saving work on Hubbard
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\begin{thebibliography}{77}%
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\begin{thebibliography}{79}%
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\makeatletter
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\providecommand \@ifxundefined [1]{%
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@ -453,6 +453,34 @@
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{\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf
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{\bibinfo {volume} {150}},\ \bibinfo {pages} {031101} (\bibinfo {year}
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{2019})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2015)\citenamefont
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{Carrascal}, \citenamefont {Ferrer}, \citenamefont {Smith},\ and\
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\citenamefont {Burke}}]{Carrascal_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
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{Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}},
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\bibinfo {author} {\bibfnamefont {J.~C.}\ \bibnamefont {Smith}}, \ and\
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\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href
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{\doibase 10.1088/0953-8984/27/39/393001} {\bibfield {journal} {\bibinfo
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{journal} {J. Phys. Condens. Matter}\ }\textbf {\bibinfo {volume} {27}},\
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\bibinfo {pages} {393001} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2018)\citenamefont
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{Carrascal}, \citenamefont {Ferrer}, \citenamefont {Maitra},\ and\
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\citenamefont {Burke}}]{Carrascal_2018}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont
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{Carrascal}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Ferrer}},
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\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Maitra}}, \ and\ \bibinfo
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{author} {\bibfnamefont {K.}~\bibnamefont {Burke}},\ }\href {\doibase
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10.1140/epjb/e2018-90114-9} {\bibfield {journal} {\bibinfo {journal} {Eur.
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Phys. J. B}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} {142}
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(\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Goodson}(2012)}]{Goodson_2012}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont
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{Goodson}},\ }\href {\doibase 10.1002/wcms.92} {\bibfield {journal}
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{\bibinfo {journal} {{WIREs} Comput. Mol. Sci.}\ }\textbf {\bibinfo {volume}
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{2}},\ \bibinfo {pages} {743} (\bibinfo {year} {2012})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Szabo}\ and\ \citenamefont
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{Ostlund}(1989)}]{SzaboBook}%
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\BibitemOpen
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@ -461,12 +489,6 @@
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{Ostlund}},\ }\href@noop {} {\emph {\bibinfo {title} {Modern quantum
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chemistry: {Introduction} to advanced electronic structure}}}\ (\bibinfo
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{publisher} {McGraw-Hill},\ \bibinfo {year} {1989})\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Goodson}(2012)}]{Goodson_2012}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont
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{Goodson}},\ }\href {\doibase 10.1002/wcms.92} {\bibfield {journal}
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{\bibinfo {journal} {{WIREs} Comput. Mol. Sci.}\ }\textbf {\bibinfo {volume}
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{2}},\ \bibinfo {pages} {743} (\bibinfo {year} {2012})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {M{\o}ller}\ and\ \citenamefont
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{Plesset}(1934)}]{Moller_1934}%
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\BibitemOpen
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%% This BibTeX bibliography file was created using BibDesk.
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%% Created for Pierre-Francois Loos at 2020-11-12 16:42:11 +0100
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%% Created for Pierre-Francois Loos at 2020-11-14 21:44:16 +0100
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@article{Carrascal_2015,
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abstract = {This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of density functional theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy\textendash{}Lieb construction to create a parametrization of the exact function with negligible errors. The symmetric case is most commonly studied, but we find a rich variation in behavior by including asymmetry, as strong correlation physics vies with charge-transfer effects. We explore the behavior of the gap and the many-body Green's function, demonstrating the `failure' of the Kohn\textendash{}Sham (KS) method to reproduce the fundamental gap. We perform benchmark calculations of the occupation and components of the KS potentials, the correlation kinetic energies, and the adiabatic connection. We test several approximate functionals (restricted and unrestricted Hartree\textendash{}Fock and Bethe ansatz local density approximation) to show their successes and limitations. We also discuss and illustrate the concept of the derivative discontinuity. Useful appendices include analytic expressions for density functional energy components, several limits of the exact functional (weak- and strong-coupling, symmetric and asymmetric), various adiabatic connection results, proofs of exact conditions for this model, and the origin of the Hubbard model from a minimal basis model for stretched H2.},
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author = {Carrascal, D J and Ferrer, J and Smith, J C and Burke, K},
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date-added = {2020-11-14 21:44:15 +0100},
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date-modified = {2020-11-14 21:44:15 +0100},
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doi = {10.1088/0953-8984/27/39/393001},
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file = {/Users/loos/Zotero/storage/LRMWNYEQ/Carrascal et al. - 2015 - The Hubbard dimer a density functional case study.pdf},
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issn = {0953-8984, 1361-648X},
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journal = {J. Phys. Condens. Matter},
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language = {en},
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month = oct,
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number = {39},
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pages = {393001},
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shorttitle = {The {{Hubbard}} Dimer},
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title = {The {{Hubbard}} Dimer: A Density Functional Case Study of a Many-Body Problem},
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volume = {27},
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year = {2015},
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Bdsk-Url-1 = {https://doi.org/10.1088/0953-8984/27/39/393001}}
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@article{Carrascal_2018,
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abstract = {The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequencydependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and relative oscillator strengths are defined appropriately. The method of Casida for extracting oscillator strengths from a frequencydependent kernel is demonstrated to yield the exact result with this kernel. An unambiguous way of labelling the nature of excitations is given. The fluctuation-dissipation theorem is proven for the groundstate exchange-correlation energy. The distinction between weak and strong correlation is shown to depend on the ratio of interaction to asymmetry. A simple interpolation between carefully defined weak-correlation and strong-correlation regimes yields a density-functional approximation for the kernel that gives accurate transition frequencies for both the single and double excitations, including charge-transfer excitations. Many exact results, limits, and expansions about those limits are given in the Appendices.},
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author = {Carrascal, Diego J. and Ferrer, Jaime and Maitra, Neepa and Burke, Kieron},
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date-added = {2020-11-14 21:44:15 +0100},
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date-modified = {2020-11-14 21:44:15 +0100},
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doi = {10.1140/epjb/e2018-90114-9},
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journal = {Eur. Phys. J. B},
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pages = {142},
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title = {Linear Response Time-Dependent Density Functional Theory of the {{Hubbard}} Dimer},
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volume = {91},
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year = {2018},
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Bdsk-Url-1 = {https://doi.org/10.1140/epjb/e2018-90114-9}}
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@article{Surjan_2018,
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author = {Surj{\'a}n,P{\'e}ter R. and Mih{\'a}lka,Zsuzsanna {\'E}. and Szabados,{\'A}gnes },
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author = {Surj{\'a}n,P{\'e}ter R. and Mih{\'a}lka,Zsuzsanna {\'E}. and Szabados,{\'A}gnes},
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date-added = {2020-11-12 16:40:48 +0100},
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date-modified = {2020-11-12 16:42:07 +0100},
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doi = {10.1007/s00214-018-2372-3},
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\newcommand{\hn}[1]{\Hat{n}_{#1}}
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\newcommand{\Dv}{\Delta v}
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\newcommand{\up}{\uparrow}
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\newcommand{\dw}{\downarrow}
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\newcommand{\updw}{\up\dw}
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\newcommand{\pt}{$\mathcal{PT}$}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France.}
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\newcommand{\UCAM}{Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, U.K.}
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@ -135,49 +157,65 @@ More importantly here, although EPs usually lie off the real axis, these singula
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%===================================%
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\subsection{An illustrative example}
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%===================================%
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In order to highlight the general properties of EPs mentioned above, we propose to consider the following $2 \times 2$ Hamiltonian commonly used in quantum chemistry
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In order to highlight the general properties of EPs mentioned above, we propose to consider the ubiquitous symmetric Hubbard dimer at half filling (\ie, with two opposite-spin fermions) whose Hamiltonian reads in the singlet configuration state function basis
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%\begin{align}
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% \ket{1\up1\dw} & \ket{1\up2\dw} & \ket{1\dw2\up} & \ket{2\up2\dw} \\
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% \uddot \quad \vac & \updot \quad \dwdot & \dwdot \quad \updot & \vac \quad \uddot \\
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%\end{align}
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\begin{equation}
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\label{eq:H_2x2}
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\label{eq:H_FCI}
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\bH =
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\begin{pmatrix}
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\epsilon_1 & \lambda \\
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\lambda & \epsilon_2
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-2t + U & 0 & U/2 \\
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0 & U & 0 \\
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U/2 & 0 & -2t + U \\
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\end{pmatrix},
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\end{equation}
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which represents two states of energies $\epsilon_1$ and $\epsilon_2$ coupled with a strength of magnitude $\lambda$.
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This Hamiltonian could represent, for example, a minimal-basis configuration interaction with doubles (CID) for the hydrogen molecule. \cite{SzaboBook}
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where $t$ is the hopping parameter and $U$ is the on-site Coulomb repulsion.
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We refer the interested reader to Refs.~\onlinecite{Carrascal_2015,Carrascal_2018} for more details about this system.
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We will consistently use this system to illustrate the different concepts discussed in the present review article.
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For real $\lambda$, the Hamiltonian \eqref{eq:H_2x2} is clearly Hermitian, and it becomes non-Hermitian for any complex $\lambda$ value.
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Its eigenvalues are
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For real $U$, the Hamiltonian \eqref{eq:H_FCI} is clearly Hermitian, and it becomes non-Hermitian for any complex $U$ value.
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The eigenvalues associated with its singlet ground state and singlet doubly-excited state are
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\begin{equation}
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\label{eq:E_2x2}
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E_{\pm} = \frac{\epsilon_1 + \epsilon_2}{2} \pm \frac{1}{2} \sqrt{(\epsilon_1 - \epsilon_2)^2 + 4\lambda^2}.
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\label{eq:E_FCI}
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E_{\pm} = \frac{1}{2} \qty( U \pm \sqrt{(4t^2) + U^2} ).
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\end{equation}
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%and they are represented as a function of $\lambda$ in Fig.~\ref{fig:2x2}.
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One notices that the two states become degenerate only for a pair of complex conjugate values of $\lambda$
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and they are represented as a function of $U$ in Fig.~\ref{fig:FCI} together with the energy of the singlet open-shell configuration of energy $U$.
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%%% FIG 1 %%%
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\begin{figure*}
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\includegraphics[height=0.3\textwidth]{fig1a}
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\hspace{0.1\textwidth}
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\includegraphics[height=0.3\textwidth]{fig1b}
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\caption{
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Exact energies for the Hubbard dimer as functions of $U/t$.
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\label{fig:FCI}}
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\end{figure*}
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One notices that these two states become degenerate only for a pair of complex conjugate values of $U$
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\begin{equation}
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\label{eq:lambda_EP}
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\lambda_\text{EP} = \pm i\,\frac{\epsilon_1 - \epsilon_2}{2},
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U_\text{EP} = \pm 4 i t,
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\end{equation}
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with energy
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\begin{equation}
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\label{eq:E_EP}
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E_\text{EP} = \frac{\epsilon_1 + \epsilon_2}{2},
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E_\text{EP} = \pm 2 i t,
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\end{equation}
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which correspond to square-root singularities in the complex-$\lambda$ plane. % (see Fig.~\eqref{fig:2x2}).
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These two $\lambda$ values, given by Eq.~\eqref{eq:lambda_EP}, are the so-called EPs and one can clearly see that they connect the ground and excited states.
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Starting from $\lambda_\text{EP}$, two square-root branch cuts run on the imaginary axis towards $\pm i \infty$.
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In the real $\lambda$ axis, the point for which the states are the closest ($\lambda = 0$) is called an avoided crossing and this occurs at $\lambda = \Re(\lambda_\text{EP})$.
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The ``shape'' of the avoided crossing is linked to the magnitude of $\Im(\lambda_\text{EP})$: the smaller $\Im(\lambda_\text{EP})$, the sharper the avoided crossing is.
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which correspond to square-root singularities in the complex-$U$ plane [see Fig.~\eqref{fig:FCI}].
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These two $U$ values, given by Eq.~\eqref{eq:lambda_EP}, are the so-called EPs and one can clearly see that they connect the ground and excited states.
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Starting from $U_\text{EP}$, two square-root branch cuts run on the imaginary axis towards $\pm i \infty$.
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In the real $U$ axis, the point for which the states are the closest ($U = 0$) is called an avoided crossing and this occurs at $U = \Re(U_\text{EP})$.
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The ``shape'' of the avoided crossing is linked to the magnitude of $\Im(U_\text{EP})$: the smaller $\Im(U_\text{EP})$, the sharper the avoided crossing is.
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Around $\lambda = \lambda_\text{EP}$, Eq.~\eqref{eq:E_2x2} behaves as \cite{MoiseyevBook}
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Around $U = U_\text{EP}$, Eq.~\eqref{eq:E_FCI} behaves as \cite{MoiseyevBook}
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\begin{equation} %\label{eq:E_EP}
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E_{\pm} = E_\text{EP} \pm \sqrt{2\lambda_\text{EP}} \sqrt{\lambda - \lambda_\text{EP}},
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E_{\pm} = E_\text{EP} \pm \sqrt{2U_\text{EP}} \sqrt{U - U_\text{EP}},
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\end{equation}
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and following a complex contour around the EP, \ie, $\lambda = \lambda_\text{EP} + R \exp(i\theta)$, yields
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and following a complex contour around the EP, \ie, $U = U_\text{EP} + R \exp(i\theta)$, yields
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\begin{equation}
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E_{\pm}(\theta) = E_\text{EP} \pm \sqrt{2\lambda_\text{EP} R} \exp(i\theta/2),
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E_{\pm}(\theta) = E_\text{EP} \pm \sqrt{2U_\text{EP} R} \exp(i\theta/2),
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\end{equation}
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and we have
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\begin{align}
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@ -269,7 +307,26 @@ Rather than solving Eq.~\eqref{eq:SchrEq}, HF theory uses the variational princi
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\hH^{\text{HF}} = \sum_{i} f(\vb{x}_i).
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\end{equation}
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Coming back to the Hubbard dimer, the HF energy is
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\begin{equation}
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E_\text{HF} = -t \qty[ \sin \theta_\alpha + \sin \theta_\beta ] + \frac{U}{2} \qty[ 1 + \cos \theta_\alpha \cos \theta_\beta ]
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\end{equation}
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where
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\begin{align}
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\psi_{1\sigma} & = \cos(\frac{\theta_\sigma}{2}) s_1 - \sin(\frac{\theta_\sigma}{2})s_2
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\\
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\psi_{2\sigma} & = \sin(\frac{\theta_\sigma}{2}) s_1 + \cos(\frac{\theta_\sigma}{2})s_2
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\end{align}
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\begin{align}
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\theta_\alpha & = \arctan (-\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U})
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\\
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\theta_\beta & = \arctan (+\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U})
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\end{align}
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%=====================================================%
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\subsection{M{\o}ller-Plesset perturbation theory}
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%=====================================================%
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The HF Hamiltonian \eqref{eq:HFHamiltonian} can be used as the zeroth-order Hamiltonian of Eq.~\eqref{eq:SchrEq-PT}. This partitioning of the Hamiltonian leads to the so-called M{\o}ller-Plesset (MP) perturbation theory. \cite{Moller_1934}
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The discovery of a partitioning of the Hamiltonian that allowed chemists to recover a large chunck of the correlation energy (\ie, the difference between the exact energy and the Hartree-Fock energy) using perturbation theory has been a major step in the development of post-HF methods.
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