We recast existing theory of ultrafast time-resolved x-ray scattering by molecules in the gas phase into a unified and coherent framework based on first-order time-dependent perturbation theory and quantum electrodynamics. The effect of the detection window is analyzed in detail and the contributions to the total scattering signal are discussed. This includes the coherent mixed component caused by interference between scattering amplitudes from different electronic states. A new, detailed, and fully converged simulation of ultrafast total x-ray scattering by excited H2 molecules illustrates the theory and demonstrates that the inelastic component can contribute strongly to the total difference scattering signal, i.e., on the same order of magnitude as the elastic component.
Since δ is related to the angular frequencies of the scattered x-ray photons by a Fourier transform of the linear coherence function C(δ) [see Eq. (12)], the adiabatic approximation solely affects the energy that is transferred between the molecule and the photons. Equation (7) is therefore valid as long as the detector is not sensitive to the corresponding changes in photon energy. In experiments in which the nonresonant scattering signal dσ/dΩ does not resolve energy transfer, this condition is usually met. See also the discussion below Eq. (16).