Partial-wave approximation

Due to the complexity of a full quantum mechanical treatment of electron impact ionization, or even a partial wave approximation, for all but relatively simple systems, a large number of semiempirical and semiclassical formulae have been developed. These often make basic assumptions which can limit their range of validity to fairly small classes of atomic or molecular systems. The more successful approaches apply to broad classes of systems and can be very useful for generating cross sections in the absence of good experimental results. The success of such calculations to reproduce experimentally determined cross sections can also give insight into the validity of the approximations and assumptions on which the methods are based. 

In the partial-wave approximation, the reaction cross-section is equal to... 

A special aspect of this description appears if one starts the orbital optimisation process with orbitals obtained by linear combinations ofRHF orbitals of the isolated atoms (LCAO approximation s.str.). Let Pn.opt and be the starting and final orbitals of such a calculation. Then the difference between c n.opi and Papt in the vicinity of each atom merely consists in a distortion of the atomic orbitals of each atom. This distortion just compensates the contribution of the orbitals of the other atoms to Pn.ctpt in order to restore the proportionality between the partial waves of ipopi and the appropriate atomic orbital. 

Continua. The wavefunctions of scattering and bound states have been calculated numerically in the close coupled approximation [358]. Converged partial wave expansions of the elastic scattering solutions have been calculated for pairs of angular momenta 71/2 = 00, 02, 22, 10, 30, 12, 11, and 13 at several hundred energy points. Rotationally inelastic... 

Several other calculations of the first few partial-wave phase shifts for positron-helium scattering have been carried out using a variety of approximation methods in all cases, however, rather simple uncorrelated helium wave functions have been used. Drachman (1966a, 1968) and McEachran et al. (1977) used the polarized-orbital method, whereas Ho and Fraser (1976) used a formulation based on the static approximation, with the addition of several short-range correlation terms, to determine the s-wave phase shifts only. The only other elaborate variational calculations of the s-wave phase shift were made by Houston and Drachman (1971), who employed the Harris method with a trial wave function similar to that used by Humberston (1973, 1974), see equation (3.77), and with the same helium model HI. Their results were slightly less positive than Humberston s HI values, and are therefore probably less... 

Fig. 4.1. The results of various calculations of the l = 0 partial-wave contribution to the positronium formation cross section in positron-hydrogen scattering in the Ore gap A, Archer, Parker and Pack (1990) B, Humberston (1982) C, Stein and Sternlicht (1972) D, Chan and Fraser (1973) E, Wakid (1973) F, Dirks and Hahn (xlO) (1971) G, Wakid and LaBahn (1972) H, Khan and Ghosh (xlO-1) (1983) I, Born approximation (xl0 3).

The d-wave contribution to erPs is relatively insensitive to the method of approximation, and even the results of the Born approximation agree quite well with the accurate variational calculations. It is therefore to be expected that reasonably accurate values can be obtained for all higher-partial-wave contributions to erPS by using the Born approximation, and this has been confirmed by the results of Gien (1997). 

The total positronium formation cross section in the Ore gap, constructed from the addition of accurate variational results for the first three partial waves and the values given by the Born approximation for all partial waves with l > 2, is plotted in Figure 4.4. On the scale of the ordinate, the s-wave contribution is too small to be visible. A very small s-wave contribution is found to be a feature of the positronium formation cross section for several other atoms. 

Fig. 4.9. The total positronium formation cross section, and the various partial-wave contributions to it, for positron-helium scattering in the Ore gap. The contributions with l < 3 are determined variationally whilst the sum of all higher partial waves is calculated in the Born approximation.

The B-spline K-matrix method follows the close-coupling prescription a complete set of stationary eigenfunctions of the Hamiltonian in the continuum is approximated with a linear combination of partial wave channels (PWCs) [Pg.286]

In Fig. 7.3 the individual terms of the given expansion are shown in Fig. 7.3(a) for ( = even which, because of the phase (F) in equ. (7.11), gives real functions in Fig. 7.3(h) for ( = odd which gives imaginary functions. Each individual contribution, shown on the left-hand side, is centred at the common origin and displays certain symmetry properties with respect to this origin. (This aspect becomes important if the emission of electrons is considered, for example, photoionization of an s-electron leads to only one partial wave with ( = 1, i.e., this pattern provides an approximate view for such an emission process (for the... 

Table (8.1) shows results of test calculations of e-He partial wave phase shifts, compared with earlier variational calculations [383], The polarization pseudostate was approximated here for He by variational scaling of the well-known hydrogen pseudostate [76]. The present method is no more difficult to implement for polarization response (SEP) than it is for static exchange (SE). 

Within the independent electron and single active electron approximations, the symmetries of the contributing photoelectron partial waves will be determined by the symmetry of the orbital(s) from which ionization occurs, and so the PAD will directly reflect the evolution of the molecular orbital configuration. Example calculations demonstrating this are shown in Fig. 3 for a model Civ molecule, where a clear difference in the PAD is observed according to whether ionization occurs from an a or an ai symmetry orbital [55] (discussed in more detail below). 

---