diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 5e91ff6..1bbca55 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -174,6 +174,7 @@ More importantly here, although EPs usually lie off the real axis, these singula %===================================% \subsection{Illustrative Example} +\label{sec:example} %===================================% %%% FIG 1 %%% @@ -201,7 +202,7 @@ Using the (localised) site basis, the (singlet) Hilbert space of the Hubbard dim & \ket{\Lup \Ldown} & & \ket{\Lup\Rdown} & & \ket{\Rup\Ldown} & & \ket{\Rup\Rdown} \end{align} where $\Lsi$ ($\Rsi$) denotes an electron with spin $\sigma$ on the left (right) site. -The exact Hamiltonian is then +The exact [or full configuration interaction (FCI)] Hamiltonian is then \begin{equation} \label{eq:H_FCI} \bH = @@ -315,7 +316,7 @@ This function is smooth for $x \in \mathbb{R}$ and infinitely differentiable in In the Hartree-Fock (HF) approximation, the many-electron wave function is approximated as a single Slater determinant $\Psi^{\text{HF}}(\vb{x}_1,\ldots,\vb{x}_N)$ [where $\vb{x} = (\sigma,\vb{r})$ is a composite vector gathering spin and spatial coordinates] defined as an antisymmetric combination of $N$ (real-valued) one-electron spin-orbitals $\phi_p(\vb{x})$, which are, by definition, eigenfunctions of the one-electron Fock operator \begin{equation}\label{eq:FockOp} - \Hat{f}(\vb{x}) \phi_p(\vb{x}) = [ \Hat{h}(\vb{x}) + \Hat{v}^\text{HF}(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}), + \Hat{f}(\vb{x}) \phi_p(\vb{x}) = [ \Hat{h}(\vb{x}) + \Hat{v}_\text{HF}(\vb{x}) ] \phi_p(\vb{x}) = \epsilon_p \phi_p(\vb{x}), \end{equation} where \begin{equation} @@ -323,7 +324,7 @@ where \end{equation} is the core Hamiltonian and \begin{equation} - \Hat{v}^\text{HF}(\vb{x}) = \sum_i^{N} \qty[ \Hat{J}_i(\vb{x}) - \Hat{K}_i(\vb{x}) ] + \Hat{v}_\text{HF}(\vb{x}) = \sum_i^{N} \qty[ \Hat{J}_i(\vb{x}) - \Hat{K}_i(\vb{x}) ] \end{equation} is the HF mean-field potential with \begin{subequations} @@ -354,7 +355,7 @@ From hereon, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied (or v Rather than solving Eq.~\eqref{eq:SchrEq}, HF theory uses the variational principle to find an approximation of $\Psi$ as a single Slater determinant. Hence a Slater determinant is not an eigenfunction of the exact Hamiltonian $\hH$. However, it is, by definition, an eigenfunction of the so-called (approximated) HF many-electron Hamiltonian defined as the sum of the one-electron Fock operators \begin{equation}\label{eq:HFHamiltonian} - \hH^{\text{HF}} = \sum_{i} f(\vb{x}_i). + \hH_{\text{HF}} = \sum_{i} f(\vb{x}_i). \end{equation} % @@ -502,8 +503,9 @@ However, the choice of the functional form of the fit remains a subtle task. This technique was first generalized by using complex scaling parameters and applying analytic continuation by solving the Laplace equation, \cite{Surjan_2018} and then further improved thanks to Cauchy's integral formula \cite{Mihalka_2019} \begin{equation} \label{eq:Cauchy} - \frac{1}{2\pi i} \oint_{\gamma} \frac{E(z)}{z - a} = E(a). + \frac{1}{2\pi i} \oint_{\gamma} \frac{E(z)}{z - a} = E(a), \end{equation} +which states that the value of the energy can be computed at $z=a$ inside the contour $\gamma$ only by the knowledge of its values on the same contour. Their method consists in refining self-consistently the values of $E(z)$ computed on a contour going through the physical point at $z = 1$ and encloses points of the ``trusted'' region (where the MP series is convergent). The shape of this contour is arbitrary but no singularities are allowed inside the contour to ensure $E(z)$ is analytic. When the values of $E(z)$ on the so-called contour are converged, Cauchy's integrals formula \eqref{eq:Cauchy} is invoked to compute the values at $E(z=1)$ which corresponds to the final estimate of the FCI energy. The authors illustrate this protocol on the dissociation curve of \ce{LiH} and the stretched water molecule showing encouraging results. \cite{Mihalka_2019} @@ -521,11 +523,11 @@ The authors illustrate this protocol on the dissociation curve of \ce{LiH} and t \label{fig:RMP}} \end{figure*} -Let us illustrate the behavior of the RMP and UMP series on the Hubbard dimer (see Fig.~\ref{fig:RMP}). -Within the RMP parition thecnique, we have +Let us illustrate the behaviour of the RMP and UMP series on the Hubbard dimer (see Fig.~\ref{fig:RMP}). +Within the RMP partition technique, we have \begin{equation} \label{eq:H_RMP} - \bH^\text{RMP} = + \bH_\text{RMP} = \begin{pmatrix} -2t + U - \lambda U/2 & 0 & 0 & \lambda U/2 \\ 0 & U - \lambda U/2 & \lambda U/2 & 0 \\ @@ -536,25 +538,24 @@ Within the RMP parition thecnique, we have which yields the ground-state energy \begin{equation} \label{eq:E0MP} - E_0(\lambda) = U - \frac{\lambda U}{2} - \frac{1}{2} \sqrt{(4t)^2 + \lambda ^2 U^2} + E_{-}(\lambda) = U - \frac{\lambda U}{2} - \frac{1}{2} \sqrt{(4t)^2 + \lambda ^2 U^2}. \end{equation} -From this expression, it is easy to identify that the radius of convergence is defined by the exceptional points at $\lambda = \pm i 4t / U$ (which is similar to the FCI exceptional point). +From this expression, it is easy to identify that the radius of convergence is defined by the EPs at $\lambda = \pm i 4t / U$ (which are actually located at the same position than the exact EPs discussed in Sec.~\ref{sec:example}). Equation \eqref{eq:E0MP} can be Taylor expanded in terms of $\lambda$ to obtain the $n$th-order MP correction \begin{equation} - E_0^{(n)} = U \delta_{0,n} - \frac{1}{2} \frac{U^n}{(4t)^{n-1}} \mqty( n/2 \\ 1/2) + E_\text{MP}^{(k)} = U \delta_{0,k} - \frac{1}{2} \frac{U^k}{(4t)^{k-1}} \mqty( 1/2 \\ k/2), \end{equation} with \begin{equation} - E_0(\lambda) = \sum_{n=0}^\infty E_0^{(n)} \lambda^n + E_{-}(\lambda) = \sum_{k=0}^\infty E_\text{MP}^{(k)} \lambda^k. \end{equation} We illustrate this by plotting the energy expansions at orders of $\lambda$ just below and above the radius of convergence in Fig.~\ref{fig:RMP}. -\titou{T2 is going to add a discussion about this figure.} At the UMP level now, we have \begin{widetext} \begin{equation} \label{eq:H_UMP} - \bH^\text{UMP} = + \bH_\text{UMP} = \begin{pmatrix} -2t^2 \lambda/U & 0 & 0 & +2t^2 \lambda/U \\ 0 & U - 2t^2 \lambda/U & +2t^2\lambda/U & +2t \sqrt{U^2 - (2t)^2} \lambda/U \\ @@ -564,7 +565,7 @@ At the UMP level now, we have \end{equation} \end{widetext} A closed-form expression for the ground-state energy can be obtained but it is cumbersome, so we eschew reporting its expression. - +The convergence of the UMP as a function of the ratio $U/t$ is depicted in Fig.~\ref{fig:UMP}. %%% FIG 3 %%% \begin{figure*}