  The Berry connection and curvature are key components of electronic structure
calculations for atoms and molecules in magnetic fields. They ensure the
correct translational behavior of the effective nuclear Hamiltonian and the
correct center-of-mass motion during molecular dynamics in these environments.
In this work, we demonstrate how these properties of the Berry connection and
curvature arise from the translational symmetry of the electronic wave function
and how they are fully captured by a finite basis set of London orbitals but
not by standard Gaussian basis sets. This is illustrated by a series of
Hartree-Fock calculations on small molecules in different basis sets. Based on
the resulting physical interpretation of the Berry curvature as the shielding
of the nuclei by the electrons, we introduce and test a series of
approximations using the Mulliken fragmentation scheme of the electron density.
These approximations will be particularly useful in ab initio molecular
dynamics calculations in a magnetic field, since they reduce the computational
cost, while recovering the correct physics and up to 95% of the exact Berry
curvature.
