From 054772aa905e2fdc2e687e42e34b5fc4ee1e41bb Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 17 Nov 2020 22:14:38 +0100 Subject: [PATCH] almost done with HF part --- Manuscript/EPAWTFT.tex | 77 +++++++++++++++++++++++------------------- 1 file changed, 42 insertions(+), 35 deletions(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 299ce01..5e91ff6 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -189,6 +189,7 @@ More importantly here, although EPs usually lie off the real axis, these singula \caption{% Exact energies for the Hubbard dimer ($U=4t$) as functions of $\lambda$ on the real axis (\subref{subfig:FCI_real}) and in the complex plane (\subref{subfig:FCI_cplx}). Only the interacting closed-shell singlets are plotted in the complex plane, becoming degenerate at the EP (black dot). + The contour followed around the EP in order to interchange states is also represented. \label{fig:FCI}} \end{figure*} @@ -205,21 +206,21 @@ The exact Hamiltonian is then \label{eq:H_FCI} \bH = \begin{pmatrix} - U & - t & + t & 0 \\ + U & - t & - t & 0 \\ - t & 0 & 0 & - t \\ - + t & 0 & 0 & + t \\ - 0 & - t & + t & U \\ + - t & 0 & 0 & - t \\ + 0 & - t & - t & U \\ \end{pmatrix}, \end{equation} where $t$ is the hopping parameter and $U$ is the on-site Coulomb repulsion. We refer the interested reader to Refs.~\onlinecite{Carrascal_2015,Carrascal_2018} for more details about this system. The parameter $U$ dictates the correlation regime. In the weak correlation regime (\ie, small $U$), the kinetic energy dominates and the electrons are delocalised over both sites. -For large $U$ (or strong correlation regime), the electron repulsion term drives the physics and the electrons localise on opposite sites to minimise their Coulomb repulsion. +In the large-$U$ (or strong correlation) regime, the electron repulsion term drives the physics and the electrons localise on opposite sites to minimise their Coulomb repulsion. This phenomenon is sometimes referred to as a Wigner crystallisation. \cite{Wigner_1934} -To illustrate the formation of an exceptional points, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$. -When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) eigenvalues +To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$. +When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) (eigen)energies \begin{subequations} \begin{align} E_{\mp} &= \frac{1}{2} \qty(U \mp \sqrt{ (4 \lambda t)^2 + U^2 } ), @@ -240,12 +241,12 @@ with energy \label{eq:E_EP} E_\text{EP} = \frac{U}{2}. \end{equation} -These $\lambda$ values correspond to so-called EPs and connect the ground and excited state in the complex plane. +These $\lambda$ values correspond to so-called EPs and connect the ground and excited states in the complex plane. Crucially, the energy surface becomes non-analytic at $\lambda_{\text{EP}}$ and a square-root singularity forms with two branch cuts running along the imaginary axis from $\lambda_{\text{EP}}$ to $\pm \i \infty$ (see Fig.~\ref{subfig:FCI_cplx}). On the real $\lambda$ axis, these EPs lead to the singlet avoided crossing at $\lambda = \Re(\lambda_{\text{EP}})$. The ``shape'' of this avoided crossing is related to the magnitude of $\Im(\lambda_{\text{EP}})$, with smaller values giving a ``sharper'' interaction. -Remarkably, the existence of these square-root singularities means that following a complex contour around an EP in the complex $\lambda$ plane will interconvert the closed-shell ground and excited states. +Remarkably, the existence of these square-root singularities means that following a complex contour around an EP in the complex $\lambda$ plane will interconvert the closed-shell ground and excited states (see Fig.~\ref{subfig:FCI_cplx}). This behaviour can be seen by expanding the radicand in Eq.~\eqref{eq:singletE} as a Taylor series around $\lambda_{\text{EP}}$ to give \begin{equation} E_{\pm} \approx E_{\text{EP}} \pm \sqrt{32t^2 \lambda_{\text{EP}}} \sqrt{\lambda - \lambda_{\text{EP}}}. @@ -328,25 +329,25 @@ is the HF mean-field potential with \begin{subequations} \begin{gather} \label{eq:CoulOp} - \Hat{J}_i(\vb{x})\phi_j(\vb{x})=\qty[\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_i(\vb{x}') \dd\vb{x}' ] \phi_j(\vb{x}) + \Hat{J}_i(\vb{x})\phi_j(\vb{x})=\qty[\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_i(\vb{x}') \dd\vb{x}' ] \phi_j(\vb{x}), \\ \label{eq:ExcOp} - \Hat{K}_i(\vb{x})\phi_j(\vb{x})=\qty[\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_j(\vb{x}') \dd\vb{x}'] \phi_i(\vb{x}) + \Hat{K}_i(\vb{x})\phi_j(\vb{x})=\qty[\int \phi_i(\vb{x}')\frac{1}{\abs{\vb{r} - \vb{r}'}}\phi_j(\vb{x}') \dd\vb{x}'] \phi_i(\vb{x}), \end{gather} \end{subequations} being the Coulomb and exchange operators (respectively) in the spin-orbital basis. \cite{SzaboBook} The HF energy is then defined as \begin{equation} \label{eq:E_HF} - E_\text{HF} = \sum_i^{N} h_i + \frac{1}{2} \sum_{ij}^{N} \qty( J_{ij} - K_{ij} ) + E_\text{HF} = \sum_i^{N} h_i + \frac{1}{2} \sum_{ij}^{N} \qty( J_{ij} - K_{ij} ), \end{equation} with \begin{align} - h_i & = \mel{\phi_i}{h}{\phi_i} + h_i & = \mel{\phi_i}{\Hat{h}}{\phi_i}, & - J_{ij} & = \mel{\phi_i}{\Hat{J}_j}{\phi_i} + J_{ij} & = \mel{\phi_i}{\Hat{J}_j}{\phi_i}, & - K_{ij} & = \mel{\phi_i}{\Hat{K}_j}{\phi_i} + K_{ij} & = \mel{\phi_i}{\Hat{K}_j}{\phi_i}. \end{align} If the spatial part of the spin-orbitals are restricted to be the same for spin-up and spin-down electrons, one talks about restricted HF (RHF) theory, whereas if one does not enforce this constrain it leads to the so-called unrestricted HF (UHF) theory. From hereon, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied (or virtual) orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. @@ -369,38 +370,44 @@ Rather than solving Eq.~\eqref{eq:SchrEq}, HF theory uses the variational princi Coming back to the Hubbard dimer, the HF energy is [see Eq.~\eqref{eq:E_HF}] \begin{equation} - E_\text{HF} = -t \qty[ \sin \theta_\alpha + \sin \theta_\beta ] + \frac{U}{2} \qty[ 1 + \cos \theta_\alpha \cos \theta_\beta ] + E_\text{HF} = -t \qty[ \sin \theta_\alpha + \sin \theta_\beta ] + \frac{U}{2} \qty[ 1 + \cos \theta_\alpha \cos \theta_\beta ], \end{equation} where \begin{align} - \mathcal{B}^{\sigma} & = \cos(\frac{\theta_\sigma}{2}) \Lsi - \sin(\frac{\theta_\sigma}{2}) \Rsi + \mathcal{B}^{\sigma} & = \cos(\frac{\theta_\sigma}{2}) \Lsi + \sin(\frac{\theta_\sigma}{2}) \Rsi, \\ - \mathcal{A}^{\sigma} & = \sin(\frac{\theta_\sigma}{2}) \Lsi + \cos(\frac{\theta_\sigma}{2}) \Rsi + \mathcal{A}^{\sigma} & = - \sin(\frac{\theta_\sigma}{2}) \Lsi + \cos(\frac{\theta_\sigma}{2}) \Rsi \end{align} -are the bonding $\mathcal{B}^{\sigma}$ and anti-bonding $\mathcal{A}^{\sigma}$ molecular orbitals for the spin-$\sigma$ electrons and the angles which minimises the HF energy, \ie, $\pdv*{E_\text{HF}}{\theta_\sigma} = 0$, are +are the bonding $\mathcal{B}^{\sigma}$ and anti-bonding $\mathcal{A}^{\sigma}$ molecular orbitals for the spin-$\sigma$ electrons. +The angles which minimises the HF energy, \ie, $\pdv*{E_\text{HF}}{\theta_\sigma} = 0$, are \begin{equation} - \theta_\text{RHF}^\alpha = \theta_\text{RHF}^\beta = \pi/4 + \theta_\text{RHF}^\alpha = \theta_\text{RHF}^\beta = \pi/4, \end{equation} -for $0 \le U \le 4t$, and +for $0 \le U \le 4t$, yielding \begin{align} - \theta_\text{UHF}^\alpha & = \arctan (-\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U}) - \\ - \theta_\text{UHF}^\beta & = \arctan (+\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U}) + \mathcal{B}_\text{RHF}^{\sigma} & = \frac{\Lsi + \Rsi}{\sqrt{2}}, + & + \mathcal{A}_\text{RHF}^{\sigma} & = \frac{\Lsi - \Rsi}{\sqrt{2}}, \end{align} -otherwise. -In the RHF formalism, the two electrons are restricted to ``live'' in the same spatial orbital. -The RHF ground-state energy is then +and the following RHF ground-state energy: \begin{equation} E_\text{RHF} = -2t + \frac{U}{2} \end{equation} - +In the RHF formalism, the two electrons are restricted to ``live'' in the same spatial orbital. The RHF wave function cannot model properly the physics of the system at large $U$ because the spatial orbitals are restricted to be the same, and, \textit{a fortiori}, it cannot represent two electrons on opposite sites. -Within the HF approximation, at the critical value of $U = 4t$, famously known as the Coulson-Fischer point, \cite{Coulson_1949} a symmetry-broken UHF solution appears with lower in energy than the RHF one. -The UHF ground-state energy is + +Within the HF approximation, at the critical value of $U = 4t$, famously known as the Coulson-Fischer point, \cite{Coulson_1949} a symmetry-broken UHF solution appears with a lower energy than the RHF one. +Indeed, for $U \ge 4t$, we have +\begin{align} + \theta_\text{UHF}^\alpha & = \titou{\arctan (-\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U})}, + \\ + \theta_\text{UHF}^\beta & = \titou{\arctan (+\frac{\sqrt{U^2 - 4t^2}}{U},\frac{2t}{U})}, +\end{align} +and the corresponding UHF ground-state energy is \begin{equation} - E_\text{UHF} = - \frac{2t^2}{U} + E_\text{UHF} = - \frac{2t^2}{U}. \end{equation} -for $U \ge 4t$. +Note that, for $U \ge 4t$, the RHF wave function remains a genuine solution of the HF equations but corresponds to a saddle point, not a minimum. %=====================================================% \subsection{M{\o}ller-Plesset perturbation theory} @@ -419,14 +426,14 @@ This yields + \lambda\sum_{i