Invariance of magnetic-field induced current density to a continuous transformation of the origin of the coordinate system

The quantum mechanical current density induced by a magnetic field in the electrons of a molecule evaluated via theoretical schemes based on 'continuous transformation of the origin of the current density' (CTOCD), is independent of the origin of the coordinate system, irrespective of other approximations retained in the calculation. Nuclear magnetic shieldings determined by continuous transformation of the origin of the current density are intrinsically origin independent. These statements are also proven for damped CTOCD schemes. Integral conditions for charge-current conservation, equivalent to those for gauge invariance, are satisfied exactly within CTOCD methods for symmetry reasons, e.g., for centrosymmetric molecules.