Sum rule for a Schiff-like dipole moment

TL;DR

This paper derives a modified energy-weighted sum rule for a Schiff-like dipole operator, accounting for correction terms dependent on the number of particles, and applies it to sodium clusters using RPA.

Contribution

It introduces a new sum rule correction for Schiff-like dipole moments and compares analytical and RPA-based approaches, with application to sodium clusters.

Findings

RPA results obey the modified sum rule for Na clusters

Analytical expansion provides explicit N dependence

Corrections depend on the number of system components

Abstract

The energy-weighted sum rule for an electric dipole transition operator of a Schiff type differs from the Thomas-Reiche-Kuhn sum rule by several corrective terms which depend on the number of system components, ${\cal N}$. The deviations are evaluated by two distinct approaches. One of them is based on the charge density expansion around an uniform distribution and provides an analytical ${\cal N}$ dependence for the sum rule while, the other one hinges of the RPA approach and yields compact expressions for corrections as function of the RPA amplitudes. Although the formalism might be considered for any many body system, we applied it for illustration, to the case of $Na$ clusters. One concludes that the RPA results for $Na$ clusters obey the modified TRK sum rule.