ADAPTIVE ANALYTIC MAPPING PROCEDURES FOR SIMPLE AND ACCURATE CALCULATION OF SCATTERING LENGTHS AND PHOTOASSOCIATION ABSORPTION INTENSITIES
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Date
2009
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Publisher
Ohio State University
Abstract
We have shown that one and two-parameter analytical mapping functions such as $~r(y;\bar{r}, \alpha)=\bar{r}\left[1\,+\,\frac{1}{\alpha}~\tan(\pi y/2)\right]~$ and $~r(y;\bar{r})=\bar{r} \left[ \frac{1+ y}{1-y} \right]~$ transform the conventional radial Schr{\" o}dinger equation into equivalent alternate forms \vspace{-2mm} $$ \frac{d^2\phi(y)}{dy^2}~=~ \left[\frac{\pi^2}{4}+\left(\frac{2\mu} {\hbar^2} \right) g^2(y) \,[E - U(r(y))]\right]\phi(y) \hspace{7mm} {\rm and} \hspace{7mm} \frac{d^2\phi(y)}{dy^2}~=~\left(\frac{2\mu}{\hbar^2}\right) g^2(y)\left[E - U(r(y)) \right]\phi(y) \vspace{-2mm} $$ % respectively, in which $\,g(y)=dr(y)/dy\,$. \textbf{78}, 052510 (2008).}~ 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 $\phi(y)$ at the right-hand boundary point: \vspace{-2mm} $$ a_s~=~\bar{r}\left[\frac{2} {\pi\alpha}~\frac{1} {\phi(y)}~\frac{d\phi(y)} {dy}+1\right]_{y=1} \hspace{7mm} {\rm and} \hspace{7mm} a_s~=~\bar{r}\left[\,2~\frac{1}{\phi(y)}~\frac{d\phi(y)}{dy}~-1\right]_{y=1} $$ % The required logarithmic derivative of $\phi(y)$ 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 $\,r\to \infty$. 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. \smallskip VVM is grateful to INTAS grant 06-1000014-5964 for support.
Description
\,V.V.\ Meshkov, A.V.\ Stolyarov, and R.J.\ Le Roy, \textit{Phys.~Rev.~A
Author Institution: Guelph-Waterloo Centre for Graduate Work in Chemistry and; Biochemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; Department of Chemistry, Moscow State University, GSP-2 Leninskie Gory; 1/3, Moscow 119991, Russia
Author Institution: Guelph-Waterloo Centre for Graduate Work in Chemistry and; Biochemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; Department of Chemistry, Moscow State University, GSP-2 Leninskie Gory; 1/3, Moscow 119991, Russia