  Laying a basis for molecularly specific theory for the mobilities of ions in
solutions of practical interest, we report a broad survey of velocity
autocorrelation functions (VACFs) of Li$^+$ and PF$_6{}^-$ ions in water,
ethylene carbonate, propylene carbonate, and acetonitrile solutions. We extract
the memory function, $\gamma(t)$, which characterizes the random forces
governing the mobilities of ions. We provide comparisons, controlling for
electrolyte concentration and ion-pairing, for van~der~Waals attractive
interactions and solvent molecular characteristics. For the heavier ion
(PF$_6{}^-$), velocity relaxations are all similar: negative tail relaxations
for the VACF and a clear second relaxation for $\gamma\left(t\right)$, observed
previously also for other molecular ions and with \emph{n}-pentanol as solvent.
For the light Li$^+$ ion, short time-scale oscillatory behavior masks simple,
longer time-scale relaxation of $\gamma\left(t\right)$. But the corresponding
analysis of the \emph{solventberg} Li$^+\left(\mathrm{H}_2\mathrm{O}\right)_4$
does conform to the standard picture set by all the PF$_6{}^-$ results.

Soft Condensed Matter;Chemical Physics