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.
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 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 βD = 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 βD = 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 βD = 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 ω1, −1 peak in the MT response. Dots indicate the area associated with each peak relative to the peak at ω2, 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/(βm 2ω4) = 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 ω2, 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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