Higher-order cusp relations are derived for the wave function and the electron density of the ground and the excited states of atoms, ions or molecules. The total energy is expressed in terms of the electronic potential and density terms defined at the nucleus. It is proved that the linear term of the spherical part in the expansion of the Kohn-Sham potential, the classical Coulomb arid the exchange correlation potentials around a nucleus are all equal to zero. A relationship involving the values of the density and its second and third derivatives at the nucleus is derived.