Density-potential relation via the single-particle kinetic energy density for one-dimensional two-level fermion systems with application to Be-like atomic ions in the ${\left(1s\right)}^2{\left(2s\right)}^2$ configuration

Using the Dawson-March transformation of two-level wave functions in terms of density amplitude and phase, a differential equation is constructed relating the single-particle kinetic energy to the fermion density. Employing then the differential form of the virial theorem derived by March and Young [Nucl. Phys. 12, 237 (1959)], a nonlinear electron-density–one-body potential relation emerges. Application to the Be-like atomic ions is briefly considered.