The present study examines the mathematical properties of the free-free (f − f) matrix elements of the full electric field operator, OE(κ, r̅), of the multipolar Hamiltonian. κ is the photon wavenumber. Special methods are developed and applied for their computation, for the general case where the scattering wavefunctions are calculated numerically in the potential of the term-dependent (N − 1) electron core, and are energy-normalized. It is found that, on the energy axis, the f − f matrix elements of OE(κ, r̅) have singularities of first order, i.e., as ε′ → ε, they behave as (ε − ε′)-1. The numerical applications are for f − f transitions in hydrogen and neon, obeying electric dipole and quadrupole selection rules. In the limit κ = 0, OE(κ, r̅) reduces to the length form of the electric dipole approximation (EDA). It is found that the results for the EDA agree with those of OE(κ, r̅), with the exception of a wave-number region k′ = k ± κ about the point k′ = k.
Graphical abstract