  The physically correct parametrization of the energy levels of transition
ions in crystals in terms of crystal-field (CF) Hamiltonians ${\cal H}_{\rm
CF}=\sum_{k}\sum_{q}B_{kq}C_{q}^{(k)}$ has to be based on the CF parameters
$B_{kq}$ that lead to the correct CF splitting second moments, both the global
one $\sigma$ and the partial ones $\sigma_{k}$. Only such parametrizations
correspond to the appropriate multipolar structure of the surrounding CF. Each
parametrization being characterized by its own multipolar crystal-field
strengths $S_{k}=(\frac{1}{2k+1}\sum\limits_{q}|B_{kq}|^{2})^{1/2}$, for
$k=2,4$ and 6, yields a definite second moment $\sigma$, which can be derived
from the additivity relationship $\sigma^{2}=\sum_{k}\sigma_{k}^{2}$ and the
known asphericities $< \Psi||C^{(k)}||\Psi >$ of the central-ion eigenfunctions
$\Psi$. The condition $\sigma=\sigma_{\rm exp}$ must be satisfied to ensure the
parametrization's correctness. However, our survey of literature indicates that
there exists many other well-fitted crystal-field parameter sets that do not
obey this condition. Therefore, such sets are erratic and non-physical, and
should be re-examined or rejected. Having $\sigma$ for several $(\geq3)$
eigenstates $|\Psi>$ along with the relevant $< \Psi||C^{(k)}||\Psi >$
asphericities, one can estimate $\sigma_{k}$ and $S_{k}$, which are
well-founded experimentally. The above findings set up the parametrization
process properly. Lack of consistency between the second moments representing
various parametrizations and the pertinent second moments observed in
experiments is presumably the main reason for deluge of formally accurate but
accidental and inequivalent parametrizations.
