The limits of the total crystal-field splittings

The crystal-fields causing $|J>$ electron states splittings of the same second moment $\sigma^{2}$ can produce different total splittings $\Delta E$ magnitudes. Based on the numerical data on crystal-field splittings for the representative sets of crystal-field Hamiltonians ${\cal H}_{\rm CF}=\sum_{k}\sum_{q}B_{kq}C_{q}^{(k)}$ with fixed indexes either $k$ or $q$, the potentials leading to the extreme $\Delta E$ have been identified. For all crystal-fields the admissible ranges $(\Delta E_{min},\Delta E_{max})$ have been found numerically for $1\leq J\leq 8$. The extreme splittings are reached in the crystal-fields for which ${\cal H}_{\rm CF}s$ are the definite superpositions of the $C_{q}^{(k)}$ components with different rank $k=2,4$ and 6 and the same index $q$. Apart from few exceptions, the lower limits $\Delta E_{min}$ occur in the axial fields of ${\cal H}_{\rm CF}(q=0)=B_{20}C_{0}^{(2)}+B_{40}C_{0}^{(4)}+B_{60}C_{0}^{(6)}$, whereas the upper limits $\Delta E_{max}$ in the low symmetry fields of ${\cal H}_{\rm CF}(q=1)=B_{21}C_{1}^{(2)}+B_{41}C_{1}^{(4)}+B_{61}C_{1}^{(6)}$. Mixing the ${\cal H}_{\rm CF}$ components with different $q$ yields a secondary effect and does not determine the extreme splittings. The admissible $\Delta E_{min}$ changes with $J$ from $2.00\sigma$ to $2.40\sigma$, whereas the $\Delta E_{max}$ from $2.00\sigma$ to $4.10\sigma$. The maximal gap $\Delta E_{max}-\Delta E_{min}=2.00\sigma$ has been found for the states $|J=4>$. Not all the nominally allowed total splittings, preserving $\sigma^{2}=const$ condition, are physically available, and in consequence not all virtual splittings diagrams can be observed in real crystal-fields.