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"Subsection", CellChangeTimes->{{3.814269865336405*^9, 3.814269873696*^9}},ExpressionUUID->"682f5d84-4017-4e58-bb21-1f874ccd820f"], Cell[TextData[{ 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]] }], "Text", CellChangeTimes->{{3.8142698778026943`*^9, 3.814270002740933*^9}, { 3.8142701039070168`*^9, 3.8142701163949423`*^9}, {3.81427015808629*^9, 3.814270160550354*^9}, {3.8142702008455763`*^9, 3.814270503562158*^9}, { 3.814270542836125*^9, 3.814270975747213*^9}},ExpressionUUID->"9b51c24b-654d-48dc-878d-\ 1415e41df166"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Summary", "Section"]], "Subsection", CellChangeTimes->{{3.8142698234669323`*^9, 3.8142698246701813`*^9}},ExpressionUUID->"555b6565-ed9c-41db-8747-\ 3eab70a65674"], 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]], "\n" }], "Text", CellChangeTimes->{{3.814269827673455*^9, 3.814269857755002*^9}, { 3.814271151755376*^9, 3.814271392564*^9}, {3.814271434020362*^9, 3.8142716695547*^9}},ExpressionUUID->"c8622b61-f1b5-4100-9c90-7559df35a50a"] }, Open ]] }, Open ]] }, WindowSize->{1912, 1391}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, FrontEndVersion->"12.1 for Mac OS X x86 (64-bit) (March 13, 2020)", StyleDefinitions->"Default.nb", ExpressionUUID->"92a1c26d-f273-40fc-a18e-954c81c40723" ] (* End 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