  We determine the number of complex solutions to a nonlinear eigenvalue
problem on the Grassmannian in its Pl\"ucker embedding. This is motivated by
quantum chemistry, where it represents the truncation to single electrons in
coupled cluster theory. We prove the formula for the Grassmannian of lines
which was conjectured in earlier work with Fabian Faulstich. This rests on the
geometry of the graph of a birational parametrization of the Grassmannian. We
present a squarefree Gr\"obner basis for this graph, and we develop connections
to toric degenerations from representation theory.
