The decomposition of (non)linear optical properties (NLOPs, e.g., polarizability, hyperpolarizabilities...) is not straightforward because it runs into the issue of origin-dependence: all obtained atomic contributions will depend on your choice of origin. This means the obtained values will change if the arbitrary origin of coordinates is moved, and also the contribution of a given functional group is modulated based on its distance from the origin, rather than just its nature and environment.

These NLOPs are derivatives of the energy. In our work, we decompose the energy into atomic contributions (to then obtain the atomic NLOP contributions), and we show that, without any assumptions or approximations, one can consider only the origin-independent energy terms, and the obtained value, which is fully origin-independent, will be related to the total property by a specific exact factor that depends exclusively on the property, as can easily be shown through a simple mathematical demonstration. This makes it possible to perform a decomposition completely avoiding the terms that introduce origin-dependence (i.e., the dipole moment operator).

In our current studies, we consider the possibility of achieving full origin-independence through other means. Here we do a systematic study and a comparison of the different methods, across a broad number of systems and molecular sets.