*s*-wave scattering length

^{1}, Andrey V. Stolyarov

^{1,a)}and Robert J. Le Roy

^{2,b)}

### 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 *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* = ∞. In particular, for an arbitrary interaction potential that dies off as fast as 1/*r* ^{ n } for *n* ⩾ 4, the modified wave function ϕ(*y*) obtained by using the two-parameter mapping function has no singularities, and For a well bound potential with equilibrium distance *r* _{ e }, the optimal mapping parameters are and . 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 *a* _{ s }-values both for model Lennard-Jones (2*n*, *n*) potentials and for realistic published potentials for the Xe–e^{−}, ), and ^{3, 4} systems. Use of this same transformed Schrödinger equation was previously shown [V. V. Meshkov *et al.*, Phys. Rev. A78, 052510 (2008)] to ensure the efficient calculation of all bound levels supported by a potential, including those lying extremely close to dissociation.

This work has been supported by the Russian Foundation for Basic Research by grant 10-03-00195a, and by Natural Sciences and Engineering Research Council (NSERC) Canada. The Moscow team is also grateful for partial support from the Federal Program “Scientists and Educators for an Innovative Russia 2009-2013” (Contract No. P 2280).

I. INTRODUCTION

II. ADAPTIVE MAPPING PROCEDURE FOR SCATTERING LENGTH CALCULATIONS

A. Reduced variable transformation of the radial equation

B. Developing a formula for the scattering length

C. Introduction of mapping functions

D. Determination of optimal mapping parameters

III. IMPLEMENTATION, TESTING, AND DISCUSSION

A. Model-potential applications and tests

B. Applications to “real” systems

IV. CONCLUSIONS

### Key Topics

- Wave functions
- 28.0
- Atom scattering
- 5.0
- Dissociation
- 4.0
- Helium-4
- 4.0
- Interatomic potentials
- 4.0

## Figures

Comparison of the original ψ(*r*) and modified unbound wave functions at zero energy for the state of Cs_{2} (see Ref. 15), as calculated using the Numerov method (see Refs. 21 and 29). The scaling function *g* _{tg}(*r*) was calculated from Eq. (35) using mapping parameters and β = 2.

Comparison of the original ψ(*r*) and modified unbound wave functions at zero energy for the state of Cs_{2} (see Ref. 15), as calculated using the Numerov method (see Refs. 21 and 29). The scaling function *g* _{tg}(*r*) was calculated from Eq. (35) using mapping parameters and β = 2.

Mapping density functions ρ(*r*) = *dy*/*dr* = 1/*g*(*r*) for the state of Cs_{2} (see Ref. 15) calculated: (i) within the framework of the conventional WKB approximation (39), (ii) using the one-parameter mapping function (24) with , and (iii) using the two-parameters mapping function (31) with and β = 2. ψ(*r*) is the associated zero-energy wavefunction.

Mapping density functions ρ(*r*) = *dy*/*dr* = 1/*g*(*r*) for the state of Cs_{2} (see Ref. 15) calculated: (i) within the framework of the conventional WKB approximation (39), (ii) using the one-parameter mapping function (24) with , and (iii) using the two-parameters mapping function (31) with and β = 2. ψ(*r*) is the associated zero-energy wavefunction.

Convergence tests for *a* _{ s }-values calculated in double precision arithmetic for the model 15-level LJ(2*n*, *n*) potentials of Table I using the one-parameter mapping function (24) with [Å]. Calculations performed using Johnson's log-derivative method alone are denoted “JLD,” while the results labeled “JLD-RE” were obtained by also applying a (*N*,*N*/2)-Richardson extrapolation procedure to those results.

Convergence tests for *a* _{ s }-values calculated in double precision arithmetic for the model 15-level LJ(2*n*, *n*) potentials of Table I using the one-parameter mapping function (24) with [Å]. Calculations performed using Johnson's log-derivative method alone are denoted “JLD,” while the results labeled “JLD-RE” were obtained by also applying a (*N*,*N*/2)-Richardson extrapolation procedure to those results.

Convergence tests for *a* _{ s }-values calculated by the JLD-RE(*N*,*N*/2) method in normal double-precision arithmetic (solid curves) for the five model LJ(2*n*, *n*) potentials of Table I using the two-parameters mapping function of Eq. (31) with the optimal parameters β_{opt} listed there. The dashed lines present results obtained by using quadruple-precision arithmetic.

Convergence tests for *a* _{ s }-values calculated by the JLD-RE(*N*,*N*/2) method in normal double-precision arithmetic (solid curves) for the five model LJ(2*n*, *n*) potentials of Table I using the two-parameters mapping function of Eq. (31) with the optimal parameters β_{opt} listed there. The dashed lines present results obtained by using quadruple-precision arithmetic.

Relative errors (42) in *a* _{ s }-values obtained for the 15-level LJ(12, 6) potential (41) as functions of the mapping parameter β, where *N* is the number of grid points used in the JLD method.

Relative errors (42) in *a* _{ s }-values obtained for the 15-level LJ(12, 6) potential (41) as functions of the mapping parameter β, where *N* is the number of grid points used in the JLD method.

Optimal mapping parameter β_{opt} as a function of the dissociation energy of 15-level LJ(2*n*, *n*) potentials (41) for *n* = 4, 5, and 6.

Optimal mapping parameter β_{opt} as a function of the dissociation energy of 15-level LJ(2*n*, *n*) potentials (41) for *n* = 4, 5, and 6.

Optimal mapping parameters β_{opt} determined for the bound vibrational levels *v* ∈ [0, 14] of the model 15-level LJ(2*n*, *n*) potentials (41) for *n* = 4, 5, and 6. The dashed horizontal lines correspond to the β_{opt} values implied by Eq. (43).

Optimal mapping parameters β_{opt} determined for the bound vibrational levels *v* ∈ [0, 14] of the model 15-level LJ(2*n*, *n*) potentials (41) for *n* = 4, 5, and 6. The dashed horizontal lines correspond to the β_{opt} values implied by Eq. (43).

Comparison of convergence behavior for scattering lengths implied by the ^{3} potential (see Ref. 28) as calculated by the JLD(*N*) and JLD-RE(*N*, *N*/2) procedures using smooth vs. non-smooth long-range damping functions of Eq. (47).

Comparison of convergence behavior for scattering lengths implied by the ^{3} potential (see Ref. 28) as calculated by the JLD(*N*) and JLD-RE(*N*, *N*/2) procedures using smooth vs. non-smooth long-range damping functions of Eq. (47).

Convergence of calculated scattering lengths for the Xe–e^{−}, ^{3, 4} , and systems with respect to the lower bound of the integration interval, *r* _{min}, where *r* _{0} is the distance where *U*(*r*) = 0.

Convergence of calculated scattering lengths for the Xe–e^{−}, ^{3, 4} , and systems with respect to the lower bound of the integration interval, *r* _{min}, where *r* _{0} is the distance where *U*(*r*) = 0.

## Tables

The *s*-wave scattering wavelengths *a* _{ s } (in Å) calculated for the LJ(2*n*, *n*) potentials defined by Eq. (41) (Ref. 29). All models have equilibrium distance *r* _{ e } = 1 [Å] and the reduced mass is set as μ = 16.85762920 [a.u.] so that in “spectroscopists units,” the scaling factor ℏ^{2}/2μ = 1 [cm^{−1} Å^{2}]. is the well depth in cm^{−1}, while *v* _{max} is the vibrational quantum number of the last bound level supported by the potential.

The *s*-wave scattering wavelengths *a* _{ s } (in Å) calculated for the LJ(2*n*, *n*) potentials defined by Eq. (41) (Ref. 29). All models have equilibrium distance *r* _{ e } = 1 [Å] and the reduced mass is set as μ = 16.85762920 [a.u.] so that in “spectroscopists units,” the scaling factor ℏ^{2}/2μ = 1 [cm^{−1} Å^{2}]. is the well depth in cm^{−1}, while *v* _{max} is the vibrational quantum number of the last bound level supported by the potential.

Comparison of *s*-wave scattering lengths *a* _{ s } (in ) obtained by applying the JLD-RE(*N*,*N*/2) procedure (Ref. 29) to realistic published potentials for Xe–e^{−} (Ref. 27), (Refs. 12 and 13 and 15), and ^{3, 4} (Ref. 28) interactions, using the two-parameter mapping function of Eq. (31). The optimal mapping parameters β_{opt} were determined by minimization the functional (36), while the values (in .) were fixed at the equilibrium distance of the relevant potential, .

Comparison of *s*-wave scattering lengths *a* _{ s } (in ) obtained by applying the JLD-RE(*N*,*N*/2) procedure (Ref. 29) to realistic published potentials for Xe–e^{−} (Ref. 27), (Refs. 12 and 13 and 15), and ^{3, 4} (Ref. 28) interactions, using the two-parameter mapping function of Eq. (31). The optimal mapping parameters β_{opt} were determined by minimization the functional (36), while the values (in .) were fixed at the equilibrium distance of the relevant potential, .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content