A symmetry-lowering change separating two axial metal–ligand distances from four equatorial ones. Occupancy of the split orbitals determines the electronic energy gain, while elastic cost and pre-existing ligand asymmetry affect the observed structure.

For a real nonmagnetic electronic multiplet, a vibrational coordinate can couple linearly through a real symmetric electronic operator only if its representation occurs in the symmetric square. Remove the fully symmetric component when asking about symmetry-lowering distortion. In $T_d$, both ${\mathrm{Sym}}^2{T}_1$ and ${\mathrm{Sym}}^2{T}_2$ equal $A_1+E+T_2$, so $E$ and $T_2$ modes are symmetry-eligible for either triplet; coupling strengths are not determined by this decomposition.

stable isotope labeling uses nonradioactive isotopes to distinguish molecular populations. Paired labeled and unlabeled samples can be compared by mass spectrometry, subject to labeling completeness, mixing and measurement-response controls.

A molecular species differing from another only in its isotopic composition. Different reduced masses alter vibrational and rotational spacings even when the underlying electronic potential is nearly the same.

Measurement of isotopic composition using non-radioactive isotopes. Food-web and dietary interpretations require an appropriate baseline and allowance for tissue formation and physiological fractionation.

Oxygen-18 is the stable oxygen isotope with mass number 18. oxygen from enriched water can trace the source of an oxygen atom in a reaction product, provided subsequent oxygen exchange is excluded.

A radioactive carbon isotope. Its short half-life makes radioactive decay storage relevant to waste-gas management; no biological mechanism is needed for the retention calculation.

Integrating a canonical partition function over volume gives the displayed continuous-volume normalization. For a temperature-independent measure and Hamiltonian, $-{\partial }_{\beta }\ln \Delta =⟨E+PV⟩$. Multiplying by a volume-independent prefactor leaves normalized volume probabilities unchanged, but a temperature-dependent prefactor changes this derivative.