It was shown that a direct use of the theory of special points, developed previously for calculating the electron density and local exchange potentials, can lead to the violation of the idempotent property of the density matrix and can give rise to a nonphysical divergence when the nonlocal exchange potential is accounted for. A scheme of interpolation of the density matrix over the Brillouin zone in the reciprocal space was proposed to overcome these difficulties. The electron structures of diamond, rutile, and MgO ideal crystals were calculated by the Hartree-Fock and CNDO methods. The convergence of the results with increasing the number of special points in the Brillouin zone was analyzed, and the efficiency of the proposed modification of the special point theory was illustrated. The idempotent property of the density matrix was verified quantitatively.