  We examine the effects of the additional term of the type $\sim e^{- \lambda'
N_pN_n}$ on the recently proposed empirical formula for the lowest excitation
energy of the $2^+$ states in even-even nuclei. This study is motivated by the
fact that this term carries the favorable dependence of the valence nucleon
numbers dictated by the $N_pN_n$ scheme. We show explicitly that there is not
any improvement in reproducing $E_x(2_1^+)$ by including the extra $N_pN_n$
term. However, our study also reveals that the excitation energies
$E_x(2_1^+)$, when calculated by the $N_pN_n$ term alone (with the mass number
$A$ dependent term), are quite comparable to those calculated by the original
empirical formula.

Nuclear Theory