We investigate localized states in one-dimensional Frenkel exciton systems that are created by a shift in the optical transition frequency of a single chromophore. In this paper, we focus on localized states lying below the exciton band that can act as exciton traps. A comparison is made between systems with infinite-range (r(-n), n = 2, 3, ...) transfer and those with nearest-neighbor (n = infinity) transfer. A distinction is also made between normal bands (m minimum exciton energy at k = 0) and inverted bands (minimum energy at k = pi). The position of the localized state relative to the bottom of the band is calculated as a function of the shift in the single-chromophore transition frequency. The nature of the localized state is displayed in calculations of the participation ratio and the effective oscillator strength. Similarities and differences in localized states between normal and inverted band systems and between infinite-range and nearest-neighbor transfer are analyzed.