64953 lines
3.6 MiB
64953 lines
3.6 MiB
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for the convergence behaviour of MP theories.\n\n1) Singlet FCI on real axis \
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and complex plane\n2) RMP expansion\n3) UMP expansion",
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PlotRange -> {{0., 46.120000000000005`}, {0., 16.34}}, AspectRatio ->
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Automatic}]},
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"PointLegend",
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DisplayFunction->(FormBox[
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StyleBox[
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StyleBox[
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PaneBox[
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TagBox[
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GridBox[{{
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TagBox[
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GridBox[{{
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GraphicsBox[{{
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Directive[
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EdgeForm[
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Directive[
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Opacity[0.3],
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GrayLevel[0]]],
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PointSize[0.5],
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CapForm["Butt"],
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AbsoluteThickness[1.6],
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RGBColor[0,
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NCache[
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Rational[2, 3], 0.6666666666666666], 0]], {
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LineBox[{{0, 6.272}, {20, 6.272}}]}}, {
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Directive[
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EdgeForm[
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Directive[
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Opacity[0.3],
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GrayLevel[0]]],
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PointSize[0.5],
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CapForm["Butt"],
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AbsoluteThickness[1.6],
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RGBColor[0,
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NCache[
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Rational[2, 3], 0.6666666666666666], 0]], {
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InsetBox[
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GraphicsBox[{
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EdgeForm[],
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DiskBox[{0, 0}]}, {DefaultBaseStyle -> {"Graphics", {
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AbsolutePointSize[6]},
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Directive[
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EdgeForm[
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Directive[
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Opacity[0.3],
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GrayLevel[0]]],
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PointSize[0.5],
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CapForm["Butt"],
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AbsoluteThickness[1.6],
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RGBColor[0,
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Rational[2, 3], 0]]}}],
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NCache[
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Scaled[{
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Rational[1, 2],
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Rational[1, 2]}],
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Scaled[{0.5, 0.5}]], Automatic,
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Scaled[1]]}}}, AspectRatio -> Full,
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|
ImageSize -> {20, 6.272}, PlotRangePadding -> None,
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ImagePadding -> Automatic,
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BaselinePosition -> (Scaled[-0.08426020408163262] ->
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Baseline)], #}, {
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GraphicsBox[{{
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Directive[
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EdgeForm[
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Directive[
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Opacity[0.3],
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GrayLevel[0]]],
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PointSize[0.5],
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CapForm["Butt"],
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AbsoluteThickness[1.6],
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RGBColor[1, 0, 0]], {
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LineBox[{{0, 6.72}, {20, 6.72}}]}}, {
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Directive[
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EdgeForm[
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Directive[
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Opacity[0.3],
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GrayLevel[0]]],
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PointSize[0.5],
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CapForm["Butt"],
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AbsoluteThickness[1.6],
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RGBColor[1, 0, 0]], {
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InsetBox[
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GraphicsBox[{
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EdgeForm[],
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PolygonBox[{{-1, -1}, {1, -1}, {1, 1}, {-1, 1}}]}, {
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DefaultBaseStyle -> {"Graphics", {
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AbsolutePointSize[6]},
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Directive[
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EdgeForm[
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|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
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|
RGBColor[1, 0, 0]]}}],
|
|
NCache[
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|
Scaled[{
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|
Rational[1, 2],
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|
Rational[1, 2]}],
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|
Scaled[{0.5, 0.5}]], Automatic,
|
|
Scaled[1]]}}}, AspectRatio -> Full,
|
|
ImageSize -> {20, 6.72}, PlotRangePadding -> None,
|
|
ImagePadding -> Automatic,
|
|
BaselinePosition -> (Scaled[-0.051309523809523805`] ->
|
|
Baseline)], #2}},
|
|
GridBoxAlignment -> {
|
|
"Columns" -> {Center, Left}, "Rows" -> {{Baseline}}},
|
|
AutoDelete -> False,
|
|
GridBoxDividers -> {
|
|
"Columns" -> {{False}}, "Rows" -> {{False}}},
|
|
GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}},
|
|
GridBoxSpacings -> {
|
|
"Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}},
|
|
GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}},
|
|
AutoDelete -> False,
|
|
GridBoxItemSize -> {
|
|
"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}},
|
|
GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}],
|
|
"Grid"], Alignment -> Left, AppearanceElements -> None,
|
|
ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction ->
|
|
"ResizeToFit"], LineIndent -> 0, StripOnInput -> False], {
|
|
FontFamily -> "Times"}, Background -> Automatic, StripOnInput ->
|
|
False], TraditionalForm]& ),
|
|
Editable->True,
|
|
InterpretationFunction:>(RowBox[{"PointLegend", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"Directive", "[",
|
|
RowBox[{
|
|
RowBox[{"PointSize", "[", "0.009166666666666668`", "]"}],
|
|
",",
|
|
RowBox[{"CapForm", "[", "\"Butt\"", "]"}], ",",
|
|
RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",",
|
|
InterpretationBox[
|
|
ButtonBox[
|
|
TooltipBox[
|
|
GraphicsBox[{{
|
|
GrayLevel[0],
|
|
RectangleBox[{0, 0}]}, {
|
|
GrayLevel[0],
|
|
RectangleBox[{1, -1}]}, {
|
|
RGBColor[0,
|
|
Rational[2, 3], 0],
|
|
RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle ->
|
|
"ColorSwatchGraphics", AspectRatio -> 1, Frame -> True,
|
|
FrameStyle -> RGBColor[0., 0.4444444444444444, 0.],
|
|
FrameTicks -> None, PlotRangePadding -> None, ImageSize ->
|
|
Dynamic[{
|
|
Automatic, 1.35 CurrentValue["FontCapHeight"]/
|
|
AbsoluteCurrentValue[Magnification]}]],
|
|
StyleBox[
|
|
RowBox[{"RGBColor", "[",
|
|
RowBox[{"0", ",",
|
|
FractionBox["2", "3"], ",", "0"}], "]"}], NumberMarks ->
|
|
False]], Appearance -> None, BaseStyle -> {},
|
|
BaselinePosition -> Baseline, DefaultBaseStyle -> {},
|
|
ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]},
|
|
If[
|
|
Not[
|
|
AbsoluteCurrentValue["Deployed"]],
|
|
SelectionMove[Typeset`box$, All, Expression];
|
|
FrontEnd`Private`$ColorSelectorInitialAlpha = 1;
|
|
FrontEnd`Private`$ColorSelectorInitialColor =
|
|
RGBColor[0,
|
|
Rational[2, 3], 0];
|
|
FrontEnd`Private`$ColorSelectorUseMakeBoxes = True;
|
|
MathLink`CallFrontEnd[
|
|
FrontEnd`AttachCell[Typeset`box$,
|
|
FrontEndResource["RGBColorValueSelector"], {
|
|
0, {Left, Bottom}}, {Left, Top},
|
|
"ClosingActions" -> {
|
|
"SelectionDeparture", "ParentChanged",
|
|
"EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator ->
|
|
Automatic, Method -> "Preemptive"],
|
|
RGBColor[0,
|
|
Rational[2, 3], 0], Editable -> False, Selectable ->
|
|
False]}], "]"}], ",",
|
|
RowBox[{"Directive", "[",
|
|
RowBox[{
|
|
RowBox[{"PointSize", "[", "0.009166666666666668`", "]"}],
|
|
",",
|
|
RowBox[{"CapForm", "[", "\"Butt\"", "]"}], ",",
|
|
RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",",
|
|
InterpretationBox[
|
|
ButtonBox[
|
|
TooltipBox[
|
|
GraphicsBox[{{
|
|
GrayLevel[0],
|
|
RectangleBox[{0, 0}]}, {
|
|
GrayLevel[0],
|
|
RectangleBox[{1, -1}]}, {
|
|
RGBColor[1, 0, 0],
|
|
RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle ->
|
|
"ColorSwatchGraphics", AspectRatio -> 1, Frame -> True,
|
|
FrameStyle -> RGBColor[0.6666666666666666, 0., 0.],
|
|
FrameTicks -> None, PlotRangePadding -> None, ImageSize ->
|
|
Dynamic[{
|
|
Automatic, 1.35 CurrentValue["FontCapHeight"]/
|
|
AbsoluteCurrentValue[Magnification]}]],
|
|
StyleBox[
|
|
RowBox[{"RGBColor", "[",
|
|
RowBox[{"1", ",", "0", ",", "0"}], "]"}], NumberMarks ->
|
|
False]], Appearance -> None, BaseStyle -> {},
|
|
BaselinePosition -> Baseline, DefaultBaseStyle -> {},
|
|
ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]},
|
|
If[
|
|
Not[
|
|
AbsoluteCurrentValue["Deployed"]],
|
|
SelectionMove[Typeset`box$, All, Expression];
|
|
FrontEnd`Private`$ColorSelectorInitialAlpha = 1;
|
|
FrontEnd`Private`$ColorSelectorInitialColor =
|
|
RGBColor[1, 0, 0];
|
|
FrontEnd`Private`$ColorSelectorUseMakeBoxes = True;
|
|
MathLink`CallFrontEnd[
|
|
FrontEnd`AttachCell[Typeset`box$,
|
|
FrontEndResource["RGBColorValueSelector"], {
|
|
0, {Left, Bottom}}, {Left, Top},
|
|
"ClosingActions" -> {
|
|
"SelectionDeparture", "ParentChanged",
|
|
"EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator ->
|
|
Automatic, Method -> "Preemptive"],
|
|
RGBColor[1, 0, 0], Editable -> False, Selectable ->
|
|
False]}], "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{#, ",", #2}], "}"}], ",",
|
|
RowBox[{"LegendMarkers", "\[Rule]",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
GraphicsBox[{
|
|
EdgeForm[],
|
|
DiskBox[{0, 0}]}], ",",
|
|
RowBox[{"Offset", "[", "6.272`", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
GraphicsBox[{
|
|
EdgeForm[],
|
|
PolygonBox[{{-1, -1}, {1, -1}, {1, 1}, {-1, 1}}]}], ",",
|
|
RowBox[{"Offset", "[", "6.72`", "]"}]}], "}"}]}], "}"}]}],
|
|
",",
|
|
RowBox[{"Joined", "\[Rule]",
|
|
RowBox[{"{",
|
|
RowBox[{"True", ",", "True"}], "}"}]}], ",",
|
|
RowBox[{"LabelStyle", "\[Rule]",
|
|
RowBox[{"{",
|
|
RowBox[{"FontFamily", "\[Rule]", "\"Times\""}], "}"}]}], ",",
|
|
RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& )],
|
|
Scaled[{0.2, 0.85}], ImageScaled[{0.5, 0.5}],
|
|
BaseStyle->{FontSize -> Larger},
|
|
FormatType->StandardForm]},
|
|
AspectRatio->0.7,
|
|
Axes->{False, False},
|
|
AxesLabel->{None, None},
|
|
AxesOrigin->{0, 0},
|
|
DisplayFunction->Identity,
|
|
Frame->{{True, True}, {True, True}},
|
|
FrameLabel->{{
|
|
FormBox[
|
|
GraphicsBox[{
|
|
Thickness[0.02440214738897023],
|
|
StyleBox[{
|
|
FilledCurveBox[CompressedData["
|
|
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|
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|
|
"], CompressedData["
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"]],
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|
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FilledCurveBox[{{{1, 4, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3, 3}, {1, 3,
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"Droplines" -> {
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"freeformCursorMode" -> True,
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"placement" -> {"x" -> "All", "y" -> "None"}}}}, "RotationControl" ->
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"Globe"},
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PlotRange->{{-1.2, 1.2}, {0, 2.4}, {-2, 10}},
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PlotRangePadding->{
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Scaled[0.02],
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Scaled[0.02], Automatic},
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Ticks->{Automatic, Automatic, Automatic},
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PaneBox[
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TagBox[
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GridBox[{{
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EdgeForm[
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GrayLevel[0]]],
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PointSize[0.5],
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CapForm["Butt"],
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AbsoluteThickness[1.6],
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RGBColor[0,
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NCache[
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Rational[2, 3], 0.6666666666666666], 0]], {
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InsetBox[
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GraphicsBox[{
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EdgeForm[],
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DiskBox[{0, 0}]}, {DefaultBaseStyle -> {"Graphics", {
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AbsolutePointSize[6]},
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Directive[
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EdgeForm[
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Directive[
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Opacity[0.3],
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GrayLevel[0]]],
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PointSize[0.5],
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CapForm["Butt"],
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AbsoluteThickness[1.6],
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RGBColor[0,
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Rational[2, 3], 0]]}}],
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NCache[
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Scaled[{
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Rational[1, 2],
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Rational[1, 2]}],
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Scaled[{0.5, 0.5}]], Automatic,
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Scaled[1]]}}}, AspectRatio -> Full,
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ImageSize -> {20, 6.272}, PlotRangePadding -> None,
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ImagePadding -> Automatic,
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BaselinePosition -> (Scaled[-0.08426020408163262] ->
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Baseline)], #}, {
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GraphicsBox[{{
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Directive[
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EdgeForm[
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Directive[
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GrayLevel[0]]],
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PointSize[0.5],
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CapForm["Butt"],
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AbsoluteThickness[1.6],
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RGBColor[1, 0, 0]], {
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LineBox[{{0, 6.72}, {20, 6.72}}]}}, {
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Directive[
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EdgeForm[
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Directive[
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GrayLevel[0]]],
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PointSize[0.5],
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CapForm["Butt"],
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AbsoluteThickness[1.6],
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RGBColor[1, 0, 0]], {
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InsetBox[
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GraphicsBox[{
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EdgeForm[],
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PolygonBox[{{-1, -1}, {1, -1}, {1, 1}, {-1, 1}}]}, {
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DefaultBaseStyle -> {"Graphics", {
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AbsolutePointSize[6]},
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Directive[
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EdgeForm[
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Directive[
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Opacity[0.3],
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GrayLevel[0]]],
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PointSize[0.5],
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CapForm["Butt"],
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AbsoluteThickness[1.6],
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RGBColor[1, 0, 0]]}}],
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NCache[
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Scaled[{
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Rational[1, 2],
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Rational[1, 2]}],
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Scaled[{0.5, 0.5}]], Automatic,
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Scaled[1]]}}}, AspectRatio -> Full,
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ImageSize -> {20, 6.72}, PlotRangePadding -> None,
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ImagePadding -> Automatic,
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BaselinePosition -> (Scaled[-0.051309523809523805`] ->
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Baseline)], #2}},
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GridBoxAlignment -> {
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"Columns" -> {Center, Left}, "Rows" -> {{Baseline}}},
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AutoDelete -> False,
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GridBoxDividers -> {
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"Columns" -> {{False}}, "Rows" -> {{False}}},
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GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}},
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GridBoxSpacings -> {
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"Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}},
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GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}},
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|
AutoDelete -> False,
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GridBoxItemSize -> {
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"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}},
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|
GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}],
|
|
"Grid"], Alignment -> Left, AppearanceElements -> None,
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|
ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction ->
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|
"ResizeToFit"], LineIndent -> 0, StripOnInput -> False], {
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|
FontFamily -> "Times"}, Background -> Automatic, StripOnInput ->
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|
False], TraditionalForm]& ),
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|
Editable->True,
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InterpretationFunction:>(RowBox[{"PointLegend", "[",
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RowBox[{
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|
RowBox[{"{",
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RowBox[{
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RowBox[{"Directive", "[",
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RowBox[{
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RowBox[{"PointSize", "[", "0.007333333333333334`", "]"}],
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",",
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RowBox[{"CapForm", "[", "\"Butt\"", "]"}], ",",
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|
RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",",
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InterpretationBox[
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ButtonBox[
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|
TooltipBox[
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|
GraphicsBox[{{
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|
GrayLevel[0],
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|
RectangleBox[{0, 0}]}, {
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|
GrayLevel[0],
|
|
RectangleBox[{1, -1}]}, {
|
|
RGBColor[0,
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|
Rational[2, 3], 0],
|
|
RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle ->
|
|
"ColorSwatchGraphics", AspectRatio -> 1, Frame -> True,
|
|
FrameStyle -> RGBColor[0., 0.4444444444444444, 0.],
|
|
FrameTicks -> None, PlotRangePadding -> None, ImageSize ->
|
|
Dynamic[{
|
|
Automatic, 1.35 CurrentValue["FontCapHeight"]/
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|
AbsoluteCurrentValue[Magnification]}]],
|
|
StyleBox[
|
|
RowBox[{"RGBColor", "[",
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|
RowBox[{"0", ",",
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|
FractionBox["2", "3"], ",", "0"}], "]"}], NumberMarks ->
|
|
False]], Appearance -> None, BaseStyle -> {},
|
|
BaselinePosition -> Baseline, DefaultBaseStyle -> {},
|
|
ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]},
|
|
If[
|
|
Not[
|
|
AbsoluteCurrentValue["Deployed"]],
|
|
SelectionMove[Typeset`box$, All, Expression];
|
|
FrontEnd`Private`$ColorSelectorInitialAlpha = 1;
|
|
FrontEnd`Private`$ColorSelectorInitialColor =
|
|
RGBColor[0,
|
|
Rational[2, 3], 0];
|
|
FrontEnd`Private`$ColorSelectorUseMakeBoxes = True;
|
|
MathLink`CallFrontEnd[
|
|
FrontEnd`AttachCell[Typeset`box$,
|
|
FrontEndResource["RGBColorValueSelector"], {
|
|
0, {Left, Bottom}}, {Left, Top},
|
|
"ClosingActions" -> {
|
|
"SelectionDeparture", "ParentChanged",
|
|
"EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator ->
|
|
Automatic, Method -> "Preemptive"],
|
|
RGBColor[0,
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|
Rational[2, 3], 0], Editable -> False, Selectable ->
|
|
False]}], "]"}], ",",
|
|
RowBox[{"Directive", "[",
|
|
RowBox[{
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BaselinePosition -> Scaled[0.32439307852814453`],
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ImageSize -> {38., 17.}, PlotRange -> {{0., 37.02}, {0., 16.34}},
|
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AspectRatio -> Automatic}]},
|
|
"PointLegend",
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|
DisplayFunction->(FormBox[
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StyleBox[
|
|
StyleBox[
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|
PaneBox[
|
|
TagBox[
|
|
GridBox[{{
|
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TagBox[
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GridBox[{{
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|
GraphicsBox[{{
|
|
Directive[
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|
EdgeForm[
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|
Directive[
|
|
Opacity[0.3],
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|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[0,
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|
NCache[
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|
Rational[2, 3], 0.6666666666666666], 0]], {
|
|
LineBox[{{0, 6.272}, {20, 6.272}}]}}, {
|
|
Directive[
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|
EdgeForm[
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|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[0,
|
|
NCache[
|
|
Rational[2, 3], 0.6666666666666666], 0]], {
|
|
InsetBox[
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|
GraphicsBox[{
|
|
EdgeForm[],
|
|
DiskBox[{0, 0}]}, {DefaultBaseStyle -> {"Graphics", {
|
|
AbsolutePointSize[6]},
|
|
Directive[
|
|
EdgeForm[
|
|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[0,
|
|
Rational[2, 3], 0]]}}],
|
|
NCache[
|
|
Scaled[{
|
|
Rational[1, 2],
|
|
Rational[1, 2]}],
|
|
Scaled[{0.5, 0.5}]], Automatic,
|
|
Scaled[1]]}}}, AspectRatio -> Full,
|
|
ImageSize -> {20, 6.272}, PlotRangePadding -> None,
|
|
ImagePadding -> Automatic,
|
|
BaselinePosition -> (Scaled[-0.08426020408163262] ->
|
|
Baseline)], #}, {
|
|
GraphicsBox[{{
|
|
Directive[
|
|
EdgeForm[
|
|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[1, 0, 0]], {
|
|
LineBox[{{0, 6.72}, {20, 6.72}}]}}, {
|
|
Directive[
|
|
EdgeForm[
|
|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[1, 0, 0]], {
|
|
InsetBox[
|
|
GraphicsBox[{
|
|
EdgeForm[],
|
|
PolygonBox[{{-1, -1}, {1, -1}, {1, 1}, {-1, 1}}]}, {
|
|
DefaultBaseStyle -> {"Graphics", {
|
|
AbsolutePointSize[6]},
|
|
Directive[
|
|
EdgeForm[
|
|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[1, 0, 0]]}}],
|
|
NCache[
|
|
Scaled[{
|
|
Rational[1, 2],
|
|
Rational[1, 2]}],
|
|
Scaled[{0.5, 0.5}]], Automatic,
|
|
Scaled[1]]}}}, AspectRatio -> Full,
|
|
ImageSize -> {20, 6.72}, PlotRangePadding -> None,
|
|
ImagePadding -> Automatic,
|
|
BaselinePosition -> (Scaled[-0.051309523809523805`] ->
|
|
Baseline)], #2}},
|
|
GridBoxAlignment -> {
|
|
"Columns" -> {Center, Left}, "Rows" -> {{Baseline}}},
|
|
AutoDelete -> False,
|
|
GridBoxDividers -> {
|
|
"Columns" -> {{False}}, "Rows" -> {{False}}},
|
|
GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}},
|
|
GridBoxSpacings -> {
|
|
"Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}},
|
|
GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}},
|
|
AutoDelete -> False,
|
|
GridBoxItemSize -> {
|
|
"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}},
|
|
GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}],
|
|
"Grid"], Alignment -> Left, AppearanceElements -> None,
|
|
ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction ->
|
|
"ResizeToFit"], LineIndent -> 0, StripOnInput -> False], {
|
|
FontFamily -> "Times"}, Background -> Automatic, StripOnInput ->
|
|
False], TraditionalForm]& ),
|
|
Editable->True,
|
|
InterpretationFunction:>(RowBox[{"PointLegend", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"Directive", "[",
|
|
RowBox[{
|
|
RowBox[{"PointSize", "[", "0.009166666666666668`", "]"}],
|
|
",",
|
|
RowBox[{"CapForm", "[", "\"Butt\"", "]"}], ",",
|
|
RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",",
|
|
InterpretationBox[
|
|
ButtonBox[
|
|
TooltipBox[
|
|
GraphicsBox[{{
|
|
GrayLevel[0],
|
|
RectangleBox[{0, 0}]}, {
|
|
GrayLevel[0],
|
|
RectangleBox[{1, -1}]}, {
|
|
RGBColor[0,
|
|
Rational[2, 3], 0],
|
|
RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle ->
|
|
"ColorSwatchGraphics", AspectRatio -> 1, Frame -> True,
|
|
FrameStyle -> RGBColor[0., 0.4444444444444444, 0.],
|
|
FrameTicks -> None, PlotRangePadding -> None, ImageSize ->
|
|
Dynamic[{
|
|
Automatic, 1.35 CurrentValue["FontCapHeight"]/
|
|
AbsoluteCurrentValue[Magnification]}]],
|
|
StyleBox[
|
|
RowBox[{"RGBColor", "[",
|
|
RowBox[{"0", ",",
|
|
FractionBox["2", "3"], ",", "0"}], "]"}], NumberMarks ->
|
|
False]], Appearance -> None, BaseStyle -> {},
|
|
BaselinePosition -> Baseline, DefaultBaseStyle -> {},
|
|
ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]},
|
|
If[
|
|
Not[
|
|
AbsoluteCurrentValue["Deployed"]],
|
|
SelectionMove[Typeset`box$, All, Expression];
|
|
FrontEnd`Private`$ColorSelectorInitialAlpha = 1;
|
|
FrontEnd`Private`$ColorSelectorInitialColor =
|
|
RGBColor[0,
|
|
Rational[2, 3], 0];
|
|
FrontEnd`Private`$ColorSelectorUseMakeBoxes = True;
|
|
MathLink`CallFrontEnd[
|
|
FrontEnd`AttachCell[Typeset`box$,
|
|
FrontEndResource["RGBColorValueSelector"], {
|
|
0, {Left, Bottom}}, {Left, Top},
|
|
"ClosingActions" -> {
|
|
"SelectionDeparture", "ParentChanged",
|
|
"EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator ->
|
|
Automatic, Method -> "Preemptive"],
|
|
RGBColor[0,
|
|
Rational[2, 3], 0], Editable -> False, Selectable ->
|
|
False]}], "]"}], ",",
|
|
RowBox[{"Directive", "[",
|
|
RowBox[{
|
|
RowBox[{"PointSize", "[", "0.009166666666666668`", "]"}],
|
|
",",
|
|
RowBox[{"CapForm", "[", "\"Butt\"", "]"}], ",",
|
|
RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",",
|
|
InterpretationBox[
|
|
ButtonBox[
|
|
TooltipBox[
|
|
GraphicsBox[{{
|
|
GrayLevel[0],
|
|
RectangleBox[{0, 0}]}, {
|
|
GrayLevel[0],
|
|
RectangleBox[{1, -1}]}, {
|
|
RGBColor[1, 0, 0],
|
|
RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle ->
|
|
"ColorSwatchGraphics", AspectRatio -> 1, Frame -> True,
|
|
FrameStyle -> RGBColor[0.6666666666666666, 0., 0.],
|
|
FrameTicks -> None, PlotRangePadding -> None, ImageSize ->
|
|
Dynamic[{
|
|
Automatic, 1.35 CurrentValue["FontCapHeight"]/
|
|
AbsoluteCurrentValue[Magnification]}]],
|
|
StyleBox[
|
|
RowBox[{"RGBColor", "[",
|
|
RowBox[{"1", ",", "0", ",", "0"}], "]"}], NumberMarks ->
|
|
False]], Appearance -> None, BaseStyle -> {},
|
|
BaselinePosition -> Baseline, DefaultBaseStyle -> {},
|
|
ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]},
|
|
If[
|
|
Not[
|
|
AbsoluteCurrentValue["Deployed"]],
|
|
SelectionMove[Typeset`box$, All, Expression];
|
|
FrontEnd`Private`$ColorSelectorInitialAlpha = 1;
|
|
FrontEnd`Private`$ColorSelectorInitialColor =
|
|
RGBColor[1, 0, 0];
|
|
FrontEnd`Private`$ColorSelectorUseMakeBoxes = True;
|
|
MathLink`CallFrontEnd[
|
|
FrontEnd`AttachCell[Typeset`box$,
|
|
FrontEndResource["RGBColorValueSelector"], {
|
|
0, {Left, Bottom}}, {Left, Top},
|
|
"ClosingActions" -> {
|
|
"SelectionDeparture", "ParentChanged",
|
|
"EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator ->
|
|
Automatic, Method -> "Preemptive"],
|
|
RGBColor[1, 0, 0], Editable -> False, Selectable ->
|
|
False]}], "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{#, ",", #2}], "}"}], ",",
|
|
RowBox[{"LegendMarkers", "\[Rule]",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
GraphicsBox[{
|
|
EdgeForm[],
|
|
DiskBox[{0, 0}]}], ",",
|
|
RowBox[{"Offset", "[", "6.272`", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
GraphicsBox[{
|
|
EdgeForm[],
|
|
PolygonBox[{{-1, -1}, {1, -1}, {1, 1}, {-1, 1}}]}], ",",
|
|
RowBox[{"Offset", "[", "6.72`", "]"}]}], "}"}]}], "}"}]}],
|
|
",",
|
|
RowBox[{"Joined", "\[Rule]",
|
|
RowBox[{"{",
|
|
RowBox[{"True", ",", "True"}], "}"}]}], ",",
|
|
RowBox[{"LabelStyle", "\[Rule]",
|
|
RowBox[{"{",
|
|
RowBox[{"FontFamily", "\[Rule]", "\"Times\""}], "}"}]}], ",",
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BaselinePosition -> Scaled[0.32439307852814453`],
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ImageSize -> {38., 17.}, PlotRange -> {{0., 37.02}, {0., 16.34}},
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AspectRatio -> Automatic}],GraphicsBox[{
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ImageSize -> {55.52742216687422, 24.508064757160646`},
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ImageSize -> {37.01828144458281, 16.338709838107096`},
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BaselinePosition -> Scaled[0.32439307852814453`],
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ImageSize -> {38., 17.}, PlotRange -> {{0., 37.02}, {0., 16.34}},
|
|
AspectRatio -> Automatic}]},
|
|
"PointLegend",
|
|
DisplayFunction->(FormBox[
|
|
StyleBox[
|
|
StyleBox[
|
|
PaneBox[
|
|
TagBox[
|
|
GridBox[{{
|
|
TagBox[
|
|
GridBox[{{
|
|
GraphicsBox[{{
|
|
Directive[
|
|
EdgeForm[
|
|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[0,
|
|
NCache[
|
|
Rational[2, 3], 0.6666666666666666], 0]], {
|
|
LineBox[{{0, 6.272}, {20, 6.272}}]}}, {
|
|
Directive[
|
|
EdgeForm[
|
|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[0,
|
|
NCache[
|
|
Rational[2, 3], 0.6666666666666666], 0]], {
|
|
InsetBox[
|
|
GraphicsBox[{
|
|
EdgeForm[],
|
|
DiskBox[{0, 0}]}, {DefaultBaseStyle -> {"Graphics", {
|
|
AbsolutePointSize[6]},
|
|
Directive[
|
|
EdgeForm[
|
|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[0,
|
|
Rational[2, 3], 0]]}}],
|
|
NCache[
|
|
Scaled[{
|
|
Rational[1, 2],
|
|
Rational[1, 2]}],
|
|
Scaled[{0.5, 0.5}]], Automatic,
|
|
Scaled[1]]}}}, AspectRatio -> Full,
|
|
ImageSize -> {20, 6.272}, PlotRangePadding -> None,
|
|
ImagePadding -> Automatic,
|
|
BaselinePosition -> (Scaled[-0.08426020408163262] ->
|
|
Baseline)], #}, {
|
|
GraphicsBox[{{
|
|
Directive[
|
|
EdgeForm[
|
|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[1, 0, 0]], {
|
|
LineBox[{{0, 6.72}, {20, 6.72}}]}}, {
|
|
Directive[
|
|
EdgeForm[
|
|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[1, 0, 0]], {
|
|
InsetBox[
|
|
GraphicsBox[{
|
|
EdgeForm[],
|
|
PolygonBox[{{-1, -1}, {1, -1}, {1, 1}, {-1, 1}}]}, {
|
|
DefaultBaseStyle -> {"Graphics", {
|
|
AbsolutePointSize[6]},
|
|
Directive[
|
|
EdgeForm[
|
|
Directive[
|
|
Opacity[0.3],
|
|
GrayLevel[0]]],
|
|
PointSize[0.5],
|
|
CapForm["Butt"],
|
|
AbsoluteThickness[1.6],
|
|
RGBColor[1, 0, 0]]}}],
|
|
NCache[
|
|
Scaled[{
|
|
Rational[1, 2],
|
|
Rational[1, 2]}],
|
|
Scaled[{0.5, 0.5}]], Automatic,
|
|
Scaled[1]]}}}, AspectRatio -> Full,
|
|
ImageSize -> {20, 6.72}, PlotRangePadding -> None,
|
|
ImagePadding -> Automatic,
|
|
BaselinePosition -> (Scaled[-0.051309523809523805`] ->
|
|
Baseline)], #2}},
|
|
GridBoxAlignment -> {
|
|
"Columns" -> {Center, Left}, "Rows" -> {{Baseline}}},
|
|
AutoDelete -> False,
|
|
GridBoxDividers -> {
|
|
"Columns" -> {{False}}, "Rows" -> {{False}}},
|
|
GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}},
|
|
GridBoxSpacings -> {
|
|
"Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}},
|
|
GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}},
|
|
AutoDelete -> False,
|
|
GridBoxItemSize -> {
|
|
"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}},
|
|
GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}],
|
|
"Grid"], Alignment -> Left, AppearanceElements -> None,
|
|
ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction ->
|
|
"ResizeToFit"], LineIndent -> 0, StripOnInput -> False], {
|
|
FontFamily -> "Times"}, Background -> Automatic, StripOnInput ->
|
|
False], TraditionalForm]& ),
|
|
Editable->True,
|
|
InterpretationFunction:>(RowBox[{"PointLegend", "[",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"Directive", "[",
|
|
RowBox[{
|
|
RowBox[{"PointSize", "[", "0.009166666666666668`", "]"}],
|
|
",",
|
|
RowBox[{"CapForm", "[", "\"Butt\"", "]"}], ",",
|
|
RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",",
|
|
InterpretationBox[
|
|
ButtonBox[
|
|
TooltipBox[
|
|
GraphicsBox[{{
|
|
GrayLevel[0],
|
|
RectangleBox[{0, 0}]}, {
|
|
GrayLevel[0],
|
|
RectangleBox[{1, -1}]}, {
|
|
RGBColor[0,
|
|
Rational[2, 3], 0],
|
|
RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle ->
|
|
"ColorSwatchGraphics", AspectRatio -> 1, Frame -> True,
|
|
FrameStyle -> RGBColor[0., 0.4444444444444444, 0.],
|
|
FrameTicks -> None, PlotRangePadding -> None, ImageSize ->
|
|
Dynamic[{
|
|
Automatic, 1.35 CurrentValue["FontCapHeight"]/
|
|
AbsoluteCurrentValue[Magnification]}]],
|
|
StyleBox[
|
|
RowBox[{"RGBColor", "[",
|
|
RowBox[{"0", ",",
|
|
FractionBox["2", "3"], ",", "0"}], "]"}], NumberMarks ->
|
|
False]], Appearance -> None, BaseStyle -> {},
|
|
BaselinePosition -> Baseline, DefaultBaseStyle -> {},
|
|
ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]},
|
|
If[
|
|
Not[
|
|
AbsoluteCurrentValue["Deployed"]],
|
|
SelectionMove[Typeset`box$, All, Expression];
|
|
FrontEnd`Private`$ColorSelectorInitialAlpha = 1;
|
|
FrontEnd`Private`$ColorSelectorInitialColor =
|
|
RGBColor[0,
|
|
Rational[2, 3], 0];
|
|
FrontEnd`Private`$ColorSelectorUseMakeBoxes = True;
|
|
MathLink`CallFrontEnd[
|
|
FrontEnd`AttachCell[Typeset`box$,
|
|
FrontEndResource["RGBColorValueSelector"], {
|
|
0, {Left, Bottom}}, {Left, Top},
|
|
"ClosingActions" -> {
|
|
"SelectionDeparture", "ParentChanged",
|
|
"EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator ->
|
|
Automatic, Method -> "Preemptive"],
|
|
RGBColor[0,
|
|
Rational[2, 3], 0], Editable -> False, Selectable ->
|
|
False]}], "]"}], ",",
|
|
RowBox[{"Directive", "[",
|
|
RowBox[{
|
|
RowBox[{"PointSize", "[", "0.009166666666666668`", "]"}],
|
|
",",
|
|
RowBox[{"CapForm", "[", "\"Butt\"", "]"}], ",",
|
|
RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}], ",",
|
|
InterpretationBox[
|
|
ButtonBox[
|
|
TooltipBox[
|
|
GraphicsBox[{{
|
|
GrayLevel[0],
|
|
RectangleBox[{0, 0}]}, {
|
|
GrayLevel[0],
|
|
RectangleBox[{1, -1}]}, {
|
|
RGBColor[1, 0, 0],
|
|
RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle ->
|
|
"ColorSwatchGraphics", AspectRatio -> 1, Frame -> True,
|
|
FrameStyle -> RGBColor[0.6666666666666666, 0., 0.],
|
|
FrameTicks -> None, PlotRangePadding -> None, ImageSize ->
|
|
Dynamic[{
|
|
Automatic, 1.35 CurrentValue["FontCapHeight"]/
|
|
AbsoluteCurrentValue[Magnification]}]],
|
|
StyleBox[
|
|
RowBox[{"RGBColor", "[",
|
|
RowBox[{"1", ",", "0", ",", "0"}], "]"}], NumberMarks ->
|
|
False]], Appearance -> None, BaseStyle -> {},
|
|
BaselinePosition -> Baseline, DefaultBaseStyle -> {},
|
|
ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]},
|
|
If[
|
|
Not[
|
|
AbsoluteCurrentValue["Deployed"]],
|
|
SelectionMove[Typeset`box$, All, Expression];
|
|
FrontEnd`Private`$ColorSelectorInitialAlpha = 1;
|
|
FrontEnd`Private`$ColorSelectorInitialColor =
|
|
RGBColor[1, 0, 0];
|
|
FrontEnd`Private`$ColorSelectorUseMakeBoxes = True;
|
|
MathLink`CallFrontEnd[
|
|
FrontEnd`AttachCell[Typeset`box$,
|
|
FrontEndResource["RGBColorValueSelector"], {
|
|
0, {Left, Bottom}}, {Left, Top},
|
|
"ClosingActions" -> {
|
|
"SelectionDeparture", "ParentChanged",
|
|
"EvaluatorQuit"}]]]], BaseStyle -> Inherited, Evaluator ->
|
|
Automatic, Method -> "Preemptive"],
|
|
RGBColor[1, 0, 0], Editable -> False, Selectable ->
|
|
False]}], "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{#, ",", #2}], "}"}], ",",
|
|
RowBox[{"LegendMarkers", "\[Rule]",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
GraphicsBox[{
|
|
EdgeForm[],
|
|
DiskBox[{0, 0}]}], ",",
|
|
RowBox[{"Offset", "[", "6.272`", "]"}]}], "}"}], ",",
|
|
RowBox[{"{",
|
|
RowBox[{
|
|
GraphicsBox[{
|
|
EdgeForm[],
|
|
PolygonBox[{{-1, -1}, {1, -1}, {1, 1}, {-1, 1}}]}], ",",
|
|
RowBox[{"Offset", "[", "6.72`", "]"}]}], "}"}]}], "}"}]}],
|
|
",",
|
|
RowBox[{"Joined", "\[Rule]",
|
|
RowBox[{"{",
|
|
RowBox[{"True", ",", "True"}], "}"}]}], ",",
|
|
RowBox[{"LabelStyle", "\[Rule]",
|
|
RowBox[{"{",
|
|
RowBox[{"FontFamily", "\[Rule]", "\"Times\""}], "}"}]}], ",",
|
|
RowBox[{"LegendLayout", "\[Rule]", "\"Column\""}]}], "]"}]& )],
|
|
Scaled[{0.2, 0.85}], ImageScaled[{0.5, 0.5}],
|
|
BaseStyle->{FontSize -> Larger},
|
|
FormatType->StandardForm]},
|
|
AspectRatio->0.7,
|
|
Axes->{False, False},
|
|
AxesLabel->{None, None},
|
|
AxesOrigin->{0, -2.3025850929940455`},
|
|
DisplayFunction->Identity,
|
|
Frame->{{True, True}, {True, True}},
|
|
FrameLabel->{{
|
|
FormBox[
|
|
GraphicsBox[{
|
|
Thickness[0.024336821611097587`],
|
|
StyleBox[{
|
|
FilledCurveBox[CompressedData["
|
|
1:eJxTTMoPymNmYGBgBGIdIIaxWZDYzATYDFA+AxnqSdVLTzfQ022U2EWJ34kR
|
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ulfJ\
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\>"],ExpressionUUID->"dda8e76c-085c-4ef4-a2f8-ad287a24dfe0"]
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}, Open ]]
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Cell[CellGroupData[{
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Cell["Will the UMP ground-state always converge?", "Subsection",
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CellChangeTimes->{{3.814269865336405*^9,
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3.814269873696*^9}},ExpressionUUID->"682f5d84-4017-4e58-bb21-1f874ccd820f"],
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Cell[TextData[{
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StyleBox["This important question can be paraphrased as \
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\[OpenCurlyDoubleQuote]does the ground-state EP ever fall within the radius \
|
|
of convergence?\[CloseCurlyDoubleQuote]. \nTo answer this, we can consider \
|
|
the fixed FCI point on the corresponding Riemann surface\n\nFirstly the \
|
|
nature of the symmetry-breaking in UHF will always give an energy that is an \
|
|
increasingly accurate estimate of the exact energy. \nThis is because UHF can \
|
|
capture the strong correlation effects by spatially separating the two \
|
|
electrons.\nAt the fixed point \[Lambda]=1, the energies of the states will \
|
|
always be the FCI energy, and this is independent of the reference orbitals.\n\
|
|
\nIf the ground-state is a good FCI estimate, it will cross from \[Lambda]=0 \
|
|
to 1 in a \[OpenCurlyDoubleQuote]flat\[CloseCurlyDoubleQuote] fashion. This \
|
|
makes it far less likely to interact with \na higher energy state, so it is \
|
|
less likely to encounter an exceptional point. Equally, the first excited \
|
|
state will tend towards the \nexact first-excited singlet FCI energy. It is \
|
|
possible for these states to cross on the real-\[Lambda] line, but any \
|
|
crossing must occur\nbeyond \[Lambda]=1 because otherwise they would not \
|
|
connect to the correct fixed point.\n\n",
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|
FontColor->RGBColor[1, 0, 0]],
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|
StyleBox["So what happens to the RMP exceptional point?",
|
|
FontWeight->"Bold",
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|
FontColor->RGBColor[1, 0, 0]],
|
|
StyleBox["\n\nIn the RMP case, the convergence is controlled by an \
|
|
exceptional point between the ground- and doubly-excited RHF configurations.\n\
|
|
These two states interact strongly in the \[Lambda] expansion in the large \
|
|
U/t limit because the ground RHF state is heavily \ncontaminated by the 3rd \
|
|
FCI singlet state. It is this strong interaction that causes the exceptional \
|
|
point, and ultimately\nthe divergence. \n\nIn the UMP case, this \
|
|
\[OpenCurlyDoubleQuote]energetic\[CloseCurlyDoubleQuote] contamination from \
|
|
the 3rd FCI state is reduced (removed?) by replacing it with triplet \
|
|
contamination.\nThe strong interaction is then shifted from the ground \
|
|
configuration to the \
|
|
\[OpenCurlyDoubleQuote]open-shell\[CloseCurlyDoubleQuote] configurations in \
|
|
the UHF framework, \nsince the 3rd FCI singlet state contributions are \
|
|
rotated into the virtual orbitals.\nThis explains why we see the severe \
|
|
excited-state exceptional point in the UMP Riemann surface.\nIn a way, we can \
|
|
think of this as the \
|
|
\[OpenCurlyDoubleQuote]open-shell\[CloseCurlyDoubleQuote] configurations \
|
|
\[OpenCurlyDoubleQuote]shielding\[CloseCurlyDoubleQuote] the ground-state \
|
|
from the strong interaction (and\nexceptional point), hence we have more \
|
|
robust convergence in ground-state, but less robust convergence in \
|
|
excited-state.",
|
|
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Cell[CellGroupData[{
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Cell[TextData[StyleBox["Summary", "Section"]], "Section",
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CellChangeTimes->{{3.8142698234669323`*^9,
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Cell[TextData[{
|
|
StyleBox["In this notebook, we have investigated the convergence of the RMP \
|
|
and UMP series in the Hubbard dimer through visualisation \non Riemann \
|
|
surfaces. \n\n",
|
|
FontColor->RGBColor[1, 0, 0]],
|
|
StyleBox["Our main conclusions:",
|
|
FontWeight->"Bold",
|
|
FontColor->RGBColor[1, 0, 0]],
|
|
StyleBox["\n1) Exact FCI problem has intrinsic exceptional points in the U/t \
|
|
complex plane, corresponding to an avoided crossing on the real axis.\n2) \
|
|
When RMP converges, it does so fast. Both the ground- and excited-states have \
|
|
the same radius of convergence as they are controlled\n by the same \
|
|
exceptional point. However, this exceptional point is prone to moving within \
|
|
the radius of convergence, causing divergence.\n3) Allowing the orbitals to \
|
|
break symmetry in UMP gives a more robust convergence for ground-state by \
|
|
removing energetic contamination\n in the reference at the expense of \
|
|
adding triplet contamination. This convergence is at the expense of a \
|
|
massively divergent excited-state\n perturbation series.\n \n",
|
|
FontColor->RGBColor[1, 0, 0]],
|
|
StyleBox["Moving Forwards...",
|
|
FontWeight->"Bold",
|
|
FontColor->RGBColor[1, 0, 0]],
|
|
StyleBox["\nOne of the more significant ideas could be the effect of orbital \
|
|
optimisation on the radius of convergence. In the UHF case we saw that \n\
|
|
ground-state orbital optimisation massively improved the robustness of the \
|
|
ground-state MP expansion. ",
|
|
FontColor->RGBColor[1, 0, 0]],
|
|
StyleBox["Would excited-state\norbital optimisation would do the same for \
|
|
excited-state MP expansions at the expense of ground-state convergence? \n",
|
|
FontWeight->"Bold",
|
|
FontColor->RGBColor[1, 0, 0]],
|
|
StyleBox["If so, this would provide reassurance in the computation of \
|
|
dynamic correlation for state-specific excited states, and would suggest\n\
|
|
that we should expect reliable behaviour in orbital-optimised coupled-cluster \
|
|
expansions.",
|
|
FontColor->RGBColor[1, 0, 0]],
|
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"\n"
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*)
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