  It has been shown recently that a simple formula in terms of the valence
nucleon numbers and the mass number can describe the essential trends of
excitation energies of the first $2^+$ states in even-even nuclei. By
evaluating the first order energy shift due to the zero-range residual
interaction, we find that the factor which reflects the effective particle
number participating in the interaction from the Fermi orbit governs the main
dependence of the first $2^+$ excitation energy on the valence nucleon numbers.
