Static and optical anapole magnetizabilities and polarizabilities

The anapolar response of a molecule exposed to a nonhomogeneous magnetic field B with spatially uniform curl C = ∇ × B is rationalized via second-rank tensors, a nonsymmetric a_αβ, and a symmetric b_αβ, referred to as static anapole magnetizabilities, which can be evaluated by quantum-mechanical Rayleigh-Schrödinger perturbation theory or allowing for the definitions of electronic current densities J^B(r) and J^C(r) induced in the electron cloud. The isotropic part of b_αβ is even under the fundamental symmetry operations of charge conjugation C, parity P, and time reversal T and does not vanish for all matter and antimatter. The isotropic part of a_αβ is even under C and T, but odd under P, and is exhibited only by chiral compounds. In the presence of optical fields, represented for simplicity by a monochromatic plane wave of frequency ω, dynamic anapole magnetizabilities and various anapole polarizabilities are taken into account. Assuming, within the electric quadrupole approximation for the impinging wave, that the electric field at the origin of the coordinate system is E(0), with uniform gradient ∇E, and magnetic field is B, the anapolar response is interpreted by second-rank a_αβ(ω), a_αβ ^'(ω), f_αβ(ω), f_αβ ^'(ω) and third-rank g_α,βγ(ω), g_α,βγ ^'(ω) frequency-dependent tensors. The same basic definitions are arrived at introducing frequency-dependent electronic current densities J^B(r, ω) and J^C(r, ω). As the frequency-dependent anapole susceptibilities depend on the origin of the coordinate system, relationships connecting them for two different origins are reported.