Classical perturbation theory of good action-angle variables. Applications to semiclassical eigenvalues and to collisional energy transfer in polyatomic molecules

George C Schatz, Thomas Mulloney

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45 Citations (Scopus)

Abstract

In this paper, a classical perturbation theory is presented for determining good action-angle variables for nonseparable molecular vibrational motions, and applied to the calculation of semiclassical eigenvalues and to classical trajectory studies of state-resolved collisional energy transfer. Classical perturbation theory provides an approximate method for determining the coefficients in a Fourier series representation of the generating function for the canonical transformation between good action-angle variables and harmonic ones. It assumes that the anharmonic contributions to the total energy are small in comparison with the harmonic contributions, and is similar in many respects to quantum perturbation theory. Indeed, we show that the second-order semiclassical and quantum eigenvalues are identical within an additive constant for a model of two coupled oscillators. Numerous applications to semiclassical eigenvalues for different cubic and quartic force field problems show that the second-order theory typically accounts for all but 5% of the anharmonic contribution to the total energy, while third-order perturbation theory is always equal to or better than second order in accuracy, predicting eigenvalues for the (000) and (100) states of CO2 within 0.0001 eV of the exact semiclassical values. The applications to collisional energy transfer using classical trajectory methods consider a linear model of Kr + CO2(001). Agreement of moments of the final symmetric and asymmetric stretch good actions obtained using second-order perturbation theory with the analogous exact semiclassical results is generally to better than 15%, although some more serious errors are encountered at low collision energies when the errors in determining these moments become comparable to the moments themselves. The accuracy of perturbation theory in determining energy transfer information is, however, generally acceptable, and this coupled with the great simplicity of the approach means that it should be capable of widespread use in studies of state-resolved collision phenomena.

Original languageEnglish
Pages (from-to)989-999
Number of pages11
JournalJournal of Physical Chemistry
Volume83
Issue number8
Publication statusPublished - 1979

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polyatomic molecules
Energy transfer
eigenvalues
perturbation theory
energy transfer
Molecules
Trajectories
Fourier series
moments
trajectories
harmonics
collisions
field theory (physics)
energy
oscillators
coefficients

ASJC Scopus subject areas

  • Physical and Theoretical Chemistry

Cite this

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title = "Classical perturbation theory of good action-angle variables. Applications to semiclassical eigenvalues and to collisional energy transfer in polyatomic molecules",
abstract = "In this paper, a classical perturbation theory is presented for determining good action-angle variables for nonseparable molecular vibrational motions, and applied to the calculation of semiclassical eigenvalues and to classical trajectory studies of state-resolved collisional energy transfer. Classical perturbation theory provides an approximate method for determining the coefficients in a Fourier series representation of the generating function for the canonical transformation between good action-angle variables and harmonic ones. It assumes that the anharmonic contributions to the total energy are small in comparison with the harmonic contributions, and is similar in many respects to quantum perturbation theory. Indeed, we show that the second-order semiclassical and quantum eigenvalues are identical within an additive constant for a model of two coupled oscillators. Numerous applications to semiclassical eigenvalues for different cubic and quartic force field problems show that the second-order theory typically accounts for all but 5{\%} of the anharmonic contribution to the total energy, while third-order perturbation theory is always equal to or better than second order in accuracy, predicting eigenvalues for the (000) and (100) states of CO2 within 0.0001 eV of the exact semiclassical values. The applications to collisional energy transfer using classical trajectory methods consider a linear model of Kr + CO2(001). Agreement of moments of the final symmetric and asymmetric stretch good actions obtained using second-order perturbation theory with the analogous exact semiclassical results is generally to better than 15{\%}, although some more serious errors are encountered at low collision energies when the errors in determining these moments become comparable to the moments themselves. The accuracy of perturbation theory in determining energy transfer information is, however, generally acceptable, and this coupled with the great simplicity of the approach means that it should be capable of widespread use in studies of state-resolved collision phenomena.",
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AB - In this paper, a classical perturbation theory is presented for determining good action-angle variables for nonseparable molecular vibrational motions, and applied to the calculation of semiclassical eigenvalues and to classical trajectory studies of state-resolved collisional energy transfer. Classical perturbation theory provides an approximate method for determining the coefficients in a Fourier series representation of the generating function for the canonical transformation between good action-angle variables and harmonic ones. It assumes that the anharmonic contributions to the total energy are small in comparison with the harmonic contributions, and is similar in many respects to quantum perturbation theory. Indeed, we show that the second-order semiclassical and quantum eigenvalues are identical within an additive constant for a model of two coupled oscillators. Numerous applications to semiclassical eigenvalues for different cubic and quartic force field problems show that the second-order theory typically accounts for all but 5% of the anharmonic contribution to the total energy, while third-order perturbation theory is always equal to or better than second order in accuracy, predicting eigenvalues for the (000) and (100) states of CO2 within 0.0001 eV of the exact semiclassical values. The applications to collisional energy transfer using classical trajectory methods consider a linear model of Kr + CO2(001). Agreement of moments of the final symmetric and asymmetric stretch good actions obtained using second-order perturbation theory with the analogous exact semiclassical results is generally to better than 15%, although some more serious errors are encountered at low collision energies when the errors in determining these moments become comparable to the moments themselves. The accuracy of perturbation theory in determining energy transfer information is, however, generally acceptable, and this coupled with the great simplicity of the approach means that it should be capable of widespread use in studies of state-resolved collision phenomena.

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