EPAWTFT/Notebooks/HughHubbarrdDimer.nb
2020-11-18 20:14:29 +00:00

65793 lines
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SelectionMove[Typeset`box$, All, Expression];
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FrontEnd`Private`$ColorSelectorInitialColor =
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RowBox[{"0", ",",
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BaselinePosition -> Baseline, DefaultBaseStyle -> {},
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Not[
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SelectionMove[Typeset`box$, All, Expression];
FrontEnd`Private`$ColorSelectorInitialAlpha = 1;
FrontEnd`Private`$ColorSelectorInitialColor =
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FrontEnd`Private`$ColorSelectorUseMakeBoxes = True;
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FrontEndResource["RGBColorValueSelector"], {
0, {Left, Bottom}}, {Left, Top},
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AbsoluteCurrentValue[Magnification]}]],
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Cell[TextData[{
StyleBox["Let\[CloseCurlyQuote]s take a quick look at the exceptional points \
we have found so far...\n\nFCI: 1 = \[PlusMinus] i 4t / U\nRMP: \
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StyleBox[" \[PlusMinus] i 4t / U\nCFP: 1 = 2t / U\n\nSo we see \
that the magnitudes of each exceptional point are all closely related.\nIs it \
possible that each of these exceptional point is dictated by the fundamental \
exceptional point in the FCI energy?\nOr is this an artefact of our \
incredible simple model?",
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Cell[TextData[StyleBox["We can now consider the MP expansion from the \
symmetry-broken UHF reference state.\nThe optimal rotation angles are \
computed in the HF section and are given as...",
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PaneBox[
TagBox[
GridBox[{{
TagBox[
GridBox[{{
GraphicsBox[{{
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GrayLevel[0]]],
PointSize[0.5],
CapForm["Butt"],
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RGBColor[0,
NCache[
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DiskBox[{0, 0}]}, {DefaultBaseStyle -> {"Graphics", {
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Directive[
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GrayLevel[0]]],
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NCache[
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ImagePadding -> Automatic,
BaselinePosition -> (Scaled[-0.08426020408163262] ->
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GraphicsBox[{{
Directive[
EdgeForm[
Directive[
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GrayLevel[0]]],
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RGBColor[1, 0, 0]], {
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ImagePadding -> Automatic,
BaselinePosition -> (Scaled[-0.051309523809523805`] ->
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GridBoxAlignment -> {
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AutoDelete -> False,
GridBoxDividers -> {
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GridBoxItemSize -> {"Columns" -> {{All}}, "Rows" -> {{All}}},
GridBoxSpacings -> {
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GridBoxAlignment -> {"Columns" -> {{Left}}, "Rows" -> {{Top}}},
AutoDelete -> False,
GridBoxItemSize -> {
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GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}],
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GraphicsBox[{{
GrayLevel[0],
RectangleBox[{0, 0}]}, {
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RectangleBox[{1, -1}]}, {
RGBColor[0,
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RectangleBox[{0, -1}, {2, 1}]}}, DefaultBaseStyle ->
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FrameTicks -> None, PlotRangePadding -> None, ImageSize ->
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StyleBox[
RowBox[{"RGBColor", "[",
RowBox[{"0", ",",
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ButtonFunction :> With[{Typeset`box$ = EvaluationBox[]},
If[
Not[
AbsoluteCurrentValue["Deployed"]],
SelectionMove[Typeset`box$, All, Expression];
FrontEnd`Private`$ColorSelectorInitialAlpha = 1;
FrontEnd`Private`$ColorSelectorInitialColor =
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Not[
AbsoluteCurrentValue["Deployed"]],
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FrontEnd`Private`$ColorSelectorInitialAlpha = 1;
FrontEnd`Private`$ColorSelectorInitialColor =
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exceptional point this close to the radius of convergence gives an \
increasingly slow convergence of the UMP series, but it will converge \
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correcting the spin-contamination in the wave function.\n\nOn the other hand, \
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symmetry-broken UHF wave function can retain a convergent ground-state \
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series. (Note: the orbitals are not optimised for excited-state here).\nIn \
contrast, the RMP expansion was always convergent for the open-shell excited \
state (which was a single CSF) while\nthe radius of convergence for the \
doubly-excited state was identical to the ground-state as this was the only \
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}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["U = 3t", "Subsubsection",
CellChangeTimes->{{3.814269128149407*^9, 3.814269131479203*^9},
3.8142694324399776`*^9},ExpressionUUID->"15c444d1-fa26-43c5-84a4-\
246e5de0c8ed"],
Cell[TextData[StyleBox["If we now consider a case within the radius of \
convergence for RMP, we see that the ground-state exceptional point has moved\
\nslightly further beyond the radius of convergence. However, this is still \
much closer than the RMP exceptional point, so the convergence\nis now slower \
than the RMP convergence.\n\nOn the other hand, the excited-state exceptional \
point is still within the radius of convergence and its expansion will \
therefore diverge.",
FontColor->RGBColor[1, 0, 0]]], "Text",
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3.8142695537684793`*^9, 3.814269748903685*^9}, {3.8142698084262543`*^9,
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Cell["Will the UMP ground-state always converge?", "Subsection",
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StyleBox["This important question can be paraphrased as \
\[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",
FontColor->RGBColor[1, 0, 0]],
StyleBox["So what happens to the RMP exceptional point?",
FontWeight->"Bold",
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.",
FontColor->RGBColor[1, 0, 0]]
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