# On the use of algebraic approximants to sum divergent series
for Fermi resonances in vibrational spectroscopy

## David Z. Goodson and Alexei V. Sergeev

### Abstract

Pade summation of large-order perturbation theory can often yield highly accurate energy eigenvalues for molecular vibrations. However, for eigenstates involved in Fermi resonances the convergence of the Pade approximants can be very slow. This is because the energy is a multivalued function of the perturbation parameter while Pade approximants are single valued, and Fermi resonances occur when a branch point lies close to the physical value of the parameter. Algebraic approximants are multivalued generalizations of Pade approximants. Using the (200) state of H/sub 2/S and the (400) state of H/sub 2/O as examples of Fermi resonances, it is demonstrated here that algebraic approximants greatly improve the summation convergence.

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