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Pierre-Francois Loos 2020-12-03 16:30:16 +01:00
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@ -1730,7 +1730,7 @@ In a later study by the same group, they used analytic continuation techniques
to resum a divergent MP series such as a stretched water molecule.\cite{Mihalka_2017a} to resum a divergent MP series such as a stretched water molecule.\cite{Mihalka_2017a}
Any MP series truncated at a given order $n$ can be used to define the scaled function Any MP series truncated at a given order $n$ can be used to define the scaled function
\begin{equation} \begin{equation}
E_{\text{MP}n}(\lambda) = \sum_{k=0}^{n} \lambda^{k} E^{(k)}. E_{\text{MP}n}(\lambda) = \sum_{k=0}^{n} \lambda^{k} \titou{E_\text{MP}^{(k)}}.
\end{equation} \end{equation}
Reliable estimates of the energy can be obtained for values of $\lambda$ where the MP series is rapidly Reliable estimates of the energy can be obtained for values of $\lambda$ where the MP series is rapidly
convergent (\ie, for $\abs{\lambda} < \rc$), as shown in Fig.~\ref{fig:rmp_anal_cont} for the RMP10 series convergent (\ie, for $\abs{\lambda} < \rc$), as shown in Fig.~\ref{fig:rmp_anal_cont} for the RMP10 series
@ -1749,8 +1749,9 @@ It was then further improved by introducing Cauchy's integral formula\cite{Mihal
which states that the value of the energy can be computed at $\lambda_1$ inside the complex which states that the value of the energy can be computed at $\lambda_1$ inside the complex
contour $\mathcal{C}$ using only the values along the same contour. contour $\mathcal{C}$ using only the values along the same contour.
Starting from a set of points in a ``trusted'' region where the MP series is convergent, their approach Starting from a set of points in a ``trusted'' region where the MP series is convergent, their approach
self-consistently refines estimates of the $E(\lambda)$ values on a contour around the physical point self-consistently refines estimates of the $E(\lambda)$ values on a contour \titou{around} the physical point
$\lambda = 1$. $\lambda = 1$.
\titou{T2: actually this is not true as the point $\lambda = 1$ is chosen to be on the contour.}
The shape of this contour is arbitrary, but there must be no branch points or other singularities inside The shape of this contour is arbitrary, but there must be no branch points or other singularities inside
the contour. the contour.
Once the contour values of $E(\lambda)$ are converged, Cauchy's integral formula Eq.~\eqref{eq:Cauchy} can Once the contour values of $E(\lambda)$ are converged, Cauchy's integral formula Eq.~\eqref{eq:Cauchy} can