  Ultrafast x-ray scattering experiments are routinely analyzed in terms of the
isotropic scattering component. Here we present an analytical method for
calculating total isotropic scattering directly from ab initio two-electron
densities of ground and excited electronic states. The method is generalized to
compute isotropic elastic, inelastic, and coherent mixed scattering. The
computational results focus on the potential for differentiating between
electronic states and on the composition of the total scattering in terms of
elastic and inelastic scattering. By studying the umbrella motion in the first
excited state of ammonia, we show that the associated electron density
redistribution leaves a comparably constant fingerprint in the total signal
that is similar in magnitude to the contribution from the changes in molecular
geometry.
