Based on the Kato's cusp condition of the electron density and our recent relations for local strongly decaying properties in an electronic system, necessary conditions for trial electron densities of atomic and molecular systems are derived. These conditions take the form of integral-differential equations, and their validity is verified numerically. The relevance of these conditions to the Thomas–Fermi problem in the orbital-less density functional approach is discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006