Difficulties in operator-based formulation of the bulk quadrupole moment

The electric multipole moments are the most fundamental properties of insulating materials. However, the general formulation of bulk multipoles has been a long standing problem. The solution for the electric dipole moment was provided decades ago by King-Smith, Vanderbilt, and Resta. Recently, there have been attempts of generalizing Resta's formula to higher-order multipoles. In this work, we point out several issues in the recent proposals.