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Exact Solutions to the Coupled Hartree‐Fock Perturbation Equations
1.M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 2997 (1963).
2.A. D. McLachlan and M. A. Ball, Rev. Mod. Phys. 36, 844 (1964).
3.A. Dalgarno and G. A. Victor, Proc. Roy. Soc. (London) A291, 291 (1966).
4.A. Dalgarno, in Perturbation Theory and its Applications in Quantum Mechanics, edited by Wilcox (Wiley, New York, 1966), pp. 145–183.
5.(a) R. M. Stevens, R. M. Pitzer, and W. N. Lipscomb, J. Chem. Phys. 38, 550 (1963).
5.(b) P. W. Langhoff, M. Karplus, and R. P. Hurst, J. Chem. Phys. 44, 505 (1966).
6.T. C. Caves and M. Karplus, J. Chem. Phys. 50, 3649 (1969).
7.M. J. Jamieson, thesis, Queen’s University of Belfast (1969);
7.Intern. J. Quantum. Chem. 4, 103 (1971).
8.D. J. Thouless, Quantum Mechanics of Many Body Systems (Academic, New York, 1961).
9.T. H. Dunning and V. McKoy, J. Chem. Phys. 47, 1735 (1967).
10.M. H. Alexander and R. G. Gordon, J. Chem. Phys. 55, 4889 (1971).
11.E. Isaacson and H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966), pp. 102–108.
12.R. G. Gordon, J. Chem. Phys. 51, 14 (1969).
13.R. G. Gordon, J. Chem. Phys. 52, 6211 (1970).
14.R. G. Gordon, Methods in Computational Physics, edited by B. Alder, S. Fernlach, and M. Rotenberg (Academic, New York, 1971) pp. 81–109.
15.In what follows we shall adhere closely to the notation of Dalgarno and Victor (Refs. 3, 4, 7). All quantities are in atomic units, unless indicated otherwise.
16.This formulation of the CPHF equations resembles Omidvar’s treatment of the analogous close‐coupled equations for electronhelium scattering (Ref. 17).
17.K. Omidvar, Phys. Rev. 133, A970 (1964).
18.W. H. Miller, J. Chem. Phys. 50, 407 (1969).
19.G. Wolken, thesis, Harvard University, 1971.
20.I. C. Percival and M. J. Seaton, Proc. Camb. Phil. Soc. 53, 654 (1957).
21.P. G. Burke, V. M. Burke, I. C. Percival, and R. McCarrol, Proc. Phys. Soc. (London) 80, 413 (1962).
22.R. Peterkop and V. Veldre, Advan. At. Mol. Phys. 2, 263 (1966).
23.P. G. Burke, D. D. McVicar, and K. Smith, Proc. Phys. Soc. (London) 84, 749 (1964).
24.G. F. Drukarev, The Theory of Electron‐Atom Collisions (Academic, New York, 1965), pp. 69–78.
25.See R. Marriott, Proc. Phys. Soc. (London) 72, 121 (1958).
26.H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, Princeton, N.J., 1956), pp. 520–532.
27.M. H. Alexander (unpublished).
28.D. R. Hartree, The Calculation of Atomic Structures (Wiley, New York, 1957) p. 40.
29.K. Smith, R. J. W. Henry, and P. G. Burke, Phys. Rev. 147, 21 (1966).
30.Caves and Karplus (Ref. 6) used an iterative scheme to solve the matrix equations which arise in the variational treatment of the time‐independent CPHF equations.
31.J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970);
31.B. Noble, Nonlinear Integral Equations, edited by P. M. Anselone (University of Wisconsin Press, Madison, Wise, 1964), pp. 215–319.
32.H. B. Keller, “Some Elements of Numerical Analysis,” New York University Courant Institute Report, 1967, p. 317.
33.D. Shanks, thesis, University of Maryland, 1954;
33.J. Math. & Phys. 34, 1 (1955).
34.G. A. Petersson and V. McKoy, J. Chem. Phys. 46, 4362 (1967);
34.D. Diestler and V. McKoy, J. Chem. Phys. 48, 2941 (1968);
34.N. W. Winter and T. H. Dunning, Jr., Chem. Phys. Letters 8, 169 (1971);
34.J. H. Renken, Phys. Rev. A 3, 510 (1971).
35.Nonlocal matrix operators do not arise in the treatment of inelastic molecular collisions (Ref. 12). It is then much simpler to solve directly a set of coupled equations, rather than resorting to an iterative procedure. Furthermore, for molecular collisions at moderate energies, the functions oscillate rapidly, making it difficult to approximate the quantities by a polynomial; a step which underlies the solution technique of Sec. IV. In the case of the CPHF equations or the close‐coupled equations in electron‐atom scattering, the functions are much more slowly varying, leading themselves readily to polynomial approximation.
36.M. Abramowitz and I. A. Stegun (Eds.), Natl. Bur. Std. (U.S.) Appl. Math. Ser. 55, 448 (1965).
37.M. Karplus, J. Chem. Phys. 37, 2723 (1962);
37.M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 2493 (1963).
38.Isaacson and Keller, Ref. 11, pp. 192–193.
39.For an error analysis of the technique of Hermite interpolation see G. Birkhoff and A. Priver, J. Math. & Phys. 46, 440 (1967);
39.P. G. Ciarlet, M. H. Schultz, and R. S. Varga, Num. Math. 9, 394 (1967).
40.Most of the additional computer time is used in evaluating the contribution to the Karplus‐Kolker functional of the difference between the true and approximate representations of This requires very accurate two dimensional numerical quadrature.
41.Jamieson (Ref. 7) also encountered poor convergence behavior of the simple iterative scheme for values of the frequency near the poles of
42.A. Dalgarno and G. A. Victor, Proc. Phys. Soc. (London) 90, 605 (1967).
43.V. G. Kaveeshwar, K. T. Chung, and R. P. Hurst, Phys. Rev. 172, 35 (1968).
44.P. W. Langhoff and M. Karplus, J. Chem. Phys. 52, 1435 (1970).
45.We have used a double‐precision version of the accurate 5‐term analytic representation of the Hartree‐Fock orbital given by E. Clementi, J. Chem. Phys. 38, 996 (1963).
46.Similar results were obtained by Alexander and Gordon (Ref. 10) in the case of the uncoupled Hartree‐Fock approximation.
47.P. K. Mukherjee, S. Sengupta, and A. Mukherji, J. Chem. Phys. 51, 1397 (1969).
48.Most probably the elaborate variational calculation of Mukherjee et al. (Ref. 47) would yield good values for the higher oscillator strengths. Unfortunately, these authors did not calculate any oscillator strengths.
49.For example, to obtain CPHF polarizabilities for the Be atom using our method would require the iterative solution of four uncoupled inhomogeneous equations; whereas the Hartree method (Refs. 7 and 28) would require the solution of 12 coupled inhomogeneous equations.
50.A. Allison, J. Comp. Phys. 6, 378 (1970).
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