  The relationship between the energy total splitting $\Delta E$ of the
free-ion electron states in the axial crystal-fields and the second moment of
that splitting $\sigma^{2}$ is thoroughly investigated. The non-Kramers and
Kramers states with the quantum number $1\leq J \leq 8$ in the axial
crystal-fields of any multipolar composition but fixed $\sigma^{2}$ are
considered. Since the crystal-field Hamiltonian ${\cal H}_{\rm CF}$ is a
superposition of the three effective multipoles various $\Delta E$ can
correspond to a fixed $\sigma^{2}$ according to the resultant combination of
the independent contributions. This $\Delta E$ variation range is the subject
of the study. For the states under examination $\Delta E$ can take the values
from $2.00\sigma$ to $3.75\sigma$, whereas the difference $\Delta E_{max}-
\Delta E_{min}$, except the states with $J\leq 5/2$, amounts roughly to
$\sigma$. For comparison, the one-multipolar ${\cal H}_{\rm CF}$s yield
accurately defined $\Delta E$ ranging from $2.50\sigma$ to $3.00\sigma$. The
limitations of the allowed $\Delta E$ values exclude rigorously a number of
virtually possible splitting diagrams. The documentary evidence for this
restriction has been supplied in the paper collating the nominally admissible
total energy splittings $\Delta {\cal E}$ (i.e. those preserving the
$\sigma^{2}$) with the $(\Delta E_{min}, \Delta E_{max})$ ranges occurring in
the actual axial crystal-fields. Although the $\Delta E$ unlike the
$\sigma^{2}$ is not an essential characteristic and depends on the reference
frame orientation, it is useful to know its dispersion range, particularly
attempting to assign or verify complex electron spectra.
