# Energy-weighted sum rule for nuclear density functional theory

## TL;DR

This paper derives a gauge-invariant expression for the energy-weighted sum rule in nuclear density functional theory, applicable without assuming a Hamiltonian, and verifies it through numerical calculations for various nuclei.

## Contribution

It presents a novel derivation of the energy-weighted sum rule that does not rely on the Hamiltonian and includes gauge symmetry considerations, extending previous methods.

## Key findings

- Derived gauge-invariant sum rule expressions for nuclear density functionals.
- Numerical verification for multipole operators up to L=3 in selected nuclei.
- Identified the importance of local gauge invariance of certain densities.

## Abstract

The expressions for the energy-weighted sum rule of the isoscalar and isovector coordinate operators are derived based on the second-order fluctuation of the local densities. Conventional derivation of the Thouless theorem for the energy-weighted sum rule is based on the double commutator of the Hamiltonian, while the present derivation does not assume a Hamiltonian operator and is applicable to nuclear energy density functionals. The expressions include the contribution of the local gauge symmetry breaking of the energy density functional. It is shown that the local gauge invariance of the kinetic and current densities and kinetic pair density is important, while all the other local densities do not contribute to the energy-weighted sum rule of the coordinate operators. The finite-amplitude method calculations are performed and the expressions for the energy-weighted sum rule are numerically examined for the isoscalar and isovector multipole operators up to $L=3$ for selected spherical and axially deformed nuclei.