where the summation is over the number of electrons and r and p are the position and momentum operators. The two representations are denoted length gauge (L in Eq. 3) and velocity gauge (V in Eq. 4). The trace of the tensors in the two choices of gauge is equivalent in exact calculations: 14 3-4 lead to the same OR value, i.e., gauge invariance refers to the equality in Eq. 5. This equivalence is also satisfied for approximate methods with variationally optimized orbitals in the CBS limit. However, achieving gauge invariance in approximate variational methods does not guarantee that the OR is numerically the same as that obtained in an exact calculation. For non-variational methods, like in standard approximations of CC theory, and in general with incomplete basis sets, one obtains numerically distinct values of the OR in Eq. 1 with different choices of gauge.