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Hugh Burton 2020-11-30 21:34:39 +00:00
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Manuscript/EPAWTFT.tex
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@ -1295,11 +1295,11 @@ orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \the
%\begin{equation} %\begin{equation}
% E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon). % E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon).
%\end{equation} %\end{equation}
With this representation, the parametrised RMP Hamiltonian becomes With this representation, the parametrised \hugh{atomic} RMP Hamiltonian becomes
\begin{widetext} \begin{widetext}
\begin{equation} \begin{equation}
\label{eq:H_RMP} \label{eq:H_RMP}
\bH_\text{RMP}\qty(\lambda) = \hugh{\bH_\text{atom}\qty(\lambda)} =
\begin{pmatrix} \begin{pmatrix}
2(U-\epsilon) - \lambda U & -\lambda t & -\lambda t & 0 \\ 2(U-\epsilon) - \lambda U & -\lambda t & -\lambda t & 0 \\
-\lambda t & (U-\epsilon) - \lambda U & 0 & -\lambda t \\ -\lambda t & (U-\epsilon) - \lambda U & 0 & -\lambda t \\
@ -1337,7 +1337,8 @@ In contrast, smaller $\epsilon$ gives a weaker attraction to the atomic site,
representing strong screening of the nuclear attraction by core and valence electrons, representing strong screening of the nuclear attraction by core and valence electrons,
and again a less negative $\lambda$ is required for ionisation to occur. 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}, 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 \ce{O}, \ce{Ne}.
Molecules containing these atoms are therefore often class $\beta$ systems with
a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006} a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006}
% EXACT VERSUS APPROXIMATE % EXACT VERSUS APPROXIMATE
@ -1384,10 +1385,9 @@ For $\lambda>1$, the HF potential becomes an attractive component in Stillinger'
Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction
becomes increasingly repulsive. becomes increasingly repulsive.
Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent
points where the two-electron repulsion overcomes the attractive HF potential and an electron points where the two-electron repulsion overcomes the attractive HF potential and a single electron dissociates from the molecule.\cite{Sergeev_2006}
are successively expelled from the molecule.\cite{Sergeev_2006}
Symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons. In contrast, symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
Consider the reference UHF solution where the spin-up and spin-down electrons are localised on the left and Consider the reference UHF solution where the spin-up and spin-down electrons are localised on the left and
right sites respectively. right sites respectively.
The spin-up HF potential will then be a repulsive interaction from the spin-down electron The spin-up HF potential will then be a repulsive interaction from the spin-down electron
@ -1413,7 +1413,7 @@ occurs exactly at $\lambda = 1$.
In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided
crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}). crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}).
We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes
a new type of MP critical and represents a QPT in the perturbation approximation. a new type of MP critical point and represents a QPT in the perturbation approximation.
Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the
radius of convergence (see Fig.~\ref{fig:RadConv}). radius of convergence (see Fig.~\ref{fig:RadConv}).