= Solution
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> is
$$
K=\begin{pmatrix}
k&-k&0&0\\
-k&k+\gamma&-\gamma&0\\
0&-\gamma&k+\gamma&-k\\
0&0&-k&k
\end{pmatrix}.
$$
\b[The 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.
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