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Pierre-Francois Loos 2020-12-01 16:58:48 +01:00
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Manuscript/EPAWTFT.tex
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@ -221,8 +221,8 @@ $\Nn$ (clamped) nuclei is defined for a given nuclear framework as
- \sum_{i}^{\Ne} \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} - \sum_{i}^{\Ne} \sum_{A}^{\Nn} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}}
+ \sum_{i<j}^{\Ne}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}}, + \sum_{i<j}^{\Ne}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}},
\end{equation} \end{equation}
where $\vb{r}_i$ defines the position of the $i$-th electron, $\vb{R}_{A}$ and $Z_{A}$ are the position where $\vb{r}_i$ defines the position of the $i$th electron, $\vb{R}_{A}$ and $Z_{A}$ are the position
and charge of the $A$-th nucleus respectively, and $\vb{R} = (\vb{R}_{1}, \dots, \vb{R}_{\Nn})$ is a and charge of the $A$th nucleus respectively, and $\vb{R} = (\vb{R}_{1}, \dots, \vb{R}_{\Nn})$ is a
collective vector for the nuclear positions. collective vector for the nuclear positions.
The first term represents the kinetic energy of the electrons, while The first term represents the kinetic energy of the electrons, while
the two following terms account for the electron-nucleus attraction and the electron-electron repulsion. the two following terms account for the electron-nucleus attraction and the electron-electron repulsion.
@ -845,7 +845,7 @@ gradient discontinuities or spurious minima.
The behaviour of the RMP and UMP series observed in \ce{H2} can also be illustrated by considering The 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. 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 electron correlation $U/t$. 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} $U/t$.
Using the ground-state RHF reference orbitals leads to the parametrised RMP Hamiltonian Using the ground-state RHF reference orbitals leads to the parametrised RMP Hamiltonian
\begin{widetext} \begin{widetext}
\begin{equation} \begin{equation}
@ -945,15 +945,15 @@ While a closed-form expression for the ground-state energy exists, it is cumbers
Instead, the radius of convergence of the UMP series can be obtained numerically as a function of $U/t$, as shown Instead, the radius of convergence of the UMP series can be obtained numerically as a function of $U/t$, as shown
in Fig.~\ref{fig:RadConv}. in Fig.~\ref{fig:RadConv}.
These numerical values reveal that the UMP ground-state series has $\rc > 1$ for all $U/t$ and always converges. These numerical values reveal that the UMP ground-state series has $\rc > 1$ for all $U/t$ and always converges.
However, in the strong correlation limit (large $U$), this radius of convergence tends to unity, indicating that However, in the strong correlation limit (large $U/t$), this radius of convergence tends to unity, indicating that
the corresponding UMP series becomes increasingly slow. the convergence of 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 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 $U/t \to \infty$, and thus the of $U/t$, reaching the limiting value of $1/2$ for $U/t \to \infty$. Hence, the
excited-state UMP series will always diverge. excited-state UMP series will always diverge.
% DISCUSSION OF UMP RIEMANN SURFACES % DISCUSSION OF UMP RIEMANN SURFACES
The convergence behaviour can be further elucidated by considering the full structure of the UMP energies The convergence behaviour can be further elucidated by considering the full structure of the UMP energies
in the complex $\lambda$-plane. in the complex $\lambda$-plane (see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}).
These Riemann surfaces are illustrated for $U = 3t$ and $7t$ alongside the perturbation terms at each order These Riemann surfaces are illustrated for $U = 3t$ and $7t$ alongside the perturbation terms at each order
in Fig.~\ref{subfig:UMP_cvg}. in Fig.~\ref{subfig:UMP_cvg}.
At $U = 3t$, the RMP series is convergent, while RMP becomes divergent for $U=7t$. At $U = 3t$, the RMP series is convergent, while RMP becomes divergent for $U=7t$.
@ -961,7 +961,7 @@ 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}). for larger $U/t$ as the radius of convergence becomes increasingly close to one (Fig.~\ref{fig:RadConv}).
% EFFECT OF SYMMETRY BREAKING % EFFECT OF SYMMETRY BREAKING
As the UHF orbitals break the molecular symmetry, new coupling terms emerge between the electronic states that As the UHF orbitals break the \trash{molecular} \titou{spin} symmetry, new coupling terms emerge between the electronic states that
cause fundamental changes to the structure of EPs in the complex $\lambda$-plane. 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 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 the doubly-excited state (Fig.~\ref{fig:RMP}), the UMP energy has two EPs: one connecting the ground state with the