In this paper we study localized vibrational modes in a chain of atoms with the same mass, but with two alternating force constants between nearest neighbors. This model simulates a row of atoms in the 〈111〉 direction of a diamond or silicon crystal. Cubic and quartic anharmonicity are included in the potential. The harmonic and quartic anharmonic terms produce intrinsic localized modes with frequencies above the optical branch and in the gap. The inclusion of cubic terms with negative coefficients reduces the frequencies, leads to new localized modes in the gap, and introduces static distortions of the chain. Modes of both even and odd parity occur. Surface modes occur when the localization is near a free end.