The time-independent view

Raman scattering is a two-photon process and must be described by second-order perturbation theory. The cross section for a transition from state ot(Et) with energy Ei to state of(Ef)) with energy Ef (in the following the indices 0 and 1 will label the lower and the upper electronic state, respectively) is given by the Kramers-Heisenberg-Dirac formula (Kramers and Heisenberg 1925 Dirac 1927 for a sufficiently detailed derivation see, for example, Weissbluth 1978 ch.24) 

The nonresonant term tnonres is identical to the resonant one but with the denominator Ei — tuo — E + T it is naturally much smaller than the resonant term and therefore it will be omitted in the subsequent discussion. In order to simplify the presentation, all unimportant prefactors are disregarded in (14.1). 

The matrix elements (4 i(.E,n) pio f of(Ef)) are just the partial photodissociation amplitudes (2.68) which are required in the calculation of dissociation cross sections for vibrationally excited parent molecules. The actual calculation proceeds in the following way  

1) Calculation of all bound-state wavefunctions of(Ef) in the electronic ground state. 

2) Calculation of continuum wavefunctions 4 i(.E, n) in the excited electronic state for all possible final states n and their overlap with all bound-state wavefunctions for a sufficiently narrow grid of energies. 

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The main characteristics of indirect dissociation are resonances in the time-independent picture and recurrences in the time-dependent approach. Resonances and recurrences are the two sides of one coin they reveal the same dynamical information but provide different explanations and points of view. To begin this chapter we discuss in Section 7.1, on a qualitative level, indirect photodissociation of a one-dimensional system. A more quantitative analysis follows in Section 7.2. The time-dependent and the time-independent views of indirect photodissociation are outlined and illustrated in Sections 7.3 and 7.4, respectively, with emphasis on vibrational excitation of the NO moiety in the photodissociation of CH30N0(S i). Section 7.5 accentuates the relation between... 

The time-independent and time-dependent approaches merely provide different views of the dissociation process and different numerical tools for the calculation of photodissociation cross sections. The time-independent approach is a boundary value problem, i.e., the stationary wavefunction... 

Which approach offers the better interpretation of the photodissociation dynamics in general and of the dissociation cross sections in particular The answer to this question depends very much on the particular system and cannot be decided in a unique way. There are cases which are better described by the time-independent picture while other systems may be better understood within the time-dependent view. In the course of this monograph we will always attempt to combine both views in order to reveal all facets of a particular system. 

Figure 3.2 shows the same trajectory as in Figure 5.1 in a different representation together with the time-independent total dissociation wave-function StotiR,7)- The energy for the trajectory is the same as for the wavefunction. It comes as no surprise that, on the average, the wave-function closely follows the classical trajectory. Another example of this remarkable coincidence between quantum mechanical wavefunctions on one hand and classical trajectories on the other hand is shown in Figure 4.2. There, the time-dependent wavepacket (f) follows essentially the trajectory which starts at the equilibrium geometry of the ground electronic state. This is not unexpected in view of Equations (4.29) which state that the parameters of the center of the wavepacket obey the same equations of motion as the classical trajectory provided anharmonic effects are small. Figures 3.2 and 4.2 elucidate the correspondence between classical and quantum mechanics, at least for short times and fast evolving systems ...

Inherent to the optical matrix description of the reflectivity of neutron beams is the assumption that the neutron beam is a wave. Alternatively and entirely equivalently, the neutron could be viewed as a quantum mechnical particle with the reflection process the result of the interaction of the neutron with a potential energy barrier, V. The reflectivity is then the modulus squared of the reflected wave amplitude, the latter quantity being related to the potential energy barrier by the time-independent Schrodinger wave equation. 

The time-dependent theory of spectroscopy bridges this gap. This approach has received less attention than the traditional time-independent view of spectroscopy, but since 1980, it has been very successfully applied to the field of coordination chemistry.The intrinsic time dependence of external perturbations, for example oscillating laser fields used in electronic spectroscopy, is also expKdtly treated by modern computational methods such as time-dependent density functional theory, a promising approach to the efficient calculation of electronic spectra and exdted-state structures not based on adjustable parameters, as described in Chapter 2.40. In contrast, the time-dependent theory of spectroscopy outlined in the following often relies on parameters obtained by adjusting a calculated spectrum to the experimental data. It provides a unified approach for several spectroscopic techniques and leads to intuitive physical pictures often qualitatively related to classical dynamics. The concepts at its core, time-dependent wave functions (wave packets) and autocorrelation functions, can be measured with femtosecond (fs) techniques, which often illustrate concepts very similar to those presented in the following for the analysis of steady-state spectra. The time-dependent approach therefore unifies spectroscopic... 

In the Schrodinger representation of the latter matrix element in (8.10), the molecular states are regarded as time-dependent basis functions exp( — i fcf/ft) fe> and exp( —i t/ft) w>, and the operator /t is considered to be time-independent. For present purposes, it is more illuminating to use the Heisenberg representation, in which the molecular states are the time-independent basis functions lfc>, m> and the operator is viewed as time-dependent. Since H i> = , i> for each of the molecular states i>> we have... 

The generalized transport equation, equation 17, can be dissected into terms describing bulk flow (term 2), turbulent diffusion (term 3) and other processes, eg, sources or chemical reactions (term 4), each having an impact on the time evolution of the transported property. In many systems, such as urban smog, the processes have very different time scales and can be viewed as being relatively independent over a short time period, allowing the equation to be "spht" into separate operators. This greatly shortens solution times (74). The solution sequence is... 

Clearly, /t0(f) is the total number and is the total volume of the agglomerates in the charge. The latter is approximately independent of time in view of the quasi-steady-state assumption in Eq. (46). Kapur (K3) has shown that an asymptotic similarity solution to Eq. (47) exists in the following form ... 

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