Toward a Unique Definition of the Local Spin

In this work, we demonstrate that there is a continuum of different formulations for the decomposition of ⟨Ŝ(2)⟩ that fulfills all physical requirements imposed to date. We introduce a new criterion based upon the behavior of single-electron systems to fix the value of the parameter defining that continuum, and thus we put forward a new general formula applicable for both single-determinant and correlated wave functions. The numerical implementation has been carried out in the three-dimensional physical space for several atomic definitions. A series of representative closed-shell and open-shell systems have been used to illustrate the performance of this new decomposition scheme against other existing approaches. Unlike other decompositions of ⟨Ŝ(2)⟩, the new scheme provides very small local-spin values for genuine diamagnetic molecules treated with correlated wave functions, in conformity with the physical expectations.