15min:

ROBERT J. LE ROY, Guelph-Waterloo Centre for Graduate Work in Chemistry and Biochemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; VLADIMIR V. MESHKOV AND ANDREJ V. STOLYAROV, Department of Chemistry, Moscow State University, GSP-2 Leninskie Gory 1/3, Moscow 119991, Russia.

We have shown that one and two-parameter analytical mapping functions such as ~r(y;\barr, )=\barr\left[1 + \frac1 ~\tan( y/2)\right]~ and ~r(y;\barr)=\barr \left[ \frac1+ y1-y \right]~ transform the conventional radial Schr\" odinger equation into equivalent alternate forms \vspace-2mm \fracd² (y)dy²~=~ \left[\frac ²4+\left(\frac2µ \hbar² \right) g²(y) [E - U(r(y))]\right] (y) \hspace7mm and \hspace7mm \fracd² (y)dy²~=~\left(\frac2µ\hbar²\right) g²(y)\left[E - U(r(y)) \right] (y) \vspace-2mm

respectively, in which g(y)=dr(y)/dy .~ Such transformed equations are defined on the finite domain y\in [-1,1], and they may be solved routinely using standard numerical methods at all energies up to and including the potential asymptote. At the energy of the potential asymptote, the s-wave scattering length a_s can be expressed in terms of the logarithmic derivative of the wave function (y) at the right-hand boundary point:

\vspace-2mm a_s~=~\barr\left[\frac2 ~\frac1 (y)~\fracd (y) dy+1\right]_y=1 \hspace7mm and \hspace7mm a_s~=~\barr\left[ 2~\frac1 (y)~\fracd (y)dy~-1\right]_y=1

The required logarithmic derivative of (y) can be obtained efficiently by direct outward integration of the differential equation all the way to the end point y\!=\!1, which corresponds to the limit r\to . This zero-energy wavefunction may also be combined with wavefunctions for ordinary bound states generated in the same manner^a to calculate photoassociation absorption matrix elements using any appropriately modified Franck-Condon computer program.

skip VVM is grateful to INTAS grant 06-1000014-5964 for support.