The time-dependent picture

The theory discussed until now is based on the Kramers-Heisenberg-Dirac dispersion relation for the transition polarizability tensor as given in Eq. (6.1-1). The expression shown in this equation describes a steady state scattering process and contains no explicit reference to time. Therefore, the resonance Raman theory which is based on the KHD dispersion relation is sometimes also termed as time-independent theory (Ganz et al., 1990). 

In order to facilitate deduction of excited-state properties (Johnson and Peticolas, 1976 Warshel, 1977) considerable activity ensued in the analysis of resonance Raman spectra and excitation profiles. However, progress was limited by the conceptual and computational clumsiness of Eq. (6.1-1). Only for small molecules it has been possible to calculate resonance Raman spectra numerically by means of Eq. (6.1-1) (see for example Strempel and Kiefer, 1991 a-c and references therein). The main problem arises through the summation over many states which for large polyatomic molecules is nearly impossible to perform. 

The time-dependent formulation of Raman (and particularly resonance Raman) scattering has been introduced by Hizhnyakov and Tehver (1967), Lee and Heller (1979), Heller et al. (1982), Tannor and Heller (1982), and Myers et al. (1982). Its derivation is strikingly simple. Through a straightforward mathematical transformation (see for instance Myers et al., 1982), the KHD relation can be transformed into an integral over time which we will show here for the vibronic part of the polarizability in the Condon approximation. First we rewrite the A-term as given in Eq. (6.1-8) to be 

The denominator s = Ey , E( F in the expression of Eq. (6.1-11) can now be transformed into a time-dependent expression using the mathematical relation 

If the propagator e is considered to operate on the right hand side, then 

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The slowness of the nuclear motion results (in the time-dependent picture) in a small change of the electron-nuclear interaction during the electronic time Te = (ACg )- -, where Aee is the separation of electronic states. The probability of a transition from a nondegenerate electronic state to an excited state is exponentially small. Thus, the quantum numbers n and v are essentially integrals of motion. 

In connection with the time-dependent picture of electronic transition a missing mode effect (MIME) has been postulated [75] trying to explain the vibrational progressions when they are measured in quanta which do not occur in the set of normal vibrational modes in the molecule. It has been shown that the total wave packet < being a product of overlap factors, Eq. (47), of several displaced modes can lead, when Fourier transformed, to a spectrum with a progressional interval which is a mixture of the original normal modes. The spectrum of W(CO)5(py) on which this effect has been exemplified is, however, insufficiently resolved [75] to be used as a proof that the MIME in view of the uncertainty of the damping factor exists in reality. 

Raman Spectroscopy The time-dependent picture of Raman spectroscopy is similar to that of electronic spectroscopy (6). Again the initial wavepacket propagates on the upper excited electronic state potential surface. However, the quantity of interest is the overlap of the time-dependent wavepacket with the final Raman state 4>f, i.e. < f (t)>. Here iff corresponds to the vibrational wavefunction with one quantum of excitation. The Raman scattering amplitude in the frequency domain is the half Fourier transform of the overlap in the frequency domain,... 

In the time-dependent picture, the overall width of the total spectrum is determined by the two slopes dV/dR and dV/dr at the transition point (Re,re). 

In the time-dependent picture, resonances show up as repeated recurrences of the evolving wavepacket. Resonances and recurrences reveal, in different ways, the same dynamical effect, namely the temporary excitation of internal motion within the complex. In the context of classical mechanics, the existence of quantum mechanical resonances is synonymous with trapped trajectories performing complicated Lissajou-type motion before they finally dissociate. The larger the lifetime, the more frequently the wavepacket recurs to its starting position, and the narrower are the resonances. 

The great asset of the time-dependent picture rests on the fact that no reference is made to the (many) stationary continuum states in the excited electronic state. The propagation of a single wavepacket, which contains the entire history of the dynamics in the upper state, thus yields the absorption cross section, all Raman cross sections, and the final state distributions of the photofragments. 

So far we have invoked the time-independent formulation to describe electronic transitions. In the same manner as described in Section 4.1 we can also derive the time-dependent picture of electronic transitions, using either the adiabatic or the diabatic representation. In the following we feature the latter which is more convenient for numerical applications (Coalson 1985, 1987, 1989 Coalson and Kinsey 1986 Heather and Metiu 1989 Jiang, Heather, and Metiu 1989 Manthe and Koppel 1990a,b Broeckhove et al. 1990 Schneider, Domcke, and Koppel 1990 Weide, Staemmler, and Schinke 1990 Manthe, Koppel, and Cederbaum 1991 Heumann, Weide, and Schinke 1992). 

In the present context, it is relevant to consider the barrier penetration that is associated with the traditional (one-dimensional) picture of tunneling. When we consider the time-dependent description of tunneling where a (broad) wave packet hits, e.g., a rectangular barrier, one finds that the center of the wave packet moves as a classical particle. The part of the packet that penetrates the barrier and tunnels through is not slowed down, i.e., it has exactly the same position and velocity as a wave packet that did not experience a barrier (see also [7] for a general discussion of the time-dependent picture of tunneling). The classical transmission probability of Eq. (6.15) is now replaced by the quantum mechanical transmission probability Pqm(E) (see Fig. 6.4.2). Thus, as a natural extension of the conventional formulation based on classical mechanics, in the derivation above we replace Pc by Pqm. That is, we can replace Eq. (6.21) by... 

Figure 6.1-2 The time-dependent picture of resonance Raman scattering panel A interaction of the incident photon with the electronic transition moment sends the initial vibrational state / > to the excited electronic surface, where it is propagated by the excited-state vibrational Hamiltonian panel B the Raman overlap < /li(i) as a...

In the time-dependent picture, since F=0, Eq. (6.1-17) then reduces to... 

The quantum dynamics of photodissociation processes can be performed within the time-independent or time-dependent framework. In the time-dependent picture used in the applications presented in the next section, the time-dependent Schrodinger equation is solved... 

Photodissociation processes can be studied in the time-independent as well as the time-dependent picture. In the first approach, one solves the time-independent Schrodinger equation for the nuclear motion at a fixed energy E. The correct boundary conditions for large separations between the products are fulfilled and different wave functions are obtained for each quantum state a of the products.In the second approach, the time-dependent Schrodinger equation for the nuclear motion is solved for a wave packet xd) with the appropriate initial condition for time t = 0. Both approaches are equivalent, provided the Hamiltonian is time-independent. 

For the time-dependent picture, Equation 7.36 serves quite well. The probability for a certain continuum state, Cm(t), increases with time. This means that the probability of a state of type (Is oos) also increases with time. The latter state is a so-called Auger state. Thus, the (2s) state decays by moving one electron to Is and ejecting the other. The total energy of the states before and after is the same. 

On the theoretical side, especially for molecular systems, a major step forward was made when E. Heller reformulated the time dependent picture for applications in spectroscopy and J. Kinsey and D. Imre described their novel dynamical Raman experiments in terms of the wave packet theory. The progress was greatly helped by advances made in the theoretical execution and speed of computation... 

The general understanding of molecular dynamics rests mainly upon classical mechanics this holds true for full bimolecular collisions (see Trajectory Simulations of Molecular Collisions Classical Treatment) as well as half-collisions, i.e., the dissociation of a parent molecule into different products. The classical picture of photodissociation closely resembles the time-dependent picture the electronic transition from the ground to the excited electronic state is assumed to take place instantaneously so that the internal coordinates (Qi) and corresponding momenta (/, ) of the parent molecule remain unchanged during the excitation step (vertical transition). After the molecule is promoted to the PES of the upper state it starts to move subject to the classical equations of motion (Hamilton s equations)... 

Having decided to work within the time-dependent picture, we notice that solving the quantum mechanical molecular dynamics problem involves three steps ... 

It is intriguing to think about isotope effects in the time-dependent picture of a grazing trajectory. In substituting a heavier isotope, the potential surface remains the same, but the vibrational wavepacket will contract in certain dimensions, making (j) a smaller "target" for <()(t), which is itself smaller and spreading slower. 

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