Basis pseudostates

A partial acknowledgment of the influence of higher discrete and continuum states, not included within the wavefunction expansion, is to add, to the tmncated set of basis states, functions of the fomi T p(r)<6p(r) where dip is not an eigenfiinction of the internal Flamiltonian but is chosen so as to represent some appropriate average of bound and continuum states. These pseudostates can provide fiill polarization distortion to die target by incident electrons and allows flux to be transferred from the the open channels included in the tmncated set. 

This method simply involves the solution of the Lippmann—Schwinger equations (6.73) or (6.87) with the potential matrix elements (7.35). The states i) are not eigenstates of the target Hamiltonian. They are configuration-interaction states or pseudostates obtained by diagonalising the target Hamiltonian in a square-integrable basis as described in section 5.6. 

The diagonalisation of the hydrogen Hamiltonian in a Slater-function (4.38) basis has been reviewed by Callaway (1978) in the context of variational solutions of the integrodifferential equations. This basis has useful features. The inclusion of all the Slater functions necessary for the radial eigenstate u r) produces exact eigenstates for principal quantum numbers up to n in the / manifold. The remainder are pseudostates, which represent the higher discrete states and the continuum. Since Slater functions are not orthonormal there are linear-dependence difficulties that severely limit the size of the basis for which the diagonalisation is numerically feasible. 

With such pseudostate expansions the T-matrix elements are basis-dependent. Choosing a basis is an art, which can be assisted by fitting the results of approximate calculations that are relevant to the problem under consideration without actually treating the basis parameters as phenomenological fitting parameters for the experiment to be described. 

Madison and Callaway (1987) compared the results of pseudostate calculations with those of explicit distorted-wave second-Born calculations, omitting exchange amplitudes. They concluded that it is possible to find basis sets of a managable size whose results are quite close to the second-Born results at the energy of detailed investigation and which give close results also at diflFerent energies. 

Pseudostate calculations have the advantage over Born and optical-potential methods that they constitute a numerically-exact solution of a problem. The problem is not identical to a scattering problem but can be made quite realistic for useful classes of scattering phenomena by an appropriate basis choice. The state vectors, or equivalently the set of half-off-shell T -matrix elements, for such a calculation contain quite realistic information about the ionisation space. 

An alternative method based on a discrete variational representation of the continuum in terms of pseudostates has been developed by Drake and Goldman [11]. The method is simplest to explain for the case of hydrogen. The key idea is to define a variational basis set containing a huge range of distance scales according to ... 

We can numerically solve for Xk > th continuum state at an energy that corresponds to the eigenvalue of a pseudostate. Recall toat Xk satisfies scattering boundary conditions, and Xn zero for large r, square-integrable and normalised. If however we project Xk O " sis subspace, P)(P, we find for a good choice of basis that... 

---