EPAWTFT/Notebooks/HughHubbarrdDimer.nb
2020-11-19 11:14:19 +00:00

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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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FrontEnd`Private`$ColorSelectorInitialColor =
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BaselinePosition -> Baseline, DefaultBaseStyle -> {},
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If[
Not[
AbsoluteCurrentValue["Deployed"]],
SelectionMove[Typeset`box$, All, Expression];
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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UHF energy is a pretty good estimate. \nMost of the UMP expansion is actually \
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 \
perturbation series\nat the expense of a divergent excited-state perturbation \
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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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",
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StyleBox["So what happens to the RMP exceptional point?",
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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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