An optimized semiclassical approximation for vibrational response functions

Abstract

The observables of multidimensional infrared spectroscopy may be calculated from nonlinear vibrational response functions. Fully quantum dynamical calculations of vibrational response functions are generally impractical, while completely classical calculations are qualitatively incorrect at long times. These challenges motivate the development of semiclassical approximations to quantum mechanics, which use classical mechanical information to reconstruct quantum effects. The mean-trajectory (MT) approximation is a semiclassical approach to quantum vibrational response functions employing classical trajectories linked by deterministic transitions representing the effects of the radiation-matter interaction. Previous application of the MT approximation to the third-order response function R(3)(t3, t2, t1) demonstrated that the method quantitatively describes the coherence dynamics of the t3 and t1 evolution times, but is qualitatively incorrect for the waiting-time t2 period. Here we develop an optimized version of the MT approximation by elucidating the connection between this semiclassical approach and the double-sided Feynman diagrams (2FD) that represent the quantum response. Establishing the direct connection between 2FD and semiclassical paths motivates a systematic derivation of an optimized MT approximation (OMT). The OMT uses classical mechanical inputs to accurately reproduce quantum dynamics associated with all three propagation times of the third-order vibrational response function.

Thermal weights for semiclassical vibrational response functions

“…The OMT approximation 28,[45][46][47] was developed to improve the dependence on the waiting time of the thirdorder response function calculated with the MT approximation. 28,33,41 The third-order response function in both approximations is evaluated from semiclassical paths consisting of classical trajectories linked by discontinuous transitions.…”

“…28,33,41 The third-order response function in both approximations is evaluated from semiclassical paths consisting of classical trajectories linked by discontinuous transitions. At four classical states along the path, corresponding to the radiation-matter interactions, phase-space information is collected to compute the response function.…”

“…These are the OMT approximation 28,[45][46][47] based on the consideration of double-sided Feynman diagrams [48][49][50][51] and the semiclassical Wigner transform (SWT) approximation 26 based on the application of Wigner transforms with action-angle variables.…”

“…The optimized mean-trajectory (OMT) approximation was developed 28,41,[45][46][47] to improve upon the MT result while preserving its essential features. This approach was derived from consideration of the double-sided Feynman diagrams [48][49][50][51] (2FDs) conventionally used to represent the time course of a nonlinear spectroscopic measurement.…”

“…[7][8][9][10][11] Semiclassical approximations to quantum dynamics [12][13][14][15][16][17][18][19][20][21][22][23][24][25] have the capacity to incorporate quantum effects with a computational effort comparable to that of a purely classical calculation. One semiclassical strategy applied to the evaluation of spectroscopic response functions [26][27][28] draws on the old quantum theory 29 by propagating classical trajectories at quantized action values. 30,31 The calculation of a canonical spectroscopic response function may be divided into two related parts: determining the thermal weights for initial conditions and performing the time propagation.…”