  With the help of newly developed X-ray free-electron laser (XFEL) sources,
creating double core holes simultaneously at the same or different atomic sites
in a molecule has now become possible. Double core hole (DCH) X-ray emission is
a new form of X-ray nonlinear spectroscopy that can be studied with a XFEL.
Here we computationally explore the metal K-edge valence-to-core (VtC) X-ray
emission spectroscopy (XES) of metal/metal and metal/ligand double core hole
states in a series of transition metal complexes with time-dependent density
functional theory. The simulated DCH VtC-XES signals are compared with
conventional single core hole (SCH) XES signals. The energy shifts and
intensity changes of the DCH emission lines with respect to the corresponding
SCH-XES features are fingerprints of the coupling between the second core hole
and the occupied orbitals around the DCHs that contain important chemical
bonding information of the complex. The core hole localization effect on DCH
VtC-XES is also briefly discussed. We theoretically demonstrate that DCH XES
provides subtle information on the local electronic structure around metal
centers in transition metal complexes beyond conventional linear XES. Our
predicted changes from calculations between SCH-XES and DCH-XES features should
be detectable with modern XFEL sources.
