Analyzing Wavepacket Dynamics in Three Dimensions

The 2D resonance Raman spectra presented in Section 4.4.2 do not take full advantage of the experimental implementation, because the signal is integrated over the detection frequency. This third dimension distinguishes the information content of the present experiment from that of FSRS, where the vibrational resonances are directly imprinted on the spectrum of the signal field.27,34,35,37,39 This section explores a representation of the data that takes into account the detection frequency, _t, in addition to the two experimentally controlled delay times. It is envisioned that this perspective on wavepacket motion will be particularly useful for studies of reactive system such as I3- and/or systems with highly anharmonic potentials.

The present experiment can be viewed as a sequence in which the first pulse-pair initiates a wavepacket that is probed with the final three pulses.40-42 Knowledge of the maxima and minima in the signal intensity shown in Figure 4.7a does not give unambiguous information about the wavepacket’s position, because the sign of the gradient on the excited state potential energy surface is undetermined. Dynamics in the signal spectrum must be examined to determine the phase of wavepacket motion. To access this information, we employ a

representation, where the signal is Fourier transformed with respect to ₂ (but not ₁) in Figure 4.10a. As in the data presented above, a baseline is subtracted from the signal in ₂ before Fourier transformation in order to isolate the oscillatory part of the response. Therefore, the (time-evolving) Raman spectrum of the system appears in the conjugate domain, ₂ (i.e.

incoherent relaxation does not contribute). In addition to ₂, the signal is displayed with respect to the detection frequency,_t. This two-dimensional representation of the signal in ₂ and _t

will hereafter be referred to as a “correlation spectrum” and will be denoted as S



 2, t



. The

correlation spectra analyzed in this work focus only on the region of the Raman spectrum associated with the symmetric stretch, because the low-frequency intermolecular mode (near 20 cm-1) observed in the transient grating signals (see Figure 4.7) is below the detection threshold of the six-wave mixing experiment.

The isosurface in Figure 4.10b displays recurrences in the magnitude of the correlation spectrum as a function of ₁. Mean vibrational and emission frequencies are computed at each delay time, ₁, using the formula









2 2 2 2 , , t t j j t t d d S d d S               









(4.19)

where _j represents either ₂ or _t (i.e. motions are characterized in both dimensions). The dynamics in the emission and vibrational frequencies are presented in Figure 4.10c and 4.10d, respectively. The data are fit to a damped sinusoidal function

 

0 1sin





exp





F t A A  t t (4.20)

Figure 4.10. (a) Correlation spectrum, S



 2, t



, measured at 1=100 fs. (b) Isosurface of

signal is drawn at 40% of the maximum intensity. (c) Mean detection frequency, _t , and fit to Equation 4.20. (d) Mean vibrational frequency, ₂ , and fit to Equation 4.20. Noise associated with t and 2 increases with 1, because the signal magnitude decreases. Fitting

parameters are given in Table 4.2.

Recurrences in the mean emission frequency, _t , are readily observed in all data sets. In analogy with related third-order experiments, we suggest that such oscillations reflect the energy gap between electronic states at the position of the wavepacket, which is dynamic in ₁. Transient absorption studies of several systems have demonstrated that recurrences with 180º- phase-shifts may be observed at high and low detection frequencies when the spectral width of a

probe pulse is comparable to the total absorbance line width.90,95 Under these conditions, high and low detection frequencies correspond to opposite turning points on the potential energy surface. Such phase-shifted oscillations in _t are not very pronounced in the current data, because the spectral width of the laser pulse is narrower than the total absorption line width. Nonetheless, as shown in Figure 4.10c, we detect a beat in _t with an amplitude near 20 cm-1. Figure 4.11 presents calculated signals, which are treated using the same data-processing

software used to handle the experimental data shown in Figure 4.10. The calculations confirm that it is indeed reasonable to observe recurrences in _t under our experimental conditions. Moreover, the calculated amplitude of the recurrence (15 cm-1) is similar to that observed experimentally (20 cm-1).

Oscillations in the mean vibrational frequency, 2 , are also observed, although the

amplitude marginally exceeds the detection threshold. With inspiration provided by related techniques,45 we hypothesize that recurrences in ₂ with small amplitudes should be anticipated in systems, such as I3-, with weak ground-state anharmonicities. That is, the

amplitude of 2 reflects variation in the curvature of the potential energy surface at

nonequilibrium geometries traversed by the wavepacket in the delay time, ₁. For example, the second derivative of a one-dimensional potential energy function varies linearly in the mode displacement if a cubic expansion is considered. Consistent with this interpretation, calculations based on a purely harmonic model presented in Figure 4.11d predict an oscillation amplitude in

2

 that is roughly two times smaller than that observed experimentally (see Table 4.2). Key differences between the experiment and model are further illustrated in Figure 4.13, where

(correlated) dynamics in 2 and t are plotted in three dimensions. The “spiral” shape

associated with the measurement is broader in the ₂ dimension than that predicted with the harmonic model. Therefore, we tentatively attribute the larger amplitude of ₂ found in the experimental data to a small amount of anharmonicity of the ground state potential.

Figure 4.11. (a) Correlation spectrum, S



 2, t



, calculated at 1=100 fs. (b) Isosurface of

signal is drawn at 60% of the maximum intensity. (c) Mean detection frequency, _t , and fit to Equation 4.20. (d) Mean vibrational frequency, 2 , and fit to Equation 4.20. Signals are

Correlations in 2 and t are illustrated using the representation shown in Figure

4.12. Anharmonicity contributes in a straightforward way to amplitude in ₂ . However, as in traditional third-order experiments,90,95 oscillatory amplitude in _t depends on the magnitude of the mode displacement, the inhomogeneous line width, and the pulse duration. The interplay between these three factors is fairly well-understood. Amplitude in _t increases with both the mode displacement and heterogeneity in the absorbance line shape. Short (impulsive) laser pulses are required to observe oscillations; however, coherent dynamics must also vanish in the impulsive limit, so an intermediate pulse width is most desirable for this type of technique. Although the present experiment is not sensitive to such effects, we anticipate that more complex (and informative) trajectories in ₂ and _t can be derived from two-color measurements conducted on reactive systems.

Figure 4.12. Dynamics in mean vibrational and emission frequencies, ₂ and _t , adapted from the fits shown in Figures 4.10 and 4.11. The average values of the two variables are shifted by small amounts between (a) experimental and (b) theory. The shapes of the spirals can still be directly compared because the magnitudes of the ranges are identical in the two panels. It is predicted that anharmonicity, which is absent in the model, causes the spiral to expand in the

2

Table 4.2. Dynamics in Correlation Spectra (a) Parameter Measured t  Measured 2  Calculated t  Calculated 2  (b) 0 A 37759±2 cm-1 113.3±0.5 37795±1 cm-1 113.7±0.3 (b) 1 A 26±7 cm-1 2.1±0.3 15±1 cm-1 1.2±0.3  114±3 cm-1 114±5 cm-1 113±1 cm-1 113±1 cm-1

 1.5±0.3 rad. 0.2±0.4 rad. -3.9±0.1 rad. -4.5±0.1 rad.

t 4.3±1.0 ps 2.3±1.0 ps ∞ ∞

(a)

Parameters of Equation 4.20.

(b)

The signal amplitude has arbitrary units.

4.5. Relative Magnitudes of Cascaded Third-Order and Direct Fifth-Order Signals in 2D

In document Molesky_unc_0153D_16297.pdf (Page 153-159)