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Manuscript Title: HSTERM, a program to calculate potential curves and radial matrix elements for two-electron systems within the hyperspherical adiabatic approach.
Authors: A.G. Abrashkevich, D.G. Abrashkevich, M. Shapiro
Program title: HSTERM
Catalogue identifier: ADBZ_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 90(1995)311
Programming language: Fortran.
Computer: DEC 3000 ALPHA AXP 800.
Operating system: DEC OSF/1 V2.0, AIX 3.2.5, SunOs 4.1.3, HP/UX 9.
RAM: 860K words
Word size: 64
Peripherals: disc.
Keywords: Atomic physics, Two-electron systems, Hyperspherical Coordinates, Schrodinger equation, Hyperspherical adiabatic representation, Diabatic-by-sector approach, Potential curves, Dipole transition Amplitudes, Finite element method, High-order accuracy Approximations, Eigensolutions, Ordinary Differential equations.
Classification: 2.1, 2.4.

Nature of problem:
The purpose of this program is to calculate potential curves and matrix elements of radial coupling for two-electron systems within the hyper- spherical adiabatic [4] and diabatic-by-sector [5] approaches. The program computes also the dipole transition matrix elements in the length and acceleration forms and calculates overlap matrices necessary for integration of close-coupling equations in the diabatic-by-sector approach [5].

Solution method:
The solution of the five-dimensional eigenvalue problem for a given set of hyperradial points is reduced after the averaging over the tensor product of two spherical harmonics (bipolar harmonics) to the solution of a system of coupled second order ordinary differential equations. The solution of this system is carried out using the following two-step procedure: (i) the high-order accuracy finite element method [6] is used to obtain the eigensolutions of the decoupled equations obtained from abovementioned system of equations by omitting all coupling term; (ii) the total solution of the system is obtained by expanding it over the linear combinations of these eigensolutions of proper atomic symmetry and solving an algebraic eigenvalue problem for the expansion coefficients.

The computer memory requirements depend on: a) the number of equations to be solved (it depends on the maximum value of electron orbital momentum considered), b) the order of shape functions and the number of finite elements chosen; and c) the number of eigensolutions required. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long Write-Up and listing for details). External subprograms used D01GAF, F01AAF, F02ABF and F02FJF [1], GAULEG [2], SPLINE and SEVAL [3]

Running time:
The running time depends critically upon: a) the number of channels considered; b) the number of required eigensolutions; c) the order and number of finite elements. The test run which accompanies this paper took 56.2 s on the DECstation 3000 Model 800.

[1] NAG Fortran Library Manual, Mark 15 (The Numerical Algorithms Group Limited, Oxford, copyright, 1991).
[2] W.H. Press, B.F. Flanery, S.A. Teukolsky and W.T. Vetterley, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1986).
[3] G.E. Forsythe, M.A. Malcolm and C.B. Moler, Computer Methods for Mathematical Computations (Englewood Cliffs, Prentice Hall, New Jersey, 1977).
[4] J. Macek, J. Phys. B 1(1968)831; U. Fano, Rep. Progr. Phys. 46(1983)97; C.D. Lin, Adv. Atom. Mol. Phys. 22(1986)77.
[5] J.C. Light and R.B. Walker, J. Chem. Phys. 65(1976)4272; B. Lepetit, J.M. Launay and M. LeDourneuf, Chem. Phys. 106(1986)103; D.M. Hood and A. Kuppermann, in Theory of Chemical Reaction Dynamics, ed. D.C. Clary (D. Reidel Publ. Co., Boston, 1986), pp.193-214.
[6] A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev and I.V. Puzynin, Comput. Phys. Commun. 85(1995)40.