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Pierre-Francois Loos 2020-12-02 21:20:03 +01:00
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@ -1581,7 +1581,7 @@ While the diagonal quadratic approximants provide significanty improved estimate
ground-state energy, we can use our knowledge of the UMP singularity structure to develop ground-state energy, we can use our knowledge of the UMP singularity structure to develop
even more accurate results. even more accurate results.
We have seen in previous sections that the UMP energy surface We have seen in previous sections that the UMP energy surface
contains only \titou{square-root branch cuts} that approach the real axis in the limit $U/t \to \infty$. contains only square-root branch cuts that approach the real axis in the limit $U/t \to \infty$.
Since there are no true poles on this surface, we can obtain more accurate quadratic approximants by Since there are no true poles on this surface, we can obtain more accurate quadratic approximants by
taking $d_q = 0$ and increasing $d_r$ to retain equivalent accuracy in the square-root term [see Eq.\eqref{eq:QuadApp}]. taking $d_q = 0$ and increasing $d_r$ to retain equivalent accuracy in the square-root term [see Eq.\eqref{eq:QuadApp}].
Figure~\ref{fig:nopole_quad} illustrates this improvement for the pole-free [3/0,4] quadratic Figure~\ref{fig:nopole_quad} illustrates this improvement for the pole-free [3/0,4] quadratic
@ -1633,7 +1633,8 @@ that can be used to extract further information about the exact result.
The Shanks transformation presents one approach for extracting this information The Shanks transformation presents one approach for extracting this information
and accelerating the rate of convergence of a sequence.\cite{Shanks_1955,BenderBook} and accelerating the rate of convergence of a sequence.\cite{Shanks_1955,BenderBook}
\hugh{Consider the partial sums \titou{T2: $N$ is defined as the number of electrons. Maybe we should use $n$ instead?}
Consider the partial sums
$S_N = \sum_{k=0}^{N} a_k$ $S_N = \sum_{k=0}^{N} a_k$
defined from the truncated summation of an infinite series defined from the truncated summation of an infinite series
$ S = \sum_{k=0}^{\infty} a_k$. $ S = \sum_{k=0}^{\infty} a_k$.
@ -1642,7 +1643,7 @@ If the series converges, then the partial sums will tend to the exact result
\lim_{N\rightarrow \infty} S_N = S. \lim_{N\rightarrow \infty} S_N = S.
\end{equation} \end{equation}
The Shanks transformation attempts to generate increasingly accurate estimates of this The Shanks transformation attempts to generate increasingly accurate estimates of this
limt by defining a new series as limit by defining a new series as
\begin{equation} \begin{equation}
T(S_N) = \frac{S_{N+1} S_{N-1} - S_{N}^2}{S_{N+1} + S_{N-1} - 2 S_{N}}. T(S_N) = \frac{S_{N+1} S_{N-1} - S_{N}^2}{S_{N+1} + S_{N-1} - 2 S_{N}}.
\end{equation} \end{equation}
@ -1655,7 +1656,7 @@ the approximate form
\end{equation} \end{equation}
Furthermore, while this transformation can accelerate the convergence of a series, Furthermore, while this transformation can accelerate the convergence of a series,
there is no guarantee that this acceleration will be fast enough to significantly there is no guarantee that this acceleration will be fast enough to significantly
improve the accuracy of low-order approximations.} improve the accuracy of low-order approximations.
\hugh{To the best of our knowledge, the Shanks transformation has never previously been applied \hugh{To the best of our knowledge, the Shanks transformation has never previously been applied
to the acceleration of the MP series. to the acceleration of the MP series.