  A molecular cluster's inner valence ionized state undergoes autoionization,
which is nonlocal by nature. In a molecular system, when the inner valence's
ionization potential (IP) is higher than the double ionization energy (DIP), it
is energetically favorable for the initially ionized system to emit a secondary
electron and reach a final state which is lower in energy. This relaxation
usually happens via intermolecular coulombic decay (ICD) or electron
transfer-mediated decay (ETMD). We have choosen the Na$^+$-(H$_2$O)$_{n=1-5}$
and Mg$^{2+}$-(H$_2$O)$_{m=1-5}$ cluster as the test systems. These systems are
also found in the human body, which makes this study important. We have
calculated the IP, DIP values, and the lifetime of Na-2s and Mg-2s temporary
bound states (TBSs) in these clusters to study the effect of solvation on IP,
DIP, and the lifetime of Na-2s and Mg-2s TBSs. We observe a considerable
increase (96\%) in the lifetime of the Na-2s TBS in the second solvated shell
structure in Na$^+$-(H$_ 2$O)$_{n=2}$ compared to the first solvated one.
However, the increase in the lifetime of the Mg-2s state in the second
solvation shell is only 33\%. We have revealed the different factors that
affect the lifetime of TBSs and which type of decay process (ICD or ETMD) is
dominant. We have shown how the charge of metal ions and increased water
molecules affect the decay rate. We have shown that the decay of Mg-2p is also
possible in all magnesium-water clusters, but it is not valid for the decay of
Na-2p.
