^{1}and Roger F. Loring

^{1,a)}

### 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)}(*t* _{3}, *t* _{2}, *t* _{1}) demonstrated that the method quantitatively describes the coherence dynamics of the *t* _{3} and *t* _{1} evolution times, but is qualitatively incorrect for the waiting-time *t* _{2} 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.

Mallory Gerace acknowledges support from the National Institutes of Health (NIH) through Cornell's Molecular Biology Training Grant T32GM008267.

I. INTRODUCTION

II. MEAN TRAJECTORY APPROXIMATION

III. OPTIMIZED MT APPROXIMATION

IV. CALCULATING OMT RESPONSE FUNCTIONS

V. NUMERICAL RESULTS

VI. CONCLUSIONS

### Key Topics

- Oscillators
- 17.0
- Coherence
- 11.0
- Correlation functions
- 9.0
- Melt texturing
- 9.0
- Optical phase matching
- 7.0

## Figures

The four paths associated with the mean-trajectory approximation for the third-order vibrational response function are shown. The action of the first trajectory is quantized to integer multiples of ℏ and interactions with the electric field are represented by jumps of ±ℏ/2 in the action at constant angle.

The four paths associated with the mean-trajectory approximation for the third-order vibrational response function are shown. The action of the first trajectory is quantized to integer multiples of ℏ and interactions with the electric field are represented by jumps of ±ℏ/2 in the action at constant angle.

Correspondence between double-sided Feynman diagrams and semiclassical paths from Fig. 1 is illustrated.

Correspondence between double-sided Feynman diagrams and semiclassical paths from Fig. 1 is illustrated.

Two double-sided Feynman diagrams (2FD) are shown with the corresponding optimized mean trajectory (OMT) diagrams. Both 2FD correspond to the same semiclassical path but classical states are collected at different points along the path in the OMT diagrams as indicated by red dots.

Two double-sided Feynman diagrams (2FD) are shown with the corresponding optimized mean trajectory (OMT) diagrams. Both 2FD correspond to the same semiclassical path but classical states are collected at different points along the path in the OMT diagrams as indicated by red dots.

The double-sided Feynman diagrams contributing to signal wavevectors (a) and (b) are shown with their corresponding OMT diagrams. The MT diagrams corresponding to each row are shown in (c). Red dots on the semiclassical diagrams indicate the collection of phase-space information.

The double-sided Feynman diagrams contributing to signal wavevectors (a) and (b) are shown with their corresponding OMT diagrams. The MT diagrams corresponding to each row are shown in (c). Red dots on the semiclassical diagrams indicate the collection of phase-space information.

The phase-space path for a Morse oscillator corresponding to the OMT diagram on the left is shown. Solid lines show portions of classical trajectories used in the calculation, with the remainder of each period indicated with dotted curves. Red dots show classical states along the semiclassical path used to calculate the spectroscopic response.

The phase-space path for a Morse oscillator corresponding to the OMT diagram on the left is shown. Solid lines show portions of classical trajectories used in the calculation, with the remainder of each period indicated with dotted curves. Red dots show classical states along the semiclassical path used to calculate the spectroscopic response.

The real parts of (left-hand panel) and (right-hand panel) are shown for a thermal ensemble of Morse oscillators with βℏω = 2 and β*D* = 40. Quantum mechanical results are shown in plots (a) and (b), the OMT approximation is shown in plots (c) and (d), and the MT approximation is given in plots (e) and (f).

The real parts of (left-hand panel) and (right-hand panel) are shown for a thermal ensemble of Morse oscillators with βℏω = 2 and β*D* = 40. Quantum mechanical results are shown in plots (a) and (b), the OMT approximation is shown in plots (c) and (d), and the MT approximation is given in plots (e) and (f).

The real part of for a thermal ensemble of Morse oscillators with βℏω = 2 and β*D* = 40 is shown. The quantum mechanical result is shown in (a), the OMT approximation in (b), and the MT approximation in (c).

The real part of for a thermal ensemble of Morse oscillators with βℏω = 2 and β*D* = 40 is shown. The quantum mechanical result is shown in (a), the OMT approximation in (b), and the MT approximation in (c).

The frequency dependence of for a thermal ensemble of Morse oscillators with βℏω = 2 and β*D* = 40 is shown on a semilogarithmic plot. The quantum mechanical result is shown in (a), the OMT calculation in (b), and the MT result in (c). Gray dashed vertical lines indicate peaks present in the quantum mechanical response and the red dashed vertical line indicates the position of the spurious ω_{1, −1} peak in the MT response. Dots indicate the area associated with each peak relative to the peak at ω_{2, 0}. Peaks associated with n-quantum coherences are labeled nQ.

The frequency dependence of for a thermal ensemble of Morse oscillators with βℏω = 2 and β*D* = 40 is shown on a semilogarithmic plot. The quantum mechanical result is shown in (a), the OMT calculation in (b), and the MT result in (c). Gray dashed vertical lines indicate peaks present in the quantum mechanical response and the red dashed vertical line indicates the position of the spurious ω_{1, −1} peak in the MT response. Dots indicate the area associated with each peak relative to the peak at ω_{2, 0}. Peaks associated with n-quantum coherences are labeled nQ.

The real part of for a thermal ensemble of quartically perturbed harmonic oscillators with and *a*/(β*m* ^{2}ω^{4}) = 0.025 is shown in the left-hand panel. The frequency spectrum is shown in the right-hand panel. Dots indicate the area of each peak relative to the peak at ω_{2, 0}. Dashed lines indicate the frequencies (vertical) and areas (horizontal) of quantum mechanical peaks in (b). The quantum mechanical system response is shown in plot (a) and the corresponding frequency spectrum in plot (b). The OMT results are shown in plots (c) and (d) and the MT calculations are shown in plots (e) and (f).

The real part of for a thermal ensemble of quartically perturbed harmonic oscillators with and *a*/(β*m* ^{2}ω^{4}) = 0.025 is shown in the left-hand panel. The frequency spectrum is shown in the right-hand panel. Dots indicate the area of each peak relative to the peak at ω_{2, 0}. Dashed lines indicate the frequencies (vertical) and areas (horizontal) of quantum mechanical peaks in (b). The quantum mechanical system response is shown in plot (a) and the corresponding frequency spectrum in plot (b). The OMT results are shown in plots (c) and (d) and the MT calculations are shown in plots (e) and (f).

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