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Evaluating the latest expression at ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", "\[Epsilon]"}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "b6a3c5d7-f350-4a35-992f-303ee61e416e"], ", we get" }], "Text", CellChangeTimes->{{3.795578345755261*^9, 3.795578373504217*^9}, { 3.79558342932677*^9, 3.795583438677984*^9}, 3.795583796485182*^9, { 3.795584068562727*^9, 3.7955840745069227`*^9}},ExpressionUUID->"1e019b3f-0ec6-4996-b433-\ 436bfa68249a"], Cell[BoxData[ FrameBox[ RowBox[{"\[Alpha]", "=", RowBox[{ FractionBox[ RowBox[{ SuperscriptBox[ OverscriptBox["E", "~"], RowBox[{"w", "=", "\[Epsilon]"}]], "-", SuperscriptBox["E", RowBox[{"w", "=", "\[Epsilon]"}]]}], RowBox[{" ", RowBox[{"\[Epsilon]", " ", SubsuperscriptBox["E", "x", RowBox[{"w", "=", "\[Epsilon]"}]]}]}]], "=", RowBox[{ FractionBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["E", RowBox[{"w", "=", "1"}]], "-", SuperscriptBox["E", RowBox[{"w", "=", "0"}]]}], ")"}], "\[Epsilon]"}], "+", 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Easy." }], "Item", CellChangeTimes->{{3.7955835646032867`*^9, 3.795583632216599*^9}, { 3.795583676432597*^9, 3.795583681226728*^9}, {3.795583759742305*^9, 3.795583761044952*^9}, {3.7955841440139723`*^9, 3.795584145624457*^9}},ExpressionUUID->"af84314f-3568-4b2a-86ae-\ 79f56f67a074"], Cell[TextData[{ "a MOM-style calculation at ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", "1"}], TraditionalForm]],ExpressionUUID-> "ee22702f-91b4-436d-91e6-74d7ec2b66b3"], " to get ", Cell[BoxData[ FormBox[ SuperscriptBox["E", RowBox[{"w", "=", "1"}]], TraditionalForm]],ExpressionUUID-> "2fa89da7-986c-488f-9250-02cb37d26e5e"], ". Less easy but doable with the new algorithms developed in Gill\ \[CloseCurlyQuote]s group and Head-Gordon\[CloseCurlyQuote]s group (MOM, SGM, \ etc)." }], "Item", CellChangeTimes->{{3.7955835646032867`*^9, 3.7955836633523397`*^9}, { 3.795583762585924*^9, 3.7955837819945183`*^9}, {3.795584157782454*^9, 3.795584176310916*^9}},ExpressionUUID->"8a822d39-2341-4596-ada7-\ e75d9e6690e9"], Cell[TextData[{ "a GOK-DFT calculation at ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", "\[Epsilon]"}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "9999f8ce-914f-402d-9fd6-55f4678521c9"], " to get ", Cell[BoxData[ FormBox[ SuperscriptBox["E", RowBox[{"w", "=", "\[Epsilon]"}]], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "f72c2cb9-9c03-48f4-85f7-b28b6d915ec5"], " and ", Cell[BoxData[ FormBox[ SubsuperscriptBox["E", "x", RowBox[{"w", "=", "\[Epsilon]"}]], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "93920de6-9927-4bf9-a7af-530961d43aa6"], ". Because ", Cell[BoxData[ FormBox["\[Epsilon]", TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "f0dad6c5-1ee2-4393-941b-dadaa8701da1"], " is small, in practice, the calculation is going to converge easily, which \ is a very important point, especially if one starts with the converged \ ground-state density matrix." }], "Item", CellChangeTimes->{{3.7955835646032867`*^9, 3.795583705466729*^9}, { 3.795583798079362*^9, 3.7955838291067142`*^9}},ExpressionUUID->"6fce318d-b27e-4732-8652-\ d9af60c9d126"] }, Open ]], Cell[TextData[{ "In such a way, one can approximate efficiently the ensemble derivative at \ ", Cell[BoxData[ FormBox[ RowBox[{"w", "=", "0"}], TraditionalForm]], FormatType->"TraditionalForm",ExpressionUUID-> "39cdb037-a197-48f8-805e-9092aacf0854"], ", and obtains excitation energies." }], "Text", CellChangeTimes->{{3.795583713553678*^9, 3.795583746528973*^9}},ExpressionUUID->"2122a45a-faf8-43b9-ad55-\ a691c6aeb453"], Cell[TextData[StyleBox["What do you guys 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