©1999 American Institute of Physics. All rights reserved.

^{ }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 H_{2}S and the (400) state of H_{2}O 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]

*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*() of an anharmonic oscillator, considered as a^{ }function of the perturbation parameter , has a complicated singularity^{ }at the origin in the complex plane,^{2}^{,}^{3} and therefore^{ }has a zero radius of convergence. Nevertheless, íek *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} ^{ }

^{ }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*() 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*(), they can have serious convergence problems^{ }in such cases. ^{ }

^{ }to use *algebraic* summation approximants. Consider an expansion *E*() = *E*_{n}^{n}. The^{ }conventional ``linear'' Padé approximant is a function *E*_{[L,M]}() = *P*_{L}()/*Q*_{M}() in terms^{ }of the polynomials *P*_{L} and *Q*_{M}, of degrees *L* and^{ }*M*, respectively, defined by the linear equation

Similarly, algebraic approximants^{ }*E*_{[p0,p1,,pm]} of arbitrary degree *m* can be defined by

The^{ }*A*_{k}() are polynomials of degree *p*_{k} that satisfy

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 approximants^{8}^{,}^{9} and have analyzed some of their mathematical^{ }properties.^{9} ^{ }

*m* solutions for *E*(), an^{ }algebraic approximant of degree *m* > 1 is a multiple-valued function with^{ }*m* branches. Square-root branch points occur at those values of^{ } 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* – 4*A*_{0}*A*_{2}. If none of the^{ }branch points are close to the origin or close to^{ }the physical value of , 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*. ^{ }

^{ }have computed perturbation series through 40th order for the molecules^{ }H_{2}O and H_{2}S, with the anharmonic oscillator Hamiltonians used by^{ }íek *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 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. 1.^{ }There seems to be an advantage to using *m*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 = 0.600 96 ± 0.288 37*i*. (The physical solution corresponds to^{ } = 1.) For H_{2}O 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*2, with an advantage for *m*3 beginning at^{ }26th order. ^{ }

^{ }able to model complicated singularities at the 0 and ^{ }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*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 0 or but rather the accuracy^{ }with which they model the resonance branch points. ^{ }

^{ }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 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 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^{ }(), *m* = 4 ().
First citation in article

Full figure (12 kB)

Fig. 2. Summation convergence vs order for the (400) state^{ }of H_{2}O. 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 (), *m* = 4 ().
First citation in article

Table I. Harmonic vibrational frequencies, in cm^{ – 1}. | |||

_{1}^{ } | _{2} | _{3} | |

H_{2}O | 3832.0 | 1648.9 | 3942.6 |

H_{2}S | 2721.9 | 1214.5 | 2733.3 |

^{a}Current address:^{ }Department of Chemistry, Purdue University, West Lafayette, Indiana 47907.

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