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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[1734643, 31641, 1148853, 18888, 499, 972730, 16001, "CachedBoxData", "BoxData", "Output",ExpressionUUID->"d50be571-0f86-44a3-8622-8f155cc25fa5"]
|
||||
Cell[1729418, 31509, 4497, 105, 296, "Input",ExpressionUUID->"d61acf55-1480-4c5a-9923-50bf417b3f76"],
|
||||
Cell[1733918, 31616, 1148853, 18888, 499, 972730, 16001, "CachedBoxData", "BoxData", "Output",ExpressionUUID->"d50be571-0f86-44a3-8622-8f155cc25fa5"]
|
||||
}, Open ]]
|
||||
}, Open ]],
|
||||
Cell[CellGroupData[{
|
||||
Cell[2883545, 50535, 182, 3, 45, "Subsubsection",ExpressionUUID->"15c444d1-fa26-43c5-84a4-246e5de0c8ed"],
|
||||
Cell[2883730, 50540, 751, 11, 127, "Text",ExpressionUUID->"a332f586-96f4-4001-827b-1319e53ac3b0"],
|
||||
Cell[2882820, 50510, 182, 3, 45, "Subsubsection",ExpressionUUID->"15c444d1-fa26-43c5-84a4-246e5de0c8ed"],
|
||||
Cell[2883005, 50515, 751, 11, 127, "Text",ExpressionUUID->"a332f586-96f4-4001-827b-1319e53ac3b0"],
|
||||
Cell[CellGroupData[{
|
||||
Cell[2884506, 50555, 4032, 100, 275, "Input",ExpressionUUID->"c7eb1c72-bdc6-4577-90fe-9b9777d50eef"],
|
||||
Cell[2888541, 50657, 903533, 14859, 543, 681187, 11213, "CachedBoxData", "BoxData", "Output",ExpressionUUID->"dda8e76c-085c-4ef4-a2f8-ad287a24dfe0"]
|
||||
Cell[2883781, 50530, 4032, 100, 275, "Input",ExpressionUUID->"c7eb1c72-bdc6-4577-90fe-9b9777d50eef"],
|
||||
Cell[2887816, 50632, 903533, 14859, 508, 681187, 11213, "CachedBoxData", "BoxData", "Output",ExpressionUUID->"dda8e76c-085c-4ef4-a2f8-ad287a24dfe0"]
|
||||
}, Open ]]
|
||||
}, Open ]]
|
||||
}, Open ]],
|
||||
Cell[CellGroupData[{
|
||||
Cell[3792135, 65523, 185, 2, 54, "Subsection",ExpressionUUID->"682f5d84-4017-4e58-bb21-1f874ccd820f"],
|
||||
Cell[3792323, 65527, 3126, 49, 587, "Text",ExpressionUUID->"9b51c24b-654d-48dc-878d-1415e41df166"]
|
||||
Cell[3791410, 65498, 185, 2, 54, "Subsection",ExpressionUUID->"682f5d84-4017-4e58-bb21-1f874ccd820f"],
|
||||
Cell[3791598, 65502, 3126, 49, 587, "Text",ExpressionUUID->"9b51c24b-654d-48dc-878d-1415e41df166"]
|
||||
}, Open ]]
|
||||
}, Open ]],
|
||||
Cell[CellGroupData[{
|
||||
Cell[3795498, 65582, 187, 3, 67, "Section",ExpressionUUID->"555b6565-ed9c-41db-8747-3eab70a65674"],
|
||||
Cell[3795688, 65587, 69, 0, 426, "Text",ExpressionUUID->"c8622b61-f1b5-4100-9c90-7559df35a50a"]
|
||||
}, Closed]]
|
||||
Cell[3794773, 65557, 187, 3, 67, "Section",ExpressionUUID->"555b6565-ed9c-41db-8747-3eab70a65674"],
|
||||
Cell[3794963, 65562, 69, 0, 35, "Text",ExpressionUUID->"c8622b61-f1b5-4100-9c90-7559df35a50a"],
|
||||
Cell[CellGroupData[{
|
||||
Cell[3795057, 65566, 428, 11, 51, "Input",ExpressionUUID->"be816819-a54c-429b-81a6-87e2bd4bddc9"],
|
||||
Cell[3795488, 65579, 173, 2, 69, "Output",ExpressionUUID->"af287abc-d325-4b3d-b738-17f8150381b7"]
|
||||
}, Open ]]
|
||||
}, Open ]]
|
||||
}
|
||||
]
|
||||
*)
|
||||
|
||||
Loading…
Reference in New Issue
Block a user