A shortcut to quantum-mechanical absolute scattering phase-shift computations in van der Waals systems

We discuss the advantages of a single-valued function (and related formula) for the absolute definition and computation of scattering phase shifts in spherically symmetric van der Waals potentials. Although the expression, as such, is known since the 2000s [K. Chadan, R. Kobayashi, and T. Kobayashi, J. Math. Phys. <b>42</b>, 4031 (2001)], only little numerical evidence of its effectiveness has been available so far. Our effort, here, is to give this device the recognition it deserves and to make it more widely known as an alternative to standard methods. This is all the more interesting as in the standard approaches the access to absolute is not straightforward but needs additional operations to be performed. We show how the formula can be derived, as a consequence of variable-phase approaches from the two broadly accepted methods for , and compare its performance with these methods. He–He and two of its keynote thermophysical properties, namely, the ⁴He and ³He second virial and acoustic virial coefficients are being studied for the purpose. The use of absolute is mandatory for those coefficients. Other important points related to the concept of phase function and its connection with Volterra equation are given in Supplementary material.

我们探讨了单值函数（single-valued function）及其相关公式，用于球对称范德瓦尔斯势（spherically symmetric van der Waals potentials）下散射相移（scattering phase shifts）的绝对定义与计算。尽管该形式自2000年代起便已为人所知[K. Chadan、R. Kobayashi与T. Kobayashi，《J. Math. Phys.》第42卷，4031页（2001年）]，但截至目前，支撑其有效性的数值证据仍较为有限。本文旨在为这一方法正名，将其作为标准计算方法的替代方案加以推广。尤为值得关注的是，传统标准方法难以直接获取绝对相移，需执行额外运算步骤方可实现。我们阐述了该公式的推导路径：其可通过变相法（variable-phase approaches）由两种广泛认可的散射相移计算方法导出，并对比了该公式与这两种方法的性能表现。本文以氦-氦相互作用体系为研究对象，针对其两项关键热物理性质展开分析：即⁴He与³He的第二维里系数及声学维里系数。对于这类维里系数的计算而言，绝对相移的使用具有强制性要求。补充材料中还阐述了与相函数概念及其与沃尔泰拉积分方程（Volterra equation）关联相关的其他重要内容。