Rapid Accurate Calculation of the s-Wave Scattering Length

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 $r(y;\bar{r},β) = \bar{r}[1+\frac{1}β\tan(πy/2)]$ has no singularities, and $$a_s=\bar{r}[1+\frac{2}{πβ}\frac{1}{ϕ(1)}\frac{dϕ(1)}{dy}].$$ For a well bound potential with equilibrium distance $r_e$, the optimal mapping parameters are $\,\bar{r}\approx r_e\,$ 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 ($2n,n$) potentials and for realistic published potentials for the Xe--e$^-$, Cs$_2(a\,^3Σ_u^+$) and $^{3,4}$He$_2(X\,^1Σ_g^+)$ 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.