[This is a copy of the original document!] The Journal of Chemical Physics, Vol. 110, No. 16, pp. 8205–8206, 22 April 1999
©1999 American Institute of Physics. All rights reserved.
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On the use of algebraic approximants to sum divergent series for Fermi resonances in vibrational spectroscopy

David Z. Goodson and Alexei V. Sergeeva)

Department of Chemistry, Southern Methodist University, Dallas, Texas 75275

Received: 12 January 1999; accepted: 27 January 1999

Padé 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 Padé approximants can be very slow. This is because the energy is a multivalued function of the perturbation parameter while Padé 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 Padé approximants. Using the (200) state of H2S and the (400) state of H2O as examples of Fermi resonances, it is demonstrated here that algebraic approximants greatly improve the summation convergence. © 1999 American Institute of Physics.[S0021-9606(99)01216-7]


(C, hacek)í(z, hacek)ek et al.1 have suggested large-order Rayleigh–Schrödinger perturbation expansions as an 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,2,3 and therefore has a zero radius of convergence. Nevertheless, (C, hacek)í(z, hacek)ek et al. found in practice that the expansions could be summed with Padé approximants. Recently the computational cost of the perturbation theory was compared with that of variational diagonalization of the Hamiltonian for a model two-mode oscillator problem.4 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.5

However, there is a class of eigenstates for which the perturbation theory appears to fail: eigenstates involved in Fermi resonances, for which the wave functions show strong mixing of two or more of the unperturbed harmonic eigenfunctions. In the function E( lambda ) the resonant states are connected by branch points, with the different eigenvalues residing on different Riemann sheets.3 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 Padé 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 algebraic summation approximants. Consider an expansion E( lambda ) = (summation)[sub n=0][sup (infinity)]En lambda n. The conventional ``linear'' Padé approximant is a function E[L,M]( lambda ) = PL( lambda )/QM( lambda ) in terms of the polynomials PL and QM, of degrees L and M, respectively, defined by the linear equation

[dformula P( lambda )-Q( lambda )E( lambda )=O( lambda [sup L+M+1]).]

Similarly, algebraic approximants E[p0,p1,...,pm] of arbitrary degree m can be defined by

[dformula [subformula (summation)[sub k=0][sup m]]A[sub k]( lambda )E[sub [p[sub 0],p[sub 1],...,p[sub m]]][sup k]( lambda )=0.]

The Ak( lambda ) are polynomials of degree pk that satisfy

[dformula [subformula (summation)[sub k=0][sup m]]A[sub k]( lambda )E[sup k]( lambda )=O( lambda [sup q]), q=m+[subformula (summation)[sub k=0][sup m]]p[sub k].]

These approximants were proposed by Padé,6 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,4,7 but higher-degree approximants have rarely been applied to physical problems. We have recently developed an algorithm for computing high-degree approximants8,9 and have analyzed some of their mathematical properties.9

Since Eq. (3) has m solutions for E( lambda ), an algebraic approximant of degree m > 1 is a multiple-valued function with m branches. Square-root branch points occur at those values of  lambda for which two of the solutions become equal. For quadratic approximants, for example, the singular points are simply the zeros of the discriminant polynomial A[sub 1][sup 2] – 4A0A2. If none of the branch points are close to the origin or close to the physical value of  lambda , then linear approximants should be adequate. However, if it is necessary to model branch points and the number of resonant states connected by the branch points is N, then the degree of the approximants should chosen to be equal to or greater than N.

We have computed perturbation series through 40th order for the molecules H2O and H2S, with the anharmonic oscillator Hamiltonians used by (C, hacek)í(z, hacek)ek et al.1 Table I 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 H2S, 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. 1. There seems to be an advantage to using m(greater-than-or-equal-to)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 = 0.600 96 ± 0.288 37i. (The physical solution corresponds to  lambda = 1.) For H2O the m = 1 approximants for the (200) state show no convergence problems. However, for the (400) state, in Fig. 2, which seems to involve a resonance of at least three states, they converge relatively slowly. The convergence is better for m(greater-than-or-equal-to)2, with an advantage for m(greater-than-or-equal-to)3 beginning at 26th order.

Figure 1. Figure 2.

High-degree approximants have the additional advantage of being able to model complicated singularities at the  lambda (right-arrow)0 and  lambda (right-arrow)(infinity) limits,9,10 which are generic features of anharmonic oscillator energies, but this advantage is realized only if the perturbation series is known to rather high order.9 For the 40th-order expansions considered here we find that the high-degree (m(greater-than-or-equal-to)3) approximants noticeably 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 (right-arrow)0 or  lambda (right-arrow)(infinity) 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.


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Full figure (13 kB)

Fig. 1. Summation convergence vs order k of the perturbation expansion for the (200) state of H2S. The ordinate is – log10|(SkE)/E|, which is a continuous measure of the number of converged digits, where Sk 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 theory seems to converge. (The last digit in E is uncertain.) The degrees of the approximants are as follows: m = 1 (– – –), m = 2 (—), m = 3 ((open triangle)), m = 4 ((open diamond)). First citation in article

Full figure (12 kB)

Fig. 2. Summation convergence vs order for the (400) state of H2O. The ordinate is defined as in Fig. 1 and the converged energy is E = 19 538.4 cm – 1. (The last digit is uncertain.) The degrees of the approximants are m = 1 (– – –), m = 2 (—), m = 3 ((open triangle)), m = 4 ((open diamond)). First citation in article


Table I. Harmonic vibrational frequencies, in cm – 1.
  omega 1  omega 2  omega 3
H2O 3832.0 1648.9 3942.6
H2S 2721.9 1214.5 2733.3
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aCurrent address: Department of Chemistry, Purdue University, West Lafayette, Indiana 47907.

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