Rapid Accurate Calculation of the s-Wave Scattering Length

Vladimir V. Meshkov; Andrey V. Stolyarov; Robert J. Le Roy

doi:10.1063/1.3649946

paper

Rapid Accurate Calculation of the s-Wave Scattering Length

arXiv:1109.6522 · doi:10.1063/1.3649946

Abstract

Transformation of the conventional radial Schrödinger equation defined on the interval $r \in [0, \infty)$ into an equivalent form defined on the finite domain $y (r) \in [a, b]$ 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 $r = \infty$ . In particular, for an arbitrary interaction potential that dies off as fast as $1/ r^{n}$ for $n \geq 4$ , the modified wave function $ϕ (y)$ obtained by using the two-parameter mapping function $r (y; r ˉ, β) = r ˉ [1 + \frac{1}{β} tan (π y /2)]$ has no singularities, and $a_{s} = r ˉ [1 + \frac{2}{π β} \frac{1}{ϕ ( 1 )} \frac{d ϕ ( 1 )}{d y}] .$ For a well bound potential with equilibrium distance $r_{e}$ , the optimal mapping parameters are $r ˉ \approx r_{e}$ and $β \approx \frac{n}{2} - 1$ . An outward integration procedure based on Johnson's log-derivative algorithm [B.R.\ Johnson, J.\ Comp.\ Phys., \textbf{13}, 445 (1973)] combined with a Richardson extrapolation procedure is shown to readily yield high precision $a_{s}$ -values both for model Lennard-Jones ( $2 n, n$ ) potentials and for realistic published potentials for the Xe--e $^{-}$ , Cs $_{2} (a^{3} Σ_{u}^{+}$ ) and $^{3, 4}$ He $_{2} (X^{1} Σ_{g}^{+})$ systems. Use of this same transformed Schr{ö}dinger equation was previously shown [V.V. Meshkov et al., Phys.\ Rev.\ A, {\bf 78}, 052510 (2008)] to ensure the efficient calculation of all bound levels supported by a potential, including those lying extremely close to dissociation.

12 pages, 9 figures, to appear in J. Chem. Phys

Rapid Accurate Calculation of the s-Wave Scattering Length

Abstract

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