The J-Matrix Method Development and Application potx

[...]... shifts Section 4 presents the natural generalization of the J -matrix method to multi-channel scattering Finally, Section 5 contains a brief discussion of the over all results and suggestions for applications and areas of further theoretical interest 2 The H0 Problem The problem examined in this section is the “solution” of the equation, H0 − k 2 2 ⌿ = 0 (9) within the framework of the L 2 function space... 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 expansion 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. .. light nuclei The method is applied to 6 He and 6 Be nuclei for scattering and reaction problems Finally, in Part V, Johnson and Holder present a generalization of the density functional theory, which is widely used in chemical physics applications, using a theoretical framework whose structure parallels that of the J -matrix method It is the hope of the editors that this volume will provide the interested... 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 H0 − E To this inner matrix we add one extra row and column, for each asymptotic channel, containing matrix elements of H0 and the cos(n + 1)θ terms The right-hand side “driving”... obtaining the exact solution ψ E and the exact tan δ = t for the Hamiltonian H0 + V N at energy E Compared to the wave function and the phase shift for the exact Hamiltonian H0 + V , ψ E and t in general contain first-order errors However, we may reduce these errors to second order by employing ψ E as a trial function in the Kato [2] formula If we write ψt for ψ E of Eq (17) and tanδt for tanδ of Eq (25), then... 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 Section 3 contains the application of the results thus obtained to potential scattering problems This section presents a formula which allows for the computation... transformation 5 Discussion The comparison between the approach taken in this chapter and the R-matrix method is considered first In R-matrix theory [9], Hilbert space is divided into two parts, an inner coordinate-space up to a radius A and the remaining space from A to infinity In the present work and in the spirit of Feshbach’s generalization of R-matrix theory [6], we have divided the Hilbert space into... large inner block One extra row and column are added to this block for each 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... scattering In the following chapter [1], we will apply the method to electronhydrogen elastic s-wave scattering below the n = 2 threshold, and to inelastic radiallimit scattering calculations above and below the ionization threshold Our basic approach is to treat an uncoupled Hamiltonian H0 exactly in the space spanned by the complete L 2 basis The remaining part of the Hamiltonian (i.e., the E.J Heller... Quantum Scattering: Theory 15 ⌫ N −1,n = φ N −1 | ψn Note that ⌫ N −1,n and γn 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 . approach in the J-matrix method and demonstrate the resulting improvements. On the other hand, a method for the accurate evaluation of the S-matrix for multi-channel analytic and non-analytic. The J-Matrix Method Abdulaziz D. Alhaidari · Eric J. Heller · Hashim A. Yamani · Mohamed S. Abdelmonem Editors The J-Matrix Method Developments and Applications Foreword by Hashim A. Yamani and. course, this should not be the case. The correction of this anomalous behavior is still an outstanding problem for the J-matrix method and for similar methods that approximate the target ionization