ADAPTIVE ANALYTIC MAPPING PROCEDURES FOR SIMPLE AND ACCURATE CALCULATION OF SCATTERING LENGTHS AND PHOTOASSOCIATION ABSORPTION INTENSITIES

\underline{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; r̄, α) = r̄[1 + \frac{1}{α} tan(πy/2)] and r(y; r̄) = r̄[\frac{1+y}{1−y} ] transform the conventional radial Schrödinger equation into equivalent alternate forms

\frac{d^{2}ϕ(y)}{dy^{2}} = [\frac{π^{2}}{4} + (\frac{2μ}{h̄^{2}}) g^{2}(y) [E − U(r(y))]] ϕ(y)  and  \frac{d^{2}ϕ(y)}{dy^{2}} = (\frac{2μ}{h̄^{2}}) g^{2}(y) [E − U(r(y))] ϕ(y)

respectively, in which g(y) = dr(y)/dy.^{a} Such transformed equations are defined on the finite domain y ∈ [−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 ϕ(y) at the right-hand boundary point:

a_{s} = r̄[\frac{2}{πα} \frac{1}{ϕ(y)} \frac{dϕ(y)}{dy} + 1]_{y=1}  and  a_{s} = r̄[ 2 \frac{1}{ϕ(y)} \frac{dϕ(y)}{dy} − 1]_{y=1}

The required logarithmic derivative of ϕ(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 → ∞. 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. VVM is grateful to INTAS grant 06-1000014-5964 for support.

a V.V. Meshkov, A.V. Stolyarov, and R.J. Le Roy, Phys. Rev. A 78, 052510 (2008).