Wavepacket width

Rq and Pq are the same quantities as those defined in Sec. 6.6.1 for the path-branching representation. The wavepacket width in coordinate space is set to AR = 1. The number of grid points for FFT is 512 on the space... 

We now proceed to some examples of this Fourier transfonn view of optical spectroscopy. Consider, for example, the UV absorption spectnun of CO2, shown in figure Al.6.11. The spectnuu is seen to have a long progression of vibrational features, each with fairly unifonu shape and width. Wliat is the physical interpretation of tliis vibrational progression and what is the origin of the width of the features The goal is to come up with a dynamical model that leads to a wavepacket autocorrelation fiinction whose Fourier transfonn... 

Here and are real tirne dependent quantities that specify the average position and momentum of the wavepacket p = (p), x = (v)) and and are complex fiinctions which detennine the width, phase and nomialization of the wavepacket. 

In an electron scattering or recombination process, the free center of the incoming electron has the functions Wi = ui U u, and the initial state of the free elechon is some function v/ the width of which is chosen on the basis of the electron momentum and the time it takes the electron to aiTive at the target. Such choice is important in order to avoid nonphysical behavior due to the natural spreading of the wavepacket. 

The fundamental method [22,24] represents a multidimensional nuclear wavepacket by a multivariate Gaussian with time-dependent width niaUix, A center position vector, R, momentum vector, p and phase, y,... 

The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

One drawback is that, as a result of the time-dependent potential due to the LHA, the energy is not conserved. Approaches to correct for this approximation, which is valid when the Gaussian wavepacket is narrow with respect to the width of the potential, include that of Coalson and Karplus [149], who use a variational principle to derive the equations of motion. This results in replacing the function values and derivatives at the central point, V, V, and V" in Eq. (41), by values averaged over the wavepacket. 

Figure 38. Time variation of the wavepacket population on the ground X state and the excited A state of NaK. The system is excited by a quadratically chirped pulse with parameters otm = 3.13 X lO eVfs, (5 = 1.76eV, and I = 0.20TWcm . The pulse is centered at r = 0 and has a temporal width t = 20 fs. Taken from Ref. [37].

Figure 39. Pump-dump control of NaK molecule by using two quadratically chirped pulses. The initial state taken as the ground vibrational eigenstate of the ground state X is excited by a quadratically chirped pulse to the excited state A. This excited wavepacket is dumped at the outer turning point at t 230 fs by the second quadratically chirped pulse. The laser parameters used are = 2.75(1.972) X 10-2 eVfs- 1.441(1.031) eV, and / = 0.15(0.10)TWcm-2 for the first (second) pulse. The two pulses are centered at t = 14.5 fs and t2 = 235.8 fs, respectively. Both of them have a temporal width i = 20 fs. (See color insert.) Taken from Ref. [37].

In this expression, Erf denotes the error function, while M2 is the full width at half maximum (FWHM) of the Gaussian probe pulse. The calculation of the total ionization probability S(t) therefore only requires the knowledge of the excited state wavepacket x e(r,R, f) at time t = T. Note that the origin of time (t = 0) is chosen here as the peak intensity of the pump pulse, and consequently the quantity Tin Eq. (1) corresponds to the pump-probe delay. 

We used short broadband pump pulses (spectral width 200 cm 1, pulse duration 130 fs FWHM) to excite impulsively the section of the NH absorption spectrum which includes the ffec-exciton peak and the first three satellite peaks [4], The transient absorbance change signal shows pronounced oscillations that persist up to about 15ps and contain two distinct frequency components whose temperature dependence and frequencies match perfectly with two phonon bands in the non-resonant electronic Raman spectrum of ACN [3] (Fig. 2a, b). Therefore the oscillations are assigned to the excitation of phonon wavepackets in the ground state. The corresponding excitation process is only possible if the phonon modes are coupled to the NH mode. Self trapping theory says that these are the phonon modes, which induce the self localization. 

The peak shift data in Fig. 17 show oscillatory character, as is our first two examples (I2 and LH1). This arises from vibrational wavepacket motion. In addition, the very fast drop in peak shift to about 65% of the initial value in -20 fs results from the interference between the wavepackets created in different intramoleculear modes. This conclusion follows directly from obtaining the frequencies and relative coupling strengths of the intramolecular modes from transient grating studies of IR144, carried out in the same solvents (data not shown). Thus, by visual inspection of Fig. 17, an answer to a long-standing question—What fraction of the spectral width arises from intra- and intermolecular motion —is immediately apparent. 

The physical insight obtained from the time-domain point of view allows simple qualitative predictions about the widths of the progressions to be made. Two potential surfaces (such as the coupled and uncoupled surfaces in Fig. 2) can be compared in terms of the slope of steepest descent at the position of the initial wavepacket at t = 0. The steeper the slope, the faster the initial decrease of < (j) 10(t) > in the time domain and the broader the spectrum in the frequency domain. This insight is important in developing a strategy to fit experimental spectra. In the example above, a vibronic progression in an experimental spectrum that is broader than that expected from a harmonic potential surface (Fig. 3a) requires that the wavepacket be displaced in the positive Qx direction in the coupled potential surface. 

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