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1: 33.22 Particle Scattering and Atomic and Molecular Spectra
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►At positive energies , , and:
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►Both variable sets may be used for attractive and repulsive potentials: the set cannot be used for a zero potential because this would imply for all , and the set cannot be used for zero energy because this would imply always.
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►The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity (Mott and Massey (1956, pp. 63–65)).
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Attractive potentials: | , . |
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Repulsive potentials: | , . |
2: Bibliography Y
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Coulomb wave functions in repulsive fields.
Phys. Rev. 49 (2), pp. 174–189.
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3: 18.40 Methods of Computation
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►Equation (18.40.7) provides step-histogram approximations to , as shown in Figure 18.40.1 for and , shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein.
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►The example chosen is inversion from the for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50).
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►Further, exponential convergence in , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate for these OP systems on and respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a).
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4: 33.1 Special Notation
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►The main functions treated in this chapter are first the Coulomb radial functions , , (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions , , , (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions.
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5: 18.39 Applications in the Physical Sciences
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►As in this subsection both positive (repulsive) and negative (attractive) Coulomb interactions are discussed, the prefactor of in (18.39.43) has been set to , rather than the of (18.39.28) implying that is an attractive interaction, being repulsive.
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►For these are the repulsive CP OP’s with corresponding to the continuous spectrum of , , and for we have the attractive CP OP’s, where the spectrum is complemented by the infinite set of bound state eigenvalues for fixed .
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►Given that in both the attractive and repulsive cases, the expression for the absolutely continuous, , part of the function of (18.35.6) may be simplified:
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►The weight functions for both the attractive and repulsive cases are now unit normalized, see Bank and Ismail (1985), and Ismail (2009).
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6: Bibliography O
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A closed form solution of the -wave Bethe-Goldstone equation with an infinite repulsive core.
J. Math. Phys. 27 (4), pp. 1154–1158.
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7: Bibliography R
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Computation of electron repulsion integrals using the Rys quadrature method.
J. Comput. Chem. 4 (2), pp. 154–175.
8: Bibliography H
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Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion.
Internat. J. Quantum Chem. 88 (6), pp. 701–734.
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9: Bibliography B
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Stability of repulsive Bose-Einstein condensates in a periodic potential.
Phys. Rev. E (3) 63 (036612), pp. 1–11.
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10: Bibliography C
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Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity.
Phys. Rev. A 62 (063610), pp. 1–10.
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