diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index e7ebe04..36f3505 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -235,6 +235,7 @@ the two following terms account for the electron-nucleus attraction and the elec % EXACT SCHRODINGER EQUATION The exact many-electron wave function at a given nuclear geometry $\Psi(\vb{R})$ corresponds +of using adjacent partial sums to the solution of the (time-independent) Schr\"{o}dinger equation \begin{equation} \hH(\vb{R})\, \Psi(\vb{R}) = E(\vb{R})\, \Psi(\vb{R}), @@ -507,7 +508,7 @@ the total spin operator $\hat{\mathcal{S}}^2$, leading to ``spin-contamination'' % HF energies as a function of U/t %%%%%%%%%%%%%%%%% \begin{figure} - \includegraphics[width=\linewidth]{HF_real.pdf} + \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 $U/t$. The symmetry-broken UHF solution emerges at the coalescence point $U=2t$ (black dot), often known as the Coulson-Fischer point.} @@ -551,15 +552,15 @@ modelling the correct physics with the two electrons on opposite sites. %%%%%%%%%%%%%%%%% \begin{figure*}[t] \begin{subfigure}{0.49\textwidth} - \includegraphics[height=0.65\textwidth,trim={0pt 0pt 0pt -35pt},clip]{HF_cplx_angle} + \includegraphics[height=0.65\textwidth,trim={0pt 0pt 0pt -35pt},clip]{fig3a} \subcaption{\label{subfig:UHF_cplx_angle}} \end{subfigure} \begin{subfigure}{0.49\textwidth} - \includegraphics[height=0.65\textwidth]{HF_cplx_energy} + \includegraphics[height=0.65\textwidth]{fig3b} \subcaption{\label{subfig:UHF_cplx_energy}} \end{subfigure} \caption{% - (\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$ \titou{in the Hubbard dimer for $U/t = ??$}. + (\subref{subfig:UHF_cplx_angle}) Real component of the UHF angle $\ta^{\text{UHF}}$ for $\lambda \in \bbC$ in the Hubbard dimer for $U/t = 2$. Symmetry-broken solutions correspond to individual sheets and become equivalent at the \textit{quasi}-EP $\lambda_{\text{c}}$ (black dot). The RHF solution is independent of $\lambda$, giving the constant plane at $\pi/2$. @@ -794,17 +795,17 @@ gradient discontinuities or spurious minima. %%% FIG 2 %%% \begin{figure*} \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{fig2a} + \includegraphics[height=0.75\textwidth]{fig4a} \subcaption{\label{subfig:RMP_3.5} $U/t = 3.5$} \end{subfigure} % \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{fig2b} + \includegraphics[height=0.75\textwidth]{fig4b} \subcaption{\label{subfig:RMP_cvg}} \end{subfigure} % \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{fig2c} + \includegraphics[height=0.75\textwidth]{fig4c} \subcaption{\label{subfig:RMP_4.5} $U/t = 4.5$} \end{subfigure} \caption{ @@ -849,7 +850,7 @@ The Taylor expansion of the RMP energy can then be evaluated to obtain the $k$th % RADIUS OF CONVERGENCE PLOTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] - \includegraphics[width=\linewidth]{RadConv} + \includegraphics[width=\linewidth]{fig5} \caption{ Radius of convergence $r_c$ for the RMP ground state (red), the UMP ground state (blue), and the UMP excited state (orange) series as functions of the ratio $U/t$. @@ -871,17 +872,17 @@ for the two states using the ground-state RHF orbitals is identical. %%% FIG 3 %%% \begin{figure*} \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{fig3a} + \includegraphics[height=0.75\textwidth]{fig6a} \subcaption{\label{subfig:UMP_3} $U/t = 3$} \end{subfigure} % \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{fig3b} + \includegraphics[height=0.75\textwidth]{fig6b} \subcaption{\label{subfig:UMP_cvg}} \end{subfigure} % \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{fig3c} + \includegraphics[height=0.75\textwidth]{fig6c} \subcaption{\label{subfig:UMP_7} $U/t = 7$} \end{subfigure} \caption{ Convergence of the UMP series as a function of the perturbation order $n$ for the Hubbard dimer at $U/t = 3$ and $7$. @@ -1153,17 +1154,17 @@ states which share the symmetry of the ground state,\cite{Goodson_2004} and are %------------------------------------------------------------------% \begin{figure*}[t] \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{rmp_cp} + \includegraphics[height=0.75\textwidth]{fig7a} \subcaption{\label{subfig:rmp_cp}} \end{subfigure} % \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{rmp_cp_surf} + \includegraphics[height=0.75\textwidth]{fig7b} \subcaption{\label{subfig:rmp_cp_surf}} \end{subfigure} % \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{rmp_ep_to_cp} + \includegraphics[height=0.75\textwidth]{fig7c} \subcaption{\label{subfig:rmp_ep_to_cp}} \end{subfigure} \caption{% @@ -1253,17 +1254,17 @@ set representations of the MP critical point.\cite{Sergeev_2006} %------------------------------------------------------------------% \begin{figure*}[t] \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth,trim={0pt 5pt -10pt 15pt},clip]{ump_cp} + \includegraphics[height=0.75\textwidth,trim={0pt 5pt -10pt 15pt},clip]{fig8a} \subcaption{\label{subfig:ump_cp}} \end{subfigure} % \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{ump_cp_surf} + \includegraphics[height=0.75\textwidth]{fig8b} \subcaption{\label{subfig:ump_cp_surf}} \end{subfigure} % \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.75\textwidth]{ump_ep_to_cp} + \includegraphics[height=0.75\textwidth]{fig8c} \subcaption{\label{subfig:ump_ep_to_cp}} \end{subfigure} % \includegraphics[height=0.65\textwidth,trim={0pt 5pt 0pt 15pt}, clip]{ump_critical_point} @@ -1329,8 +1330,8 @@ radius of convergence (see Fig.~\ref{fig:RadConv}). %%%%%%%%%%%%%%%%% \begin{figure*} - \includegraphics[height=0.23\textheight]{PadeRMP35} - \includegraphics[height=0.23\textheight]{PadeRMP45} + \includegraphics[height=0.23\textheight]{fig9a} + \includegraphics[height=0.23\textheight]{fig9b} \caption{\label{fig:PadeRMP} RMP ground-state energy as a function of $\lambda$ obtained using various resummation techniques at $U/t = 3.5$ (left) and $U/t = 4.5$ (right).} @@ -1429,7 +1430,7 @@ a convergent series. %%%%%%%%%%%%%%%%% \begin{figure}[t] - \includegraphics[width=\linewidth]{QuadUMP} + \includegraphics[width=\linewidth]{fig10} \caption{\label{fig:QuadUMP} UMP energies as a function of $\lambda$ obtained using various resummation techniques at $U/t = 3$.} \end{figure} @@ -1541,18 +1542,18 @@ The remedy for this problem involves applying a suitable transformation of the c \begin{figure*} \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.85\textwidth]{ump_qa322} - \subcaption{\label{subfig:ump_ep_to_cp} [3/2,2] Quadratic} + \includegraphics[height=0.85\textwidth]{fig11a} + \subcaption{\label{subfig:322quad} [3/2,2] Quadratic} \end{subfigure} % \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.85\textwidth]{ump_exact} - \subcaption{\label{subfig:ump_cp_surf} Exact} + \includegraphics[height=0.85\textwidth]{fig11b} + \subcaption{\label{subfig:exact} Exact} \end{subfigure} % \begin{subfigure}{0.32\textwidth} - \includegraphics[height=0.85\textwidth]{ump_qa304} - \subcaption{\label{subfig:ump_cp} [3/0,4] Quadratic} + \includegraphics[height=0.85\textwidth]{fig11c} + \subcaption{\label{subfig:304quad} [3/0,4] Quadratic} \end{subfigure} \caption{% Comparison of the [3/2,2] and [3/0,4] quadratic approximants with the exact UMP energy surface in the complex $\lambda$ @@ -1716,7 +1717,7 @@ However, like the UMP series in stretched \ce{H2},\cite{Lepetit_1988} the cost of larger denominators is an overall slower rate of convergence. \begin{figure} - \includegraphics[width=\linewidth]{rmp_anal_cont} + \includegraphics[width=\linewidth]{fig12} \caption{% Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function of $\lambda$ for the symmetric Hubbard dimer at $U/t = 4$. diff --git a/Manuscript/QuadUMP.pdf b/Manuscript/fig10.pdf similarity index 100% rename from Manuscript/QuadUMP.pdf rename to Manuscript/fig10.pdf diff --git a/Manuscript/ump_qa322.pdf b/Manuscript/fig11a.pdf similarity index 100% rename from Manuscript/ump_qa322.pdf rename to Manuscript/fig11a.pdf diff --git a/Manuscript/ump_exact.pdf b/Manuscript/fig11b.pdf similarity index 100% rename from Manuscript/ump_exact.pdf rename to Manuscript/fig11b.pdf diff --git a/Manuscript/ump_qa304.pdf b/Manuscript/fig11c.pdf similarity index 100% rename from Manuscript/ump_qa304.pdf rename to Manuscript/fig11c.pdf diff --git a/Manuscript/rmp_anal_cont.pdf b/Manuscript/fig12.pdf similarity index 100% rename from Manuscript/rmp_anal_cont.pdf rename to Manuscript/fig12.pdf diff --git a/Manuscript/fig1a.pdf b/Manuscript/fig1a.pdf index 0c07b48..52cbeac 100644 Binary files a/Manuscript/fig1a.pdf and b/Manuscript/fig1a.pdf differ diff --git a/Manuscript/fig1b.pdf b/Manuscript/fig1b.pdf index b738aa5..80d55c4 100644 Binary files a/Manuscript/fig1b.pdf and b/Manuscript/fig1b.pdf differ diff --git a/Manuscript/fig2.pdf b/Manuscript/fig2.pdf new file mode 100644 index 0000000..2eab252 Binary files /dev/null and b/Manuscript/fig2.pdf differ diff --git a/Manuscript/fig3a.pdf b/Manuscript/fig3a.pdf index 1e306f3..df1badf 100644 Binary files a/Manuscript/fig3a.pdf and b/Manuscript/fig3a.pdf differ diff --git a/Manuscript/fig3b.pdf b/Manuscript/fig3b.pdf index 0945b7a..c376cbe 100644 Binary files a/Manuscript/fig3b.pdf and b/Manuscript/fig3b.pdf differ diff --git a/Manuscript/fig4a.pdf b/Manuscript/fig4a.pdf new file mode 100644 index 0000000..18bfea8 Binary files /dev/null and b/Manuscript/fig4a.pdf differ diff --git a/Manuscript/fig4b.pdf b/Manuscript/fig4b.pdf new file mode 100644 index 0000000..91b2a02 Binary files /dev/null and b/Manuscript/fig4b.pdf differ diff --git a/Manuscript/fig4c.pdf b/Manuscript/fig4c.pdf new file mode 100644 index 0000000..c6aba76 Binary files /dev/null and b/Manuscript/fig4c.pdf differ diff --git a/Manuscript/RadConv.pdf b/Manuscript/fig5.pdf similarity index 100% rename from Manuscript/RadConv.pdf rename to Manuscript/fig5.pdf diff --git a/Manuscript/fig6a.pdf b/Manuscript/fig6a.pdf new file mode 100644 index 0000000..1c9301e Binary files /dev/null and b/Manuscript/fig6a.pdf differ diff --git a/Manuscript/fig6b.pdf b/Manuscript/fig6b.pdf new file mode 100644 index 0000000..ca3ee26 Binary files /dev/null and b/Manuscript/fig6b.pdf differ diff --git a/Manuscript/fig6c.pdf b/Manuscript/fig6c.pdf new file mode 100644 index 0000000..3d8aab8 Binary files /dev/null and b/Manuscript/fig6c.pdf differ diff --git a/Manuscript/rmp_cp.pdf b/Manuscript/fig7a.pdf similarity index 100% rename from Manuscript/rmp_cp.pdf rename to Manuscript/fig7a.pdf diff --git a/Manuscript/fig7b.pdf b/Manuscript/fig7b.pdf new file mode 100644 index 0000000..d2902de Binary files /dev/null and b/Manuscript/fig7b.pdf differ diff --git a/Manuscript/rmp_ep_to_cp.pdf b/Manuscript/fig7c.pdf similarity index 100% rename from Manuscript/rmp_ep_to_cp.pdf rename to Manuscript/fig7c.pdf diff --git a/Manuscript/ump_cp.pdf b/Manuscript/fig8a.pdf similarity index 100% rename from Manuscript/ump_cp.pdf rename to Manuscript/fig8a.pdf diff --git a/Manuscript/fig8b.pdf b/Manuscript/fig8b.pdf new file mode 100644 index 0000000..8049cfb Binary files /dev/null and b/Manuscript/fig8b.pdf differ diff --git a/Manuscript/ump_ep_to_cp.pdf b/Manuscript/fig8c.pdf similarity index 100% rename from Manuscript/ump_ep_to_cp.pdf rename to Manuscript/fig8c.pdf diff --git a/Manuscript/PadeRMP35.pdf b/Manuscript/fig9a.pdf similarity index 100% rename from Manuscript/PadeRMP35.pdf rename to Manuscript/fig9a.pdf diff --git a/Manuscript/PadeRMP45.pdf b/Manuscript/fig9b.pdf similarity index 100% rename from Manuscript/PadeRMP45.pdf rename to Manuscript/fig9b.pdf diff --git a/Manuscript/rmp_cp_surf.pdf b/Manuscript/rmp_cp_surf.pdf deleted file mode 100644 index 8ecdb40..0000000 Binary files a/Manuscript/rmp_cp_surf.pdf and /dev/null differ