Qaiyoom, Shabana; (1990) R-matrix Theory for Atomic and Molecular Systems. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Calculations using the R-matrix theory involve solving the Schrodinger equation within two regions of space. The two solutions obtained are matched at the boundary to give the wave function for all space. The inner region can often be required to be quite large which means that a large basis set is needed to accurately represent the wave function. This can involve the diagonalisation of very large matrices which may require considerable amounts of computer time and memory. In the present work, a propagation method is developed in which the radial basis functions spanning a given region, a ≤ r ≤ b say, are expanded in terms of Legendre polynomials that are orthogonal on the range [a, b]. The method has the considerable advantage that the elements of the Hamiltonian matrix for this region can be generated exactly and extremely rapidly using recurrence relations and that also the use of Legendre polynomials leads to easy propagation of the physical solution from one region to the next. Thus the whole space can be subdivided into as many regions as required and the wave function can be generated for all space by matching at each range boundary. This greatly increases the flexibility of the R-matrix technique. In order to assess the accuracy and the convergence properties of this method, test calculations for the electronic energy levels and oscillator strengths of the hydrogen atom, the hydrogen molecular ion and the HeH2+ ion have been carried out and some encouraging results obtained.

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