Asymptotic wavefunctions

Determination of the symmetry properties of the asymptotic wavefunction, however, is not sufficient to treat SIKIE, because the critical question is whether the symmetry of a particular affects the propensity to connect with a particular... 

Figure 1. Cartoon depicting symmetry considerations of the asymptotic wavefunction that influences which potential energy surface is accessed. Representative angular momentum quantum numbers relevant to the process are the total molecular angular momentum exclusive of nuclear spin (f) and the relative orbital angular momentum (/).

The first part of the review deals with aspects of photodissociation theory and the second, with reactive scattering theory. Three appendix sections are devoted to important technical details of photodissociation theory, namely, the detailed form of the parity-adapted body-fixed scattering wavefunction needed to analyze the asymptotic wavefunction in photodissociation theory, the definition of the initial wavepacket in photodissociation theory and its relationship to the initial bound-state wavepacket, and finally the theory of differential state-specific photo-fragmentation cross sections. Many of the details developed in these appendix sections are also relevant to the theory of reactive scattering. 

RESONANCE ANALYSIS INFORMATION FROM ASYMPTOTIC WAVEFUNCTIONS... 

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. 

Given this, the asymptotic wavefunction must be a linear combination of the form... 

This is a special solution for the full scattering problem in the trivial case where the asymptotic scattering wavefunction is not changed at all by the interaction potential, i.e., for the case of zero final state interaction. This implies that the full scattering solution can be replaced by the asymptotic wavefunction f>. This yields the state to state photodissociation cross section in the FC-limit ... 

With the scheme of PSANB, one can calculate the nonadiabatic transition probability as follows. For the sake of simpler explanation, we describe the procedure for the systems of two electronic states. Suppose we have an asymptotic wavefunction irrespective of their history of branching such that... 

Fig. 2. Schematic configuration space for the reaction AB + CD — A + BCD. R is the radial coordinate between the center-of-mass of the two diatoms, and r is the vibrational coordinate of the reactive AB diatom. I denotes the interaction region and II denotes the asymptotic region. The shaded regions are the absorption zones for the time-dependent wavefunction to avoid boundary reflections. The reactive flux is evaluated at the r = rB surface.

An application of the variational principle to an unbounded from below Dirac-Coulomb eigenvalue problem, requires imposing upon the trial function certain conditions. Among these the most important are the symmetry properties, the asymptotic behaviour and the relations between the large and the small components of the wavefunction related to the so called kinetic balance [1,2,3]. In practical calculations an exact fulfilment of these conditions may be difficult or even impossible. Therefore a number of minimax principles [4-7] have been formulated in order to allow for some less restricted choice of the trial functions. There exist in the literature many either purely intuitive or derived from computational experience, rules which are commonly used as a guidance in generating basis sets for variational relativistic calculations. 

Although numerical values of and (p >, as well as p(0) can be easily obtained for near Hartree-Fock wavefunctions [16,17], some effort has been done in order to study the general features of these quantities as a function of the atomic number [18,19,20], The easiest way to estimate these comes from the TF model [21,22], and has the important property of providing the exact asymptotic behaviour at large Z,... 

We have recently discovered [19] a new exact formula for the asymptotic behavior of the wavefunction of an atom or molecule, when one coordinate... 

The asymptotic form of the body-fixed radial scattering wavefunction,... 

The S matrix elements, which occur in Eq. (4.8), contain information about the dynamics or scattering on the final or upper-state electronic energy surface. As they refer only to the asymptotic form of the continuum wavefunction, they do not contain information about the probability of the photodissociation... 

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