# Difficulties in operator-based formulation of the bulk quadrupole moment

## TL;DR

This paper discusses the challenges in formulating the bulk quadrupole moment in insulators, highlighting issues with recent operator-based approaches and emphasizing the complexity of extending dipole formulations to higher multipoles.

## Contribution

The paper critically analyzes recent proposals for operator-based formulations of the bulk quadrupole moment, identifying key issues and limitations.

## Key findings

- Recent operator-based formulas for quadrupole moments have unresolved issues.
- The extension from dipole to quadrupole moments is non-trivial and problematic.
- The paper clarifies the fundamental difficulties in defining higher-order multipole moments.

## Abstract

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 at generalizing Resta's formula to higher-order multipoles. We point out several issues in the recent proposals.