  The aim of this article is to propose a new reduced-order modelling approach
for parametric eigenvalue problems arising in electronic structure
calculations. Namely, we develop nonlinear reduced basis techniques for the
approximation of parametric eigenvalue problems inspired from quantum chemistry
applications. More precisely, we consider here a one-dimensional model which is
a toy model for the computation of the electronic ground state wavefunction of
a system of electrons within a molecule, solution to the many-body electronic
Schr\"odinger equation, where the varying parameters are the positions of the
nuclei in the molecule. We estimate the decay rate of the Kolmogorov n-width of
the set of solutions for this parametric problem in several settings, including
the standard L2-norm as well as with distances based on optimal transport. The
fact that the latter decays much faster than in the traditional L2-norm setting
motivates us to propose a practical nonlinear reduced basis method, which is
based on an offline greedy algorithm, and an efficient stochastic energy
minimization in the online phase. We finally provide numerical results
illustrating the capabilities of the method and good approximation properties,
both in the offline and the online phase.
