Rapid Accurate Calculation of the s-Wave Scattering Length

TL;DR

This paper introduces a transformation of the radial Schrödinger equation that allows for rapid, accurate calculation of the s-wave scattering length using boundary derivatives, applicable to various potentials.

Contribution

The authors develop a new finite domain transformation and an optimized numerical method for precise s-wave scattering length calculations, improving efficiency over previous approaches.

Findings

Accurate s-wave scattering lengths computed for Lennard-Jones and real atomic potentials.

The method achieves high precision with simple boundary derivative evaluations.

Optimal parameters are identified for different potential types.

Abstract

Transformation of the conventional radial Schr\"odinger 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 $\phi(y)$ 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 $\phi(y)$ obtained by using the two-parameter mapping function $r(y;\bar{r},\beta) = \bar{r}[1+\frac{1}{\beta}\tan(\pi y/2)]$ has no singularities, and $$a_s=\bar{r}[1+\frac{2}{\pi\beta}\frac{1}{\phi(1)}\frac{d\phi(1)}{dy}].$$ For a well bound potential with equilibrium distance $r_e$, the optimal mapping parameters are $\,\bar{r}\approx r_e\,$ and $\,\beta\approx \frac{n}{2}-1$. 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 ($2n,n$) potentials and for realistic published potentials for the Xe--e$^-$, Cs$_2(a\,^3\Sigma_u^+$) and $^{3,4}$He$_2(X\,^1\Sigma_g^+)$ systems. Use of this same transformed Schr{\"o}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.