Wavepacket autocorrelation function

To illustrate the potential of the hybrid method in describing the role of an intramolecular bath in the decay dynamics induced by a conical intersection, we consider the model of Ref. [7,8] for the S2-S1 Cl in pyrazine. Fig. 1 shows the wavepacket autocorrelation function C(t) = ( k(O)l (t) for an increasing number of bath modes. G-MCTDH hybrid calculations for 4 core (primary) modes plus nb bath (secondary) modes are compared with reference calculations by the standard MCTDH method. 

D. J. Tannor Prof. Rabitz, from the time-domain formulation of conventional electronic absorption spectroscopy, we know that the information content in the wavepacket autocorrelation function is identical to that in the high-resolution spectrum. Yet it is clear that the wavepacket autocorrelation function only directly probes the times at... 

Many of the previously described conclusions were anticipated by Jortner and Kommandeur,23 using a different formalism. Jortner and Kommandeur identify the initial decay of the wavepacket autocorrelation function with free induction decay and note that later recurrences of the autocorrelation... 

We now proceed to some examples of this Fourier transform view of optical spectroscopy. Consider, for example, the UV absorption spectrum of CO2, shown in figure Al. 6,11. The spectrum is seen to have a long progression of vibrational features, each with fairly uniform shape and width. What is the physical interpretation of this vibrational progression and what is the origin of the width of the features The goal is to come up with a d5mamical model that leads to a wavepacket autocorrelation function whose Fourier transform... 

Fig. 15.3 (a) Time dependence of the S2 state (diabatic) population for 4(system)+20(bath)-mode models based on the lowest-order bath spectral densities M = 1,2,3, as weU as the reference spectral density of Ref. [39]. The 4-mode (system) dyntimics is also shown for comparison (slowly decaying trace), (b) Time dependence of the wavepacket autocorrelation function, (v (0) vr(t)). At the level of the M = 3 approximation, the dynamics is indistinguishable from the reference dynamics (Adapted from Ref [39])... 

The time-dependent quantity in the integrand of Eq. (4.24), (,(r, R, f = 0) ,(r, R, f)), is called the autocorrelation function. It is the integral over all space of the product of the initial wavepacket with the wavepacket at time t. Rama Krishna and Coalson [86] have shown that the Fourier transform over time in Eq. (4.24) can be replaced by twice the half-Fourier transform where the time integral mns from f = 0 to f = oo. Using this result we obtain the final expression ... 

The first relevant quantity required to obtain the rates is the autocorrelation function which are shown in Fig.8 for the ground vibrational level of the two excited electronic states. The two cases present a very similar behavior. Simply, for the A case its decay seems much faster. What is notorious is the large difference between the EP halfwidths as a function of the energy for the two electronic states, of approximately 2-3 orders of magnitude, as shown in Fig.9. This is explain by the norm of the initial wavepackets, which is much smaller for the B state, because its well is at larger R and shorter r, where the non-adiabatic couplings are much smaller. 

Fig. 1. The autocorrelation function C(t) = (U (O)l I (i) is shown for a wavepacket initially prepared on the upper diabatic surface [7]. Panels (a) and (b) C(t) for the four core modes calculated by the standard MCTDH method for the model Hamiltonian Hy of Eq. (9), shown on different scales in the two panels. Panel (c) G-MCTDH calculation (bold line) as compared with standard MCTDH calculation (dotted line) for the composite system with four core modes (combined into two 2-dimensional particles

Recent advances in ultrashort laser technology has enabled us to investigate dynamics of molecules in a time domain, and furthermore, the success of a theoretical interpretation of the results of time-domain experiments by a moving wavepacket on a potential energy surface (PES) impressively demonstrated the importance of time-domain experiments [1]. On the other hand, it is well-known that a spectrum in a frequency domain and an autocorrelation function in the time domain can be transferred with each other via a Fourier transformation [2]. Therefore, it can be said that the spectrum... 

The decrease in < 0 < (t) > depends on the slope of the potential surface at the point at which the wavepacket is initially placed. The slope of the potential in the Qy direction is steeper on the positive Qx side of the surface than on the negative Qx side. When the slope in the Qy dimension is large, the Qy part of the two-dimensional wavepacket will rapidly change its shape and < 010 (t) > will decrease rapidly. Therefore, for a positive Qx displacement in the coupled potential, the autocorrelation function will decrease more rapidly than it would for a negative displacement. 

In practice one does not proceed as we did in the above derivation. Instead of calculating first all stationary wavefunctions and then constructing the wavepacket according to (4.3), one solves the time-dependent Schrodinger equation (4.1) with the initial condition (4.4) directly. Numerical propagation schemes will be discussed in the next section. Since 4 /(0) is real the autocorrelation function fulfills the symmetry relation... 

Fig. 4.3. Time-dependence of the autocorrelation function S(t) (real part) of the wavepacket shown in Figure 4.2...

Each rotational state is coupled to all other states through the potential matrix V defined in (3.22). Initial conditions Xj(I 0) are obtained by expanding — in analogy to (3.26) — the ground-state wavefunction multiplied by the transition dipole function in terms of the Yjo- The total of all one-dimensional wavepackets Xj (R t) forms an R- and i-dependent vector x whose propagation in space and time follows as described before for the two-dimensional wavepacket, with the exception that multiplication by the potential is replaced by a matrix multiplication Vx-The close-coupling equations become computationally more convenient if one makes an additional transformation to the so-called discrete variable representation (Bacic and Light 1986). The autocorrelation function is simply calculated from... 

In the time-independent approach one has to calculate all partial cross sections before the total cross section can be evaluated. The partial photodissociation cross sections contain all the desired information and the total cross section can be considered as a less interesting by-product. In the time-dependent approach, on the other hand, one usually first calculates the absorption spectrum by means of the Fourier transformation of the autocorrelation function. The final state distributions for any energy are, in principle, contained in the wavepacket and can be extracted if desired. The time-independent theory favors the state-resolved partial cross sections whereas the time-dependent theory emphasizes the spectrum, i.e., the total absorption cross section. If the spectrum is the main observable, the time-dependent technique is certainly the method of choice. 

The rapid decay of the autocorrelation function at very short times is mainly due to a dephasing of the wavepacket rather than a displacement in coordinate space. 

The periodic motion of the wavepacket in the potential well naturally shows up in the autocorrelation function S(t) as depicted schematically in Figure 7.3. Each return of (<) to its origin leads to a maximum in the autocorrelation function, a so-called recurrence. Since the part that is temporarily trapped in the inner region gradually diminishes the overall amplitude of S(t) decays in time. Eventually, the entire wavepacket leaks... 

Fig. 7.3. Autocorrelation function for the temporarily trapped wavepacket illustrated in Figure 7.2.

In this section we consider indirect photodissociation of systems with more than one degree of freedom in the time-dependent approach. We will use the results of Section 7.2 to derive approximate expressions for the wavepacket evolving in the upper electronic state, the corresponding autocorrelation function, and the various photodissociation cross sections. 

If we switch on the coupling to the continuum at t = 0 the excited bound states begin to decay with the consequence that the wavepacket and therefore the autocorrelation function decay too. In order to account for this we multiply, according to Equation (7.14), each term in (7.18) by... 

The transition from direct to indirect photodissociation proceeds continuously (see Figure 7.21) and therefore there are examples which simultaneously show characteristics of direct as well as indirect processes the main part of the wavepacket (or the majority of trajectories, if we think in terms of classical mechanics) dissociates rapidly while only a minor portion returns to its origin. The autocorrelation function exhibits the main peak at t = 0 and, in addition, one or two recurrences with comparatively small amplitudes. The corresponding absorption spectrum consists of a broad background with superimposed undulations, so-called diffuse structures. The broad background indicates direct dissociation whereas the structures reflect some kind of short-time trapping. 

The quantum mechanical wavepacket closely follows the main classical route. It slides down the steep slope, traverses the well region, and travels toward infinity. A small portion of the wavepacket, however, stays behind and gives rise to a small-amplitude recurrence after about 40-50 fs. Fourier transformation of the autocorrelation function yields a broad background, which represents the direct part of the dissociation, and the superimposed undulations, which are ultimately caused by the temporarily trapped trajectories (Weide, Kiihl, and Schinke 1989). A purely classical description describes the background very well (see Figure 5.4), but naturally fails to reproduce the undulations, which have an inherently quantum mechanical origin. 

To understand the development or the absence of reflection structures one must imagine — in two dimensions — how the continuum wavefunction for a particular energy E overlaps the various ground-state wave-functions and how the overlap changes with E. This is not an easy task Figure 9.9 shows two examples of continuum wavefunctions for H2O. Alternatively, one must imagine how the time-dependent wavepacket, starting from an excited vibrational state, evolves on the upper-state PES and what kind of structures the autocorrelation function develops as the wavepacket slides down the potential slope. 

The wavepacket in the lBi state, 3>b, performs large-amplitude symmetric stretch motion leading to recurrences in the autocorrelation function the recurrences in turn cause vibrational structures in the absorption spectrum. 

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