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Hugh Burton
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@ -497,10 +497,10 @@ UHF solution appears with a lower energy than the RHF one.
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This critical point is analogous to the infamous Coulson--Fischer point identified in the hydrogen dimer.\cite{Coulson_1949}
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For $U \ge 2t$, the optimal orbital rotation angles for the UHF orbitals become
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\begin{align}
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\ta^\text{UHF} & = \arctan (-\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U}),
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\ta^\text{UHF} & = \arctan (-\frac{2t}{\sqrt{U^2 - 4t^2}}),
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\label{eq:ta_uhf}
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\\
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\tb^\text{UHF} & = \arctan (+\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U}),
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\tb^\text{UHF} & = \arctan (+\frac{2t}{\sqrt{U^2 - 4t^2}}),
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\label{eq:tb_uhf}
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\end{align}
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with the corresponding UHF ground-state energy
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@ -516,6 +516,18 @@ of the HF energy rather than a minimum.
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\subsection{Complex adiabatic connection}
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%============================================================%
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Self-consistency in HF approximations leads to the inherently non-linear Fock eigenvalue
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problem that is normally solved using an iterative approach.
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Alternatively, the non-linear terms arising from the Coulomb and exchange can be considered
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as a perturbation from the reference core Hamiltonian problem by introducing the parameterised Fock operator
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\begin{equation}
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\Hat{f}_{\lambda}(\vb{x}) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\vb{x}).
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\end{equation}
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The orbitals in the reference problem correspond to the symmetry-pure eigenfunctions of the one-electron core
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Hamiltonian, while self-consistent solutions at $\lambda = 1$ represent the orbitals of the exact HF solution.
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\titou{In a recent paper, \cite{Burton_2019} using holomorphic Hartree-Fock (h-HF) \cite{Hiscock_2014,Burton_2018,Burton_2016} as an analytic continuation of conventional HF theory, we have demonstrated, on a simple model, that one can interconvert states of different symmetries and natures by following well-defined contours in the complex $\lambda$-plane, where $\lambda$ is the strength of the electron-electron interaction (see Fig.~\ref{fig:iAC}).
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In particular, by slowly varying $\lambda$ in a similar (yet different) manner to an adiabatic connection in density-functional theory, \cite{Langreth_1975,Gunnarsson_1976,Zhang_2004} one can ``morph'' a ground-state wave function into an excited-state wave function via a stationary path of HF solutions. \cite{Seidl_2018}
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In such a way, we could obtain a doubly-excited state wave function starting from the ground state wave function, a process which is not as easy as one might think. \cite{Gilbert_2008,Thom_2008,Shea_2018}
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Cell[CellGroupData[{
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Cell[423001, 8063, 754, 23, 47, "Input",ExpressionUUID->"66bc836c-9fc9-47ad-ab86-9e609d819ce9"],
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Cell[423758, 8088, 237, 7, 42, "Output",ExpressionUUID->"aea98ab3-5b9b-452e-ba12-30ff9d7d7042"]
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Cell[422276, 8038, 754, 23, 47, "Input",ExpressionUUID->"66bc836c-9fc9-47ad-ab86-9e609d819ce9"],
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Cell[423033, 8063, 237, 7, 42, "Output",ExpressionUUID->"aea98ab3-5b9b-452e-ba12-30ff9d7d7042"]
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Cell[CellGroupData[{
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Cell[424032, 8100, 4786, 117, 248, "Input",ExpressionUUID->"900c20e6-7495-4b35-b098-6a8279e0abef"],
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Cell[428821, 8219, 79689, 1498, 427, "Output",ExpressionUUID->"3181322c-be4a-4ea0-94e8-9ba2452f85ae"]
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Cell[423307, 8075, 4786, 117, 248, "Input",ExpressionUUID->"900c20e6-7495-4b35-b098-6a8279e0abef"],
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Cell[428096, 8194, 79689, 1498, 427, "Output",ExpressionUUID->"3181322c-be4a-4ea0-94e8-9ba2452f85ae"]
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Cell[507800, 9695, 317, 5, 35, "Text",ExpressionUUID->"5c754450-1b09-4800-bcc0-9052c82f8a81"],
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Cell[CellGroupData[{
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Cell[508867, 9729, 6600, 167, 327, "Input",ExpressionUUID->"c68d16b8-ddf2-4ed4-88c2-e682495595d7"],
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Cell[515470, 9898, 385704, 6371, 577, 147770, 2470, "CachedBoxData", "BoxData", "Output",ExpressionUUID->"673148aa-d9dd-48bc-b4ff-3ceaa38d4fee"]
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Cell[508142, 9704, 6600, 167, 327, "Input",ExpressionUUID->"c68d16b8-ddf2-4ed4-88c2-e682495595d7"],
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Cell[514745, 9873, 385704, 6371, 577, 147770, 2470, "CachedBoxData", "BoxData", "Output",ExpressionUUID->"673148aa-d9dd-48bc-b4ff-3ceaa38d4fee"]
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Cell[901346, 16277, 152, 3, 30, InheritFromParent,ExpressionUUID->"66339502-9d62-4d8b-8698-b21b86842a75"],
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Cell[CellGroupData[{
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Cell[901523, 16284, 6685, 167, 348, "Input",ExpressionUUID->"4259ed50-c390-40a9-957e-aa8bae6464ff"],
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Cell[900798, 16259, 6685, 167, 348, "Input",ExpressionUUID->"4259ed50-c390-40a9-957e-aa8bae6464ff"],
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Cell[907486, 16428, 403198, 6654, 556, 159627, 2661, "CachedBoxData", "BoxData", "Output",ExpressionUUID->"600380b8-2ea1-4d1d-b0fa-743ea99d8dbe"]
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Cell[CellGroupData[{
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Cell[1311446, 23112, 178, 3, 54, "Subsection",ExpressionUUID->"c2053f1f-6868-41f1-b542-bad7e2dbf717"],
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Cell[1310902, 23092, 1063, 22, 219, "Text",ExpressionUUID->"07e56b25-3c15-4626-9e46-6c77981db19f"]
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Cell[CellGroupData[{
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Cell[1312739, 23145, 199, 4, 67, "Section",ExpressionUUID->"3854b603-a008-4f0e-b49e-08c3ef01c6e9"],
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Cell[1312941, 23151, 361, 6, 58, "Text",ExpressionUUID->"32b5cf5e-aab9-4bf0-abae-2b6923b2e903"],
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Cell[1312014, 23120, 199, 4, 67, "Section",ExpressionUUID->"3854b603-a008-4f0e-b49e-08c3ef01c6e9"],
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Cell[1312216, 23126, 361, 6, 58, "Text",ExpressionUUID->"32b5cf5e-aab9-4bf0-abae-2b6923b2e903"],
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Cell[CellGroupData[{
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Cell[1313327, 23161, 181, 3, 30, "Input",ExpressionUUID->"2e011f46-99c5-4217-9435-469a096a3541"],
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Cell[1313511, 23166, 1045, 34, 60, "Output",ExpressionUUID->"a537d7f5-aeb6-4c60-bc00-6fefc3c86d43"]
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Cell[1312602, 23136, 181, 3, 30, "Input",ExpressionUUID->"2e011f46-99c5-4217-9435-469a096a3541"],
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Cell[1312786, 23141, 1045, 34, 60, "Output",ExpressionUUID->"a537d7f5-aeb6-4c60-bc00-6fefc3c86d43"]
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Cell[1314571, 23203, 351, 6, 35, "Text",ExpressionUUID->"ee57a3c2-1718-4ad7-9bba-a3b0b944b469"],
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Cell[1313846, 23178, 351, 6, 35, "Text",ExpressionUUID->"ee57a3c2-1718-4ad7-9bba-a3b0b944b469"],
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Cell[CellGroupData[{
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Cell[1314947, 23213, 1671, 47, 82, "Input",ExpressionUUID->"69c5aaef-072f-4d84-bdc3-8a653dd87893"],
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Cell[1316621, 23262, 2985, 91, 156, "Output",ExpressionUUID->"46edaaf4-6210-4097-95e3-9431990ac15a"]
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Cell[1314222, 23188, 1671, 47, 82, "Input",ExpressionUUID->"69c5aaef-072f-4d84-bdc3-8a653dd87893"],
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Cell[1315896, 23237, 2985, 91, 156, "Output",ExpressionUUID->"46edaaf4-6210-4097-95e3-9431990ac15a"]
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Cell[1319621, 23356, 520, 8, 58, "Text",ExpressionUUID->"a5ef3f43-7b1a-40f7-bc5e-3884a79db05d"],
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Cell[1318896, 23331, 520, 8, 58, "Text",ExpressionUUID->"a5ef3f43-7b1a-40f7-bc5e-3884a79db05d"],
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Cell[CellGroupData[{
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Cell[1320166, 23368, 3464, 95, 145, "Input",ExpressionUUID->"af7c449f-e708-4dec-9a95-e2a226c7965a"],
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Cell[1323633, 23465, 19689, 536, 533, "Output",ExpressionUUID->"b0e836df-80be-4362-8e71-80b19b38dba8"]
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Cell[1319441, 23343, 3464, 95, 145, "Input",ExpressionUUID->"af7c449f-e708-4dec-9a95-e2a226c7965a"],
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Cell[1322908, 23440, 19689, 536, 533, "Output",ExpressionUUID->"b0e836df-80be-4362-8e71-80b19b38dba8"]
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}, Open ]],
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Cell[1343337, 24004, 20541, 535, 455, "Input",ExpressionUUID->"2fb4d73f-58eb-41c5-8b49-bf0fd006c5c0"],
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Cell[1363881, 24541, 547, 8, 81, "Text",ExpressionUUID->"5398c3b2-d083-4063-bf47-7124d8ed3084"],
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Cell[1364431, 24551, 4750, 121, 174, "Input",ExpressionUUID->"1120755f-bdd0-4db6-b62e-c8ce52edcbf1"],
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Cell[1342612, 23979, 20541, 535, 455, "Input",ExpressionUUID->"2fb4d73f-58eb-41c5-8b49-bf0fd006c5c0"],
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Cell[1363156, 24516, 547, 8, 81, "Text",ExpressionUUID->"5398c3b2-d083-4063-bf47-7124d8ed3084"],
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Cell[1363706, 24526, 4750, 121, 174, "Input",ExpressionUUID->"1120755f-bdd0-4db6-b62e-c8ce52edcbf1"],
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Cell[CellGroupData[{
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Cell[CellGroupData[{
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Cell[1369001, 24665, 177, 3, 45, "Subsubsection",ExpressionUUID->"87aeb388-e717-4e85-afe9-385f2f418096"],
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Cell[1369181, 24670, 615, 9, 81, "Text",ExpressionUUID->"96ec60cc-3a71-4407-b2a0-6a68cf68aaf7"],
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Cell[CellGroupData[{
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Cell[1370546, 24708, 7126, 174, 285, "Input",ExpressionUUID->"1f6626fd-ad7d-48ac-93e2-72ebe0bf57c2"],
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Cell[1377675, 24884, 113030, 2074, 434, "Output",ExpressionUUID->"34acc1c9-b25d-4f05-99cd-11b76ee92514"]
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Cell[1369821, 24683, 7126, 174, 285, "Input",ExpressionUUID->"1f6626fd-ad7d-48ac-93e2-72ebe0bf57c2"],
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Cell[1376950, 24859, 113030, 2074, 434, "Output",ExpressionUUID->"34acc1c9-b25d-4f05-99cd-11b76ee92514"]
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}, Open ]]
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}, Open ]],
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Cell[CellGroupData[{
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Cell[1490754, 26964, 181, 3, 45, "Subsubsection",ExpressionUUID->"6262f130-07d2-49f4-8990-db089589021f"],
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Cell[1490938, 26969, 400, 6, 58, "Text",ExpressionUUID->"f7218a51-6ba0-46a1-94d4-eac08b273c62"],
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Cell[1490029, 26939, 181, 3, 45, "Subsubsection",ExpressionUUID->"6262f130-07d2-49f4-8990-db089589021f"],
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Cell[1490213, 26944, 400, 6, 58, "Text",ExpressionUUID->"f7218a51-6ba0-46a1-94d4-eac08b273c62"],
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Cell[CellGroupData[{
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Cell[1491363, 26979, 6988, 173, 285, "Input",ExpressionUUID->"4943dd3f-0811-4c76-9e70-3d37f6d5006f"],
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Cell[1498354, 27154, 109993, 2042, 441, "Output",ExpressionUUID->"4e37242e-2f9a-4545-9032-8ab412b5cae6"]
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Cell[1490638, 26954, 6988, 173, 285, "Input",ExpressionUUID->"4943dd3f-0811-4c76-9e70-3d37f6d5006f"],
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Cell[1497629, 27129, 109993, 2042, 441, "Output",ExpressionUUID->"4e37242e-2f9a-4545-9032-8ab412b5cae6"]
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}, Open ]]
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}, Open ]],
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Cell[CellGroupData[{
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Cell[1608396, 29202, 182, 3, 45, "Subsubsection",ExpressionUUID->"e71df111-b9c0-4fcb-a9d1-d6b16b02676a"],
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Cell[1607671, 29177, 182, 3, 45, "Subsubsection",ExpressionUUID->"e71df111-b9c0-4fcb-a9d1-d6b16b02676a"],
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Cell[1607856, 29182, 612, 9, 81, "Text",ExpressionUUID->"77a3d959-22f6-4251-9ded-781e16688ac1"],
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Cell[CellGroupData[{
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Cell[1609218, 29220, 7109, 174, 285, "Input",ExpressionUUID->"53291c95-01a4-4ec5-95f9-fda312a1f585"],
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Cell[1616330, 29396, 109077, 2024, 441, "Output",ExpressionUUID->"5086d734-903c-42b0-9a32-753e369a9d15"]
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Cell[1608493, 29195, 7109, 174, 285, "Input",ExpressionUUID->"53291c95-01a4-4ec5-95f9-fda312a1f585"],
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Cell[1615605, 29371, 109077, 2024, 441, "Output",ExpressionUUID->"5086d734-903c-42b0-9a32-753e369a9d15"]
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}, Open ]]
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}, Open ]]
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}, Open ]],
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Cell[CellGroupData[{
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Cell[1725468, 31427, 269, 4, 54, "Subsection",ExpressionUUID->"88abadb0-1d05-421c-b4b3-14793c0012a1"],
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Cell[1725740, 31433, 449, 7, 35, "Text",ExpressionUUID->"0a1530fa-55e8-44ea-8d84-88d4b1fd62fa"],
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Cell[1724743, 31402, 269, 4, 54, "Subsection",ExpressionUUID->"88abadb0-1d05-421c-b4b3-14793c0012a1"],
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Cell[1725015, 31408, 449, 7, 35, "Text",ExpressionUUID->"0a1530fa-55e8-44ea-8d84-88d4b1fd62fa"],
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Cell[CellGroupData[{
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Cell[1726214, 31444, 208, 4, 45, "Subsubsection",ExpressionUUID->"7622ddd0-a3ff-4d4b-b73e-3215ff17337a"],
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Cell[1726425, 31450, 1642, 23, 288, "Text",ExpressionUUID->"ead47e6e-2cb5-4d9b-a984-d08011c4950f"],
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Cell[1728070, 31475, 2048, 55, 184, "Input",ExpressionUUID->"0b00d05d-2ec1-4d27-8811-b2d0e7c5ad74"],
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Cell[1725489, 31419, 208, 4, 45, "Subsubsection",ExpressionUUID->"7622ddd0-a3ff-4d4b-b73e-3215ff17337a"],
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Cell[1725700, 31425, 1642, 23, 288, "Text",ExpressionUUID->"ead47e6e-2cb5-4d9b-a984-d08011c4950f"],
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Cell[1727345, 31450, 2048, 55, 184, "Input",ExpressionUUID->"0b00d05d-2ec1-4d27-8811-b2d0e7c5ad74"],
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Cell[CellGroupData[{
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Cell[1730143, 31534, 4497, 105, 296, "Input",ExpressionUUID->"d61acf55-1480-4c5a-9923-50bf417b3f76"],
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Cell[1734643, 31641, 1148853, 18888, 499, 972730, 16001, "CachedBoxData", "BoxData", "Output",ExpressionUUID->"d50be571-0f86-44a3-8622-8f155cc25fa5"]
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Cell[1729418, 31509, 4497, 105, 296, "Input",ExpressionUUID->"d61acf55-1480-4c5a-9923-50bf417b3f76"],
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Cell[1733918, 31616, 1148853, 18888, 499, 972730, 16001, "CachedBoxData", "BoxData", "Output",ExpressionUUID->"d50be571-0f86-44a3-8622-8f155cc25fa5"]
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}, Open ]]
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}, Open ]],
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Cell[CellGroupData[{
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Cell[2883545, 50535, 182, 3, 45, "Subsubsection",ExpressionUUID->"15c444d1-fa26-43c5-84a4-246e5de0c8ed"],
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Cell[2883730, 50540, 751, 11, 127, "Text",ExpressionUUID->"a332f586-96f4-4001-827b-1319e53ac3b0"],
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Cell[CellGroupData[{
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Cell[2884506, 50555, 4032, 100, 275, "Input",ExpressionUUID->"c7eb1c72-bdc6-4577-90fe-9b9777d50eef"],
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Cell[2883781, 50530, 4032, 100, 275, "Input",ExpressionUUID->"c7eb1c72-bdc6-4577-90fe-9b9777d50eef"],
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Cell[2887816, 50632, 903533, 14859, 508, 681187, 11213, "CachedBoxData", "BoxData", "Output",ExpressionUUID->"dda8e76c-085c-4ef4-a2f8-ad287a24dfe0"]
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}, Open ]]
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}, Open ]]
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}, Open ]],
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Cell[CellGroupData[{
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Cell[3792135, 65523, 185, 2, 54, "Subsection",ExpressionUUID->"682f5d84-4017-4e58-bb21-1f874ccd820f"],
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Cell[3792323, 65527, 3126, 49, 587, "Text",ExpressionUUID->"9b51c24b-654d-48dc-878d-1415e41df166"]
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Cell[3791410, 65498, 185, 2, 54, "Subsection",ExpressionUUID->"682f5d84-4017-4e58-bb21-1f874ccd820f"],
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Cell[3791598, 65502, 3126, 49, 587, "Text",ExpressionUUID->"9b51c24b-654d-48dc-878d-1415e41df166"]
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}, Open ]]
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}, Open ]],
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Cell[CellGroupData[{
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Cell[3795498, 65582, 187, 3, 67, "Section",ExpressionUUID->"555b6565-ed9c-41db-8747-3eab70a65674"],
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Cell[3795688, 65587, 69, 0, 426, "Text",ExpressionUUID->"c8622b61-f1b5-4100-9c90-7559df35a50a"]
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}, Closed]]
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Cell[3794773, 65557, 187, 3, 67, "Section",ExpressionUUID->"555b6565-ed9c-41db-8747-3eab70a65674"],
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Cell[3794963, 65562, 69, 0, 35, "Text",ExpressionUUID->"c8622b61-f1b5-4100-9c90-7559df35a50a"],
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Cell[CellGroupData[{
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Cell[3795057, 65566, 428, 11, 51, "Input",ExpressionUUID->"be816819-a54c-429b-81a6-87e2bd4bddc9"],
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Cell[3795488, 65579, 173, 2, 69, "Output",ExpressionUUID->"af287abc-d325-4b3d-b738-17f8150381b7"]
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||||
}, Open ]]
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||||
}, Open ]]
|
||||
}
|
||||
]
|
||||
*)
|
||||
|
||||
Loading…
Reference in New Issue
Block a user