A computer code for the computation of the phase shift in atom-atom and electron-atom potential scattering is presented. The phase shift is the central quantity required for the calculation of all types of scattering cross sections. The program uses the Variable Phase Approach (VPA). This is the only exact method for the direct calculation of the scattering phase shift. All other methods are based on examining the large distance behavior of the exact solution of the Schrödinger equation. Such methods yield the phase shift only modulo $\pi$. The absolute value of the phase shift and its variation with scattering energy is, however, needed for a full understanding of the scattering process, such as for instance in the study of shape resonances and Glory oscillations. The VPA has been sparingly used owing to the instability of the underlying equations and the consequent difficulty of writing computer code to solve them. We present a computer code for the efficient implementation of the VPA method for atom-atom scattering problems over a wide range of scattering energies. The code works for potentials which are singular and for those that are non-singular at the origin. An example of the implementation of the code is given for both an interaction potential with an attractive well and for a purely repulsive potential.

Program summary

Program title: VPA:Variable Phase Approach

Program files doi:

Code Ocean capsule: https://doi.org/VPA: Variable Phase Approach

Licensing provisions: GNU General Public License 3 (GPL)

Programming language: Fortran 90

Nature of problem: To compute the quantum mechanical phase shift in an atom-atom or an electron-atom potential scattering problem. The interaction potential, dependent on a single radial coordinate, is defined by the user.

Solution method: The Variable Phase Approach is used. This is the only available method for the direct calculation of the phase shift. The initial internuclear separation for the integration is chosen with the help of the analytical solution for a hard sphere core for all orbital angular momentum quantum numbers L of interest. The first-order nonlinear equation is integrated numerically with a modified LSODA${}^{\mathrm{19,20}}$ program. The analytic formula for calculation of the Jacobian of the equation is also provided.

Restrictions and unusual features of the program: Computer codes for the atom-atom interaction potential, POTENT1, and for its derivative with respect to the inter nuclear separation, DPOTENT1, must be supplied by a user. The calculation of the spherical Bessel and Neumann functions required is performed explicitly for large l angular momentum quantum numbers where the standard upward iterative generation of the functions becomes unstable.

中文翻译：

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VPA：用于使用可变相位​​方法计算原子-原子电势散射中的相移的计算机程序

介绍了用于计算原子-原子和电子-原子电势散射中相移的计算机代码。相移是计算所有类型的散射截面所需的中心量。该程序使用可变相位​​方法（VPA）。这是直接计算散射相移的唯一精确方法。所有其他方法均基于检查Schrödinger方程精确解的远距离行为。这样的方法仅产生相移模数$\pi$。然而，为了充分理解散射过程，例如在研究形状共振和荣耀振动中，需要相移的绝对值及其随散射能量的变化。由于基本方程式的不稳定性以及随之而来的编写用于解决这些问题的计算机代码的困难，因此很少使用VPA。我们提出了一种计算机代码，用于在广泛的散射能量范围内有效实施VPA方法以解决原子-原子散射问题。该代码适用于在原点处为奇数的电势和对于在源极处为非奇数的电势。给出了代码实现的一个示例，该示例既针对具有吸引力的井的交互作用电位，也针对纯粹的排斥电位。

计划摘要

计划标题： VPA：可变阶段法

程序文件doi：

代码海洋舱：https://doi.org/VPA：可变相位方法

许可条款：GNU通用公共许可证3（GPL）

编程语言： Fortran 90

问题的性质：计算原子原子或电子原子电势散射问题中的量子机械相移。取决于单个径向坐标的交互电位由用户定义。

解决方法：使用可变相位​​方法。这是直接计算相移的唯一可用方法。借助对于所有感兴趣的轨道角动量量子数L的硬球核的解析解决方案，选择了用于积分的初始核间分离。一阶非线性方程与改进的LSODA进行数值积分${}^{\mathrm{19,20}}$程序。还提供了用于计算方程式的雅可比行列式的解析公式。

该程序的限制和异常功能：用户必须提供有关原子-原子相互作用势POTENT1及其有关核间分离的衍生物DPOTENT1的计算机代码。对于大的l角动量量子数，需要进行明确的球形Bessel和Neumann函数的计算，在该函数中，标准的向上迭代生成函数变得不稳定。