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explicit diagonalization of Eq. (18) (or the equivalent expression with thespin-orbit interaction included). This subset usually contains the bands ofinterest (e.g., the valence band and conduction band). Then, the influence ofoutlying bands is included within the second-order k p approach. Due to theexact treatment of the important subset, the dispersion of each band is nolonger strictly quadratic, as in Eq. (21). Therefore, the Kane model betterdescribes band nonparabolicities. In particular, this approach is necessary fornarrow-gap semiconductors, where significant coupling between the valenceand conduction bands occurs.For semiconductor quantum dots, a Kane-like treatment was firstdiscussed by Sercel and Vahala [39,40]. More recently, such a descriptionhas been used to successfully describe experimental data on narrow-bandgapInAs nanocrystals [13]. Furthermore, even wide-bandgap semiconductornanocrystals, such as CdSe, may require a more sophisticated Kane treatmentof the coupling of the valence and conduction bands [41]. These issues will bediscussed in Section IV.III.CADMIUM SELENIDE NANOCRYSTALSA. SamplesAlthough we focus here on nanocrystal spectroscopy, we cannot overemphasizethe importance of sample quality in obtaining useful optical information.Indeed, a thorough understanding of the size dependence of the electronicstructure in semiconductor quantum dots could not be achieved until samplepreparation was well under control. Early spectroscopy (e.g., on II–VIsemiconductor nanocrystals [5,42–51]) was constrained by distributions inthe size and shape of the nanocrystals, which broaden all spectroscopicfeatures, conceal optical transitions, and inhibit a complete investigation.Later, higher-quality samples became available in which many of the electronicstates could be resolved [52–55]. However, the synthetic methodsutilized to prepare these nanocrystals could not produce a complete seriesof such samples. Therefore, the optical studies were limited to one [52–54] or afew sizes [55].Fortunately, this situation has dramatically changed since the introductionof the synthetic method of Murray et al. [7]. This procedure andsubsequent variations [12,56–60] use a wet-chemical (organometallic) synthesisto fabricate high-quality nanocrystals. From the original synthesis,highly crystalline, nearly monodisperse [