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.

本文探讨了单值函数（single-valued function）及其相关公式在球对称范德瓦尔斯势（spherically symmetric van der Waals potentials）下散射相移（scattering phase shifts）的绝对定义与计算中的优势。尽管该表达式本身早在21世纪初便已为人所知[K. Chadan、R. Kobayashi与T. Kobayashi，《数学物理杂志》（J. Math. Phys.），第42卷，4031页（2001年）]，但截至目前，关于其有效性的数值证据仍寥寥无几。本文旨在赋予该方法应有的学术认可，并使其作为标准计算方法的替代方案得到更广泛的传播。这一点尤为引人关注，因为在传统标准方法中，获取绝对散射相移（absolute scattering phase shifts）并非易事，需执行额外操作方可实现。本文阐述了该公式如何通过变相位法（variable-phase approaches），从两种被广泛接受的相关方法中推导得出，并将其计算性能与这两种方法进行了对比。本文以氦-氦（He–He）相互作用体系为研究对象，针对其两项核心热物理性质——即⁴He与³He的第二维里系数（second virial coefficients）及声学维里系数（acoustic virial coefficients）展开研究，以验证该公式的实际效果。对于这些维里系数而言，绝对散射相移的使用是不可或缺的。与相函数概念及其与沃尔泰拉积分方程（Volterra equation）的关联相关的其他重要内容，详见补充材料。