  In this study, two novel improvements for the theoretical calculation of the
neutron distributions are presented. First, the available experimental proton
distributions are used as a constraint rather than inferred from the
calculation. Second, the recently proposed distribution formula, d3pF, is used
for the neutron density, which is more detailed than the usual shapes, for the
first time in nuclear structure calculation. A semi-microscopic approach for
binding energy calculation is considered in this study, however, the proposed
improvements can be introduced to any other approach. The ground state binding
energy and neutron density distribution of $^{208}$Pb nucleus are calculated by
optimizing the binding energy considering three different distribution
formulae. The implementation of the proposed improvements leads to a
qualitative and quantitative improvement in the calculation of the binding
energy and neutron density distribution. The calculated binding energy agrees
with the experimental value, and the calculated neutron density shows
fluctuations within the nuclear interior, which agrees with the predictions of
self-consistent approaches.
