002180661*^9}, {
|
|
3.579916151076499*^9, 3.579916154291819*^9}, {3.6560326719657087`*^9,
|
|
3.656032705698762*^9}, {3.656032784400991*^9, 3.6560327883205967`*^9}, {
|
|
3.656034171959223*^9, 3.6560341725918818`*^9}, 3.6560345909276237`*^9,
|
|
3.656034741857669*^9, {3.6574192551059093`*^9, 3.657419257808338*^9}, {
|
|
3.65743255880517*^9, 3.657432564418058*^9}, {3.779526877419273*^9,
|
|
3.7795268990686483`*^9}, 3.790614203831818*^9},
|
|
CellLabel->"In[10]:=",ExpressionUUID->"517a10dc-802a-4ead-9f44-0905089379ba"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"\[ScriptCapitalU]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"\[Alpha]_", ",", "A_"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Beta]_", ",", "B_"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Gamma]_", ",", "C_"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Delta]_", ",", "D_"}], "}"}]}], "]"}], ":=",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"2",
|
|
SuperscriptBox["\[Pi]",
|
|
RowBox[{"5", "/", "2"}]], " "}],
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}], ")"}],
|
|
RowBox[{"(",
|
|
RowBox[{"\[Gamma]", "+", "\[Delta]"}], ")"}],
|
|
SqrtBox[
|
|
RowBox[{
|
|
"\[Alpha]", "+", "\[Beta]", "+", "\[Gamma]", "+", "\[Delta]"}]]}]],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"2", "\[Alpha]"}], "\[Pi]"], ")"}],
|
|
RowBox[{"3", "/", "4"}]],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"2", "\[Beta]"}], "\[Pi]"], ")"}],
|
|
RowBox[{"3", "/", "4"}]],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"2", "\[Gamma]"}], "\[Pi]"], ")"}],
|
|
RowBox[{"3", "/", "4"}]],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"2", "\[Delta]"}], "\[Pi]"], ")"}],
|
|
RowBox[{"3", "/", "4"}]],
|
|
RowBox[{
|
|
SubscriptBox["F", "0"], "[",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}], ")"}],
|
|
RowBox[{"(",
|
|
RowBox[{"\[Gamma]", "+", "\[Delta]"}], ")"}]}],
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]", "+", "\[Gamma]", "+", "\[Delta]"}]],
|
|
SuperscriptBox[
|
|
RowBox[{"Norm", "[",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", " ", "A"}], "+",
|
|
RowBox[{"\[Beta]", " ", "B"}]}],
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}]], "-",
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{"\[Gamma]", " ", "C"}], "+",
|
|
RowBox[{"\[Delta]", " ", "D"}]}],
|
|
RowBox[{"\[Gamma]", "+", "\[Delta]"}]]}], "]"}], "2"]}], "]"}],
|
|
RowBox[{"Exp", "[",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox[
|
|
RowBox[{"\[Alpha]", " ", "\[Beta]"}],
|
|
RowBox[{"\[Alpha]", "+", "\[Beta]"}]]}], " ",
|
|
SuperscriptBox[
|
|
RowBox[{"Norm", "[",
|
|
RowBox[{"A", "-", "B"}], "]"}], "2"]}], "]"}],
|
|
RowBox[{"Exp", "[",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox[
|
|
RowBox[{"\[Gamma]", " ", "\[Delta]"}],
|
|
RowBox[{"\[Gamma]", "+", "\[Delta]"}]]}], " ",
|
|
SuperscriptBox[
|
|
RowBox[{"Norm", "[",
|
|
RowBox[{"C", "-", "D"}], "]"}], "2"]}], "]"}]}]}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.5764509073310003`*^9, 3.576451028887642*^9}, {
|
|
3.576460736353737*^9, 3.576460738268038*^9}, 3.576464845679883*^9, {
|
|
3.5799161578550377`*^9, 3.579916159903202*^9}, {3.579916203518588*^9,
|
|
3.579916203677882*^9}, {3.656032713388176*^9, 3.6560327526888027`*^9}, {
|
|
3.656032805105885*^9, 3.656032821823473*^9}, {3.6560333882078037`*^9,
|
|
3.656033412302227*^9}, {3.6560334619334507`*^9, 3.6560334911467667`*^9}, {
|
|
3.6560341750473127`*^9, 3.6560341763679028`*^9}, {3.656034591807816*^9,
|
|
3.656034593991696*^9}, {3.656034836229966*^9, 3.656034848075869*^9}, {
|
|
3.657419260329598*^9, 3.657419274128447*^9}, {3.6574325601692133`*^9,
|
|
3.657432565287834*^9}, {3.779526885627684*^9, 3.7795269060209312`*^9}, {
|
|
3.779599173326108*^9, 3.7795991739647827`*^9}, {3.790614205920676*^9,
|
|
3.790614210485475*^9}},
|
|
CellLabel->"In[11]:=",ExpressionUUID->"440fb372-2001-4d90-ade1-e5498676702f"]
|
|
}, Open ]]
|
|
}, Closed]]
|
|
}, Open ]],
|
|
|
|
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]]
|
|
}, Closed]],
|
|
|
|
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]]
|
|
}, Closed]]
|
|
}, 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"]
|
|
}, Closed]],
|
|
|
|
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"]
|
|
}, Closed]],
|
|
|
|
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"]
|
|
}, Closed]],
|
|
|
|
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 ]]
|
|
}, Closed]],
|
|
|
|
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", "=", "10"}], "}"}], ",", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Psi]", "n_"], "[", "u_", "]"}], "=",
|
|
SuperscriptBox["u",
|
|
RowBox[{"n", "-", "1"}]]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"H", "[", "R_", "]"}], "=",
|
|
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[{
|
|
RowBox[{"S", "[", "R_", "]"}], "=",
|
|
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}},
|
|
CellLabel->
|
|
"In[173]:=",ExpressionUUID->"8d489c13-936e-4272-b8f6-29963888bb9f"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{"Eig0", "=",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Sort", "[",
|
|
RowBox[{"Eigenvalues", "[",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"N", "[",
|
|
RowBox[{
|
|
RowBox[{"H", "[", "R", "]"}], ",", "100"}], "]"}], ",",
|
|
RowBox[{"N", "[",
|
|
RowBox[{
|
|
RowBox[{"S", "[", "R", "]"}], ",", "100"}], "]"}]}], "}"}],
|
|
"]"}], "]"}], "\[LeftDoubleBracket]", "1",
|
|
"\[RightDoubleBracket]"}], "-",
|
|
RowBox[{"(",
|
|
FractionBox["8",
|
|
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", ",", "200"}], "}"}]}], "}"}]}], "]"}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"Eig2", "=",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Sort", "[",
|
|
RowBox[{"Eigenvalues", "[",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"N", "[",
|
|
RowBox[{
|
|
RowBox[{"H", "[", "R", "]"}], ",", "100"}], "]"}], ",",
|
|
RowBox[{"N", "[",
|
|
RowBox[{
|
|
RowBox[{"S", "[", "R", "]"}], ",", "100"}], "]"}]}], "}"}],
|
|
"]"}], "]"}], "\[LeftDoubleBracket]", "2",
|
|
"\[RightDoubleBracket]"}], "-",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
FractionBox["3",
|
|
SuperscriptBox["R", "2"]], "+",
|
|
FractionBox["352",
|
|
RowBox[{"105", " ", "\[Pi]", " ", "R"}]]}], ")"}]}]}], "}"}], ",",
|
|
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
FractionBox["1", "10"], ",",
|
|
FractionBox["1", "5"], ",",
|
|
FractionBox["1", "2"], ",", "1", ",", "2", ",", "5", ",", "10", ",",
|
|
"20", ",", "50", ",", "100", ",", "200"}], "}"}]}], "}"}]}], "]"}]}],
|
|
";"}]}], "Input",
|
|
CellChangeTimes->{{3.78271083027982*^9,
|
|
3.782710830463963*^9}},ExpressionUUID->"af4fe8f8-c3eb-4495-a149-\
|
|
b4bd1830693b"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Sort", "[",
|
|
RowBox[{"Eigenvalues", "[",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"N", "[",
|
|
RowBox[{
|
|
RowBox[{"H", "[", "R", "]"}], ",", "100"}], "]"}], ",",
|
|
RowBox[{"N", "[",
|
|
RowBox[{
|
|
RowBox[{"S", "[", "R", "]"}], ",", "100"}], "]"}]}], "}"}],
|
|
"]"}], "]"}], "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}],
|
|
"-",
|
|
RowBox[{"(",
|
|
FractionBox["8",
|
|
RowBox[{"3", " ", "\[Pi]", " ", "R"}]], ")"}]}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
SuperscriptBox["10",
|
|
RowBox[{"-", "n"}]], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"n", ",", "8", ",", "4", ",",
|
|
RowBox[{"-", "1"}]}], "}"}]}], "]"}]}], "}"}]}],
|
|
"]"}], "\[IndentingNewLine]",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Sort", "[",
|
|
RowBox[{"Eigenvalues", "[",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"N", "[",
|
|
RowBox[{
|
|
RowBox[{"H", "[", "R", "]"}], ",", "100"}], "]"}], ",",
|
|
RowBox[{"N", "[",
|
|
RowBox[{
|
|
RowBox[{"S", "[", "R", "]"}], ",", "100"}], "]"}]}], "}"}],
|
|
"]"}], "]"}], "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}],
|
|
"-",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
FractionBox["3",
|
|
SuperscriptBox["R", "2"]], "+",
|
|
FractionBox["352",
|
|
RowBox[{"105", " ", "\[Pi]", " ", "R"}]]}], ")"}]}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
SuperscriptBox["10",
|
|
RowBox[{"-", "n"}]], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"n", ",", "8", ",", "4", ",",
|
|
RowBox[{"-", "1"}]}], "}"}]}], "]"}]}], "}"}]}], "]"}]}], "Input",
|
|
CellChangeTimes->{{3.753773404008658*^9, 3.753773427239814*^9}, {
|
|
3.753773617591683*^9, 3.7537736405209913`*^9}, {3.7537739124809237`*^9,
|
|
3.753773921334259*^9}, {3.782710828009213*^9,
|
|
3.782710828215776*^9}},ExpressionUUID->"eb8a2378-386f-4e82-b835-\
|
|
17c5b76bd346"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{"FindFit", "[",
|
|
RowBox[{"Eig0", ",",
|
|
RowBox[{
|
|
FractionBox["a",
|
|
RowBox[{"1", "+",
|
|
RowBox[{"b", " ",
|
|
SuperscriptBox["R",
|
|
RowBox[{"1", "/", "2"}]]}], "+",
|
|
RowBox[{"c", " ", "R"}]}]], "/.",
|
|
RowBox[{"a", "->",
|
|
RowBox[{"-", "0.04763687334741862`"}]}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"b", ",", "c"}], "}"}], ",", "R", ",",
|
|
RowBox[{"MaxIterations", "\[Rule]", "10000"}]}], "]"}],
|
|
"\[IndentingNewLine]",
|
|
RowBox[{"(*",
|
|
RowBox[{"FindFit", "[",
|
|
RowBox[{"Eig1", ",",
|
|
RowBox[{
|
|
FractionBox["a",
|
|
RowBox[{"1", "+",
|
|
RowBox[{"b", " ",
|
|
SuperscriptBox["R",
|
|
RowBox[{"1", "/", "2"}]]}], "+",
|
|
RowBox[{"c", " ", "R"}]}]], "/.",
|
|
RowBox[{"a", "->",
|
|
RowBox[{"-", "0.05656280"}]}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"b", ",", "c"}], "}"}], ",", "R", ",",
|
|
RowBox[{"MaxIterations", "\[Rule]", "10000"}]}], "]"}],
|
|
"*)"}]}], "\[IndentingNewLine]",
|
|
RowBox[{"FindFit", "[",
|
|
RowBox[{"Eig2", ",",
|
|
RowBox[{
|
|
FractionBox["a",
|
|
RowBox[{"1", "+",
|
|
RowBox[{"b", " ",
|
|
SuperscriptBox["R",
|
|
RowBox[{"1", "/", "2"}]]}], "+",
|
|
RowBox[{"c", " ", "R"}]}]], "/.",
|
|
RowBox[{"a", "->",
|
|
RowBox[{"-", "0.02892648"}]}]}], ",",
|
|
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}},
|
|
CellLabel->
|
|
"In[205]:=",ExpressionUUID->"2f2c8334-f314-4d97-9abf-fd397dbace2b"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"b", "\[Rule]", "0.0077278287742736755`"}], ",",
|
|
RowBox[{"c", "\[Rule]", "0.17834979356404287`"}]}], "}"}]], "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}},
|
|
CellLabel->
|
|
"Out[205]=",ExpressionUUID->"f04e12cb-ae53-4b96-8239-58bd2665964b"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"b", "\[Rule]",
|
|
RowBox[{"-", "0.07430699598043342`"}]}], ",",
|
|
RowBox[{"c", "\[Rule]", "0.07118268293633198`"}]}], "}"}]], "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, 3.782710725917925*^9}},
|
|
CellLabel->
|
|
"Out[206]=",ExpressionUUID->"34d77878-3289-4d0f-a73e-9cb3a0e22085"]
|
|
}, 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", "=", "0.00843194"}], ",",
|
|
RowBox[{"c", "=", "0.178161"}]}], "}"}], ",",
|
|
FractionBox["a",
|
|
RowBox[{"1", "+",
|
|
RowBox[{"b", " ",
|
|
SuperscriptBox["R",
|
|
RowBox[{"1", "/", "2"}]]}], "+",
|
|
RowBox[{"c", " ", "R"}]}]]}], "]"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]c2", "[", "\[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[{"-", "0.0738888"}]}], ",",
|
|
RowBox[{"c", "=", "0.0710620"}]}], "}"}], ",",
|
|
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}},
|
|
CellLabel->"In[13]:=",ExpressionUUID->"6a07f7bc-64fd-4691-a797-042e1daf8a9c"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Show", "[",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"Plot", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{"\[Epsilon]c0", "[",
|
|
FractionBox["2",
|
|
RowBox[{"2",
|
|
SuperscriptBox["\[Pi]", "2"],
|
|
SuperscriptBox["R", "3"]}]], "]"}]}], ",",
|
|
RowBox[{"2",
|
|
RowBox[{"\[Epsilon]c1", "[",
|
|
FractionBox["2",
|
|
RowBox[{"2",
|
|
SuperscriptBox["\[Pi]", "2"],
|
|
SuperscriptBox["R", "3"]}]], "]"}]}], ",",
|
|
RowBox[{"2",
|
|
RowBox[{"\[Epsilon]c2", "[",
|
|
FractionBox["2",
|
|
RowBox[{"2",
|
|
SuperscriptBox["\[Pi]", "2"],
|
|
SuperscriptBox["R", "3"]}]], "]"}]}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",", "0", ",", "200"}], "}"}], ",",
|
|
RowBox[{"PlotRange", "\[Rule]", "All"}]}], "]"}], ",",
|
|
RowBox[{"ListPlot", "[",
|
|
RowBox[{"{",
|
|
RowBox[{"Eig0", ",", "Eig1", ",", "Eig2"}], "}"}], "]"}]}], "}"}],
|
|
"]"}]], "Input",
|
|
CellChangeTimes->{{3.744904061183731*^9, 3.744904100316234*^9},
|
|
3.7449041488210297`*^9, {3.744904416154532*^9, 3.744904417611268*^9}, {
|
|
3.744904471849855*^9, 3.744904516909964*^9}, {3.744904621486627*^9,
|
|
3.74490462983591*^9}, {3.7449047012735453`*^9, 3.74490473795422*^9}, {
|
|
3.744904777400316*^9, 3.744904809786846*^9}, {3.7451445213517923`*^9,
|
|
3.745144538159807*^9}, {3.7483211654695377`*^9, 3.7483211668756*^9}, {
|
|
3.7483213040395203`*^9, 3.74832130527619*^9}, {3.7483648013275642`*^9,
|
|
3.7483648177808647`*^9}, {3.748365116232815*^9, 3.748365179095387*^9}, {
|
|
3.74836521192799*^9, 3.7483652230795307`*^9}, {3.753773558121643*^9,
|
|
3.753773561447156*^9}, {3.753774182032125*^9, 3.7537741863590183`*^9}, {
|
|
3.7537742351091747`*^9, 3.753774248000457*^9}},
|
|
CellLabel->"In[15]:=",ExpressionUUID->"6e23a9ca-c6d3-4d4a-b7ff-3092bcade88a"],
|
|
|
|
Cell[BoxData[
|
|
GraphicsBox[{{{{}, {},
|
|
TagBox[
|
|
{RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[1.6],
|
|
Opacity[1.], LineBox[CompressedData["
|
|
1:eJwVj2k81AsbhlWE5FSo5KjGki1h0BwmzNMg+1TDWFMyln8NsmWJGGu2ZGyj
|
|
7CVbHaKkjvTQnjZJUnJkC0dZkkTJ2/vh/l1frg/XLed+jOm5XEBAAH/v/5Q/
|
|
s3ihaMtLo+Q6YbdmdiU2iqhu9SVqaUnZsq2FkpXYGlZTpkIgLXHTirbapxX4
|
|
QCvAeyXxgkZ9pJcftacCH6V1mM9699FItUJruhjlOBW/eiSsup9GM/PbFilV
|
|
jifv1qrOlQzS5MjPmUd6LmIufKuZOT1CK+wM7u33uYiPqXE3P3lP0viPVxT/
|
|
UViGlwwko3bWTtHeqaZYX+CU4XSPR3Fw9TSN8r54Um1XGUbJiAx8KZmhFdum
|
|
ys++v4B8PsN78vQcrXt0pqxR7QK28XoCRr2XaKnnp1dOj5ViUl7u4yyyABxg
|
|
DoW9vFeK6gTNWadWAM7GKndbl5RioDAvIrB6GXh81JQScirFX6a6t6dKVkDM
|
|
sGHhxpclKH33hPHn08JQkrUwJVhfjE2ZCp1pYiLQXaZzVzisGA+6P/VQTxYB
|
|
06IW4z6jYixbtiWREy8K8ZapaxaeFyEZWh+NnRCDnjWbGV7fC9GyWZjx0XsN
|
|
PNrbMNXgU4DkJffZDVNrQNzUpJNKLcA00qHnCgZrIfi0tlCQSAEGu/9z3O7e
|
|
WlDRrQubKM9Hk4/+9xo61wG/yJJTO34OBz73Hg6blYSnj0qcs3hnsSs9tSFD
|
|
XwpOvRQjtRJnsU1TX7TqpBQ40lrkA+As1gdkX3krtB5C/vtTlzGdhzGzlkvU
|
|
DRtAUGT5lluOeUhabCxYpEjD5jP7T2cY8FGqyHNqfYQ0XGWZLr6X4aMITdJE
|
|
A6XhVqOqROp8Lk5y/cYPmm0CsoxrjtCNXLwtuI3a4iADec8ufDXTz0XX1Zlv
|
|
YsNk4ZOoTnGzVQ4+E6eICrqSIJP0aNW9iizssJLIex5JghTp/24CLwu7kyeU
|
|
zhaQ4G059/PUiSwcEqow0XhPAjeykMIYIwt/Lm7kOrrIQcW1dKrSfCaqTcx/
|
|
u+wkD6S+uc3m9pmopd6VEBouD9GDzs1X6JlIOVovRT8rD7fvLu0w1sxE+sej
|
|
2m+65UE+tNxvUCQTnf7t8V3mqAAFfm6OK27zMOl58xDLXhHe8No77TV5mL76
|
|
XBApRBGUmAp+8lt4mG0Zsnw8RxGMKf3+a8V5WPxAg8R9rQj6IvMbmeMZeP12
|
|
sUu13TYgm1BWHa7KwOGamI5fTCVoObzWzGV7BtLPmLZU7FUBGz0V2lhSOnIt
|
|
r1V1+KlAiXXeXQu/dLwtpJC1eFoFZLbYVnXbpiM1UsDL9qkKHBu2vh69NR11
|
|
jjSJLZmrwl1lNYGPN06jojHZwZ6uBvPK4hV1M2m4ck52QnCnOjzhWQRwE1PR
|
|
pD71jaadOsi96Vr41z8VY30XWpyD1CFl+9OIQy6p+GvwTdaVOnXYq8jedl8r
|
|
FefaM/UPaOyA8/ds+DK9KThaLZJwVVkDpHfs+355Vwp6ai6Ik8W1YDo/rjrr
|
|
j2S07jl1UXqLFmT/J/O3wo8k1D213lBAUwvYnvwXL0eSULBPy+fFPi149SrA
|
|
9VRrEp5P927zzdYCuT287ftCkrDvc2di9Z9k+NArcH7r8Cl0uly7pKCmDW7a
|
|
ZhFSXYkIjoZ8sV3aEKyogJP3E1FF8InGjJU2DIX6t/c3JOJ3l4+ud3y1IaFu
|
|
9pxobiLmim2+5VanDTc+SVW7OyTiq6MpYYV6OmBw7ZGK5L8JaKPqOb3BTBdy
|
|
DR+UWCzE40Lq8aZee13IT7pVnPk5HisnEhLKvHRhsOym0PiHeBS8Vr5JO1EX
|
|
1LhG/McP4/Efo9HdNvd14ae+WQc1Nx6V7DiZccY7oUZiJLlrZzwKRAfoTNEo
|
|
IOGoYbsnOg5r+mMWr++lgP12hxbL4Dh0Mcl8ePIQBWSd5lNcjsThddGrLmLR
|
|
FFhjxxkutY1Dn+yvcUq3KfBFOSgQVeLwbVVo5wGDv0AviZGv/joW6ztPHn+s
|
|
pwfXXAu1bCixWJng4T5goQfKOTH08e2xWESx2vvDWQ9Kther5MjFYkqetOqO
|
|
k3rgsXGGtFo8FtkHrr7PaNUDwb4BfvxQDEoNjRo7WuqDsRn2pufEYMiMrcSI
|
|
CxUak2uVUpe46FNGXVryocId+6ju6G9cdGfJfZKOooJcA1sl6jMXGY0T9y1L
|
|
qHBZxMKT38NF5Yjk8JohKtyP9q6h3uBi93L8cNx3F8xMH2r6K4iLuyTUagWj
|
|
DWBEa9jOoS8any9zfvzsjAGQXrUkNr2IRrfp5MHcEgPInvuypNESjfHtYxtV
|
|
7xiAT9CQn3Hpbz+9imsjZAiv3Xd1bfX47YupMnNTDcGmovaP7E9RGC+oMquc
|
|
ZwSVf+4++l44Cr1/ZBZedAT4mW5lO8OOxGzGp7vhbgClYxsGPB0jsaXU9D8b
|
|
AoDT9bKr3zoSpc2/U76FAjQ3296b3RmJj7Jd2/fwAb6PGAWliUSiqobKitHX
|
|
ABP9EeGc2ggcd2sm1Gx3w48Bl8DyFRF47MGITq0NHZwDqOWzd8OxBS5z1O3p
|
|
sCYyuqfsZjiubfK/UHWQDgUsT+aB2nC8UjsvUXaMDiSvxN7Bc+H4JU/sSx6P
|
|
Djur6hzsAsMxhKN5JfY1HWw0k2Pr5cMxcm2our2rMSxuezaikByGKc4rty36
|
|
mEDTtmZLEU4oGmmt/jYWbAIB0+crJNmhOC0k8bAr0gTM//7EUXQJRce6zUeu
|
|
pJlAe3ydlL1VKG4T2fk3+7IJrMMHjPntodjSwN7ZNm4Cu8tJTdSJEPy2ttWU
|
|
f9QUDhRkZSaEhCD7YYQXmdgDMuFsBjnnOLYKb9rPcTIHnXz7BFGxYCSv4+aO
|
|
BVvCVkWd+ZBzgeg2Y6jO8LGGyvEhV8djAVhadF927jQD7ufvaOpj+uPVdGpP
|
|
B3MfGJ8M3lV+0w+D/zHcV2i9HwobvvcByRex7eKXZBcmMK7B8h4eB4+Y6Tcf
|
|
cbSFzhT8q5B+FAsqyQ2co3ZAbtxdyZsnkDfTHOMfyoKhsjs//Tq98WP18RAi
|
|
0B6cJnNa9d554b09mXmxsQ6A40ubVPo9cdnedwNnuI5g+dWMISrmiQK7ry8r
|
|
CXeCW6pKKYHmHpi0cc6m45Qz2IUaBFfEs9HfmX1nLs4F2vqEiPZ37tgfo+tB
|
|
yzkAEos7SExNd5TKOvsuudQVyGDiyMk8jOVr1cIf5h8E6r/PqEzRw9gauNVU
|
|
+dIh+JOe/aHtpBsWkUpkDe3cICEwV+bQCjdsjlwQINrdYNU6fr2c4iGUsxL+
|
|
qeBwGGKFYywEDA7itReqqW87D4N2qz3J6pgrpj34pUVydYfusazwPYUHMCto
|
|
7Kjbe3cYNZScDBt0wcj1W9bIO7Fhs4dgshHFBatUhdqCP7BhV4pqX36cM678
|
|
umxT60EPWM3YnPZtyAkrtOqiikY8QD/HtIGy2wlV3asZi76eQAqtaZevdMQe
|
|
i5v1rHFP0O4VSry11RE7LpzYQQvyApfRFJ5TpgN+v7rfQP6LF9gFWVtLSjng
|
|
0o3AhkQ/b7i89cbc3kJ7tNLaYxo56w2yAbEJKGePD101j3CCCJhzmlu++SoL
|
|
JdhEU+FxApZOSHTo1bHwIFEq3h5KQLE/f9auloXfAiXrdSMJ4PAT9dMvsVAx
|
|
aW5hMZ4AgzpevWAZC7n1mJaRS4DfRh37ZTkspArvu9JwkwCpnMkneqEsTFyd
|
|
vHy0iQD6ValfB4+z8OW6O3Yytwmw+DUZkxDEQkJWdz7qDgEZ/OCoV8dYyCdL
|
|
082fEDDQ5W0eRLDwq8uHV+/eEyBqViR+x4mFcHiTkngfATnPbfZ/dmBhqhcz
|
|
jNZPAOmWx7C0PQvlA+7Jlg3/7pMI+eXPZOH+xEpP34nf/ZThaUUrFhak9jcW
|
|
TxGwYJZ+nWnBwtEMmVUdXwhgiy60cc1+/8lPq6HM/fYfPBzoNWbh05L7Akfm
|
|
CXg7KvBxNZ2FG8t/MfN/ENAocknPAFjofumvi88WCRhsq3vHMWJhzRX/uaUl
|
|
Arb0CDzNN2Dh/wDkncNU
|
|
"]]},
|
|
Annotation[#, "Charting`Private`Tag$1749#1"]& ], {},
|
|
TagBox[
|
|
{RGBColor[0.560181, 0.691569, 0.194885], AbsoluteThickness[1.6],
|
|
Opacity[1.], LineBox[CompressedData["
|
|
1:eJwVlXk81IkfxpFQkZAirXWV4ydGYbHq+8mVaChUjmhmMPOV+0wiR+PKPWJc
|
|
M02FSkpLpZI+uZfSodoOWUvlWCUlKWK23x/P6/3H8349/z7qjFAXfzEREZHS
|
|
n/k/NfIWzvJVH28LPSjxeJmAh41Sur8Gk3WE7cV1LNXlfGyJvVypQyJxh61u
|
|
4KLJx0LTtwdkcrsIu/ot7l0GfOykhLMkyIdEvcOTQ3Jb+BivKLwhFDwhSrJo
|
|
b1o387GYY7vtR85zYq96yhmKIR/rZLPbvsX1E7E1oSLaunz8M7vPfoY1SFxP
|
|
7tytr87HKbb0aGzNEJHJ+VzHX8vHhLY63VnBG6Ih7ZhpmzQfl4u5BkVz3xFB
|
|
nNojD0R+7sPXy9M5o4R0XGrl+DQPNRNLP4Wzx4nyV7L1JiM8vNJsaTwVN0Gc
|
|
srif1/uch90Wx2++Z30kOpKfZZje5OFFS4VjJnVTRKX/DEvvAg8/9fudiqr5
|
|
RExaJny4UsLDY+ukhj8LpokL06tXVMbwkMt1Yn3MmSVMYgzFVQ152FPQHz7G
|
|
EhJmW8szznErMKOkuLvQSARivG8ICHYF6pOE55Y6EfBZK+jpDKvACMmCoxE1
|
|
ohDGFpRd3FmBi7bGd6YES8BgcEZJcr4cldrirD/kSAL/i2iej3c5OjRLOo2w
|
|
ZEGoYzZD1S5DIyFjZs2ULIh+MaM2yZdhttrBB5qWq+BSm02Z+WIpRjFuRbu1
|
|
r4KP1xOOHnpWijYjYe3XnsqBScnAmyp2KQ5/GKDHzigAl6Jj3jhSgmoLjRUL
|
|
pkpQ855rt/EWF1fz/acUjyrBnoUzHivPc1GKULAxQCXYvmHacUUxFz8mhUz4
|
|
7FAGJ3bKDesILt4R32Bxd/86sGRLDFz6Hxe9pTnPU2LXw6TzMHqfKcZeGdNl
|
|
4t5qMLeuRraZX4R9jvIlD+LVQCadccQwvwhfZE5uLK342f9l5VGfXIRvl56z
|
|
MXitBl+D/9087leEPxbWJrl7qUNARZOjsn4R6k1+/1rroQFtf5bfjmg+iRkP
|
|
mt/u3acF2u8/XEmdKMRc6bJItRgtWK2yJW3470I86RAjNlGkBT3XH0VT+wrx
|
|
VKeBWtIzLQgtkvCg3izE63dOedW4bYCIgBqt6fRCfHc5uW/RZSM4+thd6NQu
|
|
RKs827vnnHVAh8jLlw3nYJLD1Qt9ITrw7yVepTWTg3eWahYu5OiAhWOm5nEv
|
|
DlrEizBd7+tA55cSTS07Dm4JaFohtNeFH7sc49+rcFDL2mj/Pis96PiY5JTZ
|
|
XYASs+snxU30gdO+JCVHvwBt6rOeG7rpA8/95GZ5zQJMCZ676xmpD03y40kC
|
|
5QJcfPO88Mof+uC6+VDrS4kCnH3EMT9gsAl+D30a3DyUj2M1UqkN2gYw0r2P
|
|
uq00H/0N52SMZCjwr4yy5eOV+birP71KSZUCgbsc9txemo/G6YpbRQwpsHFf
|
|
wlv69zwUH6QEPdxNgZ3deyH4WR6eyWX1BJ+kgLh89AG/3Dwc/PA0rUbFCH43
|
|
of59e0keetTWCTX1NoNd2vNW//kcpOr6f1qzwxj8KYplBcrZOJcV3TSwzxgM
|
|
zV+2HZTOxvOTqamVTGNoP02cMRdmofjVauXNacawQ0zFatW7LLy1bWw7tcMY
|
|
FCNUD+/4Iws3ugVyjlubADFhbv/YMQtFEsO3TBGmcNe/JW464wTWP02I7jYz
|
|
AxflZdG31DLxfKofY3inGdDNiVo9xUzkmzo6z3uagaf0CqmqZZl4okRJd1OC
|
|
GTD/dJhr/JyBvgcaXue3mEFjoQG5tyMDV78ds3Z3MIfPlv0C36AMjJl2lR/1
|
|
soAEm3cbjrak4+/yenXiiZZgkfGiwDI1DR+Ienb35lmCodw8Ryo+DWmfMt8U
|
|
CyxBQot2vz8iDdmPxtfqtlqC+ZB6F5f208+9kERduhUWM8K++1n+9FfouhRn
|
|
bYUO0Xkf6kwqssV1ZrRLtkFLuumYWlAqsuY5vCp3gJVtS6ocgtgY2jm6pY5q
|
|
BdU+/nPfulLwhKfEhoUgG/hNsetekU0y+nYdZRqRdqC41OhX6ZVJ2CKpvCfQ
|
|
wx7UEqM3VUkdQyO5pOLxKAdY7C/uVZCNR9r0Vn2noF0w0n4v/mJIHJ7md6yf
|
|
zXGClc63i0oHYrEh16K/z2U3bKKP+949dhijbm3dzdu1B36xrzZQdo9B7Kn6
|
|
nOnlAgI4K/fNNBoDdpg3B7i7AveXxzED26Ow4rzRtcBDbmBdzbnS4ByJBdPN
|
|
yWGH9wJ5SUzlfEwEjtREx5AR+8D+qd5A8O1wbLfjlKSk7IeZxFOkoVg4ijq/
|
|
Gs5Lcofb3yOMfelhKLL9uqjgiAcIaVnnQ3ihmLF2ltqX7glj/TKniegQDPP0
|
|
bZ097gXNTlIayZHBOJRs7EcUHYDkuBBqQkgQri4sfZV52hvUdXSmliYGYvUq
|
|
vSNd5T6gysodtr10CFsifrXVvngQDAfrwPV5APLVBOu3utHAsWFd/28qAdgc
|
|
PydCPqKBwOPc2L0gEtUdJX9o7qeD/Gz/xMObLLz6UDfr5VM6GD5UoV9XYWF2
|
|
5yJFzZsBVm0tRtZHmFgYOX6I9poBFvNTlH3j/j//UlVWw8MXfnl99dUcwx8v
|
|
6C7tifrHFwbuy26zeuGHEl9ElVt8/KDYuXzlxAE/PEf54xh/1A9UQz+0xA74
|
|
oi6jxmkh2B+qhRWvjJm+2L/zZv3eCX9YUZvy49k8A/vOxm0iIpkw3zdZFZfD
|
|
wG8Neyw1PjNBw/HGx2d6DBTeiLiWFsICr3GJyoVeOjpS7GzjZ1jgtGrohEsw
|
|
Hbu8DQMCI0kQy85quLaGjvK+ZBMvmgRh9zzbQpGOPuRpmUeHSXDhHeGiAh2/
|
|
RijUG8eTMBrlldizio5aGbNzC2wSAixG3d8sp2NSPWbnF5NgoFGrpS6koYXk
|
|
7ivXbpIQyDxce3+UhmnSmWJjTSRE5XQmeozQ8LFcq9u6OyScCVTmjLylIbne
|
|
+PuxVhL+STwRKTpMQ66RkpX9PRJennytbfGahl+8/nny6jUJT9yY9o2PaAh0
|
|
5Y0ygyR8G04Wt39IwyymSywxRALbq3LxRS8NNcLb11e+I4FagLXzPTTck3be
|
|
P3iShJOvyiatO2hYkTXUeGqKBAWGeelfbTQcy1+3vO8zCSFMRnJAKw2TyrMv
|
|
m86SkCG3Znk+0vC+oEMk4DsJK/uTazTv0HBt9aJL+TwJi0yLjMbbNGRc/K2q
|
|
d4GEElfKWccmGl6+EjYrFJJgYuM2P3iThv8B7UGuXg==
|
|
"]]},
|
|
Annotation[#,
|
|
"Charting`Private`Tag$1749#3"]& ]}, {}}, {{}, {}, {{}, {}}}},
|
|
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,
|
|
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->{{3.744904066101049*^9, 3.7449041008582478`*^9},
|
|
3.744904160840914*^9, 3.7449042782036343`*^9, {3.744904508074739*^9,
|
|
3.7449045174623337`*^9}, 3.744904630372691*^9, {3.744904666640123*^9,
|
|
3.744904758220169*^9}, {3.744904810277049*^9, 3.7449048401872*^9},
|
|
3.744910399839883*^9, 3.744957252113614*^9, 3.744964218493539*^9,
|
|
3.7449721676575813`*^9, 3.745082533941929*^9, 3.745118967167205*^9,
|
|
3.745126129763678*^9, 3.74512678481641*^9, {3.745144514348254*^9,
|
|
3.74514453876085*^9}, 3.745144782321116*^9, 3.747980422075601*^9, {
|
|
3.7483211430867367`*^9, 3.748321167334876*^9}, 3.748321204346262*^9,
|
|
3.748321305891493*^9, 3.748321375957294*^9, 3.748321408146278*^9,
|
|
3.748361706344314*^9, 3.748361921194618*^9, {3.748364798036769*^9,
|
|
3.748364818312265*^9}, {3.748365084569332*^9, 3.748365232153314*^9}, {
|
|
3.7537742391103*^9, 3.753774248868038*^9}, 3.790598345376995*^9},
|
|
CellLabel->"Out[15]=",ExpressionUUID->"ec22f91b-cb1e-44dd-995e-82b7699d74af"]
|
|
}, Open ]]
|
|
}, Open ]]
|
|
}, Closed]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell["Uniform scaling", "Section",
|
|
CellChangeTimes->{{3.78488690364141*^9,
|
|
3.784886906701776*^9}},ExpressionUUID->"36f69d13-fe6b-4a04-acec-\
|
|
16703fdd0e12"],
|
|
|
|
Cell[TextData[{
|
|
"The ",
|
|
Cell[BoxData[
|
|
FormBox[
|
|
SubscriptBox["C", "x"], TraditionalForm]],
|
|
FormatType->"TraditionalForm",ExpressionUUID->
|
|
"9f63fd3f-a1cb-4258-b502-65d8e1bfb1dd"],
|
|
"coefficient which gives the exact exchange energy for the hydrogen atom is"
|
|
}], "Text",
|
|
CellChangeTimes->{{3.78488692311138*^9,
|
|
3.784886962794784*^9}},ExpressionUUID->"17ce12bb-a006-40f9-ab62-\
|
|
8a0036a5342f"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox[
|
|
RowBox[{"20", " "}], "27"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["\[Pi]", "2"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}], "//", "N"}]], "Input",
|
|
CellChangeTimes->{{3.784883266461767*^9, 3.784883266886877*^9}, {
|
|
3.7848869772383013`*^9, 3.784886979000092*^9}, 3.7848870505946007`*^9},
|
|
CellLabel->
|
|
"In[130]:=",ExpressionUUID->"44e4ae27-771e-4e10-b2a8-3f5c0700fb97"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"-", "0.8610721122293528`"}]], "Output",
|
|
CellChangeTimes->{3.7848832673151903`*^9, 3.784886980796595*^9,
|
|
3.78488705101871*^9},
|
|
CellLabel->
|
|
"Out[130]=",ExpressionUUID->"47184591-6c91-4963-aebc-73b69a443808"]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox[
|
|
RowBox[{"20", " "}], "27"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["\[Pi]", "2"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}],
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["3", "4"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["3", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}]], "//", "N"}]], "Input",
|
|
CellChangeTimes->{{3.7848870046187353`*^9, 3.784887080745246*^9}},
|
|
CellLabel->
|
|
"In[131]:=",ExpressionUUID->"31e71679-e385-4ae9-bf70-b310ce826eaf"],
|
|
|
|
Cell[BoxData["1.1658816487244295`"], "Output",
|
|
CellChangeTimes->{{3.784887019498967*^9, 3.784887040097371*^9},
|
|
3.784887081183888*^9},
|
|
CellLabel->
|
|
"Out[131]=",ExpressionUUID->"dda12046-695a-4ac3-bf5e-c8600e6dcaef"]
|
|
}, Open ]],
|
|
|
|
Cell[TextData[{
|
|
"The ratio of this coefficient with the LDA ",
|
|
Cell[BoxData[
|
|
FormBox[
|
|
SubscriptBox["C", "x"], TraditionalForm]],
|
|
FormatType->"TraditionalForm",ExpressionUUID->
|
|
"f626843a-763b-4282-8e18-57cf08a337ce"],
|
|
" coefficient does not violate the Lieb-Oxford bound (which is equal to \
|
|
1.174)."
|
|
}], "Text",
|
|
CellChangeTimes->{{3.784887103891222*^9, 3.784887131716177*^9}, {
|
|
3.784887167296002*^9,
|
|
3.784887189325076*^9}},ExpressionUUID->"0ad72096-b75c-4b05-aea4-\
|
|
1c9a3ba2d63e"],
|
|
|
|
Cell[TextData[{
|
|
"The ",
|
|
Cell[BoxData[
|
|
FormBox[
|
|
SubsuperscriptBox["C", "x",
|
|
RowBox[{"(", "0", ")"}]], TraditionalForm]],
|
|
FormatType->"TraditionalForm",ExpressionUUID->
|
|
"d4a04e61-b7ea-485b-8800-f38fa83ec53e"],
|
|
" coefficient does violate the Lieb-Oxford bound by a significant amount"
|
|
}], "Text",
|
|
CellChangeTimes->{{3.784887244991795*^9, 3.7848872555813847`*^9}, {
|
|
3.784887288097371*^9,
|
|
3.784887297712399*^9}},ExpressionUUID->"f3cf0761-27e0-47e5-819a-\
|
|
4f76e0d90b9b"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["4", "3"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["1", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}],
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["3", "4"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["3", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}]], "//", "N"}]], "Input",
|
|
CellChangeTimes->{{3.784887267423334*^9, 3.784887283145205*^9}},
|
|
CellLabel->
|
|
"In[133]:=",ExpressionUUID->"77eca9b4-30de-4a4a-8d4c-a627ab7872b0"],
|
|
|
|
Cell[BoxData["1.2326422655122395`"], "Output",
|
|
CellChangeTimes->{{3.7848872733320503`*^9, 3.7848872836149473`*^9}},
|
|
CellLabel->
|
|
"Out[133]=",ExpressionUUID->"ef1cfa21-7595-4fd6-a4fa-a0c034a8b13d"]
|
|
}, Open ]],
|
|
|
|
Cell[TextData[{
|
|
"The ",
|
|
Cell[BoxData[
|
|
FormBox[
|
|
SubsuperscriptBox["C", "x",
|
|
RowBox[{"(", "0", ")"}]], TraditionalForm]],
|
|
FormatType->"TraditionalForm",ExpressionUUID->
|
|
"326e8bc4-fe36-4aac-a45e-612cc500bac8"],
|
|
" and ",
|
|
Cell[BoxData[
|
|
FormBox[
|
|
SubsuperscriptBox["C", "x",
|
|
RowBox[{"(", "1", ")"}]], TraditionalForm]],
|
|
FormatType->"TraditionalForm",ExpressionUUID->
|
|
"60f53d5f-4d9b-4474-95f9-680905f6bdd7"],
|
|
" coefficients can be scaled in such a way that it doesn\[CloseCurlyQuote]t \
|
|
does not violate the Lieb-Oxford bound "
|
|
}], "Text",
|
|
CellChangeTimes->{{3.784887207339552*^9, 3.784887235826338*^9}, {
|
|
3.784887310960289*^9,
|
|
3.7848873193914347`*^9}},ExpressionUUID->"c212926d-9477-4fee-9f18-\
|
|
1c758debf2c7"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Solve", "[",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]",
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["4", "3"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["1", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}],
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["3", "4"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["3", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}]]}], "\[Equal]", "1.174"}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.784887340086206*^9, 3.7848873585035954`*^9}},
|
|
CellLabel->
|
|
"In[134]:=",ExpressionUUID->"40b08d32-7779-4afd-83c6-170f189f9a96"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"{",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Alpha]", "\[Rule]", "0.9524255599917547`"}], "}"}],
|
|
"}"}]], "Output",
|
|
CellChangeTimes->{3.784887359346363*^9},
|
|
CellLabel->
|
|
"Out[134]=",ExpressionUUID->"896c805b-7892-4f9b-ac07-361220bdc186"]
|
|
}, Open ]],
|
|
|
|
Cell[TextData[{
|
|
"We can also scale the ",
|
|
Cell[BoxData[
|
|
FormBox[
|
|
SubsuperscriptBox["C", "x",
|
|
RowBox[{"(", "0", ")"}]], TraditionalForm]],
|
|
FormatType->"TraditionalForm",ExpressionUUID->
|
|
"e39724df-2639-4a66-b29a-178ef1826722"],
|
|
" and ",
|
|
Cell[BoxData[
|
|
FormBox[
|
|
SubsuperscriptBox["C", "x",
|
|
RowBox[{"(", "1", ")"}]], TraditionalForm]],
|
|
FormatType->"TraditionalForm",ExpressionUUID->
|
|
"afbc343a-32e2-40e2-8ec7-adb5641ff50d"],
|
|
" coefficients in such a way that it does reproduce the hydrogen atom value"
|
|
}], "Text",
|
|
CellChangeTimes->{{3.784887207339552*^9, 3.784887235826338*^9}, {
|
|
3.784887310960289*^9, 3.7848873193914347`*^9}, {3.784887372968326*^9,
|
|
3.7848873921908484`*^9}},ExpressionUUID->"350a122c-6d8b-461e-bad5-\
|
|
e2fdbc14b8f2"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"Solve", "[",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]",
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["4", "3"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["1", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}],
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["3", "4"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["3", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}]]}], "\[Equal]",
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox[
|
|
RowBox[{"20", " "}], "27"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["\[Pi]", "2"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}],
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["3", "4"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["3", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}]]}], "]"}]], "Input",
|
|
CellChangeTimes->{{3.784887340086206*^9, 3.7848873585035954`*^9},
|
|
3.784887404053864*^9},
|
|
CellLabel->
|
|
"In[135]:=",ExpressionUUID->"ffbc7770-1424-4564-b99b-b9ad1c0639d4"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"{",
|
|
RowBox[{"{",
|
|
RowBox[{"\[Alpha]", "\[Rule]",
|
|
FractionBox[
|
|
RowBox[{"5", " ",
|
|
SuperscriptBox["\[Pi]",
|
|
RowBox[{"2", "/", "3"}]]}],
|
|
RowBox[{"9", " ",
|
|
SuperscriptBox["2",
|
|
RowBox[{"1", "/", "3"}]]}]]}], "}"}], "}"}]], "Output",
|
|
CellChangeTimes->{3.784887359346363*^9, 3.784887404575333*^9},
|
|
CellLabel->
|
|
"Out[135]=",ExpressionUUID->"27597e7b-3c41-49b8-9900-9077d931d997"]
|
|
}, Open ]]
|
|
}, Closed]]
|
|
}, Open ]],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell["Hydrogen molecule", "Title",
|
|
CellChangeTimes->{{3.7850491155949287`*^9, 3.785049118067005*^9}, {
|
|
3.790585065501623*^9,
|
|
3.790585066350246*^9}},ExpressionUUID->"d8a7ce9c-a1d9-4ac0-b8bf-\
|
|
9ca66a94ac58"],
|
|
|
|
Cell["LDA exchange functionals", "Text",
|
|
CellChangeTimes->{{3.790617745116544*^9, 3.7906177720649967`*^9}},
|
|
FontColor->RGBColor[
|
|
1, 0, 0],ExpressionUUID->"d408aeeb-1654-4525-903e-88994da645ee"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xLDA", "[", "n_", "]"}], ":=",
|
|
RowBox[{"Module", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"CxLDA", "=",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["3", "4"]}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["3", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}]}], "}"}], ",",
|
|
RowBox[{"CxLDA", " ",
|
|
SuperscriptBox["n",
|
|
RowBox[{"1", "/", "3"}]]}]}], "]"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]x0", "[", "n_", "]"}], ":=",
|
|
RowBox[{"Module", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"Cx0", "=",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["4", "3"]}], " ",
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["1", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}]}], "}"}], ",",
|
|
RowBox[{"Cx0", " ",
|
|
SuperscriptBox["n",
|
|
RowBox[{"1", "/", "3"}]]}]}], "]"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]x1", "[", "n_", "]"}], ":=",
|
|
RowBox[{"Module", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"Cx1", "=",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["176", "105"]}], " ",
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["1", "\[Pi]"], ")"}],
|
|
RowBox[{"1", "/", "3"}]]}]}], "}"}], ",",
|
|
RowBox[{"Cx1", " ",
|
|
SuperscriptBox["n",
|
|
RowBox[{"1", "/", "3"}]]}]}], "]"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xeLDA", "[",
|
|
RowBox[{"n_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xLDA", "[", "n", "]"}], "+",
|
|
RowBox[{"w",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]x1", "[", "n", "]"}], "-",
|
|
RowBox[{"\[Epsilon]x0", "[", "n", "]"}]}], ")"}]}]}]}]}], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.79059838113128*^9, 3.7905985061910954`*^9},
|
|
3.7906057050230303`*^9, {3.7906109479713078`*^9, 3.790610952959711*^9}, {
|
|
3.790617319564548*^9, 3.790617416879177*^9}, {3.790618439965136*^9,
|
|
3.7906184689663363`*^9}, {3.79066088543036*^9,
|
|
3.790660891318091*^9}},ExpressionUUID->"f51f78c8-fe8e-48f3-a562-\
|
|
1209bfc50f71"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xLDAdn", "[", "n_", "]"}], ":=",
|
|
RowBox[{"D", "[",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xLDA", "[", "n", "]"}], ",", "n"}], "]"}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]x0dn", "[", "n_", "]"}], ":=",
|
|
RowBox[{"D", "[",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]x0", "[", "n", "]"}], ",", "n"}], "]"}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]x1dn", "[", "n_", "]"}], ":=",
|
|
RowBox[{"D", "[",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]x1", "[", "n", "]"}], ",", "n"}],
|
|
"]"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xeLDAdn", "[",
|
|
RowBox[{"n_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xLDAdn", "[", "n", "]"}], "+",
|
|
RowBox[{"w",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]x1dn", "[", "n", "]"}], "-",
|
|
RowBox[{"d\[Epsilon]x0dn", "[", "n", "]"}]}], ")"}]}]}]}]}], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.79059838113128*^9, 3.7905985061910954`*^9},
|
|
3.7906057050230303`*^9, {3.7906109479713078`*^9, 3.790610952959711*^9}, {
|
|
3.790617319564548*^9, 3.790617416879177*^9}, {3.790618439965136*^9,
|
|
3.7906184689663363`*^9}, {3.79066088543036*^9, 3.7906609462132797`*^9}},
|
|
CellLabel->"In[85]:=",ExpressionUUID->"dd062623-5a91-460e-945c-d5f15e24abfb"],
|
|
|
|
Cell["LDA correlation functionals", "Text",
|
|
CellChangeTimes->{{3.790617745116544*^9, 3.790617768448262*^9}},
|
|
FontColor->RGBColor[
|
|
1, 0, 0],ExpressionUUID->"683219a6-4bcc-4fb8-93b5-7c590bc511b5"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"(*", " ",
|
|
RowBox[{"Intermediate", " ", "quantities"}], " ", "*)"}],
|
|
"\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"X", "[",
|
|
RowBox[{"x_", ",", "b_", ",", "c_"}], "]"}], "=",
|
|
RowBox[{
|
|
SuperscriptBox["x", "2"], "+",
|
|
RowBox[{"b", " ", "x"}], "+", "c"}]}], ";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"x", "[", "d_", "]"}], "=",
|
|
SqrtBox["d"]}], ";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"Q", "[",
|
|
RowBox[{"b_", ",", "c_"}], "]"}], "=",
|
|
SqrtBox[
|
|
RowBox[{
|
|
RowBox[{"4", "c"}], "-",
|
|
SuperscriptBox["b", "2"]}]]}], ";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"f", "[", "\[Zeta]_", "]"}], "=",
|
|
FractionBox[
|
|
RowBox[{
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"1", "+", "\[Zeta]"}], ")"}],
|
|
RowBox[{"4", "/", "3"}]], "+",
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "\[Zeta]"}], ")"}],
|
|
RowBox[{"4", "/", "3"}]], "-", "2"}],
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
SuperscriptBox["2",
|
|
RowBox[{"1", "/", "3"}]], "-", "1"}], ")"}]}]]}], ";"}],
|
|
"\[IndentingNewLine]",
|
|
RowBox[{"(*",
|
|
RowBox[{"interpolation", " ", "function"}], "*)"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"vwn", "[",
|
|
RowBox[{"A_", ",", "a_", ",", "b_", ",", "c_", ",", "d_"}], "]"}], "=",
|
|
|
|
RowBox[{"A",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Log", "[",
|
|
FractionBox[
|
|
SuperscriptBox[
|
|
RowBox[{"x", "[", "d", "]"}], "2"],
|
|
RowBox[{"X", "[",
|
|
RowBox[{
|
|
RowBox[{"x", "[", "d", "]"}], ",", "b", ",", "c"}], "]"}]], "]"}],
|
|
"+",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"2", "b"}],
|
|
RowBox[{"Q", "[",
|
|
RowBox[{"b", ",", "c"}], "]"}]],
|
|
RowBox[{"ArcTan", "[",
|
|
FractionBox[
|
|
RowBox[{"Q", "[",
|
|
RowBox[{"b", ",", "c"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{"x", "[", "d", "]"}]}], "+", "b"}]], "]"}]}], "-",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"b", "*", "a"}],
|
|
RowBox[{"X", "[",
|
|
RowBox[{"a", ",", "b", ",", "c"}], "]"}]],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"Log", "[",
|
|
FractionBox[
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"x", "[", "d", "]"}], "-", "a"}], ")"}], "2"],
|
|
RowBox[{"X", "[",
|
|
RowBox[{
|
|
RowBox[{"x", "[", "d", "]"}], ",", "b", ",", "c"}], "]"}]],
|
|
"]"}], "+",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"b", "+",
|
|
RowBox[{"2", "a"}]}], ")"}]}],
|
|
RowBox[{"Q", "[",
|
|
RowBox[{"b", ",", "c"}], "]"}]],
|
|
RowBox[{"ArcTan", "[",
|
|
FractionBox[
|
|
RowBox[{"Q", "[",
|
|
RowBox[{"b", ",", "c"}], "]"}],
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{"x", "[", "d", "]"}]}], "+", "b"}]], "]"}]}]}],
|
|
")"}]}]}], ")"}]}]}], ";"}], "\[IndentingNewLine]",
|
|
"\[IndentingNewLine]",
|
|
RowBox[{"(*",
|
|
RowBox[{"VWN5", " ", "correlation", " ", "functional"}], "*)"}],
|
|
"\[IndentingNewLine]", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]cVWN5", "[",
|
|
RowBox[{"rs_", ",", "\[Zeta]_"}], "]"}], "=",
|
|
RowBox[{"With", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"f2", "=",
|
|
FractionBox["4",
|
|
RowBox[{"9",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
SuperscriptBox["2",
|
|
RowBox[{"1", "/", "3"}]], "-", "1"}], ")"}]}]]}], "}"}], ",",
|
|
RowBox[{
|
|
RowBox[{"vwn", "[",
|
|
RowBox[{
|
|
FractionBox["0.0621814", "2"], ",",
|
|
RowBox[{"-", "0.10498"}], ",", "3.72744", ",", "12.9352", ",",
|
|
"rs"}], "]"}], "+",
|
|
RowBox[{
|
|
RowBox[{"vwn", "[",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
FractionBox["1",
|
|
RowBox[{"6",
|
|
SuperscriptBox["\[Pi]", "2"]}]]}], ",",
|
|
RowBox[{"-", "0.00475840"}], ",", "1.13107", ",", "13.0045", ",",
|
|
"rs"}], "]"}],
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"f", "[", "\[Zeta]", "]"}], "f2"], ")"}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "\[Zeta]"}], ")"}], "4"]}], "+",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"vwn", "[",
|
|
RowBox[{
|
|
FractionBox["0.0310907", "2"], ",",
|
|
RowBox[{"-", "0.32500"}], ",", "7.06042", ",", "18.0578", ",",
|
|
"rs"}], "]"}], "-",
|
|
RowBox[{"vwn", "[",
|
|
RowBox[{
|
|
FractionBox["0.0621814", "2"], ",",
|
|
RowBox[{"-", "0.10498"}], ",", "3.72744", ",", "12.9352", ",",
|
|
"rs"}], "]"}]}], ")"}],
|
|
RowBox[{"f", "[", "\[Zeta]", "]"}],
|
|
SuperscriptBox["\[Zeta]", "4"]}]}]}], "]"}]}], ";"}]}]}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.783426305086109*^9, 3.7834263553520613`*^9}, {
|
|
3.783426480832552*^9, 3.783426485774303*^9}, {3.783426632185308*^9,
|
|
3.783426658830944*^9}, {3.783427391840214*^9, 3.783427392389812*^9},
|
|
3.7834287156166573`*^9, {3.7836773637524137`*^9, 3.7836775311274233`*^9}, {
|
|
3.7906176573902884`*^9, 3.790617676290105*^9}},
|
|
CellLabel->"In[16]:=",ExpressionUUID->"c5e15fc2-7ce6-4ed3-bc7f-bcdc9085134d"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]cLDA", "[", "n_", "]"}], ":=",
|
|
RowBox[{"\[Epsilon]cVWN5", "[",
|
|
RowBox[{
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["3",
|
|
RowBox[{"4", "\[Pi]", " ", "n"}]], ")"}],
|
|
RowBox[{"1", "/", "3"}]], ",", "0"}], "]"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]c0", "[", "n_", "]"}], ":=",
|
|
RowBox[{"Module", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"a", "=",
|
|
RowBox[{"-", "0.0238184"}]}], ",",
|
|
RowBox[{"b", "=", "0.00540994"}], ",",
|
|
RowBox[{"c", "=", "0.0830766"}]}], "}"}], ",",
|
|
FractionBox["a",
|
|
RowBox[{"1", "+",
|
|
RowBox[{"b", " ",
|
|
SuperscriptBox["n",
|
|
RowBox[{
|
|
RowBox[{"-", "1"}], "/", "6"}]]}], "+",
|
|
RowBox[{"c", " ",
|
|
SuperscriptBox["n",
|
|
RowBox[{
|
|
RowBox[{"-", "1"}], "/", "3"}]]}]}]]}],
|
|
"]"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]c1", "[", "n_", "]"}], ":=",
|
|
RowBox[{"Module", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"a", "=",
|
|
RowBox[{"-", "0.0144633"}]}], ",",
|
|
RowBox[{"b", "=",
|
|
RowBox[{"-", "0.0506019"}]}], ",",
|
|
RowBox[{"c", "=", "0.0331417"}]}], "}"}], ",",
|
|
FractionBox["a",
|
|
RowBox[{"1", "+",
|
|
RowBox[{"b", " ",
|
|
SuperscriptBox["n",
|
|
RowBox[{
|
|
RowBox[{"-", "1"}], "/", "6"}]]}], "+",
|
|
RowBox[{"c", " ",
|
|
SuperscriptBox["n",
|
|
RowBox[{
|
|
RowBox[{"-", "1"}], "/", "3"}]]}]}]]}],
|
|
"]"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]ceLDA", "[",
|
|
RowBox[{"n_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]cLDA", "[", "n", "]"}], "+",
|
|
RowBox[{"w",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]c1", "[", "n", "]"}], "-",
|
|
RowBox[{"\[Epsilon]c0", "[", "n", "]"}]}], ")"}]}]}]}]}], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.79059838113128*^9, 3.7905985061910954`*^9},
|
|
3.7906057050230303`*^9, {3.7906109479713078`*^9, 3.790610952959711*^9}, {
|
|
3.790617319564548*^9, 3.790617516913457*^9}, {3.790617702215055*^9,
|
|
3.790617719701961*^9}, {3.7906182605153427`*^9, 3.790618262031168*^9}, {
|
|
3.790618364188952*^9, 3.790618401731085*^9}},
|
|
CellLabel->"In[22]:=",ExpressionUUID->"8db93606-ec73-4fe8-9228-51f79c4ef957"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]cLDAdn", "[", "n_", "]"}], ":=",
|
|
RowBox[{"D", "[",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]cLDA", "[", "n", "]"}], ",", "n"}], "]"}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]c0dn", "[", "n_", "]"}], ":=",
|
|
RowBox[{"D", "[",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]c0", "[", "n", "]"}], ",", "n"}], "]"}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]c1dn", "[", "n_", "]"}], ":=",
|
|
RowBox[{"D", "[",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]c1", "[", "n", "]"}], ",", "n"}],
|
|
"]"}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]ceLDAdn", "[",
|
|
RowBox[{"n_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]cLDAdn", "[", "n", "]"}], "+",
|
|
RowBox[{"w",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]c1dn", "[", "n", "]"}], "-",
|
|
RowBox[{"d\[Epsilon]c0dn", "[", "n", "]"}]}], ")"}]}]}]}]}], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.79059838113128*^9, 3.7905985061910954`*^9},
|
|
3.7906057050230303`*^9, {3.7906109479713078`*^9, 3.790610952959711*^9}, {
|
|
3.790617319564548*^9, 3.790617416879177*^9}, {3.790618439965136*^9,
|
|
3.7906184689663363`*^9}, {3.79066088543036*^9, 3.790660978615497*^9}},
|
|
CellLabel->"In[90]:=",ExpressionUUID->"3c5be8aa-305f-4c0e-a51a-4bc92cdcf171"],
|
|
|
|
Cell["LDA exchange-correlation functionals", "Text",
|
|
CellChangeTimes->{{3.790617745116544*^9, 3.790617768448262*^9}, {
|
|
3.790617800705172*^9, 3.790617802097224*^9}},
|
|
FontColor->RGBColor[
|
|
1, 0, 0],ExpressionUUID->"264d5181-32f7-492d-a01f-33c1ad90708e"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xcLDA", "[", "n_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xLDA", "[", "n", "]"}], "+",
|
|
RowBox[{"\[Epsilon]cLDA", "[", "n", "]"}]}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xc0", "[", "n_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]x0", "[", "n", "]"}], "+",
|
|
RowBox[{"\[Epsilon]c0", "[", "n", "]"}]}]}], ";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xc1", "[", "n_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]x1", "[", "n", "]"}], "+",
|
|
RowBox[{"\[Epsilon]c1", "[", "n", "]"}]}]}], ";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xceLDA", "[",
|
|
RowBox[{"n_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xeLDA", "[",
|
|
RowBox[{"n", ",", "w"}], "]"}], "+",
|
|
RowBox[{"\[Epsilon]ceLDA", "[",
|
|
RowBox[{"n", ",", "w"}], "]"}]}]}]}], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790617806453972*^9, 3.7906178574975023`*^9}},
|
|
CellLabel->"In[26]:=",ExpressionUUID->"f6517707-c164-4ea9-b57c-8d4d27eafad3"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xcLDAdn", "[", "n_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xLDAdn", "[", "n", "]"}], "+",
|
|
RowBox[{"d\[Epsilon]cLDAdn", "[", "n", "]"}]}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xc0dn", "[", "n_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]x0dn", "[", "n", "]"}], "+",
|
|
RowBox[{"d\[Epsilon]c0dn", "[", "n", "]"}]}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xc1dn", "[", "n_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]x1dn", "[", "n", "]"}], "+",
|
|
RowBox[{"d\[Epsilon]c1dn", "[", "n", "]"}]}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xceLDAdn", "[",
|
|
RowBox[{"n_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xeLDAdn", "[",
|
|
RowBox[{"n", ",", "w"}], "]"}], "+",
|
|
RowBox[{"d\[Epsilon]ceLDAdn", "[",
|
|
RowBox[{"n", ",", "w"}], "]"}]}]}]}], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790617806453972*^9, 3.7906178574975023`*^9}, {
|
|
3.790660991606427*^9, 3.790661018422545*^9}},
|
|
CellLabel->"In[94]:=",ExpressionUUID->"33dfcd59-33cc-4f88-bc22-62c17b91dbeb"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xeLDAdw", "[", "n_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]x1", "[", "n", "]"}], "-",
|
|
RowBox[{"\[Epsilon]x0", "[", "n", "]"}]}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]ceLDAdw", "[", "n_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]c1", "[", "n", "]"}], "-",
|
|
RowBox[{"\[Epsilon]c0", "[", "n", "]"}]}]}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xceLDAdw", "[", "n_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xc1", "[", "n", "]"}], "-",
|
|
RowBox[{"\[Epsilon]xc0", "[", "n", "]"}]}]}]}], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.7906610362852507`*^9, 3.790661042510214*^9}, {
|
|
3.790661095924047*^9, 3.790661108253295*^9}, {3.790661781802154*^9,
|
|
3.790661784729022*^9}},
|
|
CellLabel->
|
|
"In[131]:=",ExpressionUUID->"9663902a-8667-481b-b613-f1c2b56b8e76"],
|
|
|
|
Cell[TextData[StyleBox["Molecular coordinates",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.790585499997554*^9,
|
|
3.790585503381317*^9}},ExpressionUUID->"d38e2fe6-260d-4638-b639-\
|
|
6984dd9d14e4"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"ZA", "=", "1"}], ";", "\t",
|
|
RowBox[{"ZB", "=", "1"}], ";", "\t",
|
|
RowBox[{
|
|
RowBox[{"rA", "[", "R_", "]"}], "=",
|
|
RowBox[{"{",
|
|
RowBox[{"0", ",", "0", ",",
|
|
RowBox[{"-",
|
|
FractionBox["R", "2"]}]}], "}"}]}], ";", "\t",
|
|
RowBox[{
|
|
RowBox[{"rB", "[", "R_", "]"}], "=",
|
|
RowBox[{"{",
|
|
RowBox[{"0", ",", "0", ",",
|
|
RowBox[{"+",
|
|
FractionBox["R", "2"]}]}], "}"}]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790612571234482*^9, 3.790612583186882*^9}, {
|
|
3.790612652169979*^9, 3.790612655860166*^9}},
|
|
CellLabel->"In[30]:=",ExpressionUUID->"36b985d7-30a2-430d-8dc3-30d6b4a45727"],
|
|
|
|
Cell[TextData[StyleBox["Basis functions",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.7905855167750597`*^9,
|
|
3.790585520805128*^9}},ExpressionUUID->"e115f0cc-d9e8-4997-a5a4-\
|
|
88b0e9ed97e0"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"With", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"nG", "=", "3"}], ",",
|
|
RowBox[{"d", "=",
|
|
RowBox[{"{",
|
|
RowBox[{"0.1543289673", ",", "0.5353281423", ",", "0.4446345422"}],
|
|
"}"}]}], ",",
|
|
RowBox[{"\[Alpha]", "=",
|
|
RowBox[{"{",
|
|
RowBox[{"3.425250914", ",", "0.6239137298", ",", "0.1688554040"}],
|
|
"}"}]}]}], "}"}], ",", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Chi]", "A"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], ",", "R_"}], "]"}], "=",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{"k", "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]", "k", "\[RightDoubleBracket]"}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
"\[Alpha]", "\[LeftDoubleBracket]", "k",
|
|
"\[RightDoubleBracket]"}]}], "\[Pi]"], ")"}],
|
|
RowBox[{"3", "/", "4"}]],
|
|
RowBox[{"Exp", "[",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
"\[Alpha]", "\[LeftDoubleBracket]", "k",
|
|
"\[RightDoubleBracket]"}]}],
|
|
SuperscriptBox[
|
|
RowBox[{"Norm", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], "-",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "]"}], "2"]}], "]"}]}]}]}], ";",
|
|
"\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Chi]", "B"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], ",", "R_"}], "]"}], "=",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{"k", "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]", "k", "\[RightDoubleBracket]"}],
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox[
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
"\[Alpha]", "\[LeftDoubleBracket]", "k",
|
|
"\[RightDoubleBracket]"}]}], "\[Pi]"], ")"}],
|
|
RowBox[{"3", "/", "4"}]],
|
|
RowBox[{"Exp", "[",
|
|
RowBox[{
|
|
RowBox[{"-",
|
|
RowBox[{
|
|
"\[Alpha]", "\[LeftDoubleBracket]", "k",
|
|
"\[RightDoubleBracket]"}]}],
|
|
SuperscriptBox[
|
|
RowBox[{"Norm", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], "-",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "]"}], "2"]}], "]"}]}]}]}]}]}],
|
|
"\[IndentingNewLine]", "]"}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790585522858306*^9, 3.790585573549855*^9}, {
|
|
3.7905858144379053`*^9, 3.790585875501539*^9}, {3.7905859443497963`*^9,
|
|
3.790585990206952*^9}, {3.790586022013885*^9, 3.790586034909927*^9}, {
|
|
3.790586153134603*^9, 3.790586260399447*^9}, {3.790587429719051*^9,
|
|
3.790587430109853*^9}, 3.790590184504653*^9, {3.790611134040208*^9,
|
|
3.790611142811584*^9}, {3.790612085404666*^9, 3.790612086432859*^9},
|
|
3.790612188260309*^9, {3.7906126717632303`*^9, 3.79061267734007*^9},
|
|
3.790612782078556*^9, {3.790612892956616*^9, 3.790612905674065*^9}, {
|
|
3.790612947951182*^9, 3.790612965035768*^9}, {3.790613311831658*^9,
|
|
3.790613358728168*^9}, {3.790613507527297*^9, 3.79061350977219*^9}, {
|
|
3.790614141437229*^9, 3.7906141534618187`*^9}},
|
|
CellLabel->"In[31]:=",ExpressionUUID->"f0a2825f-92fc-42d2-aeee-57f53535b1d5"],
|
|
|
|
Cell[TextData[StyleBox["Integrals over basis functions",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.790586406973572*^9, 3.7905864081493177`*^9}, {
|
|
3.790587204829371*^9,
|
|
3.7905872084457607`*^9}},ExpressionUUID->"5353bb94-3a0f-4a57-a4c7-\
|
|
bf24da4dc996"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"With", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"nG", "=", "3"}], ",",
|
|
RowBox[{"d", "=",
|
|
RowBox[{"{",
|
|
RowBox[{"0.1543289673", ",", "0.5353281423", ",", "0.4446345422"}],
|
|
"}"}]}], ",",
|
|
RowBox[{"\[Alpha]", "=",
|
|
RowBox[{"{",
|
|
RowBox[{"3.425250914", ",", "0.6239137298", ",", "0.1688554040"}],
|
|
"}"}]}]}], "}"}], ",", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["S", "AA"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalS]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}], "]"}]}],
|
|
";", "\t",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["S", "BB"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalS]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}], "]"}]}],
|
|
";", "\t",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalS]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}], "]"}]}],
|
|
";", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["t", "AA"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalT]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}], "]"}]}],
|
|
";", "\t",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["t", "BB"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalT]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}], "]"}]}],
|
|
";", "\t",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["t", "AB"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalT]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}], "]"}]}],
|
|
";", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["v", "AA"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[ScriptCapitalV]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"ZA", ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}]}], "]"}], "+",
|
|
RowBox[{"\[ScriptCapitalV]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"ZB", ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}], ")"}]}]}]}],
|
|
"]"}]}], ";",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["v", "BB"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[ScriptCapitalV]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"ZA", ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}]}], "]"}], "+",
|
|
RowBox[{"\[ScriptCapitalV]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"ZB", ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}], ")"}]}]}]}],
|
|
"]"}]}], ";",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["v", "AB"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"\[ScriptCapitalV]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"ZA", ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}]}], "]"}], "+",
|
|
RowBox[{"\[ScriptCapitalV]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"ZB", ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}], ")"}]}]}]}],
|
|
"]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "AA"], "[", "R_", "]"}], "=",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["t", "AA"], "[", "R", "]"}], "+",
|
|
RowBox[{
|
|
SubscriptBox["v", "AA"], "[", "R", "]"}]}]}], ";", "\t",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "BB"], "[", "R_", "]"}], "=",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["t", "BB"], "[", "R", "]"}], "+",
|
|
RowBox[{
|
|
SubscriptBox["v", "BB"], "[", "R", "]"}]}]}], ";", "\t",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "AB"], "[", "R_", "]"}], "=",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["t", "AB"], "[", "R", "]"}], "+",
|
|
RowBox[{
|
|
SubscriptBox["v", "AB"], "[", "R", "]"}]}]}], ";",
|
|
"\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"AAAA", "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "3"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "4"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalU]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}]}]}],
|
|
"]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"BBBB", "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "3"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "4"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalU]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}]}]}],
|
|
"]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"ABAB", "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "3"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "4"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalU]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}]}]}],
|
|
"]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"AABB", "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "3"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "4"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalU]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}]}]}],
|
|
"]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"AAAB", "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "3"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "4"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalU]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}]}]}],
|
|
"]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"BBAB", "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "1"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "2"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "3"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
UnderoverscriptBox["\[Sum]",
|
|
RowBox[{
|
|
SubscriptBox["k", "4"], "=", "1"}], "nG"],
|
|
RowBox[{
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"d", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}],
|
|
RowBox[{"\[ScriptCapitalU]", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "1"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "2"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "3"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rA", "[", "R", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"\[Alpha]", "\[LeftDoubleBracket]",
|
|
SubscriptBox["k", "4"], "\[RightDoubleBracket]"}], ",",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "}"}]}], "]"}]}]}]}]}]}],
|
|
"]"}]}], ";"}]}], "\[IndentingNewLine]", "]"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790586409237515*^9, 3.790586416789282*^9}, {
|
|
3.790586519869673*^9, 3.790586646126953*^9}, {3.790586840542666*^9,
|
|
3.790586904158124*^9}, {3.7905869574145403`*^9, 3.790586995550918*^9}, {
|
|
3.7905870746701403`*^9, 3.790587132023671*^9}, {3.790587178278302*^9,
|
|
3.7905871883999863`*^9}, 3.790587412791628*^9, {3.790587490192586*^9,
|
|
3.790587536264368*^9}, {3.790587610680653*^9, 3.79058765873384*^9}, {
|
|
3.790587918649395*^9, 3.7905879881868057`*^9}, {3.790588031536386*^9,
|
|
3.790588064809623*^9}, {3.790588116425434*^9, 3.790588122617651*^9}, {
|
|
3.790589958211094*^9, 3.7905899595395327`*^9}, {3.790590017090168*^9,
|
|
3.7905900647156267`*^9}, {3.790590188107126*^9, 3.790590188443287*^9}, {
|
|
3.790590259300644*^9, 3.79059026249852*^9}, {3.790590572948614*^9,
|
|
3.790590598010453*^9}, {3.79061268483996*^9, 3.790612743776194*^9}, {
|
|
3.790614708170142*^9, 3.790614712227317*^9}},
|
|
CellLabel->"In[32]:=",ExpressionUUID->"341cb469-2a0b-4b3f-a5c7-4a3d8fb61885"],
|
|
|
|
Cell[TextData[StyleBox["Spatial orbitals in minimal basis:",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.790585084725306*^9,
|
|
3.790585104069684*^9}},ExpressionUUID->"5b2593a5-509e-44f9-a6a7-\
|
|
5b7dcf896d99"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Phi]", "1"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], ",", "R_"}], "]"}], "=",
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Chi]", "A"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}], "+",
|
|
RowBox[{
|
|
SubscriptBox["\[Chi]", "B"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}],
|
|
SqrtBox[
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "+",
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R", "]"}]}], ")"}]}]]]}], ";", "\t",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Phi]", "2"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], ",", "R_"}], "]"}], "=",
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Chi]", "A"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}], "-",
|
|
RowBox[{
|
|
SubscriptBox["\[Chi]", "B"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}],
|
|
SqrtBox[
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-",
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R", "]"}]}], ")"}]}]]]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.7905850270009003`*^9, 3.790585079807232*^9}, {
|
|
3.790585431622954*^9, 3.79058543356772*^9}, {3.7905858844788322`*^9,
|
|
3.7905859157992983`*^9}, {3.790587433782695*^9, 3.790587438383739*^9}, {
|
|
3.790613976434304*^9, 3.79061399579806*^9}, {3.790614159389223*^9,
|
|
3.7906141606657143`*^9}},
|
|
CellLabel->"In[33]:=",ExpressionUUID->"3bf77406-a77c-45d7-8c2f-817ad440cdff"],
|
|
|
|
Cell[TextData[StyleBox["One-electron densities",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.790585084725306*^9, 3.790585104069684*^9}, {
|
|
3.790605567798952*^9,
|
|
3.790605573521681*^9}},ExpressionUUID->"8580055f-686c-403d-8800-\
|
|
a4eab889fce4"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["n", "1"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], ",", "R_"}], "]"}], "=",
|
|
RowBox[{"2",
|
|
SuperscriptBox[
|
|
RowBox[{
|
|
SubscriptBox["\[Phi]", "1"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}], "2"]}]}],
|
|
";", "\t",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["n", "2"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], ",", "R_"}], "]"}], "=",
|
|
RowBox[{"2",
|
|
SuperscriptBox[
|
|
RowBox[{
|
|
SubscriptBox["\[Phi]", "2"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}], "2"]}]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.7905850270009003`*^9, 3.790585079807232*^9}, {
|
|
3.790585431622954*^9, 3.79058543356772*^9}, {3.7905858844788322`*^9,
|
|
3.7905859157992983`*^9}, {3.790587433782695*^9, 3.790587438383739*^9}, {
|
|
3.790605575633658*^9, 3.790605631352661*^9}, {3.790611080408037*^9,
|
|
3.790611083617523*^9}},
|
|
CellLabel->"In[34]:=",ExpressionUUID->"4bf40614-2147-47eb-9ce0-13fe949a9c00"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x_", ",", "y_", ",", "z_"}], "}"}], ",", "R_", ",", "w_"}],
|
|
"]"}], "=",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["n", "1"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}], "+",
|
|
RowBox[{"w", " ",
|
|
RowBox[{
|
|
SubscriptBox["n", "2"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}]}]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.7905850270009003`*^9, 3.790585079807232*^9}, {
|
|
3.790585431622954*^9, 3.79058543356772*^9}, {3.7905858844788322`*^9,
|
|
3.7905859157992983`*^9}, {3.790587433782695*^9, 3.790587438383739*^9}, {
|
|
3.790605575633658*^9, 3.790605631352661*^9}, {3.790611080408037*^9,
|
|
3.790611083617523*^9}, {3.790619238747375*^9, 3.790619276696437*^9}},
|
|
CellLabel->"In[35]:=",ExpressionUUID->"33c091a9-0792-4be2-bcd6-e02a256b26b5"],
|
|
|
|
Cell[TextData[StyleBox["Integrals over orbitals",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.790586406973572*^9, 3.7905864081493177`*^9}, {
|
|
3.790587204829371*^9,
|
|
3.790587219589327*^9}},ExpressionUUID->"6e302dfa-38b2-43a3-a507-\
|
|
71bf2633d8e8"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "1"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
FractionBox["1",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "+",
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R", "]"}]}], ")"}]}]],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "AA"], "[", "R", "]"}], "+",
|
|
RowBox[{
|
|
SubscriptBox["h", "BB"], "[", "R", "]"}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["h", "AB"], "[", "R", "]"}]}]}], ")"}]}], "]"}]}], ";",
|
|
"\t",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "2"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
FractionBox["1",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-",
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R", "]"}]}], ")"}]}]],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "AA"], "[", "R", "]"}], "+",
|
|
RowBox[{
|
|
SubscriptBox["h", "BB"], "[", "R", "]"}], "-",
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["h", "AB"], "[", "R", "]"}]}]}], ")"}]}], "]"}]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790587238182136*^9, 3.790587267173367*^9}, {
|
|
3.790587455877968*^9, 3.790587477805868*^9}, {3.7905875469262457`*^9,
|
|
3.790587604542864*^9}, {3.790587670334567*^9, 3.790587690230748*^9}, {
|
|
3.790590272880838*^9, 3.790590291365608*^9}, 3.7905903766007347`*^9},
|
|
CellLabel->"In[36]:=",ExpressionUUID->"41a0d75d-44b6-4653-b70e-858ab3e937b8"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["1",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "+",
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R", "]"}]}], ")"}]}]], ")"}], "2"],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"AAAA", "[", "R", "]"}], "+",
|
|
RowBox[{"BBBB", "[", "R", "]"}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{"AABB", "[", "R", "]"}]}], "+",
|
|
RowBox[{"4",
|
|
RowBox[{"ABAB", "[", "R", "]"}]}], "+",
|
|
RowBox[{"4",
|
|
RowBox[{"AAAB", "[", "R", "]"}]}], "+",
|
|
RowBox[{"4",
|
|
RowBox[{"BBAB", "[", "R", "]"}]}]}], ")"}]}], "]"}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
FractionBox["1",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-",
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R", "]"}]}], ")"}]}]], ")"}], "2"],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"AAAA", "[", "R", "]"}], "+",
|
|
RowBox[{"BBBB", "[", "R", "]"}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{"AABB", "[", "R", "]"}]}], "+",
|
|
RowBox[{"4",
|
|
RowBox[{"ABAB", "[", "R", "]"}]}], "-",
|
|
RowBox[{"4",
|
|
RowBox[{"AAAB", "[", "R", "]"}]}], "-",
|
|
RowBox[{"4",
|
|
RowBox[{"BBAB", "[", "R", "]"}]}]}], ")"}]}], "]"}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["J", "12"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
FractionBox["1",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "+",
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R", "]"}]}], ")"}]}]], ")"}],
|
|
RowBox[{"(",
|
|
FractionBox["1",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-",
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R", "]"}]}], ")"}]}]], ")"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"AAAA", "[", "R", "]"}], "+",
|
|
RowBox[{"BBBB", "[", "R", "]"}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{"AABB", "[", "R", "]"}]}], "-",
|
|
RowBox[{"4",
|
|
RowBox[{"ABAB", "[", "R", "]"}]}]}], ")"}]}], "]"}]}],
|
|
";"}]}], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790587701078045*^9, 3.790587721308972*^9}, {
|
|
3.790588079838372*^9, 3.790588105093894*^9}, 3.790588135725808*^9, {
|
|
3.790588310918952*^9, 3.790588322533786*^9}, {3.790588364656127*^9,
|
|
3.7905883758140497`*^9}, {3.790588409551832*^9, 3.7905884242861843`*^9}, {
|
|
3.790588557126391*^9, 3.7905885830952806`*^9}},
|
|
CellLabel->"In[37]:=",ExpressionUUID->"1430ab04-b05a-4d2d-bbf6-f9edc3cd5345"],
|
|
|
|
Cell[BoxData[{
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["K", "11"], "[", "R_", "]"}], "=",
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R", "]"}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["K", "22"], "[", "R_", "]"}], "=",
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R", "]"}]}],
|
|
";"}], "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["K", "12"], "[", "R_", "]"}], "=",
|
|
RowBox[{"Simplify", "[",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
FractionBox["1",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "+",
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R", "]"}]}], ")"}]}]], ")"}],
|
|
RowBox[{"(",
|
|
FractionBox["1",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-",
|
|
RowBox[{
|
|
SubscriptBox["S", "AB"], "[", "R", "]"}]}], ")"}]}]], ")"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"AAAA", "[", "R", "]"}], "+",
|
|
RowBox[{"BBBB", "[", "R", "]"}], "-",
|
|
RowBox[{"2",
|
|
RowBox[{"AABB", "[", "R", "]"}]}]}], ")"}]}], "]"}]}],
|
|
";"}]}], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790588642589957*^9, 3.790588682679491*^9}, {
|
|
3.790588776310627*^9, 3.790588778758974*^9}},
|
|
CellLabel->"In[40]:=",ExpressionUUID->"09f7d062-2fdd-45b1-8df2-0dd679b5cbc6"],
|
|
|
|
Cell[TextData[StyleBox["Nuclear energy",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.790587253093514*^9, 3.790587257373096*^9}, {
|
|
3.7906590777707577`*^9,
|
|
3.790659080098378*^9}},ExpressionUUID->"94763911-d015-4b98-a645-\
|
|
b27f631322bf"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "NN"], "[", "R_", "]"}], "=",
|
|
FractionBox[
|
|
RowBox[{"ZA", " ", "ZB"}],
|
|
RowBox[{"Norm", "[",
|
|
RowBox[{
|
|
RowBox[{"rA", "[", "R", "]"}], "-",
|
|
RowBox[{"rB", "[", "R", "]"}]}], "]"}]]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790585364582788*^9, 3.790585422341731*^9}, {
|
|
3.790588811486704*^9, 3.7905888669431877`*^9}, {3.7906190335224743`*^9,
|
|
3.7906190337111177`*^9}, {3.790619093049782*^9, 3.790619093357505*^9}, {
|
|
3.7906590832436953`*^9, 3.790659125362174*^9}},
|
|
CellLabel->"In[66]:=",ExpressionUUID->"46e4593c-f9e1-4f4a-a999-d0f67e325be6"],
|
|
|
|
Cell[TextData[StyleBox["CID energies",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.790587253093514*^9, 3.790587257373096*^9}, {
|
|
3.79060545274035*^9,
|
|
3.790605453095336*^9}},ExpressionUUID->"eeb33af8-ab09-4c5f-98a3-\
|
|
684ae529c2b6"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "CID"], "[", "R_", "]"}], "=",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], "+",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], "2"], "-",
|
|
RowBox[{
|
|
FractionBox["1", "2"],
|
|
SqrtBox[
|
|
RowBox[{
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], "-",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], ")"}],
|
|
"2"], "+",
|
|
RowBox[{"4",
|
|
SuperscriptBox[
|
|
RowBox[{
|
|
SubscriptBox["K", "12"], "[", "R", "]"}], "2"]}]}]]}]}], ",",
|
|
RowBox[{
|
|
FractionBox[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], "+",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], "2"], "+",
|
|
RowBox[{
|
|
FractionBox["1", "2"],
|
|
SqrtBox[
|
|
RowBox[{
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], "-",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], ")"}],
|
|
"2"], "+",
|
|
RowBox[{"4",
|
|
SuperscriptBox[
|
|
RowBox[{
|
|
SubscriptBox["K", "12"], "[", "R", "]"}], "2"]}]}]]}]}]}], "}"}],
|
|
"+",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "NN"], "[", "R", "]"}]}]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790605458040031*^9, 3.790605458543043*^9}, {
|
|
3.790605491525733*^9, 3.79060554005932*^9}, {3.790614460047532*^9,
|
|
3.790614470197546*^9}, 3.7906591751978273`*^9},
|
|
CellLabel->"In[69]:=",ExpressionUUID->"ab8163b6-fc27-4314-ae7e-a7f0f7d95629"],
|
|
|
|
Cell[TextData[StyleBox["Hartree-Fock energies",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.790587253093514*^9,
|
|
3.790587257373096*^9}},ExpressionUUID->"04f91dc9-a818-4dc9-a431-\
|
|
8d20ab592198"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R_", "]"}], "=",
|
|
RowBox[{
|
|
RowBox[{"{", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["h", "1"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R", "]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["K", "11"], "[", "R", "]"}]}], ",",
|
|
"\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["h", "2"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R", "]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["K", "22"], "[", "R", "]"}]}]}], "\[IndentingNewLine]",
|
|
"}"}], "+",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "NN"], "[", "R", "]"}]}]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790585364582788*^9, 3.790585422341731*^9}, {
|
|
3.790588811486704*^9, 3.7905888669431877`*^9}, {3.7906190335224743`*^9,
|
|
3.7906190337111177`*^9}, {3.790619093049782*^9, 3.790619093357505*^9}, {
|
|
3.79065915742903*^9, 3.790659168377755*^9}, {3.790660617755475*^9,
|
|
3.790660622626095*^9}},ExpressionUUID->"78388547-17f1-4781-bf29-\
|
|
2487d17b5c53"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]w", "HF"], "[",
|
|
RowBox[{"R_", ",", "w_"}], "]"}], "=",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["h", "1"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2", "w", " ",
|
|
RowBox[{
|
|
SubscriptBox["h", "2"], "[", "R", "]"}]}], "+",
|
|
RowBox[{
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}], "2"],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R", "]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["K", "11"], "[", "R", "]"}]}], ")"}]}], "+",
|
|
RowBox[{
|
|
SuperscriptBox["w", "2"],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R", "]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["K", "22"], "[", "R", "]"}]}], ")"}]}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}], "w",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "12"], "[", "R", "]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["K", "12"], "[", "R", "]"}]}], ")"}]}], "+",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "NN"], "[", "R", "]"}]}]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{3.790659173613167*^9},
|
|
CellLabel->"In[68]:=",ExpressionUUID->"1039bc21-357b-49d9-a514-fd3af5ae05a7"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalDelta]\[CapitalEpsilon]w", "HF"], "[",
|
|
RowBox[{"R_", ",", "w_"}], "]"}], "=",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "+",
|
|
RowBox[{"w",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]w", "HF"], "[",
|
|
RowBox[{"R", ",", "w"}], "]"}]}]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790660001969771*^9, 3.790660072444165*^9}},
|
|
CellLabel->"In[79]:=",ExpressionUUID->"01749816-6d54-416a-ab4f-bab2e420acb4"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Epsilon]w", "HF"], "[",
|
|
RowBox[{"R_", ",", "w_"}], "]"}], "=",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "1"], "[", "R", "]"}], "+",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R", "]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["K", "11"], "[", "R", "]"}]}], ")"}]}], "+",
|
|
RowBox[{"w",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "12"], "[", "R", "]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["K", "12"], "[", "R", "]"}]}], ")"}]}]}], ",",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "2"], "[", "R", "]"}], "+",
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "12"], "[", "R", "]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["K", "12"], "[", "R", "]"}]}], ")"}]}], "+",
|
|
RowBox[{"w",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R", "]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["K", "22"], "[", "R", "]"}]}], ")"}]}]}]}], "}"}]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790660423886979*^9, 3.790660530808771*^9}, {
|
|
3.790661586617362*^9, 3.7906615882014847`*^9}},
|
|
CellLabel->
|
|
"In[114]:=",ExpressionUUID->"36f3a3dd-e13f-4bbf-a50c-b11a31e789f8"],
|
|
|
|
Cell[TextData[StyleBox["LDA energies",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.790587253093514*^9, 3.790587257373096*^9}, {
|
|
3.79060545274035*^9, 3.790605453095336*^9}, {3.790605665401697*^9,
|
|
3.7906056655375757`*^9}},ExpressionUUID->"acf62eaa-9c57-4e21-98ab-\
|
|
4e5f44a5d9f7"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "LDA"], "[", "R_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"{", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["h", "1"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xcLDA", "[",
|
|
RowBox[{
|
|
SubscriptBox["n", "1"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}],
|
|
"]"}],
|
|
RowBox[{
|
|
SubscriptBox["n", "1"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}],
|
|
",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}], ",", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["h", "2"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xcLDA", "[",
|
|
RowBox[{
|
|
SubscriptBox["n", "2"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}],
|
|
"]"}],
|
|
RowBox[{
|
|
SubscriptBox["n", "2"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}],
|
|
",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}]}], "\[IndentingNewLine]", "}"}], "+",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "NN"], "[", "R", "]"}]}]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790605458040031*^9, 3.790605458543043*^9}, {
|
|
3.790605491525733*^9, 3.79060554005932*^9}, {3.7906056671315413`*^9,
|
|
3.790605692291338*^9}, {3.790610838410942*^9, 3.790610918068091*^9}, {
|
|
3.7906110542128553`*^9, 3.790611062835252*^9}, {3.790618152013009*^9,
|
|
3.790618153069519*^9}, {3.790618673469589*^9, 3.790618675001255*^9},
|
|
3.790659180965103*^9, {3.790660627881838*^9, 3.7906606344245253`*^9}},
|
|
CellLabel->
|
|
"In[115]:=",ExpressionUUID->"4419a1ba-532a-442b-95d2-330c205bfd7e"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]w", "LDA"], "[",
|
|
RowBox[{"R_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["h", "1"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2", "w", " ",
|
|
RowBox[{
|
|
SubscriptBox["h", "2"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2",
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}], "2"],
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2",
|
|
SuperscriptBox["w", "2"],
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"4",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}], "w", " ",
|
|
RowBox[{
|
|
SubscriptBox["J", "12"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xcLDA", "[",
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",", "w"}],
|
|
"]"}], "]"}],
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",", "w"}],
|
|
"]"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}], "]"}],
|
|
"+",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "NN"], "[", "R", "]"}]}]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.7906191044624767`*^9, 3.7906191773964367`*^9}, {
|
|
3.790619286782889*^9, 3.790619299369211*^9}, {3.790619337303144*^9,
|
|
3.790619344809649*^9}, {3.790619402147251*^9, 3.790619402226974*^9},
|
|
3.790659183868813*^9},
|
|
CellLabel->
|
|
"In[116]:=",ExpressionUUID->"c0b69368-df8e-46ab-a64b-3ea147722dd9"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalDelta]\[CapitalEpsilon]w", "LDA"], "[",
|
|
RowBox[{"R_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "LDA"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "+",
|
|
RowBox[{"w",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "LDA"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]w", "LDA"], "[",
|
|
RowBox[{"R", ",", "w"}], "]"}]}]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790660001969771*^9, 3.790660091003895*^9},
|
|
3.790660542656966*^9},
|
|
CellLabel->
|
|
"In[117]:=",ExpressionUUID->"0dd5826b-1b72-4f2c-834b-e79b419ac43e"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Epsilon]w", "LDA"], "[",
|
|
RowBox[{"R_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{"{", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "1"], "[", "R", "]"}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2", "w", " ",
|
|
RowBox[{
|
|
SubscriptBox["J", "12"], "[", "R", "]"}]}], "+",
|
|
RowBox[{
|
|
FractionBox["1", "2"],
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xcLDAdn", "[",
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}], "]"}],
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}]}], "+",
|
|
RowBox[{"\[Epsilon]xcLDA", "[",
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}], "]"}]}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["n", "1"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}],
|
|
",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}]}], ",", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "2"], "[", "R", "]"}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["J", "12"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2", "w", " ",
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R", "]"}]}], "+",
|
|
RowBox[{
|
|
FractionBox["1", "2"],
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xcLDAdn", "[",
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}], "]"}],
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}]}], "+",
|
|
RowBox[{"\[Epsilon]xcLDA", "[",
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}], "]"}]}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["n", "2"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}],
|
|
",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}]}]}], "\[IndentingNewLine]", "}"}]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790660423886979*^9, 3.790660614840477*^9}, {
|
|
3.790660673368799*^9, 3.790660734031897*^9}, {3.790661577778819*^9,
|
|
3.7906615810821047`*^9}, {3.79066238202234*^9,
|
|
3.7906624156374083`*^9}},ExpressionUUID->"61cc9c57-6cf1-464b-bd07-\
|
|
45e24fd7c9c3"],
|
|
|
|
Cell[TextData[StyleBox["eLDA energies",
|
|
FontColor->RGBColor[1, 0, 0]]], "Text",
|
|
CellChangeTimes->{{3.790587253093514*^9, 3.790587257373096*^9}, {
|
|
3.79060545274035*^9, 3.790605453095336*^9}, {3.790605665401697*^9,
|
|
3.7906056655375757`*^9}, 3.790615223845895*^9,
|
|
3.790615777404924*^9},ExpressionUUID->"1e4ce167-3a3b-4c53-9c2a-\
|
|
63a8f4ac6e4e"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "eLDA"], "[", "R_", "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"{", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["h", "1"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xceLDA", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["n", "1"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}],
|
|
",", "0"}], "]"}],
|
|
RowBox[{
|
|
SubscriptBox["n", "1"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}],
|
|
",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}], ",", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["h", "2"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xceLDA", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["n", "2"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}],
|
|
",", "1"}], "]"}],
|
|
RowBox[{
|
|
SubscriptBox["n", "2"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}],
|
|
",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}]}], "\[IndentingNewLine]", "}"}], "+",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "NN"], "[", "R", "]"}]}]}],
|
|
";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790605458040031*^9, 3.790605458543043*^9}, {
|
|
3.790605491525733*^9, 3.79060554005932*^9}, {3.7906056671315413`*^9,
|
|
3.790605692291338*^9}, {3.790610838410942*^9, 3.790610918068091*^9}, {
|
|
3.7906110542128553`*^9, 3.790611062835252*^9}, {3.790615225694908*^9,
|
|
3.7906152329132547`*^9}, {3.790618147868698*^9, 3.790618149719122*^9}, {
|
|
3.790618670193342*^9, 3.79061867932382*^9}, 3.790659187972827*^9, {
|
|
3.790660643667115*^9, 3.790660649050318*^9}},
|
|
CellLabel->
|
|
"In[119]:=",ExpressionUUID->"d4265afd-5fb1-4adc-ad27-da2413b01953"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]w", "eLDA"], "[",
|
|
RowBox[{"R_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["h", "1"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2", "w", " ",
|
|
RowBox[{
|
|
SubscriptBox["h", "2"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2",
|
|
SuperscriptBox[
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}], "2"],
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2",
|
|
SuperscriptBox["w", "2"],
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"4",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}], "w", " ",
|
|
RowBox[{
|
|
SubscriptBox["J", "12"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"\[Epsilon]xceLDA", "[",
|
|
RowBox[{
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",", "w"}],
|
|
"]"}], ",", "w"}], "]"}],
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",", "w"}],
|
|
"]"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}]}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.7906191044624767`*^9, 3.7906191943435097`*^9}, {
|
|
3.790619315024371*^9, 3.79061931729541*^9}, {3.790619349519698*^9,
|
|
3.790619356926655*^9}, {3.790619407198872*^9, 3.790619409338375*^9}, {
|
|
3.7906194502646437`*^9, 3.790619451317535*^9}, {3.7906195478586807`*^9,
|
|
3.790619547970214*^9}, 3.790659191412819*^9},
|
|
CellLabel->
|
|
"In[120]:=",ExpressionUUID->"9cd135b3-7599-482f-9e19-dfd18b561c56"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalDelta]\[CapitalEpsilon]w", "eLDA"], "[",
|
|
RowBox[{"R_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "eLDA"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], "+",
|
|
RowBox[{"w",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "eLDA"], "[", "R", "]"}],
|
|
"\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], "-",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]w", "eLDA"], "[",
|
|
RowBox[{"R", ",", "w"}], "]"}]}]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790660001969771*^9, 3.790660105027638*^9},
|
|
3.790660544962287*^9},
|
|
CellLabel->
|
|
"In[121]:=",ExpressionUUID->"99ae0dcb-f21c-46d4-84aa-7f172765d9d8"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Epsilon]w", "eLDA"], "[",
|
|
RowBox[{"R_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{"{", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "1"], "[", "R", "]"}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["J", "11"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2", "w", " ",
|
|
RowBox[{
|
|
SubscriptBox["J", "12"], "[", "R", "]"}]}], "+",
|
|
RowBox[{
|
|
FractionBox["1", "2"],
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xceLDAdn", "[",
|
|
RowBox[{
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}], ",", "w"}], "]"}],
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}]}], "+",
|
|
RowBox[{"\[Epsilon]xceLDA", "[",
|
|
RowBox[{
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}], ",", "w"}], "]"}]}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["n", "1"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}],
|
|
",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}]}], ",", "\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["h", "2"], "[", "R", "]"}], "+",
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{"1", "-", "w"}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["J", "12"], "[", "R", "]"}]}], "+",
|
|
RowBox[{"2", "w", " ",
|
|
RowBox[{
|
|
SubscriptBox["J", "22"], "[", "R", "]"}]}], "+",
|
|
RowBox[{
|
|
FractionBox["1", "2"],
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xceLDAdn", "[",
|
|
RowBox[{
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}], ",", "w"}], "]"}],
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}]}], "+",
|
|
RowBox[{"\[Epsilon]xceLDA", "[",
|
|
RowBox[{
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",",
|
|
"w"}], "]"}], ",", "w"}], "]"}]}], ")"}],
|
|
RowBox[{
|
|
SubscriptBox["n", "2"], "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R"}], "]"}]}],
|
|
",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}]}]}], "\[IndentingNewLine]", "}"}]}], ";"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790660423886979*^9, 3.790660614840477*^9}, {
|
|
3.790660673368799*^9, 3.790660753527668*^9}, {3.7906615534098883`*^9,
|
|
3.790661574889584*^9}},
|
|
CellLabel->
|
|
"In[139]:=",ExpressionUUID->"57a35adb-5f0e-4384-a38f-33a0653fb6b6"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{"With", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"Rmin", "=", "0.1"}], ",",
|
|
RowBox[{"Rmax", "=", "5"}], ",",
|
|
RowBox[{"\[CapitalDelta]R", "=", "0.1"}]}], "}"}], ",",
|
|
"\[IndentingNewLine]",
|
|
RowBox[{
|
|
RowBox[{"ScanEHF", "=",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"Flatten", "[",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "HF"], "[", "R", "]"}]}], "}"}],
|
|
"]"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",", "Rmin", ",", "Rmax", ",", "\[CapitalDelta]R"}],
|
|
"}"}]}], "]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{"ScanEwHF", "=",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"Flatten", "[",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]w", "HF"], "[",
|
|
RowBox[{"R", ",",
|
|
FractionBox["1", "2"]}], "]"}]}], "}"}], "]"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",", "Rmin", ",", "Rmax", ",", "\[CapitalDelta]R"}],
|
|
"}"}]}], "]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{"ScanECID", "=",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"Flatten", "[",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "CID"], "[", "R", "]"}]}], "}"}],
|
|
"]"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",", "Rmin", ",", "Rmax", ",", "\[CapitalDelta]R"}],
|
|
"}"}]}], "]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{"ScanELDA", "=",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"Flatten", "[",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "LDA"], "[", "R", "]"}]}], "}"}],
|
|
"]"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",", "Rmin", ",", "Rmax", ",", "\[CapitalDelta]R"}],
|
|
"}"}]}], "]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{"ScanEwLDA", "=",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"Flatten", "[",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]w", "LDA"], "[",
|
|
RowBox[{"R", ",",
|
|
FractionBox["1", "2"]}], "]"}]}], "}"}], "]"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",", "Rmin", ",", "Rmax", ",", "\[CapitalDelta]R"}],
|
|
"}"}]}], "]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{"ScanEeLDA", "=",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"Flatten", "[",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]", "eLDA"], "[", "R", "]"}]}],
|
|
"}"}], "]"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",", "Rmin", ",", "Rmax", ",", "\[CapitalDelta]R"}],
|
|
"}"}]}], "]"}]}], ";", "\[IndentingNewLine]",
|
|
RowBox[{"ScanEweLDA", "=",
|
|
RowBox[{"Table", "[",
|
|
RowBox[{
|
|
RowBox[{"Flatten", "[",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",",
|
|
RowBox[{
|
|
SubscriptBox["\[CapitalEpsilon]w", "eLDA"], "[",
|
|
RowBox[{"R", ",",
|
|
FractionBox["1", "2"]}], "]"}]}], "}"}], "]"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"R", ",", "Rmin", ",", "Rmax", ",", "\[CapitalDelta]R"}],
|
|
"}"}]}], "]"}]}], ";"}]}], "\[IndentingNewLine]", "]"}]], "Input",
|
|
InitializationCell->True,
|
|
CellChangeTimes->{{3.790658215232338*^9, 3.790658236824192*^9}, {
|
|
3.7906583897274837`*^9, 3.790658524654791*^9}, {3.790659704040538*^9,
|
|
3.790659705870874*^9}, {3.790659735886373*^9, 3.790659813232091*^9}},
|
|
CellLabel->
|
|
"In[123]:=",ExpressionUUID->"93054b36-78ec-4bb1-83ea-8fbb31835fb9"],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"d\[CapitalEpsilon]weLDAdw", "[",
|
|
RowBox[{"R_", ",", "w_"}], "]"}], ":=",
|
|
RowBox[{"NIntegrate", "[",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"d\[Epsilon]xceLDAdw", "[",
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",", "w"}],
|
|
"]"}], "]"}],
|
|
RowBox[{"nw", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",", "y", ",", "z"}], "}"}], ",", "R", ",", "w"}],
|
|
"]"}]}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"x", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"y", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{"z", ",",
|
|
RowBox[{"-", "\[Infinity]"}], ",", "\[Infinity]"}], "}"}]}],
|
|
"]"}]}]], "Input",
|
|
CellChangeTimes->{{3.7906608146381903`*^9, 3.790660820086027*^9}, {
|
|
3.790661275357267*^9, 3.790661293506763*^9}, {3.790661435266225*^9,
|
|
3.790661483849555*^9}, {3.790661526538616*^9, 3.790661526618237*^9}, {
|
|
3.790661636466195*^9, 3.790661654523027*^9}, 3.790661792604311*^9},
|
|
CellLabel->
|
|
"In[134]:=",ExpressionUUID->"9eac41d8-7253-4bcb-9a2a-4a332b730ddb"],
|
|
|
|
Cell[CellGroupData[{
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Epsilon]w", "LDA"], "[",
|
|
RowBox[{"R", ",", "w"}], "]"}], "\[LeftDoubleBracket]", "2",
|
|
"\[RightDoubleBracket]"}], "-",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Epsilon]w", "LDA"], "[",
|
|
RowBox[{"R", ",", "w"}], "]"}], "\[LeftDoubleBracket]", "1",
|
|
"\[RightDoubleBracket]"}]}], ")"}]}], "/.",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"R", "\[Rule]", "1.4"}], ",",
|
|
RowBox[{"w", "\[Rule]", "0"}]}], "}"}]}]], "Input",
|
|
CellChangeTimes->{{3.790662000166115*^9, 3.7906620185750113`*^9}, {
|
|
3.790662138430471*^9, 3.790662147334878*^9}, 3.790662401812807*^9},
|
|
CellLabel->
|
|
"In[150]:=",ExpressionUUID->"cc5802c8-5164-4d35-a4aa-ac0f06412d48"],
|
|
|
|
Cell[BoxData["1.4989461809762727`"], "Output",
|
|
CellChangeTimes->{{3.790662007156047*^9, 3.7906620189650087`*^9}, {
|
|
3.790662139159606*^9, 3.7906621611887913`*^9}, {3.7906623969111633`*^9,
|
|
3.7906624071288967`*^9}},
|
|
CellLabel->
|
|
"Out[150]=",ExpressionUUID->"6c0f8654-3af4-43e9-b5c1-7a47f72b02d4"]
|
|
}, Open ]],
|
|
|
|
Cell[BoxData[
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{"2",
|
|
RowBox[{"(",
|
|
RowBox[{
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Epsilon]w", "eLDA"], "[",
|
|
RowBox[{"R", ",", "w"}], "]"}], "\[LeftDoubleBracket]", "2",
|
|
"\[RightDoubleBracket]"}], "-",
|
|
RowBox[{
|
|
RowBox[{
|
|
SubscriptBox["\[Epsilon]w", "eLDA"], "[",
|
|
RowBox[{"R", ",", "w"}], "]"}], "\[LeftDoubleBracket]", "1",
|
|
"\[RightDoubleBracket]"}]}], ")"}]}], "+",
|
|
RowBox[{"d\[CapitalEpsilon]weLDAdw", "[",
|
|
RowBox[{"R", ",", "w"}], "]"}]}], "\[IndentingNewLine]"}]], "Input",
|
|
CellChangeTimes->{{3.790661475825005*^9, 3.790661489241352*^9}, {
|
|
3.7906618427444572`*^9, 3.790661844966864*^9}, {3.790661881697806*^9,
|
|
3.7906618992816668`*^9}, {3.7906619466147127`*^9, 3.790661955270863*^9}},
|
|
CellLabel->
|
|
"In[140]:=",ExpressionUUID->"8cbe4386-fbc6-4914-8e8c-e90d47f39535"]
|
|
}, Open ]]
|
|
},
|
|
WindowSize->{1815, 1395},
|
|
WindowMargins->{{0, Automatic}, {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, 68, "Input",ExpressionUUID->"8b7c0bbd-9f99-42dc-b8a5-a4dc3cb8cc89",
|
|
InitializationCell->True],
|
|
Cell[1316, 41, 308, 6, 46, "Input",ExpressionUUID->"b24e8243-3c38-4b03-af3a-708cb1afa004",
|
|
InitializationCell->True],
|
|
Cell[1627, 49, 341, 9, 49, "Input",ExpressionUUID->"5822ee69-a453-46a0-9a58-bd14b8a97301",
|
|
InitializationCell->True],
|
|
Cell[1971, 60, 868, 21, 71, "Input",ExpressionUUID->"43788edb-8cf4-404a-88d1-db5bf1515eb0",
|
|
InitializationCell->True],
|
|
Cell[2842, 83, 343, 8, 46, "Input",ExpressionUUID->"058853fd-911e-44b3-8eac-1c3fbeb8c4a5",
|
|
InitializationCell->True],
|
|
Cell[3188, 93, 279, 6, 46, "Input",ExpressionUUID->"5dbbd40c-d911-445f-b476-85cd91f8d014",
|
|
InitializationCell->True]
|
|
}, Closed]],
|
|
Cell[CellGroupData[{
|
|
Cell[3504, 104, 157, 3, 72, "Title",ExpressionUUID->"98d632cb-7177-481c-9b9f-d6f48d10e284"],
|
|
Cell[CellGroupData[{
|
|
Cell[3686, 111, 207, 4, 67, "Section",ExpressionUUID->"9458067c-0b7a-4f70-9a58-1f771c4b3842"],
|
|
Cell[CellGroupData[{
|
|
Cell[3918, 119, 168, 3, 54, "Subsection",ExpressionUUID->"11bf4889-b9bd-4f4e-8dbb-a1e9e8a04d2a"],
|
|
Cell[4089, 124, 1682, 43, 65, "Input",ExpressionUUID->"fd4d58e4-e6f8-4497-ac22-d536c7dc2652",
|
|
InitializationCell->True],
|
|
Cell[5774, 169, 1412, 35, 65, "Input",ExpressionUUID->"f4215905-d329-4e5c-925a-feafdeee6fce",
|
|
InitializationCell->True],
|
|
Cell[7189, 206, 2169, 59, 65, "Input",ExpressionUUID->"517a10dc-802a-4ead-9f44-0905089379ba",
|
|
InitializationCell->True],
|
|
Cell[9361, 267, 3587, 99, 72, "Input",ExpressionUUID->"440fb372-2001-4d90-ade1-e5498676702f",
|
|
InitializationCell->True]
|
|
}, Open ]]
|
|
}, Closed]]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[13009, 373, 151, 3, 98, "Title",ExpressionUUID->"cdfcc3a6-7299-4a1b-a9d0-f29078225f1e"],
|
|
Cell[CellGroupData[{
|
|
Cell[13185, 380, 159, 3, 67, "Section",ExpressionUUID->"23d29e55-92a0-4dbd-8800-76494f7702a3"],
|
|
Cell[CellGroupData[{
|
|
Cell[13369, 387, 250, 4, 54, "Subsection",ExpressionUUID->"5917ba4c-2cd7-4975-9b3b-073e72bd0e40"],
|
|
Cell[CellGroupData[{
|
|
Cell[13644, 395, 723, 23, 48, "Input",ExpressionUUID->"c8ade4e2-951f-4fce-ae16-fdd4ea72db6f"],
|
|
Cell[14370, 420, 517, 15, 62, "Output",ExpressionUUID->"e2c1e6f8-35e1-43ee-b47e-a64bb64f8585"]
|
|
}, Open ]],
|
|
Cell[14902, 438, 651, 15, 30, "Input",ExpressionUUID->"02e08ec6-7c15-4841-9609-90296edfe494"],
|
|
Cell[CellGroupData[{
|
|
Cell[15578, 457, 550, 13, 48, "Input",ExpressionUUID->"150e916f-8806-4d8e-b5d0-c0bc713b1f4d"],
|
|
Cell[16131, 472, 276, 4, 34, "Output",ExpressionUUID->"fb35d58c-512f-4f74-aa01-67c7b7b81a77"]
|
|
}, Open ]]
|
|
}, Closed]]
|
|
}, Closed]],
|
|
Cell[CellGroupData[{
|
|
Cell[16468, 483, 215, 4, 53, "Section",ExpressionUUID->"8fd0ca40-ccd3-410e-8538-798441b5cf4b"],
|
|
Cell[CellGroupData[{
|
|
Cell[16708, 491, 174, 3, 54, "Subsection",ExpressionUUID->"10e37c4d-fcc0-4a25-949c-6383c67ff9eb"],
|
|
Cell[CellGroupData[{
|
|
Cell[16907, 498, 4321, 122, 145, "Input",ExpressionUUID->"b7e33c7b-f239-4612-b339-911d66bee804"],
|
|
Cell[21231, 622, 199, 3, 34, "Output",ExpressionUUID->"3acd8e38-829c-4dd5-ac53-54c357a30852"],
|
|
Cell[21433, 627, 199, 3, 34, "Output",ExpressionUUID->"0a957f43-bd43-468d-a336-addbe42a4e30"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[21669, 635, 847, 27, 57, "Input",ExpressionUUID->"5bc098a3-a8b3-48df-b81b-4249253ebc03"],
|
|
Cell[22519, 664, 176, 2, 34, "Output",ExpressionUUID->"3597039b-63a4-48ef-b436-f5b1fb49d7fd"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[22732, 671, 1432, 45, 94, "Input",ExpressionUUID->"7707f078-0791-4cee-b4ea-44b37115dc20"],
|
|
Cell[24167, 718, 197, 3, 34, "Output",ExpressionUUID->"d4f8c9be-6e4c-40eb-ba9a-d4053731284e"],
|
|
Cell[24367, 723, 199, 3, 34, "Output",ExpressionUUID->"2e03225f-8957-4361-919c-4f6a7c36dc6e"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[24603, 731, 2671, 77, 54, "Input",ExpressionUUID->"731b5390-31ed-4914-91db-765023c263dd"],
|
|
Cell[27277, 810, 497, 7, 34, "Output",ExpressionUUID->"ed71661b-3a8d-4eee-a4d0-69211937142a"]
|
|
}, Open ]],
|
|
Cell[27789, 820, 2240, 61, 53, "Input",ExpressionUUID->"1f5927dc-663a-4f11-a2d1-7bc195f79332"]
|
|
}, Closed]]
|
|
}, Closed]]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[30090, 888, 150, 3, 98, "Title",ExpressionUUID->"e756f09e-4f67-4fb6-a4c1-1a4fcf53e013"],
|
|
Cell[CellGroupData[{
|
|
Cell[30265, 895, 259, 4, 67, "Section",ExpressionUUID->"81f07a4a-071b-4e38-8018-d116e770b20b"],
|
|
Cell[CellGroupData[{
|
|
Cell[30549, 903, 291, 4, 54, "Subsection",ExpressionUUID->"7471e1f3-36cd-497f-a3e0-06aa7035995d"],
|
|
Cell[30843, 909, 228, 6, 35, "Text",ExpressionUUID->"c0165a48-1e75-4237-86f9-15c7a0fc2c6e"],
|
|
Cell[31074, 917, 570, 16, 49, "Input",ExpressionUUID->"eeea1fbc-15c7-4b79-b5b2-0d2e682c7cb1"],
|
|
Cell[31647, 935, 230, 4, 35, "Text",ExpressionUUID->"6509cf0c-3cb5-4484-a79b-9a4e9a4c867b"],
|
|
Cell[31880, 941, 449, 10, 48, "Input",ExpressionUUID->"ea3020ca-486a-47b1-95e0-aa46f448646d"],
|
|
Cell[32332, 953, 176, 3, 35, "Text",ExpressionUUID->"dc6b5921-169b-4f2d-9120-4c7ca89c05f3"],
|
|
Cell[32511, 958, 670, 18, 48, "Input",ExpressionUUID->"12ac5e19-b8fd-477e-a85d-a74e029b6d4c"],
|
|
Cell[33184, 978, 201, 3, 35, "Text",ExpressionUUID->"21ac5bf4-2ae7-4b41-8495-ff3b6d5d2017"],
|
|
Cell[33388, 983, 794, 21, 45, "Input",ExpressionUUID->"c8865179-994b-4123-9657-507eae2fd52f"],
|
|
Cell[34185, 1006, 153, 3, 35, "Text",ExpressionUUID->"d352d13e-d96f-4f21-9cc5-32da8a2e376d"],
|
|
Cell[34341, 1011, 1177, 34, 50, "Input",ExpressionUUID->"e2a90d7c-9765-4224-b6c1-0cb3ba56719e"],
|
|
Cell[35521, 1047, 193, 3, 35, "Text",ExpressionUUID->"4256ed0c-1e05-4553-a95a-370962e94038"],
|
|
Cell[35717, 1052, 707, 21, 58, "Input",ExpressionUUID->"d7507220-76aa-479e-af51-0886ad0e4f5c"]
|
|
}, Closed]],
|
|
Cell[CellGroupData[{
|
|
Cell[36461, 1078, 301, 4, 38, "Subsection",ExpressionUUID->"77af7129-5b6f-46d1-972e-088c04a397e1"],
|
|
Cell[36765, 1084, 310, 7, 35, "Text",ExpressionUUID->"137a2815-3e41-4c54-bb77-2baab3309f67"],
|
|
Cell[37078, 1093, 570, 16, 49, "Input",ExpressionUUID->"ee1a49a3-d78d-48fd-b18b-b45c556268d6"],
|
|
Cell[37651, 1111, 318, 6, 35, "Text",ExpressionUUID->"7f6139ea-87a7-485f-96a7-072785ed5a50"],
|
|
Cell[37972, 1119, 571, 14, 48, "Input",ExpressionUUID->"e2512605-e53f-4895-bc29-b005c70f53b5"],
|
|
Cell[CellGroupData[{
|
|
Cell[38568, 1137, 570, 17, 50, "Input",ExpressionUUID->"2caf6c5b-6ae4-41ea-a5bc-6e2553b311b0"],
|
|
Cell[39141, 1156, 544, 16, 55, "Output",ExpressionUUID->"71a67885-8142-46b8-9292-410654424b59"]
|
|
}, Open ]],
|
|
Cell[39700, 1175, 176, 3, 35, "Text",ExpressionUUID->"32015b28-f5cc-4eff-8d8d-b8691b888e8f"],
|
|
Cell[39879, 1180, 847, 23, 48, "Input",ExpressionUUID->"1fcbd290-f3ea-43bd-b2ef-f31601137679"],
|
|
Cell[40729, 1205, 201, 3, 35, "Text",ExpressionUUID->"9b1e04fb-55cb-4e81-8daf-ba707c78dfce"],
|
|
Cell[40933, 1210, 827, 22, 45, "Input",ExpressionUUID->"c51ccbbc-5ee1-4cca-ade7-af230c7f6ba6"],
|
|
Cell[41763, 1234, 153, 3, 35, "Text",ExpressionUUID->"8bfb35af-a4e4-4813-ae15-3703f4aeab96"],
|
|
Cell[41919, 1239, 1240, 35, 50, "Input",ExpressionUUID->"60e340d1-6bd1-4928-9814-51c59353edd1"],
|
|
Cell[43162, 1276, 250, 4, 35, "Text",ExpressionUUID->"f5f45693-a80c-4a46-ad73-07811620c0d9"],
|
|
Cell[43415, 1282, 787, 23, 58, "Input",ExpressionUUID->"f99857fd-0430-402d-a733-929b6b359e23"]
|
|
}, Closed]],
|
|
Cell[CellGroupData[{
|
|
Cell[44239, 1310, 348, 5, 38, "Subsection",ExpressionUUID->"ef62b877-dbc0-41f4-bb50-f7bab5069a50"],
|
|
Cell[44590, 1317, 252, 6, 35, "Text",ExpressionUUID->"5b10be0f-a732-422f-9446-681c78cbd103"],
|
|
Cell[44845, 1325, 561, 17, 32, "Input",ExpressionUUID->"009be4dc-79a6-499d-b9af-ddbb9078f85a"],
|
|
Cell[45409, 1344, 918, 28, 53, "Input",ExpressionUUID->"afd8bd15-999f-411c-a673-53d5b75e5201"],
|
|
Cell[46330, 1374, 297, 4, 35, "Text",ExpressionUUID->"1fe4835f-3b82-458a-9e2a-9272922b081a"],
|
|
Cell[46630, 1380, 1329, 40, 58, "Input",ExpressionUUID->"b5e6e435-424a-4f3e-8416-709ae4356a73"],
|
|
Cell[47962, 1422, 546, 14, 58, "Text",ExpressionUUID->"d05917ae-a767-4a78-834a-c4b18c1c0b13"]
|
|
}, Closed]],
|
|
Cell[CellGroupData[{
|
|
Cell[48545, 1441, 419, 8, 38, "Subsection",ExpressionUUID->"bcd158fd-9aa7-4c7b-837e-afd37c54f55a"],
|
|
Cell[48967, 1451, 571, 11, 81, "Text",ExpressionUUID->"a0112f10-d580-4370-a6ed-bff98b24a4a6"],
|
|
Cell[49541, 1464, 2757, 86, 101, "Input",ExpressionUUID->"aabf1284-1a62-408f-8d1c-50c99b5acc35"],
|
|
Cell[52301, 1552, 278, 6, 35, "Text",ExpressionUUID->"88a4bd50-0ba7-4f7c-9fc0-e0feb01831fc"],
|
|
Cell[52582, 1560, 606, 19, 48, "Input",ExpressionUUID->"a889f096-9a50-426c-9cf3-81c363ec2eaf"],
|
|
Cell[53191, 1581, 355, 7, 35, "Text",ExpressionUUID->"d3c69495-05ef-49fe-8484-fa7fd62cc408"],
|
|
Cell[53549, 1590, 1887, 55, 58, "Input",ExpressionUUID->"0138b572-760b-4f82-a618-73197e776fa2"],
|
|
Cell[55439, 1647, 452, 7, 35, "Text",ExpressionUUID->"b83db319-04e5-42b8-ae4c-57f93abe09fd"]
|
|
}, Open ]]
|
|
}, Closed]],
|
|
Cell[CellGroupData[{
|
|
Cell[55940, 1660, 240, 4, 53, "Section",ExpressionUUID->"258ca12d-c8c5-4bc9-84ec-1711f7860786"],
|
|
Cell[CellGroupData[{
|
|
Cell[56205, 1668, 347, 5, 54, "Subsection",ExpressionUUID->"ee1991f0-6627-4afa-871c-73710071f6fd"],
|
|
Cell[56555, 1675, 324, 7, 58, "Text",ExpressionUUID->"a6c10556-4f53-4da2-8cf0-79bbf3ec1242"],
|
|
Cell[56882, 1684, 1209, 36, 64, "Input",ExpressionUUID->"1885d9ca-6bb6-490d-a0bb-a3b96576a614"],
|
|
Cell[58094, 1722, 300, 6, 35, "Text",ExpressionUUID->"ead26072-79d4-4549-aab6-46fe8cf9b6db"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[58431, 1733, 353, 5, 54, "Subsection",ExpressionUUID->"ac951d60-fbce-44ba-8f12-9e5545df5beb"],
|
|
Cell[58787, 1740, 228, 5, 35, "Text",ExpressionUUID->"dca5790f-2358-43e0-b359-677842457a7d"],
|
|
Cell[59018, 1747, 1279, 38, 64, "Input",ExpressionUUID->"cce86f90-52f7-41f4-a63d-73a6c518b4c3"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[60334, 1790, 348, 5, 54, "Subsection",ExpressionUUID->"9f2ce783-2740-403c-974c-46a7ef9321fe"],
|
|
Cell[60685, 1797, 310, 7, 35, "Text",ExpressionUUID->"6c7b6dc7-06c2-4f95-a398-fe8f7adea7db"],
|
|
Cell[60998, 1806, 610, 18, 32, "Input",ExpressionUUID->"c5451bea-41ca-4be3-9429-a6d8ddd06c4a"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[61645, 1829, 468, 8, 54, "Subsection",ExpressionUUID->"ab8ea4d9-6951-44dc-96fc-10c74ccd1a4f"],
|
|
Cell[62116, 1839, 279, 6, 35, "Text",ExpressionUUID->"f0dac186-7114-4a5e-b53c-f938d8c4fd37"],
|
|
Cell[62398, 1847, 1342, 40, 33, "Input",ExpressionUUID->"2ca8802e-361f-4487-a3f0-67d18fc8fe5f"],
|
|
Cell[63743, 1889, 386, 8, 35, "Text",ExpressionUUID->"00fbe382-6c25-4893-a74d-546919a831b8"],
|
|
Cell[64132, 1899, 368, 8, 32, "Input",ExpressionUUID->"b7e98868-257f-43e4-b057-ff1cbb083046"],
|
|
Cell[64503, 1909, 728, 11, 104, "Text",ExpressionUUID->"336e8167-7e94-4334-9956-c0652866375e"],
|
|
Cell[65234, 1922, 4284, 106, 217, "Input",ExpressionUUID->"8d489c13-936e-4272-b8f6-29963888bb9f"],
|
|
Cell[69521, 2030, 2420, 71, 88, "Input",ExpressionUUID->"af4fe8f8-c3eb-4495-a149-b4bd1830693b"],
|
|
Cell[71944, 2103, 2312, 69, 88, "Input",ExpressionUUID->"eb8a2378-386f-4e82-b835-17c5b76bd346"],
|
|
Cell[CellGroupData[{
|
|
Cell[74281, 2176, 2231, 57, 116, "Input",ExpressionUUID->"2f2c8334-f314-4d97-9abf-fd397dbace2b"],
|
|
Cell[76515, 2235, 1027, 17, 34, "Output",ExpressionUUID->"f04e12cb-ae53-4b96-8239-58bd2665964b"],
|
|
Cell[77545, 2254, 1045, 18, 34, "Output",ExpressionUUID->"34d77878-3289-4d0f-a73e-9cb3a0e22085"]
|
|
}, Open ]],
|
|
Cell[78605, 2275, 1895, 53, 93, "Input",ExpressionUUID->"6a07f7bc-64fd-4691-a797-042e1daf8a9c"],
|
|
Cell[CellGroupData[{
|
|
Cell[80525, 2332, 1934, 44, 49, "Input",ExpressionUUID->"6e23a9ca-c6d3-4d4a-b7ff-3092bcade88a"],
|
|
Cell[82462, 2378, 10269, 188, 233, "Output",ExpressionUUID->"ec22f91b-cb1e-44dd-995e-82b7699d74af"]
|
|
}, Open ]]
|
|
}, Open ]]
|
|
}, Closed]],
|
|
Cell[CellGroupData[{
|
|
Cell[92792, 2573, 159, 3, 53, "Section",ExpressionUUID->"36f69d13-fe6b-4a04-acec-16703fdd0e12"],
|
|
Cell[92954, 2578, 403, 11, 35, "Text",ExpressionUUID->"17ce12bb-a006-40f9-ab62-8a0036a5342f"],
|
|
Cell[CellGroupData[{
|
|
Cell[93382, 2593, 461, 13, 47, "Input",ExpressionUUID->"44e4ae27-771e-4e10-b2a8-3f5c0700fb97"],
|
|
Cell[93846, 2608, 236, 5, 34, "Output",ExpressionUUID->"47184591-6c91-4963-aebc-73b69a443808"]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[94119, 2618, 586, 20, 67, "Input",ExpressionUUID->"31e71679-e385-4ae9-bf70-b310ce826eaf"],
|
|
Cell[94708, 2640, 221, 4, 34, "Output",ExpressionUUID->"dda12046-695a-4ac3-bf5e-c8600e6dcaef"]
|
|
}, Open ]],
|
|
Cell[94944, 2647, 498, 13, 35, "Text",ExpressionUUID->"0ad72096-b75c-4b05-aea4-1c9a3ba2d63e"],
|
|
Cell[95445, 2662, 487, 13, 35, "Text",ExpressionUUID->"f3cf0761-27e0-47e5-819a-4f76e0d90b9b"],
|
|
Cell[CellGroupData[{
|
|
Cell[95957, 2679, 560, 19, 69, "Input",ExpressionUUID->"77eca9b4-30de-4a4a-8d4c-a627ab7872b0"],
|
|
Cell[96520, 2700, 199, 3, 34, "Output",ExpressionUUID->"ef1cfa21-7595-4fd6-a4fa-a0c034a8b13d"]
|
|
}, Open ]],
|
|
Cell[96734, 2706, 741, 21, 35, "Text",ExpressionUUID->"c212926d-9477-4fee-9f18-1c758debf2c7"],
|
|
Cell[CellGroupData[{
|
|
Cell[97500, 2731, 660, 21, 69, "Input",ExpressionUUID->"40b08d32-7779-4afd-83c6-170f189f9a96"],
|
|
Cell[98163, 2754, 256, 7, 34, "Output",ExpressionUUID->"896c805b-7892-4f9b-ac07-361220bdc186"]
|
|
}, Open ]],
|
|
Cell[98434, 2764, 765, 20, 35, "Text",ExpressionUUID->"350a122c-6d8b-461e-bad5-e2fdbc14b8f2"],
|
|
Cell[CellGroupData[{
|
|
Cell[99224, 2788, 1083, 38, 69, "Input",ExpressionUUID->"ffbc7770-1424-4564-b99b-b9ad1c0639d4"],
|
|
Cell[100310, 2828, 448, 13, 55, "Output",ExpressionUUID->"27597e7b-3c41-49b8-9900-9077d931d997"]
|
|
}, Open ]]
|
|
}, Closed]]
|
|
}, Open ]],
|
|
Cell[CellGroupData[{
|
|
Cell[100819, 2848, 211, 4, 98, "Title",ExpressionUUID->"d8a7ce9c-a1d9-4ac0-b8bf-9ca66a94ac58"],
|
|
Cell[101033, 2854, 197, 3, 35, "Text",ExpressionUUID->"d408aeeb-1654-4525-903e-88994da645ee"],
|
|
Cell[101233, 2859, 2158, 65, 171, "Input",ExpressionUUID->"f51f78c8-fe8e-48f3-a562-1209bfc50f71",
|
|
InitializationCell->True],
|
|
Cell[103394, 2926, 1370, 36, 110, "Input",ExpressionUUID->"dd062623-5a91-460e-945c-d5f15e24abfb",
|
|
InitializationCell->True],
|
|
Cell[104767, 2964, 198, 3, 35, "Text",ExpressionUUID->"683219a6-4bcc-4fb8-93b5-7c590bc511b5"],
|
|
Cell[104968, 2969, 5783, 167, 394, "Input",ExpressionUUID->"c5e15fc2-7ce6-4ed3-bc7f-bcdc9085134d",
|
|
InitializationCell->True],
|
|
Cell[110754, 3138, 2351, 69, 167, "Input",ExpressionUUID->"8db93606-ec73-4fe8-9228-51f79c4ef957",
|
|
InitializationCell->True],
|
|
Cell[113108, 3209, 1368, 36, 110, "Input",ExpressionUUID->"3c5be8aa-305f-4c0e-a51a-4bc92cdcf171",
|
|
InitializationCell->True],
|
|
Cell[114479, 3247, 256, 4, 35, "Text",ExpressionUUID->"264d5181-32f7-492d-a01f-33c1ad90708e"],
|
|
Cell[114738, 3253, 1089, 28, 110, "Input",ExpressionUUID->"f6517707-c164-4ea9-b57c-8d4d27eafad3",
|
|
InitializationCell->True],
|
|
Cell[115830, 3283, 1180, 31, 110, "Input",ExpressionUUID->"33dfcd59-33cc-4f88-bc22-62c17b91dbeb",
|
|
InitializationCell->True],
|
|
Cell[117013, 3316, 880, 21, 89, "Input",ExpressionUUID->"9663902a-8667-481b-b613-f1c2b56b8e76",
|
|
InitializationCell->True],
|
|
Cell[117896, 3339, 214, 4, 35, "Text",ExpressionUUID->"d38e2fe6-260d-4638-b639-6984dd9d14e4"],
|
|
Cell[118113, 3345, 679, 19, 63, "Input",ExpressionUUID->"36b985d7-30a2-430d-8dc3-30d6b4a45727",
|
|
InitializationCell->True],
|
|
Cell[118795, 3366, 210, 4, 35, "Text",ExpressionUUID->"e115f0cc-d9e8-4997-a5a4-88b0e9ed97e0"],
|
|
Cell[119008, 3372, 3586, 92, 187, "Input",ExpressionUUID->"f0a2825f-92fc-42d2-aeee-57f53535b1d5",
|
|
InitializationCell->True],
|
|
Cell[122597, 3466, 276, 5, 35, "Text",ExpressionUUID->"5353bb94-3a0f-4a57-a4c7-bf24da4dc996"],
|
|
Cell[122876, 3473, 28451, 690, 843, "Input",ExpressionUUID->"341cb469-2a0b-4b3f-a5c7-4a3d8fb61885",
|
|
InitializationCell->True],
|
|
Cell[151330, 4165, 227, 4, 35, "Text",ExpressionUUID->"5b2593a5-509e-44f9-a6a7-5b7dcf896d99"],
|
|
Cell[151560, 4171, 1855, 57, 68, "Input",ExpressionUUID->"3bf77406-a77c-45d7-8c2f-817ad440cdff",
|
|
InitializationCell->True],
|
|
Cell[153418, 4230, 264, 5, 35, "Text",ExpressionUUID->"8580055f-686c-403d-8800-a4eab889fce4"],
|
|
Cell[153685, 4237, 1223, 36, 46, "Input",ExpressionUUID->"4bf40614-2147-47eb-9ce0-13fe949a9c00",
|
|
InitializationCell->True],
|
|
Cell[154911, 4275, 1097, 30, 46, "Input",ExpressionUUID->"33c091a9-0792-4be2-bcd6-e02a256b26b5",
|
|
InitializationCell->True],
|
|
Cell[156011, 4307, 267, 5, 35, "Text",ExpressionUUID->"6e302dfa-38b2-43a3-a507-71bf2633d8e8"],
|
|
Cell[156281, 4314, 1612, 49, 66, "Input",ExpressionUUID->"41a0d75d-44b6-4653-b70e-858ab3e937b8",
|
|
InitializationCell->True],
|
|
Cell[157896, 4365, 2945, 90, 153, "Input",ExpressionUUID->"1430ab04-b05a-4d2d-bbf6-f9edc3cd5345",
|
|
InitializationCell->True],
|
|
Cell[160844, 4457, 1366, 45, 109, "Input",ExpressionUUID->"09f7d062-2fdd-45b1-8df2-0dd679b5cbc6",
|
|
InitializationCell->True],
|
|
Cell[162213, 4504, 258, 5, 35, "Text",ExpressionUUID->"94763911-d015-4b98-a645-b27f631322bf"],
|
|
Cell[162474, 4511, 685, 16, 65, "Input",ExpressionUUID->"46e4593c-f9e1-4f4a-a999-d0f67e325be6",
|
|
InitializationCell->True],
|
|
Cell[163162, 4529, 253, 5, 35, "Text",ExpressionUUID->"eeb33af8-ab09-4c5f-98a3-684ae529c2b6"],
|
|
Cell[163418, 4536, 2845, 78, 63, "Input",ExpressionUUID->"ab8163b6-fc27-4314-ae7e-a7f0f7d95629",
|
|
InitializationCell->True],
|
|
Cell[166266, 4616, 214, 4, 35, "Text",ExpressionUUID->"04f91dc9-a818-4dc9-a431-8d20ab592198"],
|
|
Cell[166483, 4622, 1311, 37, 110, "Input",ExpressionUUID->"78388547-17f1-4781-bf29-2487d17b5c53",
|
|
InitializationCell->True],
|
|
Cell[167797, 4661, 1512, 50, 46, "Input",ExpressionUUID->"1039bc21-357b-49d9-a514-fd3af5ae05a7",
|
|
InitializationCell->True],
|
|
Cell[169312, 4713, 884, 24, 46, "Input",ExpressionUUID->"01749816-6d54-416a-ab4f-bab2e420acb4",
|
|
InitializationCell->True],
|
|
Cell[170199, 4739, 1693, 55, 46, "Input",ExpressionUUID->"36f3a3dd-e13f-4bbf-a50c-b11a31e789f8",
|
|
InitializationCell->True],
|
|
Cell[171895, 4796, 301, 5, 35, "Text",ExpressionUUID->"acf62eaa-9c57-4e21-98ab-4e5f44a5d9f7"],
|
|
Cell[172199, 4803, 3140, 85, 110, "Input",ExpressionUUID->"4419a1ba-532a-442b-95d2-330c205bfd7e",
|
|
InitializationCell->True],
|
|
Cell[175342, 4890, 2108, 63, 46, "Input",ExpressionUUID->"c0b69368-df8e-46ab-a64b-3ea147722dd9",
|
|
InitializationCell->True],
|
|
Cell[177453, 4955, 919, 26, 46, "Input",ExpressionUUID->"0dd5826b-1b72-4f2c-834b-e79b419ac43e",
|
|
InitializationCell->True],
|
|
Cell[178375, 4983, 4206, 117, 159, "Input",ExpressionUUID->"61cc9c57-6cf1-464b-bd07-45e24fd7c9c3",
|
|
InitializationCell->True],
|
|
Cell[182584, 5102, 352, 6, 35, "Text",ExpressionUUID->"1e4ce167-3a3b-4c53-9c2a-63a8f4ac6e4e"],
|
|
Cell[182939, 5110, 3268, 88, 110, "Input",ExpressionUUID->"d4265afd-5fb1-4adc-ad27-da2413b01953",
|
|
InitializationCell->True],
|
|
Cell[186210, 5200, 2089, 61, 46, "Input",ExpressionUUID->"9cd135b3-7599-482f-9e19-dfd18b561c56",
|
|
InitializationCell->True],
|
|
Cell[188302, 5263, 923, 26, 46, "Input",ExpressionUUID->"99ae0dcb-f21c-46d4-84aa-7f172765d9d8",
|
|
InitializationCell->True],
|
|
Cell[189228, 5291, 4345, 121, 159, "Input",ExpressionUUID->"57a35adb-5f0e-4384-a38f-33a0653fb6b6",
|
|
InitializationCell->True],
|
|
Cell[193576, 5414, 3751, 102, 279, "Input",ExpressionUUID->"93054b36-78ec-4bb1-83ea-8fbb31835fb9",
|
|
InitializationCell->True],
|
|
Cell[197330, 5518, 1251, 33, 30, "Input",ExpressionUUID->"9eac41d8-7253-4bcb-9a2a-4a332b730ddb"],
|
|
Cell[CellGroupData[{
|
|
Cell[198606, 5555, 803, 22, 30, "Input",ExpressionUUID->"cc5802c8-5164-4d35-a4aa-ac0f06412d48"],
|
|
Cell[199412, 5579, 301, 5, 34, "Output",ExpressionUUID->"6c0f8654-3af4-43e9-b5c1-7a47f72b02d4"]
|
|
}, Open ]],
|
|
Cell[199728, 5587, 896, 22, 52, "Input",ExpressionUUID->"8cbe4386-fbc6-4914-8e8c-e90d47f39535"]
|
|
}, Open ]]
|
|
}
|
|
]
|
|
*)
|
|
|