Advantages of complex scaling only the most diffuse basis functions in simultaneous description of both resonances and bound states

An accurate description of both bound and resonance states is desirable in the study of molecules. However, it is not a trivial task to accomplish, as theoretical and numerical formalisms developed for describing one do not always fit for describing the other. One such formalism, mainly designed to describe resonances, is the complex scaling method, in which the Hamiltonian internal coordinates are scaled by a complex factor of eiθ. As Kaprálová-Žďánská and Šmydke recently showed in their study on helium, in order to well describe both bound and resonance states, one should use quasi-complete basis sets. Yet, the use of large and dense basis sets is ineffective and highly impractical. In addition, the above-mentioned complex scaling method is unsuitable for a molecular analysis. Not long ago, an evidence that a mixed complex-scaled basis set can serve as a solution to the problem was presented by White et al., however, the behaviour of the bound states was not investigated systematically. In this work, we demonstrate systematically that using such a mixed basis set, in which only diffuse functions are scaled, is an appropriate approach for simultaneous description. On the one hand, using this method, complex scaling can be applied on molecular systems in order to find resonance states. On the other hand, in this method, both the bound and the resonance states can be well described even in relatively small basis sets. In addition, we demonstrate the stability of the bound state in relation to the scaling parameter.在分子物理研究中，精准刻画束缚态（bound states）与共振态（resonance states）均为核心诉求。然而，实现该目标绝非易事：因专为描述其中一类态所开发的理论与数值形式体系，往往无法适配另一类态的刻画需求。其中一类专为共振态设计的形式体系为复标度法（complex scaling method），该方法通过复因子e^(iθ)对哈密顿量的内坐标进行标度。正如Kaprálová-Žďánská与Šmydke近期在氦原子相关研究中所证实的，若要同时精准刻画束缚态与共振态，需采用准完备基组（quasi-complete basis sets）。但使用大尺寸高密度基组不仅计算效率低下，且极不具备实操可行性。此外，前述复标度法并不适用于分子体系的分析工作。不久前，White等人提出混合复标度基组可作为该问题的解决方案，但尚未对束缚态的行为开展系统性探究。本研究系统性地证明，仅对弥散函数（diffuse functions）进行标度的混合基组方案，是实现两类态同时精准刻画的合理途径。一方面，该方法可将复标度法应用于分子体系以探测共振态；另一方面，即便采用相对小型的基组，亦能同时良好地刻画束缚态与共振态。此外，本研究还验证了束缚态对标度参数（scaling parameter）的稳定性。