The recent fabrication of optical traps using super-continuum light impresses the need for a theory of inter-particle interactions under such conditions. Development of the theory provides a basis for calculation of the observations expected under experimental conditions. An expression for the inter-particle potential energy induced by continuum states of light is first derived using quantum electrodynamics. This energy expression is cast as a function of the spectral irradiance of the light, and the electric susceptibility of the interacting particles. Specific results are derived for light with a Lorentzian spectrum. It is shown that by filtering part of the spectrum, it is possible to exert control over the length of linear particle chains organized along the Poynting vector. The results exhibit scope for the optical fabrication of moldable structures using broadband light.