accepted Titou changes

Manuscript/EPAWTFT.tex

@@ -510,7 +510,7 @@ the total spin operator $\hat{\mathcal{S}}^2$, leading to ``spin-contamination''
 \begin{figure}
     \includegraphics[width=\linewidth]{fig2}
     \caption{\label{fig:HF_real}
-    RHF and UHF energies \titou{in the Hubbard dimer} as a function of the correlation strength $U/t$. 
+    RHF and UHF energies in the Hubbard dimer as a function of the correlation strength $U/t$. 
     The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot), often known as the Coulson-Fischer point.}
 \end{figure}
 %%%%%%%%%%%%%%%%%
@@ -740,9 +740,9 @@ diatomics, where low-order RMP and UMP expansions give qualitatively wrong bindi
 % SLOW UMP CONVERGENCE AND SPIN CONTAMINATION
 The divergence of RMP expansions for stretched bonds can be easily understood from two perspectives.\cite{Gill_1988a}
 Firstly, the exact wave function becomes increasingly multi-configurational as the bond is stretched, and the 
-\titou{R}HF wave function no longer provides a qualitatively correct reference for the perturbation expansion.
+RHF wave function no longer provides a qualitatively correct reference for the perturbation expansion.
 Secondly, the energy gap between the bonding and antibonding orbitals associated with the stretch becomes
-increasingly small at larger bond lengths, \titou{leading to a divergence, for example, in the second-order MP correction \eqref{eq:EMP2}.}
+increasingly small at larger bond lengths, leading to a divergence, for example, in the second-order MP correction \eqref{eq:EMP2}.
 In contrast, the origin of slow UMP convergence is less obvious as the reference UHF energy remains
 qualitatively correct at large bond lengths and the orbital degeneracy is avoided.
 Furthermore, this slow convergence can also be observed in molecules with a UHF ground state at the equilibrium
@@ -753,8 +753,8 @@ Using the UHF framework allows the singlet ground state wave function to mix wit
 leading to spin contamination where the wave function is no longer an eigenfunction of the $\Hat{\cS}^2$ operator.
 The link between slow UMP convergence and this spin-contamination was first systematically investigated
 by Gill \etal\ using the minimal basis \ce{H2} model.\cite{Gill_1988}
-In this work, the authors compared \titou{the UMP series with the exact RHF- and UHF-based FCI expansions (T2: I don't understand this)}
-and identified that the slow UMP convergence arises from its failure to correctly predict the amplitude of the
+In this work, the authors %compared titou{the UMP series with the exact RHF- and UHF-based FCI expansions (T2: I don't understand this)} and
+identified that the slow UMP convergence arises from its failure to correctly predict the amplitude of the
 low-lying double excitation.
 This erroneous description of the double excitation amplitude has the same origin as the spin-contamination in the reference
 UHF wave function, creating the first direct link between spin-contamination and slow UMP convergence.\cite{Gill_1988}
@@ -816,7 +816,7 @@ gradient discontinuities or spurious minima.
 
 The behaviour of the RMP and UMP series observed in \ce{H2} can also be illustrated by considering
 the analytic Hubbard dimer with a complex-valued perturbation strength.
-In this system, the stretching of the \ce{H\bond{-}H} bond is directly mirrored by an increase in the \trash{electron correlation} \titou{ratio} $U/t$.
+In this system, the stretching of the \ce{H\bond{-}H} bond is directly mirrored by an increase in the ratio $U/t$.
 Using the ground-state RHF reference orbitals leads to the parametrised RMP Hamiltonian
 \begin{widetext}
 \begin{equation}
@@ -925,7 +925,7 @@ The ground-state UMP expansion is convergent in both cases, although the rate of
 for larger $U/t$ as the radius of convergence becomes increasingly close to one (Fig.~\ref{fig:RadConv}).
 
 % EFFECT OF SYMMETRY BREAKING
-As the UHF orbitals break the \trash{molecular} \titou{spin} symmetry, new coupling terms emerge between the electronic states that
+As the UHF orbitals break the spin symmetry, new coupling terms emerge between the electronic states that
 cause fundamental changes to the structure of EPs in the complex $\lambda$-plane.
 For example, while the RMP energy shows only one EP between the ground state and 
 the doubly-excited state (Fig.~\ref{fig:RMP}), the UMP energy has two EPs: one connecting the ground state with the
@@ -1269,7 +1269,7 @@ set representations of the MP critical point.\cite{Sergeev_2006}
     \end{subfigure}
 %    \includegraphics[height=0.65\textwidth,trim={0pt 5pt 0pt 15pt}, clip]{ump_critical_point}
 \caption{%
-    The UMP ground-state EP \titou{in the symmetric Hubbard dimer} becomes a critical point in the strong correlation limit (\ie, large $U/t$).
+    The UMP ground-state EP in the symmetric Hubbard dimer becomes a critical point in the strong correlation limit (\ie, large $U/t$).
     (\subref{subfig:ump_cp}) As $U/t$ increases, the avoided crossing on the real $\lambda$ axis
     becomes increasingly sharp.
     (\subref{subfig:ump_cp_surf}) Complex energy surfaces for $U = 5t$.