We present a new nonlinear variational framework for simultaneously computing
ground and excited states of quantum systems. Our approach is based on
approximating wavefunctions in the linear span of basis functions that are
augmented and optimized \emph{via} composition with normalizing flows. The
accuracy and efficiency of our approach are demonstrated in the calculations of
a large number of vibrational states of the triatomic H$_2$S molecule as well
as ground and several excited electronic states of prototypical one-electron
systems including the hydrogen atom, the molecular hydrogen ion, and a carbon
atom in a single-active-electron approximation. The results demonstrate
significant improvements in the accuracy of energy predictions and accelerated
basis-set convergence even when using normalizing flows with a small number of
parameters. The present approach can be also seen as the optimization of a set
of intrinsic coordinates that best capture the underlying physics within the
given basis set.

我们提出了一种新的非线性变分框架，用于同时计算量子系统的基态和激发态。通过使用归一化流将基函数的线性空间进行扩充和优化的方法，我们展示了该方法在计算三原子 H2S 分子的多个振动态以及典型单电子系统（如氢原子、分子氢离子和碳原子）的基态和几个激发态方面的准确性和效率。这些结果表明，即使使用参数较少的归一化流，能量预测的准确性和基组收敛加速也得到了显著改进。该方法也可以看作是在给定基组中最好地捕捉底层物理的一组内在坐标的优化。