Multichannel wavefunction

It is possible to implement explicitly Siegert boundary conditions in the determination of the multichannel wavefunction. An alternative approach is to transform the reaction coordinate R into a complex one R = p exp i )... 

For the propagation of the multichannel wavefunction 4>(R), in real or complex-scaled coordinates, an efficient algorithm is furnished by fhe Fox-Goodwin-Numerov method [8, 44], which results from a discrefizafion of the differential operator appearing in Eqs. (39). Given adjacent points R — h, R, and R + h on the grid, we define an inward mafrix (labeled /) and an outward matrix (labeled o) as ... 

Once the complex Floquet eigenenergy has been determined from an iter-afive resolution of the implicit energy-dependent Eq. (53), the multichannel wavefunction written as a column vector T (R) can be calculated at the matching point because it satisfies the set of homogeneous linear equations ... 

The theory presented below describes ARPA in two situations (1) an initial continuum of states and (2) an initial (dense) set of bound states. In either case the PA yield is determined by the projection of the incoming multichannel wavefunction onto a... 

Takatsuka and Gordon (21a) have developed a "full collision" formulation of photodissociation which describes a multichannel process on the repulsive surface for both direct and indirect events. The scattering wavefunctions that are used to generate the T-matrix and the FC overlaps are not zeroth-order uncoupled functions, but solutions of the coupled-channel problem. 

The asymptotic wavefunctions obtained from multichannel scattering calculations provide the S matrix, the eigenphases and eigenphase sum, and the cross sections. In principle, any of these quantities may be chosen for fitting the resonance formula to determine accurate values of the resonance parameters E, and T. 

It should be born in mind that our discussion now centers on extracting the resonance position Er and the total width T from the much richer scattering information that the S and Q matrices contain. Multichannel continuum wavefunctions are usually calculated for more general purposes of obtaining the scattering amplitudes and the cross sections for various state-to-state processes and of unraveling the dynamics of the whole continuum system including both resonance and nonresonance mechanisms and the intricate interference between them. 

An alternative method to obtain the nonadiabatic wavefunctions [Eq. (4.1.1)], the coupled equation approach, will be discussed in Section 4.4.3. It has been used for an excited 1E+ state of H2 and the error is now smaller than 1 cm-1 for the lowest vibrational levels (Yu and Dressier, 1994). Multichannel Quantum Defect Theory (MQDT), discussed in Chapter 8, has also been used with success for the same problem by Ross and Jungen (1994). Finally, a variational numerical approach (Wolniewicz, 1996), gives very good results for H2. 

The facts to which the above two paragraphs refer, suggest that, at least concerning the use of square-integrable functions for the calculations of resonance states, alternative theories are needed. Indeed, the CESE-SSA, whose basic elements and characteristics are reviewed here, is structured so as to allow the practical and efficient computation of the MEP in electronic structures, the multichannel continuum and partial widths, and, in general, the production of easily usable wavefunctions that contain the information that is relevant to the state and property of interest. 

Obviously, there is much room for further development of the basic concepts of the SSEA and for improvement of its methodology, as well as for additional applications to new and challenging TDMEPs. In all cases, the fundamental issue is how to identify and construct the wavefunctions that are considered relevant to each problem. For example, the possibility of treating correctly the contribution from two-electron continue is an open question. Even if two-electron products of energy-normalized scattering states are used as basis sets, the computational requirements of this (multichannel in general) problem are huge, and so its solution would require dedicated effort and powerful computers. 

Provided the interaction potential is sufficiently attractive (which it almost always is), the multichannel Schrodinger equation supports bound states at energies below the lowest threshold. The corresponding wavefunction for state n is... 

Propagating wavefunctions explicitly in the presence of deeply closed channels is notoriously numerically unstable. It is much more satisfactory to use a propagator that is stable in the presence of closed channels, such as one of the log-derivative propagators described above. The multichannel matching condition can be expressed very simply in terms of the log-derivative matrix F(r) of Equation 1.52. If E is an eigenvalue of the coupled equations, there must exist a wavefunction vector tn( mid) = (fmid) = t IT ( mid) for which... 

In this chapter, we have addressed a question of some controversy in the past [1,7, 11,12]. Our work elucidates what, exactly, is transferred from the incoming channels during an ARPA process. We have shown that ARPA projects the incoming wavefunction onto a certain multichannel waveform, whose shape is determined by the amplitudes and the phases of the pump and dump laser pulses (Equations 8.40 and 8.41). In this way one can choose the basis of the projective measurement performed by the PA process. We have also discussed the role of temperature of the initial sample in the ARPA process and the extent to which it can limit the process. 

We have shown that the the multichannel structure of the wavefunction can be measured with the help of a numerical parameter search based on the knowledge of the level couplings and decay rates. Alternatively, one can use intermediate levels coupled to only one (say, fl E+) incoming channel and not to the other(s). 

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