Four-Dimensional Scaling of Dipole Polarizability in Quantum Systems

Polarizability is a key response property of classical and quantum systems, which has an impact on intermolecular interactions, spectroscopic observables, and vacuum polarization. The calculation of polarizability for quantum systems involves an infinite sum over all excited (bound and continuum) states, concealing the physical interpretation of polarization mechanisms and complicating the derivation of efficient response models. Approximate expressions for the dipole polarizability, α, rely on different scaling laws α ∝ R³, R⁴, or R⁷, for various definitions of the system radius R. In this work, we consider a range of quantum systems of varying spatial dimensionality and having qualitatively different spectra, demonstrating that their polarizability follows a universal four-dimensional scaling law α=C(4μq²/h²)L⁴, where μ and q are the (effective) particle mass and charge, C is a dimensionless constant of order one, and the characteristic length L is defined via the L2-norm of the position operator. The applicability of this unified formula is demonstrated by accurately predicting molecular polarizabilities from the effective polarizability of their atomic constituents.