The electronic states of $\mathrm{Cd}\mathrm{S}∕\mathrm{Cd}\mathrm{Se}∕\mathrm{Cd}\mathrm{S}$ colloidal nanocrystal quantum-dot quantum wells are studied by large-scale pseudopotential local density approximation (LDA) calculations. Using this approach, we determine the effects of CdS core size, CdSe well thickness, and CdS shell thickness on the band-edge wave functions, band-gap, and electron-hole Coulomb interactions. We find the conduction-band wave function to be less confined to the CdSe well layer than predicted by $\mathbf{k}∙\mathbf{p}$ effective-mass theory, which accounts for the previous underestimation of the electron $g$ factor.