Time-dependent wavepacket theory

Exciting new developments, not discussed in the review are the extension of time-dependent wavepacket reactive scattering theory to full dimensional four-atom systems [137,199-201], the adaptation of the codes to use the power of parallel computers [202], and the development of new computational techniques for acting with the Hamiltonian operator on the wavepacket [138]. 

Chapter 3 treats nuclear motions on the adiabatic potential energy surfaces (PES). One of the most powerful and simplest means to study chemical dynamics is the so-called ab initio molecular dynamics (or the first principle dynamics), in which nuclear motion is described in terms of the Newtonian d3mamics on an ab initio PES. Next, we review some of the representative time-dependent quantum theory for nuclear wavepackets such as the multiconfigurational time-dependent Hartree approach. Then, we show how such nuclear wavepacket d3mamics of femtosecond time scale can be directly observed with pump>-probe photoelectron spectroscopy. 

There are two classes in applications of quantum nuclear dynamics one is the stationary-state scattering theory to treat reactive scattering (chemical reactions), and the other is time-dependent wavepacket method. Here... 

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

Time-independent and time-dependent theories are not really separate disciplines. This should be clear from the work of Kouri [188,189] and Althorpe [136,158], who use time-independent wavepacket techniques. These are easily derived from the more natural time-dependent versions by Fourier transforming the propagator over time. This is equivalent to transforming from the time domain to the energy domain at the beginning rather than the end of the calculation. 

In Section 4.1 we will use the time-independent continuum basis 4//(Q E,0), defined in Section 2.5, to construct the wavepacket in the excited state and to derive (4.2). Numerical methods are discussed in Section 4.2 and quantum mechanical and semiclassical approximations based on the time-dependent theory are the topic of Section 4.3. Finally, a critical comparison of the time-dependent and the time-independent approaches concludes this chapter. 

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. 

Using expansion (16.2) for the wavepacket in terms of the stationary wavefunctions we can derive a set of coupled equations for the expansion coefficients au(t) similar to (2.16). In the limit of first-order perturbation theory [see Equation (2.17)] the time dependence of each coefficient is then given by... 

Figure 4. Illustration of the time-dependent theory of resonance Raman scattering for a nontotally symmetric mode. (j> is the initial wavepacket, (t) is the moving wavepacket, (ft, is the final state of interest, and A is the displacement of the upper potential surface along the normal coordinate Qj.

Recently, in this laboratory, we have applied time-dependent quantum mechanics-wavepacket dynamics to several bona fide time-domain spectroscopies. Specifically, we have formulated time-dependent theories of coherent-pulse-seque nee-induced control of photochemical reaction, picosecond CARS spectroscopy, and photon echoes. These processes all involve multiple pulse sequences in which the pulses are short or comparable in time scale to the... 

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