What's the lifetime of the excited state when you shine light on molecules? Let's say, when x-ray hits a molecule, an electron from an inner shell absorbs the energy and flies away, so there's a hole waiting for an electron to come down and fill it. How long does this hole last? I know its lifetime may depend on the species of the molecule, but is there a general value for all small molecules, like a nanosecond? Or could you show me a specific way to calculate the lifetimes of excited states?
 A: Core-hole lifetimes have been tabulated by Krause and Oliver. The widths there are given as the life-time broadening in eV. The time is then given by the Heisenberg uncertainty relation. A calculator page on Hyperphysics gives 0.66 femtosecond for a 1 eV linewidth.
Core-hole decay may take more time than what it takes for a molecule to fly apart. I wrote an article about a dissociative core-excited state of the oxygen molecule, where the spectrum shows evidence of free-atom-like Auger lines. 
A: No, there isn't really a one-size-fits-all timescale. The lifetime of excited states can vary wildly from molecule to molecule and between different excited states of the same molecule.
For single electronic excitations, a good rule of thumb is about a nanosecond, but some metastable states (at least in atoms) can last all the way up to millisecond or multi-second ranges.
However, the specific process you described isn't really an excited state - it's an Auger process, which is enacted by the Coulomb interaction between the different configurations, instead of a dipole coupling (which is generally much weaker). Auger decays are typically blazingly fast, normally clocking in between a few picoseconds and a few femtoseconds, depending on the system.
And, while we're here: the measurement of Auger decay lifetimes in real time is a current and active research field. Good keywords to search for are time-resolved Auger decay and attosecond transient absorption spectroscopy, to get recent literature that probes this area.
A: The excited state lifetime varies a lot. Typically ns range for fluorescent molecules
