  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.
