Normalization is a relatively straight forward process. Before it is printed out, each of the 3 N elements of is scaled by normalization factor , for that particular vibrational mode. The difference is in the numerator of each term in the summation. Gaussian uses rather than 1. Using the elements of yields the consistent results for polyatomic cases, and automatically takes symmtery into consideration.
Gaussian uses be the normalized cartesian displacement.
In other words, in the coordinate system that Gaussian uses, the sum of the squares of the cartesian displacements is 1. In the more common coordinate system for diatomics, the unit length is a unit change in internuclear distance from the equilibrium value. One of the consequences of using this coordinate system is that force constants which you think should be equal are not. Since the Hessian depends only on the electronic part of the Hamiltonian, you would expect the force constants to be the same for these to molecules. In fact, the force constant Gaussian prints out is different.