  Quasi-saddles or inherent saddles of the potential energy surface, $U$, of a
liquid are defined as configurations which correspond to absolute minima of the
pseudo-potential surface, $W =\wf$, as identified by a multi-dimensional
minimisation procedure. The sensitivity of statistical properties of inherent
saddles to the convergence criteria of the minimisation procedure is
investigated using, as a test system, a simple liquid bound by a quadratically
shifted Lennard-Jones pair potential with continuous zeroth, first and second
derivatives at the cut-off distance. The variation in statistical properties of
saddles is studied over a range of error tolerances spanning five orders of
magnitude. The largest value of the tolerance corresponds to that used for the
unshifted LJ liquids in a previous work (J. Chem. Phys. {\bf 115}, 8784
(2001)). Based on our results, it is clear that there are no qualitative
changes in statistical properties of saddles over this range of error
tolerances and even the quantitative changes are small. The lowest magnitude
eigenvalue, $| \omega_0^2|$, of the Hessian is, however, found to be very
sensitive to the tolerance; as the tolerance is decreased, $| \omega_0^2|$ is
found to show an overall decrease. This indicates that if convergence criteria
are not strict, absolute or low-lying minima of $W(\br)$ will be diagnosed as
having no inflexion directions. The results also show that it is not possible
to set up an unambiguous numerical criterion to further classify the
quasi-saddles into true saddles which contain no zero curvature,
non-translational normal modes and inflexion points which have one or more
zero-curvature normal mode directions.
