%0 Journal Article %1 meshkov:154108 %A Meshkov, Vladimir V. %A Stolyarov, Andrey V. %A Roy, Robert J. Le %D 2011 %I AIP %J The Journal of Chemical Physics %K equation length mechanics physics quantum scattering schrodinger %N 15 %P 154108 %R 10.1063/1.3649946 %T Rapid, accurate calculation of the s-wave scattering length %U http://link.aip.org/link/?JCP/135/154108/1 %V 135 %X 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 as 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/rn for n ⩾ 4, the modified wave function ϕ(y) obtained by using the two-parameter mapping function r(y;math,β) = math1+mathtan(πy/2) has no singularities, and as = math1+mathmathmath. For a well bound potential with equilibrium distance re, the optimal mapping parameters are  math ≈ re  and  β ≈ math−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 as-values both for model Lennard-Jones (2n, n) potentials and for realistic published potentials for the Xe–e−, Cs 2(a3Σu+), and 3, 4 He 2(X1Σg+) 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.

@article{meshkov:154108, 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 as 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/rn for n ⩾ 4, the modified wave function ϕ(y) obtained by using the two-parameter mapping function r(y;math,β) = math[1+mathtan(πy/2)] has no singularities, and as = math[1+mathmathmath]. For a well bound potential with equilibrium distance re, the optimal mapping parameters are  math ≈ re  and  β ≈ math−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 as-values both for model Lennard-Jones (2n, n) potentials and for realistic published potentials for the Xe–e−, Cs 2(a3Σu+), and 3, 4 He 2(X1Σg+) 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. }, added-at = {2011-11-28T15:53:56.000+0100}, author = {Meshkov, Vladimir V. and Stolyarov, Andrey V. and Roy, Robert J. Le}, biburl = {https://www.bibsonomy.org/bibtex/24cfb143ca1fcc96594fb9f5e90d779b3/drmatusek}, doi = {10.1063/1.3649946}, eid = {154108}, interhash = {d0ee8509b03322eca94037d56b29f78d}, intrahash = {4cfb143ca1fcc96594fb9f5e90d779b3}, journal = {The Journal of Chemical Physics}, keywords = {equation length mechanics physics quantum scattering schrodinger}, month = {October}, number = 15, numpages = {11}, pages = 154108, publisher = {AIP}, timestamp = {2012-08-03T18:24:43.000+0200}, title = {Rapid, accurate calculation of the s-wave scattering length}, url = {http://link.aip.org/link/?JCP/135/154108/1}, volume = 135, year = 2011 }