Eighth order methods with minimal phase‐lag for accurate computations for the elastic scattering phase‐shift problem

Two new hybrid eighth algebraic order two‐step methods with phase‐lag of order twelve and fourteen are developed for computing elastic scattering phase shifts of the radial Schrödinger equation. Based on these new methods we obtain a new variable‐step procedure for the numerical integration of the Schrödinger equation. Numerical results obtained for the integration of the phase shift problem for the well known case of the Lennard–Jones potential show that these new methods are better than other finite difference methods.

[1]  Theodore E. Simos,et al.  A Numerov-type Method for Computing Eigenvalues and Resonances of the Radial Schrödinger Equation , 1996, Comput. Chem..

[2]  John M. Blatt,et al.  Practical points concerning the solution of the Schrödinger equation , 1967 .

[3]  Tom E. Simos.  An explicit almost P-stable two-step method with phase-lag of order infinity for the numerical integration of second-order pacific initial-value problems , 1992 .

[4]  A. D. Raptis,et al.  A variable step method for the numerical integration of the one-dimensional Schrödinger equation , 1985 .

[5]  R. Thomas,et al.  Phase properties of high order, almostP-stable formulae , 1984 .

[6]  J. W. Cooley,et al.  An improved eigenvalue corrector formula for solving the Schrödinger equation for central fields , 1961 .

[7]  L. Brusa,et al.  A one‐step method for direct integration of structural dynamic equations , 1980 .

[8]  R. L. Roy,et al.  On calculating phase shifts and performing fits to scattering cross sections or transport properties , 1978 .

[9]  G. Herzberg,et al.  Spectra of diatomic molecules , 1950 .

[10]  Ben P. Sommeijer,et al.  Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions , 1987 .

[11]  M H Chawla,et al.  A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value , 1986 .

[12]  H. A. Watts,et al.  Solving Nonstiff Ordinary Differential Equations—The State of the Art , 1976 .

[13]  A. Wagner,et al.  Comparison of perturbation and direct-numerical-integration techniques for the calculation of phase shifts for elastic scattering , 1974 .

[14]  M. M. Chawla,et al.  An explicit sixth-order method with phase-lag of order eight for y ″= f ( t , y ) , 1987 .

[15]  Tom E. Simos,et al.  A Numerov-type method for the numerical solution of the radial Schro¨dinger equation , 1991 .