Comparison of the original ψ(r) and modified unbound wave functions at zero energy for the state of Cs2 (see Ref. 15), as calculated using the Numerov method (see Refs. 21 and 29). The scaling function g tg(r) was calculated from Eq. (35) using mapping parameters and β = 2.
Mapping density functions ρ(r) = dy/dr = 1/g(r) for the state of Cs2 (see Ref. 15) calculated: (i) within the framework of the conventional WKB approximation (39), (ii) using the one-parameter mapping function (24) with , and (iii) using the two-parameters mapping function (31) with and β = 2. ψ(r) is the associated zero-energy wavefunction.
Convergence tests for a s -values calculated in double precision arithmetic for the model 15-level LJ(2n, n) potentials of Table I using the one-parameter mapping function (24) with [Å]. Calculations performed using Johnson's log-derivative method alone are denoted “JLD,” while the results labeled “JLD-RE” were obtained by also applying a (N,N/2)-Richardson extrapolation procedure to those results.
Convergence tests for a s -values calculated by the JLD-RE(N,N/2) method in normal double-precision arithmetic (solid curves) for the five model LJ(2n, n) potentials of Table I using the two-parameters mapping function of Eq. (31) with the optimal parameters βopt listed there. The dashed lines present results obtained by using quadruple-precision arithmetic.
Relative errors (42) in a s -values obtained for the 15-level LJ(12, 6) potential (41) as functions of the mapping parameter β, where N is the number of grid points used in the JLD method.
Optimal mapping parameter βopt as a function of the dissociation energy of 15-level LJ(2n, n) potentials (41) for n = 4, 5, and 6.
Optimal mapping parameters βopt determined for the bound vibrational levels v ∈ [0, 14] of the model 15-level LJ(2n, n) potentials (41) for n = 4, 5, and 6. The dashed horizontal lines correspond to the βopt values implied by Eq. (43).
Comparison of convergence behavior for scattering lengths implied by the 3 potential (see Ref. 28) as calculated by the JLD(N) and JLD-RE(N, N/2) procedures using smooth vs. non-smooth long-range damping functions of Eq. (47).
Convergence of calculated scattering lengths for the Xe–e−, 3, 4 , and systems with respect to the lower bound of the integration interval, r min, where r 0 is the distance where U(r) = 0.
The s-wave scattering wavelengths a s (in Å) calculated for the LJ(2n, n) potentials defined by Eq. (41) (Ref. 29). All models have equilibrium distance r e = 1 [Å] and the reduced mass is set as μ = 16.85762920 [a.u.] so that in “spectroscopists units,” the scaling factor ℏ2/2μ = 1 [cm−1 Å2]. is the well depth in cm−1, while v max is the vibrational quantum number of the last bound level supported by the potential.
Comparison of s-wave scattering lengths a s (in ) obtained by applying the JLD-RE(N,N/2) procedure (Ref. 29) to realistic published potentials for Xe–e− (Ref. 27), (Refs. 12 and 13 and 15), and 3, 4 (Ref. 28) interactions, using the two-parameter mapping function of Eq. (31). The optimal mapping parameters βopt were determined by minimization the functional (36), while the values (in .) were fixed at the equilibrium distance of the relevant potential, .
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