An algorithm for the solution of the pair equation for helium is described, suitable for concurrent vector pro cessors. Emphasis is placed on minimizing memory re quirements so that the technique may be extended to systems with more electrons. The pair function is ex panded in a series of partial waves whose radial factor is represented by a tensor product of B-splines. Perfor mance is analyzed for the ground and first excited 1S state, and for the lowest 3S and 1,3P states. An odd-even, column Nesbet algorithm is derived for solving the gen eralized eigenvalue problem and is shown to converge in all cases.
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References
1.
Accad, Y., Pekeris, C.L., and Schiff, B.1971. S and P states of the helium isoelectronic sequence up to Z = 10. Phys. Rev. A 4(2):516-536.
2.
de Boor, C.1978. A practical guide to splines. New York : Springer, Applied Math. Series 27.
3.
Carroll, D.P., Silverstone, H.J., and Metzger, R.M.1979. Piecewise polynomial configuration interaction natural orbital study of ls2 helium. Phys. Rev.71(10):4142-4163.
4.
Char, B.W., Geddes, K.O., Gonnet, G.H., and Watt, S.M.1985. Maple user's guide. Waterloo, Ontario , Canada: WATCOM Publications Limited.
5.
Davidson, E.R.1975. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices . J. Comput. Phys.17:87.
6.
Drake, G.W.F.1988. High precision variational calculations for the ls2 1S state of H- and the ls2 1S, ls2s 1S, and ls2s 3S states of helium. Nucl. Instr. Meth. Phys. Res. B 31:7-13.
7.
Feagin, J.M., Macek, J., and Starace, A.1985. Use of the Fock expansion for the 1S-state wave function of two-electron atoms and ions. Phys. Rev. A 32(6):3219-3230.
8.
Fletcher, A.W.1984. Computational Galerkin methods. New York: Springer.
9.
Froese Fischer, C.1990. Variadonal predictions of transition energies and electron affinities: He and Li ground states and Li, Be, Mg core-excited states . Phys. Rev. A 41(7):3481-3488.
10.
Froese Fischer, C., and Guo, W.1990. Spline algorithms for the Hartree-Fock equation for the helium ground state. J. Comput. Phys.90:486-496.
11.
Froese Fischer, C., and Idrees, I.1989. Spline algorithms for continuum functions. Comput. Phys.3:53-58.
12.
Hill, R.N.1985. Rates of convergence and error estimation formulas for the Rayleigh-Ritz variational method. J. Chem. Phys.83(3): 1173-1196.
13.
Jankowski, K., Malinowski, P., and Polasik, M.1979. Second-order correlation energies of Mg and Ar . J. Phys. B 12(19):3157-3170.
14.
Levin, F.S., and Shertzer, J.1985. Finite-element solution of the Schrödinger equation of the helium ground state. Phys. Rev. A 32(6):3285-3290.
15.
Mårtensson, A.-M.1979. An iterative, numerical procedure to obtain pair functions applied to two-electron systems. J. Phys. B 12(24):3995-4012.
16.
Meyerott, R.E., Luke, P.J., Clendenin, W., and Geltman, S.1952. A numerical variational method. Phys. Rev.85(3):393-400.
17.
Nesbet, R.K.1981. Large matrix technique in quantum chemistry and atomic physics. In Sparse matrices and their uses, edited by I. S. Duff.New York : Academic Press, pp. 139-159.
18.
Ortega, J.M.1988. Introduction to parallel and vector solutions of linear systems. New York: Plenum Press .
19.
Roothaan, C.C.J., and Weiss, A.W.1960. Correlated orbitals for the ground state of heliumlike systems. Reu. Mod. Phys.32(2): 194-205.
20.
Salomonson, S., and Öster, P.1989. Solution of the pair equation using a finite spectrum . Phys. Rev.40(10):5559-5567.
21.
Schwartz, C.1962. Importance of angular correlations between atomic electrons . Phys. Rev.126:1015-1017.
22.
Schwartz, C.1963. Estimating convergence rates of variational calculations . In Methods in computational physics, vol. 2, edited by B. Alder, S. Fernbach, and M. Rotenburg.New York: Academic Press, pp. 241-266.
23.
Sinanoglu, O.1961. Theory of electron correlation in atoms and molecules . Proc. Roy. Soc. London Series A 260:379-392.
24.
Varga, R.S.1962. Matrix iterative analysis. Englewood Cliffs, New Jersey: Prentice-Hall.
