Effect of second-nearest-neighbor forces on the vibrations of amorphous Si${\mathrm{O}}_2$

A method is presented of solving for the vibrational Green's function of a silicon-dioxide Bethe lattice when it contains second-nearest-neighbor interactions. The method is used to find the density of states of a Bethe lattice constructed with a Keating Hamiltonian. The primary effect of the second-nearest-neighbor interactions is a softening of the rocking bands below 450 ${\mathrm{cm}}^{-1\mathrm{}}$.