The J-Matrix Method by Abdulaziz doc

[...]... by approximating the potential V by its projection V N onto the finite subspace spanned by the first N basis functions That is, the exact solution ⌿ of the scattering problem, H0 + V N − k 2 2 ⌿ = 0, (3) is obtained by determining its expansion coefficients in terms of the basis set {φn } subject to the asymptotic boundary condition, ˜ ˜ ⌿ → S(r ) + tan δ C(r ), (4) where δ is the phase shift due to the. .. channel α ≤ Nc The right-hand side driving term and the solution α “vector” containing the aβn ’s and Rαα ’s have as many columns as open channels The R matrix can then be obtained by solving the resulting linear equations As before, the calculation may be facilitated by a pre-diagonalization of the inner block using an energy-independent transformation 5 Discussion The comparison between the approach... for m > Nβ in direct analogy with the single-channel case, because of the three-term recursion relation satisfied by the coefficients of Sβ and Cβ The remainder of the equations; i.e., Eq (41) for m ≤ Nβ , lead to the same number of conditions as there are unknowns The totality of these equations can be organized in a similar fashion as done in the form (22), with the L 2 matrix elements ˜ of V appearing... coefficients of the sine-like and cosine-like functions in terms of the basis sets is outlined In Section 2.2, the general method is illustrated in detail for the case of the radial kinetic energy in a Laguerre basis The analogous results for the oscillator basis and for the Coulomb problem are outlined in Sections 2.3 and 2.4, respectively The details of the Coulomb derivation are given in the Appendix... by these N basis functions, we expect the sine-like and the cosine-like solutions, derived in Section 2.1, to be valid Therefore we write our solution as ˜ ˜ ˜ ψ E = ⌽ + S + t C, (17) 8 E.J Heller, H.A Yamani N −1 0 ˜ ˜ ˜ where ⌽ = n=0 an φn , S is the sine-like expansion χ E of Eq (10), and C is the ˜ cosine-like solution of Eq (13) The unknown coefficient t, then, corresponds to the ˜ tangent of the. .. are both the components of the wave function ψn at the boundary of their respective inner spaces Recent work seems to indicate that the R-matrix method works best using eigenfunctions of the scattering H0 as a basis [10] In this basis, H0 is of course diagonal, and may be treated exactly by the addition of the Buttle correction In our basis set, H0 is tridiagonal and is also treated exactly Other types... sets will be sufficient for many problems The analytic nature of the solutions allows variational corrections to be made and provides a solid footing for further theoretical work We now summarize the steps necessary to perform a calculation with the J -matrix ˜ method First, the potential V N (or V ) is evaluated in the Laguerre basis set; and is then added to the N × N tridiagonal representation of... Laguerres Then a different transformation ⌫ will be necessary to pre-diagonalize the inner matrix In the following chapter we apply the method presented here to s-wave electronhydrogen scattering model The generalization of the method to all partial waves for both Laguerre and Hermite basis sets has been derived and will be the subject of a future publication The case where H0 , contains the term α... not connect the N terms in the expansion of ⌽ or the first N ˜ ˜ terms in the expansion of S and C with φm for m ≥ N + 1 Furthermore, for each m ≥ N + 1 the right-hand side of Eq (21) leads to the three-term recursion relation (8a) and (8c) for the coefficients cn and sn Therefore the right-hand side of Eq (21) vanishes identically Thus, we now have exactly (N + 1) equations to determine the (N + 1)... obeys the same differential equation with different boundary conditions is then constructed The fact that both J -matrix solutions obey the same recurrence scheme is essential to the success of the method as an efficient technique for solving scattering problems [1] The program of the chapter is as follows: In Section 2.1, the generalized H0 problem is considered and a general procedure for obtaining the . of the reference Hamiltonian H 0 by giving a full solution of the reference problem in the complete basis. Furthermore, in the spirit of the R-matrix method, Heller and Yamani proposed that the. loss of the advantages provided by the method [14]. (5) Vanroose et al. [15] enhanced the method, especially in the treatment of long range potentials, by introducing additional terms in the three-term. use the J-matrix method as a universal approach for the description of the process involving the ionization of atoms. They succeed in ad- dressing the difficult problem of correctly describing the