diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 0d3e73d..9845ad9 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -293,7 +293,7 @@ In the large-$U$ (or strong correlation) regime, the electron repulsion term bec and the electrons localise on opposite sites to minimise their Coulomb repulsion. This phenomenon is often referred to as Wigner crystallisation. \cite{Wigner_1934} -To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t\rightarrow \lambda t$ to give the parameterised Hamiltonian $\hH(\lambda)$. +To illustrate the formation of an EP, we scale the off-diagonal coupling strength by introducing the complex parameter $\lambda$ through the transformation $t \to \lambda t$ to give the parameterised Hamiltonian $\hH(\lambda)$. When $\lambda$ is real, the Hamiltonian~\eqref{eq:H_FCI} is Hermitian with the distinct (real-valued) (eigen)energies \begin{subequations} \begin{align} @@ -599,7 +599,7 @@ The inherent non-linearity in the Fock eigenvalue problem arises from self-consi in the HF approximation, and is usually solved through an iterative approach.\cite{Roothaan_1951,Hall_1951} Alternatively, the non-linear terms arising from the Coulomb and exchange operators can be considered as a perturbation from the core Hamiltonian \eqref{eq:Hcore} by introducing the -transformation $U \rightarrow \lambda\, U$, giving the parametrised Fock operator +transformation $U \to \lambda\, U$, giving the parametrised Fock operator \begin{equation} \Hat{f}(\vb{x} ; \lambda) = \Hat{h}(\vb{x}) + \lambda\, \Hat{v}_\text{HF}(\vb{x}). \end{equation} @@ -972,7 +972,7 @@ These numerical values reveal that the UMP ground-state series has $\rc > 1$ for However, in the strong correlation limit (large $U$), this radius of convergence tends to unity, indicating that the corresponding UMP series becomes increasingly slow. Furthermore, the doubly-excited state using the ground-state UHF orbitals has $\rc < 1$ for almost any value -of $U/t$, reaching the limiting value of $1/2$ for $U/t \rightarrow \infty$, and thus the +of $U/t$, reaching the limiting value of $1/2$ for $U/t \to \infty$, and thus the excited-state UMP series will always diverge. % DISCUSSION OF UMP RIEMANN SURFACES @@ -1264,22 +1264,20 @@ states which share the symmetry of the ground state,\cite{Goodson_2004} and are \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). - } + 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*} %------------------------------------------------------------------% % 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 +Instead, we can use an asymmetric Hubbard dimer \cite{Carrascal_2015,Carrascal_2018} 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 $\epsilon > 0$. +represent the attraction between the electrons and the model ``atomic'' nucleus [see Eq.~\eqref{eq:H_FCI}], where we define $\epsilon > 0$. %The exact Hamiltonian [Eq.~\eqref{eq:H_FCI}] then becomes %\begin{equation} %\label{eq:H_FCI_Asymm} @@ -1292,10 +1290,7 @@ represent the attraction between the electrons and the model ``atomic'' nucleus, %\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} - \theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0. -\end{equation} +orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$. %and energy %\begin{equation} % E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon). @@ -1313,18 +1308,17 @@ With this representation, the parametrised RMP Hamiltonian becomes \end{pmatrix}. \end{equation} \end{widetext} -} +\titou{Shall we change the symbol $\bH_\text{RMP}\qty(\lambda)$ to avoid confusion? Maybe $\Tilde{\bH}_\text{RMP}\qty(\lambda)$?} % 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}), +This limit corresponds to the dissociative regime in the asymmetric Hubbard dimer as discussed in Ref.~\onlinecite{Carrascal_2018}, and the RMP energies become \begin{subequations} \begin{align} - E_{-} &= 2U - 2 \epsilon - U \lambda + E_{-} &= 2U - 2 \epsilon - U \lambda, \\ - E_{\text{S}} &= U - \epsilon - U \lambda + E_{\text{S}} &= U - \epsilon - U \lambda, \\ E_{+} &= U \lambda, \end{align} @@ -1344,22 +1338,19 @@ representing strong screening of the nuclear attraction by core and valence elec 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. -} +a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006} % 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} +The critical point in the exact case $t=0$ lies on the \titou{negative} real $\lambda$ axis (Fig.~\ref{subfig:rmp_cp}: dashed lines), +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 +idealised destination for the ionised electrons with a non-zero (yet small) hopping term $t$. +Taking the 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 +In the limit $t \to 0$, these EPs approach the real axis, mirroring Sergeev's discussion on finite basis set representations of the MP critical point.\cite{Sergeev_2006} -} %------------------------------------------------------------------% % Figure on the UMP critical point @@ -1374,33 +1365,28 @@ set representations of the MP critical point.\cite{Sergeev_2006} \subcaption{\label{subfig:ump_ep_to_cp}} \end{subfigure} \caption{% -\hugh{% The UMP ground-state EP becomes a critical point in the strong correlation limit (large $U/t$). - (\subref{subfig:ump_cp}) As the $U/t$ increases, the avoided crossing on the real $\lambda$ axis + (\subref{subfig:ump_cp}) As $U/t$ increases, the avoided crossing on the real $\lambda$ axis becomes increasingly sharp. - (\subref{subfig:ump_ep_to_cp}) The convergence of the EPs at $\lep$ onto the real axis for $U/t \rightarrow \infty$ - mirrors the formation of the RMP critical point and other QPTs in the infinite basis set limit. -} + (\subref{subfig:ump_ep_to_cp}) The convergence of the EPs at $\lep$ onto the real axis for $U/t \to \infty$ + mirrors the formation of the RMP critical point and other QPTs in the complete basis set limit. \label{fig:UMP_cp}} \end{figure*} %------------------------------------------------------------------% % RELATIONSHIP BETWEEN QPT AND UMP -\hugh{% -In Section~\ref{sec:spin_cont} we showed that the slow convergence of the UMP series for the strongly correlated -Hubbard dimer was due to a complex-conjguate pair of EPs just outside the radius of convergence. +In Sec.~\ref{sec:spin_cont} we showed that the slow convergence of the UMP series for the strongly correlated +Hubbard dimer was due to a complex-conjugate pair of EPs just outside the radius of convergence. These EPs have positive real components and small imaginary components (Fig.~\ref{fig:UMP}), suggesting a potential -connection to MP critical points and QPTs. +connection to MP critical points and QPTs (see Sec.~\ref{sec:MP_critical_point}). For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's -Hamiltonian [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. Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent -points were the two-electron repulsion overcomes the attractive HF potential and an electron +points where the two-electron repulsion overcomes the attractive HF potential and an electron are successively expelled from the molecule.\cite{Sergeev_2006} -} -\hugh{% 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 right sites respectively. @@ -1413,14 +1399,12 @@ for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}). Note that, although this appears to be an avoided crossing between the ground and first-excited state, the earlier excited-state avoided crossing means that the first-excited state qualitatively represents the double excitation for $\lambda > 0.5.$ -} % SHARPNESS AND QPT -\hugh{% The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$. For small $U/t$, the HF potentials will be weak and the electrons will delocalise over the two sites, both in the UHF reference and as $\lambda$ increases. -This delocalisataion dampens the electron swapping process and leads to a ``shallow'' avoided crossing +This delocalisation dampens the electron swapping process and leads to a ``shallow'' avoided crossing that corresponds to EPs with non-zero imaginary components (solid lines in Fig.~\ref{subfig:ump_cp}). As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping term to make electron delocalisation less favourable. @@ -1428,11 +1412,10 @@ These effects create a stronger driving force for the electrons to swap sites un 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 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 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. Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the radius of convergence (see Fig.~\ref{fig:RadConv}). -} %%==================================================== %\subsection{The physics of quantum phase transitions}