### Abstract

Two methods for calculating the good action variables and semiclassical eigenvalues for coupled oscillator systems are presented, both of which relate the actions to the coefficients appearing in the Fourier representation of the normal coordinates and momenta. The two methods differ in that one is based on the exact expression for the actions together with the EBK semiclassical quantization condition while the other is derived from the Sorbie-Handy (SH) approximation to the actions. However, they are also very similar in that the actions in both methods are related to the same set of Fourier coefficients and both require determining the perturbed frequencies in calculating actions. These frequencies are also determined from the Fourier representations, which means that the actions in both methods are determined from information entirely contained in the Fourier expansion of the coordinates and momenta. We show how these expansions can very conveniently be obtained from fast Fourier transform (FFT) methods and that numerical filtering methods can be used to remove spurious Fourier components associated with the finite trajectory integration duration. In the case of the SH based method, we find that the use of filtering enables us to relax the usual periodicity requirement on the calculated trajectory. Application to two standard Henon-Heiles models is considered and both are shown to give semiclassical eigenvalues in good agreement with previous calculations for nondegenerate and 1:1 resonant systems. In comparing the two methods, we find that although the exact method is quite general in its ability to be used for systems exhibiting complex resonant behavior, it converges more slowly with increasing trajectory integration duration and is more sensitive to the algorithm for choosing perturbed frequencies than the SH based method. The SH based method is less straightforward to use in studying resonant systems, but good results are obtained for 1:1 resonant systems using actions defined in terms of the complex coordinates Q_{1}±iQ_{2}. The SH based method is also shown to be remarkably accurate in determining high energy eigenvalues (about three-quarters of the dissociation energy).

Original language | English |
---|---|

Pages (from-to) | 5913-5919 |

Number of pages | 7 |

Journal | Journal of Chemical Physics |

Volume | 81 |

Issue number | 12 |

Publication status | Published - 1984 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

*Journal of Chemical Physics*,

*81*(12), 5913-5919.

**Fourier transform methods for calculating action variables and semiclassical eigenvalues for coupled oscillator systems.** / Eaker, Charles W.; Schatz, George C; De Leon, Nelson; Heller, Eric J.

Research output: Contribution to journal › Article

*Journal of Chemical Physics*, vol. 81, no. 12, pp. 5913-5919.

}

TY - JOUR

T1 - Fourier transform methods for calculating action variables and semiclassical eigenvalues for coupled oscillator systems

AU - Eaker, Charles W.

AU - Schatz, George C

AU - De Leon, Nelson

AU - Heller, Eric J.

PY - 1984

Y1 - 1984

N2 - Two methods for calculating the good action variables and semiclassical eigenvalues for coupled oscillator systems are presented, both of which relate the actions to the coefficients appearing in the Fourier representation of the normal coordinates and momenta. The two methods differ in that one is based on the exact expression for the actions together with the EBK semiclassical quantization condition while the other is derived from the Sorbie-Handy (SH) approximation to the actions. However, they are also very similar in that the actions in both methods are related to the same set of Fourier coefficients and both require determining the perturbed frequencies in calculating actions. These frequencies are also determined from the Fourier representations, which means that the actions in both methods are determined from information entirely contained in the Fourier expansion of the coordinates and momenta. We show how these expansions can very conveniently be obtained from fast Fourier transform (FFT) methods and that numerical filtering methods can be used to remove spurious Fourier components associated with the finite trajectory integration duration. In the case of the SH based method, we find that the use of filtering enables us to relax the usual periodicity requirement on the calculated trajectory. Application to two standard Henon-Heiles models is considered and both are shown to give semiclassical eigenvalues in good agreement with previous calculations for nondegenerate and 1:1 resonant systems. In comparing the two methods, we find that although the exact method is quite general in its ability to be used for systems exhibiting complex resonant behavior, it converges more slowly with increasing trajectory integration duration and is more sensitive to the algorithm for choosing perturbed frequencies than the SH based method. The SH based method is less straightforward to use in studying resonant systems, but good results are obtained for 1:1 resonant systems using actions defined in terms of the complex coordinates Q1±iQ2. The SH based method is also shown to be remarkably accurate in determining high energy eigenvalues (about three-quarters of the dissociation energy).

AB - Two methods for calculating the good action variables and semiclassical eigenvalues for coupled oscillator systems are presented, both of which relate the actions to the coefficients appearing in the Fourier representation of the normal coordinates and momenta. The two methods differ in that one is based on the exact expression for the actions together with the EBK semiclassical quantization condition while the other is derived from the Sorbie-Handy (SH) approximation to the actions. However, they are also very similar in that the actions in both methods are related to the same set of Fourier coefficients and both require determining the perturbed frequencies in calculating actions. These frequencies are also determined from the Fourier representations, which means that the actions in both methods are determined from information entirely contained in the Fourier expansion of the coordinates and momenta. We show how these expansions can very conveniently be obtained from fast Fourier transform (FFT) methods and that numerical filtering methods can be used to remove spurious Fourier components associated with the finite trajectory integration duration. In the case of the SH based method, we find that the use of filtering enables us to relax the usual periodicity requirement on the calculated trajectory. Application to two standard Henon-Heiles models is considered and both are shown to give semiclassical eigenvalues in good agreement with previous calculations for nondegenerate and 1:1 resonant systems. In comparing the two methods, we find that although the exact method is quite general in its ability to be used for systems exhibiting complex resonant behavior, it converges more slowly with increasing trajectory integration duration and is more sensitive to the algorithm for choosing perturbed frequencies than the SH based method. The SH based method is less straightforward to use in studying resonant systems, but good results are obtained for 1:1 resonant systems using actions defined in terms of the complex coordinates Q1±iQ2. The SH based method is also shown to be remarkably accurate in determining high energy eigenvalues (about three-quarters of the dissociation energy).

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M3 - Article

VL - 81

SP - 5913

EP - 5919

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 12

ER -