The understanding of electronic structure of atomic and molecular term states involved in spectroscopic transitions is aided by projecting combinations of micro-configurations to multi-electron states with "good" quantum numbers of angular momenta. In rare-earth (RE) compounds, atomic term labels are justifiably carried over to compounds, because the 4f orbitals are localized at the RE centers, and the transitions between them are modified by crystal-field splittings that are small compared to the spin-orbit interactions within the RE atom. The source of photoluminescence in Tb(III) compounds is lowest-lying excited quintet [superscript 5]D[subscript 4] originating from the [Xe]4f[superscript 8] configuration, in contrast to other D-states originating from the [Xe]4f[superscript 7]5d configuration, sometimes confused with the [superscript 5]D[subscript 4](4f[subscript 8]) state. The [superscript 5]D[subscript 4] state not only is missing in available databases (NIST Atomic Spectra Database Levels, http://physics.nist.gov/PhysRefData/ASD/levels_form.html; Moore, C. E. Atomic Energy Levels as Derived from the Analyses of Optical Spectra; Vol. I-III; National Bureau of Standards Circular, 467, U.S. Government Printing Office: Washington, DC, 1949-1958; Martin, W. C. "Atomic Energy Levels--the Rare Earth Elements"; NSRDS-NBS 60; U.S. Department of Commerce, National Bureau of Standards: Washington, DC, 1978), but also its structure in terms of combinations of the 4f[superscript 8] microstates has so far eluded interpretation. We present a shortcut method using the shift operators arriving at the [superscript 5]D4(4f[superscript 8]) state, which turns out to be a linear combination of no fewer than 6 Slater determinants. We then compare the Tb(III) PL transitions [superscript 5]D[subscript 4] [right arrow] [superscript 7]F[subscript J], J = 6, 5, 4, 3 in compounds as diverse as TbCl[subscript 3](aq), Tb(III)-Y zeolite, and Tb(III) polystyrene sulfonate, all at similar wavelengths but showing features sensitive to chemical environment.