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\begin{document}
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\title{On the use of algebraic approximants to sum divergent
series\\ for Fermi resonances in vibrational spectroscopy}
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\author{David Z. Goodson and Alexei V. Sergeev\cite{purdue}}
\address{Department of Chemistry, Southern Methodist University,
Dallas, Texas  75275}
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\narrowtext

\v{C}\'{\i}\v{z}ek {\it et al.} \cite{cizek} have suggested that
large-order Rayleigh-Schr\"odinger perturbation expansions
are a viable alternative to variational methods for
calculating molecular vibration energy levels.  The energy
$E(\lambda)$ of an anharmonic oscillator, considered as a
function of the perturbation parameter $\lambda$, has a
complicated singularity at the origin in the complex
$\lambda$ plane \cite{anharmosc,bender}, and therefore has
a zero radius of convergence.  Nevertheless,
\v{C}\'{\i}\v{z}ek {\it et al.} found in practice that
the expansions could be summed with
Pad\'e approximants.  Recently the computational cost of the
perturbation theory was compared with that of variational
diagonalization of the Hamiltonian for a model 2-mode oscillator
problem \cite{suvernev1}.  It was found that perturbation theory
had a significant advantage over variational calculations in the
number of arithmetic operations needed to obtain a given level of
accuracy.  Scaling arguments indicate that this advantage can be
even greater for rotating oscillators \cite{suvernev2}.

However, there is a class of eigenstates for which the
perturbation theory appears to fail: eigenstates involved in
Fermi resonances, for which the wavefunctions show strong
mixing of two or more of the unperturbed harmonic eigenfunctions.
In the function $E(\lambda)$ the resonant states are connected
by a branch point, with the different eigenvalues residing
on different Riemann sheets.  The closer the degeneracy of
the harmonic energies, the closer the branch point is to the
origin, and hence the greater the effect on the convergence.
Since Pad\'e approximants are rational functions, which cannot
explicitly model the multiple-valued nature of $E(\lambda)$,
they can have serious convergence problems in such cases.

A simple solution to this problem is to use
{\it algebraic} summation approximants.
Consider an expansion $E(\lambda)=\sum_{n=0}^\infty E_n\lambda^n$.
The conventional ``linear'' Pad\'e approximant is a
function $E_{[L,M]}(\lambda)=P_L(\lambda)/Q_M(\lambda)$ in
terms of the polynomials $P_L$ and $Q_M$, of degrees $L$ and
$M$ respectively, defined by the linear equation
\begin{equation}
P(\lambda)-Q(\lambda)E(\lambda)={\rm O}\big(\lambda^{L+M+1}\big).
\end{equation}
Similarly, algebraic approximants $E_{[p_0,p_1,\dots,p_k]}$
of arbitrary degree $m$ can be defined by
\begin{equation}
\sum_{k=0}^m
A_k(\lambda)E^k_{[p_0,p_1,\dots,p_k]}(\lambda)=0.
\end{equation}
The $A_k(\lambda)$ are polynomials of degree $p_k$
that satisfy
\begin{equation}
\sum_{k=0}^m A_k(\lambda)E^k(\lambda)=
{\rm O}\big(\lambda^{N}\big),
\quad
N=m+\sum_{k=0}^m p_k.
\label{polyeq}
\end{equation}
These approximants were proposed by Pad\'e \cite{pade}, but
are not nearly as well known as the linear approximants ($m=1$).
Quadratic approximants ($m=2$) have been used occasionally,
especially for calculating the complex energies
of unstable quasibound eigenstates
\cite{suvernev1,quasibound}, but higher-degree appoximants have
rarely been applied to physical problems.  We have recently
developed an algorithm for computing high-degree approximants
\cite{sergeev,algebra} and have analyzed some of their mathematical
properties \cite{algebra}.

Since Eq.~(\ref{polyeq}) has $m$ solutions for $E(\lambda)$, an
algebraic approximant of degree $m>1$ is a multiple-valued function
with $m$ branches.  Two-sheet branch-point singularities occur at those
values of $\lambda$ for which two of the solutions become equal.
For quadratic approximants the singular points are the zeros
of the discriminant polynomial $D_2(\lambda)=A_1^2-4A_0A_2$.
For cubic approximants, they are the zeros of $D_3(\lambda)=b^3+c^2$,
where $b=3A_1A_3-A_2^2$ and $c={9\over 2}A_1A_2A_3-9A_0A_3^2-A_2^3$.
In general, as long as $m\ge 2$ the approximants should be able
to explicitly model the two-sheet branch points corresponding to
Fermi resonances.

We have computed perturbation series through 40th order for
the molecules H$_2$O and H$_2$S, with the
anharmonic oscillator Hamiltonians used by
\v{C}\'{\i}\v{z}ek {\it et al.} \cite{cizek}.  Table~\ref{tab1}
compares the harmonic frequencies.  For the ground states and
for singly excited states the rate of convergence shows no
appreciable dependence on $m$.  However, for the (200) state of
H$_2$S, which is strongly resonant with the nearly degenerate
(002) state, the convergence is much more rapid for $m>1$ than
for $m=1$, as shown in Fig.~\ref{fig200}.  There seems to be
an advantage to using $m\ge 3$ beginning at 30th order.
The convergence behavior is similar for the (002) state.  The
branch point, at which these two states become degenerate,
is at $\lambda=0.60096\pm 0.28837 i$.  (The
physical solution corresponds to $\lambda=1$.)
For H$_2$O the $m=1$ approximants for the (200) state show
no convergence problems.  However, for the (400) state, in
Fig.~\ref{fig400}, which is nearly degenerate with at least two
other eigenstates, they converge relatively slowly.  The
convergence is better for $m\ge 2$, with an advantage for
$m\ge 3$ beginning at 26th order.

High-degree approximants have the additional
advantage of being able to model the complicated singularities
at the $\lambda\to 0$ and $\lambda\to\infty$
limits \cite{algebra,intelligent}, which are generic features
of anharmonic oscillator energies, but this advantage is realized
in practice only if the perturbation series is known to rather
high order \cite{algebra}.  For the 40th-order expansions
considered here we find that the high-degree
($m\ge 3$) approximants outperform the quadratic approximants only
for states involved in Fermi resonances.  This indicates that
the source of their advantage is not the behavior at $\lambda\to 0$
or $\lambda\to\infty$ but rather the accuracy with which they
model the resonance branch points.

This work was supported by the National Science Foundation and
by the Welch Foundation.

\begin{references}

\bibitem[*]{purdue} Current address: Dept. of Chemistry,
Purdue University, West Lafayette, IN\ \ 47907

\bibitem{cizek} J. \v{C}i\v{z}ek, V. \v{S}pirko, and
O. Bludsk\'y, J. Chem. Phys. {\bf 99}, 7331 (1993).

\bibitem{suvernev1} A. A. Suvernev and D. Z. Goodson,
J. Chem. Phys. {\bf 106}, 2681 (1997).

\bibitem{anharmosc} C. M. Bender and T. T. Wu, Phys. Rev. {\bf 184},
1231 (1969); G. Alvarez, J. Phys. A: Math. Gen. {\bf 27}, 4589 (1995).

\bibitem{bender} C. M. Bender and S. A. Orszag, {\it Advanced
Mathematical Methods for Scientists and Engineers}
(McGraw-Hill, New York, 1978), pp. 350-361.

\bibitem{suvernev2} A. A. Suvernev and D. Z. Goodson,
Chem. Phys. Lett. {\bf 269}, 177 (1997); J. Chem. Phys.
{\bf 107}, 4099 (1997).

\bibitem{pade} M. H. Pad\'e, Ann. Sci. Ecole Norm. Sup.
{\bf 9}, Supplement, pp. 1-8 (1892).

\bibitem{quasibound} V. M. Va\u{\i}nberg, V. D. Mur, V. S.
Popov, and A. V. Sergeev, Pis'ma Zh. Eksp. Teor. Fiz. {\bf 44},
9 (1986) [JETP Lett. {\bf 44}, 9]; T. C. Germann and S. Kais,
J. Chem. Phys. {\bf 99}, 7739 (1993).

\bibitem{sergeev} A. V. Sergeev, Zh. vychisl. Mat. Mat. Fiz.
{\bf 26}, 348 (1986) [U.S.S.R. Comput. Maths. Math. Phys. {\bf 26},
17 (1986)].

\bibitem{algebra} A. V. Sergeev and D. Z. Goodson,
J. Phys. A: Math. Gen. {\bf 31}, 4301 (1998).

\bibitem{intelligent} F. M. Fern\'andez and R. H. Tipping,
unpublished preprint.

\end{references}

\vbox{
\begin{table}
\caption{Harmonic vibrational frequencies, in cm$^{-1}$.}
\begin{tabular}{cccc}
&$\omega_{\rm 1}$&$\omega_{\rm 2}$&$\omega_{\rm 3}$\\
\tableline

H$_2$O&3832.0&1648.9&3942.6\\
H$_2$S&2721.9&1214.5&2733.3\\
\end{tabular}
\label{tab1}
\end{table} }

\begin{figure}
\caption{Summation convergence vs. order $k$ of the perturbation
expansion, for the (200) state of H$_2$S.  The ordinate is
$-\log_{10}|(S_k-E)/E|$, which is a continuous measure
of the number of converged digits, where $S_k$ is the algebraic
approximant corresponding to order $k$ of the diagonal staircase
approximant sequence and $E=8522.5667$ cm$^{-1}$ is the result
to which 40th-order perturbation seems to converge.  The degrees
of the approximants are as
follows: $m=1$ (-$\;$-$\;$-), $m=2$ (---), $m=3$ ($\triangle$),
$m=4$ ($\Diamond$).}
\label{fig200}
\end{figure}

\begin{figure}
\caption{Summation convergence vs. order of the perturbation
expansion, for the (400) state of H$_2$O.  The ordinate is
defined as in Fig.~\protect\ref{fig200} and the converged
energy is $E=19538.4$ cm$^{-1}$.  The degrees of the approximants are
$m=1$ (-$\;$-$\;$-), $m=2$ (---), $m=3$ ($\triangle$),
$m=4$ ($\Diamond$).}
\label{fig400}
\end{figure}

\end{document}