Rapid, accurate calculation of the s-wave scattering length

Abstract

Transformation of the conventional radial Schrödinger equation defined on the interval r ∈ [0, ∞) into an equivalent form defined on the finite domain y(r) ∈ [a, b] allows the s-wave scattering length a(s) to be exactly expressed in terms of a logarithmic derivative of the transformed wave function φ(y) at the outer boundary point y = b, which corresponds to r = ∞. In particular, for an arbitrary interaction potential that dies off as fast as 1/r(n) for n ≥ 4, the modified wave function φ(y) obtained by using the two-parameter mapping function r(y; ̄r,β) = ̄r[1 + 1/β tan(πy/2)] has no singularities, and a(s) = ̄r[1 + 2/πβ 1/φ(1) dφ(1)/dy]. For a well bound potential with equilibrium distance r(e), the optimal mapping parameters are ̄r ≈ r(e) and β ≈ n/2 - 1. An outward integration procedure based on Johnson's log-derivative algorithm [J. Comp. Phys. 13, 445 (1973)] combined with a Richardson extrapolation procedure is shown to readily yield high precision a(s)-values both for model Lennard-Jones (2n, n) potentials and for realistic published potentials for the Xe-e(-), Cs(2)(aΣ(u)(+)(3)), and (3, 4)He(2)(XΣ(g)(+)(1)) systems. Use of this same transformed Schrödinger equation was previously shown [V. V. Meshkov et al., Phys. Rev. A 78, 052510 (2008)] to ensure the efficient calculation of all bound levels supported by a potential, including those lying extremely close to dissociation.

LEVEL: A computer program for solving the radial Schrödinger equation for bound and quasibound levels

“…(1) on the infinite domain r A ½0; 1Þ by integration using a dimensionless radial variable, such as y ref q ðrÞ of Eq. (17), on the finite domain y ref q A ½À1; þ 1 [66,67]. Use of this procedure allows one to readily obtain accurate solutions for levels lying extremely close to dissociation, such as sometimes observed for excited states of electronically excited alkali dimers, or to calculate the s-wave scattering length associated with any given potential energy function using a single-pass finite-domain calculation.…”

Extended Fourier-transform spectroscopy studies and deperturbation analysis of the spin-orbit coupled A1Σ+ and b3Π states in RbCs

“…31,32 The explicit energy correction for the FD5 method 10 was used to extrapolate to infinite the number of the grid points. The minimum of the functional χ 2 expt + χ 2 ab was searched by the modified Levenberg-Marquardt algorithm.…”

Low temperature scattering with the R-matrix method: argon-argon scattering

“…A further PEC, and the associated scattering length prediction (computed with the method of Meshkov et al [50]), from Myatt et al [45], is also recreated here.…”