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We calculated size-dependent single particle energy eigenvalues, wave functions, energy band gap, and dielectric constant for a semiconductor nanocrystallite, within the framework of the two band Penn model. The finite size effects are presented as size-dependent correction terms to the energy eigenvalues and the energy band gap of the bulk semiconductor. Ours is a self-consistent calculation within the Penn model [Phys. Rev. 128, 2093 (1962)], providing a much deeper understanding of the size dependences of energy band gap and the dielectric constant, as compared to those performed in the past using the so called generalized Penn model. Computed results on energy band gap and dielectric constant from our investigations exhibit very good agreement with those obtained from experimental measurements and the detailed numerical calculations performed using techniques such as density functional approach. Unlike our calculation, prior reported calculations of dielectric constant with the use of the generalized Penn model completely ignore or do not properly take into account the size dependences of oscillator strength and the energy band gap. It is shown that the size dependences of oscillator strength and the energy band gap play an important role and they lead to a fractional exponent (between 1 and 2) of size (radius) that appears in the formula for the dielectric constant of a nanocrystallite.