Continuum functions

The partial wave basis functions with which the radial dipole matrix elements fLv constructed (see Appendix A) are S-matrix normalized continuum functions obeying incoming wave boundary conditions. 

The continuum electron-phase shifts induced by the short-range scattering off the chiral molecular potential are most conveniently introduced by a third choice of continuum function, obtained by diagonalizing the K-matrix by a transformation U, resulting in a set of real eigenchannel functions (apart from normalization) [41] ... 

Because the dispersed acoustic function 3.69, the optic continuum function 3.71, and the Einstein function 3.73 may be tabulated for the limiting values of undi-mensionalized frequencies (see tables 1, 2, 3 in Kieffer, 1979c), the evaluation of Cy reduces to the appropriate choice of lower and upper cutoff frequencies for the optic continuum (i.e., X/ and limits of integration in eq. 3.71), of the three... 

Basis sets of the type discussed in this paper can only be applied to bound-state problems. It is interesting to ask whether it might be possible to constmct many-electron Sturmian basis sets appropriate for problems in reactive scattering in an analogous way, using hydrogenlike continuum functions as building-blocks. We hope to explore this question in future publications. 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

A related but somewhat different approach has been suggested by the recently introduced MS Xa method. Calculation of the continuum functions within this method is relatively straightforward,51 52 and the method has been applied to the AT-shell X-ray absorption spectrum53 of N2 and very recently to photoionization of N2 and CO near threshold.54 This last... 

Figure 1.8 The probability density of several continuum eigenstates of the Hamiltonian in Eq. (24) plotted on the baseline of their corresponding energy. The potential is also plotted for convenience. Note that most continuum states (dashed lines) are delocalized and have a very small amplitude inside the potential well between the two barriers, whereas there are continuum functions that are localized inside the well. The localized eigenstate (solid line) is the same as shown in Figure 1.1.

The dipole matrix element can be evaluated further and yields, using the expansion of the continuum function into partial waves as given in equ. (7.28b) ... 

For a calculation of this matrix element one first changes the order of orbitals in such a way that the two different orbitals in the determinantal wavefunctions are at the same positions. Since in the expansion of the continuum function into partial waves, equ. (3.5a), only S is allowed, one gets... 

The continuum function is normalized with the -function, see equ. (7.28f). Since the -function also has a dimension,... 

In order finally to derive the differential cross section of photoionization one inserts equ. (8.26) in equ. (8.24) and replaces the number nph of incident photons by nPh = ce0Alo)/2ti (see equs. (8.4b) and (8.8a) and (8.8b)) and the interaction operator by equ. (8.21). Then one removes the factor h2/m0 resulting from the normalization of the continuum function from the matrix element and incorporates it in the final prefactor (see footnote concerning equ. (7.28d)), and one introduces the fine structure constant a using a = el/4ne0hc. This leads to (for the summations over magnetic quantum numbers see below)... 

The value of coherent control experiments lies not only in their ability to alter the outcome of a reaction but also in the fundamental information that they provide about molecular properties. In the example of phase-sensitive control, the channel phase reveals information about couplings between continuum states that is not readily obtained by other methods. Examination of Eq. (15) reveals two possible sources of the channel phase—namely, the phase of the three-photon dipole operator and that of the continuum function, ESk). The former is complex if there exists a metastable state at an energy of (D or 2 >i, which contributes a phase to only one of the paths, as illustrated in Fig. 3b. In this case the channel phase equals the Breit-Wigner phase of the intermediate resonance (modulo n),... 

The other source of a channel phase is the complex continuum wave function at the final energy E. At first it would appear from Eq. (15) that the phase of ESk) should cancel in the cross term. This conclusion is valid if the product continuum is not coupled either to some another continuum (i.e., if it is elastic) or to a resonance at energy E. If the continuum is coupled to some other continuum (i.e., if it is inelastic), the product scattering wave function can be expanded as a linear combination of continuum functions,... 

Using these asymptotic continuum functions, the derivation given above implies that for Tt = —such that mio + mnTt = 0, then... 

In this section we shall discuss in some detail the formalism needed to apply the so(4, 2) algebraic methods to problems whose unperturbed Hamiltonian is hydrogenic. First a scaling transformation is applied to obtain a new Hamiltonian whose unperturbed part is just the so(2, 1) generator T3, which has a purely discrete spectrum. Next we use the scaled hydrogenic eigenfunctions of T3 as a basis for the expansion of the exact wave function. This discrete basis is complete with respect to the expansion of bound-state wave functions whereas the usual bound-state eigenfunctions do not form a complete set continuum functions must also be included to ensure completeness (cf. Section VI,A)-... 

We have written x) for the continuum function of the photoelectron. The spin and orbital parts of the ionised state function are written (SAma) as for the ground state, but we now need two extra labels r labels the molecular orbital shell which has been ionised, and a takes the value 1, 2, 3... to distinguish between repeated states with the same values of S2 and A 2 which may occur in the ionised configuration. 

---