Automatic grid construction for few-body quantum-mechanical calculations

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Abstract

An algorithm for generating optimal nonuniform grids for solving the two-body Schrödinger equation is developed and implemented. The shape of the grid is optimized to accurately reproduce the low-energy part of the spectrum of the Schrödinger operator. Grids constructed this way are applicable to more complex few-body systems where the number of grid points is a critical limitation to numerical accuracy. The utility of the grid generation for improving few-body calculations is illustrated through an application to bound states of He trimers.

Highlights

► Accurate three-body calculations require nonuniform grid optimization. ► Accurate reproduction of the two-body spectrum is essential. ► A method of constructing an optimized grid automatically is implemented and tested. ► S-wave optimization is sufficient to reproduce rotationally excited molecular states. ► The accuracy of three-body calculations is improved significantly.

Introduction

The dynamics of few-body systems remains a robust field of research with many practical applications. A number of theoretical advances, coupled with increased computational resources, have lead to significant advances in both the understanding of few-body processes and in the number of physical systems that can be successfully treated with existing, well-tested methodologies (see [1], [2], [3], [4], [5], [6], [7], [8] and references therein). We are, for example, currently developing public source code and a graphical user interface for scientists interested in solving Faddeev equations numerically for a wide array of potential three-body applications. Challenging computations that have been accessible to only a small group of specialists will soon become elementary and well characterized tools used by a large number of practitioners in different fields.

Solving the two-body Schrödinger equation numerically is an elementary exercise in the case of smooth central potentials. Typically, the power of modern computers makes it possible to use even the simplest numerical approaches to perform quite accurate calculations of the low-energy part of the two-body spectrum. Three-or-more-body calculations, however, usually require greater attention to the details of numerical technique, as the required computer resources usually scale geometrically with the growth of dimensionality of the problem, and optimizing any aspect of the solution representation leads to substantial computational savings.

When solving bound state or scattering problems for few-body systems it is important to treat the states of two-body subsystems carefully, as such states represent the asymptotic boundary conditions for the corresponding few-body states. It is also important to minimize the computational cost of reproducing every asymptote of the few-body calculations. The key feature of the Faddeev approach to few-body problem is the asymptotic factorability of the solutions, so that grids constructed for the efficient solution of the two-body problem can be immediately employed with considerable numerical advantage. In previous calculations [9], [10], [11] this optimization has been performed manually. The procedure, however, is very time consuming and difficult for an inexperienced user or a student. Therefore, we needed an automatic procedure of constructing an effective grid representation of the two-body subsystems.

Methods of refining grids automatically are often used in solving various nonlinear evolution equations (hydrodynamical equations, for example) to reproduce discontinuities and other singular features of the solutions. In contrast, our goal is to create a software package specifically designed to solve the quantum-mechanical few-body problem, fully exploiting the features intrinsic to the physical problem and the numerical techniques to achieve high-performance of the resulting code. We therefore needed a solution which is on the one hand more specific to our problem, and on the other hand allows us to construct the grids on the base of some clearly understood physical and mathematical principles, with particular emphasis on reproducing the low-energy part of the two-body Schrödinger operator for subsequent use in more complex few-body calculations.

In this paper we describe and implement a practical nonuniform grid suitable for reproducing the low-energy part of the Schrödinger operator for two-body systems. When applied to systems of more than two particles, this grid permits a several-fold reduction in the number of grid points required for a desired level of numerical accuracy. Equally important, the procedure of constructing the grid is automatic and requires only minimal interference from a user.

Section snippets

Basic principles

We are solving the Schrödinger equation for a two-body system with central potentials using S3,2 or S5,3 Hermite splines and collocations at Gaussian points. We are constructing a nonuniform grid from the requirement of uniformity of the numerical error over the entire range where the solution is constructed.

Let us start from the radial Schrödinger equation in atomic units (a.u.) chosen so that =e=me=1(12μd2dx2+l(l+1)2μx2+V(x)Ei)φ(x;ϵi)=0 where μ is the reduced mass of the two-body system,

Implementation

Before describing the grid construction algorithm itself, we shall briefly outline the discretization procedure for the simplest case of zero angular momentum l=0. For discretization we use the orthogonal collocation scheme [12] with the S5,3 splines [9]. We shall seek for a solution of Eq. (1) an expansion in terms of B-spline basis in the spline space S5,3(Δ) constructed for a given mesh Δφm(x)j=1Mfm,jBj(x). Functions Bj(x) are constructed to satisfy the appropriate boundary conditions.

Convergence of two-body bound states

How practical is the construction? Can we expect any computational savings in few-body calculations when using the optimal mappings compared to widely used simple power or exponential mappings? To answer this question let us study the convergence of the bound state energies for the two-body problem with respect to the number of grid points.

To study the convergence properties of the numerical procedure it is natural to represent the property of interest – the bound state energy for example – as

The three-body calculations with optimized grids

As mentioned in the introduction, the practical value of optimizing the grids for two-body states lies in their application to more complex few-body calculations. In this case, reducing the number of points needed to achieve the required accuracy is critical for saving computational time. The direct use of the two-body optimized grids described in the previous section is especially appropriate in the Faddeev (three-body) [9], [10], [11] or Faddeev–Yakubovsky (four-body) [1], [8] formalism. When

Conclusions

We have presented an approach to constructing an optimized nonuniform grid for use in quantum few-body calculations. The approach is based on the results of Ref. [13], where a grading function which asymptotically minimizes the L2 norm of the interpolation error is introduced. We have slightly modified the optimization criterion to have several low-lying eigenstates of the two-body Hamiltonian interpolated well.

We have studied the convergence properties of the optimized grids. For this purpose

Acknowledgements

This work is supported by the NSF grant PHY-0903956. We wish to thank Dr. Kolganova (JINR, Dubna) for stimulating discussions and independent preliminary testing of the three-body code.

References (15)

  • V.A. Roudnev

    Chem. Phys. Lett.

    (2003)
  • A.K. Motovilov

    Few-Body Systems

    (2008)
  • E.A. Kolganova et al.

    Phys. Part. Nuclei

    (2009)
  • D. Blume et al.

    J. Chem. Phys.

    (2000)
  • Hiroia Suno et al.

    Phys. Rev. A

    (2008)
  • Hiroia Suno et al.

    Phys. Rev. A

    (2010)
  • P. Barletta et al.

    Few-Body Systems

    (2009)
There are more references available in the full text version of this article.

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