Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2021 / Paper 2A / Section B / 23 / a

Let $q_i=z_i-z_i^{(0)}$ denote a small longitudinal displacement from equilibrium, and let every atomic mass be $m$. The constant equilibrium lengths disappear from the second derivatives. The force constant matrix isThe mass-weighted dynamical matrix is $D=K/m$, with equations $\ddot{\mathbf{q}}=-D\mathbf{q}$. If the phrase dynamical matrix denotes the unweighted Hessian, that object is $K$ instead; the normal-mode equation is $K\mathbf{v}=m{\omega }^2\mathbf{v}$ in either notation. Every row sums to zero, as required for a freely translating chain.