A nonlinear integro-differential equation is introduced for the density amplitude of conduction electrons in simple metals as a consequence of the results obtained within the theory of liquids. The simplicity of this equation in comparison with the usual Kohn-Sham approach gives the possibility (within the approximations involved in using a Jastrow type trial function instead of a Slater determinant) to determine the self-consistent density of conduction electrons without assuming a muffin-tin lattice potential. Calculations have been performed for lithium. Perspectives connected with this approach are discussed.