Stationary wavefunction

Instead of the time-dependent description of the photoelectron wavefunction it is easier and correct to use stationary wavefunctions P(K )(r). The boundary condition for the wave packets then transforms into a boundary condition in the distance coordinate r and requires (for details see [Sta82a])... 

Equ. (7.20) describes for an out -state the asymptotic behaviour of the stationary wavefunction. As discussed above, the characteristic property of this state is that the incoming spherical waves e Kr/r have the scattering amplitude /(-)( ). It is this minus sign in the exponential term of the incoming spherical waves which is kept as a superscript to characterize the out -state, and the relation described by equ. (7.20) is frequently called the incoming spherical waves boundary condition. Hence, one should not mix up the state with the waves. 

Configuration interaction and stationary wavefunctions In order to simplify the treatment, a correlated wavefunction P(r)corr built by Cl between two uncorrelated wavefunctions will be considered. (To shorten the notations, the tilde describing the antisymmetric character of the wavefunctions and the spin of the electron have been omitted, and the spatial vectors of all electrons are indicated by the symbol r only.) One has... 

Because the (Oj values are different for different basis functions j, the time-dependent correlated wavefunction built from such basis functions cannot be a stationary wavefunction of the full Hamiltonian. Instead, the stationary... 

The stationary wavefunctions Ea(Q,q) solve the time-independent Schrodinger equation... 

In Section 2.1 we derived the expression for the transition rate kfi (2.22) by expanding the time-dependent wavefunction P(t) in terms of orthogonal and complete stationary wavefunctions Fa [see Equation (2.9)]. For bound-free transitions we proceed in the same way with the exception that the expansion functions for the nuclear part of the total wavefunction are continuum rather than bound-state wavefunctions. The definition and construction of the continuum basis belongs to the field of scattering theory (Wu and Ohmura 1962 Taylor 1972). In the following we present a short summary specialized to the linear triatomic molecule. 

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... 

The wavepacket /(t), on the other hand, is constructed in a completely different way. In view of (4.4), the initial state multiplied by the transition dipole function is instantaneously promoted to the excited electronic state. It can be regarded as the state created by an infinitely short light pulse. This picture is essentially classical (Franck principle) the electronic excitation induced by the external field does not change the coordinate and the momentum distributions of the parent molecule. As a consequence of the instantaneous excitation process, the wavepacket /(t) contains the stationary wavefunctions for all energies Ef, weighted by the amplitudes t(Ef,n) [see Equations (4.3) and (4.5)]. When the wavepacket attains the excited state, it immediately begins to move under the influence of the intramolecular forces. The time dependence of the excitation of the molecule due to the external perturbation and the evolution of the nuclear wavepacket /(t) on the excited-state PES must not be confused (Rama Krishna and Coalson 1988 Williams and Imre 1988a,b)... 

The calculation of cross sections, which are defined in the limit of an infinitely long laser pulse, by means of a wavepacket, which is created by an infinitely short pulse, seems contradictory on the first glance. On the other hand, note that the wavepacket created by the infinitely short light pulse is merely a coherent superposition of all stationary wavefunctions with expansion coefficients proportional to the amplitudes t(Ef,n). 

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... 

Fig. 7.2. Cartoon of the evolution of a one-dimensional wavepacket, (t), in a potential with barrier. Remember that the horizontal line does not represent a particular energy The wavepacket is a superposition of stationary wavefunctions for a whole range of energies.

Fig. 7.4. Schematic illustration of the stationary wavefunction P(E) for energies off and on resonance. In contrast to Figure 7.2, here the horizontal line marks a particular energy.

In order to calculate the absorption spectrum in the time-independent approach one solves the time-independent Schrodinger equation for a series of total energies and evaluates the overlap of the total continuum wavefunction, defined in (2.70), with the bound wavefunction of the parent molecule, ( tot(E) Pio I o(Ei)). Any structures in the spectrum are thus related to the energy dependence of the stationary wavefunction "Jftot(E). As illustrated schematically in Figure 7.4 for the one-... 

The time-dependent wavepacket accumulates in the inner region of the PES while it oscillates back and forth in the shallow potential well as illustrated in Figure 7.8. This vibrational motion leads to an increase of the stationary wavefunction in the inner region, however, only if the energy E is in resonance with the energy of a quasi-bound level. If, on the other hand, the energy is off resonance, destructive interference of contributions belonging to different times causes cancelation of the wavefunction. 

Thus, the stationary wavefunction becomes on resonance essentially a bound-state wavefunction with an amplitude which is proportional to the corresponding lifetime. The larger the survival time in the well region the larger is the magnitude of the stationary wavefunction. 

Resonances in half and in full collisions have exactly the same origin, namely the temporary excitation of quasi-bound states at short or intermediate distances irrespective of how the complex was created. In full collisions one is essentially interested in the asymptotic behavior of the stationary wavefunction L(.E) in the limit R —> 00, i.e., the scattering matrix S with elements Sif as defined in (2.59). The S-matrix contains all the information necessary to construct scattering cross sections for a transition from state i to state /. In the case of a narrow and isolated resonance with energy Er and width hT the Breit- Wigner expression... 

According to Section 4.1.1 the wavepacket is a superposition of stationary wavefunctions corresponding to a relatively wide range of energies. This and the superposition of three apparently different types of internal vibrations additionally obscures details of the underlying molecular motion that causes the recurrences. A particularly clear picture emerges, however, if we analyze the fragmentation dynamics in terms of classical trajectories. 

The excited complex breaks apart very rapidly and only a minor fraction performs, on the average, one single internal vibration. Therefore, the total stationary wavefunction does not exhibit a clear change of its nodal structure when the energy is tuned from one peak to another (Weide and Schinke 1989). In the light of Section 7.4.1 we can argue that the direct part of the total wavefunction, S dir-, dominates and therefore obscures the more interesting indirect part, Sind- The superposition of the direct and the indirect parts makes it difficult to analyze diffuse structures in the time-independent approach. In contrast, the time-dependent theory allows, by means of the autocorrelation function, the separation of the direct and resonant contributions and it is therefore much better suited to examine diffuse structures. 

Periodic orbits also explain the long-lived resonances in the photodissociation of CH.30N0(S i), for example, which we amply discussed in Chapter 7. But the existence of periodic orbits in such cases really does not come as a surprise because the potential barrier, independent of its height, stabilizes the periodic motion. If the adiabatic approximation is reasonably trustworthy the periodic orbits do not reveal any additional or new information. Finally, it is important to realize that, in general, the periodic orbits do not provide an assignment in the usual sense, i.e., labeling each peak in the spectrum by a set of quantum numbers. Because of the short lifetime of the excited complex, the stationary wavefunctions do not exhibit a distinct nodal structure as they do in truly indirect processes (see Figure 7.11 for examples). 

Fig. 9.9. Contour plot of the potential energy surface of H2O in the AlB state the bending angle is fixed at 104°. Superimposed are the total stationary wavefunctions I tot( ) defined in (2.70). The total energies are —2.6 eV and -2.0 eV corresponding to wavelengths of A = 180 nm and 165 nm, respectively. Energy normalization is such that E = 0 corresponds to three ground-state atoms.

The final rotational state distributions of NO can be qualitatively interpreted as a reflection of the stationary wavefunction at the transition state mediated by the dynamics in the exit channel. Figure 10.13 depicts the total stationary wavefunctions 7 E) corresponding to the en-... 

Cuts of the stationary wavefunctions along the transition line, shown on the left-hand side of Figure 10.15 and denoted by 4 ts(70)> clearly manifest that ... 

The electronic wavefunctions depend — like the potentials — parametrically on all nuclear degrees of freedom (R, r). The stationary wavefunctions (including all electronic and all nuclear degrees of freedom) which represent the initial and the final molecular states for the situation sketched in Figure 15.1, are written in full detail as... 

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... 

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