Calculations of highly excited and delocalized molecular vibrational states is a computationally challenging task, which strongly depends on the choice of coordinates for describing vibrational motions. We introduce a new method that utilizes normalizing flows (parametrized invertible functions) to optimize vibrational coordinates to satisfy the variational principle. This approach produces coordinates specifically tailored to the vibrational problem at hand, significantly increasing the accuracy and enhancing basis set convergence of calculated energy spectrum. The efficiency of the method is demonstrated in calculations of the 100 lowest excited vibrational states of H$_2$S, H$_2$CO, and HCN/CNH. The method effectively captures the essential vibrational behavior of molecules by enhancing the separability of the Hamiltonian. We further demonstrate that the optimized coordinates are transferable across different levels of basis set truncation, enabling a cost-efficient protocol for computing vibrational spectra of high-dimensional systems.

翻译：计算高度激发和离域的分子振动态是一项计算上具有挑战性的任务，其效果在很大程度上取决于描述振动运动的坐标选择。我们提出了一种新方法，该方法利用归一化流（参数化的可逆函数）来优化振动坐标以满足变分原理。这种方法生成的坐标专门针对当前振动问题进行了定制，显著提高了计算能谱的精度并增强了基组收敛性。该方法的效率在计算H$_2$S、H$_2$CO和HCN/CNH的100个最低激发振动态时得到了验证。该方法通过增强哈密顿量的可分离性，有效地捕捉了分子的基本振动行为。我们进一步证明，优化后的坐标在不同基组截断水平之间具有可迁移性，从而为计算高维系统的振动光谱提供了一种经济高效的方案。