diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 4c8d55e..6cf56cd 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -519,6 +519,7 @@ the spin-$\sigma$ electrons as \\ \mathcal{A}^{\sigma} & = - \sin(\frac{\theta_\sigma}{2}) \Lsi + \cos(\frac{\theta_\sigma}{2}) \Rsi \end{align} +\label{eq:RHF_orbs} \end{subequations} In the weak correlation regime $0 \le U \le 2t$, the angles which minimise the HF energy, \ie, $\pdv*{E_\text{HF}}{\theta_\sigma} = 0$, are @@ -1194,6 +1195,28 @@ $\lambda$ values closer to the origin. With these insights, they regrouped the systems into new classes: i) $\alpha$ singularities which have ``large'' imaginary parts, and ii) $\beta$ singularities which have very small imaginary parts.\cite{Goodson_2004,Sergeev_2006} +%------------------------------------------------------------------% +% Figure on the RMP critical point +%------------------------------------------------------------------% +\begin{figure*}[t] + \begin{subfigure}{0.49\textwidth} + \includegraphics[height=0.65\textwidth]{rmp_critical_point} + \subcaption{\label{subfig:rmp_cp}} + \end{subfigure} + \begin{subfigure}{0.49\textwidth} + \includegraphics[height=0.65\textwidth]{rmp_critical_point_surf} + \subcaption{\label{subfig:rmp_cp_surf}} + \end{subfigure} + \caption{% + \hugh{Modelling the RMP critical point using the asymmetric Hubbard dimer. + (\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed). + (\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the + real axis, giving a sharp avoided crossing on the real axis (solid). + } + \label{fig:RMP_cp}} +\end{figure*} +%------------------------------------------------------------------% + % RELATIONSHIP TO BASIS SET SIZE The existence of the MP critical point can also explain why the divergence observed by Olsen \etal\ in the \ce{Ne} atom and \ce{HF} molecule occurred when diffuse basis functions were included.\cite{Olsen_1996} @@ -1245,100 +1268,92 @@ states which share the symmetry of the ground state,\cite{Goodson_2004} and are \label{sec:critical_point_hubbard} %======================================= -%------------------------------------------------------------------% -% Figure on the RMP critical point -%------------------------------------------------------------------% -\begin{figure*}[t] - \begin{subfigure}{0.49\textwidth} - \includegraphics[height=0.65\textwidth]{rmp_critical_point} - \subcaption{\label{subfig:rmp_cp}} - \end{subfigure} - \begin{subfigure}{0.49\textwidth} - \includegraphics[height=0.65\textwidth]{rmp_critical_point_surf} - \subcaption{\label{subfig:rmp_cp_surf}} - \end{subfigure} - \caption{% - \hugh{RMP critical point in the Hubbard dimer.} - \label{fig:RMP_cp}} -\end{figure*} -%------------------------------------------------------------------% - -\hughDraft{% -The simplified site basis of the Hubbard dimer makes explicitly modelling the ionisation continuum -impossible. -To model the auto-ionisation in the Hubbard dimer, we need to turn one of the sites into a ghost atom that will act as -a destination for ionised electrons. We do this by considering an asymmetric Hubbard dimer where the "atomic" site -has a negative diagonal term in the one-electron Hamiltonian, representing the nuclear attraction. This term is already -encoded in the Initialisation section such that the core Hamiltonian is -To model the doubly-occupied atom, we can define our reference HF state as the configuration with $\theta = 0$ and energy +% INTRODUCING THE MODEL +\hugh{% +The simplified site basis of the Hubbard dimer makes explicilty modelling the ionisation continuum impossible. +Instead, we can use an asymmetric Hubbard dimer to consider one site as a ``ghost atom'' that acts as a +destination for ionised electrons. +In this asymmetric model, we introduce a one-electron potential $-\epsilon$ on the left site to +represent the attraction between the electrons and the model ``atomic'' nucleus, where we define $\epsilon > 0$. +%The exact Hamiltonian [Eq.~\eqref{eq:H_FCI}] then becomes +%\begin{equation} +%\label{eq:H_FCI_Asymm} +%\bH = +%\begin{pmatrix} +% U-2\epsilon & -t & -t & 0 \\ +% -t & -\epsilon & 0 & -t \\ +% -t & 0 & -\epsilon & -t \\ +% 0 & -t & -t & U \\ +%\end{pmatrix}. +%\end{equation} +The reference Slater determinant for a doubly-occupied atom can be represented using the RHF +orbitals [Eq.~\eqref{eq:RHF_orbs}] with \begin{equation} - E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon) + \theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0. \end{equation} -The RMP Hamiltonian then becomes +%and energy +%\begin{equation} +% E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon). +%\end{equation} +With this representation, the parametrised RMP Hamiltonian becomes \begin{widetext} \begin{equation} \label{eq:H_RMP} \bH_\text{RMP}\qty(\lambda) = - \begin{pmatrix} - -2 \delta \epsilon + 2U(1 - \lambda/2) & -\lambda t & -\lambda t & 0 \\ - -\lambda t & - \delta \epsilon + (1-\lambda)U & 0 & -\lambda t \\ - -\lambda t & 0 & - \delta \epsilon + (1-\lambda)U & -\lambda t \\ - 0 & -\lambda t & -\lambda t & \lambda U \\ - \end{pmatrix}, +\begin{pmatrix} + 2(U-\epsilon) - \lambda U & -\lambda t & -\lambda t & 0 \\ + -\lambda t & (U-\epsilon) - \lambda U & 0 & -\lambda t \\ + -\lambda t & 0 & (U-\epsilon) -\lambda U & -\lambda t \\ + 0 & -\lambda t & -\lambda t & \lambda U \\ +\end{pmatrix}. \end{equation} \end{widetext} } -\hughDraft{% -Now let's think about the physics of the problem... -For the ghost site to truly represent ionised electrons, we need the hopping term to vanish (or become very small). -$U$ controls the strength of the HF repulsive potential. A stronger repulsion will encourage the electrons to be forced -away from the "atomic" site at a less negative value of $\lambda$. -$\epsilon$ controls the strength of attraction to the atom. A stronger attraction to the nucleus will mean that the electrons -are more tightly bound to the atom and a more negative $\lambda$ will be needed for auto-ionisation. -We therefore expect that the position of the RMP critical point will be controlled by the ratio $\epsilon / U$, with smaller ratios -making ionisation occur closer to $\lambda$ origin, and making the divergence in these cases more likely. -} - -\hughDraft{% -Taking the exact case with $t=0$, the RMP energies becomes +% DERIVING BEHAVIOUR OF THE CRITICAL SITE +\hugh{% +For the ghost site to perfectly represent ionised electrons, the hopping term between the two sites must vanish with $t=0$. +This limit corresponds to the dissociative regime in the asymmetric Hubbard dimer (see Ref.~\ref{Carrascal_2018}), +and the RMP energies become \begin{subequations} \begin{align} E_{-} &= 2U - 2 \epsilon - U \lambda \\ E_{\text{S}} &= U - \epsilon - U \lambda \\ - E_{+} &= U \lambda + E_{+} &= U \lambda, \end{align} \end{subequations} -By comparison with Fig.~\ref{fig:RMP_cp}, the critical point can be identified as by solving $E_{-} = E_{+}$, -giving +as shown in Fig.~\ref{subfig:rmp_cp} (dashed lines). +The RMP critical point then corresponds to the intersection $E_{-} = E_{+}$, giving the critical $\lambda$ value \begin{equation} -\lc = 1 - \epsilon / U. + \lc = 1 - \frac{\epsilon}{U}. \end{equation} -Thus, as expected, the critical point lies on the real axis and moves closer to the origin for larger -$U / \epsilon$. +Clearly the radius of convergence $\rc = \abs{\lc}$ is controlled directly by the ratio $\epsilon / U$, +with a convergent RMP series occurring for $\epsilon > 2 U$. +The on-site repulsion $U$ controls the strength of the HF potential localised around the ``atomic site'', with a +stronger repulsion encouraging the electrons to be ionised at a less negative value of $\lambda$. +Large $U$ can be physically interpreted as strong electron repulsion effects in electron dense molecules. +In contrast, smaller $\epsilon$ gives a weaker attraction to the atomic site, +representing strong screening of the nuclear attraction by core and valence electrons, +and again a less negative $\lambda$ is required for ionisation to occur. +Both of these factors are common in atoms on the right-hand side of the periodic table, \eg\ \ce{F}, +\ce{O}, \ce{Ne}, and thus molecules containing these atoms are often class $\beta$ systems with +a divergent RMP series due to the MP critical point. } -\hughDraft{% -The position of the critical point is controlled by the ratio $\epsilon / U$. -If the magnitude of the ratio becomes greater than one, then the series diverges. -We can interpret large $U$ as strong electron repulsion effects in electron dense molecules, -such as \ce{F-}. A small $\epsilon$ is also likely to correspond to -strong nuclear screening by the core and valence electrons. Both of these factors are -common in atoms on the right-hand-side of periodic table, -\eg\ \ce{F}, \ce{O}, \ce{Ne}, etc, as well as negatively-charged species, and so we recover the -class $\beta$ system classification. -} - -\hughDraft{% -In practical calculations, one considers the perturbation energy in a finite basis and the critical point is modelled -as a cluster of branch points close to the real axis. While we cannot change the size of our basis, we can adjust -the extent to which the second site behaves as a ghost by making the hopping term larger. -When $t$ is (slightly) non-zero, our modelled critical point becomes an EP point close to the real axis with a sharp -associated avoided crossing for real $\lambda$. If $t$ becomes larger, this avoided crossing becomes less sharp and the EPs -move away from the real axis. This mirrors the discussion of EPs approaching the real axis in the exact basis. -This effect is shown by the dashed lines in Fig.~\ref{fig:RMP_cp}. +% EXACT VERSUS APPROXIMATE +\hugh{% +The critical point in the exact case $t=0$ lies on the real $\lambda$ axis, mirroring the behaviour of a quantum +phase transition.\cite{Kais_2006} +However, in practical calculations performed with a finite basis set, the critical point is modelled as a cluster +of branch points close to the real axis. +The use of a finite basis can be modelled in the asymmetric dimer by making the second site a less +idealised destination for the ionised electrons with a non-zero hopping term $t$. +Taking the small value $t=0.1$ (Fig.~\ref{subfig:rmp_cp}: solid lines), the critical point becomes a +sharp avoided crossing with a complex-conjugate pair of EPs close to the real axis (Fig.~\ref{subfig:rmp_cp_surf}). +In the limit $t \rightarrow 0$, these EPs approach the real axis, mirroring Sergeev's discussion on finite basis +set representations of the MP critical point.\cite{Sergeev_2006} } %%==================================================== diff --git a/Manuscript/rmp_critical_point.pdf b/Manuscript/rmp_critical_point.pdf index 7215beb..97da554 100644 Binary files a/Manuscript/rmp_critical_point.pdf and b/Manuscript/rmp_critical_point.pdf differ diff --git a/Manuscript/rmp_critical_point_surf.pdf b/Manuscript/rmp_critical_point_surf.pdf index 8d0f86c..bfe57ad 100644 Binary files a/Manuscript/rmp_critical_point_surf.pdf and b/Manuscript/rmp_critical_point_surf.pdf differ