Partial wave methods

Partial wave methods have been widely used and tested on a variety of systems (see reference 19 and references therein). They are usually accurate to better than 50% for total ionization cross sections of light atoms. Very heavy atoms have been less successfully treated due to the increasing contributions of resonances and... 

The algebraic formulation of the matching condition differs for the various partial-wave methods, but in general the result is a set of linear, homogeneous equations of the form... 

Even for moderate size matrices M, the partial-wave methods therefore require orders of magnitude more computer time than the solution of the eigenvalue problem (1.19). Furthermore, the formalisms of these methods are complicated, and perturbations are difficult to include because (1.21) are not derived from the variational principle for the one-electron Hamiltonian. 

The partial-wave methods do, however, have two distinct advantages. Firstly, they provide solutions of arbitrary accuracy for a muffin-tin potential and, for close-packed systems, this makes them far more accurate than any traditional fixed-basis method. Secondly, the information about the potential enters (1.21) only via a few functions of energy, the logarithmic derivatives aln i/ (E,r) /aln r, at the muffin-tin sphere. 

These methods therefore lead to secular equations (1.21) which are linear in energy, that is to eigenvalue equations of the form (1.19). When applied to a muffin-tin potential they use logarithmic-derivative parameters and provide solutions of arbitrary accuracy in a certain energy range. The linear methods thus combine the desirable features of the fixed-basis and partial-wave methods. 

As has been said before, a partial-wave method such as APW, in spite of being physically (and chemically) advantageous, suffers from the energy-dependence of its basis functions (p E). Albeit highly accurate, this makes it computationally demanding and quite slow. In order to combine the virtues of the energy-independent approaches (speed) and those of the energy-dependent ones (accuracy), linear methods were introduced by Andersen [230]. 

At first sight, the pseudopotential approach and the different partial-wave methods do not seem to have very much in common. In the first approach, the inner, atom-like wave functions are discarded altogether and replaced by a much weaker potential. In the second group, the outer wave function augments exactly these atom-like partial waves. The projector-augmented wave (PAW) method by Blochl [237], however, combines the two ideas into a unified electronic-structure method. Without going into detail, the PAW method can be looked upon as a pseudopotential method in which the pseudopotential instantaneously adapts to the electronic environment. This is because the PAW method is, in fact, a complete all-electron method, and its internal pseu-... 

A partial wave decomposition provides the frill close-coupling quantal method for treating A-B collisions, electron-atom, electron-ion or atom-molecule collisions. The method [15] is siumnarized here for the inelastic processes... 

MoCullough E A Jr 1975 The partial-wave self-oonsistent method for diatomio moleoules oomputational formalism and results for small moleoules J. Chem. Phys. 62 3991-9... 

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. 

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... 

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 quantum dynamics for the / = 0 partial wave of the F+HCl reaction was simulated on a very fine grid of energies using the method of... 

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]

Variational methods for continuum states Table 8.1. Partial wave phase shifts for He... 

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