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exceptional point on the excited energy surface that is well \ within the radius of convergence.\nWe can therefore say that the use of a \ 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 \ exceptional point.", FontColor->RGBColor[1, 0, 0]]], "Text", CellChangeTimes->{{3.814268582576668*^9, 3.814268613074078*^9}, 3.814268644505561*^9, {3.814268676983919*^9, 3.814268698739291*^9}, { 3.8142687339944763`*^9, 3.814269075783581*^9}, {3.814269181551373*^9, 3.814269183858851*^9}},ExpressionUUID->"ead47e6e-2cb5-4d9b-a984-\ d08011c4950f"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"BoxRatios", "\[Rule]", RowBox[{"{", RowBox[{"1", ",", " ", "1", ",", " ", "1"}], "}"}]}], ",", RowBox[{"ImageSize", "\[Rule]", "Large"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"Opacity", "[", "0.7", "]"}]}], ",", RowBox[{"ViewPoint", "\[Rule]", RowBox[{"{", RowBox[{"1.3", ",", RowBox[{"-", "1.6"}], ",", 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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 \ 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