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