Repeated computations on the same molecular system, but with different geometries, are often performed in quantum chemistry, for instance, in ab-initio molecular dynamics simulations or geometry optimizations. While many efficient strategies exist to provide a good guess for the self-consistent field procedure, which is usually the main computational task to be performed, little is known on how to efficiently exploit in this direction the abundance of information generated during the many computations. In this article, we present a strategy to provide an accurate initial guess for the density matrix within the self-consistent field iterations for parametrized Hartree-Fock problems where the nuclear coordinates are parametrized by a few user-specified parameters such as normal modes or collective variables in general. Our approach is based on an offline-stage where the Hartree-Fock eigenvalue problem is solved for some particular parameter values and an online-stage where the initial guess is computed very efficiently for any new parameter value. The method is based on non-linear approximations of density matrices, which belong to a non-linear manifold that is isomorphic to the Grassmann manifold, by means of the so-called Grassmann exponential and logarithm in order to map the manifold onto the tangent space and thus providing the correct geometrical setting accounting for the manifold structure when working with subspaces rather than functions itself. Numerical tests on different amino acids show promising initial results.