Use of classical Fourier amplitudes as quantum matrix elements:  a comparison of Morse oscillator Fourier coefficients with quantum matrix elements

Observables in nonlinear spectroscopic measurements may be calculated from response functions, which have the form of averages of nested commutators involving the operator governing the radiation-matter interaction. We present a semiclassical formulation of the optical nonlinear response function, employing the Herman-Kluk frozen Gaussian approximation to the quantum propagator in the coherent states representation. This semiclassical approximation permits the response function to be computed from classical trajectories and stability matrices, and provides insight into the relationship between nonlinear response in classical and quantum mechanics. Linear response calculations for an anharmonic oscillator illustrate that the semiclassical approach reproduces the significant differences between quantum and classical results.

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“…We consider a thermal ensemble of Morse oscillators, 57,58 with the Hamiltonian and density operator,…”

Section: Numerical Resultsmentioning

confidence: 99%

Optical response functions with semiclassical dynamics

Noid¹,

Ezra

Loring

2003

The Journal of Chemical Physics

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“…The SCCF response function was calculated using analytical relations between Cartesian coordinates and momenta and action-angle variables for the Morse oscillator. 69 The correction factor for a general anharmonic potential in Eq. (13) Morse potential.…”

Section: Resultsmentioning

confidence: 99%

Thermal weights for semiclassical vibrational response functions

Moberg

Alemi

Loring

2015

The Journal of Chemical Physics

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Semiclassical approximations to response functions can allow the calculation of linear and nonlinear spectroscopic observables from classical dynamics. Evaluating a canonical response function requires the related tasks of determining thermal weights for initial states and computing the dynamics of these states. A class of approximations for vibrational response functions employs classical trajectories at quantized values of action variables and represents the effects of the radiation-matter interaction by discontinuous transitions. Here, we evaluate choices for a thermal weight function which are consistent with this dynamical approximation. Weight functions associated with different semiclassical approximations are compared, and two forms are constructed which yield the correct linear response function for a harmonic potential at any temperature and are also correct for anharmonic potentials in the classical mechanical limit of high temperature. Approximations to the vibrational linear response function with quantized classical trajectories and proposed thermal weight functions are assessed for ensembles of one-dimensional anharmonic oscillators. This approach is shown to perform well for an anharmonic potential that is not locally harmonic over a temperature range encompassing the quantum limit of a two-level system and the limit of classical dynamics. C 2015 AIP Publishing LLC. [http://dx

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“…With A = ⍀ 2 / ͑2D͒, H j is also the representation in action-angle variables 70 of the Hamiltonian of the Morse oscillator with harmonic frequency ⍀ and dissociation energy D. ͑3.2͒ represents N identical, noninteracting anharmonic oscillators, each with Hamiltonian H j = ⍀I j − ͑A /2͒I j 2 .…”

Section: ͑32͒mentioning

confidence: 99%

Classical and quantum mechanical infrared echoes from resonantly coupled molecular vibrations

Noid

Loring

2005

The Journal of Chemical Physics

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The nonlinear response function associated with the infrared vibrational echo is calculated for a quantum mechanical model of resonantly coupled, anharmonic oscillators at zero temperature. The classical mechanical response function is determined from the quantum response function by setting variant Planck's over 2pi-->0, permitting the comparison of the effects of resonant vibrational coupling among an arbitrary number of anharmonic oscillators on quantum and classical vibrational echoes. The quantum response function displays a time dependence that reflects both anharmonicity and resonant coupling, while the classical response function depends on anharmonicity only through a time-independent amplitude, and shows a time dependence controlled only by the resonant coupling. In addition, the classical response function grows without bound in time, a phenomenon associated with the nonlinearity of classical mechanics, and absent in quantum mechanics. This unbounded growth was previously identified in the response function for a system without resonant vibrational energy transfer, and is observed to persist in the presence of resonant coupling among vibrations. Quantitative agreement between classical and quantum response functions is limited to a time scale of duration inversely proportional to the anharmonicity.

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Use of classical Fourier amplitudes as quantum matrix elements: a comparison of Morse oscillator Fourier coefficients with quantum matrix elements

Cited by 32 publications

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Optical response functions with semiclassical dynamics

Optical response functions with semiclassical dynamics

Thermal weights for semiclassical vibrational response functions

Classical and quantum mechanical infrared echoes from resonantly coupled molecular vibrations

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