Imaging of transition charge densities involving carbon core excitations by all X-ray sum-frequency generation

X-ray diffraction signals from the time-evolving molecular charge density induced by selective core excitation of chemically inequivalent carbon atoms are calculated. A narrowband X-ray pulse selectively excites the carbon K-edge of the -CH3 or -CH2F groups in fluoroethane (CH3-CH2F). Each excitation creates a distinct core coherence which depends on the character of the electronic transition. Direct propagation of the reduced single-electron density matrix, using real-time time-dependent density functional theory, provides the timeevolving charge density following interactions with external fields. The interplay between partially filled valence molecular orbitals upon core excitation induces characteristic femtosecond charge migration which depends on the core-valence coherence, and is monitored by the sum-frequency generation diffraction signal.