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**ASYMPT: A program for calculating asymptotics of hyperspherical potential curves and adiabatic potentials.**
*(English)*
Zbl 0976.65075

This paper describes briefly the theoretical background, the subroutines and the numerical behaviour for some test problems of the so called program ASYMPT, intended for calculating asymptotics of potential curves and adiabatic potentials with a second order accuracy \( {\mathcal O} ( \rho^{-2})\) in the inverse of the hyperradius variable. A summary of the above program can be found in http://cpc.cs.qub.ac.uk/summaries/ADLL and it is available from CPC Program Library, Queen’s University of Belfast, Northern Ireland.

In section 2 the authors outline their method. First of all they introduce the hyperspherical coordinates describing the Schrödinger equation for a two-electron atomic system in terms of these coordinates. Then the adiabatic states are defined as the eigensolutions of an operator, and assuming that \( \rho \) is slowly varying in the infinite, their series expansions can be approximated by a finite number of terms. Next a study of second order expansions in inverse powers of \( \rho \) is carried out by using Coulomb parabolic functions, showing that the elements of the discretized operator can be easily expressed in terms of full hypergeometric functions with suitable arguments. This section is ended with some numerical experiments showing the accuracy of the asymptotic approximation in a couple of examples.

Sections 3 and 4 give a brief description of the program including the input data, a flow diagram and the output data. Finally in section 5 the results of two test problems are presented. In the first one the potential curves and potential couplings for the S state of the He atom up to the \( n=3\) threshold of the He\(^+\) ion at \( \rho = 80 a.u. \) are presented. In the second one a problem concerning a hydrogen ion is considered.

In section 2 the authors outline their method. First of all they introduce the hyperspherical coordinates describing the Schrödinger equation for a two-electron atomic system in terms of these coordinates. Then the adiabatic states are defined as the eigensolutions of an operator, and assuming that \( \rho \) is slowly varying in the infinite, their series expansions can be approximated by a finite number of terms. Next a study of second order expansions in inverse powers of \( \rho \) is carried out by using Coulomb parabolic functions, showing that the elements of the discretized operator can be easily expressed in terms of full hypergeometric functions with suitable arguments. This section is ended with some numerical experiments showing the accuracy of the asymptotic approximation in a couple of examples.

Sections 3 and 4 give a brief description of the program including the input data, a flow diagram and the output data. Finally in section 5 the results of two test problems are presented. In the first one the potential curves and potential couplings for the S state of the He atom up to the \( n=3\) threshold of the He\(^+\) ion at \( \rho = 80 a.u. \) are presented. In the second one a problem concerning a hydrogen ion is considered.

Reviewer: Manuel Calvo (Zaragoza)

### MSC:

65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |

34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |

34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |

92E20 | Classical flows, reactions, etc. in chemistry |