The J-Matrix Method Episode 2 docx

16 E.J. Heller, H.A. Yamani concerning H 0 that can be extracted decreases with the number of basis functions per channel, and this can cause difficulties when basis size is a restriction. On the other hand, approximate treatment of H 0 can be advantageous when for example channel threshold details are of no interest or are unwanted artifacts of particular models. In the J-matrix method, H 0 is accounted for exactly independent of basis size. Thus we start with a large part of the problem “diagonalized” and the full analytic structure of the S matrix is built into the problem, raising the hope that quite small basis 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 H 0 − E. To this inner matrix we add one extra row and column, for each asymptotic channel, containing matrix elements of H 0 and the cos(n + 1)θ terms. The right-hand side “driving” terms are similarly constructed with the sin(n +1)θ terms. The resulting linear equations can be solved efficiently if a pre-diagonalizing transformation ⌫ is applied to the inner matrix as in Section 2. If desired, the matrix elements of H 0 +V N − E can be eval- uatedintheSlaterset(λr) n e −λr / 2 , n = 1, 2, ,N, since these are just transformed 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 electron- hydrogen 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 H 0 , contains the term α / r (i.e., the Coulomb case) is also worked out for Laguerre sets. Acknowledgments We are grateful for helpful discussions with Professor William P. Reinhardt and his support of this work. We have also benefited greatly from conversations with L. Fishman, A. Hazi, T. Murtaugh, and T. Rescigno. This work was supported by a grant from the National Science Foundation. References 1. E.J. Heller and H.A. Yamani, following paper, Phys. Rev. A 9, 1209 (1974). 2. T. Kato, Progr. Theor. Phys. 6, 394 (1951). 3. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965). 4. C. Schwartz, Ann. Phys. (N.Y.) 16, 36 (1961). 5. E.J. Heller, W.P. Reinhardt, and Hashim A. Yamani, J. Comp. Phys. 13, 536 (1973). 6. H. Feshbach, Ann. Phys. (N.Y.) 19, 287 (1962). 7. F.E. Harris, Phys. Rev. Lett. 19, 173 (1967). 8. P.G. Burke, D.F. Gallaher, and S. Geltman, J. Phys. B 2, 1142 (1962). 9. See for example, A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30, 257 (1958); A. M. Lane and D. Robson, Phys. Rev. 178, 1715 (1969). New L 2 Approach to Quantum Scattering: Theory 17 10. P.J.A. Buttle, Phys. Rev. 160, 719 (1967); P.G. Burke and W.D. Robb, J. Phys. B 5, 44 (1972); E.J. Heller, Chem. Phys. Lett. 23, 102 (1973). 11. C. Schwartz, Phys. Rev. 124, 1468 (1961). For a comprehensive review of the algebraic meth- ods for scattering, including a discussion of pseudo-resonances, see D.G. Truhlar, J. Abdallah, and R.L. Smith, Advances in Chemical Physics (to be published). 12. R.G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 274. 13. W.P. Reinhardt, D.W. Oxtoby, and T.N. Rescigno, Phys. Rev. Lett. 28, 401 (1972); T.S. Murtaugh and W.P. Reinhardt, J. Chem. Phys. 57, 2129 (1972). 14. E.J. Heller, W.P. Reinhardt, and H.A. Yamani, J. Comp. Phys. 13, 536 (1973); E.J. Heller, T.N. Rescigno, and W.P. Reinhardt, Phys. Rev. A 8, 2946 (1973); H.A. Yamani and W.P. Reinhardt Phys. Rev. A11, 1144 (1975). 15. T.N. Rescigno and W.P. Reinhardt, Phys. Rev. A 8, 2828 (1973); J. Nuttall and H. L. Cohen, Phys. Rev. 188, 1542 (1969). J-Matrix Method: Extensions to Arbitrary Angular Momentum and to Coulomb Scattering Hashim A. Yamani ∗ and Louis Fishman Abstract The J-matrix method introduced previously for s-wave scattering is extended to treat the th partial wave kinetic energy and Coulomb Hamiltonians within the context of square integrable (L 2 ), Laguerre (Slater), and oscillator (Gaussian) basis sets. The determination of the expansion coefficients of the con- tinuum eigenfunctions in terms of the L 2 basis set is shown to be equivalent to the solution of a linear second order differential equation with appropriate boundary conditions, and complete solutions are presented. Physical scattering problems are approximated by a well-defined model which is then solved exactly. In this manner, the generalization presented here treats the scattering of particles by neutral and charged systems. The appropriate formalism for treating many channel problems where target states of differing angular momentum are coupled is spelled out in detail. The method involves the evaluation of only L 2 matrix elements and finite matrix operations, yielding elastic and inelastic scattering information over a con- tinuous range of energies. 1 Introduction In two previous publications [1,2] (referred to as I and II) the J-matrix (Jacobi ma- trix) method was introduced as a new approach for solution of quantum scattering problems. As discussed in I, the principal characteristics of the method are its use H.A. Yamani Ministry of Commerce & Industry, P.O. Box 5729, Riyadh 11127, Saudi Arabia e-mail: haydara@sbm.net.sa ∗ Supported by a fellowship from the College of Petroleum and Minerals, Dhahran, Saudi Arabia. Reprinted with permission from: J-matrix Method: Extension to Arbitrary Angular Momentum and to Coulomb Scattering, H.A. Yamani and L. Fishman, Journal of Mathematical Physics 16, 410–420 (1975). Copyright 1975, American Institute of Physics. A.D. Alhaidari et al. (eds.), The J-Matrix Method, 19–46. 19 C  Springer Science+Business Media B.V. 2008 20 H.A. Yamani, L. Fishman of only square integrable (L 2 ) basis functions and its ability to yield an exact solu- tion to a model scattering Hamiltonian, which, in a well-defined and systematically improvable manner, approximates the actual scattering Hamiltonian. The method is numerically highly efficient as scattering information is obtained over a continuous range of energies from a single matrix diagonalization. The development of the J-matrix method as presented in I is based primarily upon the observation that the s wave kinetic energy, H 0 =− 1 2 d 2 dr 2 (1) can be analytically diagonalized in the Laguerre(Slater) basis: φ n (λr) = (λr)e −λr / 2 L 1 n (λr), n = 0, 1, ,∞, (2) where λ is a scaling parameter. This follows from the fact that the infinite matrix representation of  H 0 − k 2  2  in the above basis is tridiagonal (i.e., J or Jacobi matrix) and that the resulting three-term recursion scheme can be analytically solved yielding the expansion coefficients of both a “sine-like” ˜ S(r) and a “cosine like” ˜ C(r) function. The J-matrix solutions ˜ S(r)and ˜ C(r) are used to obtain the exact solution of the model scattering problem defined 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,  H 0 + V N − k 2  2  ⌿ = 0, (3) is obtained by determining its expansion coefficients in terms of the basis set { φ n } subject to the asymptotic boundary condition, ⌿ → ˜ S(r) + tan δ ˜ C(r), (4) where δ is the phase shift due to the potential V N . This chapter is intended to generalize the formalism developed in I in three areas. First, the results of I are extended to all partial waves, in which case the uncoupled Hamiltonian becomes the th partial wave kinetic energy operator, H 0 =− 1 2 d 2 dr 2 + ( + 1) 2r 2 , (5) which has a Jacobi representation in the Laguerre basis, φ n (λr) = (λr) +1 e −λr / 2 L 2+1 n (λr), n = 0, 1, ,∞, (6) Secondly, a similar analysis of the Hamiltonian of Eq. (5) is presented within the context of the oscillator(Gaussian) basis, J-Matrix Method 21 φ n (λr) = (λr) +1 e −λ 2 r 2  2 L +1 / 2 n (λ 2 r 2 ), n = 0, 1, ,∞, (7) which preserves the Jacobi representation and is also analytically soluble. The third generalization involves the analysis of the th partial wave Coulomb Hamiltonian, H 0 =− 1 2 d 2 dr 2 + ( + 1) 2r 2 + z r (8) in the Laguerre function space of Eq. (6), which again yields a Jacobi form and is subsequently analytically soluble. It is noted that the analysis of the Coulomb Hamiltonian in the oscillator set of Eq. (7) does not lead to a Jacobi form. For the solution of these problems, a general technique is developed which re- duces the solution of the infinite recurrence problem for the asymptotically “sine- like” J-matrix eigenfunction to the solution of a linear second order differential equation with appropriate boundaryconditions. An asymptotically “cosine-like” so- lution which 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 H 0 problem is considered and a general procedure for obtaining the expansion coeffi- cients 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. Section 3 contains the application of the results thus obtained to potential scattering problems. This sec- tion presents a formula which allows for the computation of phase shifts. Section 4 presents the natural generalization of the J-matrix method to multi-channel scatter- ing. Finally, Section 5 contains a brief discussion of the over all results and sugges- tions for applications and areas of further theoretical interest. 2TheH 0 Problem The problem examined in this section is the “solution” of the equation,  H 0 − k 2  2  ⌿ = 0(9) within the framework of the L 2 function space { ⌽ n } in such a manner as to obtain both an asymptotically sine-like and asymptotically cosine-like function. The two J-matrix solutions, ˜ S(r)and ˜ C(r), form the basis for the asymptotic representa- tion of the scattering wavefunction associated with the full problem. It will also be required that the expansion coefficients of both ˜ S(r)and ˜ C(r) satisfy the same three-term recursion scheme, 22 H.A. Yamani, L. Fishman 2.1 Generalized H 0 Problem The basic differential equation  H 0 − k 2  2  ⌿ 0 = 0 (10) possesses both a regular and an irregular solution which behave near the origin as ⌿ 0 reg ∼ r→0 r +1 , (11a) ⌿ 0 irreg ∼ r→0 r − (11b) and asymptotically as ⌿ 0 reg ∼ r→∞ sin ξ, (12a) ⌿ 0 irreg ∼ r→∞ cos ξ, (12b) where ξ = kr −π  2 in the free particle case and ξ = kr +tln(2kr)−π  2+σ  in the Coulomb case. In the above, the definitions t =−z  k and σ  = arg⌫(+1−it) have been used [3]. Since the basis set { φ n } is complete for functions regular at the origin, ⌿ 0 reg can be expanded as ⌿ 0 reg ≡ ˜ S(r) = ∞  n=0 s n φ n (λr) (13) with the expansion coefficients s n being formally given by [4] s n =  ¯ φ n (λr)   ˜ S(r)  (14) with ¯ φ n satisfying  ¯ φ n   φ m  = δ nm . A differential equation satisfied by the set of coefficients { s n } can be constructed in the following manner. Since the basis set { φ n } tridiagonalizes the operator  H 0 − k 2  2  ,the { s n } satisfy a three-term recur- sion relation of the form, [a 1n + a 2n g(η)] s n + a 3n s n−1 + a 4n s n+1 = 0, n > 0, (15a) [a 10 + a 20 g(η)] s 0 + a 40 s 1 = 0, n = 0, (15b) where η is the energy variable defined by η = k  λ and g(η) is a function dependent upon the particular choice of { φ n } and H 0 . Differentiating Eq. (14) with respect to x,wherex is a function of the energy variable η appropriate to the particular case, leads to a differential difference equation of the form J-Matrix Method 23 η dx dη ds n dx = b 1n s n+1 + b 2n s n + b 3n s n−1 , n > 0 (16a) η dx dη ds 0 dx = b 10 s 1 + b 20 s 0 , n = 0. (16b) For the case of the Laguerrefunction space, b 2n = 0, while for the oscillator function space the general form of Eq. (16) is appropriate. Combining Eqs. (15) and (16) yields a linear second order differential equation of the form A(x) d 2 s n dx 2 + B(x) ds n dx + D(x)s n = 0 (17) with two linearly independent solutions χ 1 and χ 2 and a general solution of the form s n = α 1 χ 1 + α 2 χ 2 . (18) Equation (15b) determines s n to within a normalization constant. The advantage of the differential equation approach is that a cosine-like solution ˜ C(r), whose ex- pansion coefficients will also satisfy the differential equation (17), can be readily constructed. The cosine-like J-matrix solution, ˜ C(r) = ∞  n=0 c n φ n (λr), (19) is constructed to be (1) regular at the origin like ⌿ 0 reg so as to be expandable in the basis set { φ n } , (2) behave asymptotically as ⌿ 0 irreg , and (3) to have its expan- sion coefficients { c n } satisfy Eq. (15b). This immediately means that ˜ C(r) cannot satisfy the homogeneous differential equation (10). By choosing ˜ C(r) to satisfy the inhomogeneous differential equation  H 0 − k 2  2  ˜ C(r) = β ¯ φ 0 (λr), (20) the Green’s function [5] G(r, r  ) = 2⌿ 0 reg (r < )⌿ 0 irreg (r > )  W  ⌿ 0 reg , ⌿ 0 irreg  (21) maybeusedtoobtainthesolution[5] ˜ C(r) =− 2β W  ⌿ 0 irreg (r)  r 0 dr  ⌿ 0 reg (r  ) ¯ φ 0 (λr  ) + ⌿ 0 reg (r)  ∞ r dr  ⌿ 0 irreg (r  ) ¯ φ 0 (λr  )  , (22) 24 H.A. Yamani, L. Fishman where W  ⌿ 0 reg , ⌿ 0 irreg  is the Wronskian of the two independent solutions ⌿ 0 reg and ⌿ 0 irreg and is independentof r and β is a free parameter [6]. ˜ C(r) as given by Eq. (22) is regular at the origin and with the choice β =−W  2s 0 (23) goes asymptotically as ⌿ 0 irreg . The fact that the inhomogeneity of Eq. (20) is orthog- onal to the set { φ n } for n = 1, 2, ,∞, implies that, for n > 0, the { c n } satisfy the same three-term recursion relation as the { s n } , Eq. (15a). The n = 0 case has the form [a 10 + a 20 g(η)] c 0 + a 40 c 1 = l(η) = 0 (24) where l(η) is a function which depends upon the form of β and upon any terms that were divided out in the derivation of the homogeneous recursion relation, Eq. (15). Equation (24) is to be contrasted with the homogeneous initial condition [a 10 + a 20 g(η)] s 0 + a 40 s 1 = 0, (15b) which occurs in the sine-like J-matrix solution. It may be shown from Eqs. (22) and (23) that the set { c n } satisfies a differential difference equation analogous to Eq. (16), which when combined with Eqs. (15a) and (24) leads to the differen- tial equation (17). The application of the inhomogeneous initial condition given by Eq. (24), and an additional boundary condition specific to the case being considered, determine the two integration constants γ 1 and γ 2 in the solution c n = γ 1 χ 1 + γ 2 χ 2 . (25) 2.2 Radial Kinetic Energy: Laguerre Basis For the case of the radial kinetic energy and a Laguerre basis, the detailed construc- tion of the J -matrix solutions is given following the general technique out lined in Section 2.1 The Hamiltonian is H 0  =− 1 2 d 2 dr 2 + ( + 1) 2r 2 , (26) while the L 2 expansion set is given by φ n (λr) = (λr) +1 e −λr / 2 L 2+1 n (λr), n = 0, 1, ,∞, (27) In essence the infinite matrix problems  φ m |  H 0  − k 2  2    ˜ S(r)  = 0, m = 0, 1, 2, ,∞, (28a) J-Matrix Method 25 and  φ m |  H 0  − k 2  2    ˜ C(r)  = 0, m = 1, 2, ,∞, (28b) are solved where ˜ S(r) = ∞  n=0 s n φ n (λr), (13) ˜ C(r) = ∞  n=0 c n φ n (λr), (19) subject to the asymptotic boundary conditions ˜ S(r) ∼ r→∞ sin(kr − π  2) (29a) and ˜ C(r) ∼ r→∞ cos(kr −π  2). (29b) These solutions have the appropriate asymptotic forms to allow for the formula- tion of partial wave scattering problems for potentials falling off faster than 1  r 2 at infinity, where the solution of the scattering problem will have the asymptotic form [3] ⌿ ∼ r→∞ sin(kr − π  2) + tan δ cos(kr − π  2), (30) δ being the scattering phase shift. From the boundary conditions of Eq. (29), ˜ S(r) is designated the “sine-like” J-matrix solution, and ˜ C(r) the “cosine-like” J-matrix solution. The sine-like solu- tion is discussed in Section 2.2.1, where the recurrence relation for the coefficients { s n } is solved explicitly, giving closed form expressions. The discussion of ˜ C(r) is somewhat more complex: In Section 2.2.2, a function ˜ C(r) with the appropriate cosine-like behavior is constructed such that, for n > 0, the expansion coefficients { c n } obey the same recursion scheme as the set { s n } ; a fact that is an essential in- gredient of the J -matrix method as will be seen in Sections 3 and 4 and has been discussed in I and II. 2.2.1 Sine-Like Solution One of the linearly independent eigenfunctions of the radial kinetic energy, H 0  =− 1 2 d 2 dr 2 + ( + 1) 2r 2 , (26) 26 H.A. Yamani, L. Fishman may be taken to be regular at r = 0 and sine-like asymptotically, that is, ⌿ reg (r) ∼ r→0 r +1 , (31a) ⌿ reg (r) ∼ r→∞ sin(kr − π  2) (31b) where the eigenfunction satisfying Eq. (31) is referred to as the regular solution [3]. A J-matrix solution, ˜ S(r) ≡ ⌿ reg (r) = ∞  n=0 s n φ n (λr), (13  ) satisfying the boundary conditions of Eq. (31), is easily found within the context of the Laguerre set of Eq. (27). The matrix  φ n |  H 0  − k 2  2  | φ m  = J nm =  ∞ 0 drφ n (λr)  − 1 2 d 2 dr 2 + ( + 1) 2r 2 − k 2 2  φ m (λr), (32) may, upon application of the orthogonality and recursion properties of the Laguerre functions L 2+1 n (λr) [4], be reduced to the Jacobi (J-matrix) form J nm =− λη 2 ⌫(n + 2 + 2) ⌫(n +1) sin θ  2x(n +  +1)δ n,m − nδ n,m−1 − (n +2 + 2)δ n,m+1  , (33) where x = cos θ =  η 2 − 1 4  η 2 + 1 4  (34a) and η = k  λ = 1 2 cot(θ  2) (34b) The expansion coefficients { s n } which satisfy the matrix equation J · s = 0maybe determined by the solution of the three-term recursion relation 2x(n + +1)u n (x) −(n + 2 + 1)u n−1 (x) −(n +1)u n+1 (x) = 0 (35a) with the initial condition 2x( + 1)u 0 (x) −u 1 (x) = 0, (35b) where s n (x) =  ⌫(n + 1)  ⌫(n + 2 + 2)  u n (x). (36) [...]... by [7] ⌫( + 1 /2) ⌫(n + 1) 1 cn (x) = − 2 √ π⌫(n + 2 + 2) (sin θ 2) × 2 F1 −n − 2 − 1, n + 1; 1 θ − ; sin2 , 2 2 ( 62) which is a finite polynomial in (sin2 θ 2) 2. 3 Radial Kinetic Energy: Oscillator Basis Within the framework of the oscillator basis φn (λr ) = (λr ) +1 − 2 r 2 2 e L n+1 /2 ( 2 r 2 ), n = 0, 1, , ∞, (63) the J -matrix defined by ∞ 1 d2 ( + 1) k 2 φm (λr ), + − 2 dr 2 2r 2 2 0 (64) is a... solution, − + 3 − 2 u 0 2 + u 1 2 = − 2 π⌫ 2 + 3 − 2 2 , η e 2 ( 72) and use of the symmetry conditions given in Eq (60) give cn 2 = (−)n 2 π ⌫ 1 ⌫(n + 1) 2 2 η− e−η 3 ⌫ n+ + 2 + 2 1 F1 1 1 −n − − , − , 2 , 2 2 (73) where ∞ cn 2 φn (λr ) ˜ C(r ) = (74) n=0 ˜ ˜ S(r ) and C(r ) are real, regular at the origin, and have sine-like and cosine-like asymptotic forms, respectively 2. 4 Radial Coulomb Hamiltonian:... is then straightforward to derive the differential equation (x 2 − 1)vn + + x2 − 1 vn − x x2 + 1 x2 x2 − 1 x2 x2 + 1 x2 − 1 (n + )2 − ( − 1) + 2n + 1 − t 2 ( + 1) − 2i t(n + + 1) vn = 0 (A6) J -Matrix Method 43 which can be solved by standard techniques to give [7] vn = An t (sin θ ) +1 θ t −inθ e e + Bn t (sin θ )it e−i(n+ 2 F1 +1)θ 2 F1 −n, + 1 − i t; 2 + 2; 1 − e2iθ − − i t, + 1 − i t; n + + 2 −... leading to the fundamental recursion relation Jnm = φn | H 0 − k 2 2 |φm = − 2n + + dr φn (λr ) − 3 1 − 2 u n 2 + n + + u n−1 2 +(n +1)u n+1 2 = 0, n > 0, 2 2 (65) where again η = k λ, for the solution of the infinite matrix problems ˜ φm | H 0 − k 2 2 S(r ) = 0, m = 0, 1, 2, , ∞, ˜ φm | H 0 − k 2 2 C(r ) = 0, m = 0, 1, 2, , ∞, (66a) (66b) for the sine-like arid cosine-like J -matrix solutions The. .. yielding the recurrence relation x2 + 1 x2 − 1 − it vn (x) , x x −(n + 1)vn+1 (x) − (n + 2 + 1)vn−1 (x) = 0 (n + + 1) n>0 (77) where x = eiθ = − 1 − iη 2 1 + iη 2 (78) and sn (x) = 2 ⌫(n + 1) |⌫( + 1 − i t)| π t /2 e vn (x) ⌫ (2 + 2) ⌫(n + 2 + 2) (79) for the solution of the infinite matrix equations φm φm ˜ H 0,z − k 2 2 S(r ) = 0, 0 2 ˜ H ,z − k 2 C(r ) = 0, m = 0, 1, 2, , ∞, (80a) m = 0, 1, 2, ,... The set {u n } satisfies the differential equation u n 2 + 4n + 2 + 3 − 2 − ( + 1) u n 2 = 0, 2 (67) where the differentiation is with respect to η Equation (67) has the general solution [6] u gen (η) = An η n +1 − 2 2 e 1 L n+ /2 2 +Bn η− e−η 2 2 1 F1 1 1 −n − − , − , 2 , 2 2 (68) 32 H.A Yamani, L Fishman where 1 F1 (a, c, z) is the confluent hypergeometric function [7] The sine-like solution is... of the initial condition − + 3 − 2 u 0 2 + u 1 2 = 0, 2 (69) and normalization in the manner of Eq (47), giving sn 2 = √ (−)n ⌫(n + 1) (−)n ⌫(n + 1) u n 2 = 2 η ⌫(n + + 3 2) ⌫(n + + 3 2) +1 − 2 2 e 1 L n+ /2 2 , (70) where ∞ sn 2 φn (λr ) ˜ S(r ) = (71) n=0 Substitution into Eq (65), imposition of the initial condition appropriate to the construction of a cosine like solution, − + 3 − 2 u... the sine-like and cosine-like J -matrix solutions The {vn } satisfy the differential equation (x 2 − 1)vn (x) + + x2 − 1 vn (x) − x x2 + 1 x2 x2 − 1 x2 x2 + 1 x2 − 1 (n + )2 − ( − 1) + 2n + 1 − t 2 ( + 1) − 2i t(n + + 1) vn (x) = 0 (81) where the differentiation is with respect to x = exp(i θ ) and where t = −z k is considered to be independent of x The derivation of this equation is discussed in the. .. − (n + 2 + 1)vn−1 − (n + 1)vn+1 = 0, n > 0, (A2) where t = −z k = (−2z λ) tan θ 2 for −π < θ ≤ π Since ⌿reg for the Coulomb case has the form [3] + 1 − i t)| 1 F1 ( + 1 − i t, 2 + 2, −2i kr ), ⌫ (2 + 2) (A3) the Fourier projection of Eq (37) has the general form ⌿reg (k, r ) = 1 (2kr ) 2 +1 ikr π t /2 |⌫( e e ∞ vn = dr ⌽[ηr, t(k)]φn (r ) r (A4) 0 Since the k dependence in ⌿reg appears in both the variable... the framework of the Laguerre basis, Coulomb J -matrix solutions S(r ) and ˜ ) are constructed, which are regular at the origin and behave asymptotically C(r as [3] J -Matrix Method 33 ˜ S(r ) ∼ sin kr + t ln(2kr ) − π 2+ σ , (75a) ˜ C(r ) ∼ cos kr + t ln(2kr ) − π 2+ σ (75b) r→∞ r→∞ The Coulomb J matrix Jnm = φn | H 0,z − k 2 2 |φm ∞ = dr φn (λr ) − 0 k2 1 d2 ( + 1) z φm (λr ) + + − 2 2 2 dr 2r r 2 . found within the context of the Laguerre set of Eq. (27 ). The matrix  φ n |  H 0  − k 2  2  | φ m  = J nm =  ∞ 0 drφ n (λr)  − 1 2 d 2 dr 2 + ( + 1) 2r 2 − k 2 2  φ m (λr), ( 32) may, upon. c n (x)isgivenby[7] c n (x) = 2  ⌫( + 1 / 2 )⌫(n + 1) √ π⌫(n + 2 + 2) 1 (sin θ  2)  × 2 F 1  −n − 2 − 1, n +1; 1 2 − ;sin 2 θ 2  , ( 62) which is a finite polynomial in (sin 2 θ  2) . 2. 3 Radial Kinetic. and giving c n (x) =− ⌫( + 1 / 2 ) √ π ⌫(n +1) ⌫(n +2 +2) (cos θ  2) +1 (sin θ  2)  × 2 F 1  n +  + 3 2 , −n −  − 1 2 ; 1 2 − ;sin 2 θ 2  . (61) J-Matrix Method 31 A useful alternative