  Many basis sets for electronic structure calculations evolve with varying
external parameters, such as moving atoms in dynamic simulations, giving rise
to extra derivative terms in the dynamical equations. Here we revisit these
derivatives in the context of differential geometry, thereby obtaining a more
transparent formalisation, and a geometrical perspective for better
understanding the resulting equations. The effect of the evolution of the basis
set within the spanned Hilbert space separates explicitly from the effect of
the turning of the space itself when moving in parameter space, as the tangent
space turns when moving in a curved space. New insights are obtained using
familiar concepts in that context such as the Riemann curvature. The
differential geometry is not strictly that for curved spaces as in general
relativity, a more adequate mathematical framework being provided by fibre
bundles. The language used here, however, will be restricted to tensors and
basic quantum mechanics. The local gauge implied by a smoothly varying basis
set readily connects with Berry's formalism for geometric phases. Generalised
expressions for the Berry connection and curvature are obtained for a
parameter-dependent occupied Hilbert space spanned by non-orthogonal Wannier
functions. The formalism is applicable to basis sets made of atomic-like
orbitals and also more adaptative moving basis functions (such as in methods
using Wannier functions as intermediate or support bases), but should also
apply to other situations in which non-orthogonal functions or related
projectors should arise. The formalism is applied to the time-dependent quantum
evolution of electrons for moving atoms. The geometric insights provided here
allow us to propose new finite-differences time integrators, and also better
understand those already proposed.
